The
objective of this study is to develop a model of the
cardiovascular system capable of simulating the short-term (
5 min) transient and steady-state hemodynamic responses to head-up tilt and
lower body negative pressure. The model consists of a closed-loop lumped-parameter representation of the circulation connected to set-point models of the arterial and cardiopulmonary baroreflexes. Model parameters are largely based on literature values. Model verification was performed by comparing the simulation output under
baseline conditions and at different levels of orthostatic stress to
sets of population-averaged hemodynamic data reported in the
literature. On the basis of experimental evidence, we adjusted some
model parameters to simulate experimental data. Orthostatic stress
simulations are not statistically different from experimental data
(two-sided test of significance with Bonferroni adjustment for multiple
comparisons). Transient response characteristics of heart rate to tilt
also compare well with reported data. A case study is presented on how
the model is intended to be used in the future to investigate the
effects of postspaceflight orthostatic intolerance.
mathematical model; simulation; head-up tilt; lower body negative
pressure; orthostatic intolerance
 |
INTRODUCTION |
A MAJOR CARDIOVASCULAR
PROBLEM in the present life-science space program is orthostatic
intolerance (OI) seen in astronauts on their return to the normal
gravitational environment (16). Many hypotheses concerning
the mechanisms leading to OI have been the focus of in-flight and
ground-based experimental studies in the past, and some are presently
under investigation in both human and animal studies (see Table
1). Despite considerable efforts, there
remains a lack of universal consensus about the fundamental etiology of
postflight OI. To quantitatively supplement some of these studies, we
have developed a computational model capable of simulating the
short-term (5 min) response of the cardiovascular system to
gravitational challenge in normal and microgravity-adapted individuals.
The response of the cardiovascular system to both orthostatic stress
and the microgravity environment has been the focus of several
mathematical models in the past (11, 18, 42, 50, 51, 78,
79). Modeling efforts ranged from explaining observations seen
during spaceflight (78) to simulating the physiological response of ground-based experiments such as lower body negative pressure (LBNP) and head-up tilt (HUT) (11, 18, 42, 50, 51, 72,
80). Investigations of potential countermeasures to the adverse
effects of re-entry into the Earth's gravitational environment after
exposure to weightlessness have also been supplemented by computer
models (64, 65, 71). Melchior et al. (49) analyzed a number of the models previously used and summarized some of
the general requirements thought to be important in simulating the
short-term response to orthostatic stress.
Several mathematical models dedicated particularly to simulating the
systems-level response of the cardiovascular system to orthostatic
stress have appeared in the literature over the past three decades
(11, 18, 42, 50, 51, 72). In an early effort to simulate
the hemodynamic response to changes in posture, Boyers and co-workers
(11) implemented a seven-compartment steady-state model of
the cardiovascular system connected to models of the arterial
baroreflex and the cardiopulmonary baroreflex. In a later study,
Croston and Fitzjerrell (18) devised a 28-compartment lumped-parameter model that is very similar in design to the one presented in this paper: arterial, venous, and cardiac compartments are
modeled in terms of coupled ordinary differential equations that
describe the dynamics of the system. Their control model representation
is very elaborate, with total metabolites and oxygen debt being input
parameters in addition to arterial pressure. Melchior and co-workers
(50) simulated the hemodynamic response to LBNP by
employing a fourstep open-loop steady-state model in which
changes in venous blood distribution affect mean arterial pressure,
stroke volume, heart rate, and peripheral resistance. In a subsequent
extension of their work (51), the authors refined their
modeling strategy by including a pelvic venous compartment and a
Windkessel model of the systemic circulation. Sud and co-workers (72) devised an elaborate finite-element model of the
uncontrolled circulation to investigate the effects of LBNP on regional
blood flow.
The primary objective of the work presented in this paper is to develop
and test a general modular model of cardiovascular function that
contains the essential features associated with the effects of gravity.
In particular, we present a single cardiovascular model capable of
simulating the steady-state and transient response to two orthostatic
stress tests, namely HUT and LBNP, and compare the simulations to
population-averaged hemodynamic data.
By implementing HUT and LBNP interventions and comparing the
simulation results to experimental observations of the general population, we validate our simulations against a larger pool of
experimental studies than was possible in previous modeling studies. In
a case study, we show how the model may be used to demonstrate the
impact of changes in individual parameters on the heart rate dynamics
during orthostatic stress. By comparing these simulations to individual
subject data from astronauts before and after exposure to microgravity,
we can evaluate whether or not a particular hypothesis regarding the
mechanisms underlying postspaceflight OI can account for the change
seen in the data. The model thus provides a framework with which to
interpret experimental observations and to evaluate alternative
physiological hypotheses of the cause of OI.
Despite the fact that we tried as much as possible to give a detailed
account of the origin of the parameter values, we acknowledge that
uncertainties in the parameter assignments continue to persist.
