INTRODUCTION

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SEVERAL QUANTITIES are monitored today in neurosurgical intensive care units to assess the status of patients with severe brain diseases and decide on a proper management. Among others, the time pattern of blood flow velocity in the middle cerebral artery (MCA) and the time pattern of intracranial pressure (ICP) may contain important information on several aspects of intracranial dynamics, including cerebrospinal fluid (CSF) circulation, craniospinal compliance, cerebral hemodynamics, and the status of cerebrovascular regulatory mechanisms (10, 13, 21, 23, 28, 35).

Analysis of these data, however, is complicated considerably by the existence of strong nonlinear interactions among intracranial factors. It is generally accepted that ICP is not only affected by CSF circulation and the elasticity of the craniospinal system but also by acute blood volume changes occurring within the cranial space (7, 27). Cerebral blood volume (CBV) changes are modulated in turn by autoregulation and the reactivity of cerebral vessels to CO₂ tension. Furthermore, the superimposition of autoregulation and CO₂ reactivity occurs in a nonlinear fashion so that the performance of one mechanism depends crucially on the activity of the other.

Understanding the complexity of the relationships between intracranial quantities is essential in both physiological investigation, to reach a deeper understanding of how control mechanisms work to ensure cerebral blood flow (CBF) homeostasis, and clinical practice. In fact, changing the level of arterial pressure and arterial CO₂ pressure (Pa_CO2) is thought to provide useful information in the diagnosis and treatment of neurosurgical patients (1, 13, 23, 35). In a previous, related paper (32) we developed a mathematical model of intracranial dynamics aimed at clarifying the interactions among ICP, CBF, blood velocity changes, and cerebrovascular regulation. The model in question incorporates the intracranial pressure-volume relationship, CSF circulation, the biomechanics of large and small cerebral arteries, a collapsing venous vascular bed, and the active response of the pial vasculature to autoregulation and CO₂ pressure changes. Using a single basal set of parameters, we were able to simulate various experimental results taken from the physiological literature with a fair measure of success.

A more extensive model validation is now required. In the previous study (32), a comparison between simulation results and physiological data was performed in steady-state conditions by waiting until ICP and vessel caliber had reached a constant equilibrium level following arterial pressure and/or Pa_CO2 perturbations. However, testing the predictive capability of the model also is extremely important in dynamic conditions, in which all quantities are varying simultaneously. Several meaningful intracranial events, in fact (such as acute ICP changes, ICP wave generation, and paradoxical responses), become clearly visible only when the system is perturbed from its equilibrium state. To estimate parameters in a dynamic system, moreover, the analysis of transient phenomena generally provides more useful information than analysis of the static regimen (5).

In following up on these ideas, this second study extends the results of its predecessor (32) by including two important new aspects: 1) investigation of the model response in dynamic conditions (i.e., far removed from equilibrium) during perturbations that alter mean systemic arterial pressure (SAP) and blood CO₂ tension, by performing a sensitivity analysis on the parameters believed to exhibit greater changes in pathological subjects, and 2) validation of the model, based on a comparison between the dynamic time patterns of ICP and MCA velocity (V_MCA) generated by computer simulations and real tracings measured in patients with severe head injury. Moreover, through best fitting of model results to in vivo results, we sought to estimate the main parameters of intracranial dynamics in individual patients. Different combinations of SAP and Pa_CO2 are involved in this step as well. The results are then used to assess the main mechanisms affecting ICP and CBF in different conditions of physiological relevance.

	MATERIALS AND METHODS

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A thorough description of the model is presented in a previous, related work (32) in which all details on mathematical equations and parameter numerical values can be found. A qualitative sketch of the model's structure is shown in Fig. 1.

