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Steady-State Blood Flow Model of the Rhesus Monkey By Koula Quirk RET Fellow 2010 Advisor: Dr. Andreas A. Linninger Laboratory for Product and Process Design Department of Bioengineering University of Illinois at Chicago RET Program NSF EEC-0743068 (A. Linninger, PI) Abstract A static one-dimensional whole body model of the rhesus monkey was designed with the organs represented as compartments. Blood flow was simulated through these compartments as blood pressure drops and was distributed throughout the body. A steady-state computational model of drug delivery in the rhesus monkey was constructed from physiological values derived using cardiac output, organ resistance and blood pressure experimental data from the literature. The model predicts a steady-state view of regional organ blood flow and blood pressure. Background Pharmacokinetics is a branch of pharmacology that studies, in living organisms, the way a drug is absorbed, metabolized, distributed, and eliminated by the body. It is a complex process that is not well understood- thus giving rise to dangerous side effects and increasing risks of drug treatments. A steady-state analog has been completed modeling pharmacokinetics in the rat and the human model is in progress, both in LPPD. For allometric scaling, an intermediate animal model is desirable to gain a more precise understanding of drug delivery. The number of animals used in clinical drug trials is enormous and can be vastly reduced with the design of computational models to simulate drug distribution inside the body. The rhesus (Macaca mulatta) monkey is anatomically and physiologically close to humans, sharing about 93% of our genes (Ottesen, 2006). It breeds well in captivity, is widely available, and is the non-human primate of choice for conducting research on health-related topics. For these reasons we began work on modeling drug delivery in the rhesus monkey. Introduction Physiological data obtained from both invasive and non-invasive methods has grown exponentially in the past several years. The field of applied mathematical modeling in human and animal physiology continues to develop along with the increasing amount of available data. Large amounts of data can be better understood by using mathematical models to help researchers and doctors develop new insights into physiological functions and relationships. The model developed for the rhesus monkey was derived using quantitative results based on experimental measurements. Methods The main source of data was drawn from the Forsyth paper (Forsyth, 1968). They used the radioactive microsphere technique to measure the distribution of cardiac output and regional organ blood flow. This method makes repeated measurements of blood flow in 2 multiple organs by using plastic microspheres labeled with different radioactive nuclides. The male rhesus monkeys weighed between 3.1 and 6.8 kg. Using Kirchoff’s conservation (current and voltage) laws, Darcy’s law, and the HagenPoiseuille equation to derive the calculations, we modeled blood flow through compartments and constructed a static one-dimensional model to simulate blood distribution through the organs, or tubes. The appendix contains calculated values. Kirchoff’s law states, in essence, that the sum of voltages in a closed circuit is zero. That is, input must equal output, i.e., in our case, blood is conserved. Darcy’s law, analogous to Ohm’s law, describes flow of fluid (blood) through a porous medium. That is, the flow rate is directly proportional to the drop in vertical elevation between two places in a medium and indirectly proportional to the distance between them. In our case, the flow rate is a function of the pressure change divided by the resistance of the body organ. The Hagen-Poiseuille equation is a physical law that gives the resistance when you know the pressure drop in fluid flowing through a long cylindrical pipe (in our case arteries and veins). Equation 1 below is Darcy's law; Equation 2 is the Hagen-Poiseuille equation: (1) (2) where: F = blood flow (m*s-1) P = pressure (Pa) R = resistance (m-1) ν = fluid viscosity (Pa·s) L = length of tube (m) r = radius of tube (m) 3 In standard fluid dynamics notation: or (3) (4) The change in pressure is equal to the pressure leaving the organ subtracted from the pressure at the input to the organ. where: ΔP is the pressure