Much like previously reported models (21, 79), our model
is based on a closed-loop lumped-parameter hemodynamic model with
regional blood flow to major peripheral circulatory branches. Features
such as nonlinear regional venous compliances and venous valves
have been implemented in accordance with previous work (79). Dynamic change in intravascular volume during
orthostatic stress is, however, an added feature of our model that has
not previously been implemented for short-term studies and is of
critical importance in simulating tilt and LBNP maneuvers
(49). Blood pressure homeostasis is maintained in our
model with the aid of the two major neural reflex control loops: the
arterial baroreflex and the cardiopulmonary reflex. Unlike earlier
models (18, 50, 51, 79), we emphasize the separation of
sympathetic and parasympathetic reflex limbs. Furthermore, the four
effector mechanisms (heart rate, cardiac contractility, regional
peripheral resistance, and regional venous tone) have gain values that
can be specified independently. Because autonomic dysfunction is
believed to be at least in part responsible for OI (53),
the ability to modify the control of the heart and various vascular
beds independently makes this model a powerful tool to investigate
present hypotheses concerning the mechanisms underlying OI.
| |
THE HEMODYNAMIC MODEL |
Architecture
We have extended a previously published (21)
closed-loop lumped-parameter model of the cardiovascular system to
facilitate anatomically specific representation of gravitational and
LBNP stresses and of regionally specific cardiovascular compensatory mechanisms.
The hemodynamic model is mathematically formulated in terms of an
electric analog model in which inertial effects are neglected. All
resistors and most capacitors are assumed to be linear. These model
assumptions lead to governing differential equations that are of first
order. Figure 1 shows the nth
compartment.
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Fig. 1.
Single-compartment circuit representation. P, pressure;
R, resistance; C, compliance; q1, q2, and
q3, blood flow rates as indicated; n  1, n, n + 1, compartment indexes;
Pbias, external pressure.
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The flow rates (q) across the resistors (R) and capacitor (C) expressed
in terms of the pressures (P) are given by
See Fig. 1 legend for definition of subscripts.
Applying conservation of mass to the node at
Pn yields q1 = q2 + q3. Combining these expressions for
the flow rates leads to
The entire model is thus described mathematically by 12 such
first-order differential equations. An adaptive step-size fourth-order Runge-Kutta integration routine is used to integrate the system of
differential equations numerically. Integration steps range from
6.1 × 10
4 to 0.01 s with a mean step size of
5.6 × 103 s. To initiate the numerical integration
routine, a set of initial conditions for the state variables
(pressures) needs to be supplied. We estimate the initial pressures by
a linear algebraic solution of a steady-state version (i.e., all
pressures are assumed constant) of the hemodynamic system.
The entire model is shown in Fig. 2. The
peripheral circulation is divided into upper body, renal, splanchnic,
and lower extremity sections; the intrathoracic superior and inferior
vena cavae and extrathoracic vena cava are separately identified. The
model thus consists of 12 compartments, each of which is represented by
a linear resistance (R) and a capacitance (C) that can be either linear, nonlinear, or time varying.
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Fig. 2.
Circuit diagram of the hemodynamic system. lv, Left
ventricle; a, arterial; up, upper body; kid, kidney; sp, splanchnic;
ll, lower limbs; ab, abdominal vena cava; inf, inferior vena cava; sup,
superior vena cava; rv, right ventricle; p, pulmonary; pa, pulmonary
artery; pv, pulmonary vein; ro, right ventricular outflow; lo, left
ventricular outflow; th, thoracic; bias, as defined in Fig. 1.
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The pumping action of the heart is realized by varying the right and
left ventricular elastances according to a predefined function of time
[Er(t) and El(t),
respectively]
Edias and Esys represent the
end-diastolic and end-systolic elastance values, respectively,
T(i) denotes the cardiac cycle length of the
ith beat, and t denotes time measured with
respect to the onset of ventricular contraction. The systolic time
interval, TS, is determined by the Bazett
formula (2),
TS(n) 
0.3 · 
, which links the systolic time interval of the present beat,
TS(n), to the duration of the cardiac
cycle that preceded it, T(n 1). TS is determined by the duration of the previous
cardiac cycle, but the length of the present cardiac cycle,
T(n), is determined by means of an integral pulse
frequency modulation model of the sinoatrial (SA) node (see, for
example, Ref. 36). The efferent (instantaneous) heart rate
signal (Eq. 4) is considered a measure of autonomic nervous
input to the SA node and is integrated to yield the cumulative input to
the SA node over time. A new beat will be initiated once the integral
reaches a predefined threshold level as long as the absolute refractory
period (1.2 × TS) has passed since the
onset of the previous ventricular contraction.
Figure 3 compares the computed left
ventricular elastance (solid line) to experimental data from humans
[adapted from Senzaki and co-workers (62)]. The
ventricular compliances are computed according to C(t) = 1/E(t).
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Fig. 3.
