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MODELING THE DYNAMICS OF THYROID HORMONES AND RELATED DISORDERS by Oylum Şeker B.S., Industrial Engineering, Boğaziçi University, 2009 Submitted to the Institute for Graduate Studies in Science and Engineering in partial fulfillment of the requirements for the degree of Master of Science Graduate Program in Industrial Engineering Boğaziçi University 2012 ii MODELING THE DYNAMICS OF THYROID HORMONES AND RELATED DISORDERS APPROVED BY: Prof. Yaman Barlas ................... (Thesis Supervisor) Assist. Prof. Gönenç Yücel ................... Assoc. Prof. Ata Akın ................... DATE OF APPROVAL: 17.09.2012 iii ACKNOWLEDGEMENTS First and foremost, I would like to express my sincere and immense gratitude to Prof. Yaman Barlas, my thesis supervisor, for his guidance, patience and support throughout this study. Anything that I could have humbly learnt and hopefully continue to learn from him as an excellent professor, as an intellectual, and as a friend have been and will be invaluable assets to me. I would like to thank to Assist. Prof. Gönenç Yücel and Assoc. Prof. Ata Akın for taking part in my thesis committee and providing valuable comments, which I surely will benefit in my future research. I wish to extend my sincere thanks and appreciation to Prof. Faruk Alagöl for readily offering his vast knowledge and experience about thyroid, and for his deep interest in this study. I am grateful to Nükhet Barlas, who inspired the onset of this work, for kindly sharing her personal blood test results to support this study. I would like to express my gratitude to Prof. Çetin Önsel for being so kind to answer my exhaustive questions about thyroid. I wish to express special acknowledgements to my colleagues at SESDYN Laboratory. I would like to thank to Nisa Önsel for her sympathetic, soothing and warm companion, to Onur Özgün for kindly offering his never-ending help and support on all the otherwise unsolvable methodological and technological issues, to M. Emre Keskin for genially answering my endless questions about every single topic in industrial engineering courses, and finally to Can Sücüllü for cheering us up with his improvised shows. iv I owe my special thanks to O. Kaan Drağan for his irreplaceable friendship and sympathy, and for patiently being my one and only voluntary therapist. Lastly, I wish to thank to Google and to some undisclosed third parties which generously provided me numerous papers and books about the subject of my study. v ABSTRACT MODELING THE DYNAMICS OF THYROID HORMONES AND RELATED DISORDERS In this study, a dynamic simulation model for thyroid hormone system is constructed. The objective of this work is to first generate the dynamics of the hormones involved in thyroid hormone system in healthy body, and then to adapt the model to portray the dynamics of certain common thyroid disorders. The ultimate aim is to provide a platform for scenario analysis to support medical education, training and research, without risking patients’ health. Initially, the model structure is tested by standard structure validity tests. After the validation part, four common thyroid disorders are simulated. Firstly, Graves’ disease, the most common source of hyperthyroidism, is addressed. Goiter formation, effect of iodine availability on the severity of the disease, and increased T3/T4 –a commonly used diagnostic measure in hyperthyroidism– are all well captured by the model. Other typical behaviour of hormones and glands are also well mimicked by simulations. Secondly, iodine deficiency, one prevailing cause of hypothyroidism, is discussed for two different levels of daily iodine intake. The model was able to depict all the characteristic changes including the goiter formation and increase in T3/T4 in these two scenarios, both independently and comparatively. Thirdly, the transient inhibitory effect of excessive iodine intake on thyroid gland is discussed. The model is able to demonstrate the enlargement in thyroid volume and the mild decline in thyroid hormones. Lastly, a condition called subacute thyroiditis, a common disorder in which thyroid gland is exposed to inflammation, is analysed. The typical triphasic clinical course of subacute thyroiditis, comprised of thyrotoxicosis, hypothyroidism and normal thyroid functioning is well represented by the model. In conclusion, with respect to both qualitative and quantitative information in literature, and interviews with the medical doctors, the model exhibits an acceptable degree of validity and is able to cover a wide range of thyroidrelated disorders. vi ÖZET TİROİT HORMONLARININ VE İLGİLİ HASTALIKLARIN DİNAMİKLERİNİN MODELLENMESİ Bu çalışmada, tiroit hormon sistemi için dinamik bir benzetim modeli kurulmuştur. Çalışmanın amacı, öncelikle sağlıklı vücutta tiroit hormon sisteminin işleyişinde rol alan hormonların dinamiğini üretmek ve ardından yaygın görülen bazı tiroid rahatsızlıklarının dinamiklerinin gösterilebilmesi için modeli uyarlamaktır. Nihai amaç, hastaların hayatını tehlikeye atmaksızın tıbbi eğitim, çalışma ve araştırmayı senaryo analizleriyle destekleyecek bir ortam sunmaktır. Öncelikle, modelin yapısı standart geçerlilik testleri ile analiz edilmiştir. Geçerleme safhası bittikten sonra, yaygın görülen dört tane tiroid hastalığının benzetimi yapılmıştır. İlk olarak, hipertiroidizmin sık görülen sebeplerinden biri olan Graves’ hastalığı ele alınmıştır. Model, guatr oluşumunu, mevcut iyot miktarının hastalığın şiddetine etkisini, bu hastalığın teşhisinde sıklıkla kullanılan artmış T3/T4 oranını ve diğer hormon ve bezlerin tipik davranışlarını tutarlı bir biçimde sergileyebilmiştir. İkinci olarak, hipotiroidizmin sıkça görülen sebeplerinden biri olan iyot eksikliği iki farklı günlük iyot alımı seviyesi için incelenmiştir. İki iyot alım seviyesinde guatr oluşumu ve T3/T4 oranındaki değişimler gibi tipik göstergeler hem karşılaştırmalı hem de bağımsız olarak model tarafından üretilebilmiştir. Üçüncü olarak, aşırı iyot alımının geçici kısıtlayıcı etkileri ele alınmış, tiroit hacmindeki büyüme ve tiroit hormonlarındaki hafif düşüş elde edilebilmiştir. Son olarak, tiroit bezinin iltihaplanması sonucu ortaya çıkan subakut tiroidit adlı hastalık analiz edilmiştir. Model, hastalığın tirotoksikoz, hipotiroidizm ve ardından normal işleyişin kazanılmasından oluşan üç aşamalı tipik klinik gidişatını başarılı bir biçimde yansıtabilmiştir. Sonuç olarak, literatürde yer alan nitel ve nicel bilgiler ile tıp doktorlarlarıyla yapılan görüşmelerin ışığında, modelin makul seviyede geçerli olduğu ve geniş yelpazedeki tiroit hormon hastalıklarını kapsayabildiği söylenebilir. vii TABLE OF CONTENTS ACKNOWLEDGEMENTS .................................................................................................. iii ABSTRACT........................................................................................................................... v ÖZET .................................................................................................................................... vi LIST OF FIGURES .............................................................................................................. ix LIST OF ACRONYMS/ABBREVIATIONS ....................................................................... ix 1. INTRODUCTION ............................................................................................................. 1 2. LITERATURE REVIEW AND RESEARCH OBJECTIVE ............................................ 4 3. RESEARCH METHODOLOGY ...................................................................................... 7 4. OVERVIEW OF THE MODEL ...................................................................................... 10 5. DESCRIPTION OF THE MODEL ................................................................................. 13 5.1. Hypothalamus Sector ............................................................................................... 13 5.1.1. Background Information ................................................................................ 13 5.1.2. Fundamental Approach and Assumptions ..................................................... 15 5.1.3. Description of the Structure ........................................................................... 18 5.2. Pituitary Sector ......................................................................................................... 25 5.2.1. Background Information ................................................................................ 25 5.2.2. Fundamental Approach and Assumptions ..................................................... 27 5.2.3. Description of the Structure ........................................................................... 27 5.3. Thyroid Sector .......................................................................................................... 31 5.3.1. Background Information ................................................................................ 31 5.3.2. Fundamental Approach and Assumptions ..................................................... 34 5.3.3. Description of the Structure ........................................................................... 35 5.4. Iodine Sector ............................................................................................................ 52 5.4.1. Background Information ................................................................................ 52 5.4.2. Description of the Structure ........................................................................... 53 6. VALIDATION AND ANALYSIS OF THE MODEL .................................................... 58 6.1. Equilibrium Behaviour ............................................................................................. 59 viii 6.2. Base Run .................................................................................................................. 59 6.3. TRH Injection Test ................................................................................................... 61 6.4. Ten-Fold Increase in T4 Secretion for One Hour .................................................... 62 6.5. Zero T4 Secretion for One Hour .............................................................................. 65 6.6. Hypophysectomy ..................................................................................................... 67 7. THYROID DISORDERS ................................................................................................ 70 7.1. Graves’ Disease ........................................................................................................ 70 7.1.1. Graves’ Disease with Normal Daily Iodine Intake ........................................ 71 7.1.2. Graves’ Disease with Relatively High Daily Iodine Intake ........................... 76 7.2. Iodine Deficiency ..................................................................................................... 79 7.2.1. Severe Iodine Deficiency ............................................................................... 80 7.2.2. Moderate Iodine Deficiency........................................................................... 86 7.3. Iodine Excess ........................................................................................................... 89 7.4. Subacute Thyroiditis ................................................................................................ 95 8. CONCLUSION .............................................................................................................. 103 APPENDIX: Model Equations .......................................................................................... 105 REFERENCES .................................................................................................................. 118 ix LIST OF FIGURES Figure 1.1. Basic structure of the thyroid hormone system. ............................................. 2 Figure 3.1. Stock-flow diagram of a simple population model. ....................................... 9 Figure 4.1. Simplified causal loop diagram of the model. .............................................. 11 Figure 5.1. Stock-flow diagram of the hypothalamus sector. ......................................... 19 Figure 5.2. Effect of thyroid hormones on TRH secretion. ............................................ 20 Figure 5.3. Graphical function for the effect of capacity on TRH secretion. ................. 22 Figure 5.4. Graphical function for the effect of implied TRH secretion on hypothalamus weight........................................................................................................... 23 Figure 5.5. Graphical function for effect on hypothalamic adjustment time. ................. 25 Figure 5.6. Stock-flow diagram of the pituitary sector. .................................................. 28 Figure 5.7. Graphical function for the effect of TRH on TSH secretion. ....................... 30 Figure 5.8. Graphical function for the effect of thyroid hormones on TSH secretion. ... 30 Figure 5.9. Stock-flow diagram of the thyroid sector. .................................................... 36 Figure 5.10. Graphical function for the effect of thyroid hormone store capacity. .......... 38 Figure 5.11. The graphical function for the effect of TSH on thyroid hormone secretion. ..................................................................................................................... 39 Figure 5.12. Graphical function for the effect of iodine on thyroid capacity. .................. 40 Figure 5.13. Graphical function for the effect of preferential T3 synthesis on reduction in T3 synthesis. ................................................................................................ 47 Figure 5.14. Graphical function for the effect of T3 concentration on peripheral conversion. ................................................................................................... 49 Figure 5.15. Graphical function for the effect of thyroid stimulation on T3 secretion fraction. ........................................................................................................ 51 Figure 5.16. Graphical function for the fraction of T3 secretion. .................................... 52 Figure 5.17. Stock-flow diagram of the iodine sector. .................................................... 54 Figure 5.18. Graphical function for desired trapping fraction. ........................................ 55 Figure 5.19. Graphical function for the effect of TSH on iodide trapping. ..................... 56 x Figure 5.20. Graphical function for the effect of thyroid weight on iodide trapping. ..... 56 Figure 5.21. Simplified stock-flow diagram of the model. .............................................. 57 Figure 6.1. T3 (left top), T4 (right top), TRH (lower left), and TSH (lower right) concentrations at equilibrium....................................................................... 59 Figure 6.2. T4 concentration in base run. ....................................................................... 60 Figure 6.3. T3 concentration in base run. ....................................................................... 60 Figure 6.4. TSH concentration in base run. .................................................................... 60 Figure 6.5. TRH concentration in base run. .................................................................... 61 Figure 6.6. Average TSH concentration of six normal subjects when 25 µg TRH is injected at t=0 (Snyder and Utiger, 1972).................................................... 62 Figure 6.7. TSH concentration when 25 µg TRH is injected at t=0. .............................. 62 Figure 6.8. T4 concentration when its secretion is increased ten-fold for one hour. ...... 63 Figure 6.9. T3 concentration when T4 secretion is increased ten-fold for one hour. ..... 63 Figure 6.10. TRH concentration when T4 secretion is increased ten-fold for one hour. . 64 Figure 6.11. TSH concentration when T4 secretion is increased ten-fold for one hour. .. 64 Figure 6.12. T4 concentration when T4 secretion is stopped for one hour. ..................... 65 Figure 6.13. T3 concentration when T4 secretion is stopped for one hour. ..................... 66 Figure 6.14. TSH concentration when T4 secretion is stopped for one hour. .................. 66 Figure 6.15. T3 concentration in case of hypophysectomy. ............................................. 67 Figure 6.16. TRH concentration in case of hypophysectomy. .......................................... 67 Figure 6.17. Thyroid weight in case of hypophysectomy. ................................................ 68 Figure 6.18. Hypothalamus weight in case of hypophysectomy. ..................................... 69 Figure 7.1. T3 concentration in Graves’ disease with normal iodine intake. ................. 71 Figure 7.2. T4 concentration in Graves’ disease with normal iodine intake. ................. 72 Figure 7.3. Iodine in thyroid in Graves’ disease with normal iodine intake. .................. 73 Figure 7.4. T3 to T4 ratio in Graves’ disease with normal iodine intake. ...................... 73 Figure 7.5. TRH concentration in Graves’ disease with normal iodine intake. .............. 74 Figure 7.6. TSH concentration in Graves’ disease with normal iodine intake. .............. 74 Figure 7.7. Thyroid weight in Graves’ disease with normal iodine intake. .................... 75 Figure 7.8. Hypothalamus weight in Graves’ disease with normal iodine intake. ......... 75 xi Figure 7.9. Pituitary weight in Graves' disease with normal iodine intake. .................. 76 Figure 7.10. T3 concentration in Graves’ disease with relatively high iodine intake. .... 77 Figure 7.11. Iodine in thyroid in Graves’ disease with relatively high iodine intake. ..... 77 Figure 7.12. T3 to T4 ratio in Graves’ disease with relatively high iodine intake. ......... 78 Figure 7.13. TSH concentration in Graves’ disease with relatively high iodine intake. . 78 Figure 7.14. Thyroid weight in Graves’ disease with relatively high iodine intake. ....... 79 Figure 7.15. Pituitary weight in Graves’ disease with relatively high iodine intake. ...... 79 Figure 7.16. T3 concentration when daily iodine intake is 30 µg. .................................. 81 Figure 7.17. T4 concentration when daily iodine intake is 30 µg. .................................. 81 Figure 7.18. T3 to T4 ratio when daily iodine intake is 30 µg. ....................................... 82 Figure 7.19. Iodine in thyroid when daily iodine intake is 30 µg. ................................... 82 Figure 7.20. T3 store when daily iodine intake is 30 µg. ................................................ 83 Figure 7.21. T4 store when daily iodine intake is 30 µg. ................................................ 83 Figure 7.22. TSH concentration when daily iodine intake is 30 µg. ............................... 84 Figure 7.23. Thyroid weight when daily iodine intake is 30 µg. ..................................... 84 Figure 7.24. Hypothalamus weight when daily iodine intake is 30 µg. .......................... 85 Figure 7.25. Pituitary weight when daily iodine intake is 30 µg. .................................... 85 Figure 7.26. T3 concentration when daily iodine intake is 50 µg. .................................. 86 Figure 7.27. T4 concentration when daily iodine intake is 50 µg. .................................. 86 Figure 7.28. T3 to T4 ratio when daily iodine intake is 50 µg. ....................................... 87 Figure 7.29. T3 store when daily iodine intake is 50 µg. ................................................ 88 Figure 7.30. TSH concentration when daily iodine intake is 50 µg. ............................... 88 Figure 7.31. Thyroid weight when daily iodine intake is 50 µg. ..................................... 89 Figure 7.32. Average free T4 concentration of ten subjects receiving 27 mg iodine supplementation for 28 days (Namba et al., 1993). ..................................... 90 Figure 7.33. Free T4 concentration (in pmol/l) in case of 27 mg iodine supplementation for 28 days. .................................................................................................. 91 Figure 7.34. T3 concentration in case of 27 mg iodine supplementation for 28 days. ..... 91 Figure 7.35. Average TSH concentration of ten subjects receiving 27 mg iodine supplementation for 28 days (Namba et al., 1993). ..................................... 92 xii Figure 7.36. TSH concentration in case of 27 mg iodine supplementation for 28 days. .. 92 Figure 7.37. Average thyroid volume (as % of normal volume) of ten subjects receiving 27 mg iodine supplementation for 28 days (Namba et al., 1993). ............... 93 Figure 7.38. Thyroid weight in case of 27 mg iodine supplementation for 28 days. ....... 94 Figure 7.39. Average serum iodine levels of ten subjects receiving 27 mg iodine supplementation for 28 days (Namba et al., 1993). ..................................... 94 Figure 7.40. Iodine in blood in case of 27 mg iodine supplementation for 28 days. ........ 95 Figure 7.41. Modified structure of thyroid sector for subacute thyroiditis....................... 97 Figure 7.42. Modified structure of iodine sector for subacute thyroiditis. ....................... 98 Figure 7.43. The assumed course of inflammation status in subacute thyroiditis. ......... 100 Figure 7.44. TSH and T4 concentrations in subacute thyroiditis. .................................. 100 Figure 7.45. Data from a patient with subacute thyroiditis (Lazarus, 2009). ................. 101 Figure 7.46. Data from a patient with subacute thyroiditis (secondary axis: TSH). ...... 101 Figure 7.47. T3 to T4 ratio in subacute thyroiditis. ........................................................ 