A Mathematical Model of the Cardiovascular System Olga Stikoniene 1 , Raimondas Ciegis2 , Albinas Stankus 3 1
Institute of Mathematics and Informatics, Akademijos 4, LT-2600 Vilnius, Lithuania
2
Vilnius Gediminas Technical University, Sauletekio 11, LT-2040 Vilnius, Lithuania
3
Institute of Psychophysiology and Rehabilitation of Kaunas Medicine University, Vyduno 4, LT-5720 Falanga, Lithuania
[email protected]
[email protected]
[email protected]
Summary. Different approaches to the mathematical modelling of the cardiovascular system are discussed. The compartment model is used as a basis for construction of a simplified model, which can be useful in the investigation of the role of regulation mechanisms on the partition of the blood volume between the systemic and pulmonary circulations.
1 Introduction Lately mathematical modelling based on nonlinear system theory has been actively applied in the investigation of the cardiovascular system. The use of mathematical models and computer simulation techniques may give a deeper comprehension of the problem and help in physiological investigations and clinical practice. Several models of the cardiovascular system have been proposed in the past decades [HP02, Qua02, Urs98, UMOO]. An interesting approach to modelling the cardiovascular flow is based on the description of the cardiovascular system as a graph of vessels (edges) and tissues (nodes) [AGE97]. The heart has four chambers: the right atrium and ventricle, and the left atrium and ventricle. These lie at the center of the cardiovascular system. Blood flows from the heart and back towards the heart. The pulmonary circulation begins in the right ventricle and ends in the left atrium. In the pulmonary circulation oxygen is received by the blood and carbon dioxide is removed from it. The systemic circulation begins in the left ventricle which pumps the oxygenated blood. 0 2 is removed and C02 is received as blood flows through the various tissues. Then the deoxygenated blood returns back to the right atrium. A. Buikis et al. (eds.), Progress in Industrial Mathematics at ECMI 2002 © Springer-Verlag Berlin Heidelberg 2004
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0. Stikoniene, R. Ciegis, A. Stankus
2 Mathematical Models There are at least two ways of carrying out cardiovascular modelling. The first approach is based on detailed modelling of local processes in blood vessels using the Navier-Stokes equations. This approach is suggested by Quarteroni [Qua02]. A hierarchy of models is used in order to take into account the effects of the global circulatory system and at the same time to focus on specific regions. At the highest level a full three-dimensional fluid-structure interaction problem is considered. This model is used in the regions where details of local flow fields are needed. Such a situation arises when the flow in the vessel interacts mechanically with the wall structure and when blood flows in large arteries, where the vessel wall radius may vary by up to a few percent because of the forces exerted by the flowing blood stream. Then large arteries are simulated by the Navier-Stokes equations. The effects of circulation in smaller arteries are simulated by one-dimensional models described by a first order nonlinear hyperbolic system. At the lowest level parameter models are based on the solution of a system of nonlinear ordinary differential equations for averaged mass flow and pressure. This part of the model is described using an electrical circuit or a hydraulic analog for circulation in small vessels, the capillary bed, the venous system and the heart. A second approach involves modelling the leading processes using a "model of compartments". The cardiovascular system is described by models which are based on the solutions of systems of ODEs. This model is represented by a hydraulic analog via the Ursina model [Urs98]. Depending on the particular problem to be considered the vascular system is simulated as a combination of a few compartments. Some of these compartments are used to reproduce the systemic circulation. Often the differentiation among the systemic arteries (subscript sa), the splanchnic peripheral and venous circulations (subscripts sp and sv respectively) and the extrasplanchnic peripheral and venous circulations (subscripts ep and ev respectively) is used. Similarly the other compartments represent the arterial peripheral and venous pulmonary circulations (subscripts pa, pp and pv respectively). Each compartment of the model includes a hydraulic resistance R1 , which accounts for the pressure energy losses in the j-th compartment, a compliance cj' which describes the amount of stressed blood volume stored at a given pressure, and an unstressed volume Vu,j (defined as the volume at zero pressure). The heart is modelled as four compartments, which are used to reproduce the left atrium, left ventricle, right atrium and right ventricle (subscripts la, lv, ra and rv, respectively). Models with different values of the parameters are similarly used for the left and right hearts. Equations relating pressure and volumes at different points of the vascular system can be written by imposing conservation of mass at all meetings of compartments and balance of forces in the large arteries. We define P1 as the intravascular pressure in the j-th compartment, F1 as the blood flow and Lj as the inertness. Fo,r and Fo,z are the cardiac outputs from the left and
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the right ventricles respectively. So in the j-th compartmen t we consider the typical equations 1 dP. d:
d:
dF-
=
J
1
£. (Pin,j
=
(1)
C· (Fin,j- Fout,j),
- Paut,j - R1 F1),
(2)
J
d:
dV.