Time-varying elastance, E(t), of the left
ventricle during 1 cardiac cycle (solid line) compared with
experimental data ( , means ± SD; adapted from
Ref. 62). TS, systolic time
interval, TD, diastolic time interval.
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Atria are not represented; their function is, however, partially
absorbed into the function of adjacent compartments. Diodes represent
valves that ensure unidirectional flow through the ventricles and parts
of the venous system. A time-varying pressure source, Pth,
simulates changing transmural pressure across the intrathoracic compartments. We specified the magnitudes of the intrathoracic pressure
variations according to Ref. 31, which states
that during normal respiration intrathoracic pressure varies between approximately 4 and 6 mmHg. A simple sinusoidal variation between these two values was assumed at a respiratory frequency of 12 breaths/min.
Pressure sources at the abdominal venous compartment
(Pbias-3), the splanchnic (Pbias-2), and the
leg compartment (Pbias-1) simulate changes in venous
transmural pressure due to postural changes or external LBNP.
During high levels of LBNP or during quiet standing, the venous
transmural pressures in parts of the dependent vasculature can reach
levels at which the nonlinear nature of the venous pressure-volume relationship becomes important (30, 55). In accordance
with experimental observations in the legs (46), we model
the functional form of the pressure-volume relationships of the venous
compartments of the legs, the splanchnic circulation, and the abdominal
venous compartment according to Ref. 51
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(1)
|
where 
V represents the change in compartment volume due to a
change in transmural pressure, Ptrans.
Vmax is the maximal change in compartment volume, and
C0 represents the compartment compliance at baseline
transmural pressure (i.e., for Ptrans = 0). Figure
4 shows the pressure volume relationship
for the nonlinear venous compliances.
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Fig. 4.
Pressure-volume relations for changes in stressed volume
of the abdominal (Vab), leg (Vll), and
splanchnic (Vsp) venous compartments.
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Although the interstitial fluid compartment is not explicitly modeled,
total blood volume is modified as a function of time to simulate fluid
sequestration into the interstitium during orthostatic stress.
Parameter Assignments
Where possible, parameter values for the hemodynamic model are
based on literature values as indicated in Tables
2 and 3. However, certain values such as the regional systemic resistances had
to be estimated.
Resistances.
On the basis of estimates of the distribution of cardiac output (CO) to
the four circulatory branches [upper body (23% of CO), kidney (22%
of CO), splanchnic circulation (30% of CO), and lower body (25% of
CO)], we modeled arteriolar resistance values to produce a total
peripheral resistance of 1.0 peripheral resistance units (PRU = mmHg s/ml). This method generates a splanchnic arteriolar resistance of
3.0 PRU, which is similar to values previously used in cardiovascular
models (75). The resistance values on the venous side of
the circulation are largely taken from Beneken and DeWitt
(3) (see Table 2). Although the pulmonary vascular resistance is known to be nonlinear and dependent on CO, it was approximated by a constant value (52). The outflow
resistances of the right and left ventricles (Rro and
Rlo) have previously been estimated to be 0.003 and 0.01 PRU, respectively (23). The left ventricular inflow
resistance is a major determinant of left ventricular filling time
during diastole. Consistent with previous estimates (23),
we chose the inflow resistance to the left ventricle to be 0.01 PRU,
which generates a time constant for ventricular filling of 0.05 s.
This value agrees well with observations of filling times of the left
ventricle between 0.04 and 0.08 s (13).
Capacitances.
In addition to being a function of transmural pressure
(57), aortic capacitance changes dramatically with age
(33). For our purposes, however, it is sufficient to
choose the lumped arterial compliance such that the time constant for
blood flow from the arterial to the venous side of the systemic
circulation found experimentally, 1.9 s, is reproduced
(60). This is accomplished by using a lumped arterial
compliance of 2.0 ml/mmHg. The venous capacitance value of the upper
body compartment is taken from Beneken and DeWitt (3). The
combined capacitance of the splanchnic and kidney compartments is taken
to be 70 ml/mmHg with 55 ml/mmHg ascribed to the intestines, liver, and
the spleen and 15 ml/mmHg to the kidneys. The leg compartment is
assigned a combined venous capacitance of 19 ml/mmHg (34).
We modeled the pressure-volume relationships for the abdominal,
inferior thoracic, and superior thoracic venae cavae on previous models
of the cardiovascular system (19). The lower limb,
splanchnic, and abdominal venous compartments all exhibit nonlinear
pressure-volume relations according to Eq. 1. The
compliances discussed here are the compliances at normal supine
transmural pressures (C0 in Eq. 1). During
orthostatic stress, transmural pressures increase in the
dependent vascular beds, and the respective compliances
are computed by differentiating Eq. 1 with respect to
Ptrans. The pulmonary compliances are taken from Beneken
and DeWitt (3).
