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Modelling and Simulation of the Human Arterial Tree - a Combined Lumped-Parameter and Transmission Line Element Approach M

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Modelling and Simulation of the Human Arterial Tree - a Combined Lumped-Parameter and Transmission Line Element Approach M Transactions on Biomedicine and Health vol 2, © 1995 WIT Press, www.witpress.com, ISSN 1743-3525 Modelling and simulation of the human arterial tree - a combined lumped-parameter and transmission line element approach M. Karlsson Applied Thermodynamics and Fluid Mechanics Department of Mechanical Engineering, Linkoping University, Linkoping, Sweden 1 Introduction Computer models of the arterial tree has been a goal in bio-fluid dynamics ever since the advent of the digital computer. Over the years several con- tributions have been made and increasingly more elaborate models of the systemic arterial tree have been developed in order to gain a better insight into the hemodynamics of man. This paper describes a model of the arterial tree with distributed pa- rameters which has been developed in order to quantify hydro-mechanical effects in the arterial system. Any such model must resolve the characteris- tic features of the arterial tree such as distributed resistance and the ability to incorporate local variations in segrnental compliance. The topology of the arterial tree must also be included in order to resolve wave reflections adequately. As the model is intended for non-stationary analysis special attention must be paid to the proximal boundary condition, the heart. Fur- thermore, an effective and robust numerical method is necessary in order to keep the computational time as low as possible. 2 Modelling the cardiovascular system A model of the arterial system consisting of 128 segments, each described by a four-pole equation, has been derived. The geometrical and topological de- scription is based on the geometry presented by Avolio [1]. The description also includes vascular dimensions and elastic constants for the 128 segments of the human arterial tree. Figure 1 shows the topology of the model. Transactions on Biomedicine and Health vol 2, © 1995 WIT Press, www.witpress.com, ISSN 1743-3525 12 Computer Simulations in Biomedicine > C2 • Q2 ITO£V a • P2 Figure 1: Topology of the arterial tree model. The arterial tree is divided into seven group>s: aorta, left arm, right arm, left leg, right leg , left brain and right brain. Each group consists of all arterial segments in that particular section. To each group a sub-group is attached including all segments branching from that particular group. A more thorough discussion can be found in Karlsson [2]. The geometrical taper of the arterial segments is modelled by elements of a constant diameter; the segmental reflection is thus associated with the connection point between two consecutive arterial segments. Instead of calculating reflection coefficients and impedances along the arterial tree from the terminations all the way back to the heart, the complete arterial tree model is simulated within the simulation package HOPSAN, User's Manual [3]. Input data for the the model is segmental information, such as length and diameter as well as Youngs modulus, and general information about the blood: density, p = 1050 kg/m^, dynamic viscosity, // = 0.04 Ns/rn^ and effective wall viscosity, //^E = 0.4 Ns/m^. 3 Modelling an arterial segment Each arterial segment is modelled using a four-pole description, i. e. a version of the classical transmission line model traditionally used for power electricity transmission lines which enables a unification of modelling sim- plicity and computational efficiency. The telegraph equations are derived from the governing equations for flow through a pipe (the Navier-Stokes Transactions on Biomedicine and Health vol 2, © 1995 WIT Press, www.witpress.com, ISSN 1743-3525 Computer Simulations in Biomedicine 13 equations), with cylindrical coordinates and assuming an axi-symmetric flow. The derivation is given by i.a. D'Sousa & Oldenburger [4], Viersma [5] and Krus et al [6]. In matrix form (and Laplace transformed) the telegraph equations can be written as (AL BL\(Q,\_( -Qi \ / \ where where T is the time needed for a wave to travel along the line with length L. T can be calculated as T = L/a, where a is the speed of sound. Zc is the inviscid characteristic impedance of the line, defined as Zc = pa/A, where A is the cross sectional area of the line element and p the density. N(s) is the friction factor theoretically given by where JQ and J% are Bessel functions of the zeroth and second kind, respec- tively, and 7 = jJsjv, Viersma [5]. i/ is the kinematic viscosity and R the segment al radius. The distributed resistance is modelled according to Viersma [5] (in frequency domain) as 7Va(j) = --H (3) s where a = -^y. For fully developed laminar flow the total resistance of