INTRODUCTION

Top Abstract Introduction Results Discussion Appendix References

THE CEREBROVASCULAR BED in humans and animals is subjected to the action of sophisticated regulatory mechanisms that work to maintain an adequate cerebral blood flow (CBF) for functional and metabolic needs. Intracranial vessels respond promptly to changes in cerebral perfusion pressure (CPP; cerebral arterial pressure minus cerebral venous pressure); the response, called autoregulation, ensures that CBF remains approximately constant in the CPP range of 50-150 mmHg (38). Moreover, cerebral vasculature is particularly sensitive to chemical stimuli, especially O₂ and CO₂ tension in arterial blood. These mechanisms interact in complex nonlinear ways in response to perturbations affecting pressure and blood concentration levels simultaneously.

There are many factors that make the analysis of cerebrovascular regulation particularly complicated. First, consecutive vascular segments in the brain contribute differently to CBF control, a phenomenon often referred to as "segmental heterogeneity" (6). Some authors have observed that the large pial vessels play a major role in the regulatory response to moderate arterial pressure changes, whereas small pial arteries undergo massive vasodilation when CPP approaches the lower autoregulation limit (25, 29).

In addition, cerebral circulation takes place within a closed space (the skull and the craniospinal axis) with only a limited capacity to buffer blood volume changes; consequently, any modification in vessel caliber (either passive or induced by cerebrovascular regulatory mechanisms) may affect intracranial pressure (ICP) through changes in cerebral blood volume (CBV). Because ICP is the extravascular pressure of cerebral vessels and is almost equal to pressure in the large cerebral veins, its changes can have a significant impact on cerebral hemodynamics.

Because of the close relationship among the action of cerebrovascular control mechanisms, CBV, and ICP, CBF can be affected by several noncirculatory aspects of intracranial dynamics, especially craniospinal elasticity (defined by the intracranial pressure-volume relationship) and the circulation of cerebrospinal fluid (CSF).

The interaction among ICP, CBV, and cerebrovascular regulation must be taken into account in the treatment of patients with severe brain lesions. Changes in arterial CO₂ pressure (Pa_CO2) and mean systemic arterial pressure (SAP) are frequently used in neurosurgical intensive care units to assess the status of cerebrovascular control mechanisms (8, 48). Moreover, manipulation of arterial CO₂ concentration is routinely used as a therapeutic means in controlling ICP and adjusting CBF to metabolic requirements (34, 37). However, uncontrolled changes in CO₂ or in arterial pressure can have a dramatic impact on ICP and CBF, leading to acute intracranial hypertension and cerebral ischemia (12).

The complexity of the aforementioned relationships makes it difficult to achieve a clear understanding of cerebrovascular control in simple qualitative terms. For this reason, various mathematical models of cerebral hemodynamics and ICP dynamics have been presented in past years (7, 21, 22, 31, 47, 49). We recently developed a mathematical model of the interaction among cerebral autoregulation, CBV changes, and ICP in humans (50, 51). The model was able to reproduce various phenomena concerning ICP, such as the genesis of plateau waves (51), the ICP response to arterial pressure perturbations (14), or the ICP changes induced by bolus injection within the craniospinal space (PVI tests) (52).

The aim of this study was to significantly improve and extend the previous model to include the CO₂ reactivity of cerebral vessels, its nonlinear interaction with ICP and with cerebral autoregulation, and the description of the transcranial Doppler (TCD) velocity signal. The TCD technique, in fact, is routinely used today to assess cerebral hemodynamics due to its characteristics of noninvasiveness and continuous data acquisition (1).

There are two main justifications for developing this new model. We aimed to 1) demonstrate that several experimental results on cerebral autoregulation and chemical CBF regulation presented in past years can be summarized into a single theoretical setting, and 2) improve the interpretation of clinical maneuvers concerning cerebrovascular control and ICP dynamics.

The present preliminary study presents the model, its justification on the basis of current physiological knowledge, and its validation through experimental results taken from physiological literature. A second, related study (27) presents a sensitivity analysis of model parameters and validation of the model on the basis of clinical data from real patients with severe head trauma.

	QUALITATIVE MODEL DESCRIPTION

The model is qualitatively presented, with attention focused on its new aspects. A more thorough analysis can be found in previous reports (50, 51). The complete set of mathematical equations is given in the APPENDIX.