Both ventricles are characterized by time-dependent compliances that
vary between a minimum (end-systolic) and a maximum (diastolic) value
according to a predefined functional form (see Architecture, above). Previous estimates of maximum diastolic capacitance of 10 ml/mmHg for the left ventricle (20) seem compatible
with experimental observations (43). The right ventricular
maximum diastolic capacitance (C
) is
approximately twice the left ventricular value (24). Left
ventricular end-systolic capacitance (C
) was set to
0.5 mmHg/ml, consistent with recent estimates in humans
(62). We choose a value of 1.2 ml/mmHg for right
ventricular end-systolic capacitance (C
) within the
range of experimental values, 0.35 to 1.6 ml/mmHg (24).
Volume.
Total blood volume is reported to be in the range of 75-80 ml/kg
body wt for normal male subjects (28, 66). We set total blood volume to 5,700 ml, corresponding to a 71- to 75-kg normal male
subject with a body surface area of 1.7-2.1 m2.
Blood volume is reported to be distributed within the circulation
approximately as 15% in the aorta, systemic arteries, and arterioles;
69% in the capillaries, venules, and systemic veins; 9% in the
pulmonary circulation; and 7% in the heart (31). With an
unstressed arterial volume assumed to be 715 ml, the systemic arterial
tree contains roughly 900 ml when stressed to a mean arterial pressure
of 92 mmHg (715 ml + 2 ml/mmHg × 92 mmHg), which is ~15%
of total blood volume. With the assumption of right and left
ventricular filling pressures of 5 and 10 mmHg, respectively, and 50-ml
unstressed volumes for each ventricle, the total cardiac volume at end
diastole is 300 ml. This is only ~5% of total blood volume, which is
partly due to the lack of atria in our hemodynamic model. Furthermore,
there is considerable variation in cardiopulmonary blood volume,
ranging from 301 to 546 ml/m2 of body surface area, as
reported by Levinson et al. (45). We adopted Davis'
pulmonary unstressed volumes of 90 and 490 ml for the pulmonary
arteries and veins, respectively (20). This distribution
of arterial and cardiopulmonary blood volumes allocates ~3,600 ml, or
63% of total blood volume, to the capillaries and systemic venous circulation.
Previously published estimates were used for unstressed volumes of the
upper body and lower limbs (3), and estimates for the
splanchnic venous and central venous compartments were guided by
previously published models (75).
The volume and capacitance assignments are summarized in Table 3.
| |
THE REFLEX MODEL |
Architecture
We have adopted and extended a previously reported regulatory
set-point model of the arterial baroreflex that aims at maintaining mean arterial blood pressure constant by dynamically adjusting heart
rate, peripheral resistance, venous zero-pressure filling volume, and
right and left end-systolic cardiac capacitances (20, 22).
In addition to this representation of the arterial baroreflex, we have
implemented a similar reflex loop to represent the cardiopulmonary reflex, which, at the moment, only affects venous zero-pressure filling
volume and systemic arteriolar resistance (see Fig.
5). Briefly, predefined set-point
pressures are subtracted from locally sensed blood pressures to
generate error functions that are relayed to the autonomic nervous
system, where the error signals are rescaled. These error signals
subsequently dictate the efferent activity of the reflex model such
that the error signals approach zero over the next computational steps.
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Fig. 5.
Diagrammatic representation of the reflex model. CS,
carotid sinus; CP, cardiopulmonary; ANS, autonomic nervous system; SA,
sinoatrial;  , summation. See text for definitions of
pressure (P) symbols. The interaction between cardiopulmonary and
arterial baroreflex (indicated by dotted line) only affects the maximum
sensitivity of the arterial heart rate baroreflex.
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The input variables to the control system are mean arterial pressure
(
A) and mean central venous pressure
(CV) for the arterial baroreflex and the
cardiopulmonary reflex as substitutes for carotid sinus
(CS) and right atrial pressure
(RA), respectively. The respective error signals are
generated by subtracting predefined set-point values
(
and 
) from
these input variables and rescaling by an arctangent to generate effective blood pressure deviations (P
and
P
; Ref. 22)
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(2)
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(3)
|
These mappings are limited to approximately ±28 mmHg for the
arterial baroreflex and approximately ±8 mmHg for the cardiopulmonary reflex and are designed to exhibit the sigmoidicity common to stretch receptors.
To account for differences in timing of the reflex
responses, the respective effective blood pressure deviations are
weighted by impulse response functions that are characteristic of a
parasympathetic [p(t)] or sympathetic [s(t)]
response (4) (see Fig. 6).