View larger version (29K): [in this window] [in a new window]   Fig. 1.   Qualitative diagram describing main physiological factors included in model. Intracranial compliance is inversely related to intracranial pressure (ICP). Middle cerebral artery (MCA) is assumed to behave passively, hence its radius depends on transmural pressure [arterial pressure minus ICP (Pa  Pic)]. Autoregulation operates in small and large pial arteries by way of two distinct mechanisms, one sensitive to perfusion pressure changes [arterial pressure minus cerebral venous pressure (Pa  Pv)] and another to cerebral blood flow (CBF) changes. Furthermore, large and small pial arteries alike are sensitive to CO2 pressure in arterial blood (PaCO2). The terminal portion of the cerebral venous vascular bed collapses according to difference between ICP and sinus venous pressure (Pic  Pvs). Finally, cerebrospinal fluid (CSF) production at cerebral capillaries depends on difference between capillary pressure and ICP (Pc  Pic), whereas CSF outflow depends on difference between ICP and sinus venous pressure (Pic  Pvs). VMCA, MCA velocity.

Sensitivity Analysis

The sensitivities of V_MCA and ICP to changes in these parameters were tested using various combinations of mean SAP and Pa_CO2. For each set of parameters, we performed a preliminary simulation during hypocapnia (Pa_CO2 = 25 mmHg) and waited until the model settled at its steady-state equilibrium level. Hypocapnia was chosen to establish the baseline because neurosurgical patients are often continuously hyperventilated to reduce ICP (8, 13, 24). Starting from the equilibrium condition, we increased Pa_CO2 linearly from 25 to 35 mmHg (normocapnia) over a 20-min period. Finally, Pa_CO2 was lowered abruptly back to its initial level (see Fig. 2). For each set of parameters, the same maneuver was performed during normotension (mean SAP = 100 mmHg), hypotension (mean SAP = 65 mmHg), and hypertension (mean SAP = 135 mmHg) to study the interaction between autoregulation and CO₂ response. These arterial pressure levels are representative of three different points in the autoregulation region, that is, the central range and the ranges close to the lower and upper autoregulation limits (25).

View larger version (16K): [in this window] [in a new window]   Fig. 2.   Time pattern of PaCO2 used to perform sensitivity analysis. During the maneuver, which lasts for 20 min, PaCO2 increases linearly from 25 to 35 mmHg.

All simulations were performed by means of the software package SIMNON (SIMNON/PCW for Microsoft Windows, version 2.01, SSPA Maritime Consulting, Göteborg, Sweden) using the Runge-Kutta-Fehlberg method with adjustable step size to numerically integrate the set of differential equations (26).

Model Identification

Patients. We studied six patients with severe head injuries (Glasgow coma scale <8; mean age 29 yr) typified by closed multifocal contusions or diffuse axonal injuries within the first few days after head trauma. After the removal of large epidural or subdural hematomas, if present, patients were put into intensive care and positioned supine with their heads elevated 30° above the horizontal plane. Patients were sedated (midazolam and fentanyl) and paralyzed (vecuronium) at the time of the measurement. SAP (radial catheter), ICP (ventricular catheter), and end-tidal CO₂ pressure (PET_CO2) were monitored continuously. Jugular venous O₂ saturation (Sj_O2) was continuously monitored via a fiber-optic catheter (Opticath 5.5 Fr, Abbott Laboratories, North Chicago, IL) inserted retrogradely up to the jugular bulb of the dominant jugular vein. Moderate hyperventilation was induced to obtain a Pa_CO2 between 30 and 35 mmHg. According to the routine management protocol of our establishment, if cerebral perfusion pressure (CPP) was <80 mmHg and either ICP was >20 mmHg or Sj_O2 was 55%, norepinephrine was continuously infused after optimization of blood volume. In such cases, CPP was stabilized between 80 and 100 mmHg to normalize ICP and Sj_O2. If CPP was compromised between 65 and 80 mmHg but with ICP <20 mmHg and Sj_O2 >55%, no therapeutic action was initiated. Second-range therapies included bolus mannitol infusion and/or hyperventilation under Sj_O2 control. None of our patients had received mannitol in the 6 h preceding the study. Before the study, when a steady-state hemodynamic condition was achieved, all patients underwent transcranial Doppler measurement (TCD; Angiodyn DMS, France) of both middle cerebral arteries (2). Patients with V_MCA >100 cm/s at the basal state were excluded from this study because they might possibly present heterogeneous autoregulation and CO₂ reactivity in the two hemispheres due to either hyperemia or posttraumatic vasospasm. The TCD transducer was then fixed to a head holder, and V_MCA was monitored continuously through the temporal window of the more severely injured hemisphere. The spectral outline of MCA Doppler time recording, SAP, ICP, and PET_CO2 signals were sampled (analog-to-digital converter, National Instruments, Houston, TX) with a personal computer and stored for off-line analysis.