(3)
= Fin,j - Fout,j ·
The heart and the vascular system do not work independently within the whole body. In order to describe the cardiovascular system we must comprehend not only the properties of the heart and vessels but also those of the regulation system. Therefore the model also includes the action of reflex regulatory mechanisms: the arterial baroreceptors, the peripheral chemoreceptors, the hypoxic response of the central nervous system and so on. Such models are used for understandi ng the role of each regulatory mechanism in the cardiovascular system. The baroreflex system, which stabilizes the arterial pressure, is one of the most important systems involved in cardiovascular regulation. Typical equations for the regulation effectors () = Emax,rv, Emax,lv, Rsp, Rep, Vu,sv, Vu,ev are written as a system of ODE's with delay Do in the form
e(t) =eo+ 6e(t), uo(t)
= {
d6() -d t
1
= -(- 6()(t) + uo(t)), TO
G(t), !es 0,
fes
~ !es,min
< fes,min,
G(t) =Go ln(fes(t- Do)- fes,min Jf es
. (t) -- Jes,mtn
+ 1),
+ (Jes,O - Jes,mtn. )e-kesfcs(t) '
where() denotes the controlled parameter, uo is the output of the static characteristic, To is the time constant, Go is the constant gain factor and fes,min is the minimum sympathetic simulation. In the compartmen t model the cerebral, skeletal muscle and coronary peripheral conductances are directly regulated by local changes in 02. The stimulus for local regulation is assumed to be the change in 02 concentration in the venous blood leaving the compartmen t. At a constant pressure difference, the flow through many tissues (for example the brain and muscles) depends on the tissue's ability to consume the oxygen. The concentratio n of 02 in the venous blood is computed by imposing mass balance between 02 extraction and 0 2 consumption rate (Fick's principle)
(4)
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where Cao 2 is the 0 2 concentration in the arterial blood, Fj is the blood flow and MJ is the Oz consumption rate. Taking into account the process (4) we get new values of the hydraulic resistances.
3 Simplified Model As an example we consider a model which is based on the balance of two circulations. Namely, we are interested in the question of how the two sides of the heart and the two circulations (pulmonary and systemic) are coordinated. Also it is important to know what mechanisms control the partition of blood volume between the systemic and pulmonary circulations. For example, if the right output exceeds the left output by only 4 %, this disbalance leads to death after only six minutes. We propose a simplified model which is based on the following assumptions (see Fig. 1): •
• • •
Energy approach. The body's requirements of oxygen are reflected by the dynamic of oxygen consumption. One system provides the oxygen for the body and another system consumes it. Oxygen is carried by haemoglobin. There is unstable equilibrium between these two systems. The amount of haemoglobin is proportional to the blood volume of the circulatory system. The speed of oxygen consumption is proportional to the speed of blood circulation (Fick's principle).
hock volume
• ··----- ~Low pressure sy tern
. . .... -...... , .. '
Fig. 1. The simplified model
We construct this model using balance and mass conservation equations for the heart and the arteries and a more simple two circulation model for the rest of the circulatory system. Thus for the heart we use a compartment model and in the remaining part of the circulatory system we use equations based on metabolic laws instead of Newton's laws. The total blood volume is given by V = Vp + Vs, where Vp and V8 are the blood volumes in the pulmonary and systemic circulations respectively. We consider the equations
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(5) where the coefficients Ai = Ai(V8 , Vp), i = 1, 2 are unknown nonlinear functions. This system is considered as a black box with some unknown parameters. Those parameters are determined by using the least squares method and experimental data obtained from the previous compartment model. The numerical integration of differential equations with delay is performed using the fifth-order Dormand and Prince adaptive method.
4 Conclusions The advantage of compartment models is their simplicity. Such models may be successfully used by physiologists in the analysis of interaction of the cardiovascular system with regulation mechanisms. It is possible to investigate the nature and the role of every regulation component. But it is impossible to use such models for the investigation of real patients in hospitals. In this case it is better to use hierarchical models: the detailed modelling is done in the problematic section and more simplified compartment models are applied in the remaining part of the cardiovascular system. This approach is better than using detailed modelling of the whole system because the latter requires us to determine a great number of parameters. The accurate determination of these parameters is impossible due to a lack of clinical data.
References Hoppensteadt, F.S., Peskin, C.S.: Modeling and Simulation in Medicine and the Life Sciences. Springer, Berlin Heidelberg New York (2002) [AGE97] Abakumov M.V., Gavriluk K.V., Esikova N.B., Koshelev V.B., Lukshin A.V., Mukhin S.I., Sosnin N.V., Tishkin V.F., Favorsky A.P.: Mathematical model of haemodinamics of cardiovascular system. Journ. of Differential Equations 33, No. 7, 892-898 (1997) Ursina, M.: Interaction between carotid baroregulation and the pulsating [Urs98] heart: a mathematical model. Am. J. Physiol. Heart Circ. Physiol. 267, H1733-H1747 (1998) Ursina, M., Magosso E.: Cardiovascular response to isocapnic hypoxia. I. [UMOO] A mathematical model. Am. J. Physiol. Heart Circ. Physiol. 279, H149H165 (2000) [Qua02] Quarteroni, A.: Mathematical Modelling of the Cardiovascular system. In: Li Tatsien (ed.) ICM 2002. Vol. III, 839-849, Higher Education Press, Beijing China (2002)
[HP02]