The model thus provides for a rapid (~0.5 1 s)
parasympathetic reflex response and a slower acting
(~2-30 s) sympathetic reflex response. A 30-s running history of
the effective pressures is used to compute instantaneous effector
values by convolution with the appropriate effector specific
impulse responses as the following example of R-R interval (I) feedback
shows
![I(<IT>t</IT>)<IT>=</IT>I<SUB>0</SUB><IT>+∫</IT><SUP><IT>k=</IT>30</SUP><SUB><IT>k=</IT>0</SUB> P<SUP>eff</SUP><SUB>A</SUB>(<IT>t−k</IT>)<IT>·</IT>[<IT>&bgr;·</IT>p(<IT>k</IT>)<IT>+&ggr;·</IT>s(<IT>k</IT>)]d<IT>k</IT>](/content/vol92/issue3/fulltext/1239/img017.gif)
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(4)
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The contribution of the arterial baroreflex to the
instantaneous R-R interval at time t [I(t)] is
computed by adding a dynamically computed contribution to the baseline
value, I0. The impulse response functions, p(k)
and s(k), are multiplied by the static gain values, 
and
, for the respective reflex arcs.
During postural changes, hydrostatic pressure at the carotid sinus
receptor changes by an amount 
gh × sin(
), where
h is the distance of the carotid sinus to the regulated
arterial hydrostatic indifference
point1 (27), denotes the angle of tilt of the carotid artery measured from the
horizontal, is the density of blood, and g is the
gravitational acceleration. Because our model has only one lumped
arterial compartment, we have implemented a bias pressure source
(
) which modifies the sensed arterial
pressure (
) according to
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(5)
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Some experimental evidence exists that the maximum sensitivity
of the arterial heart rate baroreflex is linearly related to central
venous pressure (54). We implemented this feature by
making the R-R interval gain a function of central venous pressure according to
Parameter Assignments
Reflex latencies.
Several factors contribute to the time delay between baroreflex
stimulation and effector organ response: afferent nerve time response,
central nervous processing, efferent transmission, and effector organ
response. Borst and Karemaker (9) reported a 0.55-s delay
for heart rate response to electrical stimulation of the carotid sinus
nerve. They also noted a 2- to 3-s delay for changes in diastolic
pressure, which they attributed to reflex changes in peripheral
resistance. Berger and co-workers (4) characterized the
canine heart rate response to sympathetic and parasympathetic
stimulation and reported a reflex latency of ~1.7 s for the
sympathetic reflex limb. Presently, we use a 0.5-s delay for the
parasympathetic reflex response and 2.0 s for all sympathetic reflex arcs.
Static gain values.
We have adopted DeBoer et al.'s (22) static gain values
of 9 ms/mmHg for -sympathetic feedback and 9 ms/mmHg for
parasympathetic feedback, which are based on pharmacological
interventions. We incorporated Davis' (20) model of
contractility feedback, which states that, under maximal stimulation,
the arterial baroreflex can alter end-systolic left and right
ventricular cardiac elastances by a factor of 2. In our model, the
peripheral resistance and venous tone feedback arcs are influenced by
the arterial baroreflex and the cardiopulmonary reflex. Dog experiments
revealed a maximum change of systemic reservoir volume of ~12 ml/kg
under maximal carotid sinus stimulation, 6-7 ml/kg of which have
been shown to originate from the abdominal vessels (63).
If scaled to represent a 75-kg human, 12 ml/kg maximal deviation in
reservoir volume would suggest a maximal zero-pressure volume deviation
of 900 ml. Given the limitation of the afferent pressure signal of our arterial baroreflex model to ±28 mmHg, 900-ml maximum volume deviation would translate into a static gain value for venous tone feedback of
~31 ml/mmHg. Experiments in humans have shown splanchnic blood volume
to change by ~500 ml in response to a 1,000-ml hemorrhage without
significant changes in heart rate, CO, and arteriolar resistance
(56). Assuming this response to be mediated by the cardiopulmonary reflex only, we can get a rough estimate of the cardiopulmonary reflex gain to venous tone by assuming that the contribution of the splanchnic circulation is ~60% of the total venoconstriction response. These assumptions would suggest a static venous tone feedback gain of ~100 ml/mmHg because the cardiopulmonary afferent pressure signal is limited to ±8 mmHg. The effects of the
cardiopulmonary reflex on peripheral resistance can be estimated from
LBNP experiments. Low levels of LBNP usually elicit a vasoconstrictor response without increases in heart rate (38, 54).
Calculating resistance as (mean arterial pressure central
venous pressure)/CO, the data presented by Pawelczyk and Raven
(54) suggest a static gain of the cardiopulmonary
arteriolar reflex limb of 0.05-0.06 PRU/mmHg. We followed Davis's
(20) estimate of the peripheral resistance gain of the
arterial baroreflex.
Tables 4 and
5 summarize the gain values and the
timing of the respective reflex limbs.
| |
ORTHOSTATIC STRESS TESTS |
Tilt-table and/or stand tests and LBNP interventions are
commonly used orthostatic stress tests in both clinical and research environments. Both interventions have in common that increased transmural pressure across the dependent veins provokes a state of
central hypovolemia that elicits a sequence of reflex responses.