drop L is the length of pipe μ is the dynamic viscosity Q is the volumetric flow rate r is the radius d is the diameter π is the mathematical constant Results Figure 1 shows our predicted results for the rhesus monkey. Blood pressure drops from 110 mmHg as it leaves the heart and drops according to the resistance of each organ. Table 1 and 2 also show that most of the percentages of cardiac output values in the rhesus monkey are in the same broad range as those found in humans and rats (Forsyth, 1968). The main difference is the lower fraction of output to the large and small intestines found in the monkey (8%) compared to the rat (19%). The values for the rhesus brain (7%) are midway between man (14%) and rat (1%). Blood pressure for the rhesus is 110mmHg, much lower than humans at 120 or even rats at 116. Flow rates to the kidneys were similar between man, rat, and rhesus monkey. Percentage of cardiac output to the liver was the lowest in the rat. Discussion According to Forsyth the majority of cardiac output values found in the rhesus monkey are within the same range as those (culled from other reports) in man and rat. The main difference is the lower percentage of output to the large and small intestines in the monkey (8%) compared to the rat (19%). Additionally, there is a greater total cardiac output per body weight in monkeys than is reported in other species (Forsyth, 1968). This is reflected in the greater flows seen in most of the organs for the monkey compared to other species. The data also show a much higher total cardiac output per body weight in rhesus monkeys than is reported for other species. 4 Figure 1 shows the analogous computational model derived from experimental data in the literature. Figure 2 shows a general anatomical view of a monkey for reference purposes. The colors of the diagram correspond to changing pressure values through the whole system. Pressure is highest where the blood leaves the heart and enters each organ and drops according to organ resistance; pressure eventually drops to zero as it returns to the heart at the output of all the organs. Table 1 shows the pressure drops predicted in the model simulation. Numbers 1-45 represent the points of blood output from major organs (1=heart). Values are the predicted blood pressures of the rhesus monkey. These were calculated based on the data in the Forsyth article. Figure 3 is included as reference to show the general human circulatory system. Conclusion New and continuing developments in computer technology aid mathematical modeling. The improvement of existing mathematical models and construction of new ones can lead to better simulation environments which are necessary for education and research. Mathematical models can help avoid misunderstandings and wasted effort. Most concepts can be clearly defined only by using mathematics. Mathematical models often give rise to new and important questions that could not be asked without the use of these models. For example, under what conditions is the cardiovascular system stable? How does the topology of the vascular system influence the function of the system? This, in turn, deepens our understanding of human and animal physiology. It is hoped that the pharmacokinetic model of drug delivery in the rhesus monkey contributes to this end. 5 Figure 1: Analogous blood flow computational model (based on previous work on the rat model by Cierra M. Hall at LPPD/UIC. Figure 2: Monkey anatomy, for reference. (From www.infovisual.info, permission granted.) 6 Table 1: Simulated blood pressure values in the rhesus monkey (mmHg). % Solution from Delphi: % Blood Pressures: % x(1) = 110; % x(2) = 109.000000308445; % x(3) = 105.000000266026; % x(4) = 104.999998536621; % x(5) = 13.0000010838241; % x(6) = 108.999997482744; % x(7) = 104.999997672403; % x(8) = 13.0000020063037; % x(9) = 12.0000020490089; % x(10) = 11.9999992233083; % x(11) = 5.9999993852343; % x(12) = 19.9999974855523; % x(13) = 12.999997268887; % x(14) = 11.9999972403551; % x(15) = 103.999988470877; % x(16) = 101.239988934643; % x(17) = 12.920003674989; % x(18) = 12.0000038273516; % x(19) = 103.99998255177; % x(20) = 101.239983370683; % x(21) = 12.9200094757135; % x(22) = 12.0000097464582; % x(23) = 103.999977300554; % x(24) = 101.239978432723; % x(25) = 12.9200146244713; % x(26) = 12.000014997675; % x(27) = 103.999973305203; % x(28) = 101.239974677094; % x(29) = 