mon hydraulic model of a segment of the circulation (it could be a segment of a blood vessel or the whole arterial system) is a three-terminal segment (Fig. 1).
Annals of BiomedicalEngineering, Vol. 20. pp. 595-616, 1992 Printed in the USA. All rights reserved.
0090-6964/92 $5.00 + .00 Copyright 9 1992 Pergamon Press Ltd.
Modeling the Circulation with Three-Terminal Electrical Networks Containing Special Nonlinear Capacitors Joshua E. Tsitlik,*t Henry R. Halperin,*t" Aleksander S. Popel,t Artin A. Shoukas,t" Frank C.P. Yin,*t and Nicolaas Westerhof~ *The Peter Belfer Laboratory for Myocardial Research Cardiology Division of the Department of Medicine and tThe Department of Biomedical Engineering The Johns Hopkins Medical Institutions Baltimore, MD ~;Laboratory for Physiology Free University of Amsterdam The Netherlands
(Received3/19/91; Revised 7/23/91) Development, first of analog and later of digital computers, as well as algorithms for analysis of electrical circuits, stimulated the use o f electrical circuits for modeling the circulation. The networks used as building blocks for electrical models can provide accurate representation o f the hydrodynamic equations relating the inflow and outflow of individual segments of the circulation. These networks, however, can contain connections in which voltages and currents have no analogues in the circulation. Problems arise because (a) electrical current must flow in closed loops, whereas no such constraints exist for hydraulic models; and (b) electrical capacitors have a number of characteristics that are not analogous to those o f hydraulic compliant chambers. Disregarding these differences can lead to erroneous results and misinterpretation o f phenomena. To ensure against these errors, we introduce an imaginary electrical element, the n o n l i n e a r residual-charge capacitor (NRCC), with characteristics equivalent to those of a compliant chamber. I f one uses appropriate circuit connections and incorporates the residual-charge capacitor, then all voltages and currents in the model are proper analogues of pressures and flows in the circulation. It is shown that the capacitive current represents the rate of change of volume o f blood inside the vessel, as well as the rate of the corresponding displacement o f volume of the surrounding tissue. Keywords-Circulation, Modeling of circulation, Mathematical model, Electrical model, Hydraulic model, Modeling of physiological systems.
Acknowledgment-Support for this article was provided by Ischemic Heart Disease SCOR Grant 5P50HL-17655 from the National Heart, Lung and Blood Institute. The authors express their gratitude to Susan T. Edmunds, Ph.D. and Patricia A. Stephens, Ph.D. for editing the text. Address correspondence to Joshua E. Tsitlik, Cardiology Division, The Johns Hopkins University School of Medicine, 720 Rutland Avenue, Baltimore, MD 21205. 595
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INTRODUCTION Since the 1950s, electrical models have been a popular method for analyzing the circulation. These models are attractive because they can clearly represent different segments of the circulation. In addition, equations describing electrical networks can be easily solved. The use of electrical circuits for modeling the circulation flourished with the development of analog computers (9,21,25,29). Currently, electrical models continue to be used for modeling the circulation. They are solved with digital computers (17,18,30) or, sometimes, by building the circuitry and measuring the appropriate quantities (2). In spite of its popularity, electrical modeling of the circulation has its limitation. Electrical models are composed of networks, each network representing a separate segment of the circulation. (In the analysis below, terminals are the points of connection of networks to external circuitry. The points of connection between elements inside networks are termed nodes.) In electrical models, voltages, currents, and charges at nonground input and output terminals and nonground nodes of every network are usually appropriate analogues of pressures, flows, and volumes at the input and output of corresponding segments of the circulation. Problems can arise, however, when attempts are made to infer analogies with the currents in the ground wire. This lack of appropriate correspondence between parts of electrical circuits and circulation can occur because all electrical currents (including capacitive) must flow in closed loops, whereas no such constraints exist for the hydraulic models from which these electrical models were derived. This necessity for electrical current to flow in closed loops leads to closing the circuitry in ways which are not representative of the circulation. For example, in a recently published article (28), an electrical model is "used to describe all the basic relationships between intracranial quantities (pressures and flows)." The authors used this electrical model (28, see their figure 2) to derive a set of differential equations describing intracranial hydrodynamics and hemodynamics. In the electrical model, the pathway for blood return to the aortic pressure source (represented by a voltage source) goes through the cerebral fluid. In this model, the blood will not enter from intracranial vessels into the extracranial because there is no pathway for it to return. The blood flow in the extracranial vessels is just an exchange of volume inside the vessels (upper plate of the capacitor Cve) with the volume outside the vessels (lower plate of this capacitor). There is another limitation of electrical models. Electrical capacitors have a number of characteristics that are not analogous to those of hydraulic compliant chambers. These characteristics include (a) the linear charge-voltage relation of an electrical capacitor vs. the nonlinear volume-pressure relation of a compliant chamber; (b) the zero net charge on an electrical capacitor at zero applied voltage vs. the nonzero volume at zero transmural pressure (unstressed volume) in a compliant chamber; and (c) the negative net charge that can be present on a plate of an electrical capacitor vs. the impossibility of negative volume in a compliant chamber. In 1984, Babbs et aL (2) studied resuscitation using an analog electrical model of the circulation built from standard electrical elements. The model did not show any collapse of the vessels at the thoracic outlet, while it is well known (11,33) that collapse does take place. Collapse was not observed because the linear capacitors used in the model allowed a negative voltage (representing a negative transmural pressure) to be applied to the capacitors with no change in their characteristics, and without showing collapse. They