Tilt-table Simulation
A tilt-table intervention leads to rapid blood volume shifts
from the thoracic to the dependent vascular beds. Furthermore, the
increased transmural pressure in the dependent vasculature leads to
increased rates of fluid sequestration into the interstitium and thus a
reduction in intravascular volume (32, 37). This second
phase of volume redistribution occurs over a much larger timescale than
the first one.
To simulate the rapid blood volume shift, the bias pressures across the
lower limbs (Pbias-1), splanchnic (Pbias-2),
and abdominal venous (Pbias-3) compartments were specified
as functions of time
where (t) represents a ramp function in time from
zero to the maximal angle of tilt, max. The time to
maximal angle is ttilt, whereas
t0 represents the onset of tilt.
Pmax-i denotes the maximal bias pressure across
the respective compartment when upright posture is assumed. We
chose Pmax-1 = 40.0 mmHg, which leads to ~500 ml of
blood being pooled in the leg compartment on assumption of the upright
posture (27). We assigned Pmax-2 = 7.0 mmHg and Pmax-3 = 5.0 mmHg, which lead to ~300 ml of
blood being pooled in the abdomen and the pelvis (27).
The slower reduction in blood volume due to fluid loss into the
interstitium is modeled by using the work of Hagan and co-workers (32)
The change in blood volume, V, has been determined to
be ~600 ml after 35 min of quiet standing (32). To allow
for tilts to different angles, we hypothesized that V scales
according to
LBNP Simulation
External negative pressure applied to the lower body is
simulated by specifying the bias pressures across the lower body
compartment, Pbias-1 = PLBNP, according
to
LBNP is initiated at time t0;
Pmax denotes the magnitude of the external negative
pressure applied to the lower limbs. In addition to blood pooling in
the legs, significant blood pooling occurs in the pelvis and buttocks
(77, 81). In our model, the behavior of these two
circulatory beds is partly represented by the abdominal venous
compartment. Because the latter also represents the great abdominal
veins, we simulate blood pooling in the pelvis and buttocks by applying
a reduced bias pressure, Pbias-3 = 
Pbias-1, to the abdominal venous compartment ( 0.3).
The increased hydrostatic pressure gradient across the microcirculation
in the lower body and abdominal venous compartments leads to increased
rates of fluid sequestration into the interstitium. The total amount of
blood volume leaving the vascular system is both a function of level
and duration of LBNP. At high levels of LBNP (70 to 75 mmHg), it
has been shown that a linear net decrease of plasma volume on the order
of 500 ml can be observed over the course of 10 min (47).
We simulate this plasma volume loss by reducing overall blood volume as
a function of LBNP time, t t0, and LBNP level, Pmax, according
to
where
Matching Experimental Data
When simulating the hemodynamic response of a given
subject population, it is usually necessary to adjust the parameters of the model to be able to match a given set of experimental data. In the
present study, we chose to compare our simulations to previously published experimental results; thus minimal information about the
subject populations was at our disposal. The only variable we were able
to match was the peripheral resistance response at different levels of
tilt. From the data available, we estimated total peripheral resistance
by computing mean arterial pressure/(stroke volume × heart rate)
for different levels of tilt. We adjusted the static gain values of the
peripheral resistance feedback loops to approximate these total
peripheral resistance values at the various levels of tilt. After this
adjustment, the peripheral resistance gain values were still within
physiologically reasonable limits. Subsequently, we used the same
parameter settings to simulate the hemodynamic response to LBNP.
Testing Hypotheses
Although physiological reasoning can be used to predict
qualitatively the impact that a particular parameter might have on the
hemodynamic response at tilt, the magnitude of the impact can hardly be
reasoned. Furthermore, even the qualitative effect of a combination of
parameter variations can be difficult to infer. To demonstrate how this
model can be used to test hypotheses quantitatively, we focused our
attention on the initial heart rate response to tilt. The heart rate
response to HUT is usually excessively elevated in patients suffering
from OI (53).
We adjusted some of the model parameters (heart rate gain, peripheral
resistance gain, and venous tone gain) such that the (baseline)
simulated heart rate response to tilt mimics a particular experimental
heart rate recording of an astronaut before spaceflight. We
subsequently changed four model parameters (blood volume, sympathetic and parasympathetic heart rate gain, combined peripheral resistance gain, and combined venous tone gain) that are assumed to be effected by
spaceflight to obtain a sequence of simulations that document the
impact that each individual parameter has on the simulated heart rate
dynamics. Finally, we compared these simulations as well as a
simulation with a combination of effects to the postspaceflight heart
rate recording of the same astronaut to see whether any of these
simulations match the postflight response.
| |
RESULTS |
Baseline Simulation
Table 6 demonstrates that all major
hemodynamic parameters generated by the model are within the range of
what is considered physiologically normal in the general population
(61).
Representative simulated pressure waveforms are shown in Fig.
7. The figure also demonstrates how
systolic and end-diastolic pressures shown in Table 6 were derived from
these waveforms.
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Fig. 7.