102 xiii LIST OF ACRONYMS/ABBREVIATIONS abs absorption adj adjustment avail availability cap capacity chg change clear clearance conc concentration cons consumption conv conversion deiod deiodination del delay des desired disc discrepancy eff effect excr excretion fr fraction gr graphical function hypo hypothalamus (or hypothalamic) imp implied peri peripheral pit pituitary pos possible pot potential pref preferential prod productivity recov recovery red reduction xiv rest restricted restor restoration stim stimulation thres threshold thy thyroid (or thyroidal) trap trapping wt weight AT Adjustment Time HAT Adjustment Time for Hypothalamus PAT Adjustment Time for Pituitary TAT Adjustment Time for Thyroid TH Thyroid Hormones TRH Thyrotropin Releasing Hormone TSH Thyroid Stimulating Hormone 1 1. INTRODUCTION There are mainly two systems in the regulation of the functions of the body: (1) the nervous system, and (2) the endocrine system, or the hormonal system. In this study, a hormonal system will be of interest. For the lexical meaning, the word “hormone” is derived from the Greek word hormaein, which means to excite, arouse or stir up. As for the biological implication, a hormone is a chemical substance responsible for conveying messages to target cells. They are secreted by a cell or gland, and act as means of communication among the parts of the body. Through the actions of hormones, the endocrine system exerts physiological control on metabolic functions of the body. Therefore, endocrine system plays a vital role in the regulation, integration and coordination of various physiological processes (Rhoades and Bell, 2009). A crucial point in the functioning of endocrine system is to preserve the internal balance, or homeostasis, in the body. This is where feedback loops come into play. Feedback loops are the principal regulators of the endocrine system. They adjust the amount of hormones released by the gland and keep them at a desired level in order to guarantee a healthy maintenance of bodily functions. As far as the order of feedback loops is concerned, different forms of hormonal regulation exist. Rather than the systems that operate under the control of a single feedback loop, the ones involving higher order, complex feedback loops have more interesting dynamics to study. Production and release of thyroid hormones, which comprises the main focus in this study, is controlled by such higher order negative (balancing, compensating) feedback loops. Three tiers are involved in thyroid hormone system. First tier is the hypothalamus, second is the pituitary gland, and the third one is the thyroid gland (Kronenberg et al., 2008). Firstly, the hypothalamus secretes thyrotropin-releasing hormone (TRH) which 2 prompts the production of thyroid-stimulating hormone (TSH) from the pituitary. Then, TSH stimulates the thyroid gland. Upon stimulation, production and release of thyroid hormones, which are triiodothyronine (T3) and thyroxine (T4), is triggered. After T3 and T4 are secreted from the thyroid, they circulate in blood and reach their target tissues (Guyton and Hall, 2006; Kronenberg et al., 2008; Rhoades and Bell, 2009). Eventually, concentration of thyroid hormones in blood creates a double-armed negative feedback effect. That is, thyroid hormones in blood affect negatively both the hypothalamus and the pituitary, and consequently control the secretion of TRH and TSH to keep the thyroid hormone concentrations at a constant desired level. Pictorially, the basic structure of the thyroid hormone system looks as in Figure 1.1. - Hypothalamus TIER I TRH - Pituitary TIER II TSH Thyroid TIER III T3 and T4 Peripheral Tissues Figure 1.1. Basic structure of the thyroid hormone system. The thyroid hormones play key roles in the regulation of bodily functions. They govern the pace of metabolic functions of cells in the body by enhancing the rate of oxygen consumption, utilization of fats, carbohydrates and proteins by the cells. In this respect, the thyroid gland undertakes a managerial role in the regulation of metabolic functions; 3 depending on the intensity of hormone signals from the thyroid, the rate of metabolism in the body is adjusted. The prevalence of thyroid diseases is quite high and misdiagnosis of these diseases is not uncommon. Since these hormones affect virtually every part of the body and regulate some vital functions, it is important to gain an insight into the structure of this hormonal system, the interplay between the constituent components of the system, and the dynamics of the hormones involved under related disorders. In the following chapter, systems-theoretic research in the modelling of thyroid hormone system will be briefly reviewed and the research objective of this modelling study will be described. Then, the research methodology and the rationale behind it will be concisely explained. In the remaining chapters, the system dynamics model will be elucidated, validity test results will be presented, and the dynamics of some well-known disorders of thyroid hormone system will be generated and discussed. 4 2. LITERATURE REVIEW AND RESEARCH OBJECTIVE Modelling of physiological systems has aroused considerable interest over the past several decades. That physiological systems, in particular endocrine systems, are capable of preserving their internal balance through the actions of feedback mechanisms connects them to technological feedback control systems that are widely studied in engineering fields, in the sense that their regulation abides essentially to the same principles. The formidable complexity and large number of interactions inherent within and among endocrine systems introduce problems of quantification that well fit to the tremendous abilities of computers, and that (verbal) language mostly fails to suffice, whereas modelling and simulation succeeds for both simple and complex systems, as pointed in one of the early works of DiStefano and Chang (1971) on simulation of thyroid hormone dynamics. A number of studies have been conducted in which engineering principles are applied to model the thyroid hormone system at various levels of complexity, from various aspects, and with different research objectives. One prominent name in these studies is Joseph J. DiStefano III. There has been a number of pioneering researches conducted by DiStefano and colleagues to model the thyroid system with a systems-theoretical approach and integrate it with experimental data. Some of these studies deal with hypotheses about the underlying feedback structure (DiStefano, 1969; DiStefano and Stear, 1968), some with parameter estimation for thyroid hormone secretion, distribution, binding, conversion and metabolism (DiStefano and Chang, 1971; DiStefano and Mori, 1969; Wilson et al., 1977), some with the mathematical models for secretory output of thyroid hormones in response to TSH input (DiStefano, 1969), and some with the prescription of thyroid hormones in hypothyroidism and after thyroidectomy (Mak and DiStefano, 1977; Eisenberg et al., 2007; Ben-Shachar et al., 2012). 5 There are also other thyroid-related quantitative modelling studies that adopt a systemic perspective. In the work of Khee and Leow (2007), a mathematical model is proposed for pituitary-thyroid interaction that aims to provide a better understanding of the sensitivity of the pituitary to the feedback effect of thyroid hormones in the context of thyroid hormone excess and deficiency. Another work conducted by Liu et al. (1994) proposes a new mathematical model for the secretory system of hypothalamo-pituitarythyroid axis by revising and improving the previous two works by Liu and Peng (1990) and Liu and Liu (1992) which takes into account the interactions of the hormones in the axis and the binding characteristics of hormones to proteins in plasma and tissues. Lastly, the work of Degon et al. (2008) uses recent molecular-level and clinical observations to develop a computational thyroid model which captures the known aspects of thyroid physiology and uses it to evaluate the competing hypotheses related to the Wolff-Chaikoff escape. Unlike many others, our modelling study integrates all four aspects involved in the control of thyroid hormone system; namely the hypothalamus, pituitary, thyroid, and the essential ingredient iodine. The main focus is to depict the major macro-level causal relationships among these four components that strive for the homeostatic regulation of the system, rather than concentrating on the intracellular pathways and micro-level molecular mechanisms. The model combines sufficiently many aspects involved in the regulation of thyroid hormone system, and thus is able to cover a wide range of conditions (like hyperthyroidism, hypothyroidism, thyroiditis, goiter, etc.) and illustrate the associated overall descriptive behaviours of key variables in the system. The main goal of modelling physiological systems is to provide a platform to conduct experiments and subsequently propose policies, without any necessity to rehearse on humans. This study aims at modelling the thyroid system to capture the dynamics of thyroid hormones and some related diseases in order to facilitate the recognition of these disorders. Initial goal is to develop a system dynamics model as a valid representation of the underlying structure of thyroid hormone system so as to illustrate the dynamics of key stimulating and thyroid hormones in healthy body. The next purpose is to modify the 6 model to represent some well-known thyroid disorders. The final goal is to capture the typical course of behaviour of the key hormones under these disorders, hence to hopefully offer a platform for the recognition of these disorders and for scenario analysis to assist medical education, training and research. 7 3. RESEARCH METHODOLOGY This study aims at modelling the thyroid hormone system to portray the dynamics of key hormones under healthy and diseased states, particularly by stressing the role of functional feedback mechanisms involved. Being composed of a tripartite regulatory mechanism, the smooth functioning of thyroid hormone system can be disturbed by the malfunctioning of any of the constituent subsystems, either due to purely internal motives, or anomaly of essential external inputs to the system. The fact that thyroid hormone system operates under not a simple first order but a dual feedback control, the existence of two different thyroid hormones, one of them largely depending on the production of the other and requiring a more sensitive modulation, and the role of iodine intake render the problem complex enough that our intuition mostly remains incapable. The assistance of mathematical or simulation modelling may well provide a deep insight into the structure of the system and the behaviour of key variables under related disorders, and contribute to medical training and research. System dynamics methodology is an efficient tool to enhance the understanding of the behaviour of complex, large-scale systems and study their underlying structures. The idea is to address an issue by adopting a holistic approach, which essentially states that a system is more than the sum of its individual constituent parts and cannot be fully understood in terms of the properties of individual elements in isolation. So, this approach puts a particular emphasis to the causal relationships between the constituents of the system. In other words, it is the internal structure of a system which drives the system behaviour. The structure can be defined as the totality of relationships that exist between system variables and the behaviour of a system is essentially the operation of its internal structure over time (Barlas, 2002). Once a proper and valid model structure is constructed, the behaviour that the system would generate under various schemes can be experimented via simulation runs, and a broad appreciation can be developed about the system as a whole. 8 The fact that feedback relationships largely prevail in the regulation of endocrine systems makes this engineering discipline a natural choice in such modelling studies. As far as the dominant roles of accumulations, feedbacks, nonlinearities and time delays inherent in the system of interest are concerned, system dynamics methodology is very suitable for quantitative behavioural analysis of the disorders of thyroid system. One important feature of system dynamics approach is that it particularly emphasizes the importance of causal relations as opposed to mere statistical correlations (Barlas, 2002). It aims at developing an understanding of the overall dynamic behaviour of the system of concern, rather than concentrating on the point prediction of the future values of the variables involved. In this respect, it becomes an appropriate tool in the modelling of physiological systems, for it is usually the collection of the overall pattern of the key variables, rather than their precise point values, which characterizes a particular condition. In system dynamics methodology, two central concepts are used in modelling. The first one is stocks, which represent the accumulations in a system. Stocks can be used in the conceptualization of a wide range of notions, from physical to information entities. Some examples for stock variables can be inventory, population, knowledge level, etc. The stocks are changed merely via their flows; that is, the net flow into a stock corresponds to the rate of change of that stock. Examples of flow variables related to the above stock examples can be production, sales, births, deaths, learning, forgetting, etc. (Barlas, 2002). A stock variable and its flows together correspond to a first order differential (or difference) equation, the stock being the system variable and the flows being the rates of change over time. The mathematical description of a system only entails the stocks and flows actually. However, for the sake of clarity, a third type of variable is also used in system dynamics which is called converter, or auxiliary variable. Converters are used to explicitly define some intermediate parameters or variables, and thus can be constants or functions of stocks and/or flows. 9 In model diagrams, stocks are represented by rectangular boxes, and flows by valves on arrows that point into or out of the stock. If the arrowhead of the flow point into the stock, that flow is called an “inflow”, and if it points out, then it is named “outflow”. Clouds symbolize the sources and sinks for the flows if they originate from or discharge outside the boundary of the model, and they do not induce any capacity constraint on the related flow (Sterman, 2000). An example stock-flow diagram of a simple population model is shown in Figure 3.1. death fraction birth fraction Population birth rate death rate Figure 3.1. Stock-flow diagram of a simple population model. Population(t) = Population(t - dt) + (birth rate – death rate) × dt (3.1) birth rate = birth fraction × Population (3.2) death rate = death fraction × Population (3.3) In this simple model, the stock variable is Population. The inflow to the stock is birth rate and the outflow death rate; that is, the birth rate tends to increase and death rate tends to decrease the population from its present value. birth fraction and death fraction are auxiliary variables. The arrows that connect the variables show the causal relationships between the variables. The variable on the head of the arrow is defined as a function of the variable (or the parameter) on the tail of the arrow. 10 4. OVERVIEW OF THE MODEL The levels of all the hormones in thyroid hormone system are controlled by the properly operating feedback loops between the components of the system as it is the case in most other physiological systems to preserve a stable functioning. In thyroid hormone system, two fundamental feedback loops operate on hypothalamus-pituitary-thyroid and pituitary-thyroid axes. Both of these loops operate to keep the thyroid hormones T3 and T4 at their normal levels. The overall model basically consists of four subdivisions: the hypothalamus, the pituitary, the thyroid, and the iodine. Hypothalamus and pituitary sectors are basically the same in terms of their qualitative structure; they involve one gland, its related hormone, and relationships that have an effect on the gland and hormone. Thyroid sector, however, involves one gland, related two hormones, the stores of the two hormones, and all the means and links that affect the functioning of the sector. Lastly, the iodine sector, where iodine is the main rate-limiting ingredient in the synthesis of thyroid hormones, involves the iodine levels in blood and in the thyroid gland, and the relevant measures. A simplified causal loop diagram depicting the main variables in the model together with the key feedback loops is provided in Figure 4.1. A “+” sign on the head of an arrow indicates a positive causal relationship between the variable on the tail and the variable on the head of the arrow, and conversely a “−” sign a negative causality. A positive causal link means that a change in the variable on the tail of the arrow (cause) induces a change in the variable on the head of the arrow (effect) in the same direction by an amount more than what it would have been otherwise. Conversely, a negative causal link means that a change in the cause prompts a change in the variable on the head of the arrow in the opposite direction by an amount more than what it would have been otherwise. 11 - Implied TRH secretion + Desired hypothalamus weight Free T3 and T4 in Blood + + TRH secretion + + Hypothalamus Weight + TRH 1 3 T3 and T4 + - Implied TSH secretion + 5 + Thyroid Weight + + Desired thyroid weight T3 and T4 secretion TSH secretion + 2 + Pituitary Weight + + + Desired pituitary weight Implied T3 and T4 secretion 4 + + TSH Figure 4.1. Simplified causal loop diagram of the model. The 1st and 2nd loops demonstrate the negative feedback mechanism on hypothalamus-pituitary-thyroid axis and pituitary-thyroid axis respectively. The hormones involved in these loops are TRH, TSH and T3 and T4. These two main feedback loops represent the short-term hormone control mechanisms in the body. In addition to the shortterm effects, some delayed effects on the weights of hypothalamus, pituitary and thyroid gland may be observed. The 3rd, 4th and 5th loops display these delayed feedback effects 12 between the weight of one particular gland and the subsequent hormone secretions in the related axis. The model will be elucidated in detail in the next chapter; but, briefly the rationale behind the model is as follows: The amount of a hormone secreted basically depends on two factors; the capacity of the gland and the implied secretion rate. The capacity of the gland is mainly imposed by the weight of the gland. The implied secretion rate is determined by the relative amounts of stimulating hormone and inhibitory hormone, if any, and this happens without a delay. However, changes in gland weight take place in time. First, the body “decides” on a desired gland weight with a delay by continually considering the induced levels of hormone demand, which actually is the implied secretion rate of that gland. According to this target level, gland weight might be altered in the long run. 13 5. DESCRIPTION OF THE MODEL 5.1. Hypothalamus Sector 5.1.1. Background Information Hypothalamus, a key regulator of homeostasis, is a small region of the brain located above the brain stem. It is the central element in the regulation of endocrine function due to its connections with the pituitary gland, which is the master gland of the endocrine system (Rhoades and Bell, 2009). The hypothalamus synthesizes and secretes unique releasing and inhibitory hormones which coordinate the production and secretion of hormones from anterior pituitary, which is one of the two lobes of the pituitary gland (Melmed, 2002). The weight of the hypothalamus in adult human is about 4000 mg (Bhagavan, 2002). The hypothalamus secretes various hormones that affect the anterior pituitary hormones, one of them being the TRH. TRH is synthesized and secreted by the parvicellular neurons of the paraventricular cells (PVNparv) and the periventricular nucleus (PeriVN). TRH is a hypothalamic hormone which principally stimulates the synthesis and release of TSH. The connection between the hormones of the hypothalamus and the anterior pituitary is enabled via minute blood vessels called hypothalamic-hypophysial portal vessels. Through these portal vessels, TRH is transported to the anterior pituitary to trigger the secretion of TSH. The rate of TRH secretion is mainly determined by the level of free thyroid hormone levels in blood. Some portion of free T3 and T4 molecules impinges upon hypothalamic cells and couples with the receptors on these cells (Bhagavan, 2002; Rhoades and Bell, 2009; Guyton and Hall, 2006; Werner et al., 2005). The amount of thyroid hormone-receptor complexes is the main determinant of the rate of TRH secretion. As aforementioned, the levels of thyroid hormones in blood negatively affect TRH secretion. So, as the amount of thyroid hormone-receptor complexes increase on the 14 cells of the hypothalamus, the TRH output will decrease, and vice versa. In short, the rate of TRH secretion is inversely proportional to the amount of thyroid hormones in blood. There are two factors that affect the concentration of a hormone in blood; secretion of that hormone and rate of removal from blood (Guyton and Hall, 2006). As most other hormones do, TRH is cleared from the body with a certain half-life, where half-life is the time it takes for half of the amount of a hormone to be cleared from blood in our context. TRH has a half-life of 6.2 minutes (Motta, 1991). In short term, changes in TRH secretion rate occur as the levels of thyroid hormones in blood dictate. However, there might be cases where the stimulation persists at far below or far above the baseline values. Relying upon the fact that a hormone, which provokes or inhibits the activity of a gland, can also affect its weight over the long term in certain cases (Donovan, 1966; Melmed, 2002; Guyton and Hall, 2006), thyroid hormones can also influence the weight of the hypothalamus. There is not direct evidence that the weight of the TRH-secreting section of the hypothalamus can be altered according to the circulating thyroid hormone levels. However, there is evidence that the number of cells that secrete CRH, which is a hypothalamic hormone analogous to TRH in the regulation of hypothalamus-pituitary-adrenal (HPA) axis, substantially decline in subjects who externally receive the hormones that inhibit its secretion (Erkut et al., 1998). Extrapolating all this information to our case, the weight of the hypothalamus is taken as a variable quantity. First, if the magnitude of stimuli from thyroid hormones is persistently far above the standard levels, it means that the secretory capacity of hypothalamus is consistently underutilized. In such cases, the specific portions of hypothalamus, which are in charge of the TRH secretion, would shrink not to retain the redundant capacity in vain. Second, if thyroid hormones constantly circulate at considerably below normal concentrations, i.e. if hypothalamus is persistently understimulated, then the hypothalamus would continually operate at above-normal levels, and thereby expand to adjust its capacity. So, it adjusts its capacity in the direction that the current needs of the body necessitate. 