Electrical Modeling of Circulation
597
also did not take into account that the resistances are voltage (pressure) dependent in collapsible vessels. In 1987, Lin et al. (18) copied Babbs's model and analyzed it with a package for analysis of electrical networks, SPICE. Naturally, they have also not observed any collapse, since SPICE could easily (while wrongly) handle a negative pressure on the capacitors. In reference (12), the authors modeled the myocardial circulation by three parallel "simple three-parameter Windkessel models" containing linear compliances in the hydraulic representation and linear capacitors in the electrical representation. The paper mentions vessel collapse; however, no properties of the collapsible vessel are invoked. We developed an arrangement for electrical networks in which all of the voltages and currents are analogues of hydrodynamic variables of the circulation. This new arrangement builds on the advantages of existing electrical models, making these models more informative. We demonstrate that, with appropriate circuit connections, the capacitive currents in electrical models represent the interaction of the blood vessels with their surroundings. This article first presents a building element of the electrical model of the circul a t i o n - a three-terminal electrical model of a blood vessel. Then we discuss what is modeled by capacitive currents in the electrical models, a n d introduce a model of a compliant vessel-a nonlinear residual-charge capacitor. After this we demonstrate the importance of using three-terminal networks by identifying the deficiencies of twoand four-terminal networks. Later we give a model of left-ventricular loading as an example of an electrical model of the circulation utilizing three-terminal networks with the nonlinear residual-charge capacitors. THREE-TERMINAL ELECTRICAL MODEL OF A BLOOD VESSEL A detailed analysis of a hydraulic model is the first step in the development of an electrical model of the circulation in which currents and voltages in every terminal can be used to predict blood flow and pressure distribution in the circulation. The common hydraulic model of a segment of the circulation (it could be a segment of a blood vessel or the whole arterial system) is a three-terminal segment (Fig. 1). Terminals T1 and T2 are the input and the output to the segment, respectively, and terminal T3 is an external environment for the segment. The hydraulic model contains an elastic chamber which is filled via the hydraulic resistance R~ and hydraulic inertance L~ and emptied via the hydraulic resistance R2h and hydraulic inertance L2h (h refers to hydraulic parameters, subscripts refer to specific circulation segments, see the Nomenclature section). In the model, the inertances represent the mass of the blood undergoing acceleration (24). The compliance of the chamber is C h. In the model, the instantaneous pressures are the input Pl, output P2, and external Pe pressures. The flows in the model are the input flow f l , the output flow f2, and the rate of volume change of the elastic chamber f c (capacitive flow). (For use of lower and upper case letters in defining the variables, see the Nomenclature section.) It is necessary to emphasize that the pressures are relative values. In the circulation modeling, the ambient pressure (in most cases atmospheric pressure) is usually taken as a reference. The system of equations describing the compliant chamber is given in Appendix A;
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T1 v/ ~Pl \
\
l
/
," l 1
fl
L; R; ~ "~ \
f
'
T2 ~ 9
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L'. 4 ~ Z / 6v~=-6 v~ /
\', x
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ofT3
//
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~'[Pomb =Olin" Re f e re n c e FIGURE 1. Three-terminal hydraulic model of a segment of the circulation (see text).
the system of equations describing the three-terminal model is given in Appendix B. (For definitions of variables see Nomenclature, Table 1, and Appendices A and B.) The electrical counterpart to the three-terminal hydraulic model comprises a T-shaped three-terminal network (Fig. 2) with a resistance R~ and an inductance L~
TABLE 1. Variables and parameters of the hydraulic model of the circulation and their electrical analogs. Hydraulic Parameters
Electrical Symbols
Units
Voltage
U, u
(V)
Current
I,i
(A)
(m 3)
Charge
Q, q
(C)
~
Electrical resistance
R~
(fl)
Electrical capacitance
Ce
(F)
Inductance
,,
(HI
Symbols
Units
Pressure
P,p
(Pa)
Volume flow
F,f
Volume
V, v
Viscous resistance
Rh
Elastic compliance
Ch
Inertance
Lh
(-~/
( m4s2~ \ kg )
(.~ ~
Analogs
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L e1
Re Ufm
L e2
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\
X \ \ X
u 1
/
( ~ / NRCC
/
I
\
\
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/
~", x
\
~e \
\ \
\
/ ~2
/
Utm
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"c / ""
\
2
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\
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\
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/
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/
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/ x\
\
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\
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/
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C])o m b ----"0 FIGURE 2. Three-terminal electrical model of a segment of the circulation. A nonlinear residual-charge capacitor represents compliance. T h e utm in the resistor and capacitor symbols emphasize t h a t their resistance and compliance, respectively, are functions of transmural voltage (pressure).