Intracranial Hemodynamics and CSF Dynamics

The electric analog in Fig. 1 represents the four intracranial compartments considered in the model and their mutual relationships, together with the extracranial venous drainage pathway. These are briefly described in the following sections.

View larger version (12K): [in this window] [in a new window]   Fig. 1.   Electric analog of intracranial dynamics. G1 and C1, hydraulic conductance and compliance, respectively, of proximal cerebral arteries; G2 and C2, hydraulic conductance and compliance, respectively, of distal cerebral arteries; Pa, systemic arterial pressure (SAP); P1 and P2, intravascular pressure of large pial arteries and medium and small arteries, respectively; q, cerebral blood flow (CBF); Pc and Pv, capillary and cerebral venous pressure, respectively; Pvs and Pcv, sinus venous and central venous pressure, respectively; Pic, intracranial pressure (ICP); Cic, intracranial compliance; Gpv and Cvi, hydraulic conductance and compliance of large cerebral veins; Gvs, hydraulic conductance of terminal intracranial veins (bridge veins and lateral lacunae or lakes); Gve and Cve, hydraulic conductance and compliance, respectively, of extracranial venous pathways; Gf and Go, conductances to cerebrospinal fluid (CSF) formation and CSF outflow; qf and qo, rates of CSF formation and CSF outflow; and Ii, artificial CSF injection rate.

Pial arteries and arterioles. Pial arteries and arterioles comprise the sections of the cerebrovascular bed directly under the control of regulatory mechanisms. We make a distinction between two consecutive segments, the large pial arteries (proximal segment) and the medium and small pial arteries (distal segment). In the following equation, subscript j = 1 indicates a quantity or a parameter that belongs to the proximal arteries, whereas subscript j = 2 refers to the distal arterioles. Each segment is represented by a transverse hydraulic capacity (C_j), at which all blood volume changes are concentrated, and a longitudinal hydraulic conductance (G_j), which reproduces the pressure drop upstream and downstream of the capacity. The distinction between two segments is justified by the heterogeneity of cerebrovascular regulation (6).

View larger version (17K): [in this window] [in a new window]   Fig. 2.   Pattern of wall tension vs. inner radius in large (top) and small (bottom) pial arteries computed with model in steady-state basal conditions. As demonstrated by some authors (10), vessels of different sizes from a singular vascular bed have similar mechanical characteristics. Dotted line: elastic tension; dot-dashed line: active tension; continuous line: total tension. Dashed line represents tangential force per unit length, computed according to the Laplace law in basal conditions.  denotes basal value of inner radius at equilibrium;  denotes equilibrium level following vasodilation induced by complete smooth muscle relaxation, assuming constant intravascular and extravascular pressures. Note greater vasodilatory capacity of small pial arteries compared with large pial arteries.

Cerebral venous circulation. The circulatory pathway from cerebral capillaries down to the dural sinuses also consists of a series arrangement of venous capacity and upstream and downstream longitudinal conductances. Because autoregulation plays a negligible role in cerebral veins (4), we adopted a simpler description for venous mechanics than for the pial arterial-arteriolar vasculature. The venous compliance (C_vi) is inversely dependent on the local transmural pressure, implying a monoexponential pressure-volume relationship. The terminal portion of the intracranial venous bed (bridge veins and lateral lacunae or lakes, represented by the conductance G_vs in the model) passively collapses or narrows during intracranial hypertension, with a mechanism similar to that of a Starling resistor (4, 33). Because of this mechanism, cerebral venous pressure (P_v) is always slightly higher than ICP, and upstream large cerebral veins and venules remain open during intracranial hypertension. Consequently, conductance of the proximal venous circulation (G_pv) is constant. The remaining part of the vascular bed, from dural sinuses down to the heart, is summarized by the hydraulic conductance (G_ve) and compliance (C_ve) of the extracranial venous pathways.

CSF compartment. We have assumed that the processes of both CSF production at the cerebral capillaries and CSF outflow at the dural sinuses are passive and unidirectional. Hence, they are proportional to the corresponding transmural pressure value and fall to zero when this becomes negative.

