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KINETIC MODELLING OF NON-LINIER PHYSIOLOGY AND CARDIOVASCULAR CARTOGRAPHY Kumar RV1, Kishore AGR2, Chandra VS2, Kumar P2, Rekha BR1, Deepti A1, Imre J3, Lakotosh Y3 1 Center for Artificial Intelligence and Non-Linear Studies (CAINS), 2Department of Cardiology, Manipal Heart Foundation, Bangalore, India, 3ASKIT kft, Budapest, Hungary. of our screening tests applied to the INTRODUCTION "healthy" populations are bound to end up The era spawned by the advent, and ready with many false positives, making life acceptance of the "chaos" model, has lead to miserable for those unfortunate victims. an explosion of nascent schools of thought Similar is the experience with predicting that challenge hitherto entrenched linear death based on the left ventricular ejection statistical Euclidean fractions, in the immediate post myocardial principles in natural sciences . Dynamic infarction phase. If one goes deep into all human body does not obey the rules of this linear Euclidean mathematics. Stability in mathematical formulae could be applied to biology is a myth. It occurs only after death. any human organ. Take the heart for We have been predicting the unpredictable example; is it a triangle, square or an in medicine all along. What happens to the oblong? It has no integer measure at all; and human organism as time evolves depends on how do we apply the integer measure the total knowledge of the initial state of the formulae to calculate the various facets of organism. Since we cannot know this, with cardiac function? For want of a better the help of the reductionist science, we are measure of the non-integer human heart, we unable to predict the future. We can only still hang on to our old formulae. When the assess the human phenotype, which forms conventional science has been heading for a about 30% of the organism and the rest is crisis with increased specialisation, it is the made up of our genetype and consciousness. new science of non-linear mathematics that We need non-linear mathematics of "Chaos" will come to our help2. Fractal configuration, to predict man's future. The new wave and such other non-linear systems, has made thinking should be that there might be a the need for a second look at physiodynamic rhythm within a chaotic situation. processes of the human biology in a Attempts at predicting the future with the systemic approach. Newtonian and 1 help of the exercise ECG test (Bruce) fail to follow the positive-negative paradigm. Most one wonders how any of our It has been felt that, it is not adequate to As for the respiratory and circulatory approach the differences from reference systems, the PRESSURE (P), the VOLUME values considered normal (healthy state) as a (V), and the TIME (T) clearly characterise unique entity, and it is also not adequate to the status and certain changes in the status of make such these systems. In the above respiratory and approach. In some cases it is possible that the circulatory system, the correct unit of the abnormality found is just a compensation change is the cycle, which can be considered for latent anomaly, and so the correction of an ordinary unit of TIME (T). the change would have a negative effect on This Paper illustrates the fallacy in such the SYSTEM rather than a positive one. For systemic approaches of studying the non- instance, the residual volume of a patent linear parameters of the human physiology with obstructive emphysema is too high using a specialised complex multivariable (pathological) but this is the only way for model developed by this team. The model these patients to reduce the resistance of the was tested for it's practical utility in its corrections according to 3 small pulmonary airways . If the residual ability in predicting Coronary Artery volume gets reduced, the ventilation gets Disease (CAD) in a recently concluded- worse. The pathological heart - if it can - blinded study. get tachycardic, for in this way the load gets reduced (because as a result of increase in the heart rate, the pre-load gets reduced). If MODELLING the patient is made bradycardic, the patient’s DYNAMICS OF BLOOD FLOW OF HUMAN condition gets worse. Similarly when the cerebral blood flow falls, the perfusion is In often compensated by the elevation of the construction of models has been used as a blood pressure - If in such cases - the blood means of assisting in understanding of pressure is normalised, permanent cerebral natural phenomena. In engineering it is also emollitions can result. regular practice to simulate existing or many branches of science the planned engineering devices and systems as These examples well demonstrate that it is well as naturally occurring processes. The not enough to evaluate a single pathologic techniques of modeling have in fact become change by itself: It is necessary to consider so well developed in the engineering the function of the change in the system as a environment that this previously specialised whole. knowledge is proving useful in other fields as well. Engineers are becoming particularly Models and Model Based Reasoning have interested in modeling biological systems, been used for many years now. For many often in the context of the functioning of the kinds of problem solving tasks, it is human body. In this field the problems are necessary to model the behavior of some acute, but the potential rewards are certainly object or system. valuable enough to act as a considerable physical devices, such as say, electronic