15 5.1.2. Fundamental Approach and Assumptions Plasma levels of hormones normally fluctuate throughout the day or from one day to another because of neurological, psychological, environmental, or similar factors. Though these fluctuations and the features that influence them might count for some practical purposes like prescribing the right dose of a drug for a patient, the primary aim of the study is not to observe the dynamics of diurnal variations of the hormones in the body but rather to represent the long-term dynamics of the important elements involved in thyroid hormone system under certain conditions. Thus, the time unit of this study is taken to be one day, the base values of the variables are taken to be an average value in the model, and possible variations in hormone levels and neurological, psychological, environmental or similar other effects are considered to be outside the scope of this study. The circulating levels of all the hormones in this model are assumed to act according to set point theory. Here, the set points of the hormones are defined to be their absolute total quantity in blood. Though it might appear to be erroneous at first sight to take the absolute quantities of hormones in blood as their set points rather than their concentrations, it is not so because the blood or plasma volume cannot be too variable. Albeit so, it wouldn’t still hurt our assumption since this study does not encompass cases where the changes in the volumes of the fluid, which hormones float in, constitute the problem of interest. Secretion rates of hormones are commonly found in some mass unit (like µg, ng etc.) per unit time. Since the net change in the value of a stock variable is the integral of the net flow to the stock over time, the units of the flows of a stock is the unit of the stock divided by the time unit of the model. Therefore, defining the levels of hormones in terms of their absolute quantities renders it possible to use the secretion rates in their typically defined units. When the hypothalamic neurons are excited to secrete releasing hormones, that hormone is discharged into the hypophysial portal circulation. As mentioned before, this 16 portal system is composed of small blood vessels that link the hypothalamus and the anterior pituitary. The releasing hormones have only a small distance to travel in order to communicate with their target cells. Thus, it is enough to release just the quantity of hormone to the portal circulation to regulate the anterior pituitary hormone in this nearly isolated communication space. Hence, releasing hormones of hypothalamus circulate in almost undetectable amounts in systemic blood (Rhoades and Bell, 2009). Throughout the literature survey, secretion rate of TRH could not be found explicitly because hypothalamic-hypophyseal portal blood is an extremely difficult area to obtain blood samples in humans (Rhoades and Bell, 2009). Furthermore, direct measurement of the secretion rate of a hormone is quite a challenging task. Hence, secretion rates are commonly inferred from the blood concentration of the hormone and its clearance rate. For the case of TRH, related literature states that it also circulates in the cerebrospinal fluid (CSF), which is a serumlike fluid that essentially circulates through the ventricles of the brain. Firstly, the value for concentration of TRH in CSF, which is stated to range between 65-290 pg/ml (Werner et al., 2005) and taken approximately as 200 pg/ml, is assumed as its concentration in portal vessels. Secondly, the volume of plasma in these portal vessels is assumed to be 10 ml. According to these two assumptions, the normal absolute quantity of TRH in portal vessels is calculated as 2 ng in the model. The weight of the hypothalamus as a whole is 4000 mg, as mentioned in the previous section. The percentage of the TRH secreting cells, however, could not be found explicitly in literature. Thus, it is assumed that 1% of the hypothalamic cells are in charge of TRH secretion. So, the related weight is taken as 40 mg in the model. In general, hormones are cleared from the blood with some specific rate. Not only the hypothalamus sector but the whole model also works according to this principle; the clearance of each hormone from plasma occurs with respect to a certain fraction, which is called the “clearance fraction” (clear fr) in the model. In literature, half-lives are commonly used to quantify the clearance rate of a hormone. Thus, removal of hormones from blood (or the related fluid) is assumed to follow a first order exponential decay in the 17 model, and the respective clearance fractions are calculated from their half-lives using the following equation: where clearance fraction = ln2 / th (5.1) stands for the half-life in days. So, the clearance fractions are in units of . There are mainly three determinants of the magnitude of hormonal response of a target tissue; concentration of the hormone, sensitivity of the target cells, and number of functional target cells. The sensitivity of a target cell primarily depends on the number of its operational receptors, the affinity of the receptors for the hormone, and the capacity to amplify the post-receptor activities. Firstly, it is the binding of the hormone molecule to its specific receptor which gives rise to cellular response. And, the probability that a hormone molecule encounters a receptor molecule is induced by the abundance of both the hormone and the molecule. The availability of hormone receptors can be altered by the stimulating hormone itself or by another hormone. For instance, it is stated that T3 decreases the sensitivity of the TSHsecreting anterior pituitary cells to TRH (Goodman, 2009). Secondly, affinity is a measure of the tightness of binding or the likelihood of an encounter between a hormone and its receptor that result in binding. Some sources suggest that binding of a hormone to its receptor affects the affinity of unoccupied receptors. Thirdly, the post-receptor capacity of a target cell implies how well the cell can react to a unit magnitude of stimuli (Goodman, 2009; Rhoades and Bell, 2009). The first two ingredients of sensitivity and the circumstances that alter them are not explicitly included in the model, and are considered out of the scope of this study. The third one, however, is implicitly counted by allowing the intensity of hormonal stimulation to alter the secretory rates of the target cells to a certain extent. 18 The number of target cells is also not explicitly incorporated in the model; the weights of the glands are used as an indicator instead. As the weight of a gland increases, the competence or the capacity to respond to hormonal stimuli increases too. So, only the concentration of the hormone, the weight of the related gland or tissue, and the capacity of cells (more precisely, the capacity of a unit weight of gland or tissue) are assumed to dictate the response of the target tissue. One last remark is that when using the term hypothalamus weight, the weight of the relevant portion of the hypothalamus, i.e. the weight of the section responsible for TRH release, will be assumed. 5.1.3. Description of the Structure The structure constructed in this sector aims to depict the first tier in the regulation of the thyroid hormone system. As mentioned before, the hypothalamus and the related releasing hormone TRH are the top-level controllers of the negative feedback mechanism in the functioning of thyroid hormone system. TRH plays a chief role in the functioning of hypothalamic–pituitary–thyroid axis (HPT-axis) as the only positive effector of TSH secretion from the pituitary, TSH being the only direct positive stimulant of thyroid hormone synthesis and secretion. The stock-flow diagram of the sector is given in Figure 5.1. The sector involves two main stock variables; Hypo Wt (hypothalamus weight) and TRH. To begin with TRH, the only inflow to this stock is its secretion rate, and the only outflow from TRH is its clearance rate. The secretion rate of TRH is determined according to both the levels of the thyroid hormones in blood and current capability of the hypothalamus. But, before figuring out the actual TRH secretion from the hypothalamus, implied TRH secretion (imp TRH sec) is calculated. This implied secretion merely contains the effect of thyroid hormone concentration on secretion as if the hypothalamus has infinite capacity to secrete. That is, the implied secretion is a measure of how much TRH secretion 19 thyroid hormones would dictate regardless of the short-term secretory capability of the hypothalamus. Then, this implied secretion rate is exposed to the capacity restrictions of the hypothalamus and as much TRH as the capacity permits is secreted. It is calculated according to Equation 5.2 and 5.3. ratio of des hypo wt to hypo wt normal hypo wt gr for eff on HAT HAT des hypo wt eff of imp TRH sec on hypo wt Hypo Wt Hypo wt chg gr for eff of imp TRH sec on hypo wt normal hypo prod ratio of smth imp TRH sec to normal normal TRH sec smth imp TRH sec imp TRH sec eff of TH on TRH sec hypo cap TRH clear fr TRH TRH clear rate TRH sec rate ratio of imp TRH sec to hypo cap gr for hypo cap eff of cap on TRH sec gr for eff of TH on TRH sec MW of T3 log ratio of TH to normal free T3 in blood normal TH total free T3 molecules ratio of TH to normal TH total free T3&T4 molecules MW of T4 total free T4 molecules Figure 5.1. Stock-flow diagram of the hypothalamus sector. free T4 in blood 20 imp hypo sec = normal TRH sec × eff of TH on TRH sec (5.2) eff of TH on TRH sec = f (log(total TH / normal TH)) (5.3) where f (log(total TH / normal TH)) is defined as in Figure 5.2. 10 9 eff of TH on TRH sec 8 7 6 5 4 3 2 0 -1.1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1 log ratio TH to normal Figure 5.2. Effect of thyroid hormones on TRH secretion. When the thyroid hormone levels stay at their neutral values, the implied TRH secretion remains at the normal secretion value. However, when the circulating levels of thyroid hormones are disturbed from their set points, i.e. if the ratio of total circulating unbound thyroid hormones to the normal is different than one, the implied TRH secretion will be altered in the opposite direction of the shift in the thyroid hormone levels, as the negative feedback regulation necessitates. The work of Goodman (2009) suggests that the magnitude of the biological response of a target cell can be explained as a function of the logarithm of the concentration of the effector hormone. Thus, the logarithm of the ratio of hormone levels relative to the normal levels is used in the model when quantifying the hormonal effect on the response of the target tissue. So, not directly the ratio of circulating thyroid hormones to normal but the logarithm of it is used as an input to the graphical effect function shown in Figure 5.2. 21 According to the graphical function in Figure 5.2, if the thyroid hormone levels are far above their baseline values, the TRH secretion falls to considerably low values, though not to zero. The reason why hormone secretion does not reset is that most hormonesecreting tissues exhibit minimal (basal) secretion in the absence of stimulatory signals, as suggested by the related literature (Negi, 2009). The resistance of the hypothalamus not to reset but to remain at some basal secretion values are achieved via the nonzero right endpoint of the graphical function. The last step in the determination of TRH secretion is to expose the implied secretion rate to the capacity restrictions of the hypothalamus. The secretory capacity of hypothalamus is an upper bound to the total amount of TRH that can be maximally secreted daily. The way this capacity is defined is shown in Equation 5.4. hypo cap = Hypo Wt × normal hypo prod × 10 (5.4) Two main ingredients are involved in the definition of hypothalamic capacity; the normal productivity of the hypothalamus and the weight of the hypothalamus. The term productivity implies the production or release amount per unit weight. The normal productivity is defined to be the amount of TRH that one unit weight of the hypothalamus secretes under normal physiological conditions, and is a constant. This normal productivity is then multiplied with ten because it is assumed that one unit weight of the hypothalamus is capable of secreting ten times the normal amount maximally. And so, the overall capacity of the hypothalamus is defined as the product of the maximal secretory capacity of a unit weight with the current hypothalamus weight. TRH sec rate = eff of cap on hypo prod × hypo cap (5.5) eff of cap on TRH sec = f (imp TRH sec / hypo cap) (5.6) where f (imp TRH sec / hypo cap) is defined as in Figure 5.3. 22 Figure 5.3. Graphical function for the effect of capacity on TRH secretion. Equations 5.5 and 5.6 depict how the actual TRH secretion computed as a function of the capacity and implied secretion. According to the graphical function shown in Figure 5.3, when the implied secretion operates appreciably away from the capacity of hypothalamus, the capacity constraint does not become binding and the actual secretion equals to the implied. But, as the implied levels tend to push the capacity limit, a littler fraction of the implied secretion is allowed to realize. The secretory capacity of hypothalamus is fully utilized only if the implied secretion considerably exceeds the capacity. So, the extent at which the falls in thyroid hormone levels immediately influence the TRH secretion is confined to the short-term adaptation competence of the hypothalamic cells. If somehow a very high level of TRH secretion is demanded, the hypothalamus would only secrete as much as its existing capacity permits. So, this means that when necessary, the hypothalamus would utilize its maximum capacity to meet high TRH requests immediately, but may not suffice to conform the “orders” in short term. The formulations for the calculation of the TRH secretion provide some flexibility in the short run. As explained in section 5.1.1, the weight of the hypothalamus can change in conditions where it is persistently forced to over- or underfunction. In the model, this phenomenon is constructed as such: First, implied secretion rate is calculated. This implied 23 secretion may or may not actualize depending on the capacity limits. However, it is important to retain this piece of information because it tells how much the secretion would have been if there were no restrictions on it. This information does not immediately show its effect on the weight of the hypothalamus, and that’s what is meant by the term “persistent”. Generally, hormone-secreting tissues in a way accommodate themselves to the needs of the body after some time. That is, these tissues do not opt for weight adjustment in case of transient and drastic shifts from the baseline values, and usually show some inertia against weight changes. For this reason, the implied secretion levels are smoothed with a third order information delay structure in the model, where the overall delay duration is chosen to be 20 days. Smoothing provides a defence mechanism to preserve the normal weight against transient switches in secretion rates. Yet, smoothing is the first step. The second step is to check whether the smoothed implied secretion rates exceed some limits. This is done to ensure that the hypothalamus indeed functions at levels appreciably away from normals, and there is need for modifications in the weight. Thus, the weight of the hypothalamus is affected only if the smoothed implied secretion values relative to the normal values surpass some threshold values. The graphical function depicting this effect is shown in Figure 5.4. eff of imp TRH sec on hypo wt 3.7 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 11 12 ratio of smth imp TRH sec to normal Figure 5.4. Graphical function for the effect of implied TRH secretion on hypothalamus weight. 24 The input to this graphical function is the ratio of smoothed implied TRH secretion to the normal TRH secretion, and the output turns out to be some coefficient to be multiplied with the normal hypothalamus weight to figure the desired hypothalamus weight. The desired hypothalamus weight constitutes a target for the current hypothalamus weight. Gradually, the hypothalamus converges to that target value. Convergence to the target value is facilitated through the classical stock adjustment formulation, which is shown in Equation 5.8. des hypo wt = normal hypo wt × eff of imp TRH sec on hypo wt (5.7) Hypo wt chg = (des hypo wt −Hypo Wt) / HAT (5.8) The above equation is the formula for the flow of the stock that stands for the hypothalamus size. The adjustment time is a measure of how fast the hormone-secreting tissue tends to correct the difference between the target and current value. The adjustment time for hypothalamic weight change (HAT) is not a constant; it changes according to the ratio of the desired hypothalamus weight to the current hypothalamus weight (ratio of des hypo wt to hypo wt). It is not very reasonable to set the speed to correct the discrepancy regardless of the magnitude of that discrepancy. The variable ratio of des hypo wt to hypo wt is used as a measure of convergence speed of the hypothalamus to reach the desired level. If the desired weight is too high relative to the current weight, then it should take more time for the hypothalamus to adjust itself to the desired weight. In a sense, the current weight is regarded like the capacity of hypothalamus that determines the rapidity of approaching the desired level. Thus, as ratio of des hypo wt to hypo wt increases, the adjustment time increases too. The graphical function for HAT is shown in Figure 5.5. Ultimately, depending on the level of thyroid hormones in blood, and the capacity of the hypothalamus, a certain amount of TRH is released into the portal circulation to interact with the thyrotrophs in the anterior pituitary and trigger TSH secretion. 25 250 eff on HAT 200 150 100 30 1 1.5 2 2.5 3 3.5 4 4.5 5 ratio of des hypo wt to hypo wt Figure 5.5. Graphical function for effect on hypothalamic adjustment time. 5.2. Pituitary Sector 5.2.1. Background Information The pituitary gland, or alternatively called hypophysis, is a complex endocrine organ positioned in the sella turcica, a bony cavity at the base of the brain, and is linked to the hypothalamus by a stalk (Guyton and Hall, 2006). The weight of this gland is approximately 600 milligrams in adult human (Donovan, 1966; Sodeman and Sodeman, 1985; Kronenberg et al., 2008; Guyton and Hall, 2006). The pituitary secretes many hormones, which take part in various physiological processes by either acting directly on the target cells, or stimulating other endocrine glands to secrete hormones leading to alterations in body function. The human pituitary is comprised of two morphologically and functionally distinct glands that are connected to the hypothalamus. These two glands are called the neurohypophysis and the adenohypophysis, also known as the anterior pituitary. The anterior lobe of the pituitary comprises 75% of the pituitary gland. The cells of the anterior lobe secrete six different hormones, and a distinct specialized cluster of cells secretes each of these hormones. TSH is one of the hormones that the anterior pituitary is in charge of synthesizing and secreting, and the cells specialized for TSH are called 26 thyrotrophs. The thyrotrophs, i.e. TSH-secreting cells, compose 5% of the anterior pituitary cells (Bhagavan, 2002; Guyton and Hall, 2006; Kronenberg et al., 2008; Rhoades and Bell, 2009). TSH, also known as thyrotropin, is the principal regulator of thyroid hormone synthesis and secretion for it is the eventual messenger in the stimulation of the thyroid gland. As noted earlier, TRH plays the major role in the positive regulation of TSH secretion. Upon secretion, TRH reaches the anterior pituitary through the portal blood system, impinges upon the thyrotrophs, and binds to specific receptors on these cells. Binding of TRH with its receptors on thyrotrophs activates a number of intracellular mechanisms which ultimately lead to TSH release. However, it is not only TRH that influences the rate of TSH secretion, but also the levels of thyroid hormones in blood. Thyroid hormones exert a suppressive, negative feedback effect on the thyrotrophs to prevent the oversecretion of TSH, as opposed to the augmenting effect of TRH. Some portion of circulating free thyroid hormones binds with the unique thyroid hormone receptors, TR’s, on the thyrotrophs, and exerts a suppressive effect on TSH release. This means that an increase in circulating thyroid hormone concentrations would lead to a reduction in the rate of TSH secretion; and a decrease would result in a rise in TSH secretion. Consequently, the magnitude of TSH secretion is induced by the opposing signals to the anterior pituitary, one by TRH and the other by the thyroid hormones (Bhagavan, 2002; Guyton and Hall, 2006). The time it takes for the contrasting effects of TRH and thyroid hormones on TSH release to be revealed are different indeed. TRH elicits a prompt release of TSH within minutes (~15 minutes), while the inhibitory effect of thyroid hormones becomes evident after several hours (Bhagavan, 2002). Still, the time lag between the effect of TRH and thyroid hormones on TSH does not make much difference because the time unit of our model is one day and all these occur within a day anyway. Upon stimulation of the anterior pituitary by TRH, TSH is released into the circulation. 27 TSH is typically measured in “microunits” (µU) or “milliunits” (mU). Normal range for TSH secretion rate is 40-150 µU/day, and for circulating TSH in plasma 0.3-4 µU/ml (Kronenberg et al., 2008; Oertli and Udelsman, 2007). And, the half-life of TSH is about one hour (Negi, 2009). It is stated that long-standing hypothyroidism may lead to pituitary enlargement (Melmed, 2002), and increasing the negative feedback by any mechanism may result in atrophy of the thyrotrophs (Tucker, 1999). In other words, prolonged overstimulation may lead to expansion in pituitary size, and conversely sustained understimulation to shrinkage. 5.2.2. Fundamental Approach and Assumptions In this study, when speaking of changes in the weight of the pituitary, it will always be referred to the thyrotrophs, i.e. cells that secrete the TSH. The assumptions about the sensitivity of target cells explained in hypothalamus sector are valid for the pituitary too. As aforesaid, the set points for the levels of hormones in blood are taken to be the absolute quantities of the hormones rather than their concentrations. This is simply done by multiplying the relative concentration of the hormone with the total plasma volume. In this model, plasma volume is taken as 3 litres (Rhoades, 2009). This approach is qualitatively and quantitatively valid both for TSH and the two thyroid hormones. 5.2.3. Description of the Structure The structure of this sector is nearly the same as that of the hypothalamus sector. Two main stock variables are involved; TSH and Pituitary Weight (see the stock-flow diagram in Figure 5.6). The distinction is that TSH secretion has two effectors as opposed to thyroid hormones being the single effector of TRH secretion. Here, it is not only the circulating thyroid hormones that act on the secretion rate of TSH, but also TRH from hypothalamus. As mentioned earlier, the functioning of the thyroid hormone system is governed by a double-armed negative feedback mechanism. So, both the stimulant effect 28 of TRH and the inhibitory effect of thyroid hormones ought to be taken into account when figuring the rate of TSH secretion. The impact of these two factors is formulated as the product of two distinct effect functions. The one that counts for thyroid hormone inhibition is a decreasing function, and the one for TRH is an increasing one. gr for eff on PAT ratio of des pit wt to pit wt normal pit wt PAT des pit wt thres gr for pit <TRH> Pit Wt eff of smth imp TRH on pit wt Pit wt chg normal pit prod ratio of smth imp TSH sec to normal normal TRH smth imp TSH sec ratio of TRH to normal normal TSH sec eff of TRH on TSH sec gr for eff of TRH on TSH sec TSH TSH clear rate TSH sec rate eff of cap on TSH sec imp TSH sec log ratio of TRH to normal TSH clear fr pit cap ratio of imp TSH sec to pit cap gr for pit cap eff of TH on TSH sec log ratio of TH to normal gr for eff of TH on TSH sec Figure 5.6. Stock-flow diagram of the pituitary sector. The work of Guyton and Hall (2006) states that when the blood flow in portal vessels from hypothalamus to pituitary is completely hindered, TSH secretion rate diminishes 29 substantially but isn’t cut back to zero, remains at basal levels. The qualitative and quantitative structure of Equation 5.9 is constructed based on this statement. First, it is ensured that the graphical function for the effect of TRH on TSH secretion does not become zero, but yields a considerably small value when TRH stimulus is non-existent. Second, from this statement it can be inferred that the prerequisite for significant TSH secretion is TRH stimulation; low levels of thyroid hormones alone wouldn’t help enhance TSH secretion. Thus, effect of thyroid hormones should in a way depend on TRH effect; it should not be allowed to act independently. The dependency is provided by the multiplicative formulation. By enforcing the graphical function for thyroid hormone effect to yield the value one for all subnormal levels of thyroid hormones, they are only allowed to abate the existing stimulatory impact of TRH on TSH secretion. The implied secretion rates are calculated as a function of the logarithm of the ratio of the stimulant or inhibitory hormone level to its normal level, as in the hypothalamus sector. Here again, implied secretion is computed before the actual. The calculation of implied secretion includes both the effect of thyroid hormones and TRH, and excludes the capacity restriction. imp TSH sec = eff of TRH on TSH sec × eff of TH on TSH sec × normal TSH sec (5.9) eff of TRH on TSH sec = f (log(TRH / normal TRH) ) (5.10) eff of TH on TSH sec = f (log(total TH / normal TH)) (5.11) where f (log(total TH / normal TH)) and f (log(TRH / normal TRH)) are defined as in Figure 5.7 and Figure 5.8. And, in the exact same manner as in the hypothalamus sector, the implied secretion rate is constrained with capacity limit according to the following set of equations: 30 TSH sec rate = eff of cap on TSH sec × pit cap (5.12) eff of cap on TSH sec = f (imp TSH sec / pit cap) (5.13) 10.1 9 eff of TRH on TSH sec 8 7 6 5 4 3 2 0 -1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1.1 log ratio TRH to normal Figure 5.7. Graphical function for the effect of TRH on TSH secretion. eff of TH on TSH sec 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 log ratio TH to normal Figure 5.8. Graphical function for the effect of thyroid hormones on TSH secretion. Variations in secretion amounts happen in a very short while, almost immediately (Melmed, 2002). In the long run, the pituitary may adjust its weight in case over- or understimulation lingers for a sufficiently long period of time, in the same fashion as the 31 hypothalamus does. Here again, the implied secretion rates are smoothed via a third order information delay structure with a delay time of 20 days. In the same manner as clarified in section 3.1.3, the ratio of these smoothed values to its normal is filtered through a graphical function. Using the output of that function, a desired weight is calculated at every step. As mentioned before, the weight of the entire pituitary gland is around 600 mg, and the anterior lobe constitutes 75% of it. So, the anterior lobe weighs 450 mg. Since the TSH-secreting cells compose 5% of the anterior lobe, the normal weight of the pituitary is taken as 22.5 mg. The normal secretion rate of TSH is set to nearly 110 mU/day, and circulating TSH levels, i.e. the value of the stock, to 6.6 mU (i.e., 2.2 mU/l or µU/ml). 