in one arm of the circuit, and a resistance R~ and an inductance L~ in the second arm. A special element, a nonlinear residual-charge capacitor (NRCC) (described below) represents the compliant chamber in the third arm of the model. The electrical model can be described by a system of equations analogous to that describing its hydraulic equivalent. Specifically, hydraulic resistances, inertances, and compliance are replaced by their electrical analogues. Pressures, flows, and volumes are replaced, respectively by voltage, currents, and electrical charges (see Table 1 and Appendices A and B). Voltages, like pressures, are relative values. Indeed, the voltages ul, u2, and Ue are the differences (3) between the respective potentials ~ , q~2, and ~e and the ground potential ~amb (Fig. 2), representing the ambient (atmospheric) pressure, Pamb. The symbols for resistors and nonlinear residual-charge capacitors in the electrical model (Fig. 2) emphasize that their parameters are functions of the transmural voltage (1,4,6,10,14-16,20-22,25) (for definition of the transmural voltage, see below). MODELING A COMPLIANT CHAMBER
Interpretation o f Capacitive Current in Electrical Model In electrical models of circulation, the current through the resistances and inductances represents blood flow through the lumens of vessels. It is known that the current into the upper plate of the capacitor (Fig. 2) represents the rate of the volume change in the vessel (19). However, it is not known what is represented by the current into the lower plate of the capacitor. To distinguish between the plates of a capacitor, we introduce the following definitions:
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1. internal plate (the upper plate in Fig. 2) contains charge, which represents the volume inside the chamber; 2. external plate (the lower plate in Fig. 2) contains charge which represents the volume outside the chamber; and 3. transmural voltage, the voltage between the plates, represents the transmural pressure. While expanding by volume 6vi (Fig. 1), the vessel will displace some neighboring volume (-bYe = 6vi) from the surrounding environment (thoracic volume for the aorta, volume of the surrounding myocardium for intramyocardial vessels, the space in the cranium for the cerebral circulation). This decrease of the volume next to the vessel will be compensated either by compression of the environment (e.g., compression of lungs in the thorax) or by the shift of the contents of the surrounding matter. Thus, while the electrical current into the internal plate of the residual-charge capacitor (Fig. 2) represents the rate of filling the compliant chamber (Fig. 1), the current into the external plate of that capacitor represents a rate of volume shift of the surrounding matter, i.e., an interaction of the chamber with the surrounding matter. Comparison o f the Volume-Pressure Characteristic o f the Compliant Chamber and the Charge-Voltage Characteristic o f the Electrical Capacitor For blood vessels, compliance is a nonlinear function of transmural pressure. Katz et al. (13) showed that for a hydraulic model of the vessel, a latex Penrose tube, the volume-pressure curve has a sigmoidal shape (Fig. 3a). The compliance of the vessel at a particular transmural pressure is the slope of this relationship at that pressure Ch(Ptm) = do/dptm
VOLUME(ml) 14 12 10 c:~ 8 6 4 !
t
CHARGE
i
r
!
i
i Utrn Utm ~ ~ N R C C
!
- 3 0 - 1 0 10 30 50 70 TRANSMURAL PRESSURE
(a)
CHARGEON "INTERNAL"PLATE
/ !
(1)
9
VOLTAGE (b)
"TRANSMURAL"VOLTAGE (c)
(d)
FIGURE 3. (a) Volume in a collapsible tube as a function of transmural pressure. Typical transverse cross sections are shown at various points on the curve (reproduced from Biophysical Journal, 1969, 9 : 1 2 6 1 - 1 2 7 9 by copyright permission of the Biophysical Society); (b) charge-voltage characteristics of an electdcal capacitor; (c) a typical charge-voltage characteristics of a nonlinear residual-charge capacitor; (d) the symbol for a residual-charge capacitor. Letters i and e denote internal and external plates of the capacitor, respectively.
Electrical Modeling of Circulation
601
There are many different techniques for modeling the nonlinearity of the pressurevolume relationship. One popular approach is to take into account the elastic modulus of the vessel wall material (1,10,20). Examples of other techniques where pressure-volume relations are used directly can be found in the references (4,14-17). Appendix A presents a comparison of the physical processes of filling-emptying a compliant chamber and charging-discharging an electrical capacitor, systems of equations describing the chamber and the capacitor as well as a comparison of the volume-pressure relationship of the chamber (Fig. 3a) and the charge-voltage relationship of the capacitor (Fig. 3b). This discussion reveals significant differences in the characteristics of the elements. The following is a summary of the differences between the properties of a compliant chamber and an electrical capacitor: 1. The volume-pressure relationship of a compliant chamber is nonlinear with different values of compliance for different ranges of transmural pressure. The electrical capacitance has a constant value for all ranges of transmural voltage. 2. When the transmural pressure is equal to zero, the chamber contains an unstressed volume of liquid 1Io, while the charge on the plates of the electrical capacitor is equal to zero with zero transmural voltage. 3. When the transmural pressure of the chamber changes sign, the chamber is constricted and its volume decreases to values smaller than ~ (but not below zero). When the transmural voltage changes sign, the polarity of the charges on the plates reverses, i.e., negative charge appears on the internal plate of the capacitor. 4. With negative transmural pressure reaching the magnitude of the chamber closing pressure Pc~, the chamber collapses totally and its volume reaches zero, so that no more volume can be squeezed out of the chamber. With an increase in the magnitude of the negative transmural voltage, the negative charge on the internal plate of the capacitor increases. While being charged negatively, the plate gives away the positive charges and the number of these positive charges is unlimited. This situation is equivalent to an unlimited volume of blood being squeezed out of the compliant chamber. In addition to the features that can be derived from the volume-pressure and charge-voltage characteristics, we can formulate one more property: 5. The volume outside the chamber is positive but not defined for the chamber. Negative volume either inside or outside the chamber is a physical impossibility. The charge on the external plate of the electrical capacitor is, however, negative and equal in magnitude to the positive charge on the internal plate. Thus, a standard electrical capacitance is an inappropriate model for the hydraulic compliance of a blood vessel. Nonlinear Residual-Charge Capacitor We suggest to model a compliant chamber with an imaginary capacitor, which has a nonlinear charge-voltage relationship similar to the pressure-volume relationship of a compliant chamber. The example of such a characteristic is shown in Fig. 3c. We
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J.E. Tsitlik et al.