stimulus. First of all there is the possibility circuits or electric motors, it is necessary to of gaining some knowledge during the model the behavior of both the correctly investigation which will shed light upon the functioning device and some number of ill reasons for malfunctioning of the human functioning variants of it. physiological system and thereby hopefully potential designs of these devices requires suggest new means of treatment. It can be the same capability. When we think about reasonably expected that the modeling constructing a model of some entity in the procedure might result in new clinical tests real world, the issue of what we mean by a being proposed in order to promote better model soon arises. To what extend should understanding of the biological system. It is the structure of the model mirror the also a possibility that investigation of this structure of the object being modeled? type may suggest new approaches to Some representational techniques tend to engineering problems. The least one can support models whose structure is very expect from modeling or simulating a different from the structure of the object human using being modeled. For example, in predicate engineering technique is that the medical logic we write wff's (well-formed formulas) and engineering professionals involved will such as x: raven (x) black (x). In the be learning real world, though, this single fact has no procedure as the model serves as a single realisation, such as casual networks, continuing basis for discussion. There are in which the physical structure of the world many forms of model, that can take; - from is scaled representation. physiological engaged in physical mathematical an function efficient reconstruction equation. a in the To, evaluate structure of the particular There are arguments in favour of both ends modeling procedure to be discussed here is of the spectrum (and many points in the the use of a very specialised multivariable middle). model that was designed to study the human knowledge structure closely matches the physiological problem structure, then the frame problem haemodynamics. The to closely To diagnose faults in variability, particularly Take a simple example, if the may be easier to solve. Suppose; that we have a planning program and we want to is basically a Multi variable complex Model. know if we move a table in to another room, Dynamics of blood flow involves multiple what other objects also change location. A changes in multiple parameters. model that closely matches the structure of multiple parameters can be grouped as the world as shown below, (in A) will make pressure related, volume related, time answering related, and flow related, in all we have this question easy, while These alternative representations as shown (in B) about 25 different parameters. will not. There are, however, arguments for parameters are derived from a combination representations whose structures do not of closely Such transthoracic bio impedance, obtained non- representations typically do a better job of Invasively (one can also use an invasive capturing generalisations and thus of making technique to determine the volume related predictions about some kind of novel parameters). First of all the model elements situations. (or subjects) are to be created. This is done model the world. ECG, These phonocardiography and by conducting measurements on a number of ‘A’ subjects who are angiographically normal (Living room 1; individuals, who are free of coronary artery Contains; disease (in this case) for generating the (Table 1; model variables. A specially designed Made - of: wood Has - on: (vase 1: made - of glass) (Lamp 1: …)) (Table 2 : Has - on (vase 2: …))) computer program would pickup 25 qualified subjects out of double the quantity of measurements obtained, individuals having non-consistent variability will be rejected by the program as subjects suspected to have other cardiovascular ‘B’ disorders. Twenty-five consistent qualified in (Table 1, Living room) subjects are adequate for construction of a made of (Table 1, wood) multi-variable dynamic model3. on (vase 1, Table 2) K-model has a structure, which can be made of (Vase 1, glass) discussed in simple terms as follows: on (vase 2, Table 2) Consider a variable z that can take 2 extreme on (lamp 1, Table 1) values Qmax and Qmin. Qmax is the highest Applications of the present day modeling are value z can assume and Qmin is the lowest more complex: K- model (for kinetic model) value the variable z can assume: Now z as a variable does not exist beyond Since we are talking about a set of variables, Qmax and Qmin and these extremes are named all of them continuously varying either Q1 and Q2. When the variable z-tends to dependent or Independent, linearly or non- move from Q1 to Q2 it crosses a plane of linearly, this factor of deviation due to optimal values Oopt irrespective of the direction of change. Now consider that; we have ‘n’ number of variables, each variable z1, z2, z3 ………. zn . (All are tending to max from min or min to max within the frame Q1, Q2). At the core of Q1, Q2 exists O1 and O2. When our variable z transverses through the frame Q1 and Q2 it has to transverse through the natural forces called N1 and N2 has to be taken into consideration. Natural forces can be any element that can effect the circulatory status in this case. However the deviations due to natural forces N1 and N2 should always exists in the neighborhood of the optimal plane O1 and O2. The behavior of the