Mathematical for simulation FRANCOIS
modeling of human cardiovascular of orthostatic response M. MELCHIOR,
R. SRINI
SRINIVASAN,
system
AND JOHN B. CHARLES
Laboratoire de Mkdecine Akrospatiale, Centre d’Essais en Vol, 91228 Brgtigny S/Orge Cedes France; KRUG Life Sciences; and National Aeronautics and Space Administration Johnson Space Center, Houston, Texas 77058 Melchior, Frangois M., R. Srini Srinivasan, and John B. Charles. Mathematical modeling of human cardiovascular system for simulation of orthostatic response. Am. J. Physiol. 262 (Heart Circ. Physiol. 31): H1920-H1933,1992.-This paper deals with the short-term response of the human cardiovascular system to orthostatic stresses in the context of developing a mathematical model of the overall system. It discussesthe physiological issuesinvolved and how these issues have been handled in published cardiovascular models for simulation of orthostatic response. Most of the models are stimulus specific with no demonstrated capability for simulating the responses to orthostatic stimuli of different types. A comprehensive model incorporating all known phenomena related to cardiovascular regulation would greatly help to interpret the various orthostatic responses of the system in a consistent manner and to understand the interactions among its elements. This paper provides a framework for future efforts in mathematical modeling of the entire cardiovascular system. mathematical models; computer simulation; short-term response; orthostatic stresses NUMEROUS MATHEMATICAL MODELS ofthehumancardiovascular (CV) system (CVS) have appeared in the literature (2, 4-6, 8, 11, 14, 15, 19, 23-27, 30-32, 35, 38-40, 51,54,55,65,66,69, 71, 73, 74, 78,86,87,91,96, 99, 102, 105) since the publication of the classic work of
Womersley (104) in 1957 on analysis of blood flow in elastic vessels. These models vary considerably in complexity and, depending on the purpose, range from very simple ones (consisting only of a resistance and a compliance) to complex multisegment representations of the vascular tree. Simplified descriptions of the CVS with lumped parameters often form part of a model of a larger physiological system. An example is the Guyton model of overall circulation (33), which includes a three-compartment representation for the CV function to permit calculation of mean values of CV variables. Simple CV models have also served as teaching aids (19, 50). More elaborate mathematical models of the CVS are intended to characterize and provide a better understanding of the pressure-flow relationships in the system and how they are affected by external stimuli. They may be broadly divided into three categories, if we exclude from consideration detailed biomechanical models of the heart (left ventricle in particular), models of cardiac electrophysiology, and those of the microvasculature. The first includes models of the vascular network, which do not generally address control aspects of circulation
(4, 5, 11, 19, 25, 27, 31, 35, 39, 51, 65, 66, 71, 74,91, 96).
The second group consists of models of specific vascular beds (15, 78, 99, 105), such as the coronary circulation. Finally, there are overall models of the CVS (2, 6, 8, 14, 23, 24, 26, 30, 32, 38, 40, 54, 55, 69, 73, 86, 87, 102),
which by design incorporate at least some elements of central nervous system (CNS) control to enable simulation of responses to stresses and thus a study of mechanisms of CV regulation. The purpose of this paper is to discuss the physiological and methodological issues involved in the development of CV models in the third category aimed at simulating the short-term response of the human CVS to orthostatic stresses. Exercise response is not considered because of its complex nature involving several systems in addition to the CVS. Also, the fundamentals of lumped-parameter modeling of the CVS are not covered, as they may be found elsewhere (28, 47, 76, 103). The emphasis is on the requirements for constructing realistic models of the CVS in accordance with available principles and data. It is within this context that the published models are compared here. Even with such a narrow focus, the literature is quite voluminous, and therefore attention is restricted to selected articles that we have judged as germane to mathematical modeling of the overall CVS. APPROACH
Short-term CV models for simulation of orthostatic stress response have limitations, as defined by the qualifiers short term and orthostatic. Short-term response denotes the immediate reaction of the CVS, lasting typically up to 15 min, to an external stimulus. The CVS is modeled as a closed system without any fluid movement through the capillary walls and without any connection to extravascular fluid spaces. Control is exerted entirely by the pressure receptors and autoregulatory mechanisms. Such an approach is justified when the loss of blood volume due to transcapillary filtration is judged to not alter the CV response to the applied stimulus significantly (102). The term orthostatic stress refers to CV stress caused by a change of pressure in different vessels of the system. There are a variety of external stimuli that will bring about an abrupt change of pressure within the CVS, but only a few of them may be considered to act exclusively on the CVS in the short term. The stimuli used
H1920
Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (163.015.154.053) on September 8, 2018. Copyright © 1992 American Physiological Society. All rights reserved.
MATHEMATICAL
MODELING
OF HUMAN
frequently in CV studies include exposure to weightlessness, +G, (head-to-foot) acceleration, lower body negative pressure (LBNP), lower body positive pressure (LBPP), head-up tilt, and blood loss. For purposes of modeling, it is convenient to represent the overall CVS as a feedback system consisting of two major blocks (103), as shown in Fig. 1. One is a collection of distensible vessels in serial and parallel arrangement, with the heart acting as the pump. It is the controlled system, which we refer to as the hydraulic system (HS). The other is the regulating system or the reflex control system (RCS) for control of arterial pressure and blood flow to various organs. We consider the autoregulated local circulations as part of the HS, since, as their definition implies, the regulation is mostly local and much less dependent on CNS control. The interaction between the two systems is through the control signals (carotid, aortic, and central venous pressures) and the controlled parameters (cardiac frequency and contractility and vessel tone of arteries and veins). Next we discuss the physiological aspects of the HS and the RCS from the point of view of developing mathematical models of the subsystems involved in the shortterm regulation during an orthostatic stress. We point out where knowledge is lacking and broad assumptions are necessary and then compare the features of the published short-term CV models in light of this discussion. A detailed review of some of these models may be found in the paper by Coleman (21). HYDRAULIC
SYSTEM
Pressure-Flow
Relationship
in Blood Vessels
Mathematical description. Cardiovascular models, for which the outputs are the local pressures and flows, are normally constructed by dividing the HS into a number of smaller elements and determining the pressure-flow relationship for each of these elements. Thus a necessary first step in model development is a mathematical description to calculate flow for any given time-depen-
I
CARDIOVASCULAR
SYSTEM
H1921
dent pressure gradient across a vessel of given length. The Navier-Stokes equations describing the blood flow, the Lamb’s equations describing the motion of the vessel wall, and their simplifications provide such a description. Milnor (67) has described the key steps and assumptions involved in obtaining a mathematical solution of the pertinent equations. The Navier-Stokes equations are in effect a description of the balance among the various forces at play, that is, (local inertial force) + (convective inertial force) = (viscous force) + (driving force). The dimensionless Reynolds (Re) and Womersley numbers (W) permit evaluation of the relative importance of these forces in different vessels. Re evaluates the ratio of the convective inertia over the viscous force, whereas W is an indicator of the ratio of the local acceleration over the viscous force. For most vessels, except the large arteries such as the aorta, the values of Re and W are small compared with unity, and therefore the inertia terms can be neglected for these vessels. For large arteries, Re and W are greater than unity, and it is necessary to take into account the nonlinear convective inertia in the calculation of flow to account for the influence of tapering of arteries (59, 90). Model of arterial segments. There are two approaches to obtain the flow in a vessel given the pressure gradient across it as a function of time, namely, the Fourier transform method and the difference-differential equations method (for review see Ref. 20). The Fourier transform method, first used by Womersley (104), enables calculation of the velocity profile in the vessel for a periodic pressure gradient. The profile is both time and frequency dependent. Therefore, electrical analog models of arterial segments based on this approach will contain frequencydependent parameters (39). Such models are useful in studying the propagation of pressure pulse through arterial branches (71). The difference-differential equations method, developed by Rideout and Dick (77), considers the velocity
I t
Heart
+
j 1 Aiz
h
i
Fig. 1. Decomposition of overall cardiovascular systern into hydraulic system and reflex control system. Interaction of the 2 systems determines short-term cardiovascular response to orthostatic stress. P, pressure.
lOCal circulations
Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (163.015.154.053) on September 8, 2018. Copyright © 1992 American Physiological Society. All rights reserved.
