Cardiovascular Modeling Software Tools in Medicine

 Abstract— The high prevalence of cardiovascular diseases has provoked a raising interest in the development of mathematical models in order to evaluate the cardiovascular function both under physiological and pathological conditions. In this paper, a physical model of the cardiovascular system with intrinsic regulation is presented and implemented by using the object-oriented MODELICA simulation software tools. For this task, a multi-compartmental system previously validated with physiological data has been built, based on the interconnection of cardiovascular elements such as resistances, capacitances and pumping among others, by following an electrohydraulic analogy. The results obtained under both physiological and pathological scenarios provide an easy interpretative key to analyze the hemodynamic behavior of the patient. The described approach represents a valuable tool in the teaching of physiology for graduate medical and nursing students among others. Keywords— cardiovascular system, MODELICA simulation software, physical modeling, teaching tool I. INTRODUCTION ARDIOVASCULAR modeling presents a particular challenge due to the highly nonlinear characteristics of this physiological system and entails a multi-domain approach since hydraulics, electrical and chemical dynamics are combined to define the complete system behavior [1]. Several mathematical models of the cardiovascular system have been described in scientific publications [2-4] and different simulation software has appeared to help the clinician the understanding of the underlying processes which explain the operation of the cardiovascular system under both physiological and pathological conditions [5-6]. Mathematical models in physiology follow the causal approach [7], traditionally presented by either employing a set of differential and algebraic equations or else by using a block diagram representation between the interconnections of simpler subsystems as SIMULINK [8], far apart from the common representation of physiological knowledge. In contrast, the acausal modelling approach can offer many advantages in physiological system modelling when dynamics are given as a set of individual components interconnected so as to describe the inherent structure of the physical system [9] as it happens with the MODELICA object-oriented approach [10], suitable to describe system, subsystem, and component levels of a whole physical system with increasing detail. In this paper we describe the use of the MODELICA objectoriented modelling language for the acausal modelling of the cardiovascular system, where a multi-compartmental system constituted by arterial circuit, venous circuit and heart pump J. Fernandez is with the System Engineering and Automation Dpt. University of Malaga, SPAIN. (e_mail: [email protected]). R. Fernandez de Canete collaborates with the System Engineering and Automation Dpt, University of Malaga, SPAIN (e_mail: [email protected]) under Frank-Starling regulation mechanism has been employed to characterize the cardiovascular dynamics. Results obtained during simulation both in normal and abnormal physiological conditions provide an interpretative key for the clinician to understand easily the patient cardiovascular dynamics. II. MODELING OF THE CARDIOVASCULAR SYSTEM The cardiovascular system under study will be described by seven compartments, namely, two cardiac chambers, aorta, arterial and venous systemic circulation and arterial and venous pulmonary circulation all resulting into a nonlinear set of differential equations as is referred in [11] whose schematic diagram is depicted in Fig.1. Fig. 1. Scheme of compartmental model of cardiovascular system An electrical analogy model has been derived taking into account the similarity between the electrical equations of the RC circuit below and each of the vascular compartments, distinguishing between linear and non-linear (with valves) compartments. (Fig. 2). Fig. 2. Electrical analogy of cardiovascular compartment J. Perea.Paizal collaborates with the System Engineering and Automation Dpt. University of Malaga, SPAIN. (e_mail: julia_[email protected]). J. C. Ramos-Diaz is with Pediatric Service, Antequera District Hospital, Malaga, SPAIN (e_mail; [email protected]) Cardiovascular Modeling Software Tools in Medicine J. Fernandez, R. Fernandez de Canete, J. Perea-Paizal, J.C. Ramos-Diaz C Therefore, each component of the cardiovascular system in spite of its hydraulic nature will be characterized by an RC electrical component with (valve) or without diode, described by the set of equations 𝑑𝑉𝑖(𝑡) 𝑑𝑡 = 𝑓𝑖𝑛(𝑡) − 𝑓𝑜𝑢𝑡(𝑡) (1) 𝑃𝑖(𝑡)=𝑉𝑖(𝑡) 𝐶𝑖 (2) 𝑓𝑖𝑛(𝑡) = 1 𝑅𝑖𝑚𝑎𝑥(𝑃𝑖−1(𝑡) − 𝑃𝑖(𝑡),0) (valve) (3) 𝑓𝑖𝑛(𝑡) = 𝑃𝑖−1(𝑡)−𝑃𝑖(𝑡) 𝑅𝑖 (4) being 𝑓𝑖𝑛(𝑡) input flow rate, 𝑓𝑜𝑢𝑡(𝑡) output flow rate, 𝑃𝑖(𝑡) internal pressure relative to the ith compartment, while 𝐶𝑖 and 𝑅𝑖 stand for compliance and resistance. The heart contraction and ejection processes are described in each ventricle according to variable elastances 𝐶𝐿𝑉(𝑡) and 𝐶𝑅𝑉(𝑡) as is explicated described in [11]. III. THE MODELICA SIMULATION ENVIRONMENT The simulation software here used is the MODELICA which is an object-oriented language for modeling and simulation of complex physical systems in general, endowed with a multidomain modeling capability, so that several different physical domains can be used and interconnected [10]. MODELICA contains different physical components libraries ranging from electrical, mechanical, hydraulic and many others. In comparison to traditional block-oriented simulation tools as SIMULINK [8], MODELICA is based on defining special-purpose class connectors so as model variables are shared with other sub-models, so no computational order is strictly followed. Then, connections are thought of as one of