Middle cerebral artery blood flow velocity. Additional equations in the model have been included to estimate an approximate value for blood flow velocity in the middle cerebral artery (MCA) and to compare it with experimental data as recorded with the TCD technique. On the basis of studies conducted on healthy subjects and neurosurgical patients (13, 36), we assumed that the behavior of the MCA is essentially passive in nature; the transmural pressure (SAP  ICP) is a monoexponential function of the inner radius (Fig. 3). The parameters of this relationship have been set to reproduce the pressure-radius curves concerning human intracranial basal arteries (20). Finally, the MCA velocity (V_MCA) is calculated as the ratio of blood flow to vessel cross-sectional area. To this end, we assumed that about one-third of total CBF passes through each MCA. Based on these assumptions, the basal value of V_MCA in the model is as high as 60 cm/s, which agrees with the values reported by Aaslid et al. (1) (normal range 33-90 cm/s, mean 62 cm/s). Furthermore, during maneuvers that affect SAP or CO₂ pressure, the V_MCA changes are in agreement with those measured with the TCD technique (see RESULTS). This aspect has been more deeply analyzed in a second, related paper (27).

View larger version (18K): [in this window] [in a new window]   Fig. 3.   Theoretical pattern of transmural pressure (SAP  ICP) vs. inner radius in middle cerebral artery (rMCA). To qualitatively reproduce data reported in Ref. 20, we assumed a monoexponential pressure-radius relationship.  denotes basal working point at equilibrium, characterized by a transmural pressure of ~90 mmHg and an inner radius of 0.15 cm.

Cerebrovascular Regulation Mechanisms

The block diagram in Fig. 4 shows how feedback mechanisms modulate the activation factor. The upper branch refers to cerebrovascular autoregulation mechanisms, whereas the lower branch refers to CO₂ reactivity. The contributions of autoregulation and CO₂ reactivity to smooth muscle tension in the jth segment are described by two state variables, x_aut,j and x_CO2,j, respectively. The dynamics of each is reproduced by means of a first-order low-pass filter, characterized by a gain factor (G_aut,j or G_CO2,j) and a time constant (_aut,j or _CO2,j).

View larger version (13K): [in this window] [in a new window]   Fig. 4.   Block diagram shows how different regulatory mechanisms modify smooth muscle activation factor (Mj) of large (j = 1) and small (j = 2) pial arteries. Upper branch describes autoregulation mechanisms, whereas lower branch indicates CO2 response. Input quantity for CO2 mechanisms in both segments is logarithm of arterial CO2 tension (PaCO2), i.e., PaCO2 = log10 (PaCO2/PaCO2n), where PaCO2n represents PaCO2 value that does not affect basal smooth muscle tension. In contrast, input quantity for autoregulation is different in the two segments: large pial arteries actively respond to changes in cerebral perfusion pressure (CPP), whereas small pial arteries are sensitive to CBF percent changes (CBF). Dynamics of each mechanism are simulated by means of a gain factor (G) and a first-order low-pass filter with time constant . Gain factor of CO2 mechanisms is multiplied by corrective factor ACO2, which lowers CO2 reactivity at low CBF levels (see inset) as a consequence of tissue ischemia. Finally, autoregulation and CO2 mechanisms interact nonlinearly through a sigmoidal static relationship.

As shown in Fig. 4, we have assumed that autoregulation acts on large pial arteries and small arterioles by means of two distinct feedback mechanisms: the mechanism on large arteries depends on changes in CPP, reflecting a myogenic or neurogenic mechanism; and the mechanism on small arteries is triggered by alterations in blood flow to cerebral tissue, according to a metabolic or endothelium-dependent response. This distinction aims at reproducing the heterogeneity of autoregulation observed by Kontos et al. (25) and MacKenzie et al. (29) in cats.

In contrast, CO₂ is assumed to affect smooth muscle tension, in both large and small pial arteries, through a similar mechanism. Because the prevalent hypothesis ascribes the vasoactive effect of CO₂ to pH changes in the perivascular space (24, 34), we have assumed that model response depends on the logarithm of CO₂ changes rather than on Pa_CO2 by itself. Hence, the following input quantity is used in the block diagram in Fig. 4 The parameter Pa_CO2 n denotes the level of CO₂ pressure that does not affect the basal smooth muscle tension. By changing this parameter, adaptation of cerebrovascular reactivity to prolonged hypercapnia or hypocapnia can be simulated (34, 39).