H1922
MATHEMATICAL
MODELING
OF HUMAN
profile to be flat in the inner two-thirds and zero in the remaining outer one-third of the artery. The parameters of the electrical analogue model are independent of frequency in this case. Except for the addition of an inertance element, the model is structurally similar to a twoelement windkessel model. The resistance and compliance values are calculated from the geometric dimensions of the vessel (Fig. 2). The model has two advantages over that based on Womersley’s approach in that it can account for nonlinearity of vessel compliance and the input pressure gradient need not be a periodic function. The effect of gravity or acceleration is easily accounted for by inclusion of an electromotive force of appropriate strength, as indicated in Fig. 2. In modeling the arterial bed, there is no difficulty in representing the local acceleration effects by an inertance term that appears linearly in the equations. On the other hand, inclusion of the convective term and nonlinear compliance results naturally in nonlinear equations. One way to simulate the effects of the nonlinear convective term is by changing the value of blood viscosity (40, 90). A more realistic approach is to take into account the nonlinearity of arterial compliance by specifying the pressure-volume (P-V) relationship of the arterial segment based on available experimental data, as discussed in P- V Relationship in Blood Vessels. Model of uenous segments. Models of arterial segments
discussed above cannot readily be applied to venous segments because of the special features of the veins distinguishing them from arteries. These are 1) the veins may change cross section and collapse when the transmural pressure decreases to very low levels; 2) there are valves in the venous bed that prevent the flow from reversing and thus help maintain pressure in the upper body during orthostasis; 3) the effect of changes in external vascular pressure is more pronounced on the veins than on the arteries because of the higher compliance of the venous vessels; and 4) a minor consideration is that the flow in veins has little pulsation, which implies that the inertance term in the arterial model reflecting the local inertial force may be neglected. The inclusion of the above venous characteristics in CV models poses no problem except possibly for the first
h(t)
P2(f)
Resistance R
w =&is
Inductance (Inertance) L
w =&$
Capacitance (Compliance) c- 3m3l -2hE
IL (External Pressure) Fig. 2. Electrical analog of arterial segment. Voltages P1 (t) and Pz( t) represent end pressures and ground, the pressure outside artery. Inductance (L) and capacitance (C) correspond, respectively, to inertance and compliance of segment. As indicated, resistance (R), L, and C are related to vessel dimensions [length (I), radius (r), and thickness (h)] , blood viscosity (p), blood density (p), and Young’s modulus of elasticity for segment (E).
CARDIOVASCULAR
SYSTEM
feature. The arterial model may be modified, as suggested by Snyder and Rideout (86), to take into account venous collapse by postulating that during collapse the vessel cross section has an ellipsoidal shape with a constant circumference. This permits analytical expressions for the resistance and inertance of the vessel dependent on the axes dimensions of the ellipse. In any case, it is necessary to increase the resistance significantly at very low levels of transmural pressure to simulate near cessation of flow during collapse. It should be mentioned that the phenomenon of venous collapse should not be ignored in any realistic model of the CVS if it is to faithfully reproduce observations such as the decrease in the arterialvenous pressure differential during +G, acceleration (36). P-V Relationship
in Blood Vessels
Figure 3A shows the steady-state relationship between the transmural pressure across the wall of an arterial segment and the volume of the segment over a wide range of pressures from 10 to 180 mmHg. Results of studies to correlate the observed P-V characteristics and the physical properties of the arterial wall (18, 59, 97) show that the P-V curve has a constant slope (linear compliance) with a slight concavity in the pressure range of lo-80 mmHg. Accordingly, the expression for C included in Fig. 2 is a good approximation of the compliance of the artery as function of its radius in this range. Above 80 mmHg, the stiffening of the collagen matrix increases the Young’s modulus of the wall and produces a reversal of the curvature of the P-V curve. The thinning of the arterial wall also has an effect on this reversal (59). The compliance begins to decrease rapidly toward zero. The compliance at 130 mmHg is twice as small as the value at 80 mmHg. It should be noted that it is not unusual to find arterial pressures much higher than 80 mmHg under normal circumstances. For example, the pressure in the arteries of the leg reaches -180 mmHg on shifting from a supine to a standing position due to the additional hydrostatic component (80). Figure 3B shows the P-V relationship of a collapsible tube used as a mechanical model of the vein (46). The large compliance of veins at low positive pressures is clearly evident; the increase or decrease in vein capacity for a small change of pressure of -3 mmHg is considerably larger than that for the same pressure change of -0 or 10 mmHg. The collapsible tube mimics fairly well the behavior of veins except at very high pressures. For pressures >50 mmHg, the P-V curve displays a curvature toward the volume axis, which is not compatible with stiffening of the vessel wall with further decrements in distensibility at these pressures (18). The foregoing brief discussion of the P-V relationship of arteries and veins shows that the compliance of blood vessels can be approximated by a constant only under supine conditions. A simple stress, such as tilt, tends to shift the pressure in the vessels out of the linear range; the arteries may stiffen significantly in the lower body, and the veins may collapse in the upper body. Because veins are much more compliant than arteries on the average and hold well over 60% blood volume, we can predict that in a mathematical model of the CVS for orthostatic stress response, an error in venous compliance would lead
Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (163.015.154.053) on September 8, 2018. Copyright © 1992 American Physiological Society. All rights reserved.
MATHEMATICAL
A
MODELING
OF HUMAN
4
: Unoar : : : :. : :.
>
1
i
!: . ::
I
/
i
1 i I
:
1
I L/M
: 3
Nonllnoar Stdning of the collrgon Docnrdng compllmco
of Ehotkity
I
;
1
180
80
10
PRESSURE VOLUME
H1923
SYSTEM
a simpler expression with relatively few parameters is a tangent function in which the transmural pressure approaches infinity from -30 mmHg as the vessel volume increases from zero to a maximum value. It was originally proposed by Langewouters et al. (52) for the aorta and was used recently by Braakman et al. (15) for the muscle veins and arteries. It should be noted that the sum of blood volumes in the various segments of a CV model at any instant gives the total circulating blood volume contained in the CVS. The total volume remains a constant in a closed model with no provision for capillary filtration, although its distribution may vary with time under the influence of a stimulus. Any reduction of total volume (blood loss) is an orthostatic stress by itself, as mentioned earlier. It has a profound effect on CV response by lowering the preload on the heart, as has been shown in experimental as well as model studies (2, 9, 14). The effect is magnified when blood loss is combined with another orthostatic stimulus, such as +G, acceleration (88). The important question from the point of view of modeling is how the redistribution of a change in total blood volume should be initiated in the model. Clearly, altering the blood volumes of the vascular segments in different proportions (to account for the total change) will produce different transient CV responses. However, the steady-state response, with pressures and flows appropriately adjusted to correspond to the altered total volume, is unlikely to be greatly affected by changes in initial redistribution of the volume change. This conclusion 1s based on our simulation studies usingthe Croston model (23).
Compli8nce = AWAP
Y 3 0
CARDIOVASCULAR
(mmHg)
(ml)
The Heart as a Pump
I
I 20
I 30
I 40
I 50
I 60
I 70
I 80
? = ( Pint- Pa 1 Fig. 3. Pressure-volume (P-V) relationship in blood vessels. A: relationship in arterial segment. B: changes in volume as a function of transmural pressure in a collapsible tube used as a mechanical model of a venous segment. Changes in cross section are also indicated. P, transmural pressure; pint, internal pressure; pe, external pressure. [B: from Katz et al. (46) reproduced from Biophys. J., 1969, ~019, p. 1261-1271, by copyright permission of the Biophysical Society.]
to an error in the amounts of blood shifts, whereas an error in arterial compliance would result only in an error in the wave shape of pressure pulse. Thus any model of the CVS aimed at simulating a pressure stress should take into account the nonlinear behavior of the veins first and, then to a lesser extent, that of the arteries. A number of mathematical functions are available to describe the P-V relationships in blood vessels in mathematical models of the CVS. These are based on fitting experimental P-V data from human arteries and veins (24, 34, 48, 52). Most are empirical, the only exception being the complex expression derived by King (48) based on the theory of elastomers. The author applied it to explain the change in the P-V characteristic of human aorta with age, and Srinivasan and N udelman (89) used it later in a model of the baroreceptor. Although” empirical. I I --I
There have been numerous studies on quantitative characterization of the heart as a pump, the left ventricle in particular, as reviewed recently by Maughan and Kass (64). These studies began with the Starling law of the heart shown in Fig. 4. The relationship is for a constant systolic pressure, peripheral resistance, heart rate, and cardiac contractility and does not provide an adequate description of the pump function if any of these factors is changed. The studies of Suga and Sagawa (92,93) on the isolated dog left ventricle shed some light for the first time on the performance of the heart under varying afterload (peripheral resistance and arterial impedance) and
-4
0 Mean Right
4 Atrial Pressure
8 (mmHg)
Fig. 4. Starling law of heart as described by relationship between right atria1 pressure and cardiac output. Higher pressures result in larger filling volumes at end diastole.
Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (163.015.154.053) on September 8, 2018. Copyright © 1992 American Physiological Society. All rights reserved.
