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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 contributions 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 parameters which has been developed in order to quantify hydro-mechanical
effects in the arterial system. Any such model must resolve the characteristic 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. Furthermore, 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 description 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
ITO£V
a
> C2
• Q2
• 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 simplicity 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 B L \ ( 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, respectively, 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 = —s + 1
(5)
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 arterial 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
\ (tlv\t=tee - Elvb)e~^ + EM
0 <Z < ^
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, —
at
dt
_
dt ~
0;
mv
av
ZC(l) * 10-" * g,., - 133.3 *p,,- 10*
Caa[ZC(l) * 10-« + 133.3 *
~ Paa ~ RaaC aa^f"] /'L av\ ^av > 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. The following
procedure is used to calculate the input to the arterial tree
10-«[9«,-Caa]
(12)
where Q(l) is the flow input to the arterial tree model from the state-space
model. Inside the arterial tree the pressure, P(n), and the flow, Q(n), is
calculated at each connecting point (node), n, together with the characteristic, C(n), and the characteristic impedance, ZC(n). The pressure at node
1 is thus given by
Transactions on Biomedicine and Health vol 2, © 1995 WIT Press, www.witpress.com, ISSN 1743-3525
16 Computer Simulations in Biomedicine
8 OO
Time (s)
Figure 2: Pressure at 4 different locations along the aorta; the ascending
aorta, the aortic arch, aorta abdominalis and the femoral artery.
(13)
The aortic pressure p*, (calculated as p^ = (f (1) - KH/133.3) is then used
by the state-space model to calculate the input flow for the next time-step.
A 2nd-order Runge-Kutta method is used to solve the state-space model
numerically with a 0.5 ms time-step. The resulting model is a hybrid with
a heart described using a state-space approach and an arterial tree using
transmission line elements.
5
Initial tests of the hybrid model
The pressure and flow waves can be calculated in every node as a function
of time with HOPSAN. Figure 2 shows, as an example, the pressure wave at
4 different locations along the aorta: the ascending aorta, the aortic arch,
aorta abdominalis and in the femoral artery. The heart model is HEART-2.
The delay of the occurance of the peak pressure is clearly seen as well as the
pressure amplification due to the geometrial tapering of the arterial tree.
6
Discussion and Conclusion
The model exhibits some of the essential features of the human arterial tree
with respect to vascular impedance and spatial distribution of pressure and
Transactions on Biomedicine and Health vol 2, © 1995 WIT Press, www.witpress.com, ISSN 1743-3525
Computer Simulations in Biomedicine
17
flow waveforms. However, due to the limited amount of experimental data
available in the literature some parameters are difficult to estimate. Almost
all measurements available in the literature are performed on patients suffering from one or more vascular diseases. It is thus difficult to perform a
complete validation of the model. Regulatory feed-back loops may be simulated by an addition of this information. In Engvall et al [11], the non-linear
visco-elastic behaviour of the aorta during high stress levels were imposed.
This paper describes development of a multi-branched model of the human arterial tree. Arterial segments are represented by uniform tubes with
distributed friction and visco-elastic walls. A model of the heart is included
in order to handle the ventriculo- arterial coupling which is prominent during
systole.
The model is intended as a platform for future research into the circulatory mechanics of man. Initial tests show that the hybrid model is a
powerful tool for analysis of the circulatory system. One specific interest
is the coupling between the heart and the arterial tree, to find out how
disturbances in the pump function are propagated through the arteries.
7
Acknowledgement
The author wish to thank Dr Ying Sun, Department of Electrical Engineering, University of Rhode Island, USA and Dr Fetter Krus, Dr Arne Jansson
and Prof D Loyd, Department of Mechanical Engineering, Linkoping University, SWEDEN for valuable suggestions during the course of this work.
References
[1] Avolio A. P. Multi-Branched Model of the Human Arterial Tree, Med
Comp, 18:709-718, 1980.
[2] Karlsson M. Modelling and Simulation of the Human Arterial Tree,
Manuscript intended for publication, 1995.
[3] HOPSAN - A Simulation Package, Users's Guide, LiTH-IKP-R-704,
Linkoping University, Linkoping, Sweden, 1991.
[4] D'Sousa A. F. and Oldenburger R. Dynamic Response of Fluid Lines,
Journal of Basic Engineering, Transactions ASME, Ser. D, 86:589-598,
1964.
[5] Viersma T. J. Analysis, Synthesis and Design of Hydraulic Servosystems and Pipelines, Elsevier Scientific Publishing Company, Amsterdam, 1980.
Transactions on Biomedicine and Health vol 2, © 1995 WIT Press, www.witpress.com, ISSN 1743-3525
18 Computer Simulations in Biomedicine
[6] Krus P., Weddfelt K. and Palmberg J.-O. Past Pipelines Models for
Simulation of Hydraulic Systems, In Proc. of 1991 Winter Annual
, Atlanta, USA, 1991.
[7] Krus P., Karlsson M. and Engvall J. Modelling and Simulation of the
Human Arterial Tree Using Transmission Line Elements with ViscoElastic Walls, In Proc. o/ J44J MWer AmmW Meefrn^ of fAe A^ME,
Atlanta, USA, 1991.
[8] Cited in Smit C. H. On the Introduction of Viscoelasticity into OneDimensional Models of Arterial Blood Flow, Ada Mechanica 43:15-26,
1982.
[9] Pedley T. J. The Fluid Mechanics of Large Blood Vessels, Cambridge
Monographs on Mechanics and Applied Mathematics, Cambridge University Press, Cambrigde, UK, 1980.
[10] Jansson A. Fluid Power System Design • A Simulation Approach, PhDthesis, Linkoping Studies in Science and Technology. Dissertation No.
356, Linkoping University, Linkoping, Sweden, 1994.
[11] Engvall J., Karlsson M., Ask P., Loyd D., Nylander E. and Wranne
B. Importance of Collateral Vessels in Aortic Coarctation: Computer
Simulation at Rest and Exercise Using Transmission Line Elements,
0 Compwf 32:S115-S122, 1994.
[12] Sun Y. Modeling the Dynamic Interaction Between Left Ventricle and
Intra-Aortic Balloon Pump, Am J Physiol 261:H1300-H1311, 1991.