INTRODUCTION

TOP ABSTRACT INTRODUCTION QUALITATIVE MODEL DESCRIPTION PARAMETER ASSIGNMENT AND... RESULTS DISCUSSION APPENDIX REFERENCES

THE CARDIOVASCULAR RESPONSE to hypoxia is the result of a multitude of concomitant mechanisms that superimpose in vivo in complex nonlinear ways. These include local vasodilation, the peripheral chemoreflex, the lung inflation reflex, and the hypoxic response of the central nervous system (CNS) (1, 24, 34).

It is well known that the vascular beds in organs with the higher metabolic rate (especially the heart, brain, and skeletal muscle) dilate in response to severe hypoxia. Blood flow to these organs may disproportionately increase when arterial PO₂ (Pa_O2) is decreased below 30 mmHg (7, 24). This local vasodilatory effect would result in a significant arterial hypotension if it is not counteracted by vasoconstriction in other vascular beds. This is provided by activation of the peripheral chemoreflex, which causes vasoconstriction in the splanchnic and renal circulation as well as in the skeletal muscle. Because of the previous responses, hypoxia causes a significant redistribution of the available blood flow from the organs with a lower metabolic need toward those with a higher metabolic demand.

The pattern described above, however, is further complicated by changes in ventilation, which activate stretch receptors in the lung (13), and by the hypoxic response of the CNS (19). Moreover, cardiovascular and ventilatory adjustments to hypoxia modify other hemodynamic and metabolic quantities [especially systemic arterial pressure (SAP) and arterial PCO₂ (Pa_CO2)], the change of which elicits the secondary action of additional control mechanisms. Arterial pressure variations trigger the baroreflex system, whereas changes in PCO₂ activate the central chemoreceptors located in the medulla, modulate the response of peripheral chemoreceptors, and have a local effect on blood vessels. Of course, the final cardiovascular response depends on the integration of all these components, in part excitatory and in part inhibitory, and on their relative magnitude (12).

Because of the multiplicity of factors that superimpose nonlinearly, the final result of hypoxia is not easily predictable by use of only qualitative reasoning. The use of mathematical models and of computer simulation techniques may be of great value to deepen our comprehension of the problem in physiological investigation and in clinical practice. Mathematical models permit the nonlinear interactions among the different mechanisms and the effect of their individual variability to be accounted for in rigorous quantitative terms.

Several models of short-term cardiovascular regulation have been proposed in the past decades with different purposes (3, 21, 22). However, we are not aware of any model that brings together all mechanisms triggered by hypoxia and their complex mutual interactions into a single theoretical setting. In recent years, we formulated a mathematical model of the baroreflex control in pulsating conditions that is able to account for many aspects of the short-term arterial pressure regulation (46, 47). The model includes a pulsatile heart, systemic and pulmonary circulations, the afferent and efferent neural pathways, and four effectors for the regulation. With that model, we were able to address several important aspects of the baroregulation, such as the importance of venous unstressed volume control, the role of pulsatility, and the dependence of the Starling law on baroreceptor activation.

The aim of this work is to significantly extend the previous model to include some of the main cardiovascular mechanisms involved in the acute response to isocapnic hypoxia. The new mechanisms include the local vasodilatory effect of O₂, the effect of peripheral chemoreceptors and lung stretch receptors on the heart and blood vessels, and hypoxia of the CNS. Furthermore, all these mechanisms interact with the baroreflex response included in the previous model.

In this version of the model (46, 47), we did not consider changes in Pa_CO2; i.e., the only input quantity for the model is Pa_O2. For this reason, all the experimental conditions refer to isocapnic hypoxia only. The reason is that analysis of the interaction between hypoxia and PCO₂ changes would confound interpretation of physiological results and require a more complex model, including description of gas exchange at the lungs and tissues, and the inclusion of additional control loops (such as central chemoreceptors in the medulla).

In the present report the main aspects of the model are presented, and its capacity to reproduce the pattern of the cardiovascular response to hypoxia is shown. In a subsequent related study, a sensitivity analysis on the role of the main parameters will be performed to underline the role of each mechanism and the impact of individual parameter variations (48).