12.9200185399147; % x(30) = 12.0000189930253; % x(31) = 103.999970565719; % x(32) = 101.239972101602; % x(33) = 12.9200212184552; % x(34) = 12.0000217325093; % x(35) = 103.999968454168; % x(36) = 101.239970109742; % x(37) = 12.9200232992409; % x(38) = 12.0000238440601; % x(39) = 103.999995057873; % x(40) = 77.0000006413871; % x(41) = 104.00000214443; % x(42) = 102.000002032651; % x(43) = 108.999998424644; % x(44) = 12.0000011071087; % x(45) = 4; Numbers 1-45 represent the points of blood output from major organs (1=heart). Values are the predicted blood pressures of the rhesus monkey. These were calculated based on the data in the Forsyth article. This corresponds to Figure 1 diagram: That is, #1 represents the blood pressure output from the heart, and #45 represents the pressure at the input to the heart. 7 Figure 3: Main circulatory routes, arterial and venous systems, showing blood flow to major organs. (From Tortora, 1990, permission granted) Appendix Figure 4: Cardiac output values at start of radioactivemicrosphere method. (From Forsyth, 1968) Figure 5: Organ blood flow and resistance in the rhesus monkey showing significantly higher percentage of CO to liver. (From Forsyth, 1968) 9 Figure 6: Comparison of CO values in organs of man, dog, rat, baboon, and rhesus monkey, showing lower fraction of output to the large and small intestine in the rhesus monkey. (From Forsyth, 1968) 10 Cardiac Output (L/min) Mean Arterial Pressure (mmHg) Average Weight (kg) Cardiac Output/wt (L/(kg*min)) Rat 0.1060 116.0000 0.3790 0.2797 Rat 1.0068 0.0528 5.1011 1.5156 2.7000 0.9000 13.8000 22.4000 0.7600 4.4000 0.7500 4.3000 Rhesus 1.0860 110.0000 4.1000 0.2649 0.5 Rhesus 0.6272 0.2827 4.8084 0.7496 6.5000 4.6000 12.3000 11.0000 2.8000 4.4000 0.7000 4.1000 (Delp, 1998) (Forsyth, 1968) 0 Organ Flow Rates Brain (L/(kg*min)) Liver (L/(kg*min)) Kidneys (L/(kg*min)) GI Tract (L/(kg*min)) Brain (% CO) Liver (% CO) Kidneys (% CO) GI Tract (% CO) Brain (% wt) Liver (% wt) Kidneys (% wt) GI Tract (% wt) Man 5.2000 120.0000 70.0000 0.0743 1.0000 Man 0.5600 0.2500 3.7500 2.9000 11.4000 4.6000 17.5000 18.1000 2.0000 2.6000 0.4000 1.4000 Average 2.1307 115.3333 24.8263 0.2063 SD 2.702907 5.033223 39.16576 0.114552 Average 0.7313 0.1952 4.5532 1.7217 6.8667 3.3667 14.5333 17.1667 1.8533 3.8000 0.6167 3.2667 SD 0.240923 0.124391 0.710802 1.089938 4.361575 2.136196 2.67644 5.757025 1.027878 1.03923 0.189297 1.619671 (Brown, 1994) t23 *GI Tract: large intestine, smallsmall intestine, cecum,cecum, pancreas, spleen,spleen, stomachstomach, and sometimes the colon, the rectum and eso *GI Tract: large intestine, intestine, pancreas, and sometimes colon, rectum and esophagus. *GI Tract: large intestine, small intestine, cecum, pancreas, spleen, stomach, and Table 2: the Comparison of values for cardiac output and organ blood flow for rat, rhesus monkey, sometimes colon, rectum and esophagus. and human. (Compiled byof Cierra M. Hall, REU 2010 at LPPD/UIC) Comparison values for Cardiac output and organ blood flow for Rat, hum Acknowledgements Financial support by NSF RET Grant – EEC 0743068 (Andreas Linninger, PI) is gratefully acknowledged. For their generosity in patiently sharing their skills and knowledge I also thank: Prof. Andreas A. Linninger Cierra M. Hall, REU 2010 Sukhi Basati, PhD. Candidate The Laboratory for Product and Process Design staff 11 References Normal distribution of cardiac output in the unanesthetized, restrained rhesus monkey. Am. J. Physiol. 25: 736-741, 1968, Forsyth, Nies, Wyler, Neutze, and Melmon Physiological parameters in laboratory animals and humans. Pharm Res 10(7): 10931095, 1993, Davies, B. and T. Morris Dynamic pressure-flow relationships of brain blood flow in the monkey. J Neurosurg 41(5): 590-596, 1974, Early, C. B., R. C. Dewey, et al. Redistribution of cardiac output during hemorrhage in the unanesthetized monkey. Circ Res 27(3): 311-320, 1970, Forsyth, R. P., B. I. Hoffbrand, et al Modelling of individual pharmacokinetics for computer-aided drug dosage. Comput Biomed Res 5(5): 411-459, 1972, Sheiner, L. B., B. Rosenberg, et al. Applied Mathematical Models in Human Physiology. Series: Siam Monographs on Mathematical Modeling and Computation 9, 2006, J. T. Ottesen, M. S. Olufsen, and J. K. Larsen Principles of Anatomy and Physiology, 12th ed. 2009, G. J. Tortora and B. H. Derrickson biosystems.okstate.edu www.wikipedia.org www.thefreedictionary.com 12