Furthermore, we have included two kinds of nonlinear interaction between autoregulation and CO₂ reactivity. First, on the basis of several experimental studies (8, 18), we have assumed that the strength of the CO₂ mechanism is not independent on CBF but, rather, is attenuated during severe brain ischemia. Ischemia, in fact, causes tissue acidosis, which has a buffer effect on the influence of CO₂ on perivascular pH. Accordingly, the CO₂ reactivity in the model decreases exponentially when CBF falls below normal (see inset in Fig. 4).

Second, Fig. 4 shows that autoregulation and CO₂ reactivity are not simply summed to obtain the activation factor (M_j); rather, they interact through a sigmoidal relationship, which accounts for the existence of lower and upper limits for cerebrovascular reactivity.

A value characterizing the parameters in the sigmoidal relationship has been set so that smooth muscle tension falls to zero during maximal vasodilation and can rise to twice the basal value during maximum vasoconstriction.

The autoregulation time constants were set at the same values used in previous studies (50, 51). In particular, we assumed that the pressure-dependent mechanisms acting on large pial arteries exhibit rapid dynamics (_aut,1 = 10 s), whereas the flow-dependent response of pial arterioles is slightly slower (_aut,2 = 20 s).

The choice of time constants for CO₂ reactivity deserves further comment. Kontos et al. (24) found that arteriolar dilation in response to a pH decrease takes place rapidly, with a time constant of <10 s. Schneider et al. (43), during experiments on feline pial arteries with the use of a microelectrode, observed that pH and vascular diameter varied with a delay of ~30 and 45 s, respectively, after a CO₂ change; Severinghaus and Lassen (45) reported a time constant of 20 s for CO₂ reactivity in humans, whereas Auer (3) in his study on cats measured a delay between 7 and 48 s. It is worth noting that, in the last three cases, the measured quantity was the end-tidal CO₂ pressure; the observed delays therefore also included the transit time for blood to pass from the lung to the brain. Because in our model the input quantity is the arterial CO₂ pressure, the time constant includes only the time necessary for Pa_CO2 to alter perivascular pH and that required for the vessel to react to a pH change. Hence, we have chosen a value of _CO2,j = 20 s for both large and small pial arteries. In the analysis of neurosurgical patients reported in our related paper (27), we have used a greater value for the parameter _CO2,j, because the monitored quantity is end-tidal rather than arterial CO₂ pressure.

Values for the gain factors of regulation mechanisms have been set to reproduce results obtained in animals when experiments are executed at constant ICP (see RESULTS).

A list of all model parameters and their values at basal conditions can be found in Table 1.

	RESULTS

Top Abstract Introduction Results Discussion Appendix References

Model simulations have been performed to analyze 1) cerebral autoregulation, 2) CO₂ reactivity, 3) the interactions between these two mechanisms, and 4) the effect of acute changes in CO₂ on ICP. With regard to the first three topics, only results obtained under steady-state conditions are presented and compared with experimental data. Moreover, to simulate experiments performed with the dura opened or at constant ICP, during these trials we used an extremely low value for the intracranial elastance coefficient. Hence, ICP changes were negligible throughout the simulations (ICP  9.5 mmHg). In contrast, the last trials concern dynamic conditions, with ICP free to change as a consequence of blood volume and CSF volume alterations.

Cerebral Autoregulation

View larger version (27K): [in this window] [in a new window]   Fig. 5.   Percent changes in inner radius of large (top left) and small (bottom left) pial arteries and percent changes in CBF (top right) and MCA velocity (VMCA; bottom right) evaluated with model in steady-state conditions at different mean SAP levels. Throughout simulations, ICP was set at a constant level (~9.5 mmHg). Model results (continuous lines) are compared with experimental data reported by Kontos et al. (25) in cats (), MacKenzie et al. (29) in cats (), Harper et al. (19) in rats (×), and Nelson et al. (35) in rabbits (+).

CO₂ Reactivity

View larger version (26K): [in this window] [in a new window]   Fig. 6.   Percent changes in inner radius of large (top left) and small (bottom left) pial arteries and percent changes in CBF (top right) and VMCA (bottom right) evaluated with model in steady-state conditions at different levels of PaCO2. Throughout simulations, ICP was set at a constant level (~9.5 mmHg). Model results (continuous lines) are compared with experimental data reported by Wei et al. (53) in cats (+), Raper et al. (40) in cats (), and Levasseur and Kontos (26) in rabbits (, ×) for radii; with data from Reivich (41) in monkeys () and Harper and Glass (18) in dogs () for CBF; and with data from Markwalder et al. (30) in healthy human subjects (×) and Seiler and Nirkko (44) in clinical patients () for VMCA.