termed this imaginary capacitor a nonlinear residual-charge capacitor (NRCC). While the term does not reflect all the features of the element, it does reflect its main properties: the nonlinearity of the volume-pressure relationship and the existence of the nonzero residual charge on the internal plate when the transmural voltage is equal to zero. A symbol suggested for the residual-charge capacitor is shown in Fig. 3d. The symbol emphasizes that, although they have something in common, this element is not a standard electrical capacitor, that its capacitance is a function of the transmural voltage. Appendix A presents the system of equations describing the residual-charge capacitor, as well as the process of charging-discharging that capacitor. The characteristic of the nonlinear residual-charge capacitor can be easily implemented with computer modeling. Moreover, the linearized characteristic of the residual-charge capacitor can be modeled with standard electrical capacitors and batteries (Appendix C). LIMITATIONS OF TWO-TERMINAL AND FOUR-TERMINAL ELECTRICAL NETWORKS IN MODELING THE CIRCULATION Because of some difficulties of constructing the electrical models of the circulation with the three-terminal networks, investigators have used simplified models containing two- and four-terminal networks. (Below, in the next main section: A Model of Left Ventricular Loading . . . . we will show methods for closing the circuitry while still preserving the validity of all terminals.) The four-terminals networks contain two terminals for input voltage and two for output voltage and are usually terminated by a two-terminal network (e.g., [21]). Similar two-terminal networks are often used for modeling left ventricular load (e.g., [5,27,31,32]). There are, however, limitations in both two-terminal and four-terminal electrical networks. Two- Terminal N e t w o r k
To demonstrate the limitations of two-terminal networks, we will analyze the electrical analogue of the very popular hydraulic model, the three-element Windkessel, which is a model of ventricular loading (Fig. 4a) (5,27,31,32). Two-terminal networks present limitations in modeling the circulation because of problems with (a) equality of aortic and venous flows, and (b) unrealistic capacitive currents. The hydraulic three-element Windkessel model of the aorta (Fig. 4a) contains an elastic chamber (aorta), which is filled via the aortic valve and the aortic hydraulic characteristic resistance RA h and emptied via the systemic hydraulic resistance R~. The compliance of the chamber is C h. In the model, the instantaneous pressures are the left ventricular PLV, aortic PA, and outflow Pout pressures. The external pressure is assumed to be atmospheric pressure. The outflow pressure is usually approximated by zero (19,21), i.e., assumed to be equal to the external pressure and close to the venous pressure. The flows in the model are the aortic input flowfA, the systemic flow f s , and the rate of volume change in the aorta f c (capacitive flow). Let us note that the Windkessel model is a simplified version of the general hydraulic model of a circulation segment as shown in Fig. 1. A direct electrical analogue of the hydraulic model in Fig. 4a is shown in Fig. 4b. In this model the hydraulic resistances are replaced by their electrical analogues, the aortic characteristic resistance R~, and the systemic resistance R$. The hydraulic
Electrical Modeling of Circulation fx Nocle ~
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TI
T2
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fs
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, ~
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R~
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i.~A ~od. U ~ i s
'
icL
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~
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!
Pout
Ik 9 T3
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9
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(b) T1
U LV
I/
~
RA
UA
i A Node u
is
-
I \ 9,L T2
Uv-'~Uout--Uolm
iv
Node N
Capacitor
Discharge Current (c)
FIGURE 4. (a) Hydraulic Windkessel model of the aorta; (b) three-terminal electrical analogue of Windkessel model: one lead of the capecitor is connected to Node M as in hydraulic model; (c) common two-terminal electrical analogue of Windkessel model. The connection of the second lead (Node N) has no obvious analogue in the hydraulic Windkessel (Fig. la), since there is no Node N in the latter model.
compliance is represented by a standard electrical capacitor C e. Above, we asserted that it would be more accurate to model a compliant chamber with a nonlinear residual-charge capacitor, but here we want to preserve correspondence to the twoterminal models reported in the literature. The left ventricle is represented by voltage ULv, and the aortic valve is modeled by a diode. Terminals T1 and T2 and the node M in the electrical model correspond to terminals T1 and T2 and node M in the hydraulic model. The instantaneous aortic iA, systemic is, and capacitive ic currents are equivalent to the corresponding flows in the hydraulic model. The considerations that led to a two-terminal model were the following. Since the venous uv and the outflow Uoutvoltages were assumed to be approximately equal to the reference (atmospheric) potential U a t m , it was decided to tie together the terminals T2, T3, and the reference point for the ventricular voltage. This decision resulted in the two-terminal electrical equivalent of the Windkessel (Fig. 4c) without terminal T3, but with a new node N. The two-terminal model has been successfully used for defining the current iA and loading impedance for the terminal T1 (5,27,31,32). Problems arise, however, if attempts are made to use the currents in node N and terminal T2 for interpreting flows in the circulation. The current that enters terminal T1 returns via ground terminal T2. Therefore, the current iv in the conductor connecting node N and terminal T2 could be interpreted as venous current. As it is shown below, such interpretation is incorrect.
604
J.E. Tsitlik et al.