	QUALITATIVE MODEL DESCRIPTION

TOP ABSTRACT INTRODUCTION QUALITATIVE MODEL DESCRIPTION PARAMETER ASSIGNMENT AND... RESULTS DISCUSSION APPENDIX REFERENCES

The present model represents an extension of a baroreflex mathematical model described in a recent report (46). The major changes are in the analysis of the cardiovascular response to hypoxia. They include 1) a separate description of the vascular beds in organs with higher metabolic need (brain, coronary, and skeletal muscle circulation) and the local effect of hypoxia on these vascular beds, 2) the afferent information from peripheral chemoreceptors and its effect on the efferent (sympathetic and parasympathetic) neural pathways, 3) the effect of peripheral chemoreceptors on tidal volume (VT) and the consequent response of lung stretch receptors, further affecting the efferent neural pathways, and 4) the direct effect of hypoxia of the CNS.

A block diagram showing the main relationships between afferent and efferent neural information, the effector responses, and the local role of O₂ is presented in Fig. 1. The main characteristics of the model are summarized in qualitative terms, with particular emphasis on the new aspects. Model equations are reported in the APPENDIX; parameter values can be found in Tables 1-3. All parameters have been computed with reference to a 70-kg subject.

View larger version (39K): [in this window] [in a new window]   Fig. 1.   Block diagram describing relationships among afferent information, efferent neural activities, and effector responses according to the present mathematical model. Pb, baroreceptor pressure; PaO2, arterial PO2; VT, tidal volume; fab, fac, and fap, afferent activities from arterial baroreceptors, peripheral chemoreceptors, and lung stretch receptors, respectively; sh and sp, offset terms for the cardiac and peripheral sympathetic neurons describing the effect of the central nervous system (CNS) hypoxic response; fsp and fsh, activity in efferent sympathetic fibers directed to the vessels and heart, respectively; fv, activity in the vagal efferent fibers; Rbp, Rhp, Rmp, Rsp, and Rep, peripheral resistance in the brain, heart, skeletal muscle, splanchnic, and remaining extrasplanchnic systemic vascular beds (see Fig. 2); Vu,mv, Vu,sv, and Vu,ev, unstressed volume in the skeletal muscle, splanchnic, and remaining extrasplanchnic venous circulation (see Fig. 2); Emax,rv and Emax,lv, end-systolic elastance of the right and left ventricle, respectively; T, heart period.

Vascular Compartments

View larger version (36K): [in this window] [in a new window]   Fig. 2.   Hydraulic analog of the cardiovascular system. P, pressures; R, hydraulic resistances; C, compliances; L, inertances; F, flows; sa, systemic arteries; sp and sv, splanchnic peripheral and splanchnic venous circulation; ep and ev, extrasplanchnic peripheral and extrasplanchnic venous circulation; mp and mv, peripheral and venous circulation in the skeletal muscle vascular bed; bp and bv, peripheral and venous circulation in the brain vascular bed; hp and hv, peripheral and venous circulation in the heart (coronary vascular bed); la, left atrium; lv, left ventricle; pa, pulmonary arteries; pp and pv, pulmonary peripheral and pulmonary venous circulation; ra, right atrium; rv, right ventricle.

Each compartment includes a hydraulic resistance (R), a hydraulic compliance (C), and the unstressed volume (V_u). The inertial effects (L) have been included only in the large artery compartments, where blood acceleration is maximal. Equations relating pressures and flows in the vascular system have been written by enforcing the conservation of mass at all nodes in Fig. 2 and the equilibrium of forces at L.

Heart

Local Vasodilatory Effect of Hypoxia

Afferent Pathways

Baroreceptor afferent pathway. The afferent information from the arterial baroreceptors is described by means of a first-order dynamic block, which exhibits a static and a rate-dependent gain, in series with a sigmoidal static characteristic. As shown in previous work (46), this arrangement allows several experimental results on baroreceptor stimulation (9) to be reproduced fairly well.