5.3. Thyroid Sector 5.3.1. Background Information Thyroid hormones are the primary regulators of metabolic functions in the body, and almost all cells of the body are regularly subjected to the actions of thyroid hormones. They are vital for proper development and differentiation of the cells of the human body, and their deficiency result in serious or even life-threatening diseases. The human thyroid gland, located just below the larynx, is one of the largest endocrine glands, and is comprised of two lobes attached to either side of and anterior to the trachea. The thyroid gland in a healthy adult normally weighs about 20 grams. The gland is composed of spherical follicles filled with a gel-like substance called colloid, and surrounded by follicular cells (alternatively called thyrocytes). The primary ingredient of colloid is a large protein called thyroglobulin, which is the site where the thyroid hormones are formed and stored (Guyton and Hall, 2006; Rhoades and Bell, 2009). The synthesis and secretion of thyroid hormones initiates upon the stimulation of the thyroid by TSH. Binding of TSH with TSH receptors on follicular cells triggers a series of intracellular functions in these cells to synthesize and release thyroid hormones. For the 32 synthesis of thyroid hormones, iodide is indispensable. So, the first step in the formation of these hormones is the uptake of available iodide into follicular cells. There, iodide is converted into an active form called iodine, which is the constituent part of thyroid hormones. Then, iodine attaches to tyrosine residues within thyroglobulin molecules. This process, i.e. binding of iodine to tyrosine residues within thyroglobulin, is called organification. Coupling of one iodine atom with a tyrosine molecule creates monoiodotyrosine (MIT). When MIT is iodized once more, a diiodotyrosine (DIT) molecule is formed. So, tyrosine is first iodized to monoiodotyrosine and then to diiodotyrosine. And, thyroid hormones are formed from these two kinds of molecules. When one MIT and one DIT molecule couples, one molecule of triiodothyronine, T3, is formed; and when two DIT’s come together, one molecule of thyroxine, T4, is created (Bhagavan, 2002; Guyton and Hall, 2006; Rhoades and Bell, 2009). So, formation of one unit of T4 necessitates double the amount of iodine used for the production of one unit of T3. The molecular weights of T3 and T4 are 651 and 777 Da, respectively (Bhagavan, 2002), where Da (Dalton) is a unit commonly used to measure mass on atomic or molecular scale, and has a value of about 1.66054 × 10-24 g (Raymond, 2009). After being synthesized, thyroid hormones reside in thyroglobulins, unless cleaved from them on demand. Stimulation of thyroid glandular cells by TSH not only promotes the synthesis of thyroid hormones, but also the secretion of them. In fact, the most rapid impact of TSH stimulation on the thyroid is the initiation of breakdown of thyroglobulin, which results in the secretion of T3 and T4 within half an hour. For this to happen, follicular cells engulf bits of the colloid, internalizes the colloid droplet into the cells, disintegrates it by the help of enzymes, allow the hormones to discharge from the storage element thyroglobulin, and release them into the blood circulation. So, TSH stimulation simultaneously activates all the secretory mechanisms of thyroid glandular cells, but the fastest among them is the secretion process. Though synthesis requires a relatively longer time compared to the secretion process, it still happens within one day under normal conditions (Guyton and Hall, 2006; Werner et al., 2005; Rhoades and Bell, 2009). 33 One unique characteristic of thyroid gland, in contrast to most endocrine glands, is that it has a considerably large capacity to store thyroid hormones in itself. In literature, it is stated that the thyroid gland is able to store approximately two months’ supply of thyroid hormones in it. Thus, if synthesis of thyroid hormone ceases, the physiologic effects may not be recognized for about two months (Molina, 2004; Guyton and Hall, 2006; Kronenberg et al., 2008). In a sense, these stores serve as buffers to guard against sudden and transient dysfunctions in the thyroid system and thus helps preserve the healthy state of the body. Normally, T4 constitutes 93% of the thyroid hormones released daily from the thyroid gland, and only 7% is T3. The functions of these two hormones are qualitatively the same; however, they differ from each other in terms of rapidity and intensity of action. T3 is said to be physiologically active form of the thyroid hormones (Guyton and Hall, 2006; Rhoades and Bell, 2009). T4 can only be synthesized in the thyroid gland and is the major secretory product of the thyroid gland. The normal production and secretion rate of T4 from the thyroid is approximately 90 µg/day. On the other hand, only 20% of T3 is produced directly by the thyroid; the rest is generated by enzymatic removal of one iodine atom (deiodination) of T4 molecules in peripheral tissues. The production rate of T3, including the peripheral conversions, is 35 µg/day (Braunwald et al., 2001; Bhagavan, 2002). Upon stimulation by TSH, T3 and T4 are released into the blood stream. Most of the T3 and T4 molecules become bound to plasma proteins, only less than 1% of them circulate in free form. Normally, the range for total plasma concentration is 4-11 µg/dl for T4, and 75-220 ng/dl for T3. Specifically, 0.02% of T4 and 0.3% of T3 circulates in unbound form. Merely the free portion of the hormones is biologically active and able to interact with target cells (Rhoades and Bell, 2009). And so, it is the circulating free hormones that feed back to the hypothalamus and pituitary to reduce the secretion of TRH and TSH (Bhagavan, 2002; Rhoades and Bell, 2009). 34 In general, hormones are “cleared” from the plasma by various means involving metabolic destruction by the tissues, binding with the tissues, excretion by the liver into the bile, and excretion by the kidneys into the urine (Guyton and Hall, 2006). For our case, it was recognized early that T4 and T3 were dispersed widely into tissues in addition to the blood (Hays, 2009). It is stated that about 40% of plasma T4 is converted to T3 and about 40% to reverse triiodothyronine (rT3), which is a metabolically inactive form. The half-life of T4 in the bloodstream is approximately 7 days, whereas that of circulating T3 is about 1 day (Rhoades and Bell, 2009). Like the hypothalamus and pituitary, the weight of thyroid gland may also be altered under certain circumstances. The work of Vassart and Dumont (1992) asserts that hypophysectomy, hypopituitarism, or an isolated TSH deficiency, i.e. the conditions that clearly abate the thyroid function, leads to thyroid atrophy. Conversely, chronic stimulation of the thyroid for some reason is stated to enhance thyroid growth. 5.3.2. Fundamental Approach and Assumptions The stock variables for the blood levels of thyroid hormones represent the total circulating amount of them, both bound and unbound (free). Bound and unbound hormones are in a dynamic equilibrium with each other. However, only the free portion is biologically active. So, it is merely the small unbound portion of thyroid hormones that is able to diffuse into peripheral tissues, induce metabolic effects, and that undergoes deiodination or degradation. When free hormones disperse out of the blood stream, the equilibrium is disturbed. Thus, the carrier proteins free additional thyroid hormones until the equilibrium state is restored (Kronenberg et al., 2008; Rhoades and Bell, 2009; Martini, 2007). As far as the scope and the objective of this study are concerned, this process is not explicitly modeled. Instead, for each case, e.g. diffusion of the hormone into peripheral tissues, conversion into some other form etc., it is simply assumed that some fraction of the total circulating thyroid hormones leave the blood stream and diffuse into relevant parts of the body. 35 Related literature suggests that under certain physiological or pathological conditions the total circulating hormone content and amount of plasma transport proteins may change, while the free hormone concentration may remain relatively normal. In the model, however, the quantity of each thyroid hormone that is unbound is assumed some given constant fraction of total circulating hormones. As mentioned earlier, under certain circumstances the weight of the thyroid may change. There are studies, however, where the volume of the thyroid is measured rather than its weight (probably because volume measurement does not require the ablation of the gland as in the assessment of the weight). It is assumed that changes in the volume of the thyroid go parallel with the changes in the weight. These two notions will be used interchangeably when necessary. 5.3.3. Description of the Structure This sector tries to illustrate the structure of the third and last tier in the control of thyroid hormone system. The stock-flow diagram of thyroid sector is given in Figure 5.9. It involves five main stock variables; Thy Wt (thyroid weight), T4 Store, T3 Store, T4 in Blood and T3 in Blood. As mentioned before, the thyroid synthesizes and secretes two hormones, T3 and T4. The structures that represent the mechanisms in the synthetic and secretory regulation of T3 and T4 are quite similar, though not the same. To begin with T4, there are two stocks and their associated flows that involve the main measures about T4 in the model. The stock variable named T4 in Blood represents the overall amount, both bound and unbound, of circulating T4 in blood. It is subject to a single inflow, and four different outflows. The single inflow to this stock is the secretion rate of T4. Under normal conditions, T4 secretion rate is set to 90 µg/day. 36 gr for eff of imp TH sec on thy wt <TSH> TAT ratio of smth imp TH sec to normal normal TSH ratio of TSH to normal smth imp TH sec log ratio of TSH short smth imp TH sec <normal TH sec> ratio of short smth imp TH sec to normal <I in Thy> gr for eff on TAT I inhib thres des thy wt normal I in thy Thy Wt Thy wt chg normal TH sec imp TH sec <imp TH sec> ratio of des thy wt to thy wt eff of imp TH sec on normal thy wt thy wt eff of TSH on TH sec ratio of imp TH sec to thy cap gr for thy cap normal thy prod thy cap eff of cap on TH sec ratio of I in thy to thres eff of I on thy cap pot TH sec gr for eff of I on cap fr of T4 sec <fr of T3 sec> pot T4 sec gr for eff of TSH ratio of on TH sec pos to pot gr for TH <disc btw pot TH syn T4 sec store cap gr forand I rest TH syn> Conv thy stim pos T4 Abs rate <T3 syn sec rate to on T3 fr of T4 by <pot TH syn> rate> eff of T4 rT3 T4 tissues store cap clear fr eff of normal thy stim total free T3 ratio of T3 T4 in Blood T4 Store on T3 fr to T4 molecules T4 syn rate T4 clear rate T4 sec rate pot fr of gr for fr ratio of T4 to T3 des T4 syn disc from T3 sec T3 to T4 of T3 sec normal conv rate des T4 store <fr of T4 sec> T4 to T3 conv fr T4 store adj total free T4 thy syn cap T4 to T3 for T4 conv fr molecules fr of T3 sec des T4 store <fr of T4 gr for eff sec> ratio of of T3 on <gr for AT for des T4 syn total normal TH peri conv thy cap> to cap <thy cap> store restor store eff of T3 conc on peri conv ratio of <fr of <thy syn cap thy syn cap des T3 eff of thy normal T3 syn to cap T3 sec> for T4> cap on T4 for T3 syn in blood T3 store des T3 syn adj des T3 store eff of thy pot T4 syn ratio of T3 pot T3 syn T3 from cap on to normal deiod of T3 syn T4 disc from <gr for pot I cons pot TH syn thy cap> for T4 des T3 store T3 clear fr pot I cons for T3 T3 in Blood T3 Store T3 clear rate T3 syn rate disc btw pot TH syn T3 sec rate and I rest TH syn disc btw pot eff of T3 T3 syn and I Abs rate pot T3 sec store cap rest T3 syn eff of pref T3 ratio of of T3 by syn on red in pos T3 sec pot to pos ratio of cap rest I tissues T3 syn I cons cons to pot <fr of <pot TH ratio of disc btw T3 sec> sec> I rest syn and eff of thy total pot syn to pot total I gr for eff of I cap total pot syn cons ratio of <gr for TH gr for I under I pref T3 syn pos to pot store cap> store cap rest T3 sec cap <I in Thy> pos I cons pot total I cons Figure 5.9. Stock-flow diagram of the thyroid sector. 37 As the stock-flow diagram reveals, the amount of T4 to be secreted is withdrawn from the T4 Store stock. In general, it is not very reasonable to let the outflow of a stock to act independent from the value of that stock. And also, the body does not utilize its resources regardless of its present status. The thyroid hormone stores are high when compared to the daily normal requirements. The work of Bürgi (2010) suggests that when thyroid hormone synthesis is obstructed somehow, hormone secretion diminishes only after significant amount of the stores is depleted, which takes several weeks in human. Considering this proposition, it is assumed that there has to be some limit to the amount of thyroid hormones that could maximally be withdrawn and secreted from these stores within a day. This limit is based on the current level of the stores in the model. If the demand dictated by the hormonal stimuli somehow exceeds or is sufficiently close to the amount of hormone that the target tissue is willing to maximally release, it would adjust its sensitivity according to its existing state and respond as a function of both the imposed requirements and its current status. As the stores diminish, the amount that the gland would be willing to release declines too. So, the potential rate of T4 secretion is constrained by an effect function which counts for the capacity of hormone stores. The equation for T4 secretion rate is formulated as shown in Equation 5.14. T4 sec rate = pot T4 sec × eff of T4 store cap (5.14) eff of T4 store cap = f (pos T4 sec / pot T4 sec) (5.15) where f (pos T4 sec / pot T4 sec) is defined as in Figure 5.10. The input to the graphical effect function in Figure 5.10 is the ratio of possible T4 secretion (pos T4 sec) to potential T4 section (pot T4 sec) where possible T4 secretion is computed by multiplying some coefficient with the current value of T4 store. Possible T4 secretion means that given the current value of T4 store, the thyroid would avoid to release more than a certain fraction of its supplies, as explained in the preceding paragraph. This fraction is selected to be 0.15 in the model. According to this graphical function, the entire 38 potential amount is allowed for secretion if it is not sufficiently close to the upper limit, i.e. possible secretion. But as the potential value tends toward this maximum possible, the thyroid becomes reluctant to push its limits, and releases the whole possible amount only when fairly more than the maximum possible is desired. Consistent with the findings suggested by Bürgi (2010), when daily demands are approximately at normal levels, the constraint on daily secretion imposed by possible T4 secretion does not become a binding one in the model until an appreciable portion of stores is exhausted. 1 eff of TH store cap 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.5 ratio of pos to pot T4 sec Figure 5.10. Graphical function for the effect of thyroid hormone store capacity. The second factor that participates in the calculation of T4 secretion is the potential T4 secretion (pot T4 sec). It is some target amount for T4 secretion rate and is calculated using Equation 5.16. pot T4 sec = pot TH sec × fr of T4 sec (5.16) pot TH sec = eff of cap on TH sec × thy cap (5.17) To begin with the first element in the equation, potential thyroid hormone secretion (pot TH sec) is some raw value for total thyroid hormone secretion not yet exposed to hormone store limitation. It is calculated in the same way as the TRH and TSH secretions 39 have been; that is, by subjecting the implied total thyroid hormone secretion to the effect of thyroid capacity. Here, since some additional constraints exist to possibly limit this rate further, it is attributed to a potential value rather than actual. imp TH sec = normal TH sec × eff of TSH on TH sec (5.18) eff of TSH on TH sec = f (log(TSH / normal TSH)) (5.19) where f (log(TSH / normal TSH)) is depicted in Figure 5.11. 10.1 9 eff of TSH on TH sec 8 7 6 5 4 3 2 0 -1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1.1 log ratio TSH to normal Figure 5.11. The graphical function for the effect of TSH on thyroid hormone secretion. The calculation of the potential thyroid hormone secretion is done by constraining the implied thyroid hormone secretion with thyroid capacity, as just mentioned, but the capacity of the thyroid is computed in a slightly modified way in this case (see Equation 5.20). First, as in the previous two sectors, ten times the normal thyroid productivity is multiplied with the weight of the thyroid to find the maximal amount for secretion or synthesis that the thyroid is capable of. Then, this term is further multiplied with the effect of iodide on thyroid capacity (eff of I on thy cap) to count for the mild impairment in thyroidal secretion in high intrathyroidal iodide concentrations. There are a number of 40 studies where high doses of iodine are administered to subjects for a short period of time, and mild but significant decreases in serum T3 and T4 levels and increase in TSH levels are observed (Vagenakis et al., 1973; Gardner et al., 1988; Paul et al., 1988; Philippou et al., 1992; Georgitis et al., 1993; Namba et al., 1993; Lemar et al., 1995; McKonigal et al., 2000). The decrease in thyroid hormone concentrations is attributed to high intrathyroidal iodine concentrations. So, depending on the concentration, intrathyroidal iodine concentration may contract the capacity of the thyroid. thy cap = normal thy prod × Thy Wt × 10 × eff of I on thy cap (5.20) eff of I on thy cap = f (I in Thy / I inhib thres) (5.21) where f(I in Thy / I inhib thres) is defined as in Figure 5.12. 1 eff of I on thy cap 0.8 0.6 0.4 0.2 0.085 0.9 1 1.1 1.2 1.3 1.4 1.5 ratio of I in thy to thres Figure 5.12. Graphical function for the effect of iodine on thyroid capacity. According to the graphical function in Figure 5.12, as the iodine concentration within the gland tends to the threshold value, the capacity of the thyroid declines. But the real inhibitory effect takes place when the iodine levels surpass the threshold value because only then the total capacity of the thyroid falls below the value that is necessitated for normal hormone production. As the iodine levels exceed the threshold more, the inhibitory 41 effect on thyroid capacity rises too. To find the inhibitory threshold, firstly the thyroidal iodine levels are smoothed with a third order information delay structure using a delay time of 30 days, and then multiplied by 1.1. Then, the maximum of this value and 400 will be assigned to I inhib thres because it is assumed that this threshold cannot fall below 400 µg. The graphical function for the effect of capacity on thyroid hormone secretion (eff of cap on TH sec) is not explicitly shown because it is the same as the ones in the previous two sectors. To figure only the amount of T4 secretion out of the total thyroid hormone secretion, the potential total secretion is multiplied with the fraction of T4 secretion (fr of T4 sec). As mentioned before, a specific fraction of secretion is reserved for each of the two thyroid hormones. Normally, this fraction is approximately 93% (90/97) for T4 in the model. However, it is not a constant and can be altered under certain conditions. The way this fraction is revised will be elucidated later in this section. As explained in section 5.3.1, the majority of T3 is obtained via deiodination of T4 molecules. Under normal conditions, the glandular secretion of T3 comprises only 20% of the total T3 release. So, of the daily 35 µg T3 secretion, 28 µg is acquired through the conversion of T4 to T3. The outflow named T4 to T3 conv rate serves for this purpose. As is the fraction of T4 productivity for secretion, the fraction of T4 to be deiodinated is variable too. The conditions under which this fraction changes will be clarified later in this section. Other two outflows from T4 in Blood are conversion rate to rT3 (Conv rate to rT3) and (Abs rate of T4 by tissues). The values of these flows are computed by multiplying the current stock value with some constant fraction. Normally, 32 µg of T4 is assumed to be deiodinated into rT3, which comprises about 36% of the daily T4 secretion. The last outflow, called T4 clear rate, stands for the rate of (renal) clearance of T4 (with the halflife of 7 days) from the blood. 