| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
|
|
|
|||||||
|
|||||||||
1Department of Mathematics, North Carolina State University, Raleigh, North Carolina; 2Department of Mathematics and Physics, Roskilde University, Roskilde, Denmark; and 3Hebrew SeniorLife, Research and Training Institute, and 4Division of Gerontology, Beth Israel Deaconess Medical Center and Harvard Medical School, Boston, Massachusetts
Submitted 14 February 2005 ; accepted in final form 26 April 2005
 |
ABSTRACT |
|---|
 TOP ABSTRACT GlossaryMODELING BLOOD PRESSURE AND...MODELING AUTONOMIC REGULATION...PARAMETER ESTIMATIONEXPERIMENTAL DATARESULTSCONCLUSIONAPPENDIXACKNOWLEDGMENTSREFERENCES |
|---|
cardiovascular system; mathematical modeling; cerebral blood flow; gravitational effect; autonomic regulation; cerebral autoregulation
On the transition from sitting in a chair to standing, blood is pooled in the lower extremities as a result of gravitational forces. Venous return is reduced, which leads to a decrease in cardiac stroke volume, a decline in arterial blood pressure, and an immediate decrease in blood flow to the brain. The reduction in arterial blood pressure unloads the baroreceptors located in the carotid and aortic walls, which leads to parasympathetic withdrawal and sympathetic activation through baroreflex-mediated autonomic regulation. Parasympathetic withdrawal induces fast (within 1–2 cardiac cycles) increases in heart rate, whereas sympathetic activation yields a slower (within 6–8 cardiac cycles) increase in vascular resistance, vascular tone, and cardiac contractility and a further increase in heart rate (4, 7, 37). Simultaneously, cerebral autoregulation, mediated by changes in CO2, myogenic tone, and metabolic demand, leads to vasodilation of the cerebral arterioles (2, 18, 34, 38).
Our mathematical model includes two submodels: 1) a cardiovascular model that can predict blood pressure and blood flow velocity during sitting and 2) a control model that can predict autonomic and cerebral regulatory mechanisms during the postural change from sitting to standing. Both submodels are based on the same closed-loop model with 11 compartments that represent the heart and systemic circulation. Our previous work (27, 29) also used compartmental models to describe the dynamics of the cardiovascular system. One (27) used an open-loop (3-element windkessel) model to analyze dynamics of cardiovascular control. This model used arterial blood pressure measured in the finger as an input to predict model parameters that describe dynamics of cerebral vascular regulation for young subjects. These parameters were obtained by minimizing the error between computed and measured middle cerebral artery blood flow velocity. Consequently, no equations were used to describe possible mechanisms of the underlying regulation. To further advance this study, we recently developed a seven-compartment closed-loop model (29) that can predict the dynamics observed in the data. This model did not rely on an external input; rather, it included a submodel that describes the pumping of the left ventricle. In addition, the seven-compartment model included simple equations that describe the short-term regulation. This model was able to accurately predict dynamics of cerebral blood flow velocity and arterial blood pressure during sitting (t < 60 s) and standing (t > 80 s), as well as the mean values during the transition from sitting to standing (60 < t < 80 s), but it was not able to predict detailed dynamics during the transition from sitting to standing. Furthermore, we were not able to achieve adequate filling of the left ventricle. To obtain a more accurate model, we developed the 11-compartment model, which overcomes limitations of the 7-compartment model by 1) predicting resistances as nonlinear functions of pressure, 2) adding essential compartments, 3) devising an empirical model of autoregulation, and 4) including a new physiological model describing pooling of blood in the lower extremities due to effects of gravity.