Afferent pathway from peripheral chemoreceptors. Similar to the baroreceptor model, the model of the peripheral chemoreceptors includes a dynamic block and a static nonlinear characteristic. However, because experimental evidence suggests that chemoreceptors have no rate-dependent component in their response to PO₂ (6), we preferred to adopt simple first-order low-pass dynamics. The static curve relating chemoreceptor activity to Pa_O2 exhibits a hyperbolic or negative exponential trend (5). However, chemoreceptor activity cannot increase indefinitely but flattens at a saturation level. This behavior has been reproduced using a combination of exponential functions.

Afferent information from lung stretch receptors. Inflation of lungs at low pressure activates slowly adapting lung stretch receptors with vagal myelinated A fibers (1). In the model the afferent activity of these receptors depends on changes in VT, which, in turn, are elicited by the peripheral chemoreceptor activation (see Effect of Peripheral Chemoreceptors on Ventilation). The response includes a static relationship that is quite linear at low inflation levels (1, 26, 38) and low-pass dynamics.

Efferent Neural Pathways

The vagal fibers are directed to the heart only and participate in the heart period control. By contrast, sympathetic activity is directed to the heart and blood vessels. An important modification of the present work, compared with the previous study, is that we used two different equations to describe the sympathetic activity to the heart (f_sh) and to blood vessels (f_sp). This choice is justified by the observation that cardiac and peripheral sympathetic activities do not change in parallel but exhibit disparate variations in response to afferent information from chemoreceptors and lung stretch receptors, as well as in response to CNS hypoxia.

As justified in a previous study (46), we assumed that sympathetic activity decreases with a negative monoexponential relationship in response to inhibitory afferent information, whereas it increases exponentially up to a saturation level in response to excitatory inputs. Similarly, inhibitory afferent information causes an increase in vagal activity, thus slowing heart rate, whereas excitatory information causes a reduction in the vagus frequency.

As described in PARAMETER ASSIGNMENT AND SIMULATION OF THE INDIVIDUAL COMPONENTS, the weighting factors and offset term have been given to mimic the effect of receptor stimulation on heart rate and systemic resistance observed experimentally.

Effect of CNS Hypoxia

Effect of Peripheral Chemoreceptors on Ventilation

Cardiovascular Effectors for Reflex Control

According to the previous description, peripheral resistance in the skeletal muscle circulation (R_mp) is subjected to a double control: the local effect of O₂ and the activity of efferent sympathetic nerves. In the model the interaction between these two mechanisms on R_mp is assumed to be multiplicative in nature. This choice is justified by the observation that skeletal muscle blood flow during deep hypoxia increases significantly, despite sympathetic activation (28).

	PARAMETER ASSIGNMENT AND SIMULATION OF THE INDIVIDUAL COMPONENTS

TOP ABSTRACT INTRODUCTION QUALITATIVE MODEL DESCRIPTION PARAMETER ASSIGNMENT AND... RESULTS DISCUSSION APPENDIX REFERENCES

Here we discuss how a value has been assigned to all parameters that characterize the new portions of the model. For each new subsystem (Fig. 1), a comparison is presented with experimental results to illustrate the input-output relationships and to reveal agreement between model curves and real data. The effect of the nonlinear interactions among the different parts is studied in RESULTS and, to a greater extent, in the companion article (48).

Vascular System

Heart

Local Vasodilatory Effect of Hypoxia

View larger version (16K): [in this window] [in a new window]   Fig. 3.   Response of the cerebral vascular bed to hypoxia. Normalized cerebral blood flow vs. PaO2 is shown in steady-state conditions. , Experimental data of McPherson et al. (36); , experimental data of McPherson et al. (37); , experimental data of Ulatowski et al. (45); continuous line, pattern of normalized model blood flow in segment b of Fig. 2.

The time constant of the local regulation has been assigned by assuming that O₂ exerts its action on cerebral, skeletal muscle, and coronary vessels mainly through the release of vasodilatory factors (such as adenosine) in the perivascular space (49). Values derived from the previous work suggest a time constant of ~10 s.