H1924
MATHEMATICAL
MODELING
OF HUMAN
the preload (diastolic volume) conditions. The results of their studies have led directly to a new type of models described by the authors as “models of the ventricle as a contracting chamber” (94). Further studies have improved their model by including the effect of left ventricular flow and resulted in the following relationship for the canine heart (85, 94) hv(t) = &mi,E,(t/T,,,) [v,,(t) - v,] [I - o~ool~(-dbddt)l where PLV(t) and V&t) are the instantaneous pressure and volume of the left ventricle; E,,, is the end-systolic slope or the maximum elastance, which is directly related to the cardiac contractility; E&IT,,,) is the time-varying normalized elastance of the left ventricle, which is independent of the afterload, the preload, and the state of cardiac contractility; Tmax is the duration of contraction; Vd is the unstressed end-systolic volume; and -dV&dt is the left ventricular outflow. The use of the above equation in human CV models may be justified to some extent by the recent demonstration of a linear relationship between end-systolic pressure and volume of the left ventricle in human subjects (42). Such a representation for the left ventricular P-V relationship has recently been extended to other chambers of the heart (10). There is evidence from animal studies to support the use of similar elastance characteristics for the two ventricles (17). Thus, although not strong, there seems to be a sufficient basis for modeling the entire heart with appropriately scaled P-V curves for the four chambers. It may be necessary to consider the interaction among the chambers, especially between the two ventricles, but this needs further analysis (82). The question at this point is whether we need to represent the heart in an overall model of the circulation by a contracting chamber (pulsatile model) rather than by the Starling law. The contracting chamber model is preferable for a number of reasons. First, the Starling law curve can be easily deduced from the model of the heart as a contracting chamber. It can be explained by the enddiastolic P-V relationship of the left ventricle, its sigmoid shape being a direct consequence of the concavity toward positive pressure of the end-diastolic curve (limit of elasticity of the passive left ventricle). Second, the pulsatile model of the heart will help improve our understanding of the ventricular-vascular coupling during orthostasis as it has done for characterizing the heart under normal unstressed conditions. Indeed, based on the pulsatile model, a new reliable index of contractility (E,,,) independent of preload and afterload has been defined, and the effects of some systemic parameters (aortic inertance, peripheral resistance, arterial compliance) on the P-V cycle of the left ventricle have been quantitatively evaluated (16). Third, because the carotid baroreceptor response is pulse dependent (3, 83), the pulsatile models are more accurate to simulate the feedback neurogenic control during orthostasis. Finally, it should be pointed out that with a pulsatile pump representation of the heart, the influence of parameters such as cardiac contractility on stroke volume may be readily tested (19,38).
CARDIOVASCULAR
SYSTEM
This is not possible using a nonpulsatile model in which the stroke volume is a direct function of the atria1 pressure (14, 27, 30). Autoregulated
Local Circulations
Most vascular beds of the mammalian CV system are known to be autoregulated; that is, for a wide range of changes in perfusion pressure, the flow is maintained at a steady level by myogenic and metabolic reflexes (for review see Ref. 22). The autoregulatory response time varies in the different vascular beds. The restoration of blood flow following a change of perfusion pressure occurs within seconds in vital organs such as the brain. Therefore ignoring the phenomenon of autoregulation would lead to an incomplete representation of CV control in any model of the CVS. Also, an error in the calculated value of an autoregulated flow is likely to lead to errors in the calculation of flow in other parts of the CVS, especially when it is modeled as a closed system with interdependent pressures and flows. The inclusion of autoregulation of cerebral and coronary circulations is particularly important in the development of CV models, since the CV tolerance to any stress, orthostatic or otherwise, depends heavily on the maintenance of flows in these two circulations. The flow through the brain is unaffected even with severe reductions in head-level arterial pressure, such as those that occur during exposure to high levels of acceleration stress, because of concomitant decreases in cerebral venous pressure (36,58). Likewise, the coronary flow is well regulated to support the metabolic needs of the heart (53). It is essential to incorporate in a CV model local control of cerebral circulation, which accounts for -15% of the cardiac output in humans. On the other hand, the autoregulation of coronary circulation may be neglected, since coronary blood flow is ~5% of the cardiac output, and experimental evidence from animal centrifuge studies shows that its upper limit is not reached during exposure to high +G, acceleration (53). This means that considering the coronary circulation as passive in CV models is not likely to introduce significant errors in simulated results. Fluid Filtration
There is a continuous flow of fluid across the capillary wall between the intra- and the extravascular space that depends on the magnitude of the capillary pressure. The net flow is zero under steady conditions. There is a net outward filtration with an elevation of capillary pressure. This occurs, for example, in the vascular bed of the leg muscles during passive standing (37)) leading to a transient decrease of total blood volume. As noted earlier, a reduction in total circulating blood volume of sufficient magnitude (X0%) can significantly impact circulatory pressures and flows and thus the CV tolerance to an imposed stress. Therefore it may be necessary to take this loss into account in a model of the CVS if the model is to be used for analysis of the effects of high levels of orthostatic stress. However, the dynamics of capillary fluid filtration tending to return the total blood volume to its normal value may be ignored in a short-term model, since
Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (163.015.154.053) on September 8, 2018. Copyright © 1992 American Physiological Society. All rights reserved.
MATHEMATICAL
MODELING
OF HUMAN
they are slower compared with those of the nervous reflex mechanisms of arterial pressure regulation (84). Extravascular
Pressures
Models of the Hydraulic
System
We now examine published overall models of the CVS in light of the discussion of the physiological aspects and modeling approaches of the preceding sections. The pertinent questions are 1) How many vascular elements should be used in the model to represent the total system, including the heart chambers? 2) What assumptions are made to solve the Navier-Stokes equations for the arterial tree? Does the model take into account the convective term and the nonlinearity of the P-V relationship in arteries? 3) For the venous bed, does the model consider venous collapse, venous valves, and external vascular pressure? 4) Is the heart model based on the Starling law or the time-varying elastance concept? 5) Does the model take into account the autoregulation of flow in vascular beds where it is known to be present? Table 1 summarizes the HS attributes of the published models. Because models are generally constructed with specific, often narrow, objectives in mind, it is not sur-
Table 1. Comparison of hydraulic system portion of overall cardiovascular models for simulating
Investigators
orthostatic
Reference
8
Beneken and DeWit Snyder and Rideout Boyers et al.
86 14
Hyndman Croston et al.
38 24
Green and Miller 30 Avula and Oestreicher 6 Leaning et al. 54 Jaron et al. 40 Al-Dahan et al. 2 +G, acceleration, acceleration
H1925
SYSTEM
prising that none of the models listed in Table 1 incorporates all phenomena of significance pertaining to shortterm CV regulation. This is not to say that these models are unsatisfactory. Quite the contrary, they have had varying degrees of success in simulating the experimental data for which they were developed. Their success may be limited if one attempts to simulate results from a variety of experiments dealing with orthostatic stimuli using the same model. We believe that any such limitation is largely attributable to the exclusion of some key aspects of the CVS in the model. It should be noted that some of these CV aspects have come to light only recently through experimental studies. As an example, we mention the systematic study of the nonlinear P-V characteristics of large arteries (52). One of the key modeling considerations that has hitherto not been touched on is the total number of model CV elements, which includes the number of heart chambers, arterial, capillary, and venous segments. It is dictated primarily by the stress response of interest, which determines the model structure. For example, a model for study of G-stress, LBNP, and LBPP response must differentiate between leg and abdominal circulations (23, JO), whereas a model for simulation of cardiopulmonary resuscitation needs to emphasize venous return (11). Another consideration is the number of venous segments. Many models divide the arterial tree into a large number of segments but use relatively fewer venous segments (40). Such a representation may be inadequate in the studies of CV effects of weightless exposure and +G, acceleration where the venous pooling plays a dominant role (23). Other factors that need to be considered in choosing the number of CV elements are the availability of data on their characteristics and the level of complexity or simplification that the investigator wishes to introduce in the model (70). Most CV models are derived from anatomic and physiological considerations, and therefore one would expect a more realistic model with a high degree of isomorphism. However, the limited data available, especially on human CV parameters, preclude elaborate segmentation. A minor consideration is the increase in
Because blood vessels are not rigid tubes, any variation of the pressure outside will have an influence on the flow inside the vessel through a modification of the vessel diameter. As mentioned earlier, this external pressure has a more significant effect on the veins than on the arteries, since veins are relatively much more compliant. The different extravascular pressures that may vary during an orthostatic stress are the intrathoracic pressure (when respiratory maneuvers are involved), the abdominal pressure (with G-suit compression), and the extravascular leg pressure (during LBNP application or G-suit compression). External vascular pressures are easily included in CV models, but only some them may be needed depending on the response studied: influence of muscle contraction (23,87), abdominal or intrathoracic pressure (23,38,87), external pressure applied to the chest during a cardiopulmonary resuscitation (1 l), or external pressure applied to the legs by means of a G-suit (40, 69). Mathematical
CARDIOVASCULAR
system
response
Orthostatic Stress Studied
Total Segments
Bl .ood loss Ti ilt Ti ilt Bl .ood loss Ti ilt LI 3NP Ti ilt +( 2, acceleration +( >, acceleration Bl ood loss +( $ acceleration Bl ood loss in head to foot direction;
Autoregulated Local Circulation
Arterial Pulsatile Heart Pump
Inertance
X X
X X
10 36
X X
X X
4 5 19 32 8
X X X
X X X X
19 31 7
Convective term (NavierStokes)
Bed
Venous
Bed
Nonlinear arterial compliance
X
Venous collapse
~$J~~~ pressure
Venous valves
X X
X X
X X
X X
X X
X
X X X X X X
X X X
X X X
X
LBNP, lower body negative pressure. X, Factors included in model.
Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (163.015.154.053) on September 8, 2018. Copyright © 1992 American Physiological Society. All rights reserved.
H1926
MATHEMATICAL INPUT PARAMRERS
MODELING
OF HUMAN
FIRING RATE (AFFERENT)
RECEPTORS
FIRING RATE (EFFERENT)
-m-II
SYSTEM OUTPUT PARAMETERS
EFFECTOR SITES
III-
Sympathetic
Pcarotid Paortlc
CARDIOVASCULAR
sinus
Parasympathetic
I
arch
ARTERIAL BARORECEPTORS
v-T
Patria
I
4
HEART RATE
.
4
CARDIAC CONTRACTILITY
PBrB?)ymy)athetiC-I HEART
CENTRAL
I
NERVOUS
. Sympathetic
+
ARTERIOLES
PERIPHERAL RESISTANCE (Local modifications arteriolar resistance)
1
)I
Sympathetic
w
VEINS
VENOUS COMPLIANCE (Local modifications of venous compliance)
Fig. 5. Cardiovascular
+
I
I--
SYSTEM
1
HEART
*
Sympathetic
I
m
Ppulmonary
*
of
w
reflex control system.
computations with a large number of elements. More importantly, it may defeat the very purpose of modeling, which is to simplify, both structu.rally and functionally, so as to extract the essential features of the real system.
and accordingly much of the modeling of the RCS is empirical in nature. We discuss next what we know and what we need to know about the various RCS elements from a modeling perspective.
CARDIOVASCULAR
Noninvasive Techniques for Study of RCS in Humans The development of two noninvasive techniques has helped enhance our understanding of the roles of carotid and cardiopulmonary receptors in the short-term control of human CVS (61,62). One is the neck cuff technique in which the carotid transmural pressure is modified through rapid application of positive or negative pressure around the neck, and data on the open loop characteristic of the carotid baroreceptor control system are obtained. The use of the neck cuff together with other techniques’ of measuring various circulatory parameters has enabled study of the control of these parameters by the carotid baroreceptors (1, 75). The second noninvasive technique is LBNP, which entails the application of subatmospheric pressures to the lower extremities of the body. The resulting movement of the blood away from the thoracic region alters the activity of cardiopulmonary receptors. For low values of LBNP up to -20 mmHg, there are no significant changes in mean and pulse arterial pressures, indicating an inhibition of the cardiopulmonary receptors without any modification of arterial baroreceptor activity (41). Simultaneous use of neck cuff and LBNP provides information on the interaction of the two receptors. For example, the cardiopulmonary receptors have been shown to have an inhibitory effect over the carotid baroreceptor control of forearm vascular resistance using this approach (72, 98). Table 2 summarizes the available results from human studies on the control of some circulatory parameters by the carotid and cardiopulmonary receptors. The skin and muscle vascular beds of the limb are not identified sepa-
RCS
Elements of RCS Figure 5 shows the cardiovascular RCS schematically in terms of a number of interconnected black boxes. The components are the receptors at the afferent end, the CNS, and the controlled elements at the effector sites, which bring about changes in heart rate, cardiac contractility, and vessel tone. The carotid and aortic baroreceptors and the cardiopulmonary receptors are the dominant sensors in the elicitation of a short-term CV response to an orthostatic stimulus. The response is also influenced to some extent by changes in arterial pressures of oxygen and carbon dioxide, PO, and Pco~, sensed by the chemoreceptors. The effect is secondary to changes in respiratory variables, such as breathing frequency, and its magnitude depends on the type and intensity of the imposed orthostatic stimulus. For example, both breathing frequency and ventilatory volume have been shown to increase during +G, acceleration (12). In turn, these can result in higher levels of cardiac output, independent of the acceleration effects (13). Marked changes in respiratory variables have also been observed during LBNP application (60). However, we believe it is not necessary to have an explicit representation of the chemoreceptors in an overall CV model focused on short-term orthostatic response. The possible respiratory effects may be accounted for by appropriately varying parameters, such as intrathoracic pressure as a function of breathing frequency (23). The CNS is indicated by a single box in Fig. 5, although the sympathetic and parasympathetic efferent outputs controlling the output parameters may originate from different areas of the CNS. Knowledge of the relationship between the input and the output of each of these black boxes should enable one to construct a precise overall model of the RCS. However, our knowledge is limited,
l These include venous occlusion plethysmography to measure limb vascular resistance, dye-dilution technique to determine splanchnic vascular resistance, isolated limb technique to assess limb vascular capacitance, and electrocardiogram to obtain heart period and other time intervals.
Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (163.015.154.053) on September 8, 2018. Copyright © 1992 American Physiological Society. All rights reserved.
MATHEMATICAL
MODELING
OF HUMAN
CARDIOVASCULAR
H1927
SYSTEM
Table 2. Summary of results from studies on control of cardiovascular parameters by carotid and cardiopulmonary Organ/ Parameter
Heart
Vascular resistance vessels Vascular capacitance vessels
Carotid Receptors
Heart rate Cardiac contractility Arm Leg Splanchnic
-
?
?
+
++
++ ++ +
Leg Splanchnic
+ ?
rately in Table have addressed two regions (7). sites of carotid different.
Cardio pulmonary Receptors
++
Arm
++, Significant
receptors
effect; +, less significant
+ +
Investigators
Observations
Controlled only by carotid receptors No modulation by cardiopulmonary receptors No information on influence by cardiopulmonary or carotid baroreceptors in humans Principally controlled by cardiopulmonary receptors Principally controlled by carotid receptors Principally controlled by carotid receptors
Carotid and cardiopulmonary receptors exert a transient effect (90 s) over limb vascular capacitance No sustained influence A variation of splanchnic vascular capacitance has been shown after arterial baroreceptor involvement in animals, but no such observation in humans effect; -, no effect; ?, effect unknown.
+
2, although there are some studies that the difference in the responses of these The results clearly show that the effector and cardiopulmonary receptors are quite
Receptors
The carotid baroreceptor is perhaps the best understood among the receptors pertinent to orthostatic response. Taher et al. (95) have recently reviewed the mathematical models of the baroreceptor and have proposed a new model that takes into account its essential characteristics, i.e., the sigmoidal shape of the curve relating carotid pressure and receptor firing rate, resetting, and dependency on pulse pressure. The pressure receptors from the aortic region have been studied to a lesser extent than the carotid baroreceptors because of the experimental difficulties involved in isolating them. Comparison of the two types of baroreceptors in dogs shows their range of pressure to be the same, but there is a lack of effect of pulse pressure on the aortic baroreceptor response (3). It is necessary to differentiate between aortic and carotid baroreceptors because the intraluminal pressure in the aorta and carotid arteries can be very different for tilt, LBNP and +G, acceleration. Although their influences on the CNS may not be the same, they are believed to be summatory. Accordingly, for modeling purposes, the afferent signals from the two receptors may be added with different weighting factors (54). With regard to the cardiopulmonary receptors, the observations summarized in Table 2 underscore their importance in CV regulation during an orthostatic stress. Some published models for simulating the short-term CV response following a loss of blood volume show the necessity to include these receptors to produce model responses that are more accurate and in greater agreement with experimental data (2,14). Concepts regarding their interaction with baroreceptors are presently speculative (80) and need further clarification with results from experi-
ments such as the simultaneous and LBNP alluded to earlier. Central Nervous
Johnson et al.
Reference
41
Abboud et al. Johnson et al. Essandoh et al. Abboud et al. Johnson et al. Samueloff et al.
41 81
Samueloff et al. Mancia and Mark
81 61
application
of neck cuff
1 41
29 1
System
The CNS is the integrating element of the overall control system, which gathers the input information coming from receptors through afferent nerves and sends out command information to the heart and blood vessels of the different parts of the circulation through efferent sympathetic and parasympathetic pathways. Detailed knowledge of the functional characteristics of the CNS is needed to answer questions such as, Is the sigmoidal shape of the curve of heart rate versus carotid pressure only due to the sigmoid curve relating pressure and firing rate of the carotid baroreceptor, or is there a limiting factor in the CNS? An examination of available data and models on the control of heart rate by carotid baroreceptors (Fig. 6) indicates where we stand with regard to modeling the CNS control. For the sake of simplicity, we display only the steady-state curves in Fig. 6. One could say that the mathematical combination of subsystem D - (A + C) would lead to the model of the CNS indicated by subsystem B in Fig. 6 if we know the structures of models of subsystems A, C, and D. However, available models for A, C, and D were validated for different species and under different experimental conditions. Hence an acceptable model for B cannot be obtained as the combination D (A + C) until models for A, C, and D are properly validated using experimental data from the same species. Consequently, only model D is the best we have at present. Effector Mechanisms Neural control of heart rate. It has long been known that the change in heart rate in response to a sympathetic or a parasympathetic stimulation depends on the frequency of stimulation (56). More importantly, because of interaction, combined sympathetic and parasympathetic
Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (163.015.154.053) on September 8, 2018. Copyright © 1992 American Physiological Society. All rights reserved.