Application of Kirchhoff's current law (8) to node M (Fig. 4c) gives: iA = is + i c ,
(2)
iv=is+it
(3)
and application to node N gives: 9
It follows that in this model iA = i v . Put another way, the forward and return instantaneous currents are equal and have the same phase. The model thus predicts that the instantaneous aortic and venous flows are equal, that their magnitude and phase are the same at any instant of time. This prediction cannot be correct since the aortic flow is extremely pulsatile, whereas the amplitude of pulsations is small in the venous flow. Therefore, the current through terminal T2 does not adequately represent the venous flow in the circulation. We can demonstrate another unrealistic feature of the two-terminal electrical model of Fig. 4c: the unrealistic pathway for the capacitor discharge. In the original hydraulic model (Fig. 4a), the compliant chamber is filled with the blood during systole and partially emptied during diastole. During emptying of the chamber, the blood flows out of the chamber partly via the resistance R~ (if the valve does not prevent this flow), but mostly via the resistance R h into some reservoir not shown explicitly in the model. The blood never reaches the external side of the chamber, either through the chamber wall or via the resistor R h. During the discharge of the electrical capacitance in the two-terminal model (Fig. 4c), however, the current flows from one side of the capacitor to the other via the resistance R~, i.e., positive charges move from the upper plate of the capacitor to the lower plate. This current is equix,alent to the unrealistic phenomenon of blood moving from the inside of the vessel to the outside of the vessel. Thus, the parallel connection of the resistor R~ and capacitance C h creates an unrealistic pathway for current that discharges this capacitance. Four- Terminal Model
In multisegment electrical models of the circulation, two or more networks are connected in series. A typical four-terminal model of the segment of a vessel (1,21) is shown in Fig. 5. There, L e is an inductor representing the inertance of blood, and R~ represents a leakage resistor accounting for outflow through small branches. The four-terminal network has two input terminals: T1 and T2, and two output terminals: T3 and T4. The voltages and currents in terminals T1 and T3 and node M are appropriate representations of pressures and flows in the circulation, but the currents and voltages in the return, or ground wire connecting T2, N, and T4 do not have analogues in the circulation. Indeed, similar to the two-terminal model, when the four-terminal network is powered by one generator or terminated by a two-terminal network, the instantaneous forward input flow il and return input flow i2 are equal in magnitude and phase. Also the instantaneous forward i3 and return /4 output flows are equal. Thus, the currents i 2 and/4 do not represent the venous circulation. The parallel connection of the leakage resistor R~ and the capacitance C e in node N present the same problem as the connection of the systemic resistor and the capacitance in the two-terminal model. This connection makes it appear that, rather than moving to the veins, the blood leaks to the outside of the vessel.
Electrical Modeling of Circulation
605
Re
T1
~
Le
I ~ . ii
Node
?
Ivl
i3
[
-]-i2 T2
"
"'T3,..
.
RL [
1
,-c
i4 ~:
A T4
Node N FIGURE 5. Four-terminal electrical model of a segment of a blood vessel.
Thus, the electrical models with two- and four-terminal networks contain terminals which are not representative of the circulation. Therefore, it is preferable to use only three-terminal networks in the models. A MODEL OF LEFT VENTRICULAR LOADING WITH THREE-TERMINAL NETWORKS
As illustrations of building electrical models with the three-terminal networks, we show a simplified closed-circuit model of ventricular loading (Fig. 6a) and an opencircuit model of the isolated left-ventricular toad preparation (Fig. 6b). In these models inertia of the blood is disregarded (i.e., inductances are not included), and it is accepted that the resistances have constant values. Like blood vessels, the heart chambers themselves should be modeled as threeterminal devices containing the nonlinear residual-charge capacitors with their individual residual charges. However, their capacitances should also be functions of time (time-varying capacitances) because of the time varying properties of the pressurevolume relationships of the heart chambers. In general, the capacitance (compliance) of a heart chamber is a function of both time and the transmural voltage (transmural pressure). In Fig. 6a, the left ventricular chamber is represented by a three terminal model (terminals T1LV, T2LV, and T3Lu with the nonlinear residual-charge capacitor NRCCLv which has an initial charge QOLVand time-varying electrical capacitance C~z(t) = 1/EZv(t). Here, EZv(t) is an electrical equivalent of a timevarying elastance (5,26). The charging current comes via the MV diode representing the mitral valve; the discharge occurs via the AV diode (the aortic valve). The left ventricle is loaded by a three-terminal three-element Windkessel (terminals T1Ao, T2Ao, and T3Ao). The systemic outflow is collected by the nonlinear residual-charge capacitor, NRCCv, which represents the venous system (terminals T1V, T2V, and T3V). The blood coming into the left ventricle is drawn from the left atrial nonlinear residual-charge capacitor NRCCLA (terminals T1LA, T2LA, and T3LA).
606
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J.E. Tsitlik et al.
),;2 ;
;
T2t.A TILV
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R~Ao
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;
i
T2Ao T1V r
T2V
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NRCC,~ ~ I>~
e
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Ao
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I
(b) FIGURE 6. Model of the left ventdcular loading, utilizing the three-terminal networks. (a) Closed-loop model; (b) open-circuit model of the isolated heart preparation.