Afferent Pathways

Arterial baroreceptors. The parameters that describe the afferent pathway from arterial baroreceptors have been given the same values used in a previous work (46). These were assigned to reproduce experimental results by Chapleau and Abboud (9) and Kubota et al. (31) in dogs. Model dependence of baroreceptor afferent activity (spikes/s) on mean arterial pressure is shown in Fig. 4 and compared with experimental data.

View larger version (17K): [in this window] [in a new window]   Fig. 4.   Afferent information from arterial baroreceptors (spikes/s vs. mean baroreceptor pressure). Continuous line, model response in static (nonpulsatile) conditions; , results of Chapleau and Abboud (9).

Peripheral chemoreceptors. The parameters that describe the afferent activity from peripheral chemoreceptors in static conditions (see Eq. 17) have been given to reproduce experimental results by Biscoe et al. (5) (Fig. 5). These data suggest that afferent activity remains quite constant as long as Pa_O2 is higher than ~100 mmHg; then it increases.

View larger version (19K): [in this window] [in a new window]   Fig. 5.   Afferent information from chemoreceptors (spikes/s vs. PaO2) in isocapnic conditions. Continuous line, model results in steady-state conditions;  and , results of Biscoe et al. (5) in 2 different experiments.

Lung stretch receptors. In the model the activity of lung stretch receptors is a function of changes in VT. Most experimental data, however, relate the activity in vagal afferents to pulmonary inflation pressure (26, 38). To permit assignment of model parameters starting from these in vivo data, we assumed that the relationship between VT and inflation pressure is rather linear at moderate levels of lung inflation. Parameters in the static relationship (Fig. 6) have been assigned by considering the response of the so-called "slowly adapting receptors with myelinated A-fibers," since these are responsible for cardiovascular adjustments (tachycardia and vasodilation) at moderate levels of lung inflation.

View larger version (19K): [in this window] [in a new window]   Fig. 6.   Afferent information from slowly adapting lung stretch receptors with myelinated A fibers (spikes/s vs. VT percent changes). Continuous line, model results; , results by Kaufman et al. (26). In the experiment by Kaufman et al., lung inflation was expressed as percent change in pulmonary inflation pressure. Pulmonary inflation pressure changes were converted to VT changes with the assumption of a linear relationship between these quantities in the physiological range of lung inflation.

Efferent Neural Pathways

Sympathetic fibers to blood vessels. To characterize the activity of efferent sympathetic nerves directed to vessels, we need to assign four parameters: the synaptic weights (excitatory or inhibitory) connecting baroreceptors, chemoreceptors, and lung stretch receptors to sympathetic neurons and the offset term. Of course, baroreceptor and lung stretch receptor weights are inhibitory, since activation of these fibers causes vasodilation, whereas the chemoreceptor weight is excitatory. The weight connecting baroreceptors to sympathetic neurons was given the same value as in the previous work (46). The weight connecting chemoreceptors was assigned to reproduce the effect of chemoreceptor activation on systemic resistance reported elsewhere (2, 16, 17). According to these authors, activation of carotid and aortic chemoreceptors with hypoxic blood causes a 60-70% increase in total systemic resistance, in conditions when all other mechanisms are excluded. Finally, the weight connecting lung stretch receptors to sympathetic neurons was assigned to reproduce the experimental data by Daly et al. (13) showing a progressive decrease in total systemic resistance with increasing VT during lung inflation. A comparison between the effect of lung stretch receptors on systemic resistance in the model and in the experimental work by Daly et al. is shown in Fig. 7.

View larger version (19K): [in this window] [in a new window]   Fig. 7.   Response of the systemic vascular resistance to activation of lung stretch receptors. The steady-state percent decrease in total systemic resistance vs. percent changes in VT was evaluated when all other regulatory mechanisms were inoperative. Continuous line, model results; , experimental results of Daly et al. (13).