A large body of work that describes cardiovascular control modeling (9–11, 30, 44) is based on predictions of mean values for arterial blood pressure and cerebral blood flow velocity. Consequently, these models cannot predict the pulsatile dynamics of the cardiovascular system. These models use optimal control to minimize the deviation between some observed quantity (e.g., arterial blood pressure) and a given set point. Although this strategy can provide good parameter estimates, optimal control models do not describe the underlying physiological mechanisms. Other modeling strategies have been proposed by Melchior et al. (19, 20) and Heldt et al. (8), who devised pulsatile models that include pulsatility, autonomic regulation, and effects of gravity. The latter was done by changing the reference pressure outside the compartments. However, these models do not include effects of autoregulation. One way to model the effect of autoregulation is to let the cerebrovascular resistance be a function of time, as suggested by Ursino and Lodi (39). However, this work does not include the effects of autonomic regulation. A second group of models described parts of the control system without validation against experimental data (5, 19–21, 31, 32, 35, 40–43). These models used a closed-loop compartmental description of the cardiovascular system combined with physiological descriptions of the control. Although these models can provide qualitative analysis of the system, they cannot be used for quantitative comparisons with data. Furthermore, most of the models in the second group describe the effects of autonomic regulation without including the effects of cerebral autoregulation. In contrast, our model includes autonomic and cerebrovascular regulations and provides quantitative comparisons with physiological data.
|
Glossary |
|---|
|
TOPABSTRACTGlossaryMODELING BLOOD PRESSURE AND...MODELING AUTONOMIC REGULATION...PARAMETER ESTIMATIONEXPERIMENTAL DATARESULTSCONCLUSIONAPPENDIXACKNOWLEDGMENTSREFERENCES |
|---|
|
MODELING BLOOD PRESSURE AND BLOOD FLOW VELOCITY |
|---|
|
TOPABSTRACTGlossaryMODELING BLOOD PRESSURE AND...MODELING AUTONOMIC REGULATION...PARAMETER ESTIMATIONEXPERIMENTAL DATARESULTSCONCLUSIONAPPENDIXACKNOWLEDGMENTSREFERENCES |
|---|
|
The major system not included in our model is the pulmonary circulation. Addition of compartments that represent the pulmonary circulation would require more parameters, which would increase the computational complexity. Instead, the pulmonary circulation is represented as a resistance between the vena cava and the left atrium.
To study dynamics of postural change from sitting to standing, it is not important to know how blood is distributed among various inner organs. Hence, the upper body is simply represented by an arterial and a venous compartment. Each compartment is represented by a compliance element (inverse elasticity) and is separated by resistance to flow. The design of the systemic circulation with arteries and veins separated by capillaries provides some resistance and inertia to the volumetric flow rate. In our model, we include effects of resistance between compartments but neglect effects due to inertia. The major resistance to flow is located in peripheral regions between compartments that represent arteries and veins. Compartments that represent large conduit vessels are also separated by resistances that represent the overall resistance of the compartment. Resistances between conduit vessels are very small compared with peripheral resistances.
The description of blood pressure and volumetric flow in a system consisting of compliant compartments (capacitors) and resistors is equivalent to that of an electrical circuit (Fig. 1), where blood pressure plays the role of voltage and volumetric flow rate plays the role of current. To compare our model with data, we assume that the diameter of the middle cerebral artery remains constant, such that blood flow velocity can be obtained by scaling volumetric blood flow by a constant factor that represents the area of the vessel. Recent measurements of middle cerebral artery diameter by magnetic resonance imaging combined with transcranial Doppler assessment of cerebral blood flow velocity have demonstrated that the middle cerebral artery diameter does not change, despite large changes in cerebral blood flow velocity elicited by stimuli such as lower body negative pressure and CO2 changes (36).
To predict blood pressure and blood flow within and between the compartments, we base our model on volume conservation laws (41). Blood pressure and volumetric blood flow can be found by computing the volume and change in volume for each compartment. The equations that represent the arterial and venous compartments are similar. For each of these compartments, the stressed volume V = Cp (cm3, volume pumped out during 1 cardiac cycle), where C (cm3/mmHg) is compliance and p (mmHg) is blood pressure. The cardiac output (CO) from the heart is given by CO = HVstroke (cm3/s), where H (beats/s) is heart rate and Vstroke (cm3/beat) is stroke volume. For each compartment, the net change of volume is given by
 | (1) |
To model the left ventricle as a pump, the position of the mitral and aortic valves must be included. During diastole, the mitral valve is open, while the aortic valve is closed, allowing blood to enter the left ventricle. Then isometric contraction begins, increasing the ventricular pressure. Once the ventricular pressure exceeds the aortic pressure, the aortic valve opens, propelling the pulse wave through the vascular system. For healthy young people, both valves cannot be open simultaneously. To incorporate the state of the valves, we have modeled the resistances (Rav and Rmv; Fig. 1) as follows
 |
A system of differential equations is obtained by differentiating the volume equation V = Cp and inserting Eq. 1
 | (2) |
Ventricular and atrial contraction.