Parameter estimation. Estimation of model parameters was carried out by adopting an automatic procedure. Starting from an initial guess, we modified certain model parameters iteratively by a numerical algorithm to minimize a cost function of the difference between model and in vivo results. Because statistical information on the measurement errors was not available, we adopted a weighted least-squares cost function [F()] (5)

(1)

where P^(s)_ic and V^(s)_MCA represent the in vivo ICP and V_MCA values at the time instant t_i, P^(m)_ic and V^(m)_MCA are the corresponding model predictions at the same instant, P_a and PET_CO2 are systemic arterial pressure and end-tidal CO₂ tension (to be used as inputs for the model), N is the number of available data points, W_ICP and W_VMCA are weighting factors, and  = [₁, ₂ ... _p]^T is a vector of model parameters to be identified through the minimization algorithm. Throughout this work, superscript T denotes vector or matrix transpose. The weighting factors were chosen so that, at the end of the minimization procedure, the two terms in the right-hand member of Eq. 1 would have comparable values.

	RESULTS

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Sensitivity Analysis

View larger version (34K): [in this window] [in a new window]   Fig. 3.   Sensitivity analysis of role of autoregulation gain. Time pattern of ICP (A) and nonlinear dynamic relationship linking ICP to VMCA (B) were computed with model during CO2 alteration maneuvers (from hypocapnia to normocapnia) at different levels of mean systemic arterial pressure (SAP). CO2 maneuver is shown in Fig. 2. Simulated patients were characterized by a moderate increase in CSF outflow resistance (Ro/Ro 0 = 5, where subscript 0 indicates basal value); a moderate decrease in craniospinal elasticity, i.e., an increase in elastance coefficient (kE/kE 0 = 1.3); and different values of autoregulation gain for small pial arteries [Gaut,2/Gaut,2 0 = 0.25 (dashed line), Gaut,2/Gaut,2 0 = 0.75 (dotted line), Gaut,2/Gaut,2 0 = 1 (continuous bold line), Gaut,2/Gaut,2 0 = 1.25 (dot-dashed line), and Gaut,2/Gaut,2 0 = 1.5 (continuous fine line)].

Autoregulation gain. Figure 3 shows the effect of the autoregulation gain in the small pial arteries on the ICP and V_MCA time patterns. Five different values for the gain have been used (see legend, Fig. 3), ranging from depressed autoregulation to increased autoregulation above normal. In each simulation, the intracranial elastance coefficient and the CSF outflow resistance were moderately increased compared with their basal values (k_E/k_{E 0} = 1.3, R_o/R_{o 0} = 5) to describe a pathological subject.

CSF outflow resistance. The role of the CSF outflow resistance on the ICP and V_MCA time patterns during CO₂ changes is presented in Fig. 4. In all these simulations we used the same value for the intracranial elastance coefficient as in Fig. 3, but we used a normal autoregulation gain.