Sympathetic fibers to the heart. Reproduction of sympathetic activity to the heart still requires the assignment of synaptic weights. The baroreceptor weight was given the same inhibitory value used previously (46). The chemoreceptor weight was assigned starting from data reported in Daly and Scott (15, 16) and Karim et al. (25). Experiments with cholinergic blockade or vagotomy suggest that the primary bradycardic response to carotid chemoreceptor stimulation is predominantly mediated by the vagus (15, 16, 34). Moreover, Karim et al. observed that stimulation of aortic chemoreceptors results in an average increase in heart rate of ~11 beats/min due to activation of cardiac efferent sympathetic nerves. Starting from these data, we assumed that chemoreceptor activation increases heart sympathetic activity, mainly through activation of aortic chemoreceptors. Hence, the weight was given a positive value, assigned according to the results of Karim et al.

Efferent vagal activity. The positive weight from baroreceptor activity to the efferent vagal response was the same as in the previous study (46). The weight connecting chemoreceptor activity to the vagus was assigned a positive value on the basis of experimental results by Karim et al. (25) and Daly and Scott (16). These authors state that activation of carotid chemoreceptors by hypoxic (venous) blood causes a 20-30% vagally mediated reduction in heart rate. Finally, the tachycardic effect of lung stretch receptors was mimicked by assigning a negative weight, thus reducing vagal activity during lung inflation; the weight value was assigned to reproduce the progressive increase in heart rate measured by Hainsworth (23) at a moderate level of lung inflation. A comparison between model and in vivo data concerning the effect of lung inflation on heart rate is shown in Fig. 8.

View larger version (15K): [in this window] [in a new window]   Fig. 8.   Heart rate response to activation of lung stretch receptors. Heart rate changes in steady-state conditions vs. percent changes in VT were evaluated when all other regulatory mechanisms were inoperative. Continuous line, model results; , , and , experimental results of 3 different experiments reported by Hainsworth (23).

Hypoxia of the CNS

Pulmonary Ventilation

View larger version (19K): [in this window] [in a new window]   Fig. 9.   VT response to a step decrease in PaO2 from 95 to 38 mmHg, performed at 2 min. A: model simulation. B: clinical trace from Reynolds and Milhorn (40) in isocapnic conditions.

To verify the model's capacity to reproduce the VT response at various PO₂ levels, Fig. 10 compares the pattern of VT vs. PO₂ in steady-state conditions according to the present model with physiological data obtained by Reynolds and Milhorn (40) in humans at various Pa_O2 levels but constant Pa_CO2. The agreement between model predictions and in vivo data is fairly good over a wide range of PO₂.

View larger version (16K): [in this window] [in a new window]   Fig. 10.   Pattern of VT vs. PaO2 in steady-state conditions. Continuous line, model results; , results of Reynolds and Milhorn (40) in isocapnic conditions.

Effector Response

The model was simulated on Pentium-based personal computers by using the Runge-Kutta-Fehlberg 4/5 algorithm with adjustable step length for the numerical integration of differential equations.

	RESULTS

TOP ABSTRACT INTRODUCTION QUALITATIVE MODEL DESCRIPTION PARAMETER ASSIGNMENT AND... RESULTS DISCUSSION APPENDIX REFERENCES

Here we present a few simulation results, performed to validate the model during deep hypoxia, as to its steady-state and dynamic characteristics. In the conditions simulated below, we assumed that all mechanisms are present and superimpose their action in a dynamic nonlinear way.

Figure 11 shows the time patterns of SAP, heart rate, blood flow in the peripheral vascular beds with higher metabolic need (brain, skeletal muscle, and heart), and blood flow in all the remaining peripheral systemic vascular beds in response to a severe protracted hypoxia. Pa_O2 is decreased from 95 to 28 mmHg between 50 and 60 s, when the model is in steady-state pulsating conditions, and is maintained throughout the remaining simulation period. As clearly shown in Fig. 11, hypoxia in the model evokes the classic well-defined cardiovascular changes: mean SAP and heart rate increase within 1 min from the beginning of the perturbation. Furthermore, hypoxia causes a significant redistribution of the available blood flow: blood flow increases dramatically in the organs with higher metabolic need, whereas it initially decreases in the remaining systemic vascular beds, mainly as a consequence of chemoreflex vasoconstriction. As a consequence, cardiac output remains quite constant during the first 10-20 s after the perturbation. The subsequent gradual blood flow recovery in the "nonautoregulated" vascular bed is a consequence of the response of the lung inflation reflex and the CNS hypoxia, which increase heart rate and cardiac output, and of the vasodilatory action of the baroreflex and the lung inflation reflex, which reduce splanchnic and extrasplanchnic resistances.