variable due to abnormal forces such as a diseased condition can go beyond the deviation due to natural forces, but within Q1 and Q2. Figure 2 optimal plane and as long as this point is within this plane it assumes an optimal or near perfect behavior. Figure 1 illustrates the illustrates the behavior of z in a Cartesian coordinate for a normal subject (zn) and diseased subject (zd). behavior of z within the K-model in normal subject and diseased subject. When the model for a given population is Constructed, each of the model subject’s Such behavior may be ideal, this is because of the deviations due to natural forces. data are sampled, to determine if all the data sets are having the same pattern of change (Variability), the one’s that don’t fit to the contour curve f (x, y) = constant in the xy- pattern of the majority of the subject group plane. is rejected as possible abnormality, the rejected subject is then replaced with a new This interpretation is illustrated in figure 3 subject, till such time the model is and 4 for the example completely constructed (this is done by a computer Programme specially designed for z=100-x2-y2………………..(2) the purpose). Determination of Model Optimal (O) The parameter z (figure 2) can be determined in terms of a number of other parameters. For simplicity, consider a function of two independent variables x and y and denote the dependant variable by z. the equation z= f(x,y) …………..(1) In figure 3, the surface is illustrated, which is a paraboloid of revolution, and indicating may be interpreted as representing an elevation of points on a hill above the plane a cutting plane z=75. The corresponding level curve of the circle z=0. Drawing “level curves" then yields “contour lines” in the xy-plane. In this interpretation, we imagine a base region G in the xy-plane and at every point in G we imagine a marker bearing the z value associated, by (1), with that point. If we connect the values in G which have the same z values, z = constant, then we have a x2+y2 = 25…………………(3) in the xy-plane. This is the circle which, in figure 4, carries the marker z = 75. Equation (3) may represent any haemodynamic Parameter. For example, z may represent the stroke volume z at each point (x, y) taking x and y as time and pressure respectively. z = fy(x,y) = lim f(x,y+y) - f(x,y)… . (5) y y0 y The rate of change of any parameter z with respect to x and y may be calculated using equation (4) and (5). For example equation (2) z=100-x2-y2 can represent the stroke volume (z=SV, in ml), where x is the time (here, time represents the inter-beat interval, essentially the rate of change of RR-Interval between subsequent beats in msec) and y is the interbeat pressure difference( in mm hg). Applying equation (4) and (5) we have z = -2x ……………….. (6) x z = -2y y Suppose, now, that z is a function of x and y, defined for values of (x, y) in some region G of the xy-plane. Let do(xo,yo) and d1(x1,y1) be two points of G (figure 5). Then from partial differential equations, we have: z = fx(x,y) = lim f(x+x,y) - f(x,y) …(4) x x0 x ………..…… (7) Suppose, for example; lets consider that there is a change in time of 3msec (say; beat 1= 875ms and beat 2= 878ms, IBI= -3ms). Than the resulting rate of change in SV for a corresponding change in x = -3 ms is then z = -2 x -3 = 6 ml…………. x from (7) Similarly the rate of change in stroke volume for a corresponding change in pressure can be calculated. Here the derivative of z = f(x, y), which is defined in same magnitude. In either of these cases, the (4) and (5). But the rate of change in z does partial derivative fx would not exist. If, not depend only on pressure and time, it also however, fx does exist, then it gives the depends and directional derivative to the right, while -fx Contractility4 . Thus we need to define the gives the directional derivative to the left variable z (in this case stroke volume) as the (the change in sign is due to the fact that function of pressure, time, preload, afterload (x)-1/2 =-x if x is negative). That is if fx and contractility and z = f (x, y, p, a, c) exists at the point, then both the right and which can be defined in the similar way as left directional derivatives exist at that point pressure and time, and we will than have the and have the same magnitude but opposite collective change in z, that can be signs. on Preload, Afterload considered as the optimal value z can attain, that is, a value in the rate of change of z that CARTOGRAPHY CONSTRUCTION has no external forces acting on it. In this way we derive the optimal values of all the The next stage is to super impose the other variables with respect to its dependent suspected patient’s parameters on the model variables. to generate a resultant Cardiovascular Cartogram. In assigning to the limit in eqn (4) and (5) it it understood that x may be either positive or negative. If, on the other hand, we calculate the directional derivative in the direction of the positive x-axis, then x is Cardiovascular Cartography is a process of converting the relationship of many variants from its original form to a more useful one. The simplest form of cartography is to deal with is one to one (i.e. each different restricted to positive values x0+x, x>0, statement y=0. The directional derivative and the representation that is different from that partial derivative fx, differ in that in the arising from any other statement). directional derivative the point maps to a single target d1 approaches d0 always from the same side, Although one-to-one