Atrial and ventricular contraction leads to an increase in blood pressure from the low values observed in the venous system to the high values observed in the arterial system. Our model is based on the work by Ottesen and coworkers (6, 33), which predicts atrial (pla) and ventricular (plv) pressure as a function volume and cardiac activation of the form
 | (3) |
The activation function g(t), which is defined over the length of one cardiac cycle, is described by a polynomial of degree (n;m): g(t) = f(t)/f(tp) with
 | (4) |
= mod(t;T), s], 
(H) (s) denotes the onset of relaxation, H = 1/T (1/s) is heart rate, n and m characterize the contraction and relaxation phases, and pp is the peak value of the activation. The ability to vary heart rate is included in the isovolumic pressure equation (Eq. 3) by scaling time and peak values of the activation function f. The time for peak value of the contraction [tp (s)] is scaled by introducing a sigmoidal function, which depends on the heart rate (H), of the form
 | (5) |
represents the median, represents steepness, and tm (s) and tM (s) denote the minimum and maximum values, respectively. The peak ventricular pressure [pp (mmHg)] is scaled similarly using a sigmoidal function of the form
 | (6) |
represents the median, represents steepness, and pm (mmHg) and pM (mmHg) denote minimum and maximum values, respectively. Finally, the time for onset of relaxation is modeled by
 | (7) |
in the isovolumic pressure model (3). Initial values for all parameters were obtained from the work by Ottesen and Danielsen (33), in which parameters were based on data from dogs. To obtain human values for the young subject studied in this work, we identified the parameters in Table 1 during our model validation.
|
To our knowledge, previous modeling contributions (see the introduction) assume that, during steady state (i.e., sitting, for t 
60 s), the small resistances between compartments that represent large conduit vessels are constant. Nevertheless, from the theory of fluid mechanics, it is well known that the resistance depends on the radii of the vessels and that the radii themselves depend on the corresponding transmural pressure.
Our investigation has shown that such dependencies are important to include in regions that represent vessels with large diameters and high blood pressure (i.e., large arteries), whereas they are less important in regions of low blood pressure (i.e., the venous system). Furthermore, these "passive" changes in diameters are also negligible in regions with small vessels (i.e., small arteries and arterioles), where autonomic responses are active and dominate the change in vessel diameters. Our previous work (29) did not include nonlinear arterial resistances; therefore, we were not able to obtain a sufficiently wide pulse pressure immediately after postural change from sitting to standing.
To model nonlinearities for these resistances, we base our derivation on Poiseuille's law. For flow in a cylinder with circular cross-sectional area, Poiseuille's law predicts the resistance to flow (14) as
 |
(mmHg·s) is viscosity of blood, and l (cm) is length of the cylindrical vessel. If it is assumed that length of the vessel is constant
 | (8) |
 | (9) |
(t) (mmHg).] As shown in Fig. 2, the mean arterial blood pressure oscillates with the same frequency but with smaller amplitude than pa; k represents the steepness of the sigmoid, and the parameter 
2 is calculated to ensure that R returns to the value of the controlled parameter found during steady state. For k = 2, the slope of the sigmoid approximates the relation in Eq. 8. However, the relation in Eq. 8 is valid only for a steady flow. Blood flow in arteries is unsteady, and the flow through a given vessel depends on the state of the vessel. Consequently, as shown in Table 2, we should not expect that k = 2.
|
|
au), Rac(a), and Raf(a). The resistance of the aorta (Rau) could also be modeled using this method. Initial investigations showed that other mechanisms, e.g., autoregulation or autonomic regulation, may also affect Rau. As a consequence, we have used an empirical model to estimate Rau (see MODELING AUTONOMIC REGULATION AND CEREBRAL AUTOREGULATION. Cerebral autoregulation).
Gravitational effect.