The model of Fig. 6a is appropriate when an investigator is interested only in the analysis of the left ventricular loading and is not concerned with the pulmonary circulation and the right heart. Therefore, in the model in Fig. 6a, right heart and pulmonary circulation are represented together by a single voltage source e~ connected between the output terminal of the venous system T2V and the input of the left atrium T1LA. This connection reflects the fact that the circulation is a closed system. If the pulmonary circulation is included as well, a circuit similar to one given in Fig. 6a should replace the er.. All residual-charge capacitors considered in the model of left ventricular loading are located in the thorax. The common assumption is that the pleural pressure is the same in all of the thorax and that the lungs do not present resistance to volume shifts between the vessels. Therefore, the external plates of all the intrathoracic residualcharge capacitors can be connected and referred to the ambient potential via a voltage source, eintrathoracic,representing the intrathoracic pressure. In the open-circuit model (Fig, 6b), the model of the left ventricle and of the ventricular loading are the same as in the previous example (Fig. 6a). However, the systemic outflow is collected now not by the venous nonlinear residual-charge capacitor but by an element representing the limiting case of the venous compliance, i.e., the
Electrical Modeling o f Circulation
607
capacitor that has capacitance C e Oo and transmural voltage Utr = 0. We termed such an imaginary element the infinite residual-charge capacitor (IRCC). It can be viewed as an infinite reservoir. The left ventricle is filled by a constant voltage source eLA via constant resistance RL4. The voltage source draws its current from the other infinite residual-charge capacitor representing the limiting case of the left atrial and pulmonary venous compliance. All residual-charge capacitors in the model are referenced to the ambient potential because in isolated ventricle preparation the circulation is exposed to atmosphere. The rationale behind introducing the infinite residual-charge capacitors follows. If the circuitry is not closed as it is in Fig. 6a, then the blood has to be drawn from somewhere and to be discarded somewhere without affecting the voltages (pressures) at the input and output to the circuitry. This implies an infinite source at the input and an infinite drain at the output. Such infinite source and drain can be easily modeled by infinite residual-charge capacitors that produce a zero transmural voltage across them at any charge. These capacitors present an infinitely large resistance for the constant flow and negligible impedance for alternating flow of any frequency. The model in Fig. 6b is similar to the one described in (5) in three respects. First, t h e left ventricle is represented as a three-terminal device; second, a left ventricle is loaded by an electrical equivalent of Windkessel; and third, the left ventricle is filled through a constant resistance from a left atrium having a constant voltage (pressure). However, in Burkhoff's model the aortic load and the left atrium are modeled by twoterminal devices, thus creating ambiguity in understanding how the blood returns from the aorta to the left atrium. Both of the models reported in Figs. 6a and 6b give the same results for the terminals T2LV and T1Ao as the conventional models (5,27,31,32). In addition, the new models give more complete conceptual representation of the circulation and also allow analysis of the currents (flows) and voltages (pressures) at all terminals and nodes in the entire network. ----
DISCUSSION In hydraulic models, the circulation is conceived as consisting of different segments modeled by three-terminal hydraulic elements. These models are described by sets of linear, and sometimes nonlinear, hydrodynamic equations relating pressures and flows at the terminals where different elements are connected to one another and to sources. The equations are then programmed on a digital computer. Electrical models of circulation were developed as models of the hydrodynamic equations. In these models, the hydrodynamic equations are modeled by electrical networks (e.g., [21]). Often this modeling is only graphical with electrical drawings used to conceptualize the hydraulic models (4,7,10,17,20,31,32) while the analysis is done with the hydrodynamic equations. Moreover, the variables in these electrical models are often denoted as pressures and flows and not as voltages and currents. Sometimes, however, these hydraulic equations are derived directly from the electrical models (28). In the 1950s and 1960s, electrical models were analyzed either using an analog computer (9,21,25,29) or were built as analog networks (23,30). Starting from the 1970s, most analysis of electrical models was performed on digital computers (e.g.,
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[5,14,17]). For analysis of models, some researches utilize standard packages like SPICE (18). In addition, analog electrical circuits are still used occasionally (2). The commonly used electrical models of circulation employ the two- and fourterminal networks. While giving correct results for the equations modeled, these networks contain many terminals where voltages and currents are irrelevant to circulation. While not necessarily causing the errors in the calculations, these models can be extremely misleading. In the Introduction we mentioned the model of intracranial hydrodynamics and hemodynamics in (28), in which blood returns back to aorta via the intracranial space without entering the extracranial circulation. The way to construct an electrical model in which all of the terminals and nodes carry information relevant to circulation, is to use a three-terminal representation of both the circulation segments and the sources. It is common to encounter threeterminal representation of the circulation segments terminated by the two-terminal voltage sources. Sometimes the left ventricle is modeled as a three-terminal element (e.g., [5]), but the two-terminal device is connected as the load, thereby creating ambiguity in the interpretation of the return currents. In (12) the myocardial circulation is modeled by three parallel three-terminal Windkessels; however they are supplied from a two-terminal aortic pressure (voltage) source thus making the model confusing. The concept of a three-terminal hydraulic model was introduced by Brower and Noordergraaf (6) in their analysis of flow through a collapsible tube. We extended this approach to the modeling of all segments of the circulation with three-terminal electrical networks, and we demonstrated that with an appropriate connection of these networks, all of the terminals carried relevant information about the circulation. There is an inherent problem in using electrical capacitors to represent the pressure-volume characteristics of the blood vessels: the voltage-charge characteristics of a capacitor are very different from the pressure-volume characteristics of compliant and elastic blood vessels. Most investigators take into account the real characteristics of compliant vessels in either hydraulic models or electrical models. However, the use of standard linear electrical capacitors is not uncommon, especially by users of software packages for analysis of electrical networks. The linear capacitors have been used even in studies dealing with situations where the vessels are exposed to significant external pressure, which could cause collapse, like in the myocardial circulation (12), or in studies of cardiopulmonary resuscitation (2,18). We propose not using standard electrical capacitors in circulatory models at all, but instead using an imagined element, a NCRR to which the appropriate characteristics are ascribed. Substitution of NCRRs by standard capacitors can be done for particular linear cases, but these substitutions should be well-justified. In summary, in this article we have explicitly stated the limitations of using the electrical models of circulation and demonstrated that the way to avoid errors is to use three-terminal networks with NCRRs. Moreover, using this model provides an additional benefit. It was shown in this article that the capacitive current in a threeterminal model represents volume change in the vessels as well as volume shifts of matter surrounding the vessels. Analysis of this current would aid in the study of interaction of the circulation with the surrounding tissue, in particular in the study of heart-lung interaction. REFERENCES 1. Attinger,E.O.; Anne, A. Simulationof the cardiovascularsystem.Ann. N.Y. Acad. Sci. 128:810829; 1966.