View larger version (39K): [in this window] [in a new window]   Fig. 4.   Sensitivity analysis of role of CSF outflow resistance. Time pattern of ICP (A) and nonlinear dynamic relationship linking ICP to VMCA (B) were computed with model during CO2 alteration maneuvers (from hypocapnia to normocapnia) at different levels of mean SAP. CO2 maneuver is shown in Fig. 2. Simulated patients were characterized by normal autoregulation gains (Gaut,1/Gaut,1 0 = 1; Gaut,2/Gaut,2 0 = 1), a moderate decrease in craniospinal elasticity, i.e., an increase in elastance coefficient (kE/kE 0 = 1.3), and different values of CSF outflow resistance [Ro/Ro 0 = 1 (continuous bold line), Ro/Ro 0 = 2.5 (dashed line), Ro/Ro 0 = 5 (dotted line), Ro/Ro 0 = 7.5 (dot-dashed line), and Ro/Ro 0 = 10 (continuous fine line)].

Intracranial elastance coefficient. Figure 5 shows the time pattern of ICP and the dynamic "ICP-V_MCA" relationship during Pa_CO2 changes at different values of the parameter k_E (the range of values extends from k_E/ k_{E 0} = 0.5, which means good intracranial compliance, to k_E/k_{E 0} = 2, which represents a very rigid craniospi- nal system with seriously reduced compliance). The simulations were performed with the assumption of a moderate increase in CSF outflow resistance (R_o/R_{o 0} = 5, the same value as in Fig. 3) but with a normal autoregulation gain. It should be noted that an increase in k_E is reflected in a more rapid ICP rise during the maneuver (Fig. 5A), hence also in a steeper ICP-V_MCA curve (Fig. 5B). In extreme conditions (i.e., rigid craniospinal compartment with arterial hypotension), ICP rises uncontrolled, with V_MCA deteriorating. The main difference between these curves and those of Fig. 4 is that, conversely to the situation with changes in R_o, changes in k_E have no effect on the initial equilibrium point.

View larger version (35K): [in this window] [in a new window]   Fig. 5.   Sensitivity analysis of role of intracranial elastance coefficient. Time pattern of ICP (A) and nonlinear dynamic relationship linking ICP to VMCA (B) were computed with model during CO2 alteration maneuvers (from hypocapnia to normocapnia) at different levels of mean SAP. CO2 maneuver is shown in Fig. 2. Simulated patients were characterized by normal autoregulation gains (Gaut,1/Gaut,1 0 = 1; Gaut,2/Gaut,2 0 = 1), a moderate increase in CSF outflow resistance (Ro/Ro 0 = 5), and different values of intracranial elastance coefficient [kE/kE 0 = 0.5 (dashed line), kE/kE 0 = 1 (continuous bold line), kE/kE 0 = 1.3 (dotted line), kE/kE 0 = 1.5 (dot-dashed line), and kE/kE 0 = 2 (continuous fine line)].

Model Identification

Figure 6 shows the results concerning tracing 1 of patient 2. The quantities varied artificially during the clinical observation (PET_CO2 and SAP, which have been used as inputs to the model) are plotted in Fig. 6A, whereas Fig. 6B shows the comparison between monitored and simulated outputs, i.e., ICP and V_MCA. The model proves able to reproduce both clinical tracings quite well. The subject starts from a condition of hypocapnia. After some minutes, a CO₂ maneuver is performed, consisting of a initial mild hyperventilation that further decreases CO₂ pressure; the CO₂ level is then gradually increased by lowering the ventilator rate. SAP remains at a nearly constant low value during this maneuver. It can be seen that, with PET_CO2 decreasing, vasoconstriction occurs and ICP and V_MCA diminish; the subsequent rise in CO₂ causes an increase in CBV, with a consequent rise in ICP. V_MCA increases rapidly as well, rising to a maximal value, after which it remains substantially constant or even falls, despite the continuing upward trend in CO₂ and ICP values.