42 The flows into and from the stock variable for the level of T3 in blood resemble to those of T4 in blood. There are four flows that influence the level of T3 in blood. The first inflow to T3 in blood is its secretion rate. T3 secretion rate is formulated in the same manner as T4 secretion rate. The fraction of secretion for T3 is taken approximately 7% (7/97) under normal conditions. This fraction may vary under certain circumstances, as does the fraction of productivity for T4. The second inflow to T3 in Blood is the rate of deiodination of T4 into T3. The reason why the related outflow from T4 doesn’t directly enter into T3 in Blood stock is that some numerical adjustment was required to account for the mass change due to departure of iodine atoms from T4 molecules in the conversion process. As mentioned in T4 case, the value of the outflow named Abs rate of T3 by tissues is calculated by multiplying some coefficient with the stock value. Lastly, T3 clearance rate (T3 clear rate) is the renal clearance rate of T3 from the blood (with a half-life of 1 day). So, the only difference from T4 is that T3 in blood does not have an outflow analogous to the rate of conversion of T4 to rT3; all other flows are qualitatively the same as those of T4 in Blood. As mentioned in detail in section 3.3.1, the thyroid has the capacity to store appreciable amounts of preformed thyroid hormones. The two stocks in the model, T4 Store and T3 Store, stand for these hormone stores. Normally, these stocks contain two months’ supply of thyroid hormones. The set point is 5400 µg for T4 store, and 420 µg for T3 store. The only inflow to these stocks is the synthesis rate of the related hormone. To determine the synthesis rate of either thyroid hormone, a desired synthesis rate is calculated at first. The classical stock adjustment formulation is used when constructing the equation for desired synthesis rate. The associated equations for T3 are shown in Equations 5.22 - 5.24. The related equations for T4 are of the same form as T3, and therefore will not be shown explicitly anew. The first ingredient in desired T3 synthesis is its secretion rate which acts to replenish the usual daily release rate from the stores. The second term is the stock adjustment term that helps close the gap between the desired and the current level of the 43 store. So, if the T3 store were valued below the desired level, then this formulation would not only compensate for the regular daily losses but also for the gap between the normal and current stock value. Conversely, if the stock levels were sufficiently above the desired levels, then the adjustment term would take a negative value to facilitate the convergence to normal stock values. des T3 syn rate = T3 sec rate + pot T3 store adj (5.22) pot T3 store adj = disc from des T3 store / AT for store restor (5.23) disc from des T3 store = des T3 store − T3 Store (5.24) The desired levels of hormone stores are not constant; they change according to the changes in the secretory fractions of thyroid hormones. In literature, it is stated that intrathyroidal fractions of hormone stores are in concordance with the fractional secretion rates of T3 and T4 from the thyroid when preferential secretion of T3 is the case (Brent, 2010). Thus, the total normal amount of hormone store within the thyroid (5820 μg) is multiplied by the related secretory fraction to figure the desired level for that hormone store. des T4 store = (total normal TH store) × fr of T4 sec (5.25) des T3 store = (total normal TH store) × fr of T3 sec (5.26) In cases where hormone stores are above their desired level, it is possible for the desired synthesis rate to take negative values. Though it is conceptually not illogical for a “desired” quantity to take a negative value, a negative actual synthesis rate is neither plausible nor possible. In other words, the gland wouldn’t freely discharge its excessive content just to get rid of it. Hence, when subjecting this desired value to some capacity constraints, the maximum of this desired value and zero will be taken. Lastly, as mentioned earlier, the adjustment time for store restoration (AT for store restor) is an indicator of how fast the gland is thought to be likely to close the gap between the target and current stock value, and is taken as 30 days here. 44 After the desired synthesis rate is set, it is firstly filtered through some capacity constraint, as done in the determination of secretion rates of thyroid hormones. This capacity is an upper bound for synthesis rate of T4 that the thyroid gland is thought to be capable of, as it was for the secretion rates. One should note that reducing the capacity of the thyroid only in extremely high concentrations of iodine does not mean that the cases where the intraglandular quantity of iodine itself doesn’t suffice to meet the requirements for the synthesis of hormones are ignored. The potential synthesis rate will be further constrained at the time when the iodine availability is checked. The corresponding set of equations involved in the calculation of the potential synthesis is depicted below. pot T3 syn rate = thy syn cap for T3 × eff of thy cap on T3 syn (5.27) thy syn cap for T3 = thy cap × fr of T3 sec (5.28) eff of thy cap on T3 syn = f (max{des T3 syn , 0} / thy syn cap for T3) (5.29) As Equation 5.26 reveals, by multiplying the fraction of T3 secretion with the total capacity, the relevant portion of thyroidal capacity reserved for T3 synthesis (thy syn cap for T3) is taken to restrain the (nonnegative) desired amount of T3 synthesis. And, the nominator of the input to the effect of thyroid capacity on T3 synthesis is taken as the maximum of the desired T3 synthesis and zero in order to be able to discard negative synthesis rates. The graphical function for this capacity effect is the same as the others, and thus not presented over again. The last step that needs to be taken before deciding on the actual synthesis rates is to check the iodine availability. If the iodine demand for thyroid hormone synthesis cannot be fully met, the available amount of iodine is redistributed for the synthesis of two thyroid hormones not according to the former fractions but rather in favor of T3. The maintenance of normal levels of T3 in the body is of primary importance for its being the biologically active form, i.e. the one that is utilized by the cells, of thyroid hormones. So, the deficiency of T3 becomes more of an issue as compared to that of T4, 45 and necessitates effective balancing mechanisms to guard against life-threatening situations. Related literature suggests that unless the impairment of thyroid hormone production is severe, T3 levels in blood remain within normal limits (Werner et al., 2005). Thus, the measurement of blood T3 levels does not provide differentiating information in the diagnosis of hypothyroidism. It is stated that numerous intrathyroidal and extrathyroidal mechanisms act in concert in order to retain T3 availability. Within the thyroid gland, both the secretion and synthesis of T3 is favored to T4. Moreover, peripheral conversion of T4 to T3 increases in the hypothyroid state. So, at the expense of decreasing the circulating T4 concentrations, the mentioned compensatory mechanisms operate to preserve normal, healthy T3 levels (Brent, 2010; Werner et al., 2005; Greenstein and Wood, 2011). One of the most common causes of hypothyroidism is iodine deficiency. So, if an implication of iodine deficiency is detected during the availability check, synthesis of T3 will be favored. But the point where iodine availability checked is not the only one where T3 is favored; the preferential T3 synthesis will again become a current issue when the secretory fractions of thyroid hormones are revised. After determining the potential synthesis rate, the amount of iodine necessitated is calculated by simply multiplying that amount with the weight percentage of iodine in T3 molecule. The same procedure that is explained up to this point is applied for determining T4 synthesis rate too. So, at this point, the total amount of iodine required for total potential thyroid hormone synthesis is at hand. To check the adequacy of iodine supplies, the overall amount of necessary iodine is compared to the possible amount. If consumption of this potential quantity is not fully permitted, the fractions of productivity reserved for each hormone is revised in favor of T3 to mitigate the vital implications of iodine deficiency. The rationale behind the constrained utilization of intraglandular hormone stores is extrapolated to the consumption of iodine stores too. The first step to examine the availability of iodine supplies is to send the ratio of the potential iodine consumption (pot 46 total I cons) to the maximum possible amount (pos I cons) to a graphical effect function as input. The graphical function is the same as the functions that stand for the effect of hormone store capacities, and thus is not shown explicitly. The maximum possible amount of daily iodine consumption is presumably set as 15% of the existing intrathyroidal iodine stock. The value that this function yields is then multiplied with the maximum permissible amount to obtain the potential iodine consumption under iodine restriction (pot total I cons under I cap rest), as illustrated in Equation 5.30. pot total I cons under I cap rest = pos I cons × eff of thy I cap (5.30) If the potential iodine consumption lies decently below the maximum tolerable amount, then all the iodine demanded can be delivered. As the sought quantity approaches to the limit, the gland tends to act more economical and agrees to give the daily maximum only when fairly higher quantities are requested. This kind of control on the outflow of the stock is also implemented in the thyroid hormone stores, as explained before. Following the determination of accessible quantity of iodine, the synthesis rate allocated to each hormone is to be revised. For this purpose, the discrepancy between the synthesis with the permitted amount of iodine and the potential synthesis is calculated. Then, this discrepancy is divided to the value of potential synthesis rate (ratio of disc btw I rest TH syn and pot TH syn to pot TH syn). In the model, this ratio is regarded as an indicator of the severity of iodine deficiency, and thus the potential hypothyroidism. A graphical effect function is defined that uses this ratio as input (see Figure 5.13). The decreasing function in Figure 5.13 is used to attenuate the amount that will be subtracted from the potential T3 synthesis due to iodine availability constraint. If the preferential synthesis of T3 were not the case, the potential synthesis rates of each hormone would be multiplied with ratio of cap rest I cons to pot to find the actual synthesis rates. Since T3 is the critical hormone, in cases where the iodine supply is a binding constraint, the reduction in T3 synthesis should be relatively smaller than that of 47 T4. The function eff of pref T3 syn on red in T3 syn is created to decrease the quantity that is subtracted from the potential T3 synthesis rate. Because the value that the function in Figure 5.13 yields ranges from one to zero, the resultant fraction declines as the discrepancy increases, i.e. as the consumable quantity of iodine declines. eff of pref T3 syn on red in T3 syn 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 ratio of disc btw I rest TH syn and pot TH syn to pot TH syn Figure 5.13. Graphical function for the effect of preferential T3 synthesis on reduction in T3 synthesis. When correcting the shares of production rates for T3 and T4, two conditions must be satisfied. First, the synthesis rate of the hormone under iodine restriction should not exceed its potential synthesis rate. Second, the revised synthesis rate should not go beyond the total potential synthesis rate. Subtracting some fraction of the discrepancy from the total potential synthesis rate is enough to satisfy the first condition, but not the second one. The second situation can happen when the total iodine-constrained synthesis rate is below the initial one necessitated by T3. The denominator of Equation 5.31 helps hinder this situation. The nominator of Equation 5.31 will always be less than or equal to the potential T3 synthesis rate (which is the rate not yet confined to iodine adequacy). And, the denominator of the equation can take a value greater than one when the second condition explained above is active. So, as the allowable total synthesis rate approaches to zero, the 48 nominator of the ratio within the maximum operator and the nominator of the equation tend to cancel out each other. Thereby, the resultant synthesis rate is prevented to exceed the feasible rate. T3 syn rate = pot T3 syn disc btw pot T3 syn and I rest T3 syn eff of pref T3 syn on red in T3 syn pot T3 syn max 1, pot TH syn - disc btw pot TH syn and I rest TH syn (5.31) Since T4 is the secondary hormone in the calculations, its synthesis rate is just the remaining of the total permissible synthesis rate. The related equation is shown below. T4 syn rate = pot TH syn − disc btw pot TH syn and I rest TH syn − T3 syn rate (5.32) One should note that after checking the possible amount, altering the potential synthesis rates in favor of T3 does not hurt the initial principle of obtaining a feasible amount of thyroidal iodine to be consumed because less iodine is required for unit T3 synthesis as compared to T4. So, the ultimate amount of iodine expended will always be less than or equal to the quantity that is found to be feasible after the availability check. Another mechanism that operates to preserve T3 availability is the adaptability of the conversion fraction of T4 to T3. To ensure the T3 availability, this fraction increases to some extent in hypothyroid state. The fraction is adjusted as a function of the circulating T3 levels relative to its set point. The related graphical effect function is illustrated in Figure 5.14. As the ratio of currently circulating T3 levels to the normal levels decrease, the value that the function yields increases and then saturates after some point. This means, if the T3 level falls below its set point, the rate of conversion rises to compensate for the gap. 49 eff of T3 conc on peri conv 1.7 1.2 0.8 0.4 0 0 0.2 0.4 0.6 0.8 1 ratio of T3 to normal Figure 5.14. Graphical function for the effect of T3 concentration on peripheral conversion. As mentioned earlier in this section, fractions of T3 and T4 secretion are not constant. The modifications in these fractions are done in three phases. Firstly, circulating T3 and T4 levels don’t necessarily differ from their set points with the same proportion. Thus, it is thought that it wouldn’t be very reasonable to allocate the total secretion rate regardless of the individual differences from the set points by using just some constant fractions. For this purpose, the overall thyroidal secretion is distributed inversely proportional (because thyroid hormones feed back negatively) to the ratio of the related hormone to its normal value. Equation 5.33 shows how the formulation for this first part is derived. For ease of demonstration, variable name x is used for the fraction of T3 secretion not yet undergone the second and third steps. As mentioned earlier in this section, 7/97 and 90/97 are the normal thyroidal secretion fractions of T3 and T4, respectively. This first part of the formulation does not explicitly lead to preferential secretion of T3 in hypothyroid state; it just fairly distributes. After redistributing the fractions according to the individual deviations of two hormones from their set points, the second step is to consider the effect of the level of thyroid stimulation on the fractions. Normally, T3 of thyroidal origin comprises 20% of 50 total T3 production in the body. However, related literature reports that in patients with thyroid hyperfunction or hypofunction, a relatively higher fraction of the total T3 is delivered by the thyroid (Laurberg, 1984; Brent, 2008; Brent, 2010). Equation 5.34 shows the second step in the calculation of fraction of T3 secretion. 7/97 normal ratio of T3 to T4 x 90/97 1 x ratio of T3 to T4 1 ratio of T3 to T4 1 7 x normal ratio of T3 to T4 90 (5.33) 7 normal ratio of T3 to T4 90 x 7 ratio of T3 to T4 normal ratio of T3 to T4 90 7 normal ratio of T3 to T4 90 pot fr of T3 sec = eff of thy stim on T3 fr 7 ratio of T3 to T4 + normal ratio of T3 to T4 90 (5.34) The findings about preferential synthesis and secretion of T3 in hypothyroid state were previously mentioned in this section. Besides, it has been documented that in the thyroids of patients with Graves’ disease, a common source of hyperthyroidism, T3/T4 was consistently higher and was not due to iodine deficiency (Izumi and Larsen, 1977), and that thyroidal secretion of T3 rises from approximately 20% to 30% in Graves’ disease (Brent, 2008). The common point for cases where thyroidal secretion of T3 is relatively higher compared to the normal physiological conditions is that the thyroid is overstimulated. Depending on the capability of the thyroid, this hyperstimulation may result in hyperthyroidism or hypothyroidism, but the “potential demand” that the thyroid faces is 51 high in either case. In this respect, the ratio of implied thyroidal secretion to the normal can be taken as an indicator of stimulation level. But, it is thought that sudden shifts in the level of thyroid stimulation should not alter these fractions instantaneously; persistence in hyperstimulation must be sought. Thus, the smoothed version of this ratio rather than its instantaneous value is allowed to affect these fractions. Smoothing is achieved through a third order information delay structure with a delay time of 10 days. eff of thy stim on T3 fr 2.5 2 1.5 1 1 1.5 2 2.5 3 3.5 4 4.5 5 ratio of short smth imp TH sec to normal Figure 5.15. Graphical function for the effect of thyroid stimulation on T3 secretion fraction. Lastly, the fraction of T3 secretion is calculated as a function the potential value for this fraction. This final step is performed to prevent fr of T3 sec from exceeding the value one in potential (unrealistically) extreme cases because of multiplication with the coefficient that the effect of thyroid stimulation yields. The related graphical function is shown in Figure 5.16. The above mentioned mechanism directly counts for preferential T3 secretion. In addition, as aforementioned, the synthesis fractions are also modified in favor of T3 in cases of iodine deficiency. Hence, when that is the case, T3 stores will surely persist longer than T4 stores and availability of hormone supplies won’t easily become a binding constraint on T3 release, which in fact is an indirect reference to preferential secretion too. 52 1 fr of T3 sec 0.98 0.96 0.94 0.92 0.9 0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1 pot fr of T3 sec Figure 5.16. Graphical function for the fraction of T3 secretion. As the hypothalamus and pituitary do, the thyroid can change its weight under certain circumstances. The structure related to thyroid weight adjustments functions in the same manner as in the previous two sectors. So, the stock that stands for the weight of the thyroid is subjected to a single adjustment flow. This adjustment flow is formulated in the same way as the adjustment flow of the hypothalamus and pituitary, and thus will not be explained anew. 5.4. Iodine Sector 5.4.1. Background Information Iodine is a requisite substrate for the synthesis of thyroid hormones. It is present in foods usually as iodide, which is the inorganic form of iodine. Iodide is available in the form of iodized table salt, and also various kinds of food exist which are rich in iodide; e.g. seafood, kelp, spinach, soybeans, garlic etc. After being ingested, iodide is absorbed nearly completely in the stomach and duodenum. Related literature suggests that the absorption of dietary iodide is greater than 90% in healthy adults (Zimmermann, 2009). The absorbed iodide then diffuses in blood. Iodide is cleared from circulation mainly by the thyroid and 53 kidney. The clearance of circulating iodide by the thyroid is called iodide trapping. Within the thyroid gland, iodides are converted into an oxidized form of iodine. Once iodides undergo oxidation process, they are ready to participate in the production of thyroid hormones (Guyton and Hall, 2006). Due to its vital role in the functioning of the whole body, the thyroid gland is equipped with buffers to preserve the healthy state in case thyroid function is somehow impaired. As mentioned before, the thyroid reserves a considerably large amount of preformed thyroid hormones in it which is available for secretion on demand. Besides the hormone stocks, the thyroid gland has a large iodine supply which allows to maintain thyroid hormone synthesis in lack of iodine intake for some time. That is, in case of inadequate iodine intake, hormone synthesis persists until the available iodine stores has been depleted, unless, of course, the thyroid itself dysfunctions. In literature, it is stated that the normal dietary iodine intake is 150 µg/day, and the amount of intrathyroidal iodine stores range from 10 to 20 milligrams (Van Vliet and Polak, 2007). Iodide uptake from the circulation into the thyroid is primarily controlled by TSH (Guyton and Hall, 2006; Nyström et al., 2011). The rate of iodide trapping by the thyroid is altered depending on the magnitude of TSH stimulation; if TSH signals increase, amount of iodide trapped also rises, and if the stimuli decline, iodide trapping diminishes too. In addition to this, related literature postulates that the thyroid is able to “sense” its iodine content and adjust its sensitivity to TSH stimulation (Yadav, 2008). So, the intrathyroidal iodine supply can be autoregulated by an internal feedback mechanism which controls the intraglandular utilization of iodine and the thyroid response to TSH stimulation (Bhagavan, 2002). 5.4.2. Description of the Structure The iodine sector is composed of three stocks. Two of these stocks have real physical counterparts in the body, but the third one serves as an intermediate step in the calculation of iodide trapping rate. The stock-flow diagram of this sector is depicted in Figure 5.17. 54 I from T3 conv I from rT3 I from T4 in tissues retained I I from T3 in tissues I from deiod abs fr of I I intake excr fr Del for trap fr I in Blood Pot Trap Fr I excr I abs rate chg in trap fr gr for eff of TSH on I trap Trap rate gr for eff of thy wt on I trap disc eff of thy wt on I trap eff of TSH on I trapping des trap fr log ratio of TSH to normal I in Thy gr for des trap fr ratio of thy I conc to normal normal I in thy I cons I cons for T4 I cons for T3 Figure 5.17. Stock-flow diagram of the iodine sector. As can be seen in Figure 5.17, the stock Iodine in Blood is altered via two inflows and two outflows. The first inflow to the stock is the absorbed portion of dietary iodine (I abs rate), and the retained iodine from the deiodination or peripheral catabolism of thyroid hormones (I from deiod) is the second one. The normal dietary iodine intake is taken as 150 µg and the absorption fraction as 95%. The outflow Excretion of I represents the renal clearance of plasma iodine, and Trap rate the rate of clearance of iodide by the thyroid. The thyroid can accommodate itself to the current iodine status of the body. If there is inadequate iodine intake, then the thyroid traps a higher percentage of iodine, and vice versa. Since the impact of TSH on synthetic and secretory activities of the thyroid is 55 evident within a short period of time as mentioned earlier, in the model TSH exerts its effect on iodide trapping rate instantaneously too. But, in case intrathyroidal iodine concentration is not at its normal, the adaptation of the trapping rate is assumed to occur with one-day delay. The stock Pot Trap Fr serves for this purpose. First, a desired trapping fraction is calculated as a function of the ratio of current intrathyroidal iodine content to the normal (see Figure 5.18). According to this desired fraction, the value of the adjustment flow to the stock is revised (as in the adjustment flows of gland weights). But, the value of the stock is some raw value for trapping fraction and is not yet exposed to the effect of TSH stimuli. Equation 5.35 shows the formulation for trapping rate (Trap rate). 0.5 des trap fr 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 ratio of thy I conc to normal Figure 5.18. Graphical function for desired trapping fraction. Trap rate = Pot Trap Fr × I in Blood × eff of TSH on I trap × eff of thy wt on trap cap (5.35) According to Equation 5.35, the trapping fraction is either amplified or contracted depending on the magnitude of TSH stimuli. TSH and intrathyroidal regulatory mechanisms alter the trapping fraction to a certain extent. In addition to these two factors, it is stated that the enlargement of the thyroid to raises its capacity to accumulate iodide from the blood (Rhoades and Bell, 2009). Bearing this in mind, the potential trapping fraction is further multiplied with an effect function (eff of thy wt on trap cap) that aims at 56 depicting the dependency of trapping capacity on the weight of the thyroid. The graphical function is depicted in Figure 5.20. 1.4 gr for eff of TSH on I trap 1.2 1 0.8 0.6 0.4 0.2 0 -1 -0.8 -0.6 -0.4 -0.2 -0 0.2 0.4 0.6 0.8 1 log ratio of TSH to normal Figure 5.19. Graphical function for the effect of TSH on iodide trapping. gr for eff of thy wt on I trap 1.3 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 Thy Wt / normal thy wt Figure 5.20. Graphical function for the effect of thyroid weight on iodide trapping. Lastly, the outflow I cons represents the total iodine loss from available stores due to its usage in thyroid hormone synthesis and is simply the sum of the iodine consumptions necessitated by T3 synthesis rate and T4 synthesis rate. 57 To summarize this chapter, a simplified stock flow diagram of the whole model is given in Figure 5.21. Pit Wt Hypo Wt Hypo wt chg des hypo wt Pit wt chg TRH clear fr des pit wt hypo cap TRH clear rate eff of TH on TSH sec eff of TSH on TH sec Thy Wt Thy wt chg des thy wt <T4 in Blood> eff of TH on TRH sec <T3 in Blood> I from deiod <pot T3 syn> fr of T3 sec des T4 store <thy cap> I in Blood TSH clear rate TSH sec rate imp TSH sec eff of TRH on TSH sec <I in Thy> fr of T4 sec imp TH sec thy cap T4 Store T4 syn rate T4 in Blood Trap rate pos I cons I cons for T3 T4 to T3 conv rate ratio of cap rest I cons to pot I in Thy I cons T4 clear fr T4 clear rate T4 sec rate pot total I cons <Thy Wt> <TSH> Abs rate of T4 by tissues Conv rate to rT3 pot T4 syn pot TH syn I abs rate pit cap TSH TRH TRH sec rate imp TRH sec I intake TSH clear fr <thy cap> <imp TH sec> eff of pref T3 syn on red in T3 syn I cons for T4 pot T3 syn <T4 syn rate> <T3 syn rate> T3 in Blood T3 Store T3 syn rate des T3 store T3 clear fr T3 clear rate T3 sec rate <fr of T3 sec> Abs rate of T3 by tissues <thy cap> Figure 5.21. Simplified stock-flow diagram of the model. 