H1928
MATHEMATICAL
ml w n
‘carotid
0
Receptor
MODELING Afferent 1 Firing Rate
OF HUMAN
CARDIOVASCULAR
SYSTEM
Sympathetic and Parasympathetic Efferent Impulse Frequencies
. CNS
. Effector Site
Heart Rate
1
E I
Black
Box A I
Black Box D
Exp: Landgmn, 1952 Modal: Taher, 1988
Black Box A
a Ia n
Sympathetic and Parasympathetic Efferent Impulse Frequencies
brotid Heart Rate
Effector Site
-
Heart Rate
*
h
t
Exp~ Mode,: Warner, 1969
9 Parasympathetic or Sympathetic Eff erent Impulse Frequency
Black Box C
Fig. 6. Data available to model control of heart rate by carotid baroreceptors. CNS, central nervous system.
stimulations do not produce a simple algebraic summation of the individual heart rate changes (57, 79). The combined effect observed in cats has been quantified in terms of multiplication of two factors, one for each neural influence (79). Although based on steady-state measurements with sinusoidal stimulation, this way of quantifying the combined effect has been useful in constructing acceptable models of the sinoatrial node that are dynamic (transients included) and are adequate to explain experimental results from dogs as well as human subjects (44, 45, 100).
Neural control of cardiac contractility. It is broadly accepted that a sympathetic stimulation causes an increase of the cardiac contractility. There are experimental observations that show changes in systolic pressure with vagal and parasympathetic efferent stimulations, implicating changes in ventricular contractility due to these stimulations (57, 63,68). Despite these findings, the role of cardiac contractility control in the overall circulation remains unclear and controversial. This is an area in which parameter sensitivity analysis using models can help delineate the control parameter’s importance in CV regulation. As an example, we point out the analyses of Campbell et al. (19) and Hyndman (38) showing that changes in the end-systolic slope has very little effect on the stroke volume compared with changes in the total peripheral resistance. Modification of vessel tone. Neural control of vessel tone
leads to a change in resistance in arteries and a change of compliance in veins. These changes are not the same in different vascular regions, as indicated in Table 2 and alluded to earlier. The mathematical relationships needed to incorporate the above control aspects in models of the CVS, i.e., functions relating peripheral resistance and venous compliance to vasomotor output from the CNS, are generally assumed to be linear in the absence of definitive experimental evidence to the contrary. However, with regard to the control of peripheral resistance, some investigators of CV models have used a combination of a linear and a nonlinear terms (2, 38). The nonlinear term was presumed to represent the relationship between arterial pressure and efferent firing rate and the linear term the smooth muscle response. The nonlinear term was an onoff (bang-bang) rather than a sigmoid function, although the choice did not seem to affect the response of the reflex control. Still, the bang-bang control may be a preferable representation in view of the experimental evidence supporting its validity (75, 101). As for the control of venous compliance, we would emphasize the need for its inclusion in CV models intended for simulation of orthostatic response. Stroke volume is particularly sensitive to changes in venous compliance, as has been shown through sensitivity analyses using models (19, 38). Also, CV models should take into consideration the fact that the efferent neurogenic
Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (163.015.154.053) on September 8, 2018. Copyright © 1992 American Physiological Society. All rights reserved.
MATHEMATICAL
MODELING
OF HUMAN
activity as a result of an inhibition or an activation of receptors is nonuniform across the resistive vessels and perhaps across compliant vessels as well. Mathematical
Modeling
of Overall RCS
It should be emphasized that much of the information we have on the elements of the RCS is not from human subjects. Applying those results to the human RCS before any validation would not be rigorous, since we know that the RCSs of humans and that of animals differ in some important details. For example, the heart rate is controlled by cardiopulmonary receptors in animals, whereas that is not the case with humans (62). These observations underscore the need for numerous assumptions to develop a detailed model of the RCS. One way to circumvent the difficulty is to model directly the relations between hemodynamic inputs of the RCS (arterial pressure and central venous pressure) and its hemodynamic outputs (heart rate, cardiac contractility, arterial resistances, and venous compliances) based on available data from human subjects. Such an approach is purely empirical, but this appears to be the best we can do with our current state of knowledge of the RCS. Table 3 provides a comparison of the control elements and their characteristics included in the published models of the RCS. The empirical nature of these models makes it impossible to pinpoint if they took into account all the important characteristics of the various RCS elements already discussed. However, we could determine if the different controls were limited by threshold and saturation of the carotid baroreceptor input. Also, we could identify any time dependencies of the afferent, CNS, and efferent pathways in the control of heart rate. These are indicated in Table 3. Some models of the CVS do not include any reflex control mechanism at all (4, 5, 11, 19, 25, 27, 31, 35, 39, 51, 65, 66, 71, 74, 91, 96). The authors of these models were more interested in studying the intrinsic ability of the HS in adjusting the pressures and flows without neu-
CARDIOVASCULAR
rogenic regulation. Others have used different approaches to modeling the RCS that are mostly empirical. The most empirical approach was used by Avula and Oestreicher (6),who considered that the feedback initiated by baroreceptors acted directly on the systemic pressure, the relationship between carotid and systemic pressures being a sigmoidal curve (Fig. 7A). This method does not enable a detailed understanding of the reflex control mechanisms, since the peripheral resistance, cardiac contractility, and heart rate variations are all lumped in a single sigmoidal curve. In the approach used by Boyers et al. (14) and Croston and co-workers (23, 24), the CNS is considered to be an integrator that evaluates the different afferent inflows, produces a unique outflow corresponding to the level of CNS stimulation, and sends it to the efferent sites (Fig. 7B). There is no evidence that the CNS acts as an integrating system in this fashion and that a unique function can describe the level of activation of the CNS. The authors did not validate their models with experimental data. Finally, we consider the development of a model of RCS containing separate controls for the heart rate, cardiac contractility, arterial resistances, and venous tone. Such a model has been used by many authors (2,8,38,40, 54,86) and is based on the model originally proposed by Katona et al. (43) for the carotid baroreceptor control of heart rate. Each overall relationship between an input and an output of the reflex system (for example, carotid pressure and heart rate) relies on experimental data correlating the two circulatory variables. Mathematically, the relationship is a combination of serial and parallel linear and nonlinear functions (Fig. 7C). A linear function is a transfer function relating the input and the output by a linear differential equation and is justified by the time dependency of the response of the overall reflex system. A nonlinear function usually represents the thresholds and saturations associated with the limits of the overall reflex control system. It is not always possible to precisely asso-
Table 3. Comparison of the reflex control system portion of overall cardiovascular
Investigators
Orthostatic Stress Studied
‘l$Fzd receptor
Beneken and DeWit Snyder and Rideout Boyers et al.
8 86
Hyndman Croston et al.
38 24
Green and Miller Avula and Oestreicher Leaning et al. Jaron et al.
30
14
6 54 40
Al-Dahan et al. 2 X, Factors included
Blood loss
X
Tilt
X
Tilt Blood loss Tilt LBNP Tilt +G acceleration +Gz acceleration Blood loss +Gz acceleration Blood loss in model.
Threshold
Saturation
Aortic baroreceptor
Cardiopulmonary
X
X X X
system models
Modification of Vessel Tone
Receptors Reference
H1929
SYSTEM
Arterial bed
Venous bed
Heart
Dependence on local circulation
X
X X
Control
of
Rate and Cardiac Contractility
heart rate
Heart rate control time lag
Control of cardiac contractility
X
X
X
X
X
X
X
X
X
X
X X
X X
X X
X
X X
X
X
X
X X X
X X
X
X
X X
X X
X X
X
X
X
X
X
X
Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (163.015.154.053) on September 8, 2018. Copyright © 1992 American Physiological Society. All rights reserved.
H1930
MATHEMATICAL
MODELING
OF HUMAN
CARDIOVASCULAR
SYSTEM
-
a
5 too L a
0
50
100 I50 200 Poi (mm Hg 1
PRESSURE
c--
IN THE
AORTIC
250 ARCH
1
1
“A
1 “C
1
*
1:
-
1 + SUB
e
REGION
A
KC kt
l
4
Fig. 7. Different approaches to modeling reflex control system. A: aortic pressure has a direct effect on systemic arterial pressure. [From Avula and Oestreicher (6).] B: combined autonomic function represents level of activation of reflex control. [Adapted from Croston et al. (24).] C: reflex control (here, control of heart rate) results from a combination of linear and nonlinear elements with time dependencies. B, CNS input; uH, output, u (with subscripts), intermediate variables; K (with subscripts), constants (threshold or gain); 7, time constant; t, time; s, Laplace transform variable. [From Leaning et al. (54).]
I,
*
B
“B
C .
1
.