Gravitational effects are essential during postural change from sitting to standing. Consider a cylindrical vessel with length 
z (cm) and time-invariant cross-sectional area A (cm2), i.e., dA/dt = 0. Assume that there is no velocity across the vessel and that the blood pressure is only a function position along the vessel. Hence, dv/dr = 0, where v (cm/s) and r (cm) denote the velocity and radii, respectively, and the volumetric flow rate becomes q = Av (cm3/s). Finally, assume that the drag force due to viscous shear is proportional to q. Thus the drag force per cross-sectional area unit is proportional to q; i.e., the drag force can be written as –RAq, where R (mmHg·s·cm–3) may be interpreted as the resistance. In steady state, the resistance R is given by Poiseuille's law (23)
 |
 |
= 1.055 (g/cm3) is the density of the fluid and M (g) is the mass of the fluid contained in a piece of the vessel with length z (cm) and cross-sectional area A (cm2; Fig. 3). Thus Newton's second law, which describes balancing of forces, gives
 |
 | (10) |
z/A (1/s2) is the inertance and h = zcos(
) = hin – hout (cm) is the vertical difference of the vessel inlet (at hin where pin and pout represent pressure at the inlet and outlet, respectively). During steady state, Eq. 10 reduces to
 | (11) |
0, Eq. 11 approaches the normal form of Kirchhoff's current law given in Eq. 1. In the case of energy conservation (R 0), Bernoulli's law for steady flow is recovered; as a result, pin + ghin = pout + ghout. Thus Kirchhoff's current law is still valid if we interpret p as the hydrostatic pressure p + gh.
|
 | (12) |
(s) is the latency for the transition to standing. In our experiments, the subjects sit with their legs elevated and the hand, where the pressure is measured, held by a sling at the level of the heart. Therefore, compartments that represent the heart and the finger are not affected by gravity. Compartments that represent the brain and the upper body are exposed to constant hydrostatic conditions, which are neglected in the current formulation. However, compartments that represent the legs are affected by gravity. Consequently, equations for the flows qal and qvl will be modified as described in Eq. 11
 |
 |
|
MODELING AUTONOMIC REGULATION AND CEREBRAL AUTOREGULATION |
|---|
|
TOPABSTRACTGlossaryMODELING BLOOD PRESSURE AND...MODELING AUTONOMIC REGULATION...PARAMETER ESTIMATIONEXPERIMENTAL DATARESULTSCONCLUSIONAPPENDIXACKNOWLEDGMENTSREFERENCES |
|---|
Autonomic regulation.
Autonomic regulation is modeled as a pressure regulation where heart rate (H, beats/s), cardiac contractility (ca and cv, mmHg/cm3), peripheral systemic resistance (Raup and Ralp, mmHg·s·cm–3), and systemic compliance (Ca, Cau, Cal, Cac, Caf, Cv, Cvu, Cvl, and Cvc, cm3/mmHg) are functions of mean arterial blood pressure (a, mmHg).
The change in the controlled parameters is modeled using a first-order differential equation with a set-point function dependent on a
 | (13) |
(s) is a time constant that characterizes the time required for the controlled variable to obtain its full effect. Different values of were used for control of cardiac contractility, compliance, and resistance (Table 2). As described earlier, autonomic regulation yields increases in peripheral vascular resistance, heart rate, and cardiac contractility. Heart rate is directly obtained from data. Hence, it is not modeled using the set-point function (13). To obtain increases in peripheral resistances (Raup, Ralp, and Rafp) and cardiac contractility (cla and clv) in response to the decrease in arterial blood pressure, the following set-point function has been used
 | (14) |
 | (15) |
2 is calculated to ensure that x(t) returns the value of the controlled parameters found during steady state. Initial values of parameters for k, xm, and xM are from Danielsen (5) (Table 2).
These control equations (Eqs. 13–15) are formulated as functions of mean arterial blood pressure. However, our model describes the instantaneous (pulsatile) pressure. Mean values are computed as weighted averages, where the present is weighted higher than the past
 | (16) |
 | (17) |
 | (18) |
Cerebral autoregulation.