58 6. VALIDATION AND ANALYSIS OF THE MODEL Model validation is a procedure to check if the model is able to adequately illustrate the real problem, as far as the purpose of the modelling study is concerned. Model validity is tested both in structural and behavioural aspects, structural being of primary importance in system dynamics models. So, the logical course of model validation is first to test the validity of the structure, and then begin to check the behaviour accuracy. Structural validity tests check if the structure of the model is able to satisfactorily reflect the actual relations that exist in the real system. Once the model succeeds in the structural tests and sufficient confidence is established in the model structure, behaviour validity tests are implemented to check if the dynamic behaviours produced by the model can sufficiently reflect the real patterns of concern (Barlas, 2002; Barlas 1996). The validity check, especially the structural one, is in fact continuously achieved during the development of the model, and should not be perceived as a completely isolated phase. In this study, a significant portion of structural validation has been done during the process of model construction. It is done by verifying the structure both with the existing information in literature and through the interviews with medical doctors. The aim of this chapter is to represent the outputs of the simulations conducted under certain scenarios to further check the structural validity of the model that is described in the preceding chapter. The model is simulated using Vensim software. As mentioned before, the time unit of the model is one day. The time horizon of the simulations is selected long enough to observe the system behaviour evidently, ranging from a few days to several years. For all the simulation runs, a sufficiently small time step is selected (DT=1/128). 59 6.1. Equilibrium Behaviour When all the variables are initially set to their equilibrium (normal) levels, all 0.2 10 0.175 9 micrograms/dL micrograms/dL hormones stay constant at their equilibrium values, as expected. 0.15 0.125 7 6 0.1 0 1 2 3 Time (Day) 4 0 5 0.3 3 0.25 2.5 microunits/ml nanograms/ml 8 0.2 0.15 1 2 3 Time (Day) 4 5 2 1.5 1 0.1 0 1 2 3 Time (Day) 4 5 0 1 2 3 Time (Day) 4 5 Figure 6.1. T3 (left top), T4 (right top), TRH (lower left), and TSH (lower right) concentrations at equilibrium. 6.2. Base Run Here, the initial value of T4 in blood is set to three times its normal value (720 µg). All the other variables are set to their normal values initially. Firstly, since T4 is above its equilibrium value initially, it gradually decays to restore the equilibrium value and stabilizes there. It takes a few days for T4 to converge to the equilibrium value because its half-life is seven days. Secondly, since T3 is largely obtained via the deiodination of T4, T3 concentration also rises with the initial shift in T4. 60 micrograms/dL 30 22.5 15 7.5 0 0 0.60 1.20 1.80 2.40 Time (Day) 3 3.60 Figure 6.2. T4 concentration in base run. micrograms/dL 0.4 0.3 0.2 0.1 0 0 0.60 1.20 1.80 2.40 Time (Day) 3 3.60 Figure 6.3. T3 concentration in base run. microunits/ml 3 2.25 1.5 0.75 0 0 0.60 1.20 1.80 2.40 Time (Day) 3 3.60 Figure 6.4. TSH concentration in base run. 61 Due to the negative feedback effect of thyroid hormones on the release of TRH and TSH, their concentrations decline too. As thyroid hormones converge to the equilibrium, TSH and TRH restore their normal levels (see Figure 6.4 and Figure 6.5). nanograms/ml 0.24 0.18 0.12 0.06 0 0 0.60 1.20 1.80 2.40 Time (Day) 3 3.60 Figure 6.5. TRH concentration in base run. 6.3. TRH Injection Test In the work of Snyder and Utiger (1972), a TRH stimulation test is performed on six normal healthy subjects. 25 µg TRH is injected to each of the six subjects at t=0, and blood samples are collected to measure the TSH levels in blood (see Figure 6.6). To replicate this test, all the variables are set to their equilibrium levels initially, and 25 µg TRH is assumed to be injected at t=0. For the model output to be comparable to real data, only the first three-hour portion of the simulation run is displayed in Figure 6.7. Comparing the two graphs in Figure 6.6 and Figure 6.7, it can be said that the overall behaviour and even the value of the peak point of TSH from the model highly matches with those of real data. The time of the hormone to reach its maximum takes somewhat longer in the model; but since the time unit of the model is one day and relatively longterm dynamics of hormones are the point of interest in this study, this slight difference does not hurt the validity of the model. 62 Figure 6.6. Average TSH concentration of six normal subjects when 25 µg TRH is injected at t=0 (Snyder and Utiger, 1972). microunits/ml 20 15 10 5 0 0 0.019 0.038 0.058 0.077 Time (Day) 0.096 0.115 Figure 6.7. TSH concentration when 25 µg TRH is injected at t=0. 6.4. Ten-Fold Increase in T4 Secretion for One Hour In literature, it is stated that if T4 secretion were increased ten-fold for one hour, we would expect the total T4 concentration in blood to increase by 12% (Goodman, 2009). 63 Here, it is assumed that from t=0 to t=1/24, T4 secretion rate is ten times its normal value, i.e. 900 µg. After t=1/24, no external intervention is applied on T4 secretion. The behaviour of T4 under this scenario is shown in Figure 6.8. micrograms/dL 9.5 9 8.5 8 7.5 0 0.20 0.40 0.60 0.80 Time (Day) 1 1.20 1.40 Figure 6.8. T4 concentration when its secretion is increased ten-fold for one hour. micrograms/dL 0.17 0.165 0.16 0.155 0.15 0 0.20 0.40 0.60 0.80 Time (Day) 1 1.20 1.40 Figure 6.9. T3 concentration when T4 secretion is increased ten-fold for one hour. Increasing the T4 secretion for one hour causes T4 levels to rise until the end of the first hour, as expected. The amount of increase in T4 in the model outputs is about 13% which is quite consistent with the data in literature. Since no component of the system is 64 interfered with after the first day, T4 levels start to decrease thereafter, and reach equilibrium in about one day. Under this scenario, it is not only T4 whose value is disturbed from equilibrium. A significant portion of T3 in the body is formed via conversion of T4 to T3. Because of this, T3 levels also rise in this case. But after a very short while, it drops to values that are below the baseline values to compensate for the high circulating T4 and then returns almost to normal at the end of the first day (see Figure 6.9). nanograms/ml 0.22 0.195 0.17 0.145 0.12 0 0.20 0.40 0.60 0.80 Time (Day) 1 1.20 1.40 Figure 6.10. TRH concentration when T4 secretion is increased ten-fold for one hour. microunits/ml 2.5 2 1.5 1 0.5 0 0.20 0.40 0.60 0.80 Time (Day) 1 1.20 1.40 Figure 6.11. TSH concentration when T4 secretion is increased ten-fold for one hour. 65 As a result of negative feedback effect of thyroid hormones on both the hypothalamus and pituitary, TRH and TSH levels decline. As seen in Figure 6.10 and Figure 6.11, TRH and TSH levels in the body first decline and then rise so as to enable the body to restore the equilibrium state. 6.5. Zero T4 Secretion for One Hour The work of Goodman (2009) suggests that if T4 secretion were stopped for one hour, we would expect its concentration to decrease by only 1%. So, in this scenario, the secretion rate of T4 is set to zero for one hour at t=0, and after one hour the model is left to operate by its own. micrograms/dL 8.050 8 7.95 7.9 7.85 0 0.20 0.40 0.60 0.80 Time (Day) 1 1.20 1.40 Figure 6.12. T4 concentration when T4 secretion is stopped for one hour. As the secretion rate of T4 cannot be modulated with the feedback loops for the first hour, its concentration rises initially inevitably. As the external intervention ceases, circulating T4 levels quickly restore to normal levels. The lowermost point in T4 concentration shown in Figure 6.12 gives a 1.45% reduction in T4 concentration. As in the previous scenario, the change in T3 concentration goes parallel with the change in T4 concentration as the primary source of circulating T3 is the deiodination of 66 T4 to form T3. Since a number of mechanisms work in concert to preserve the T3 availability in body, it starts to rise to values that are above normal and returns to normal values as T4 stabilizes (see Figure 6.13). micrograms/dL 0.17 0.169 0.168 0.167 0.166 0 0.20 0.40 0.60 0.80 Time (Day) 1 1.20 1.40 Figure 6.13. T3 concentration when T4 secretion is stopped for one hour. TRH and TSH concentrations show qualitatively the same behaviour in this scenario. Thus, only TSH is pictorially shown in Figure 6.14. As expected, TSH initially rises to compensate for the decline in thyroid hormone levels, and then gradually reaches the equilibrium. microunits/ml 2.4 2.345 2.29 2.235 2.18 0 0.20 0.40 0.60 0.80 Time (Day) 1 1.20 1.40 Figure 6.14. TSH concentration when T4 secretion is stopped for one hour. 67 6.6. Hypophysectomy Hypophysectomy, as stated before, means the complete removal of the pituitary gland. In reality, a person cannot survive long in the absence of such a critical gland without any therapeutic intervention. However, for the sake of the validation of the model, we assume that the person continues to live. micrograms/dL 0.2 0.15 0.1 0.05 0 0 6 12 18 24 30 36 Time (Day) 42 48 54 60 Figure 6.15. T3 concentration in case of hypophysectomy. nanograms/ml 2.4 1.8 1.2 0.6 0 0 6 12 18 24 30 36 Time (Day) 42 48 54 60 Figure 6.16. TRH concentration in case of hypophysectomy. 68 Since the pituitary is completely removed, no TSH exists in the body. And since there is no TSH in the body, only minimal amounts of thyroid hormones are secreted. Low levels of thyroid hormones in blood cause TRH secretion to rise. However, since the medium of communication between the hypothalamus and the thyroid gland is no longer present, thyroid hormones in blood continue to drop. The behaviours of circulating T3 and T4 in this scenario are qualitatively the same; thus, only that of T3 is shown pictorially in Figure 6.15. The work of Smith (1930) suggests that the thyroids of rats start to regress in weight soon after hypophysectomy, evident in ten days and pronounced in thirty days. The diminution in the weights of the rat thyroids is observed to be usually about by half or even more. A similar work conducted by White (1933) investigates the effect of hypophysectomy on rabbits and shows that the average shrinkage in the thyroids is 30% in male rabbits, and 20% in females. 21 grams 18.5 16 13.5 11 0 20 40 60 80 100 120 140 160 180 200 Time (Day) Figure 6.17. Thyroid weight in case of hypophysectomy. The simulation outputs regarding the thyroid weight is consistent with the findings in literature explained above. Since low levels of thyroid hormones persist because of the absence of pituitary gland, the hypothalamus persistently over-functions due to scantiness of inhibition by thyroid hormones. This, in turn, leads to an increase in the weight of the 69 hypothalamus, which turns out to be about three-fold in our case (see Figure 6.18). The opposite is seen in thyroid gland. Since no TSH exists in the body, thyroid gland is no longer stimulated. Due to prolonged “idling”, the gland shrinks (Donovan, 1966; Melmed, 2002). As seen in Figure 6.17, the weight of the thyroid is contracted almost by half. The time scale of the output graphs for thyroid weight and hypothalamus weight is taken as 200 days in order to be able to evidently depict the changes in the sizes. 130 mg 105 80 55 30 0 20 40 60 80 100 120 140 160 180 200 Time (Day) Figure 6.18. Hypothalamus weight in case of hypophysectomy. 70 7. THYROID DISORDERS Once the model is believed to be structurally and behaviourally valid, the last step is to analyse the model via simulation experiments. In this chapter, the model will be simulated under some common thyroid-related disorders, and the dynamics of the key variables will be illustrated. 7.1. Graves’ Disease Graves’ disease is an autoimmune disease which causes excessive thyroid hormone production. It affects approximately 0.5% of population and counts for the 50 to 80% of cases of hyperthyroidism (Bürgi, 2010). This disorder is caused by thyroid-stimulating immunoglobulins, TSIs, which are proteins that mimic the actions of TSH, bind to the receptors on thyroid cells and stimulate the production of thyroid hormones. In Graves’ disease, TSH and TRH concentrations are less than normal, and often essentially zero. It is the negative feedback effect of the elevated concentrations of circulating thyroid hormones that results in low levels of TRH and TSH. Yet, thyroid hormones cannot feed back to inhibit excessive TSI stimulation. Thus, despite the scarcity of TRH and TSH, oversecretion of thyroid hormones persists, for independently acting TSIs continue to trigger the formation and secretion of thyroid hormones. Due to prolonged overstimulation, the thyroid enlarges and forms a specific type of goiter, called diffuse toxic goiter (Guyton and Hall, 2006; Rhoades and Bell, 2009). In this section, the outputs of two simulation runs will be presented. Firstly, the daily iodine intake will be assumed normal, i.e. 150 µg. Secondly, the iodine intake will be set to 400 µg/day. In these two cases, the amount of TSIs will be kept at the same level. For the outputs to be comparable, the horizontal and vertical scales of the graphs will be retained throughout the two runs (except for the graph of thyroidal iodine). 71 7.1.1. Graves’ Disease with Normal Daily Iodine Intake Here, the daily iodine intake is set to 150 µg, and the values of all the variables are initially set to their equilibrium levels. In this scenario, the thyroid hormone concentrations rise with the stimulatory effect of TSIs, and persistently stay elevated since the negative feedback loop is interrupted because of the independently acting agents. However, at approximately halfway of the simulation run, the thyroid hormones start to drop and stabilize at a level above the normal. The reason behind this that the TSIs demand more than what the thyroid gland can synthesize using only the given daily iodine intake. So, consumption rate the iodine exceeds the rate of uptake which leads to the gradual exhaustion of thyroidal iodine stores. Yet, the drop in thyroid hormone concentrations is not immediately followed by that. After the depletion of these stores, the preformed hormone stores are used to meet the residuary demand of TSIs that the thyroidal synthesis rate cannot fulfil. micrograms/dL 0.6 0.45 0.3 0.15 0 0 28 56 84 112 140 168 196 224 252 280 Time (Day) Figure 7.1. T3 concentration in Graves’ disease with normal iodine intake. Related literature suggests that in hyperthyroid patients taking drugs that block thyroid hormone synthesis stores of thyroid hormones are more rapidly depleted, and the therapeutic effect of these drugs may require several weeks to become evident (Carruthers 72 et al., 2000). So, if thyroid hormone synthesis is blocked or reduced for some reason, the hormone stores are depleted before the severity of the disease attenuates. Consistent with these findings, when both iodine and preformed hormone stores are exhausted in the simulation, the thyroid hormones start to decline to a level still higher than normal and stabilize there. As an example to the levels of hormones in patients with Graves’ disease, a clinical case has been illustrated in which a 35-year-old woman with this disease presented with very high T4 concentration (total T4: 320 nmol/l, normal range: 70–150 nmol/l) and suppressed serum TSH concentration (<0.05 mU/l, normal range: 0.5–4.0 mU/l) (Nussey and Whitehead, 2001). Consistent with this case, T4 concentration is rises maximally to three times the normal value (before being suppressed by the above-mentioned constraints). micrograms/dL 28 21 14 7 0 0 28 56 84 112 140 168 196 224 252 280 Time (Day) Figure 7.2. T4 concentration in Graves’ disease with normal iodine intake. The amount of iodine in the thyroid in euthyroid individuals is suggested to normally vary between about 3 and 20 mg. In hyperthyroidism due to Graves’ disease, the amount of iodine in the thyroid is suggested to be low, rarely above 3 mg (Nyström et al., 2011). As illustrated in Figure 7.3, the thyroidal iodine drops to very low levels in Graves’ disease (to about 600 µg). 73 16,000 micrograms 12,000 8,000 4,000 0 0 42 84 126 168 Time (Day) 210 252 Figure 7.3. Iodine in thyroid in Graves’ disease with normal iodine intake. One characteristic feature of Graves’ disease is that T3/T4 is high compared to normal physiological conditions. As depicted in Figure 7.4, the ratio of serum T3 to T4 ratio rises above normal value (which is 0.0208 in the model) with the introduction of TSIs, and climbs further with the increase in TSH and depletion of iodine stores. 0.03 0.0275 0.025 0.0225 0.02 0 42 84 126 168 Time (Day) 210 252 Figure 7.4. T3 to T4 ratio in Graves’ disease with normal iodine intake. As one might expect, TRH and TSH concentrations fall to very low values because of the inhibitory effect of elevated thyroid hormones. Since measurement of TRH concentrations is impossible, only TSH concentrations are available numerically. Albeit so, 74 the output graphs for both TRH and TSH are presented here in order to demonstrate their general course of behaviour. nanograms/ml 0.2 0.15 0.1 0.05 0 0 28 56 84 112 140 168 196 224 252 280 Time (Day) Figure 7.5. TRH concentration in Graves’ disease with normal iodine intake. In the first phase of the simulation run (when the stores are not yet exhausted), the minimal value of TSH concentration is approximately 0.028 µU/ml, quite consistent with the data given above where TSH concentration was found to be <0.05 µU/ml. microunits/ml 2.4 1.8 1.2 0.6 0 0 28 56 84 112 140 168 196 224 252 280 Time (Day) Figure 7.6. TSH concentration in Graves’ disease with normal iodine intake. 75 One common finding about Graves’ disease is thyroid gland growth due to overstimulation (Melmed and Conn, 2005). When the patient with Graves’ disease whose hormone concentrations were given above was examined, it is realized that she had moderate diffuse goiter. As Figure 7.7 reveals, the thyroid enlarges to a certain extent in Graves’ disease (note that the time scale of the graphs for gland weights is longer). 23 grams 22.25 21.5 20.75 20 0 50 100 150 200 Time (Day) 250 300 350 Figure 7.7. Thyroid weight in Graves’ disease with normal iodine intake. 40 mg 36 32 28 24 0 50 100 150 200 Time (Day) 250 300 350 Figure 7.8. Hypothalamus weight in Graves’ disease with normal iodine intake. Finally, the weights of the hypothalamus and pituitary regress as a result of the low TRH and high thyroid hormones, respectively. The relative remission in the atrophy results 76 from the relative decrease in thyroid hormones and rise in TRH and TSH following the hormone store depletion. 30 mg 22.5 15 7.5 0 0 50 100 150 200 Time (Day) 250 300 350 Figure 7.9. Pituitary weight in Graves' disease with normal iodine intake. 7.1.2. Graves’ Disease with Relatively High Daily Iodine Intake All other things being the same as the previous scenario, the iodine intake is set to 400 µg/day here, and the model is simulated again. This experiment is done to show the behaviour of the system when the iodine is no longer a binding constraint for the synthesis of thyroid hormones in Graves’ disease. For the sake of brevity, not every single output of the previous case will be shown; only the descriptive ones will be selectively presented in this scenario. With the increase in daily iodine intake, no shift in thyroid hormone concentrations is observed in the middle of the run (see Figure 7.10). As implied by the stable behaviour of T3 throughout the run, the thyroidal iodine stores do not get depleted in this case because the increased daily iodine intake suffices to meet the demand imposed by the TSIs. 77 micrograms/dL 0.6 0.45 0.3 0.15 0 0 28 56 84 112 140 168 196 224 252 280 Time (Day) Figure 7.10. T3 concentration in Graves’ disease with relatively high iodine intake. 16,000 micrograms 15,500 15,000 14,500 14,000 0 42 84 126 168 Time (Day) 210 252 Figure 7.11. Iodine in thyroid in Graves’ disease with relatively high iodine intake. Figure 7.12 shows the ratio of T3 to T4 for this scenario. Because of overstimulation of the thyroid, the ratio of T3 to T4 is higher than normal in this case too. As opposed to the previous case, no further shift can be witnessed here because TSH levels do not show a relative increase and thus do not cause the ratio to grow any further. As depicted in Figure 7.13, TSH concentration ultimately stabilizes at a subnormal level (0.028 µU/ml). 78 0.03 0.0275 0.025 0.0225 0.02 0 42 84 126 168 Time (Day) 210 252 Figure 7.12. T3 to T4 ratio in Graves’ disease with relatively high iodine intake. microunits/ml 2.5 1.875 1.25 0.625 0 0 50 100 150 200 Time (Day) 250 300 350 Figure 7.13. TSH concentration in Graves’ disease with relatively high iodine intake. Finally, the weights of the thyroid and pituitary are depicted in Figure 7.14 and Figure 7.15. The time scales of these graphs are again longer than the previous ones (the same as the time scales of the graphs of gland weights in the previous section) in order to be able to depict the complete behaviour evidently. The convergence of the thyroid to the ultimate value is smoother in this case. As for the pituitary, no remission in the atrophy can be observed since no change in neither the thyroid hormones nor TRH occurs once they stabilize. 79 To summarize, it can be said that the availability of iodine may well augment the severity of Graves’ disease depending on the amount of TSIs. 23 grams 22.25 21.5 20.75 20 0 50 100 150 200 Time (Day) 250 300 350 Figure 7.14. Thyroid weight in Graves’ disease with relatively high iodine intake. 30 mg 22.5 15 7.5 0 0 50 100 150 200 Time (Day) 250 300 350 Figure 7.15. Pituitary weight in Graves’ disease with relatively high iodine intake. 7.2. Iodine Deficiency Iodine deficiency is a common cause of hypothyroidism. Though eradicated in many regions of the world with salt iodization, it still is one prevailing cause of hypothyroidism. 80 Approximately 30% of the world population is encountered with iodine deficiency, about half of them with goiter (Goldman and Hatch, 2000). Lack of iodine, which is an indispensable ingredient for thyroid hormone synthesis as mentioned earlier, hinders the production of thyroid hormones. Consequently, since no or little thyroid hormone is available to inhibit the production of TSH, excessive amounts of TSH is secreted. Excessive TSH overstimulates the thyroid gland. But, because of scarcity of iodide, formation of T3 and T4 does not occur and consequently TSH secretion cannot be suppressed. In the long run, since the overstimulation by TSH lingers, the thyroid gland grows larger. Enlarged thyroid gland due to iodine deficiency is called endemic colloid goiter (Guyton and Hall, 2006). In this section, the outputs of two simulation runs will be presented; severe iodine deficiency and moderate iodine deficiency. In the first case, the daily iodine intake will assumed to be 30 µg. And in the second case, the iodine intake will be increased to 50 µg/day. In both cases, the model is run for 800 days. For the outputs to be comparable, the horizontal and vertical scales of the graphs will be retained throughout the two runs. 7.2.1. Severe Iodine Deficiency Here, the daily iodine intake is set to 30 µg by leaving all the other variables at their normal values initially. In the initial phase of the run, the concentrations of hormones persevere at their normal equilibrium levels since the body utilizes the iodine and hormone stores before the implications of iodine deficiency become apparent. Except for thyroidal iodine, variables stay at their set-points for a long while. To be able to explicitly demonstrate the dynamics of the variables, the interval from t=350 to t=750 will be shown in all the output graphs but the thyroidal iodine. 