Interaction Between Autoregulation and CO₂ Reactivity

View larger version (29K): [in this window] [in a new window]   Fig. 7.   Bottom: percent changes in inner radius in small pial arteries (distal segment) vs. mean SAP during normocapnia (, PaCO2 = 40 mmHg) and hypercapnia (, PaCO2 = 71 mmHg). Top: percent difference between vessel radius during hypercapnia and normocapnia at same arterial pressures. Model results (left) are compared with experimental data by Wei et al. (53) in cats (right). Throughout simulations, ICP was set at a constant level (~9.5 mmHg).

Figure 8 describes the dependence of CBF on mean SAP (autoregulation curve) under steady-state conditions at various Pa_CO2 levels. Obviously, an increase in Pa_CO2 causes an increase in CBF at a given arterial pressure value. At the same time, the lower autoregulation limit shifts toward higher pressure; in fact, cerebral vessels are nearly maximally dilated because of hypercapnia and therefore are unable to dilate further in response to arterial hypotension. These findings are confirmed by several authors who studied the effects of induced hypotension at various Pa_CO2 levels (16, 17, 37, 39).

View larger version (21K): [in this window] [in a new window]   Fig. 8.   Effect of hypocapnia and hypercapnia on CBF. Pattern of CBF vs. mean SAP (autoregulation) was simulated with model in steady-state conditions at different levels of PaCO2 (dotted line: 20 mmHg; continuous fine line: 30 mmHg; continuous bold line: 40 mmHg; dashed line: 60 mmHg; dot-dashed line: 90 mmHg). Note that hypercapnia shifts lower autoregulation limit toward higher pressures.

Figure 9 describes the effect of arterial hypotension on the relationship between CBF and Pa_CO2. Model results are compared with those obtained by Harper and Glass (18) during experiments on dogs. It can be seen that arterial hypotension progressively reduces CO₂ reactivity. The most considerable effect of hypotension is visible during hypercapnia, when the arterial bed is already dilated to the maximum. On the other hand, experimental data suggest that the vasoconstrictory effect of hypocapnia is also reduced during severe hypotension (18). The model can reproduce this behavior, attributing it to a reduction in the CO₂ reactivity during cerebral ischemia (see block diagram in Fig. 4 and Eqs. 25A and 26A in APPENDIX).

View larger version (22K): [in this window] [in a new window]   Fig. 9.   Effect of hypotension on CBF. Percent changes in CBF vs. PaCO2 were simulated with model in steady-state conditions at different mean SAP levels. All simulations were performed at constant ICP (~9.5 mmHg). Model results (continuous lines) are compared with experimental data () collected by Harper and Glass (18) in dogs.

Effects of Hypocapnia and Hypercapnia on ICP

Figure 10 shows the ICP response to an acute rise in Pa_CO2. Hypercapnia causes a vasodilation with an increase in CBV, leading to a sudden ICP rise; subsequently, however, ICP returns monotonically toward baseline due to CSF reabsorption. The final ICP value remains slightly high with respect to the initial situation because, in the model, the increase in CBF causes a parallel increase in CSF production rate. As intracranial compliance diminishes, the capacity to buffer CBV variations decreases, and ICP shows a dramatic transient rise. In the extreme case, the ICP pattern resembles that of the single A wave or plateau wave observed by Lundberg (28).

View larger version (17K): [in this window] [in a new window]   Fig. 10.   Effect of hypercapnia on ICP. Time pattern of ICP (bottom) was simulated with model in response to a sharp PaCO2 increase from 40 to 60 mmHg (top) in a patient with a moderate decrease in CSF outflow conductance (Go = 6.33 × 104 mmHg1 · s1 · ml); i.e., with moderate intracranial hypertension during normocapnia. Results of 3 different simulations are shown, assuming normal intracranial compliance (continuous line; kE = 0.11 ml1), moderately reduced intracranial compliance (dashed line; kE = 0.143 ml1), and severely reduced intracranial compliance (dotted line; kE = 0.22 ml1).

In Fig. 11 the ICP reaction to induced hypocapnia is plotted. The conditions are similar to those in Fig. 10; in this case, however, CO₂ change produces a clear drop in ICP that is partially recovered by the process of CSF production. The final ICP level is lower than the initial level due to a decrease in CSF production rate. Also in this case, the lower the craniospinal storage capacity, the greater the ICP alteration.

