ISSN: ISO 9001:2008 Certified International Journal of Engineering and Innovative Technology (IJEIT) Volume 4, Issue 1, July 2014

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1 Mathematical Modelling and Simulation of Human Systemic Arterial System Ketan B. Naik, Dr. P H. Bhathawala Department of Applied Sciences and Humanities, Parul Institute of Engineering and Technology, Limda, Vadodara, Gujarat, India, Head and Professor of Department of I.T. and Mathematics, Auro University, Surat, Gujarat, India Abstract the purpose of this study is to develop a model of arterial system using lumped parameter approach. The model contains systemic arterial circulation, starting from aorta, and follows upper and lower edges vessels. Anatomic and physiological data for circuit parameters, have been taken from medical textbooks and articles and where needed calculated using appropriate formulas. The developed mathematical model has been simulated by MATLAB software. Each compartment includes Resistor-Capacitor-Inductor (RCL) elements. The normal arterial operation is characterized by the pressure curves in various part of the system. The presented model is a one of the useful tool in studying the physiology of cardiovascular system and the related diseases. We can able to calculate pressure and blood flow in particular at the given time. Index Terms Arterial Physiology, Lumped Parameter model, Modeling, Simulation. I. INTRODUCTION Human Cardiovascular diseases, which cause the majority of death, are the most disastrous health problem in industrialized nations. Modeling of such important system has become a useful tool to diagnose the diseases and recommend the appropriate way of their medical treatment. Since William Harvey established the concept of circulation of blood in 1628, numerous attempts have been made. The fact that the arterial tree transforms intermittent flow from the left ventricle to steadier outflow was recognized by Hales in He describe the arterial system as a single elastic chamber which later become known as the Windkessel model(frank, 1899)[2].A. P. Avolio had constructed multi-branch model of the human arterial system based on the anatomical branching structure of arterial tree.(1980).olfusen and Nadim derived a lumped model for the blood flow and pressure in systemic arteries using the equations of fluid dynamics(2004).hassani et al.represent a model with 42 compartment, which describe the arterial system in detail. The mathematical formulation of cardiovascular system are typically divided into two classes of models: distributed and lumped.1d and 2D models are examples of distributed model. Lumped model is also known as 0D model. In Lumped model each element is described in terms of resistance (R), inductor (L) and capacitor C. The Parameters are computed according to the corresponding vessel radius, viscosity and density of blood, Length of vessel and young module of vessel s elasticity. Using lumped model, the entire human cardiovascular system may be described as a network of compliances, resistances and inductances not reflecting anatomical properties [3]. These type of models exhibit major features of their real counterpart such as higher pressure in response to vessel constrictions, and root aortic pressure and ventricular outflow curves repetitive of observed value [3].The present lumped model described the arterial system in details with more than 60 compartment. The system has been simulated by Matlab software.. II. METHOD The model is mathematically formulated in terms of an electrical circuit. Fig. 1 describes the block diagram of the systemic arterial system starting from aorta, ends with upper and lower extremities vessels. The following table I represent the analogy of electrical circuit and cardiovascular system. Table: I Analogy of cardiovascular system with electrical circuit. Cardiovascula r System Vessel Resistance Vessel Compliance Blood inertia Electronic Circuit & Symbols Electrical Resistance( ) Capacitance Inductance Equation Relations For Electrical Circuit = I P = F Valve Diode I = F = Equation Relations For Cardiovasc ular System 142

2 The following are the assumptions for the present model: 1) The blood flow is Laminar flow. 2) The model has no connection with bar receptors, nor with the central nervous systems 3) There is no membrane connection with the body tissue, implying there is no diffusion across the membrane. 4) Arteries are simple cylindrical vessels, as in fig.2 and the walls of arteries are elastic. 5) The Unidirectional flow was secured by including diode in the circuit of the model. 6) There is no physiological bifurcation of any vessel considered. However, the model had loops representing blood vessels from different part of body. III. MODEL IN EQUATIONS For Left Ventricle: (1) When aortic valve is open and (2) When aortic valve is closed. For Systemic Arterial system: ) - (3) Where represent jth and represent successive of the jth. For Ascending Aorta (j=1) ) -, (4) Where stands for Aortic Arc-I (See fig.1) IV. SIMULATION Following formulas are taken for simulation [5] Blood vessels resistance = (5) Fig.2 Blood vessel and electrical components equivalent Blood inertia L = (6) 143

3 Vessel compliance C = (7) Where blood viscosity = gram/cm. sec., = blood density = 1.05 gram/cm 3. [6] R, L and C for each blood vessel shown in fig.1 are calculated from following table with the help of above formulas (5), (6) and (7). Table: II Length, Radius, Wall thickness and Young s module of Systemic Arteries wall Young Name of the Length Radius thickn Module Artery ess E Ascending Aorta Brachiocephalic carotid Left common carotid Right Subclavian Left Subclavian Carotid I Carotid II Carotid III Carotid I Carotid II Carotid III 16 Carotid Left Extternal Carotid Right Auxilliary I Right Auxilliary II Left Auxilliary I Left Auxilliary II I II III I II III Right ulnar I Right ulnar II Left Ulnar I Left Ulnar II Right Radial Left Radial Aortic Aarch-I Aortic Aarch-II Thoracic aorta-i Thoracic aorta-ii Abdominal aorta-i Abdominal aorta-ii coeliac Common Hepatic Splenic Right Gastric Left Gastric superior mesentric Left Renal Right Renal Inferior Mesentric atery iliac Left common iliac Right internal iliac iliac I iliac II left internal iliac Left external iliac I Left external iliac II Right femoral Left femoral I II I I Right Anterior