81 micrograms/dL 0.27 0.23 0.19 0.15 0.11 350 390 430 470 510 550 590 630 670 710 750 Time (Day) Figure 7.16. T3 concentration when daily iodine intake is 30 µg. micrograms/dL 10 7.5 5 2.5 0 350 390 430 470 510 550 590 630 670 710 750 Time (Day) Figure 7.17. T4 concentration when daily iodine intake is 30 µg. Evidence in literature suggests that serum thyroid hormone levels change in a characteristic pattern with iodine deficiency, typically showing a low T4 and a normal or increased serum T3 concentration (Braverman, 2003). It is stated that T3 concentrations also decline, but not until hypothyroidism, or the iodine deficiency is severe (Werner et al., 2005; Brent, 2010). In this case, i.e. when iodine intake is 30 µg/day, the body fails to maintain the serum T3 levels at normal levels. When the equilibrium state is disturbed with the depletion of iodine and then hormone stores, T3 levels rise above the normal levels for 82 a short while but cannot survive there with the given level of iodide. So, it falls to a subnormal level and stabilizes there (see Figure 7.16). As mentioned above, the impact of iodine deficiency on T4 concentrations is more drastic. The proportional decrease in T4 is much more than that of T3 (see Figure 7.16 and Figure 7.17). As a consequence, the ratio of serum T3 to T4 is elevated in iodine deficiency (see Figure 7.18). 0.16 0.12 0.08 0.04 0 350 410 470 530 590 Time (Day) 650 710 Figure 7.18. T3 to T4 ratio when daily iodine intake is 30 µg. 15,000 micrograms 11,250 7,500 3,750 0 0 100 200 300 400 500 Time (Day) 600 700 Figure 7.19. Iodine in thyroid when daily iodine intake is 30 µg. 83 If iodine intake diminishes, hormone secretion remains constant until available stores of the mineral are depleted (Garrison and Somer, 1995). Consistent with this statement, in the model, the effects of iodine deficiency become apparent shortly after the depletion of these stores. The time lag between the depletion of iodine stores results from the fact that the thyroid makes use of the preformed hormone stores, and is able to release sufficient amount of thyroid hormones in that interval before a significant portion of these stores are consumed. Figure 7.19 depicts the dynamics of thyroidal iodine (note that the horizontal scale begins from t=0). 440 micrograms 360 280 200 120 350 390 430 470 510 550 590 630 670 710 750 Time (Day) Figure 7.20. T3 store when daily iodine intake is 30 µg. 6,000 micrograms 4,500 3,000 1,500 0 350 410 470 530 590 Time (Day) 650 710 Figure 7.21. T4 store when daily iodine intake is 30 µg. 84 As mentioned before, both the synthesis and secretion of T3 is favoured in hypothyroid state. Hence, the degree of diminution in T3 reserves is less severe than that of T4 (see Figure 7.20 and Figure 7.21). One should note that the decrease in hormone stores shows itself after thyroidal iodine is considerably emptied. microunits/ml 24 18 12 6 0 350 390 430 470 510 550 590 630 670 710 750 Time (Day) Figure 7.22. TSH concentration when daily iodine intake is 30 µg. 62 grams 51 40 29 18 350 390 430 470 510 550 590 630 670 710 750 Time (Day) Figure 7.23. Thyroid weight when daily iodine intake is 30 µg. As the nature of the negative feedback necessitates, TSH concentrations rise in parallel with the diminishment in thyroid hormones, as presented in Figure 7.22. Since the 85 behaviour of TRH is qualitatively the same as that of TSH, the related output graph is not shown separately in this case. The last thing to mention is the alterations in the weights of the glands. The thyroid gland starts to be stimulated more than normally when the thyroid hormone levels drop and TSH levels rise subsequently. As seen in Figure 7.23, the weight of the thyroid increases to about three times its normal weight. 44 mg 42.75 41.5 40.25 39 350 410 470 530 590 Time (Day) 650 710 Figure 7.24. Hypothalamus weight when daily iodine intake is 30 µg. 48 mg 41 34 27 20 350 390 430 470 510 550 590 630 670 710 750 Time (Day) Figure 7.25. Pituitary weight when daily iodine intake is 30 µg. 86 In parallel with the persistent drop in thyroid hormones and rise in TRH and TSH, both the hypothalamus and the pituitary also grow in size (see Figure 7.24 and Figure 7.25). 7.2.2. Moderate Iodine Deficiency In this case, the daily iodine intake is set to 50 µg. Again, for the sake of brevity, not all the outputs shown in the severe deficiency case, but only some descriptive ones will be selected and presented. micrograms/dL 0.27 0.23 0.19 0.15 0.11 350 390 430 470 510 550 590 630 670 710 750 Time (Day) Figure 7.26. T3 concentration when daily iodine intake is 50 µg. micrograms/dL 10 7.5 5 2.5 0 350 390 430 470 510 550 590 630 670 710 750 Time (Day) Figure 7.27. T4 concentration when daily iodine intake is 50 µg. 87 It was previously explained that unless the iodine deficiency is severe, the characteristic pattern of serum thyroid hormones is normal or elevated T3, and low T4 concentrations. In the case where iodine intake was 30 µg/day, T3 levels couldn’t hold on to the normal levels and dropped below the set point. Here however, since the deficiency is not as severe, T3 succeeds to stabilize at a level above the normal. The diminution in the severity is reflected to T4 concentration too; it becomes stable at a level higher than the previous case. 0.16 0.12 0.08 0.04 0 350 410 470 530 590 Time (Day) 650 710 Figure 7.28. T3 to T4 ratio when daily iodine intake is 50 µg. Though milder than the previous case, 50 µg iodine intake is still inadequate and causes insufficient synthesis and secretion of thyroid hormones. So, the hypothyroid state still exists and therefore T3/T4 is higher than normal in this case. As can be observed in Figure 7.28, this ratio starts to rise with the decrease in thyroid hormone concentrations but stabilizes at a point which is smaller than the one in the 50 µg-case. So, it can be said that this ratio is proportional to the severity of the deficiency. Another important thing to note is that the implications of the deficiency become evident later than the previous case for the relatively higher intake of iodine helps the preexisting stores to bear longer here. Also, the hormone stores equilibrate at a higher level as compared to the previous case (see Figure 7.29). 88 440 micrograms 360 280 200 120 350 390 430 470 510 550 590 630 670 710 750 Time (Day) Figure 7.29. T3 store when daily iodine intake is 50 µg. Upon the depletion of hormone stores, TSH concentration starts to rise to compensate for the thyroid hormone deficiency (see Figure 7.30). 6,000 micrograms 4,500 3,000 1,500 0 350 410 470 530 590 Time (Day) 650 710 Figure 7.30. TSH concentration when daily iodine intake is 50 µg. The behaviours of the weights of the glands are basically the same as in the previous case; the only difference is that they stabilize at a lower level in this case. As a representative one, only the graph for thyroid weight is shown in Figure 7.31. 89 62 grams 51 40 29 18 350 390 430 470 510 550 590 630 670 710 750 Time (Day) Figure 7.31. Thyroid weight when daily iodine intake is 50 µg. 7.3. Iodine Excess As mentioned earlier, sufficient iodine intake is crucial for the maintenance of healthy thyroid functioning and severe iodine deficiency obstruct thyroid hormone production and induces goiter formation. Interestingly, functioning of the thyroid is also impaired when the dietary iodine intake is far above the physiological needs. It is postulated that identical iodine excess may cause hyperthyroidism in some persons and hypothyroidism in others (Bürgi, 2010). Iodine-induced hyperthyroidism is suggested to happen often due to the autonomy in thyroid function, because of some therapeutic intervention for some pre-existing thyroid disease, or as a result of the disease itself. In this study, the effects of iodine excess on thyroids with prior pathological conditions are considered out of scope; only the impacts on normal thyroids will be of interest. A number of studies are conducted to demonstrate the inhibitory effect of excessive iodine intake on thyroids of healthy subjects, as mentioned before. In one of such studies, Namba et al. (1993) investigate the effect of 27 mg iodine administration to ten normal male volunteers on thyroid function and volume. Thyroid volume was measured before 90 treatment, on the day of the last treatment, and 1 month after the treatment. It is found that there was a significant rise in serum TSH levels, with a small decline in serum free T4 concentration during iodide administration; the values remained within the normal range except for two subjects. The volume of the thyroid gland is found to be significantly enlarged after 28 days of iodide intake. And it is stated that when iodide was discontinued, thyroid volume and function returned to baseline levels within one month for all subjects. Here, it is assumed that 27 mg iodine is supplemented for 28 days starting at t=0. As in all the previous scenarios, the variables are initially set to their equilibrium levels. For the model outputs to be comparable to real data graphs, the model is simulated for 56 days. Figure 7.32. Average free T4 concentration of ten subjects receiving 27 mg iodine supplementation for 28 days (Namba et al., 1993). The average free T4 levels of the ten subjects are depicted in Figure 7.34. T4 levels decline after the introduction of excessive iodine supplementation and start to rise soon before the withdrawal of the iodine supplementation. The dynamics of free T4 in the simulation run is demonstrated in Figure 7.33 and is highly consistent with that of the real data (note that the horizontal scales do not linearly increase). 91 Figure 7.33. Free T4 concentration (in pmol/l) in case of 27 mg iodine supplementation for 28 days. micrograms/dL 0.19 0.1775 0.165 0.1525 0.14 0 8 16 24 32 Time (Day) 40 48 56 Figure 7.34. T3 concentration in case of 27 mg iodine supplementation for 28 days. The behaviour of T3 is shown in Figure 7.34. As opposed to T4, the rising trend in T3 concentration becomes apparent in T3 after about t=7. The early improvement of T3 is a consequence of the mechanisms that operate to preserve T3 availability in body. These mechanisms even lead to elevated T3 levels transiently, to compensate for the subnormal levels of T4. So, consistent with the findings of Namba et al., the concentrations of both T3 and T4 return to normal within one month after the discontinuation of iodine supplementation. 92 Figure 7.35. Average TSH concentration of ten subjects receiving 27 mg iodine supplementation for 28 days (Namba et al., 1993). Figure 7.36. TSH concentration in case of 27 mg iodine supplementation for 28 days. In concordance with the lowered levels of thyroid hormones, the average TSH concentration of the subjects increases (see Figure 7.35), and restores the normal level within one month after the excessive iodine intake is stopped. The model output shown in Figure 7.36 matches well with the dynamics of the real data (note that the horizontal scales do not linearly increase). The maximum level that the average TSH of the subjects reaches 93 is about 2-2.5 times the normal levels, whereas that of the simulation model is a bit higher, about three times the normal level. It was mentioned earlier that the changes in the weight of the thyroid will be presumably accompanied by the changes in the volume of the thyroid. The average thyroid volume of ten subjects measured before, during and one month after the supplementation is shown in Figure 7.37. Since only three data points are available, the data is not exactly comparable to the model output. Though not significantly different, the average thyroid volume of the ten subjects is slightly higher than the volume before the supplementation. In our simulation run, the weight of the thyroid gland does not restore to normal as much as the average thyroid volume of the subjects does after one month (at t=56), but it does so in about three to four months (which is not explicitly shown here). Figure 7.37. Average thyroid volume (as % of normal volume) of ten subjects receiving 27 mg iodine supplementation for 28 days (Namba et al., 1993). 94 25 grams 23.5 22 20.5 19 0 8 16 24 32 Time (Day) 40 48 56 Figure 7.38. Thyroid weight in case of 27 mg iodine supplementation for 28 days. Lastly, serum iodine levels of the subjects remain elevated until the iodine supplementation is discontinued. After the cessation of excessive iodine intake (day 28), the serum iodine starts to drop. The related data and simulation output are depicted in Figure 7.39 and Figure 7.40. The units of the real data and the model output are different; but the overall dynamics of the two match well. Figure 7.39. Average serum iodine levels of ten subjects receiving 27 mg iodine supplementation for 28 days (Namba et al., 1993). 95 Figure 7.40. Iodine in blood in case of 27 mg iodine supplementation for 28 days. 7.4. Subacute Thyroiditis The term thyroiditis refers to the inflammation of the thyroid gland. Inflammation is the response of tissues to harmful stimuli, and can be caused by viral infections or autoimmune processes. Here, we are not interested in the causes of the inflammation, but rather in the consequences of it. Inflamed cells of the thyroid lose their secretory and synthetic abilities. Moreover, the inflammatory reaction within the gland causes the follicles to lose their integrity by disrupting them and results in the release of preformed hormones and iodine into the peripheral circulation (Grossman, 1998, Werner et al., 2005). The release of the preformed hormones is not a controlled discharge like the secretion process, but rather a leakage. The inflammatory reaction in the thyroid gland might be temporary or persistent. In this section, an instance of temporary thyroiditis, namely subacute thyroiditis, will be of interest. Subacute thyroiditis is the most common reason of the painful thyroid gland and may account for up to 5% of clinical thyroid abnormalities. In subacute thyroiditis, the thyroid gland is exposed to a transient course of inflammation, which usually lasts several weeks and then ameliorates. 96 Subacute thyroiditis demonstrates a triphasic clinical course of thyrotoxicosis, hypothyroidism and restoration of normal thyroid functioning. Thyrotoxicosis, which means an excess of thyroid hormones in the body, is a result of destruction of thyroid cells and uncontrolled release of hormone stores into the circulation. Since thyroid cells cannot synthesize hormones during the inflammation phase and leakage of preformed hormones persist due to inflammation, hormone stores get depleted after some time. As a result, hypothyroidism is observed. As the name suggests, hypothyroidism is a condition in which too little thyroid hormone is circulating throughout the body. After the inflammation subsides, levels of circulating hormones restore their normal levels and hormone stores are replenished gradually (Van den Berghe, 2008; Grossman, 1998). In the model, the notion of inflammation is quantified by using a stock which is allowed to vary between zero and one. The name of that stock is Inflammation Status. Inflammation Status being one means that the gland is completely inflamed, and being zero means that the gland is functioning properly. In a sense, this variable gives the inflamed proportion of the gland. In this case, two separate sources of thyroid hormone release exist; secretion by the normally functioning cells, and leakage from the inflamed cells. Having defined the inflammation status stock as the percentage of dysfunctioning gland, some distinction will be made for the utilization of hormone stores for secretion and leakage purposes. If some portion of the thyroid is inflamed, then that portion should leak out thyroid hormones according to the amount of preformed thyroid hormones covered by that portion, but not secrete any because it is not capable of doing it yet. Conversely, recovered or healthy portion of the gland should only be in charge of secreting, not leaking, and be allowed to consume the amount of preformed hormones that they “own”. For this purpose, three new stocks are defined in the model; two of them for the thyroid hormones, and one for thyroidal iodine. For all three substructures, the rationale behind is the same. To begin with the thyroid hormones, these stocks serve as the amount of hormone stores that the inflamed portion of the gland encapsulates. And, the old stocks for hormone stores stand for the amount of hormone stores enveloped by the recovered portion. Two things count for the 97 loss from the new stocks; one is the leakage of hormones, the other is the transition from the coverage area of inflamed cells to the healthy ones with the recovery of those inflamed cells. The modified portion of stock flow diagram and related equations for T3 are shown in Figure 7.41. Recov of T4 cont portion Leaking T4 Store Conv rate to rT3 T4 Store T4 syn rate T4 leak Abs rate of T4 by tissues T4 in Blood T4 clear rate T4 sec rate T4 to T3 conv rate Recov rate Inflammation Status T3 from deiod of T4 T3 Store T3 in Blood T3 clear rate Abs rate of T3 by tissues T3 sec rate T3 syn rate Recov of T3 cont portion Leaking T3 Store T3 leak Figure 7.41. Modified structure of thyroid sector for subacute thyroiditis. Recov of T3 cont portion = Leaking T4 Store × Recov rate (7.1) T3 leak = Leaking T3 Store × 0.02 (7.2) 98 As the above equations reveal, a certain fraction of preformed hormones that are covered by the inflamed portion of cells leak out of the gland into the circulation, and some fraction of preformed hormones flows into the regular hormone store stock with the recovery of inflamed cells. Also, since only healthy cells are capable of synthesizing, the newly synthesized thyroid hormones flow into the regular hormone store and do not get mixed with the hormones enclosed by the inflamed portion. I from deiod I excr I in Blood I abs rate I leak Trap rate Leaking I Recov rate Inflammation Status I in Thy Recov of cells I cons Figure 7.42. Modified structure of iodine sector for subacute thyroiditis. The iodine sector is modified in the same manner as explained above so as to distinguish the portion of intrathyroidal iodine that is to be consumed for synthetic purposes and that leaks out of the gland into the blood. The old stock for the thyroidal 99 iodine stands for the amount of iodine to be consumed in a controlled manner, and the new one for iodine enclosed by the inflamed cells (see Figure 7.42). There are few last remarks that have to be mentioned to fully elucidate the revised structure of the model under subacute thyroiditis. Firstly, the stock for thyroid weight (Thy Wt) represents not the whole but the properly functioning part of the gland (i.e. not exposed to inflammatory reaction). Secondly, it is assumed that the gland cannot expand in size during the course of the simulation run. Having zeroed the weight adjustments in the thyroid, the only inflow to Thy Wt becomes the recovery of the inflamed portions of the gland (Recov of inflamed thy). Recov of inflamed thy = 20 × Recov rate (7.3) Since the gland is completely inflamed at the beginning, the normal weight of the thyroid is multiplied with the rate of recovery of inflammation to figure the correspondent recovery in thyroid weight (see Equation 7.3). Thirdly, the capacity of one unit weight of the properly functioning part of the gland is taken as one fourth of the normal capacity, i.e. 2.5 times the normal productivity level. The second and third assumptions are related to the idea that the cells that newly recover from the inflammatory reaction may not perform as well as in the healthy state, at least over the course of the simulation run. Lastly, the effect of iodine on thyroid capacity is not taken into account in this case because the presumed mechanism of impairment for iodine in healthy subjects may not be the same in an inflamed gland. Having done all the modifications in model structure and parameters, the model is run for 150 days assuming a course of inflammation status as depicted in Figure 7.43. The simulation output showing the TSH and T4 concentrations are shown in Figure 7.44. Figure 7.45 and Figure 7.46 shows the real data from patients with subacute thyroiditis. 100 1 0.75 0.5 0.25 0 0 20 40 60 80 100 Time (Day) 120 140 Figure 7.43. The assumed course of inflammation status in subacute thyroiditis. 20 10 10 5 0 0 0 20 40 60 80 100 Time (Day) 120 140 TSH conc : Current T4 conc : Current Figure 7.44. TSH and T4 concentrations in subacute thyroiditis. The curves depicting FTI and TSH levels in Figure 7.45 are the ones that we basically compare our results to. The curve showing TG levels is commonly used as an indicator for thyroid damage. TG being high implies that the thyroid gland is damaged (Rubin, 2006). In this case, it returns to normal range (shaded region) near the end of the time horizon. This is consistent with our presumed Inflammation Status for this case. FTI is an indicator of free T4 levels in blood, but not the direct amount of it. In the model output shown in Figure 7.44, the total T4 level is shown. Since the real data uses a 101 different measure for T4 levels in blood, the output of out model is numerically not comparable to the real data. In subacute thyroiditis, not the behaviour or the course, but the levels of hormones may show variability from patient to patient. So, even if the units did match, it would not be very reasonable to try to exactly match to the data points of only one patient. And, since the ultimate aim of this study is not point prediction, it can be said that the model gives reasonable results by matching the typical dynamical behaviour. Figure 7.45. Data from a patient with subacute thyroiditis (Lazarus, 2009). 6 5 TSH 4 3 2 1 0 FT4 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 Figure 7.46. Data from a patient with subacute thyroiditis (secondary axis: TSH). 102 The name FT4 in Figure 7.46 refers to the free T4 concentration. Though not as obvious as the data shown in Figure 7.45, the overall course of behaviour that TSH and FT4 follows is essentially the same. The patient first presents with very low TSH and elevated FT4 values. With the decline in FT4 concentrations, TSH rises and then stabilizes (except for the last data point) at a normal level. Lastly, the work of Werner et al. (2005) states that in the thyrotoxic phase of subacute thyroiditis, the T3 to T4 ratio is lower than in Graves' thyrotoxicosis. Since the reason behind the thyrotoxic phase is the uncontrolled leakage of preformed hormones, the effect of thyroidal stimulation on the preferential secretion of T3 is non-existent in this case. Therefore, the ratio of T3 to T4 is lower in the thyrotoxic phase of subacute thyroiditis, as can be seen in Figure 7.47. 0.03 0.0275 0.025 0.0225 0.02 0 20 40 60 80 100 Time (Day) 120 140 Figure 7.47. T3 to T4 ratio in subacute thyroiditis. 103 8. CONCLUSION In this study, a model for the thyroid hormone dynamics is constructed. Thyroid hormones are the primary regulators of metabolic functions in the body, and disorders related to thyroid hormone system are commonly seen. The aim of this study is first to model the dynamics of the thyroid hormones and the stimulating hormones in healthy body, then to adapt the model to portray some common abnormalities/disorders, finally to capture the characteristic dynamics of the hormones involved under these circumstances and to provide a platform to test possible scenarios. For model validation, standard model structure and behaviour validity tests have been applied. In this study, validity tests are illustrated by four runs to demonstrate the consistency of the model outputs with the information in literature. First, the equilibrium behaviour is shown to depict the equilibrium state in the body under normal conditions. Then, the equilibrium state is disturbed with the administration of TRH and the outputs are compared to the real data. Thereafter, the model is run under two scenarios where the secretion rate of T4 is increased ten-fold and zeroed for one hour. The outputs are highly consistent with the numerical data suggested by the related literature. Finally, the effects of hypophysectomy, the complete removal of the pituitary gland, are shown. The behaviour of the model under all these scenarios reasonably matches the qualitative and quantitative data in literature. After the validation tests, four different conditions related to the thyroid are addressed. Firstly, Graves’ disease, the most common cause of hyperthyroidism, is addressed in two different levels of iodine intake. In Graves’ disease; the formation of goiter, effect of iodine availability on the severity of the disease, and other typical changes in hormones and glands are well mimicked by the model. Increased T3/T4, which is often used as a diagnostic criterion in Graves’ disease, is also captured by the simulations. Secondly, iodine deficiency, one prevailing cause of hypothyroidism, is discussed for two 104 different levels of daily iodine intake. The model was able to depict all the characteristic changes including the goiter formation and increase in T3/T4 in these two scenarios, both independently and comparatively. Thirdly, the transient inhibitory effect of excessive iodine intake on thyroid gland is discussed. The increase in thyroid volume and the mild decline in thyroid hormones are captured well. Lastly, a disorder called subacute thyroiditis is analysed. Subacute thyroiditis is a common disorder in which thyroid gland is exposed to inflammation. The model is shown to reproduce well the behaviour of hormones during the typical triphasic clinical course of subacute thyroiditis, composed of thyrotoxicosis, hypothyroidism and normal thyroid functioning. As far as the information in literature and interviews with the medical doctors are concerned, the model structure exhibits a reasonable degree of validity. As future work, more real data can be collected, parameters can be adjusted to reflect more precisely quantitative and qualitative real data, and some extensions in the model structure can be done to be able to model medical interventions and drug therapy. 