“D
1 3
J
1
* U
1
1 + ST L
*
“E
1
F,
1
1
REGION
B
J
ciate a physiological entity or attach a physiological meaning to each of the functions in the representation of the reflex control system. Nevertheless an attempt has sometimes been made by authors of CV models. For example, Leaning et al. (54) identified in their model a term corresponding to the baroreceptor; Hyndman (38) considered the peripheral resistance control function to be the serial combination of a linear and a nonlinear term, the linear term corresponding to the response of the smooth muscle and the nonlinear term corresponding to the relationship between carotid pressure and efferent nerve impulse frequency. The last approach discussed above is perhaps the best for modeling the RCS at the present time, despite its shortcomings. CONCLUDING
+z
1 + ST 4
3
REMARKS
This study was aimed at providing an analysis and overview of the available principles and data required for the development of a mathematical model of the overall CVS for simulating its short-term response to orthostatic stresses. An overall model should comprise models of its four major components, namely the arterial tree, the venous return, the cardiac pump, and the RCS. Mathematical modeling of these elements is progressively more difficult mainly because of lack of knowledge and data, especially in the area of the RCS. Accordingly, broader assumptions are required for mathematical descriptions of pump characteristics and CV control signals. For the HS, the data available on humans limit the number of vascular segments in the model if the model is
to remain realistic and meaningful. Improvement can be achieved through better mathematical description of P-V relationships that are in accord with experimental findings. Also, the P-V relationship for each vascular segment should be nonlinear and preferably have the same general shape. It is essential to take into account the time-varying elastance characteristics of the heart chambers to accurately portray the P-V changes therein. Another important factor to consider is the autoregulation of local circulations that have a strong influence on flow adjustments during orthostasis. For the RCS, numerous experimental data show the behavior of RCS to be different in different vascular regions, and this should be taken into account in any global model of the CVS. Also, the transfer function associated with each reflex arc should contain threshold and saturation, representing the limits of the operative range of the reflex. Given the current status of knowledge about the CVS, it is not hard to formulate a model having an adequate number of vascular components to analyze and interpret a specific set of experimental data. The basis for mathematical formulation exists for most of the CV elements, although the requisite details are sketchy in many instances. What is difficult and challenging is the development of an overall CV model of wider applicability. This is because of the nature of the CVS, designed to respond to a wide variety of situations with controls of different sensitivities in different circulatory regions. A comprehensive model must address the issues discussed above, incorporating all known factors of signifi-
Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (163.015.154.053) on September 8, 2018. Copyright © 1992 American Physiological Society. All rights reserved.
MATHEMATICAL
MODELING
OF HUMAN
cance if it is to succeed in simulating the CV responses to orthostatic stimuli of different types. Such a model can be used as a framework for piecing together results from different studies. It will help to assess the influence of the various controlling factors in overall CV regulation and thus sort out the differences in CV responses to different orthostatic stresses. Like established models in other areas of physiology, it could serve as an effective tool to confirm or refute physiological hypotheses on mechanisms of CV regulation, besides providing quantitative information that is difficult or even impossible to obtain experimentally. F. M. Melchior acknowledges the support received from the Direction des Recherches Etudes et Techniques, French Ministry of Defense, and the Space Biomedical Research Institute, NASA Johnson Space Center, Houston, TX, for conducting this study. Address for reprint requests: R. S. Srinivasan, KRUG Life Sciences, 1290 Hercules Dr., Suite 120, Houston, TX 77058. Received 5 February 1990; accepted in final form 21 November 1991.
D. L. Eckberg,
U. J. Johannsen,
and
A. I.
Carotid and cardiopulmonary baroreceptor control of splanchnic and forearm vascular resistance during venoAs pooling in man. J. Physiol. Lond. 286: 173-184, 1979.
2. Al-Dahan, M. I., and L. Finkelstein.
3.
4. 5. 6.
M. S. Leaning,
E. R. Carson,
D. W. Hill,
The validation of complex, unidentifiable models of the cardiovascular system. In: Proc. IFAC Symposium Identification and System Parameter Estimation. New York: Pergamon, 1985, p. 1213-1218. Angell-James, J. E., and M. de B. Daly. Comparison of the reflex vasomotor to separate and combined stimulation of the carotid sinus and aortic arch baroreceptors by pulsatile pressure in the dog. J. Physiol. Lond. 209: 257-293, 1970. Attinger, E. O., and A. Anne. Simulation of the cardiovascular system. Ann. NY Acad. Sci. 128: 810-829, 1966. Avolio, A. P. Multi-branched model of the human arterial system. Med. Biol. Eng. Comput. 18: 709-718, 1980. Avula, X. J. R., and H. L. Oestreicher. Mathematical model of the cardiovascular system under acceleration stress. Aviat.
Space Environ. Med. 49: 279-286, 1978. 7. Beiser, G. D., R. Zelis, S. E. Epstein, D. T. Mason, and E. Braunwald. The role of skin and muscle resistance vessels in reflexes mediated by the baroreceptor system. J. Clin. Invest. 49: 225-231, 1970. 8. Beneken, J., and B. DeWit. A physical approach to hemodynamic aspects of the human cardiovascular system. In: Physical Bases of Circulatory Transport: Regulation and Exchange, edited
by E. Reeve and A. Guyton. Philadelphia, l-45. 9. Bergenwald, Sjostrand.
L.,
U.
Freyschuss,
PA: Saunders, 1967, p. A.
Melcher,
and
Circulatory and respiratory adaptation acute withdrawal and reinfusion of blood. Pfluegers
nonlinear circulation.
T.
in man to Arch.
Physiol. 17. Burkhoff, Maughan,
18. 19.
20.
21. 22.
CO, response during +G,
Physiol.
62: 141-
Influence of breathing quency and tidal volume on cardiac output. Respir. Physiol.
123-133, 14. Boyers,
U.,
1986. D. G.,
and
J.
L.
E.
Farhi.
G. Cuthbertson,
and
J.
A.
and fre66:
Luetscher.
Simulation of the human cardiovascular system: a model with normal responses to change of posture, blood loss, transfusion,
A dynamic skeletal muscle
17: 593-616, 1989. Jr., and J. Schipke.
Assessas a model of aortic impedance. Am. J. Circ. Physiol. 24): H742-H753, 1988.
255 (Heart D., M. W. W. C. Hunter,
Kronenberg, D. and K. Sagawa.
T.
Yue,
W.
L.
IEEE Trans. Biomed. Eng. 32: 289-294, Cowley, A. W., Jr., C. Hinojosa-Laborde, R. Harder, J. H. Lombard, and A.
1985. B. J. Barber,
D.
S. Greene. Short-term of systemic blood flow and cardiac output. News
Sci. 4: 219-225, 1989. R. C., and D. G. Fitzjerrell.
Systems Measurement Control 95: 301-307, 1973. J. Pulsatile mechanical and mathematical model of the cardiovascular system. Med. Biol. Eng. Comput. 20: 601-607, 1982. R. W., J. M. Karemaker, and J. Strackee. Hemo26. DeBoer,
dynamic fluctuations and baroreflex sensitivity in humans: a beat-to-beat model. Am. J. Physiol. 253 (Heart Circ. Physiol. 22): 27.
H680-H689, Dickinson,
1987.
C. J. A digital computer model of the effects of stress upon the heart and venous system. Med. Biol.
gravitational
Eng. 7: 267-275, 1969. Dinnar, U. Analog models of circulation. In: Cardiovascular Fluid Dynamics. Boca Raton, FL: CRC, 1981, p. 139-159. 29. Essandoh, L. K., D. S. Houston, P. M. Vanhoutte, and J. T. Shepherd. Differential effects of lower body negative pressure on forearm and calf blood flow. J. Appl. Physiol. 61: 994-998, 1986. 30. Green, J. F., and N. C. Miller. A model describing the response of the circulatory system to acceleration stress. Ann. Biomed. Eng. 1: 455-467, 1973. 31. Grodins, F. S. Integrative cardiovascular physiology: a mathe28.
32.
matical synthesis of cardiac and blood vessel hemodynamics.
Q.
Rev. Biol. 34: 93-116, 1959. Guerrisi, M., A. Magrini, DeWit, and K. H. Wesseling.
B.
C. Franconi,
J.
J. Settels,
Pulsatile model of the cardiovascular system with neural reflexes. In: Physics in Environmental and Biomedical Research, edited by S. Onori and E. Tabet. Singapore: World Scientific, 1986, p. 459-462.
35.
Ventilation
E. Farhi. acceleration. Respir.
for
Dynamic 25. Dagan,
22: 449-506, 12. Boutellier,
L.
model
1972.
Cardiovascular model for the simulation of exercise, lower body negative pressure, and tilt experiments. Modeling Simulation 5: 471-476, 1974. 24. Croston, R. C., J. A. Rummel, and F. J. Kay. Computer model of cardiovascular control system responses to exercise. J.