View larger version (18K): [in this window] [in a new window]   Fig. 11.   Effect of hypocapnia on ICP. Time pattern of ICP (bottom) was simulated with model in response to a sharp PaCO2 decrease from 40 to 20 mmHg (top) in a patient with a moderate decrease in CSF outflow conductance (Go = 3.80 × 104 mmHg1 · s1 · ml); i.e., with moderate intracranial hypertension during normocapnia. Results of 3 different simulations are shown, assuming normal intracranial compliance (continuous line; kE = 0.11 ml1), moderately reduced intracranial compliance (dashed line; kE = 0.143 ml1), and severely reduced intracranial compliance (dotted line; kE = 0.22 ml1).

	DISCUSSION

Top Abstract Introduction Results Discussion Appendix References

The present work improves our previous model significantly, including the analysis of CO₂ cerebrovascular reactivity and the interaction among CO₂, autoregulation, and ICP. The model aims at representing an acceptable compromise between physiological accuracy and clinical usefulness. In fact, even though the emphasis of this study is on the modeling of physiological experiments, an understanding of the pathophysiology of intracranial dynamics is an essential prerequisite in the treatment of severe brain diseases.

An important methodological aspect of this study is that many different experimental results, concerning various phenomena of intracranial dynamics often considered separately in the literature, can be summarized into a single theoretical setting. The main aspects elucidated in this work are discussed separately below.

Cerebrovascular Segmental Heterogeneity

Moreover, in this study, as in a previous one (51), we assumed that autoregulation of the large pial artery is dominated by pressure-dependent mechanisms, whereas the response of small pial arteries mainly depends on blood flow changes. This hypothesis permitted us to reproduce not only the active response of the vessels quite well but also all CBF and V_MCA changes at many levels of arterial pressure and CO₂. Hence, we are confident that this subdivision into two regulated segments actually represents the best compromise between accuracy and simplicity and allows cerebrovascular reactivity to be reproduced with a limited number of parameters. Moreover, the good correlation between model and experimental CBF data confirms the minor role that the venous cerebrovascular bed plays in CBF regulation.

Cerebrovascular Response to CO₂

It is well known that CO₂ reactivity tests are frequently conducted on neurosurgical patients to assess cerebral vessel vasomotor capacity (8, 12, 48). Usually CO₂ reactivity is quantified by an index equal to CBF percent changes per mmHg of Pa_CO2 change. We assessed the sensitivity of CBF to CO₂ variations in the model by computing the central slope of the curve shown in Fig. 6. The value obtained is ~2.5% CBF change/mmHg Pa_CO2 change. This value quite closely concurs with those obtained during physiological experiments on animals [2.2%/mmHg by Jones et al. (23) in rats; 2.1%/mmHg by Göbel et al. (15) in rats; and 2.87%/mmHg by D'Alecy et al. (9) in dogs]. With regard to human subjects, Tenjin et al. (48) suggested the value 2.0%/mmHg as a threshold to discriminate between patients with preserved and impaired CO₂ reactivity.

Interaction Between Autoregulation and CO₂ Response

Several experimental results show that arterial hypotension can critically interfere with the cerebrovascular response to CO₂ changes. Harper and Glass (18), in a classic study, demonstrated that reducing arterial pressure to about one-third that of normal completely eliminates the vasodilatory response to hypercapnia and seriously weakens the vasoconstrictory response to hypocapnia. The latter result was further confirmed by others (2, 3).

A thorough analysis of the interaction between autoregulation and hypercapnia on pial vessel response can be found in the study by Wei et al. (53). According to their findings (see Fig. 7), the interaction between the two mechanisms is significantly nonlinear, i.e., the superimposition of the effects does not hold. In particular, as blood pressure decreases starting from a basal level, the responsiveness to CO₂ becomes initially greater, suggesting a "facilitatory" overlapping of the two mechanisms. However, when arterial pressure approaches the autoregulation lower limit, CO₂ responsiveness progressively disappears, indicating an "inhibitory" overlapping of autoregulation and hypercapnia.