cartograms are, in while in fx, d1 may approach d0 either from general, the simplest to perform, they are the left or from the right. In certain rare in interesting input systems for several "pathological" cases, a function may have a reasons. directional derivative from the right but not many domains, inputs must be interpreted from the left or may both directional not absolutely, but relatively (like in derivatives but the two may fail to have the Cardiovascular cartography), with respect to One important reason is that in some reference model (in this case K - clusters extending outward from the null model). For example, when images are cluster has positive value and the clusters being interpreted, size and perspective will extending inwards has a negative value. The change as a function of disease and to the center of the cartogram has the lowest extent of the disease. negative value and the outmost cluster has A second reason that many-to-one the highest positive value of deviation. cartograms are used, is that free variations is The often allowed, either because of the physical Cartography are obtained on patients at rest limitations of the system that produces the in supine position. Six measurements are inputs or because such variation simply taken in the interval of one minute (this make the task of generating the inputs protocol is used simply to make the large manageable. amount Both these factors help to measurements of data for Cardiovascular manageable). Each explain why such physiodynamic variables measurement constitutes one sub-cartogram require many-to-one cartography. At micro and these are super imposed again on one level no two people’s disease is exactly another, thus the resultant cartogram has identical, but at macro levels they are, different colours. Darkest colours represent identical disease influences the physiology changes that occurred in the initial part of in a characteristic manner, this is why the the measurement and lighter colours are interpreter cartograms later part of the measurement, showing a require to know all the ways that a target pattern of change. The dark line and dot representation can be made. As a result, it is combination shows the end point or the last important, that analysis of these cartograms measurement taken. typically require a structured analysis of the averaged situation. input, rather than a simple, exact pattern Along the circumference of the outermost match. cluster, we have all the 24 parameters The Cardiovascular cartogram is a set of designated along 24 axes, dividing the concentric circles called "clusters". There concentric circles into slices of 15 degrees. are 9 clusters and each of these clusters has When a patch of colour points outwardly 5 minor circles called "sectors". These along a given parameter, it means that this clusters are scaled depending on the parameter has been deviated by a percentage parametric deviation from the K - model. (in This scale is named K-scale and has no (Example; units. The 5th cluster is designated as ‘0’ projecting outwardly up to the 2nd cluster and that is called the null cluster. from the null cluster and if the k-scale is 10 of many-to-one The time Yellow shows the domain) from if (stroke SV the k-model volume) is than SV has deviated by +20 on k-scale factors that affect these changes. In such a away from the Model). If the deviation is continuously changing situation it is very inward, then the value is negative. difficult to make any meaningful diagnosis Cardiovascular cartography is hence the out collective representation of PRESSURE, haemodynamic VOLUME, TIME, FLOW, PRELOAD, analysis AFTERLOAD haemodynamic parameters, when mapped thus the and fluid CONTRACTILITY, mechanics generally of obtained discrete parameters. Collective carefully acquired blood against a multivariable mathematical model circulation. It is a technique that is truly a representing a set of carefully selected systemic normal approach. At of of the moment subjects, resultant cartogram studied with respect to coronary artery haemodynamic variability and deviations disease only, where the ability of the heart to (from the model), that has significant meet the basic demands of the body at rest is applications in studying the haemodynamic analysed and this is seen to be different in variability in patients with cardiovascular patients with and without coronary artery disorders. The technique was used in a disease. Unlike in stress ECG, where blinded multicentred study for reliable electrical changes can occur only in elevated detection of coronary artery diseases. body 50 South Asian male and female (36:14) in cardiovascular occur at the basic body demands at rest. Angiographically normal were recruited. Stress ECG looks at one single parameter, Haemodynamic parameters (Stroke volume, the electrical variation, thus a large scope of Cardiac errors. But in cardiovascular cartographic vascular resistance and index and systolic study, it is the variations of multi-parameter, time intervals) were carefully obtained in which is why the sensitivity and specificity supine position at rest, from the recruited are above 90%. We are sure future subjects, using a non-invasive impedance researchers will work on other areas of cardiogram (ASKIT: ICG-M-501) that was cardiovascular precisely time related with phonocardiogram using this technique. and 54, who the subjects Output age: indicates cartographic study haemodynamic changes disorders mean clearly a variability of haemodynamics at rest