105 APPENDIX: MODEL EQUATIONS abs fr of I = 0.95 {dimensionless} Abs rate of T3 by tissues = T3 in Blood × (35 − 5 × LN(2)) / 5 {µg / day} Abs rate of T4 by tissues = T4 in Blood × ((5334 − 7440 × LN(2)) / 217) / 240 {µg / day} AT for store restor = 30 {day} chg in trap fr = disc / del for trapping fr {dimensionless} Conv rate to rT3 = T4 in Blood × 32 / 240 {µg / day} del for trapping fr = 1 {day} des hypo wt = normal hypo wt × eff of imp TRH sec on hypo wt {mg} des pit wt = normal pit wt × eff of imp TSH sec on pit wt {mg} des T3 store = total normal TH store × fr of T3 sec {µg} des T3 syn = T3 sec rate + T3 store adj {µg / day} des T4 store = total normal TH store × fr of T4 sec {µg} des T4 syn = T4 sec rate + T4 store adj {µg / day} des thy wt = normal thy wt × eff of imp TH sec on thy wt {g} des trap fr = LOOKUP EXTRAPOLATE(gr for des trap fr, ratio of thy I conc to normal) {dimensionless} disc = des trap fr − Pot Trap Fr {dimensionless} disc btw pot T3 syn and I rest T3 syn = pot T3 syn × (1 − ratio of cap rest I cons to pot) {µg / day} disc btw pot TH syn and I rest TH syn = pot TH syn × (1 − ratio of cap rest I cons to pot) {µg / day} disc from des T3 store = des T3 store − T3 Store {µg} disc from des T4 store = des T4 store − T4 Store {µg} eff of cap on TH sec = LOOKUP EXTRAPOLATE(gr for thy cap, ratio of imp TH sec to thy cap) {dimensionless} eff of cap on TRH sec = LOOKUP EXTRAPOLATE(gr for hypo cap, ratio of imp TRH sec to hypo cap) {dimensionless} 106 eff of cap on TSH sec = LOOKUP EXTRAPOLATE(gr for pit cap, ratio of imp TSH sec to pit cap) {dimensionless} eff of I on thy cap = LOOKUP EXTRAPOLATE(gr for eff of I on thy cap, ratio of I in thy to thres) {dimensionless} eff of imp TH sec on thy wt = LOOKUP EXTRAPOLATE(gr for imp TH sec on thy wt, ratio of smth imp TH sec to normal) {dimensionless} eff of imp TRH sec on hypo wt = LOOKUP EXTRAPOLATE(gr for eff of imp TRH sec on hypo wt, ratio of smth imp TRH sec to normal) {dimensionless} eff of imp TSH sec on pit wt = LOOKUP EXTRAPOLATE(gr for eff of imp TSH sec on pit wt, ratio of smth imp TSH sec to normal) {dimensionless} eff of pref T3 syn on red in T3 syn = LOOKUP EXTRAPOLATE(gr for eff of pref T3 syn on red in T3 syn, ratio of disc btw I rest TH syn and pot TH syn to pot TH syn) {dimensionless} eff of T3 conc on peri conv = LOOKUP EXTRAPOLATE(gr for eff of T3 on peri conv, ratio of T3 to normal) {dimensionless} eff of T3 store cap = LOOKUP EXTRAPOLATE(gr for eff of TH store cap, ratio of pos to pot T3 sec) {dimensionless} eff of T4 store cap = LOOKUP EXTRAPOLATE(gr for eff of TH store cap, ratio of pos to pot T4 sec) {dimensionless} eff of TH on TRH sec = LOOKUP EXTRAPOLATE(gr for eff of TH on TRH sec, log ratio of TH to normal) {dimensionless} eff of TH on TSH sec = LOOKUP EXTRAPOLATE(gr for eff of TH on TSH sec, log ratio of TH to normal) {dimensionless} eff of thy cap on T3 syn = LOOKUP EXTRAPOLATE(gr for thy cap, ratio of des T3 syn to cap) {dimensionless} eff of thy cap on T4 syn = LOOKUP EXTRAPOLATE(gr for thy cap, ratio of des T4 syn to cap) {dimensionless} eff of thy I cap = LOOKUP EXTRAPOLATE(gr for I store cap, ratio of pot to pos I cons) {dimensionless} 107 eff of thy stim on T3 fr = LOOKUP EXTRAPOLATE(gr for eff of thy stim on TH fr, ratio of short smth imp TH sec to normal) {dimensionless} eff of thy wt on I trap = LOOKUP EXTRAPOLATE(gr for eff of thy wt on I trap, Thy Wt / 20) {dimensionless} eff of TRH on TSH sec = LOOKUP EXTRAPOLATE(gr for eff of TRH on TSH sec, log ratio of TRH to normal) {dimensionless} eff of TSH on I trapping = LOOKUP EXTRAPOLATE(gr for eff of TSH on I trap, log ratio of TSH to normal) {dimensionless} eff of TSH on TH sec = LOOKUP EXTRAPOLATE(gr for eff of TSH on TH sec, log ratio of TSH to normal) {dimensionless} excr fr = 103.358 / 150 {1 / day} fr of T3 sec = LOOKUP EXTRAPOLATE(gr for T3 sec fr, pot fr of T3 sec) {dimensionless} fr of T4 sec = 1 − pot fr of T3 sec{dimensionless} free T3 in blood = 0.003 × T3 in Blood {µg} free T4 in blood = T4 in Blood × 0.0002{µg} gr for des trap fr = ([(0, 0) − (5, 0.6)], (0, 0.5), (0.1, 0.5), (0.386965, 0.491429), (0.661914, 0.47619), (0.824847, 0.460952), (1, 0.419131), (1.16208, 0.329825), (1.22324, 0.245614), (1.2844, 0.154386), (1.43731, 0.0877193), (1.67006, 0.0333333), (1.96538, 0.0152381), (2.27088, 0.0104762), (2.59674, 0.00761904), (2.99389, 0.0057143), (3.5336, 0.001), (4.95927, 0.0005), (5, 0.0005)) {dimensionless} gr for eff of I on thy cap = ([(0.9, 0) − (1.5, 1)], (0.9, 1), (0.91, 1), (0.920183, 0.964912), (0.925662, 0.866667), (0.936697, 0.70614), (0.958656, 0.447619), (0.970876, 0.319048), (0.977064, 0.263158), (0.98554, 0.195238), (0.997248, 0.140351), (1.01009, 0.098), (1.02661, 0.0894737), (1.05596, 0.087), (1.09817, 0.086), (1.49, 0.085), (1.5, 0.085)) {dimensionless} gr for eff of imp TRH sec on hypo wt = ([(0, 0) − (1, 4)], (0.02, 0.3), (0.021, 0.3), (0.030581, 0.403509), (0.04058, 0.5316), (0.0814664, 0.723809), (0.224033, 0.8), (0.366972, 0.9), (0.550459, 1), (0.733198, 1), (1, 1), (2, 1), (4.86762, 1.02857), (5.23014, 1.06667), (5.74338, 1.08571), (6.13442, 1.12381), (6.5499, 1.21905), (6.89206, 1.29524), 108 (7.18534, 1.37143), (7.55193, 1.48571), (7.96741, 1.58095), (8.28513, 1.69524), (8.77393, 1.90476), (9.28717, 2.15238), (9.75153, 2.4381), (10.1181, 2.7619), (10.3381, 3.00952), (10.6558, 3.29524), (10.9491, 3.46667), (11.3157, 3.65714), (11.7067, 3.69524), (12, 3.7)) {dimensionless} gr for eff of imp TSH sec on pit wt = ([(0, 0) − (12, 4)], (0.02, 0.3), (0.021, 0.3), (0.030581, 0.403509), (0.04058, 0.5316), (0.0814664, 0.723809), (0.224033, 0.8), (0.366972, 0.9), (0.550459, 1), (0.733198, 1), (1, 1), (2, 1), (4.86762, 1.02857), (5.23014, 1.06667), (5.74338, 1.08571), (6.13442, 1.12381), (6.5499, 1.21905), (6.89206, 1.29524), (7.18534, 1.37143), (7.55193, 1.48571), (7.96741, 1.58095), (8.28513, 1.69524), (8.77393, 1.90476), (9.28717, 2.15238), (9.75153, 2.4381), (10.1181, 2.7619), (10.3381, 3.00952), (10.6558, 3.29524), (10.9491, 3.46667), (11.3157, 3.65714), (11.7067, 3.69524), (12, 3.7)) {dimensionless} gr for eff of pref T3 syn on red in T3 syn = ([(0, 0) − 1, 1)], (0, 1), (0.0001, 1), (0.0203666, 0.980952), (0.0305499, 0.966667), (0.0610998, 0.909524), (0.0916497, 0.838095), (0.120163, 0.766667), (0.167006, 0.661905), (0.211813, 0.566667), (0.258656, 0.490476), (0.297352, 0.438095), (0.329939, 0.4), (0.372709, 0.352381), (0.423625, 0.3), (0.484725, 0.252381), (0.560081, 0.195238), (0.623218, 0.147619), (0.672098, 0.114286), (0.729124, 0.0857143), (0.784114, 0.0619048), (0.828921, 0.047619), (0.881874, 0.0285714), (0.936864, 0.0190476), (0.9999, 0), (1, 0)) {dimensionless} gr for eff of T3 on peri conv = ([(0, 0.8) − (1, 1.7)], (0.0112016, 1.64762), (0.0492872, 1.648), (0.0940937, 1.64333), (0.147862, 1.62714), (0.199389, 1.58667), (0.264358, 1.53), (0.329328, 1.47333), (0.387576, 1.41905), (0.445825, 1.36), (0.501833, 1.31143), (0.55112, 1.26286), (0.613849, 1.20619), (0.676578, 1.14952), (0.737067, 1.10095), (0.802037, 1.06333), (0.858045, 1.03619), (0.923014, 1.01333), (0.9999, 1), (1, 1)) {dimensionless} gr for eff of TH on TRH sec = ([(−1.1, 0) − (1, 10)], (−1.09572, 10), (−1.04012, 9.90476), (−0.997248, 9.7807), (−0.952294, 9.69298), (−0.900917, 9.38597), (− 0.860489, 9.14286), ( − 0.813442, 8.7619), (−0.759633, 8.42105), 7.04762), (−0.566972, 6.31579), (−0.451376, (−0.693686, 5.13158), 7.71429), (−0.393578, (−0.629532, 4.51754), (−0.284404, 3.50877), (−0.200917, 2.7193), (−0.143119, 2.2807), (−0.0660551, 1.53509), 109 (0, 1), (0.0174312, 0.868421), (0.0366972, 0.719298), (0.0489297, 0.574561), (0.0733945, 0.434211), (0.0948012, 0.315789), (0.134557, 0.22807), (0.180428, 0.153509), (0.248624, 0.118421), (0.33211, 0.100877), (0.440367, 0.0833333), (0.577982, 0.0701754), (0.730887, 0.0657895), (0.852294, 0.0614035), (0.922936, 0.0570175), (0.999, 0.05), (1, 0.05)) {dimensionless} gr for eff of TH on TSH sec = ([(0, 0) − (1, 1)], (0, 1), (0.0001, 1), (0.0366972, 0.982456), (0.0519878, 0.969298), (0.0642202, 0.938596), (0.0764526, 0.903509), (0.088685, 0.850877), (0.0997963, 0.780952), (0.110092, 0.70614), (0.124236, 0.609524), (0.140673, 0.508772), (0.16208, 0.403509), (0.186544, 0.320175), (0.2263, 0.223684), (0.262997, 0.166667), (0.308868, 0.131579), (0.389002, 0.104762), (0.468432, 0.0857143), (0.553517, 0.0745614), (0.629969, 0.0657895), (0.691131, 0.064), (0.776758, 0.063), (0.83792, 0.0622), (0.902141, 0.0614035), (0.941896, 0.0570175), (0.99, 0.05), (1, 0.05)) {dimensionless} gr for eff of TH store cap = ([(0, 0) − (1.5, 1)], (0, 0), (0.75, 0.75), (0.876147, 0.833333), (0.947047, 0.87619), (1.00509, 0.919048), (1.05092, 0.942857), (1.09633, 0.960526), (1.14679, 0.97807), (1.23, 1), (1.5, 1)) {dimensionless} gr for eff of thy stim on TH fr([(1, 0) − (5, 2.5)], (1, 1), (1.001, 1), (1.1, 1.02), (1.39144, 1.0636), (1.67278, 1.12939), (1.96636, 1.18421), (2.3211, 1.27193), (2.54128, 1.34868), (2.85933, 1.45833), (3.16514, 1.57895), (3.39755, 1.72149), (3.62997, 1.84211), (3.80122, 1.95175), (3.99694, 2.08333), (4.14373, 2.20395), (4.30275, 2.2807), (4.47401, 2.35746), (4.64526, 2.42325), (4.823, 2.47368), (4.999, 2.5), (5, 2.5)) {dimensionless} gr for eff of thy wt on I trap = ([(0, 0) − (3, 1.5)], (0, 0), (1e−005, 0), (0.0651731, 0.0285714), (0.126273, 0.0714285), (0.207739, 0.142857), (0.274949, 0.228571), (0.360489, 0.333333), (0.427699, 0.428571), (0.519348, 0.580952), (0.629328, 0.71426), (0.761711, 0.814286), (0.849287, 0.895238), (1, 1), (1.16701, 1.05714), (1.33198, 1.1), (1.5096, 1.13571), (1.71079, 1.16429), (2.01629, 1.20714), (2.389, 1.25), (2.66395, 1.27143), (2.99, 1.3), (3, 1.3)) {dimensionless} gr for eff of TRH on TSH sec = ([(−1, 0) − (1.1, 10.1)], (−1, 0.05), (−0.999, 0.05), (−0.790428, 0.053), (−0.640733, 0.058), (−0.550917, 0.065), (− 0.478208, 0.0714286), (− 0.428135, 0.0833333), (−0.379205, 0.0921053), (−0.324159, 0.122807), (−0.272171, 110 0.153509), (−0.223242, 0.201754), (−0.180428, 0.245614), (−0.140673, 0.311404), (−0.100917, 0.403509), (−0.0733945, 0.517544), (−0.0519878, 0.614035), (−0.0366972, 0.697368), (−0.0244648, 0.77193), (−0.0152905, 0.855263), (−0.00917431, 0.934211), (0, 1), (0.053211, 1.44737), (0.111009, 1.92982), (0.194495, 2.67544), (0.244603, 3.27048), (0.308758, 3.94381), (0.355804, 4.56905), (0.406422, 5.30702), (0.449898, 6.1081), (0.492668, 6.68524), (0.526884, 7.18857), (0.573931, 7.77143), (0.629532, 8.36857), (0.680855, 8.84), (0.732179, 9.27714), (0.766395, 9.47476), (0.804888, 9.61905), (0.864766, 9.81143), (0.924644, 9.90762), (0.984521, 10.027), (1.09572, 10.1)) {dimensionless} gr for eff of TSH on I trap = ([( − 1, 0) − (1, 1.5)], ( − 1, 0.02), ( − 0.9999, 0.02), ( − 0.885947, 0.0357143), ( − 0.751527, 0.0785714), ( − 0.649695, 0.15), ( − 0.551935, 0.257143), ( − 0.412844, 0.440789), ( − 0.29052, 0.598684), ( − 0.155963, 0.776316), (0, 1), (0.120163, 1.1), (0.211009, 1.16429), (0.340122, 1.22143), (0.462322, 1.27143), (0.565749, 1.308), (0.669725, 1.341), (0.828746, 1.38), (0.9999, 1.4), (1, 1.4)) {dimensionless} gr for eff of TSH on TH sec = ([( − 1, 0) − (1.1, 10.1)], (−1, 0.05), (−0.999, 0.05), (−0.790428, 0.053), (−0.640733, 0.058), (−0.550917, 0.065), (−0.478208, 0.0714286), (−0.426884, 0.0904762), (−0.37556, 0.119048), (−0.327902, 0.144285), (−0.277189, 0.190476), (−0.23442, 0.228571), (−0.195927, 0.271429), (−0.161711, 0.338095), (−0.131772, 0.419048), (−0.098778, 0.52381), (−0.0761711, 0.614286), (−0.0633401, 0.68), (−0.0498981, 0.761905), (−0.0291243, 0.847619), (−0.0162933, 0.92381), (0, 1), (0.0606924, 1.34286), (0.129124, 1.77143), (0.214664, 2.38095), (0.278819, 3.06), (0.334419, 3.64286), (0.381466, 4.46857), (0.419959, 5.24571), (0.462729, 6.02286), (0.492668, 6.68524), (0.526884, 7.18857), (0.573931, 7.77143), (0.629532, 8.36857), (0.680855, 8.84), (0.732179, 9.27714), (0.766395, 9.47476), (0.804888, 9.61905), (0.864766, 9.81143), (0.924644, 9.90762), (0.984521, 10.027), (1.09572, 10.1)) {dimensionless} gr for eff on HAT = ([(1, 30) − (5, 250)], (1, 30), (1.1, 30), (1.33028, 35.0877), (1.62385, 49.2982), (1.81957, 62.1053), (2.00306, 79.2105), (2.13761, 95.614), (2.27217, 113.684), (2.3945, 137.632), (2.66361, 178.158), (2.84709, 200.263), (3.11621, 218.684), (3.45872, 111 228.816), (3.75229, 236.184), (4.10703, 242.632), (4.42508, 245.395), (5, 250)) {dimensionless} gr for eff on PAT = ([(1, 30) − (5, 250)], (1, 30), (1.1, 30), (1.33028, 35.0877), (1.62385, 49.2982), (1.81957, 62.1053), (2.00306, 79.2105), (2.13761, 95.614), (2.27217, 113.684), (2.3945, 137.632), (2.66361, 178.158), (2.84709, 200.263), (3.11621, 218.684), (3.45872, 228.816), (3.75229, 236.184), (4.10703, 242.632), (4.42508, 245.395), (5, 250)) {dimensionless} gr for eff on TAT = ([(1, 30) − (5, 250)], (1, 30), (1.1, 30), (1.33028, 35.0877), (1.62385, 49.2982), (1.81957, 62.1053), (2.00306, 79.2105), (2.13761, 95.614), (2.27217, 113.684), (2.3945, 137.632), (2.66361, 178.158), (2.84709, 200.263), (3.11621, 218.684), (3.45872, 228.816), (3.75229, 236.184), (4.10703, 242.632), (4.42508, 245.395), (5, 250)) {dimensionless} gr for hypo cap = ([(0, 0) − (1.5, 1)], (0, 0), (0.75, 0.75), (0.855397, 0.838095), (0.934827, 0.9), (0.986762, 0.92381), (1.02953, 0.947619), (1.06925, 0.961905), (1.11813, 0.980952), (1.23, 1), (1.5, 1)) {dimensionless} gr for I store cap([(0, 0) − (1.1, 1)], (0, 0), (1e − 005, 0), (0.5, 0.5), (0.7, 0.7), (0.806517, 0.780952), (0.907332, 0.87619), (0.943177, 0.914286), (0.976782, 0.942857), (1.01039, 0.971429), (1.05, 1), (1.1, 1)) {dimensionless} gr for imp TH sec on thy wt = ([(0, 0) − (12, 4)], (0.02, 0.3), (0.021, 0.3), (0.030581, 0.403509), (0.04058, 0.5316), (0.0814664, 0.723809), (0.224033, 0.8), (0.366972, 0.9), (0.550459, 1), (0.733198, 1), (1, 1), (1.5, 1), (2.7156, 1.07), (3.52294, 1.19298), (4.55046, 1.36842), (5.50459, 1.54386), (6.45872, 1.73684), (7.52294, 2.03509), (8.51376, 2.40351), (9.43119, 2.73684), (9.87156, 2.96491), (10.2018, 3.12281), (10.6055, 3.31579), (10.9725, 3.54386), (11.3157, 3.65714), (11.7067, 3.69524), (12, 3.7)) {dimensionless} gr for pit cap = ([(0, 0) − (1.5, 1)], (0, 0), (0.75, 0.75), (0.855397, 0.838095), (0.934827, 0.9), (0.986762, 0.92381), (1.02953, 0.947619), (1.06925, 0.961905), (1.11813, 0.980952), (1.23, 1), (1.5, 1)) {dimensionless} gr for T3 sec fr = ([(0.9, 0.9) − (1.1, 1)], (0.9, 0.9), (0.97, 0.97), (0.97844, 0.976754), (0.988685, 0.982456), (1.00153, 0.988596), (1.01437, 0.991667), (1.02477, 0.99386), (1.04434, 0.99693), (1.07, 1), (1.1, 1)) {dimensionless} 112 gr for thy cap = ([(0, 0) − (1.5, 1)], (0, 0), (0.75, 0.75), (0.855397, 0.838095), (0.934827, 0.9), (0.986762, 0.92381), (1.02953, 0.947619), (1.06925, 0.961905), (1.11813, 0.980952), (1.23, 1), (1.5, 1)) {dimensionless} HAT = LOOKUP EXTRAPOLATE(gr for eff on HAT, ratio of des hypo wt to hypo wt) {days} hypo cap = Hypo Wt × normal hypo prod × 10 {ng / day} Hypo Wt = INTEG (Hypo wt chg, 40) {mg} Hypo wt chg = (des hypo wt − Hypo Wt) / HAT {mg / day} I abs rate = I intake × abs fr of I {µg / day} I cons = I cons for T3+I cons for T4 {µg / day} I cons for T3 = T3 syn rate × 3 × 126.904 / 651 {µg / day} I cons for T4 = T4 syn rate × 4 × 126.904 / 777 {µg / day} I excr = I in Blood × excr fr {µg / day} I from deiod = 0.8 × retained I {µg / day} I from rT3 = (126.904 / 777) × Conv rate to rT3 {µg / day} I from T3 conv = (126.904 / 777) × T4 to T3 conv rate {µg / day} I from T3 in tissues = 3 × (126.904 / 651) × Abs rate of T3 by tissues {µg / day} I from T4 in tissues = 4 × (126.904 / 777) × Abs rate of T4 by tissues {µg / day} I in Blood = INTEG (I abs rate+I from deiod − I excr − Trap rate, 150) {µg} I in Thy = INTEG (Trap rate − I cons, 15000) {µg} I inhib thres = MAX(SMOOTH3I( I in Thy, 30, 15000 ) × 1.1, 400) {µg} I intake = 150 {µg / day} imp TH sec = normal TH sec × eff of TSH on TH sec {µg / day} imp TRH sec = normal TRH sec × eff of TH on TRH sec {ng / day} imp TSH sec = eff of TRH on TSH sec × eff of TH on TSH sec × normal TSH sec {mU / day} log ratio of TH to normal = LOG(ratio of TH to normal TH, 10) {dimensionless} log ratio of TRH to normal = LOG( ratio of TRH to normal, 10 ) {dimensionless} log ratio of TSH to normal = LOG(ratio of TSH to normal, 10) {dimensionless} MW of T3 = 651 × 1.66054 × 10^( − 18) {µg} 113 MW of T4 = 777 × 1.66054 × 10^( − 18) {µg} normal amount of T3 mol = 1.38759e+013 {dimensionless} normal amount of T4 mol = 3.72024e+013 {dimensionless} normal hypo prod = 24 × (LN(2) / (6.2 / 60)) × 2 / 40 {ng / day / mg} normal hypo wt = 40 {mg} normal I in thy = 15000 {µg} normal pit prod = 24 × 6.6 × LN(2) / 22.5 {mU / day / mg} normal pit wt = 22.5 {mg} normal ratio of T3 to T4 = normal amount of T3 mol / normal amount of T4 mol {dimensionless} normal T3 in blood = 5 {µg} normal T4 to T3 conv fr = (28 / 240) × 777 / 651 {dimensionless} normal TH = 5.10783e+013 {dimensionless} normal TH sec = 97 {µg / day} normal thy prod = 97 / 20 {µg / day / g} normal thy wt = 20 {mg} normal TRH = 2 {ng} normal TRH sec = 24 × (LN(2) / (6.2 / 60)) × 2 {ng / day} normal TSH = 6.6 {mU} normal TSH sec = 24 × 6.6 × LN(2) {mU / day} PAT = LOOKUP EXTRAPOLATE(gr for eff on PAT, ratio of des pit wt to pit wt) {day} pit cap = normal pit prod × Pit Wt × 10 {mU / day} Pit Wt = INTEG (Pit wt chg, 22.5) {mg } Pit wt chg = (des pit wt − Pit Wt) / PAT {mg / day} pos I cons = I in Thy × 0.15 {µg / day} pos T3 sec = T3 Store × 0.15 {µg / day} pos T4 sec = T4 Store × 0.15 {µg / day} pot fr of T3 sec = (normal ratio of T3 to T4 × 7 / 90) / (ratio of T3 to T4 + normal ratio of T3 to T4 × 7 / 90) × eff of thy stim on T3 fr {dimensionless} pot I cons for T3 = pot T3 syn × 3 × 126.904 / 651 {µg / day} 114 pot I cons for T4 = pot T4 syn × 4 × 126.904 / 777 {µg / day} pot T3 sec = pot TH sec × fr of T3 sec {µg / day} pot T3 syn = eff of thy cap on T3 syn × thy syn cap for T3 {µg / day} pot T4 sec = pot TH sec × fr of T4 sec {µg / day} pot T4 syn = eff of thy cap on T4 syn × thy syn cap for T4 {µg / day} pot TH sec = eff of cap on TH sec × thy cap {µg / day} pot TH syn = pot T3 syn+pot T4 syn {µg / day} pot total I cons = pot I cons for T3 + pot I cons for T4 {µg / day} pot total I cons under I cap rest = pos I cons × eff of thy I cap {µg / day} Pot Trap Fr = INTEG (chg in trap fr, (62.8697 / 150)) {1 / day} ratio of cap rest I cons to pot = pot total I cons under I cap rest / pot total I cons {dimensionless} ratio of des hypo wt to hypo wt = des hypo wt / Hypo Wt {dimensionless} ratio of des pit wt to pit wt = des pit wt / Pit Wt {dimensionless} ratio of des T3 syn to cap = MAX( des T3 syn , 0) / thy syn cap for T3 {dimensionless} ratio of des T4 syn to cap = MAX( des T4 syn , 0) / thy syn cap for T4 {dimensionless} ratio of des thy wt to thy wt = des thy wt / Thy Wt {dimensionless} ratio of disc btw I rest TH syn and pot TH syn to pot TH syn = 1 − ratio of cap rest I cons to pot {dimensionless} ratio of I in thy to thres = I in Thy / I inhib thres {dimensionless} ratio of imp TH sec to thy cap = imp TH sec / thy cap {dimensionless} ratio of imp TRH sec to hypo cap = imp TRH sec / hypo cap {dimensionless} ratio of imp TSH sec to pit cap = imp TSH sec / pit cap {dimensionless} ratio of pos to pot T3 sec = pos T3 sec / pot T3 sec {dimensionless} ratio of pos to pot T4 sec = pos T4 sec / pot T4 sec {dimensionless} ratio of pot to pos I cons = pot total I cons / pos I cons {dimensionless} ratio of short smth imp TH sec to normal = short smth imp TH sec / normal TH sec {dimensionless} ratio of smth imp TH sec to normal = smth imp TH sec / normal TH sec {dimensionless} 115 ratio of smth imp TRH sec to normal = smth imp TRH sec / normal TRH sec imp TSH sec / normal TSH sec {dimensionless} ratio of smth imp TSH sec to normal = smth {dimensionless} ratio of T3 to normal = T3 in Blood / normal T3 in blood {dimensionless} ratio of T3 to T4 = total free T3 molecules / total free T4 molecules {dimensionless} ratio of TH to normal TH = "total free T3&T4 molecules" / normal TH {dimensionless} ratio of thy I conc to normal = I in Thy / normal I in thy {dimensionless} ratio of TRH to normal = TRH / normal TRH {dimensionless} ratio of TSH to normal = TSH / normal TSH {dimensionless} retained I = I from rT3+I from T3 in tissues+I from T4 in tissues+I from T3 conv {µg / day} short smth imp TH sec = SMOOTH3I( imp TH sec, 10, 97 ) {µg / day} smth imp TH sec = SMOOTH3I( imp TH sec, 20, 97 ) {µg / day} smth imp TRH sec = SMOOTH3I(imp TRH sec, 20, 24 × (LN(2) / (6.2 / 60)) × 2) {ng / day} smth imp TSH sec = SMOOTH3I( imp TSH sec, 20, 24 × 6.6 × LN(2) ) {mU / day} T3 clear fr = LN(2) / 1 {1day} T3 clear rate = T3 in Blood × T3 clear fr {µg / day} T3 conc = T3 in Blood / 30 {µg / dL} T3 from deiod of T4 = T4 to T3 conv rate × 651 / 777 {µg / day} T3 in Blood = INTEG (T3 from deiod of T4+T3 sec rate − Abs rate of T3 by tissues − T3 clear rate, 5) {µg} T3 sec rate = pot T3 sec × eff of T3 store cap {µg / day} T3 Store = INTEG (T3 syn rate − T3 sec rate,420) {µg} T3 store adj = disc from des T3 store / AT for store restor {µg / day} T3 syn rate = (pot T3 syn − disc btw pot T3 syn and I rest T3 syn × eff of pref T3 syn on red in T3 syn) / MAX(1, (pot T3 syn / MAX(1e − 005,(pot TH syn − disc btw pot TH syn and I rest TH syn)))) {µg / day} T4 clear fr = LN( 2 ) / (7) {1 / day} 116 T4 clear rate = T4 in Blood × T4 clear fr {µg / day} T4 conc = T4 in Blood / 30 {µg / dL} T4 in Blood = INTEG (T4 sec rate − Abs rate of T4 by tissues − Conv rate to rT3 − T4 clear rate − T4 to T3 conv rate, 240) {µg} T4 sec rate = pot T4 sec × eff of T4 store cap {µg / day} T4 Store = INTEG (T4 syn rate − T4 sec rate, 5400) {µg} T4 store adj = disc from des T4 store / AT for store restor {µg / day} T4 syn rate = pot TH syn − disc btw pot TH syn and I rest TH syn − T3 syn rate {µg / day} T4 to T3 conv fr = eff of T3 conc on peri conv × normal T4 to T3 conv fr {1 / day} T4 to T3 conv rate = T4 in Blood × T4 to T3 conv fr {µg / day} TAT = LOOKUP EXTRAPOLATE(gr for eff on TAT, ratio of des thy wt to thy wt) {day} thy cap = normal thy prod × Thy Wt × 10 × eff of I on thy cap {µg / day} thy syn cap for T3 = thy cap × fr of T3 sec {µg / day} thy syn cap for T4 = thy cap × fr of T4 sec {µg / day} Thy Wt = INTEG (Thy wt chg, 20) {g} total free T3 molecules = free T3 in blood / MW of T3 {dimensionless} "total free T3&T4 molecules" = total free T3 molecules+total free T4 molecules{dimensionless} total free T4 molecules = free T4 in blood / MW of T4{dimensionless} total normal TH store = 5400 + 420 {µg} Trap rate = Pot Trap Fr × I in Blood × eff of TSH on I trapping × eff of thy wt on I trap {µg / day} TRH = INTEG (TRH sec rate − TRH clear rate, 2) {ng} TRH clear fr = LN(2) / (6.2 / (60 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