Simulated central arterial and left ventricular pressure
wave-forms. P , end-diastolic left ventricular
pressure; P , systolic left ventricular pressure;
P , diastolic arterial pressure; P ,
systolic arterial pressure.
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Tilt Table Simulation
We simulated the experimental protocol by Smith and co-workers
(67), who tilted male subjects to various angles for 5 min with a 5-min supine recovery period between tilts. In accordance with
this protocol, we report in Fig.
8 the simulated response (solid line) of
mean arterial pressure, heart rate, and stroke volume each averaged
over the last 3 min at each level of tilt. Figure 8 also depicts the
experimental data recomputed from Ref. 67 to present
absolute change and standard error.
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Fig. 8.
Comparison of tilt simulations (solid line) to
experimental results in 2 age groups ( , men age
40-49; , men age 20-29) of mean arterial
pressure (A), heart rate (B), and stroke volume
(C). Data are means ± SE; adapted from Ref.
67.
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Heart rate and stroke volume simulations reproduce the general trend of
the experimental data and, within the error bounds of each experiment,
match the data at almost every tilt angle. The experimental results of
the arterial blood pressure are less conclusive than those reported for
heart rate and stroke volume, such that a general trend is hard to
discern. The simulation predicts a blood pressure response that is
capable of matching the data at most, but not all, levels of tilt.
Figure 9 displays the simulated transient
response of heart rate to a rapid HUT (70° over 2 s). It also
introduces a number of features of the transient heart rate response
measured by Rossberg and Martinez (59) during rapid HUTs
(70° over 1.7 s) as a function of respiratory phase in 20 male
subjects. Their experimental results are reported in Table
7 along with the feature values extracted from Fig. 9. Although the tilt simulations are not synchronized to the
respiratory cycle, the simulated transient response of heart rate
generally matches the features reported in Ref. 59 within the error bounds of their experiments, the duration of the
initial heart rate complex (features 7 and 8)
being an exception.
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Fig. 9.
Features of the simulated heart rate response to a
tilt-table experiment (70° head-up tilt over 2 s). 1,
Baseline heart rate; 2, peak heart rate; 3,
transient increase in heart rate; 4, heart rate trough after
peak; 5, drop in heart rate after peak; 6, heart
rate at 180 s after initiation of tilt; 7, time to
maximal heart rate; 8, duration of initial heart rate
transient.
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Animal and most human studies agree that the hydrostatic indifference
point on the venous side is located at the level of or slightly below
the diaphragm (see, e.g., Ref. 27). When a HUT to 90° is
simulated, the transmural pressure in the inferior vena cava
compartment drops whereas the transmural pressure in the abdominal
venous compartment rises, indicating that the venous hydrostatic
indifference point in our model is located between these two
compartments. This result implies that, as in the human circulation,
cardiac filling pressure in our model drops on HUT and therefore
cardiac performance is dependent on posture.
LBNP Simulation
We based simulations of LBNP interventions on the experimental
investigation by Ahn and co-workers (1). In accordance
with their experimental protocol, we simulated each level of LBNP for 5 min with a sufficient equilibration period (30 min in the experimental study) at normal atmospheric pressures between the various levels. Figure 10 depicts the simulation and
experimental results (taken from Ahn and co-workers, Ref.
1) of step changes in LBNP to three levels of negative
external pressure. The data and simulations are reported 90 s
after the onset of each level of LBNP.
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Fig. 10.
Comparison of lower body negative pressure (LBNP)
simulations (solid line) to experimental results () of
mean arterial pressure (A), heart rate (B), and
stroke volume (C). Data are means ± SE; adapted from
Ref. 1.
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The simulation results of heart rate match the experimental data well
whereas the simulation results for mean arterial pressure and stroke
volume reproduces the general trend of the data.
Statistical Analysis
It is our goal to simulate the dynamic response of the
cardiovascular system to HUT and LBNP. To test whether our model is capable of representing the experimental data presented in Figures 8
and 10, we used a two-sided test of significance with Bonferroni correction for multiple comparisons. The test indicates that the tilt
and LBNP simulations are not statistically different from the 39 data
points we presented in Figs. 8 and 10 (15 tilt data points in young
subjects; 15 tilt data points in older subjects; 9 LBNP data points).
The level of significance used is 0.05.
Testing Hypotheses: An Illustrative Case Study
The heart rate responses to a stand test of one astronaut 120 days
before spaceflight and on landing day are shown in Fig. 11. The postflight heart rate response
shows the drastic heart rate increase on
assumption of upright posture that is characteristic of OI. We
simulated the preflight recording by adjusting the combined heart rate
gain, combined peripheral resistance gain, and combined venous tone
gain, within physiologically reasonable limits. Figure 12 depicts the simulated and actual
preflight heart rate responses. Although the match between the two
responses is not perfect, the general features and amplitudes are in
good agreement. Using the simulated response of Fig. 12 as our
baseline, we repeated the simulations for different values of total
blood volume (Fig. 13A), combined (parasympathetic and sympathetic) heart rate gain (Fig. 13B), combined peripheral resistance gain (Fig.