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2. Babbs, C.E; Weaver, C.; Ralston, S.H.; Geddes, L.A. Cardiac, thoracic, and abdominal pump mechanisms in cardiopulmonary resuscitation: Studies in an electrical model of the circulation. Am. J. Emerg. Med. 2:299-308; 1984. 3. Blum, R.; Roller, D.E. Physics: Volume two. Electricity, magnetism, and light. San Francisco, CA: Holden-Day; 1980. 4. Braakman, R.; Sipkema, P.; Westerhof, N. A dynamic nonlinear lumped parameter model for skeletal muscle circulation. Ann. Biomed. Eng. 17:593-616; 1989. 5. Burkhoff, D.; Alexander, Jr., J.; Schipke, J. Assessment of Windkessel as a model of aortic input impedance. Am. J. Physiol. 255:H742-H753; 1988. 6. Brower, R.W.; Noordergraaf, A. Pressure-flow characteristics of collapsible tubes: A reconciliation of seemingly contradictory results. Ann. Biomed. Eng. 1:333-355; 1973. 7. Bruinsma P.; Arts, T.; Dankelman, J.; Spaan, J.A.E. Model of the coronary resistance and compliance. Basic Res. Cardiol. 83:510-524; 1988. 8. Chua, L.O.; Desoer, C.A.; Kuh, E.S. Linear and nonlinear circuits. New York: McGraw-Hill Book Co.; 1987. 9. De Pater, L.; van den Berg, J. An electrical analog of the entire human circulatory system. Med. Electron. Biol. Eng. 2:161-166; 1964. 10. Jaron, D.; Moore, T.W.; Chu, C.-L. A cardiovascular model for studying impairment of cerebral function during +Gz stress. Aviat. Space Environ. Med. 55:24-31; 1984. 11. Halperin, H.R.; Tsitlik, J.E.; Beyar, R.; Chandra, N.; Guerci, A.D. Intrathoracic pressure fluctuations move blood during CPR: Comparison of hemodynamic data with predictions from a mathematical model. Ann. Biomed. Eng. 15:385-403; 1987. 12. Holenstein, R.; Nerem, R.M. Parametric analysis of flow in the intramyocardial circulation. Ann. Biomed. Eng. 18:347-365; 1990. 13. Katz, A.I.; Chen, Y.; Moreno, A.H. Flow through a collapsible tube. Experimental analysis and mathematical model. Biophys. J. 9:1261-1279; 1969. 14. Kresh, J.Y.; Brockman, S.Y.; Noordergraaf, A. Model-based analysis of transmural vessel impedance and myocardial dynamics. Am. J. Physiol. 258:H262-H276; 1990. 15. Leaning, M.S.; Pullen, H.E.; Carson, E.R.; Finkelstein, L. Modeling a complex biological system: The human cardiovascular system- 1. Methodology and model description. Trans. Inst. M. 5:71-86; 1983. 16. Leaning, M.S.; Pullen, H.E.; Carson, E.R.; A1-Dahan, M.; Rajkumar, N.; Finkelstein, L. Modeling a complex biological system: The human cardiovascular system-2. Model validation, reduction and development. Trans. Inst. M. 5:87-98; 1983. 17. Li, J.K.-J.; Cui, T.; Drzewiecki, G.M. A nonlinear model of the arterial system incorporating a pressure-dependent compliance. IEEE Trans. Biomed. Eng. BME-37:673-678; 1990. 18. Lin, C.-K.; Levenson, H.; Yamashiro, S. Optimization of coronary blood flow during cardiopulmonary resuscitation (CPR). IEEE Trans. Biomed. Eng. BME-34:473-481; 1987. 19. Milnor, W.R. Hemodynamics. Baltimore: Williams & Wilkins; 1982. 20. Moore, T.W.; Jaron, D.; Chu, C.-L.; Dinnar, U.; Hrebien, L.; White, M.J.; Hendler, E.; Dubin, S. Synchronized external pulsation for improved tolerance to acceleration stress: Model studies and preliminary experiments. IEEE Trans. Biomed. Eng. BME-32:I58-164; 1985. 21. Noordergraaf, A.; Verdow, P.D.; van Brummelen, A.G.W.; Wiegel, F.W. Analog of the arterial bed. In: Attinger, E.O., ed. Pulsatile Blood Flow. New York: McGraw-Hill; 1964; pp. 373-386. 22. Noordergraaf, A. Circulatory system dynamics. New York: Academic Press, Inc.; 1978. 23. Pollack, G.H.; Reddy, R.V.; Noordergraaf, A. Input impedance, wave travel, and reflections in the human pulmonary arterial tree: Studies using an electrical analog. IEEE Trans. Biomed. Eng. BME15:151-164; 1968. 24. Remington, J.W. The physiology of the aorta and major arteries. In: Hamilton, W.F., ed. Handbook of Physiology. Circulation. Washington, DC: Am. Physiol. Sot.; 1963: pp. 799-838. 25. Snyder, M.F.; Rideout, V.C. Computer simulation studies of the venous circulation. IEEE Trans. Biomed. Eng. BME-16:325-334; 1969. 26. Sunagawa, K.; Sagawa, K. Models of ventricular contraction based on time-varying elastance. CRC Critical Reviews in Biomed. Eng. 7:193-228; 1982. 27. Sunagawa, K.; Maughan, W.L.; Sagawa, K. Stroke volume effect of changing arterial input impedance over selected frequency ranges. Am. J. Physiol. 248:H477-H484; 1985. 28. Ursino, M.; Di Giammarco, P. A mathematical model of the relationship between cerebral blood volume and intracranial pressure changes: The generation of plateau waves. Ann. Biomed. Eng. 19:1542; 1991.