On the transition to standing, cerebral autoregulation mediates a decline in cerebrovascular resistance (Racp) in response to the decrease in arterial blood pressure. In addition, the autonomic system may also play a role, by decreasing the cerebrovascular resistance due to cholinergic vasodilation or by increasing the resistance due to release of norepinephrine (7). Consequently, it is not trivial to develop an accurate physiological model that describes cerebral autoregulation. Our strategy in this work has been to use a piecewise linear function with unknown coefficients to obtain a representative function that describes the time-varying response of the cerebrovascular resistance. Once such a function is obtained, we can interpret the result in terms of the underlying physiology. To obtain such a function, we have parameterized the cerebrovascular resistance using piecewise linear functions of the form
 | (19) |
 | (20) |
i will be estimated together with the other control parameters in Table 2. As described above, we have used a similar method to estimate the resistance Rau, which may be affected by passive nonlinear resistances and autonomic regulation. |
PARAMETER ESTIMATION |
|---|
|
TOPABSTRACTGlossaryMODELING BLOOD PRESSURE AND...MODELING AUTONOMIC REGULATION...PARAMETER ESTIMATIONEXPERIMENTAL DATARESULTSCONCLUSIONAPPENDIXACKNOWLEDGMENTSREFERENCES |
|---|
a and au. Finally, we estimated a constant factor used to calculate cerebral blood flow velocity vacp = qacp/fact (cm/s). We have used a constant factor (fact), because we assume that the cross-sectional area of the middle cerebral artery does not change significantly (36). These equations involve a total of 53 parameters that were estimated using a nonlinear optimization method, the Nelder-Mead algorithm, which is based on function information computed on sequences of simplexes (13). Estimated parameter values are shown together with initial values in Table 1. To obtain the best possible parameter values, we used the following cost function to minimize the difference between measured and computed values of cerebral blood flow velocity and finger pressure
 |
t 90 s.
|
|
|
 |
 |
 |
|
|
EXPERIMENTAL DATA |
|---|
|
TOPABSTRACTGlossaryMODELING BLOOD PRESSURE AND...MODELING AUTONOMIC REGULATION...PARAMETER ESTIMATIONEXPERIMENTAL DATARESULTSCONCLUSIONAPPENDIXACKNOWLEDGMENTSREFERENCES |
|---|
30 mmHg on the transition to standing was used as a challenge for cerebral autoregulation. Subjects sat in a straight-backed chair with their legs elevated at 90° in front of them. They were then asked to stand. Standing was defined as the moment both feet touched the floor. Subjects performed two 5-min trials in the sitting position followed by standing for 1 min and one 5-min trial in the sitting position followed by 6 min of standing. |
RESULTS |
|---|
|
TOPABSTRACTGlossaryMODELING BLOOD PRESSURE AND...MODELING AUTONOMIC REGULATION...PARAMETER ESTIMATIONEXPERIMENTAL DATARESULTSCONCLUSIONAPPENDIXACKNOWLEDGMENTSREFERENCES |
|---|
|
60 s). We applied initial parameter values from physiological considerations (see above). Then we fitted our model [without including equations that describe resistances of large arteries as nonlinear functions of pressure (Eq. 9) and those that describe active control (Eqs. 13 and 19)] to the data set. The duration of the cardiac cycles was obtained from the ECG (Fig. 6). Simulation results in Fig. 7 show that we obtained an excellent agreement between our model and the data during steady state. However, our model is not able to resolve details of the secondary oscillations observed within each cardiac cycle (Fig. 7B), a feature that is not included in our heart model. The second step in validating our model is to illustrate that we can model effects of venous pooling after the transition to standing. Venous pooling results in dramatic reductions of cerebral blood flow velocity and arterial pressure (Fig. 8): with the parameters listed in Tables 1 and 2, it is possible to decrease blood flow velocity and pressure. Two observations should be noted: 1) although we did not include effects of the control, we still see an increase in heart rate, because heart rate information is obtained from the data (Fig. 6), and 2) although blood flow velocity and pressure drop immediately after standing (at 60 s), the pulse amplitude for blood flow velocity and pressure remains very narrow.
|
t 65 s); thus the model better represents measured values (cf. dark lines in Fig. 8, A and B, in the transition region, for 60 t 65 s). The third step involved incorporation of all active control mechanisms. Results that include effects of autonomic regulation and autoregulation are shown in Fig. 9. Our model is able to predict the change in the overall profile during the transition from sitting to standing. The only minor difference is that the data include a slight overshoot in pressure in the transition to standing.
|
The resistance of the upper body (Rau) was also modeled using a piecewise linear model with unknown parameters, as described elsewhere (19). We expected that Rau may depend on autonomic regulation and may be a nonlinear passive function of pressure. This resistance follows trends predicted by remaining resistances that represent the large arteries (Fig. 5).