13C), and combined venous tone gain (Fig. 13D).
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Fig. 11.
Heart rate response to standing taken from 1 astronaut
120 days before spaceflight (A) and on landing day
(B). Astronaut data provided by Janice Meck, National
Aeronautics and Space Administration (NASA), Johnson Space Center,
Houston, TX.
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Fig. 12.
Simulated (dash-dotted line) and actual heart rate
response (solid line) to the upright posture. Experimental data taken
120 days before spaceflight. Astronaut data provided by Janice Meck,
NASA, Johnson Space Center, Houston, TX.
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Fig. 13.
Simulated heart rate response to rapid tilt to the upright posture
(90°) for different parameter settings: successive reduction in blood
volume (A), heart rate gain (B), venous tone gain
(C), and resistance gain (D). Solid line
represents baseline simulation. A: tracings are generated by
successive removal of blood in 50-ml increments (solid line, total
blood volume of 5,700 ml; dashed line, 5,650 ml; dash-dotted line,
5,600 ml; dotted line, 5,550 ml; dash-dot-dotted line, 5,500 ml).
B-D: tracings are generated by successive
reduction of the respective gain values by 5%.
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Figure 14 depicts the postspaceflight
heart rate recording and a simulated response in which several
parameters were changed (blood volume, heart rate gain, venous tone
gain, and peripheral resistance gain) to approximate the experimental
tracing.
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Fig. 14.
Simulated (dash-dotted line) and actual heart rate
response (solid line) to the upright posture. Experimental data taken
on landing day. Astronaut data provided by Janice Meck, PhD, NASA,
Johnson Space Center, Houston, TX.
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Although total blood volume has the largest impact on the magnitude of
the heart rate response, none of the individual simulations is capable
of reproducing the dynamics seen in the postflight recording (Fig.
11B).
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DISCUSSION |
Mathematical models reflect our present level of understanding of
the functional interactions that determine the overall behavior of the
system under consideration. They allow us to probe the system, often in
much greater detail than is possible in experimental studies, and can
therefore help establish the cause of a particular observation.
When fully integrated into an experimental program, mathematical models
and experiments are highly synergistic in that the existence of one
greatly enhances the value of the other: models, for example, depend on
experiments for specification and refinement of their parameters, but
they also illuminate and enhance the interpretation of experimental
results and allow for testing of hypotheses.
The research presented in this paper is part of an ongoing effort to
utilize mathematical models in the investigation of the adaptation of
the cardiovascular system to a sudden redistribution of blood volume as
experienced during a rapid tilt or a LBNP intervention. Our focus rests
specifically on the immediate transient and the short-term steady-state
response after onset of gravitational stress.
To achieve our objective of simulating the transient and steady-state
hemodynamic response to orthostatic stress, we chose to model the
entire hemodynamic system by a finite set of representative compartments, each of which captures the physical properties of a
segment of the vascular system.
In doing so, we implicitly assume that the dynamics of the system can
be simulated by restricting our analysis to relatively few
representative points within the cardiovascular system. Although this
approach is incapable of simulating pulse wave propagation, for
example, it does reproduce realistic values of beat-by-beat hemodynamic
parameters (see Table 6). Furthermore, for the same set of parameter
values, the simulated steady-state responses to HUT and LBNP agree well
with population-averaged experimental observations (see Figs. 8 and
10).
One potential limitation of the hemodynamic system in its present form
might be the lack of atria, which are thought to contribute significantly to ventricular filling at high heart rates
(6). Although the behavior of stroke volume at high heart
rates was not central to the present study, future work might
necessitate the addition of atria.
For now, we are restricted to simulating the early steady-state
response of the cardiovascular system to orthostatic stress because the
reflex control model only represents the major neural control
mechanisms that are responsible for short-term (~5 min) control of
blood pressure. Studies have shown that within minutes after the onset
of gravitational stress, hormone levels in the circulation rise
significantly (37). The addition of fast-acting hormone
loops in the future will allow us to extend the time scale of our
simulations beyond a few minutes.
Our present model does not include the possible dependence of
intrathoracic pressure on posture. There is suggestive evidence (48) that intrathoracic pressure decreases on assumption
of the upright posture. Such an effect would be expected to modify somewhat the model's response to simulated tilt, and this phenomenon should be investigated more closely in our future studies.
In addition to reproducing steady-state data of hemodynamic variables
over a wide range of orthostatic stress levels, the model is capable of
reproducing experimentally observed features of the transient heart
rate response to rapid tilt (see Table 7 and Fig. 9). Although the
amplitudes of the simulated heart rate response match experimental
observations well within the variation of the data, the simulations
generally predict larger values for the time to maximum heart rate
(feature 7 in Fig. 9) and the durat
TOP
ABSTRACT
INTRODUCTION
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