33.
and
18: 197-205, N. Westerhof.
and
Quantitative comparison of canine right and left ventricular isovolumic pressure waves. Am. J. Physiol. 253 (Heart Circ. Physiol. 22): H475-H479, 1987. Burton, A. C. Relation of structure to function of the tissues of the wall of blood vessels. Physiol. Rev. 34: 619-642, 1954. Campbell, K., M. Zeglen, T. Kagehiro, and H. Rigas. A pulsatile cardiovascular model for teaching heart-blood vessel interaction. Physiologist 25: 155-162, 1982. Chao, J. C., and N. H. C. Hwang. A review of the bases for the hydraulic transmission line equations as applied to circulatory systems. J. Biomech. 5: 129-134, 1972. Coleman, T. Mathematical analysis of cardiovascular function.
Physiol. 23. Croston,
307-318, 1975. Beyar, R., M. J. Hausknecht, H. R. Halperin, F. C. P. Yin, and M. L. Weisfeldt. Interaction between cardiac chambers and thoracic pressure in intact circulation. Am. J. Physiol. 253 (Heart Circ. Physiol. 22): H1240-H1252, 1987. 11. Beyar, R., Y. Kishon, S. Sideman, and U. Dinnar. Com-
1984. U., R. Arieli,
parameter
Ann. Biomed. Eng. D., J. Alexander,
ment of Windkessel
355:
puter studies of systemic and regional blood flow mechanisms during cardiopulmonary resuscitation. Med. Biol. Eng. Comput.
P. Sipkema,
lumped
16. Burkhoff,
10.
151, 1985. 13. Boutellier,
R.,
autoregulation F. M.,
H1931
SYSTEM
and autonomic blockade. Simulation 15. Braakman,
REFERENCES 1. Abboud, Mark.
CARDIOVASCULAR
Guyton, E. Hall.
A. C., T. G. Coleman,
R. D. Manning,
Jr.,
and
J.
Some problems and solutions for modelling overall cardiovascular regulation. Math. Biosci. 72: 141-155, 1984. 34. Hardy, H. H., and R. E. Collins. On the pressure-volume relationship in circulatory elements. Med. Biol. Eng. Comput. 20: 565-570, Hardy,
computer
1982. H. H.,
R.
E. Collins,
and
R. E.
model of the human circulatory
Eng. Comput. 20: 550-564, 1982. 36. Henry, J. P., 0. H. Gauer, S. S. Kety,
Factors maintaining stress. J. Clin. Invest. 63: 1003-1007,
and
during
K.
Kramer.
gravitational
30: 292-300, 1951. H., and J. E. Greenleaf.
Continuous of blood volume changes in humans. J. Appl. Physiol.
37. Hinghofer-Szalkay,
monitoring
cerebral circulation
Calvert. A digital system. Med. Biol.
1987.
Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (163.015.154.053) on September 8, 2018. Copyright © 1992 American Physiological Society. All rights reserved.
H1932
MATHEMATICAL
38. Hyndman,
A digital simulation
B. W.
cular system. Infor.
J. 10: 8-35, Westerhof,
MODELING
OF HUMAN
of the human cardiovas-
41.
Johnson, Eisman.
J. M.,
L. B. Rowell,
M. Niederberger,
and
M. M.
Human splanchnic and forearm vasoconstrictor responses to reductions of right atria1 and aortic pressures. Circ. 42
Res. 34: 515-524, 1974. Kass, D. A., M. Midei, W. Graves, L. Maughan. Use of a conductance
J. A. Brinker,
and
W.
(volume) catheter and transient inferior vena caval occlusion for rapid determination of Catheterization pressure-volume relationships in man. Cardiovasc. Katona,
Diagn. 15: 192-202, P. G., G. 0. Barnett,
1988. and
W. D. Jackson. Computer simulation of the blood pressure control of the heart period. In: Baroreceptors and Hypertension, edited by P. Kezdi. Oxford, UK: Pergamon, 1967, p. 191-199. 44. Katona, P. G., P. J. Martin, and F.Jih. Neural control of heart rate: a conciliation of models. IEEE Trans. Biomed. Eng. 43
45.
23: 164-166, 1976. Katona, P. G., M.
McLean,
D.
H.
Dighton,
and
A.
Guz.
Sympathetic and parasympathetic cardiac control in athletes and nonathletes at rest. J. Appl. Physiol. 52: 1652-1657, 1982. 46. Katz, A. I., Y. Chen, and A. H. Moreno. Flow through a collapsible tube. Experimental analysis and mathematical model. Biophys. Kenner,
J. 9: 1261-1279, 1969. T. Physical and mathematical systems. In: Quantitative Cardiovascular
modeling in cardiovascular Studies, edited by N. C. H. Hwang, D. R. Gross, and D. J. Patel. Baltimore, MD: University Park, 1979, p. 41-109. A. L. Pressure-volume relation for cylindrical tubes with 48. King, elastomeric walls: the human aorta. J. Appl. Physics 17: 501-505, 47.
49.
1946. Korner,
P.
I.,
M.
J.
West,
J.
Shaw,
and
J.
B.
Ughter.
Steady state properties of the baroreceptor-heart rate reflex in essential hypertension in man. CZin. Exp. Phurmacol. Physiol. 1: 50.
51.
65-76, Krogh,
1974. A.
The regulation of the supply of blood to the right heart (with a description of new circulation model). Skand. Arch. Physiol. LaCourse,
27: 227-248, 1912. J. R., G. Mohankrishnan,
and
K.
52.
Langewouters, Goedhard.
G.
J.,
K.
H.
Wesseling,
and
W.
J.
54.
Biomech. Laughlin,
17: 425-435, 1984. M. H. The effects of +G, on the coronary circulation: a review. Aviat. Space Environ. Med. 57: 5-16, 1986. Leaning, M. S., H. E. Pullen, E. R. Carson, M. Al-Dahan, N. Rajkumar, and L. Finkelstein. Modelling a complex bio-
61.
Control 5: 87-98, 1983. Leaning, M. S., H. Finkelstein. Modelling
cardiovascular
E.
Pullen,
E.
R.
Carson,
and
L.
a complex biological system: the human system. 1. Methodology and model description.
Trans. Inst. Measurement Control 5: 71-86, 1983. 56. Levy, M. N., and P. J. Martin. Neural control of Handbook of Physiology. The Cardiovascular System.
the heart. In: The Heart.
Bethesda, MD: Am. Physiol. Sot., 1983, sect. 2, vol. I, chapt. 16, p. 581-620. 57. Levy, M. N., and H. Zieske. Autonomic control of cardiac pace-maker activity and atrioventricular transmission. J. Appl. 58.
Physiol. 27: 467-470, 1969. Lindberg, E. F., and E. H. Wood. of Man in Space, edited by J. H. U. 1963, p. 61-111.
S. C., H. B. Atabek,
198-212, Loeppky,
W. G. Letzing,
and
D. J. Patel. Res. 33:
analysis of aortic flow in living dogs. Circ. 1973. J. A.,
M.
D.
and U. C. Luft. Blood responses to lower body negative
Venters,
Med. 49: 1297-1307, 1978. Mancia, G., and A. L. Mark. Arterial baroreflexes in humans. In: Handbook of Physiology. The Cardiovascular System. Peripheral Circulation and Organ Blood Flow. Bethesda, MD: Am.
Physiol. Sot., 1983, sect. 2, vol. III, pt. 2, chapt. 20, p. 755-793. Mark, A. L., and G. Mancia. Cardiopulmonary baroreflexes in humans. In: Handbook of Physiology. The Cardiovascular System. Peripheral Circulation and Organ Blood Flow. Bethesda, MD: Am. Physiol. Sot., 1983, sect. 2, vol. III, pt. 2, chapt. 21, p. 795-813. 63. Martin, P. J. Analysis of myocardial performance in the blood pressure control system. Automedica Land. 6: 175-191, 1970. 64. Maughan, W. L., and D. A. Kass. The use of the pressurevolume diagram for measuring ventricular pump function. 62.
Automedica 65. McIlroy,
Lond. 11: 317-342, 1988. M. B., and R. C. Targett.
A model of the arterial bed ventricular-systemic coupling. Am. J. Physiol. 254 (Heart Circ. Physiol. 23): H609-H616, 1988. 66. Meador, S. A. Computer simulation of cardiopulmonary resuscitation: computer simulation of a simple electrical model of the circulation. Resuscitation 13: 145-157, 1986. 67. Milnor, W. R. Hemodynamics. Baltimore, MD: William & Wilkins, 1982. 68. Mitchell, J. H., R. J. Linden, and S. J. Sarnoff. Influence of cardiac sympathetic and vagal stimulation on the relation between left ventricular diastolic pressure and myocardial segment length. Circ. Res. 8: 1100-l 106, 1960. showing
69.
Moore, Jaron.
T. W.,
J. Foley,
B. R. S. Reddy,
F. Kepics,
and
D.
Studies of cardiovascular responses during acceleration stress (Abstract). Proc. Annu. Int. Conf. IEEE Eng. Med. Biol. sot. 1988, vol. 10, pt. 1, p. 59. 70. Noordergraaf, A., and J. Melbin. The development of recognition of component significance in closed-loop cardiovascular control. Ann. Biomed. Eng. 8: 391-404, 1980. 71.