View larger version (31K): [in this window] [in a new window]   Fig. 11.   Time pattern of systemic arterial pressure (A), heart rate (B), blood flow in the metabolically autoregulated vascular beds (C; parallel arrangement of branches m, h, and b in Fig. 2), and blood flow in all the remaining systemic vascular beds [D; parallel arrangement of splanchnic (s) and extrasplanchnic (e) branches in Fig. 2] simulated with the model after an isocapnic decrease in PaO2. PaO2 was decreased from 95 to 28 mmHg between 50 and 60 s and maintained throughout the remaining simulation period.

Figure 12 shows a quantitative comparison between the steady-state percent changes in the main cardiovascular quantities obtained with the present model and the experimental results by Koehler et al. (27) in the dog during isocapnic hypoxia. The agreement between model and in vivo data is satisfactory, apart from a slight underestimation of cardiac output changes by the model.

View larger version (35K): [in this window] [in a new window]   Fig. 12.   Normalized values of the main hemodynamic quantities [mean systemic arterial pressure (SAP), cardiac output (CO), heart rate (HR), total systemic resistance (TPR), resistance in the extrasplanchnic (e) branches (Rep) in Fig. 2] evaluated with the model in steady-state conditions during deep hypoxia (PaO2 = 28 mmHg). Model results (solid bars) are compared with the experimental data of Koehler et al. (27) (gray bars). 100% denotes values at PaO2 = 80 mmHg. Experimental percent changes of extrasplanchnic resistance have been computed from data on renal resistance reported by Koehler et al.

To test the model's capacity to reproduce not only steady-state values during hypoxia but also the dynamic aspects of the response, we performed a subsequent simulation in which deep hypoxia is protracted for a few seconds only, and then PO₂ is restored to its basal level. The aim of this simulation is to verify whether the model's response to transient chemoreceptor activation exhibits the temporal pattern noticed by Rutherford and Vatner (42). These authors observed a biphasic response after intracarotid injection of cyanide or nicotine in dogs. The early phase occurred after ~6-7 s and was characterized by bradycardia, a decrease in regional (iliac) blood flow, and an increase in mean SAP. The later phase occurred after ~12-13 s and was characterized by an increase in heart rate, a relevant increase in regional (iliac) blood flow, and a return of mean SAP at the initial level. Figure 13 shows that the model provides a quite similar biphasic response after transient chemoreceptor stimulation, with time delays close to those observed in vivo. The model ascribes the early phase to the primary action of chemoreceptors, whereas the delayed phase is mainly attributed to secondary activation of lung stretch receptors.

View larger version (44K): [in this window] [in a new window]   Fig. 13.   Time pattern of systemic arterial pressure (A), heart rate (B), and blood flow (C) in the splanchnic (branch s in Fig. 2) and extrasplanchnic nonautoregulated (D) (branch e in Fig. 2) systemic vascular beds simulated with the model in response to a transient deep arterial hypoxia. PaO2 was suddenly decreased from 95 to 25 mmHg between 20 and 26 s and then restored to its basal value with a time constant of 3 s. A biphasic response, characterized by chemoreceptor activation, occurred at ~6-8 s from the beginning of the perturbation, and antagonistic activation of lung stretch receptors occurred at ~18-20 s after the beginning of the perturbation.

	DISCUSSION

TOP ABSTRACT INTRODUCTION QUALITATIVE MODEL DESCRIPTION PARAMETER ASSIGNMENT AND... RESULTS DISCUSSION APPENDIX REFERENCES

The aim of this work was to develop a mathematical model of the cardiovascular regulatory response to isocapnic hypoxia by incorporating the multitude of physiological factors that are known. In our opinion, the development of such a model is needed for several reasons. First, the enormous amount of experimental data that has been gathered in the past decades might be only superficially utilized and their mutual relationships poorly perceived, if they are not summarized into a comprehensive theoretical structure. Second, simulation of the cardiovascular behavior in response to different stimuli (including hypoxia) may be of great didactic value. Ultimately, knowledge of the interplay among the different mechanisms in pathophysiological conditions may help in the analysis and management of critical patients, in whom hypoxia is the primary perturbation or is added to other cardiovascular stresses (e.g., pulmonary embolism, heart failure, and myocardial infarction).