4 I Right Anterior II I II Right posterior Left Posterior Right Middle Left Middle Right Cerebral Left Cerebral Right Opthalmic Left Opthalmic Right facial left facial Right Superficial Left Superficial Right Maxilliary left Maxilliary Right internal mammary Right Vertebral Right profunda left profunda Right Sperior Right inferior Left inferior Right Interossea III III Above values are taken from [2] and using this data R, L and C are calculated, which summarized in the following table. Table III Calculation of R, L, and C for Systemic arteries Name of the Artery R L C Ascending Aorta E-05 Brachiocephalic E-05 carotid E-05 Left common carotid E-05 Right Subclavian E-06 Left Subclavian E-06 Carotid I E-07 Carotid II E-07 Carotid III Carotid I E-07 Carotid II E-07 Carotid III Carotid E-07 Left External Carotid E-07 Right Auxiliary I E-06 Right Auxilliary II E-06 Left Auxiliary I E-06 Left Auxiliary II E-06 I E-06 II E-06 III E-06 I E-06 II E-06 III E-06 Right ulnar I E-07 Right ulnar II E-06 Left Ulnar I E-07 Left Ulnar II E-06 Right Radial E-06 Left Radial E-06 Aortic Aarch-I E-05 Aortic Aarch-II E-05 Thoracic aorta-i Thoracic aorta-ii E

5 Abdominal aorta-i E-05 Right Opthalmic Abdominal aorta-ii E-05 coeliac E-06 Left Opthalmic Common Hepatic E-06 Splenic E-06 Right Gastric Right Superficial E-06 Left Gastric E-06 Left Superficial superior E-06 mesenteric Right Maxilliary Left Renal E-06 Right Renal E-06 left Maxilliary Inferior Mesentric E-07 atery Right internal mammary E-05 iliac Left common iliac E-05 Right Vertebral Right internal iliac E-07 iliac E-06 I Right profunda iliac E-06 II left profunda left internal iliac E-07 Right Superior Left external iliac E-06 I Left external iliac E-06 II Right inferior ulnar Right femoral collateral E-06 Left inferior ulnar Left femoral E-06 collateral Right Interossea E-07 I E-06 II E-07 I III E-06 I III Right Anterior I E-08 Right Anterior II E-07 I E-08 II E E E-08 Right facial E-08 left facial E E E E E E E E E E E E E E E-06 Using all above data in matlab Simulink (i.e. R,L,C) we have following pressure verses Time graphs. The whole circuit diagram in matlab Simulink is shown in fig.6. Right posterior Left Posterior Right Middle Left Middle Right Cerebral Left Cerebral E E E E-09 Fig.3 Calculated Pressure vs Time Graph of Left ventricle 146

6 Fig.4 calculated Pressure vs. Time graph of Brachial Arteries [6] M. S. Olufsen, A.Nadim, On deriving lumped models for blood flow and pressure in the systemic arteries. J Math Biosci Eng.;1(1):61.,2004 [7] A C. Guyton, J E. Hall, Text book of medical physiology, Eleventh edition, Elsevier saunders, pp , [8] D. A. Mohrman, L. J. Heller, Cardiovascular physiology, Lange physiology series, fifth edition, McGraw- Hill s Publication.pp [9] J. Keener, J. sneyd, Mathematical Physiology, Springer- Verlag, New York.Inc, pp , [10] L. Formaggia, A. Quarteroni, A. Veneziani, Cardiovascular Mathematics, Modeling and simulation of the circulatory system, Springer-Verlag Italia, Milano [11] Agam Kumar Tyagi, Matlab and Simulink for Engineers, Oxford University Press, pp , Fig.5 Calculated pressure vs Time graph of Ascending Aorta. V. CONCLUSION AND DISCUSSION The purpose of the study was to develop a model, based on physiological principles that can predict flow and pressure in the large as well as small systemic arteries. This goal has been achieved by developed Lumped Model for simulation scaling is necessary because not all branches (i.e. veins) in our arterial model are included, and as a result there is some blood not included in the model. Same scaling concept is adopt for arterial tree model by Ottesen and Olfusen[6] The model is useful as a tool to better understand and physiology of the systemic arteries. Model is not able to calculate flow and pressure in the last arteries in the particular sequence because of veins are not included ACKNOWLEDGMENT Authors would like to acknowledge Mr. Nirav Waghela (Master of Engineering in Electrical) for his help and support. REFERENCES [1] K. Hassani, M. Navidbakhsh, M. Abdolrazaghi Mathematical Modeling and Electrical Analog Equivalent of the Human Cardiovascular System, Cardiovascular Engineering, Springer, pp.45-51, [2] A. Avolio, Multi-branched model of the human arterial system, Medical & Biological engineering & Computing, pp. (1980), pp , November [3] J. T. Ottesen. M. S. Olufsen, J. K. Larsen Applied Mathematical Models in Human Physiology, Siam Publication, pp , [4] K. Hassani, M. Navidbakhsh, M. Rostami, Simulation of the cardiovascular system using equivalent electronics system, J biomedical papers of medical faculty of the university Palacky, Olomouc. 150(1):pp , May2006. [5] R. Vincet, Mathematical and computer modeling of physiological system New Jersey: Printice-Hall Inc, pp.27-66,

7 Fig. 6. RLC circuit of systemic arteries in Simulink 148