Finally, Fig. 10 depicts the dynamics of some of the controlled variables, e.g., arterial resistance (Raup), cardiac contractility of the left ventricle (clv), and venous compliance in the upper body (Cvu). These results display quite different dynamics of the three types of variables. In particular, the compliance and peripheral resistance do not reach a steady state during the 10 s after the transition from sitting to standing (from 80 t 90 s), perhaps because the dynamics that change the ventricular contractility occur over a much faster time scale than those that affect resistances and compliances. Finally, the dynamics of other resistances, capacitors, and atrial contractility are similar to the parameters shown in Fig. 10.
|
|
CONCLUSION |
|---|
|
TOPABSTRACTGlossaryMODELING BLOOD PRESSURE AND...MODELING AUTONOMIC REGULATION...PARAMETER ESTIMATIONEXPERIMENTAL DATARESULTSCONCLUSIONAPPENDIXACKNOWLEDGMENTSREFERENCES |
|---|
Modeling of physiological responses to standing enables a better understanding of physiological mechanisms underlying disorders related to orthostatic tolerance, e.g., orthostatic hypotension and syncope. Our model predicts that, in the absence of regulatory mechanisms (Fig. 8), blood pressure and blood flow velocity declined on the transition to standing and did not recover to baseline in the upright position. This modeling result has not been validated against data. However, similar responses have been observed clinically. For example, sustained blood pressure reduction in the upright position is seen in clinical syndromes with orthostatic hypotension associated with autonomic failure (16, 17). Different etiologies and severity of autonomic failure may lead to differences in pathophysiological responses during the transition to standing. For example, severe peripheral autonomic failure, such as pure autonomic failure or diabetic neuropathy, may be associated with orthostatic hypotension with no heart rate increment. Cerebral autoregulation, which maintains cerebral perfusion over a wide range of pressure (25), may be preserved, expanded, or reduced in orthostatic hypotension. However, cerebral blood flow would decline with impairment of autoregulation and/or when blood pressure is diminished below the autoregulated range. A transient impairment of autonomic and cerebral blood flow control is common in young people with vasodepressor syncope. This is associated with a withdrawal of sympathetic tone followed by a decline of blood pressure and cerebral perfusion (12, 22, 24).
Furthermore, our results show that, by including passive nonlinear responses of resistances in the large arteries, we can obtain sufficient widening of the pulse pressure amplitude observed immediately after the transition to standing. This response is immediate and, thus, not a regulatory response but, rather, a purely passive response that occurs because of the nature of the underlying fluid dynamics. We have described an elaborate model for predicting effects of hydrostatic changes, even though this model was only validated for the transition from sitting to standing, i.e., cos() = 1. The advantage of the model derived in the present work is that it may be applicable to prediction of hydrostatic effects observed during tilt-table experiments.
The main accomplishment of this work is that our model describes how autonomic regulation and cerebral autoregulation play a synergistic role in the control of arterial blood pressure and cerebral blood flow velocity. In particular, the cerebral resistance first decreases and then increases during active standing. This result is different from previous findings (27), which suggested an initial increase followed by a decrease. However, the new result is not surprising, because the present study was performed with a more complex closed-loop model. The main advantage of the closed-loop 11-compartment model presented in this study is that the cerebrovascular resistance offers a more accurate representation of the brain. For example, in previous work (27), the measured pressure was an input and only one compartment was included. Hence, the peripheral resistance was not distinguished between resistance of the body and the brain. Furthermore, the curve for Racp displays hysteresis effects: Immediately after standing, the decrease of Racp is faster than the increase for t 70 s during the phase where blood flow velocity is returning