the line element is defined by the Poiseuille law as (4) W It is here assumed that the resistance R^ can be used as an approximation for the frictional loss for arterial segment n. The visco-elastic behaviour of the arterial wall is modelled in the same sense as the friction term using an effective wall viscosity, //„,#, Krus et al [7], as the visco-elasticity of the arterial wall manifests itself only as a slight change in the momentum equation, Rockwell et at [8]. fi^E has the same unit of measurement as the viscosity of the fluid and gives about the same damping as the viscosity. The visco-elasticity of the arterial segment is thus defined as ^(4 = — + 1 (5) s Transactions on Biomedicine and Health vol 2, © 1995 WIT Press, www.witpress.com, ISSN 1743-3525 14 Computer Simulations in Biomedicine where #%, = R^/ZcT. R^ is "segmental wall resistance", calculated as RW = 128fiwEL/TrD*. A total friction factor N(s) including both friction and visco-elasticity is defined as, [7] N(s) = NR(S)NV(S) (6) Pedley [9] states that the damping of the attenuated pressure wave is about 10-20 times the theoretical value for viscous damping in the blood entirely. The introduction of the the method of characteristics is an effective way to connect different components for simulation of complex multi component systems. Essentially there are two kinds of components in the simulation package HOPSAN, those for calculation of characteristics, such as lines and capacitances, and components for calculation of flow and pressure from these characteristics, such as resistors and simple connectors. For a more detailed discussion on this subject see Krus et al [6]. For very large systems the method is also well suited for parallel! processing and can also handle variable time-steps, Jansson [10]. In Krus et al [6] a model suitable for the simulation of arterial segments was presented which included distributed resistance and a visco-elastic ar- terial wall behaviour. In Engvall et al [11] this kind of arterial segment was used for modelling the arterial tree in order to study the effects from an aortic coarctation with by-passing collaterals during both rest and exercise. 4 Modelling the heart The proximal end of the arterial system, i.e. the root of the ascending aorta, is connected to a model of the heart. In this study two different possible techniques are used: 1) a model only including the aortic valve, the left ventricle and the mitral valve and 2) a model based on the time-varying elastance concept. 4.1 HEART-1, a model of the left ventricle The human heart is approximated with its left ventricle and included in the model of the arterial system in order to take the ventriculo-arterial coupling into account. The wave reflections are resolved as the elastic chamber of the left ventricle is a part of the system during systole. The left atrium is modelled as a constant pressure chamber which supplies the ventricle with blood. The pumping of the heart is created by a source flow function, Qsrc, which may be thought of as the combined movement of the valvular plane and the ventricular wall. The time integral over one complete heart beat of the source flow function is zero, i.e. ff** Qsrcdt = 0, where r is the lenght of the heart-beat. As the source-flow function is zero over a heart beat, Transactions on Biomedicine and Health vol 2, © 1995 WIT Press, www.witpress.com, ISSN 1743-3525 Computer Simulations in Biomedicine 15 blood is drawn into the chamber through the mitral valve during diastole and ejected through the aortic valve during systole, Engvall et al [11]. 4.2 HEART-2, the time-varying elastance concept The time- varying elastance concept has been successfully used with lumped parameter models using a state-space approach, Sun [12]. The time- varying elastance of the left ventricle, e^, can be calculated as ^ f #W1 - e^) + Ew 0 < Z < ^ ^ \ (tlv\t=tee - Elvb)e~^ + EM tee <t < tr where E^a and £/„& are constants defining the level of the time-varying elastance. TC and Tr are time-constants defining the shape of the curve, tr is the length of the heart-beat (for the standard case of 70 BPM: ^ « 0.85-s) and tee is the duration of systole. In this case t^ = 0.3s. Deriving the state equation for each state variable gives the following four equations (where small letters are used for variables inside the heart and capitals are used for variables which belong to the arterial tree) \pia — bmv\qmv\qmv ~ ^Iv^lv ~ Rmv<}mv\l ' Lmv'i ^mv > 0 0; otherwise 7, — mv av at ZC(l) * 10-" * g,., - 133.3 *p,,- 10* dt Caa[ZC(l) * 10-« + 133.3 * _ ~ Paa ~ RaaC aa^f"] / 'L av\ ^av > 0 dt ~ 0; otherwise where C(l) is the characteristic and ZC(l) the characteristic impedance of arterial segment number 1, respectively. Due to differences in the units of measurement used in the state-space model and in the arterial tree model some constants are introduced into the governing equations.
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