the main features of the oriented-object modeling principle, so model reuse is allowed. In MODELICA the system under analysis is decomposed into single components starting from the bottom up, connecting basic components into more complicated ones, until the toplevel model is reached. It enables use of alternate integration method as is the DASSL, Runge-Kutta and Backward Difference based. The cardiovascular system under study was simulated using MODELICA starting from the seven connected compartments of the circulatory system under Frank-Starling homeostatic control mechanism. The MODELICA block diagram of the complete multi-compartmental system is illustrated in Fig. 3, where it can be seen the closed loop cardiovascular system with the two distinct pulmonary and system circuits connected through the ventricles modeling of the heart pump. It is important to highlight the use of the electrohydraulic analogy so that vascular resistances, compliances and valves have been represented as electric resistors, capacitances and diodes, along with the analogy relation between pressure and flow and the corresponding voltage and current. Besides, a pair of variable compliances have been included to generate the cyclic behavior so as to simulate the pumping function. Fig. 3. MODELICA diagram of the cardiovascular system Due the inheritance property of MODELICA which enables reuse of blocks, it is only necessary to design a component (RC circuit) and change its parameters to model each of the vascular components, connecting them afterwards to set the closed circuit. In Fig. 4 it has been included for the sake of comparison the SIMULINK diagram corresponding to the same system. Observe the complexity of this block-diagram causal approach as compared to the simplicity of the physical acausal modeling approach as depicted in Fig. 3. Fig. 4. SIMULINK diagram of the cardiovascular system IV. SIMULATION RESULTS Several experiences have been realized so as to test the performance of the cardiovascular system model, both under physiologic and pathological conditions as aortic stenosis and cava relaxation. In order to get the simulation results validation tests have been previously performed with physiological data as is referred in [11]. In first place, in Fig. 5 are shown the results obtained with the MODELICA model under physiological conditions, where it can be seen the cyclic behavior of cardiovascular variables as pressure in different compartments. Fig. 5. Pressure evolution in different compartments (see color code above) under physiological conditions in MODELICA In second place, it has been doubled the value of the aortic resistance so it can be observed an increase in the aortic pressure among others as is shown in Fig. 6. Fig. 6. Pressure evolution in different compartments (see color code above) under aortic stenosis conditions in MODELICA Fig. 7. Pressure evolution in different compartments (see color code above) under cava relaxation conditions in MODELICA Finally, a case of cava relaxation has also been included, so it can be noticed how where a reduced cava pressure vale is observed as is detailed in Fig. 7. V. CONCLUSION Modeling software tools have been presented in this paper, in particular the object-oriented modeling approach of MODELICA and a closed loop cardiovascular system model has been developed by using this particular acausal approach. The MODELICA model structure is organized as interconnected components¬, so that the system equations are embedded in the connection diagram. Besides, the model structure resembles the physical reality of the modeled system as compared to block diagram scheme in SIMULINK. The modeling effort is thus considerably reduced while reuse of components is enabled thanks to the inheritance property of MODELICA. Simulation models are more understandable in MODELICA so it can be considered as a valuable educational tool in Physiology. REFERENCES [1] A.C. Guyton, J.E. Hall, Textbook of Medical Physiology, ninth ed., W.B. Saunders, New York, NY, 1996. [2] E.B. Shim, J.Y. Sah, C.H. Youn, “Mathematical modeling of cardiovascular system dynamics using a lumped parameter method”, Jpn J Physiol., vol. 54(6), 2004, pp. 545-553. [3] M.J. Conlon, D.L. Russell, T. Mussivand, “Development of a mathematical model of the human circulatory system”, Ann Biomed Eng., vol. 34(9), 2006, pp. 400-413. [4] M. Abdolrazaghi, M. Navidbakhsh, K. Hassani, “Mathematical modelling and electrical analog equivalent of the human cardiovascular system”, Cardiovasc Eng., vol. 10(2), 2010, pp. 45-51. [5] A. Brunberg, S. Heinke, J. Spillner, R. Autschbach, D. Abel, S. Leonhardt, “ Modeling and simulation of the cardiovascular system: a review of applications, methods, and potentials”, Biomed Tech, vol. 54(5), 2009, pp. 233-244. [6] C. De Lazzari, I. Genuini, D.M. Pisanelli, A. D’Ambrosi, F. Fedele, “Interactive simulator for e-Learning environments: A teaching software for health care professionals”, Biomed Eng. Online vol. 13, 2014, pp. 172191. [7] J. Kofranek, J. Rusz, “Restoration of Guyton's diagram for regulation of the circulation as a basis for quantitative physiological model development”, Physiol. Res, vol. 59, 2010, pp. 897–908. [8] S.T. Harris, Introduction to Simulink: With Engineering Applications, Orchard Publications, 2008. [9] G. Zauner, D. Leitner, F. Breitenecker, “Modeling structural dynamic systems in Modelica/Dymola, Modelica/Mosilab and AnyLogic”, in: Proceedings of the International Workshop on Equation-Based Oriented Languages and Tools, 2007. [10] P. Fritzson, Introduction to Modeling and Simulation of Technical and Physical Systems with Modelica, Wiley-IEEE Press, 2011. [11] Avanzolini G, Barbini P, Cappello A, Cevenini G (1988). CACDS simulation of the closed-loop cardiovascular system. Int. J. Biomed. Comput. 22:39-49.