When the mathematical model was developed, the single portions of the overall system were first recognized; then each part was individually described according to physiological data to characterize its specific input-output static relationship and temporal dynamics. The ultimate system behavior was then obtained as the result of the nonlinear superimposition and mutual interactions among all these different constituents. Hence, physiological knowledge was included at two different levels in the model: the behavior of the individual portions and the way the different portions are mutually linked and interact to generate the overall response.

In the present study we analyzed the whole model response to deep hypoxia, when all parameters have a basal value assigned on the basis of physiological data. The results are in satisfactory agreement with those arising from classic physiological experiments (1, 24, 27, 30, 34): isocapnic hypoxia causes an increase in heart rate, mean SAP, and cardiac output, a decrease in venous capacity and total systemic resistance, and a significant blood flow redistribution toward the organs with higher metabolic demand. The model ascribes these well-defined cardiovascular adjustments to a complex "compromise" between various excitatory and inhibitory influences, which superimpose at different time intervals after the hypoxic perturbation.

Because the various control mechanisms exhibit a variety of temporal lags, i.e., temporal heterogeneity, the complexity of their superimposition can be better appreciated by looking at the time pattern of the hemodynamic quantities. In this regard, an important model result is that the temporal response to hypoxia displays two subsequent phases: an early phase, dominated by the activation of the peripheral chemoreflex, and a later phase, characterized by the participation of other regulatory actions. Looking at the time behavior of the different quantities (Figs. 11 and 13), we can speculate that the two phases have a different physiological meaning and respond to different protective requirements.

In the early phase (lasting ~10-20 s after a step change in Pa_O2), one can observe significant vasoconstriction and blood flow reduction in the organs subjected only to reflex control, with a moderate increase in mean SAP. In the same period, cardiac output remains rather constant; hence, a significant amount of blood flow is diverted to the organs with higher metabolic demand. This blood flow redistribution is further favored by the progressive vasodilation in the coronary, brain, and skeletal muscle circulation. Because of the moderate arterial hypertension, joined with reduced vascular resistance, blood flow to these organs increases two- or threefold above the basal value during a deep hypoxia.

The later phase of the response is characterized by various regulatory actions, which develop with a greater time constant. They include the increase in ventilation, triggered by the peripheral chemoreceptor activation, a progressive increase in heart rate, a reduction of the vasoconstriction in the reflexly regulated organs, and a further increase in mean SAP. During the later phase of the response, the main regulatory mechanisms responsible for the hemodynamic changes are the lung stretch receptors and the hypoxia of the CNS. Moreover, the baroreflex system also significantly participates in vasodilation during the later phase, responding to the gradual increase in mean SAP. As a consequence of these adjustments, blood flow in the reflexly regulated organs progressively returns toward a value close to the basal one.

The chain of events described above has a functional significance. The early response aims at maintaining adequate O₂ delivery in the organs with higher metabolic rate, despite the dramatic reduction in the O₂ content of blood, at the same time avoiding excessive work for the heart. Indeed, during the early phase, not only cardiac output remains quite constant, but also heart rate is reduced. We hypothesize that this kind of response has a protective function for the heart; an excessive increase in cardiac output and heart rate, in fact, joined with increased mean SAP, would imply a dramatic increase in power generation by the cardiac pump, with a consequent dramatic increase in its metabolic requirement. It is thus natural that cardiac output and heart rate do not increase in this phase and that arterial pressure exhibits only a moderate increase.