It is well known that hypercapnia and hypocapnia significantly modulate the autoregulation curve. Hypercapnia shifts the autoregulation plateau to higher CBF levels, and hypocapnia reduces it to a lower CBF. Moreover, data collected by several authors concur, showing that the lower autoregulation limit is fairly shifted by CO₂ changes (16, 17, 37, 39). More controversial results can be found for the effect of CO₂ on the upper limit of autoregulation (11, 16, 38).

We were able to qualitatively reproduce most of the results mentioned above (Figs. 7-9) by supposing the presence of only two significant nonlinearities in the interaction between autoregulation and CO₂ response. Both nonlinearities are confirmed by physiological considerations.

A first nonlinearity in the model derives from a sigmoidal static relationship and accounts for the natural bounds of smooth muscle tension (see last block in Fig. 4) and permits reproduction of most of the nonlinearities observed during combined stimuli. However, the presence of this sigmoidal relationship for smooth muscle tension did not allow us to explain the lack of vasoconstrictory response to hypocapnia observed during severe arterial hypotension (2, 3, 18). It has been claimed that the failure of cerebral vessels to constrict to hypocapnia during hypotension indicates that maintenance of adequate tissue O₂ supply prevails over the maintenance of adequate CO₂ and acid-base levels (18). To account for this phenomenon, we assumed that cerebrovascular CO₂ reactivity depends on CBF through a sharply nonlinear relationship (see inset in Fig. 4); when CBF falls below an alarm threshold (indicating the initiation of ischemia), CO₂ reactivity is suddenly withdrawn. As a consequence of this assumption, cerebrovascular response to hypocapnia progressively disappears when mean SAP decreases below the autoregulation lower limit (SAP < 50 mmHg) in accordance with data by Artru and Colley (2). The existence of an "ischemic threshold" for CBF, below which CO₂ reactivity drops to zero, is also confirmed by clinical observations (8).

Interaction Between CBF Control and ICP

The new simulation results presented in Figs. 10 and 11 confirm the existence of a similar vasodilatory cascade in response to maneuvers that alter CO₂ concentration in blood. Two main aspects of these simulations deserve consideration. First, the impact of CO₂ changes on ICP crucially depends on the status of intracranial compensatory mechanisms (CSF outflow and intracranial elastance). The same maneuver, with almost no effect on ICP in physiological conditions, can potentially induce dramatic ICP changes in pathological states characterized by reduced CSF outflow and poor intracranial capacity. In the worst conditions, even a mild hypercapnia can induce a disproportionate transient ICP rise, similar to that of a single plateau wave, with the risk of acute intracranial hypertension and cerebral ischemia (Fig. 10).

Furthermore, the results of Fig. 11 indicate how hyperventilation can be used to control ICP in patients at risk of intracranial hypertension. It should be noted that, in our model, the effect of CO₂ changes on ICP depends on two concurrent mechanisms that usually operate with different time constants: 1) active CBV changes, which cause rapid alterations in ICP, generally within a minute of the maneuver; and 2) alteration in CSF production rate, which takes place with a longer time constant (several minutes) and contributes to set the final steady-state ICP level. In the model, CSF production rate depends on capillary pressure and is therefore proportional to CBF.

An important new aspect of this model concerns the relationship between CBF and V_MCA. In this study we assumed that the MCA behaves passively, i.e., its inner radius is a monotonic function of transmural pressure; neither autoregulation nor CO₂ significantly affects the caliber of the MCA in the model. The consequence of such an assumption is that, during moderate variations in arterial pressure, velocity does not decrease, unlike CBF, due to variations in cross-sectional area. This behavior is supported by the experimental results by Nelson et al. (35) in the rabbit. In contrast, during CO₂ alterations, the velocity percent changes are of the same order as CBF changes, because transmural pressure and cross-sectional area in the large arteries remain almost unaffected. This model behavior agrees with data reported previously (30, 44). It is possible, however, that a different behavior for the MCA (inner radius being dependent on Pa_CO2 or on an active myogenic response) may have significant consequences on model performance.

In perspective, the possibility of simulating the time pattern of ICP and TCD velocity in response to maneuvers affecting CO₂ and/or mean SAP could be used to estimate the main parameters of intracranial dynamics from measurements taken in neurosurgical intensive care units. On the basis of knowledge of these parameters in the individual patient, a greater understanding of pathophysiological mechanisms can be achieved and a more well-founded treatment for severe brain damage developed. This model application on real velocity and ICP tracings is the subject of a second, related study (27).