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29. Warner, H.R. The use of an analog computer for analysis of control mechanism mechanisms in the circulation. Proc. of IEEE 47:1913-1915; 1959. 30. Westerhof, N.; Bosman, F.; De Vries, C.J.; Noordergraaf, A. Analog studies of the human systemic arterial tree. J. Biomech. 2:121-143; 1969. 31. Westerhof,N.; Elzinga, G.; Sipkema, P. An artificialarterial system for pumping hearts. J. Appl. Physiol. 31:776-781; 1971. 32. Westerhof, N.; Elzinga, G.; Van den Bos, G.C. Influence of central and peripheral changes on the hydraulic input impedance of the systemic arterial tree. Med. Biol. Eng. 1:710-722; 1973. 33. Yin, EC.P.; Cohen, J.M.; Tsitlik, J.E.; Zola, B.; Weisfeldt, M.L. Role of carotid artery resistanceto collapse during high-intrathoracic-pressure CPR. Am. J. Physiol. 243:H259-H267; 1982. NOMENCLATURE ~ , ~b Ee(t) F, f I, i P, p Q, q U, u
= = = = = = =
electric p o t e n t i a l t i m e - v a r y i n g elastance o f the heart c h a m b e r v o l u m e flow electric c u r r e n t pressure electric charge voltage
(Note: Lower case letters d e n o t e the i n s t a n t a n e o u s values o f the variables; u p p e r case letters d e n o t e c o n s t a n t , m e a n , a n d m a x i m u m values o f those variables.) P a r a m e t e r s or Elements: Ce
= = Le = Lh = Re = Rh = NRCC = Ch
electrical capacitance or capacitor elastic c o m p l i a n c e or c o m p l i a n t c h a m b e r inductance inertance electrical resistance or resistor viscous resistance n o n l i n e a r residual-charge capacitor Superscripts:
e h
= electrical = hydraulic Subscripts:
A amb atm C cl e i S LV RA tm
= = = = = = = = = = =
aortic ambient atmospheric capacitive c u r r e n t or flow closing pressure or voltage external internal systemic left v e n t r i c u l a r right atrial transmural
Electrical Modeling o f Circulation
V 0 1, 2
611
= venous = parameter, volume or charge, for an unstressed vessel = input and output variable or parameter in a three-terminal model, respectively APPENDIX
A:
COMPARISON O F A COMPLIANT CHAMBER WITH A STANDARD ELECTRICAL CAPACITOR AND A RESIDUAL-CHARGE CAPACITOR An essential part of the hydraulic Windkessel model of a segment of the circulation is a compliant chamber. The typical volume-pressure characteristic of the compliant chamber has a sigmoid shape (Fig. 3a). Phases of filling and emptying the compliant chamber are represented graphically in Fig. A1 (upper panel). The system of equations describing filling and emptying the compliant chamber can be readily derived from standard hydraulic equations (11,19): (A 1)
Ptm = Pc - Pe ,
Ch -
dvi
(A2)
dPtm ' f c = dv---2i = Ch dt
dPtm
oi = Do dr. A v i > 0
AU i =
~
'
dt ,
(A3) (A4)
m C h dPt m ,
(A5)
t~o
~Ve = -Sv~ , Ve > 0 .
(A6) (A7)
Here, C h = C h ( P t m ) is the chamber compliance that is a function of the transmural pressure (see volume-pressure characteristic, Fig. 3a); f c is the instantaneous rate of change of internal volume vi O.e., the volume inside the chamber); P c is the instantaneous pressure in the chamber; Pe is the instantaneous pressure outside the chamber (external pressure); Ptm is the instantaneous transmural pressure; Vo is the unstressed internal volume of the chamber; Ve is the instantaneous external volume (i.e., the volume outside the chamber); ~v denotes small changes in volume; and Av denotes total change in volume from the status of the unstretched chamber. When the chamber wall is relaxed (Fig. A1), i.e., Ptm = O, the internal volume is vi = Vo. The external volume Ve (i.e., the volume of environment surrounding the vessel) is positive; however, the magnitude of this volume is undefined (Eq. A7). Filling the chamber (Fig. Alb) means the increase of 6vi in the internal volume of the chamber. The expansion of the chamber comes at the expense of the volume external to the chamber. Consequently, the external volume (e.g., for the aorta, the neigh-
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J.E, Tsitlik et al. Internal
COMPLIANT CHAMBER:
.,=vo
.,=vo-.,
p ~ J External tr ~ v Volume: v e is not defined
~
RELAXED Pit --O
qi=~_~ ELECTRICAL CAPACITOR:
I
"J't~"fc
fc
FILUNG Pit Increasing
FILLED Pit >O
~1~ i .l~. dSq~.>O
;ll:ftesrnol*"
~
I + * 16qi=__6qe
Utr ~ , , E x t e r n o l , .
EMTYING Pit decreasing
L~jq{=Aqi >O
*****
.=--
lr ~__6qs
e
UNCHARGED
CHARGING
CHARGED
DISCHARGING
Utr =O
Utr decreasing
Utr >O
Ulr decreasing
+
NONLINEAR RESIDUALCHARGE CAPACITOR:
6qi