Noordergraaf, A., P. D. Verdouw, A. G. W. van Brummelen, and F. W. Wiegel. Analog of the arterial bed. In: Puls&e Blood Flow, edited by E. 0. Attinger. New York: McGraw-
Hill, 1964, p. 373-387. J. A., and P. B. Raven. Reduction in central venous pressure improves carotid baroreflex responses in conscious men. Am. J. Physiol. 257 (Heart Circ. Physiol. 26): H1389H1395, 1989.
72. Pawelczyk,
73. Pickering,
74.
75.
76.
logical system: the human cardiovascular system. 2. Model validation, reduction and development. Trans. Inst. Measurement 55.
SYSTEM
volume and cardiorespiratory pressure. Aviat. Space Environ.
A.
The static elastic properties of 45 human thoracic and 20 abdominal aortas and the parameters of a new model. J. 53.
60.
Sivaprasad.
Simulations of arterial pressure pulses using a transmission model. J. Biomech. 19: 771-780, 1986.
Ling,
Nonlinear
1972. and A. Noordergraaf.
Jager, G. N., N. Oscillatory flow impedance in electrical analog of arterial system: representation of sleeve effect and non-Newtonian properties of blood. Circ. Res. 16: 121-133, 1965. 40. Jaron, D., T. W. Moore, and C.-L. Chu. A cardiovascular model for studying impairment of cerebral function during +G, stress. Aviat. Space Environ. Med. 55: 24-31, 1984. 39.
59.
CARDIOVASCULAR
77.
W. D.,
P. N.
Nikiforuk,
and
J. E. Merriman.
Analogue computer model of the human cardiovascular control system. Med. Biol. Eng. 7: 401-410, 1969. Porenta, G., D. F. Young, and T. R. Rogge. A finite element model of blood flow in arteries including taper, branches, and obstructions. J. Biomech. 108: 161-167, 1986. Rea, R. F., and D. L. Eckberg. Carotid baroreceptor-muscle sympathetic relation in humans. Am. J. Physiol. 253 (Regulatory Integrative Comp. Physiol. 22): R929-R934, 1987. Rideout, V. C. Pressure-flow modeling of the cardiovascular system. In: Mathematical and Computer Modeling of Physiological Systems. Englewood Cliffs, NJ: Prentice-Hall, 1991, p. 68-130. Rideout, V. C., and D. E. Dick. Difference-differential equations for fluid flow in distensible tubes. IEEE Trans. Biomed.
Eng. 14: 171-177, 78. Rideout, V. C.,
1967. and J. Katra. Computer simulation study of the pulmonary circulation. Simulation 12: 239-245, 1969. 79. Rosenblueth, A., and F. A. Simeone. The interrelations of vagal and accelerator effects on the cardiac rate. Am. J. Physiol. 110: 42-45, 1934. 80. Rowell, L. B. Adjustments to upright posture and blood loss. In: Human Circulation Regulation During Physical Stress. Oxford,
UK: Oxford Univ. Press, 1986, p. 137-173. Acceleration. In: Physiology Brown. New York: Academic,
81.
Samueloff,
S.
F.,
N.
L.
Browse,
and
J.
T.
Shepherd.
Response of capacity vessels in human limbs to head-up tilt and suction on lower body. J. Appl. Physiol. 21: 47-54, 1966.
Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (163.015.154.053) on September 8, 2018. Copyright © 1992 American Physiological Society. All rights reserved.
MATHEMATICAL 82.
Santamore,
W.
P.,
and
quences of ventricular Am. J. Physiol. Scher, A. M.
D.
OF HUMAN
Hemodynamic conseas assessed by model analysis.
Burkhoff.
interaction
260 (Heart
MODELING
Circ.
Physiol.
29):
H146-H157,
85.
1978. Shroff,
S. G., J. S. Janicki,
quantification
and
of left ventricular
Evidence and systolic resistance. Am. J.
K. T. Weber.
Physiol. 86. Snyder,
87.
88.
249 (Heart Circ. Physiol. 18): H358-H370, 1985. M. F., and V. C. Rideout. Computer simulation studies of the venous circulation. IEEE Trans. Biomed. Eng. 16: 325-334, 1969. Snyder, M. F., V. C. Rideout, and R. J. Hillestad. Computer modeling of the human systemic arterial tree. J. Biomech. 1: 341-353, 1968. Srinivasan, R., and J. I. Leonard. Simulation of cardiovas-
cular response to acceleration exposure. Proc. Summer Simulation p. 598-603. 89. 90.
91.
stress following Conf.
Vancouver
weightless 1983, vol. 1,
and H. B. Nudelman. Modeling the carotid sinus baroreceptor. Biophys. J. 12: 1171-1181, 1972. Streeter, V. L., W. F. Keitzer, and F. Bohr. Pulsatile pressure and flow through distensible vessels. Circ. Res. 13: 3-20, 1963. Sud, V. K., and G. S. Sekhon. Analysis of blood flow through Srinivasan,
R.,
a model of the human acceleration. J. Biomech.
arterial
system under periodic
body
19: 929-941, 1986. Sagawa. Instantaneous
Suga, H., and K. pressure-volume relationships and their ratio in the excised, supported canine left ventricle. Circ. Res. 35: 117-125, 1974. 93. Suga, H., and K. Sagawa. Models of cardiac contraction. 92.
94.
Simulation 46: 181-184, Suga, H., K. Sagawa,
instantaneous 256-263,
1980.
1976. and
L. Demar. Determinants pressure in canine left ventricle. Circ. Res.
of 46:
H1933
SYSTEM
95. Taher,
1991.
The gain of carotid sinus reflex in awake animals: importance of rate of change of pressure at the receptor. In: Baroreceptors and Hypertension, edited by P. Kezdi. Oxford, UK: Pergamon, 1967, p. 89-95. 84. Schinzer, W., J. Klatt, H. Baeke, and H. Rieckert. Vergleigh von Szintigraphischen und Plethysmographischen Messungen zur Bestimmung des Kapillaren Filtrationskoeffizienten in der Menschlichen Extremitat. Basic Res. Cardiol. 73: 77-84, 83.
CARDIOVASCULAR
96. g7 l
“*
M. F., A. B. P. Cecchini, M. A. Allen, S. R. Gobran, R. C. Gorman, B. L. Guthrie, K. A. Lingenfelter, S. Y. Rabbany, P. M. Rolchigo, J. Melbin, and A. Noordergraaf. Baroreceptor responses derived from a fundamental concept. Ann. Biomed. Eng. 16: 429-443, 1988. Taylor, M. G. Wave travel in arteries and the design of the cardiovascular system. In: P&utile Blood Flow, edited by E. 0. Attinger. New York: McGraw-Hill, 1964, p. 343-372. Tozeren, A., and S. Chien. Elastic properties of arteries and their influence on the cardiovascular system. Trans. ASME 106: 182-185, 1984. Victor, R. G., and A. L. Mark. Interaction of cardiopulmon-
ary and carotid baroreflex control of vascular resistance humans. J. Clin. Invest. 76: 1592-1598, 1985.
99. Wang,
J.,
B.
Tie,
W.
Incremental
Semmlow.
Welkowitz,
J.
Biol. Eng. Comput. 27: 416-422, H. R., and R. 0. Russel. Effect
Warner,
thetic and vagal stimulation 101.
24: 567-573, Weidinger, Arch. White,
J.
1989.
of combined sympaon heart rate in the dog. Circ. Res.
1969. H.
Herznerven
and
network analogue model of the coronary
artery. Med. 100.
Kostis,
in
Aktionsstrome und deren Bedeutung
in Zentrifugalen fur den Kreislauf.
Vagalen Pluegers
276: 262-279, 1962. R. J., R. C. Croston,
and D. J. Fitzjerrell. Cardiovascular modelling: simulating the human response to exercise, lower body negative pressure, zero gravity and clinical conditions. In: Advances in Cardiovascular Physics, edited by D. N. Ghista. Basel: Karger, 1983, vol. 5, pt. I, p. 195-229. 103. White, R. J., D. J. Fitzjerrell, and R. C. Croston. Fundamentals of lumped compartmental modelling of the cardiovascular system. In: Advances in Cardiovascular Physics, edited by D. N. Ghista. Basel: Karger, 1983, vol. 5, pt. I, p. 162-184. 102.
104.
Womersley, sion and
J. R. An Elastic Tube Theory of Pulse TransmisOscillatory Flow in Mammalian Arteries. Dayton, OH:
Wright Air Development 56-614).
Center,
M., and J.-P. Marc-Vergnes. 105. Zagzoule, ical model of the cerebral circulation 1015-1022,
1957 (WADC
Report
TR
A global mathematin man. J. Biomech. 19:
1986.
Downloaded from www.physiology.org/journal/ajpheart by ${individualUser.givenNames} ${individualUser.surname} (163.015.154.053) on September 8, 2018. Copyright © 1992 American Physiological Society. All rights reserved.