Significant augmentation of heart rate, cardiac output, and mean SAP (hence, a strong increase in cardiac power) occurs in the later phase only, when pulmonary ventilation has already risen; hence, O₂ delivery to the organism is ameliorated. At this time, a certain O₂ delivery can be ensured also to the organs with lower autoregulation, enforcing greater activity and power generation for the heart.

The difference between the early and late response is evident in several experimental works performed on animals (including dogs and primates). Various authors in recent years observed that the response to chemoreceptor stimulation in spontaneously breathing animals is time variable: the skeletal muscle displays short-lasting vasoconstriction followed by later vasodilation (14, 16); heart rate consistently shows initial bradycardia followed by tachycardia. The delayed response is abolished or significantly attenuated if chemoreceptor stimulation is performed with controlled ventilation or after denervation of the lungs (42).

Even though the present model is able to account for the amplitude and time pattern of the main phenomena involved in the cardiovascular response to isocapnic hypoxia, it also exhibits some simplifications and limitations, which may become the target of future extensions.

A first limitation of the model is the absence of equations for gas exchange at the lung and in the tissue. Inclusion of these aspects will be necessary in future work to simulate conditions of real clinical relevance, such as asphyxia, respiratory diseases, or altitude hypoxia. In all these conditions, Pa_O2 is not an input quantity (as in the present work) but a controlled quantity that varies according to a mass balance in the lung and tissue. O₂ mass balance, in turn, depends on various factors, such as the O₂ fraction in the inspired air, changes in ventilation and in local blood flow, and tissue metabolism.

A second limitation in the model is the absence of a role for PCO₂ changes; i.e., we simulated isocapnic hypoxia only. By contrast, in most clinical conditions a fall in Pa_O2 is accompanied by large changes in arterial CO₂ content. For instance, during lung disease, the fall in Pa_O2 occurs along with a rise in PCO₂; conversely, at high altitude the fall in Pa_O2 is followed by a decrease in PaCO₂, consequent to the rise in ventilation. Inclusion of the cardioventilatory effect of PCO₂ changes will require analysis of many additional phenomena that are not included in this work: especially the nonlinear interaction between O₂ and CO₂ at the peripheral chemoreceptors (20), the effect of central chemoreceptors on ventilation (39), and the local CO₂ effect on the autoregulated peripheral vessels. The complexity of these aspects justifies the choice to consider only O₂ in the present preliminary report to achieve progressive and step-by-step validation of the individual aspects.

Another limitation is that sympathetic outflow in the model depends on the additive combination of the three reflexes, with different weighting factors (see Eqs. 21 and 22). This is a drastic simplification, since data in the literature suggest the existence of a more complex interaction at the central neural level (8). The introduction of more reliable equations for central processing might be the subject of future studies, perhaps by use of neural network modeling techniques. Nevertheless, it is interesting to observe that even the simple central interaction proposed in this work (Eqs. 21 and 22) permits the net final cardiovascular response to be simulated quite well, without the need for a more sophisticated neural approach.

Another aspect to be underlined is that the model was designed to simulate the cardiovascular response in large mammals. Simulation of different animals may be carried out by simply scaling all parameters per unit weight (or by considering the percent changes of hemodynamic quantities). However, the model is probably inadequate to simulate the response in small mammals (such as rodents), for which different scaling factors are necessary. For instance, rats do not exhibit an increase in arterial pressure during hypoxia, whereas they often show a decrease.

Finally, in the present model, we did not include the role of cardiopulmonary (or low-pressure) baroreceptors. The latter might be responsible for a reflex increase in heart rate with an increase in filling pressure (Bainbridge reflex). Because an increase in filling pressure occurs during hypoxia, as a consequence of venoconstriction, it is possible that the absence of the Bainbridge reflex in this work might have led us to an underestimation of the heart rate increase.

In conclusion, the present model allows a quantitative study of the cardiovascular response to isocapnic hypoxia, with particular emphasis on the temporal superimposition among the different control mechanisms at different levels of the cardiovascular system. However, to better clarify the specific role of each mechanism and the effect of its possible alterations, it will be of value to simulate the regulatory system at different levels of Pa_O2 and to test its sensitivity to parameter changes. These aspects are investigated in depth in the companion study (48).