is demands, that produces Index, were Systemic and electrocardiogram. From these acquired parameters a model was constructed using a specially designed computer program, which Methodology Haemodynamics and circulatory status generates the model extremities (Q1 and Q2), changes continuously and there are many model core (O1 and O2) and natural deviations (N1 and N2). These model including beat-to-beat stroke volume, variables are then loaded into a cartography systolic time intervals and blood pressure computer program (Scalene: CVM-ver 4.5). were obtained for 6 minutes at rest and in The acquired haemodynamic data from a supine position in 273 patients (43 females; patient, who needs to be studied, is mean age 46 years) scheduled for coronary superimposed angiography. cartography on the computer model by program the and a was found that the were made using Impedance Cardiography, that was precisely time related with simultaneously resultant cartogram is obtained. It Measurements pattern of haemodynamic variability on the resultant cartogram was similar in patients with the similar type of coronary artery disease. Cartograms of normal individuals had more harmonious and logical haemodynamic changes on the cartogram pattern and could be easily distinguished. obtained phonocardiographic, electro- cardiographic and non-invasive arterial blood pressure data. These data were superimposed on the model and an integrated CCG was obtained for each patient. A single investigator blinded to the angiographic data interpreted these maps. Results: The CCG was positive for CAD in 204 patients and negative in 69 patients. Angiographically, CAD was present in 199 CLINICAL STUDY Background: multi-variable sensitivity, specificity, positive predictive mathematical model was designed using accuracy (PPA) and negative predictive Haemodynamic variability data obtained accuracy (NPA) of this technique for from 50 control patients with normal detecting CAD respectively was 92%, 92%, coronary angiograms and left ventricular 98% and 75%. functions. Conclusion: By The patients and absent in 55 patients. The superimposing the data The technique of obtained from other patients on the model, a Cardiovascular Cartography is a reliable pattern, cardiovascular cartography (CCG) non-invasive tool to detect the presence of could be generated. In a pilot study it was and assess the severity of CAD. Preliminary observed that Coronary Artery Disease results (CAD) characteristically altered the CCG available non-invasive tests to detect CAD. pattern. This study was designed to assess Future Trends the feasibility of using such modeling and The availability of more powerful computer cartography technique to detect the presence workstations will lead to more detailed analysis of and assess the severity of CAD. of the large amount of data collected. Of Methods: particular promise is the use of the knowledge in Haemodynamic measurements compare favorably with other the cardiovascular cartograms to virtually image the coronaries. It remains to be seen whether this Chaos Theory, Bangalore, India, promise will be fulfilled in the very near future. 1996. 2 Acknowledgments The authors wish to thank Dr. Devi Prasad B.M, "Science Common Sense", Address, Proceedings National Shetty, Chairman, Manipal Heart Foundation, for and Keynote of the Conference on Biomedical Engineering, Manipal, granting permission for conducting the clinical India. studies at the heart foundation and Dr.Alok Roy, Project Director, Manipal Heart Foundation for Hegde 3 Attila Nashlady, Cardiologia Hungatia, supplement, editorial. his encouragement and great motivation. The authors are grateful to Sir. Athila Nashlady, Dr. Sabastin, Dr. Tripati, Dr. Gabor Vareski and Dr. 4 Flowers NC, Horan LG, Johnson Murali Mohan for their valuable suggestions and JC. Anterior infarctional changes Ideas. They are also thankful to the staff of occurring during mid and late Manipal Heart foundation, Scalene Health, ventricular activation detectable by ASKIT kft., for their excellent co-operation surface during the study. The authors are thankful to McGraw-Hill Book Co., for permission to reuse parts of materials from their publications. And mapping techniques. Circulation. 1976; 54: 906 5 Huesman RH, Reutter BW, Zeng GL and Gullberg GT, "Kinetic lastly they wish to thank all the patients, who parameter estimation from SPECT willfully co-operated during the study. cone beam projection measurements" Phys. Med. Biol. Plates Vol43, pp. 973-982, 1998. Plates shown explain the different types of 6 Carson RE, Lange K. "The cardiovascular cartograms and their association Parametric image reconstruction with the coronary artery disease. Note the algorithm," Amer. Ststist. Assoc., similarity of these cartograms in patients with vol. 80, pp. 20-22, 1985. Identical diseases. References 1 Kumar A. Hegde B.M and Prabhu. "The chaos theory, cardiology and electrocardiograms", Proceedings of the national conference on Plates Plate 1. Cartogram of normal subjects Plate 2. Cartogram of two patients with RCA 100% Stenosis (left) and RCA 60% (right) Plate 3. Cartograms of two patients with LAD 100% Stenosis (Left) and LAD 80% (Right) Plate 4. Cartogram of two patients with LCX disease (90% left, 100% right) Plate 5. Cartogram of patients with two vessel CAD on RCA and LAD branches. Plate 6. Cartogram of patients with disease in all three vessel branches. Note: A 20 hour physician training was required for interpreting and diagnosing these cartograms in our experience.