View larger version (20K): [in this window] [in a new window]   Fig. 6.   Best fitting between model simulation and clinical data in tracing 1 of patient 2. A: measured quantities used as inputs for model [top: mean SAP; bottom: end-tidal CO2 pressure (PETCO2)]. A CO2 maneuver, consisting of passage from hypocapnia to normocapnia and back to hypocapnia, was performed at constant mean SAP. B: time pattern of VMCA (top) and ICP (bottom) measured in patient (continuous fine line with ) and simulated with model using parameters listed in Table 3 (continuous bold line). C: dynamic relationship linking ICP to VMCA during CO2 maneuver delimited between arrows in A (, clinical data; continuous bold line, model results).

In Fig. 6C, the dynamic relationship between ICP and V_MCA is plotted with regard to the CO₂ maneuver delimited by the arrows in Fig. 6A (from minimal to maximal value of PET_CO2). The pattern is very similar to those discussed in the sensitivity analysis section. The initial portion of the curve is covered as long as the cerebrovascular dilation, caused by the CO₂ rise, is able to increase velocity; thereafter, velocity increases more slowly because of the antagonism between cerebral vasodilation and ICP rise. After an inflection point, vasodilation causes only a rise in ICP, which is accompanied by deteriorating V_MCA.

Figure 7 deals with patient 3. For each of the two tracings, a quite similar CO₂ maneuver is performed at two different SAP levels. In tracing 1 (Fig. 7, A and B), an arterial hypotension is produced, maintaining PET_CO2 at a normocapnic level. Observing the subsequent ICP rise, we can infer that an active vasodilation occurs, which is still not sufficient to prevent the accompanying fall in blood velocity. The time pattern of the following CO₂ maneuver is analogous to that of the previous patient. After the attainment of an upper limit of PET_CO2 (inflection point), blood velocity increases no further and a steeping in ICP is observed, suggesting the existence of a vasodilatory cascade.

View larger version (35K): [in this window] [in a new window]   Fig. 7.   Best fitting between model simulations and clinical data in both tracings of patient 3. A and C: measured quantities of mean SAP (top) and PETCO2 (bottom) used as inputs for model. B and D: time pattern of VMCA (top) and ICP (bottom) measured in patient (continuous fine line with ) and simulated with model using parameters listed in Table 3 (continuous bold line). A and B relate to CO2 maneuvers performed during hypotension (tracing 1), whereas C and D relate to CO2 changes during hypertension (tracing 2). E: dynamic relationships linking ICP to VMCA during CO2 maneuvers delimited between arrows in A and C during hypotension (tracing 1: , clinical data; continuous bold line, model results) and hypertension (tracing 2: , clinical data; continuous bold line, model results), respectively.

In tracing 2 (Fig. 7, C and D), the change in PET_CO2 is brought about in a condition of arterial hypertension. It can be observed in this case that an increase in CO₂ is accompanied by a parallel, monotonic increase in both ICP and V_MCA. The comparison between the ICP-V_MCA curves of the two tracings, shown in Fig. 7E, allows analysis of the effect of CPP on CO₂ maneuvers. During hypotension (tracing 1), a condition of maximal velocity is attained at a somewhat lower ICP level. With the assumption of a mean arterial pressure of ~80 mmHg, the inflection point is located at a CPP of 80  25 = 55 mmHg. In case of hypertension (tracing 2), the ICP-V_MCA relationship is shifted to the right and an inflection point is not attained, because CPP is significantly higher than in tracing 1. If we suppose that the general intracranial status is much the same in the patient, as inferred by the fair similarity between the model parameters in the two tracings (see Table 3, patient 3), we would find the inflection point at an ICP level of ~65 mmHg, with SAP ~120 mmHg.

The last example, referring to patient 6, is reported in Fig. 8. This tracing is a severe test for the model. The period of observation, in fact, is considerably long and contains four consecutive maneuvers, two involving PET_CO2 and two involving SAP (Fig. 8A). The simulated data seem to reproduce the main dynamic events with sufficient adequacy (Fig. 8B). The only significant discrepancy concerns the pattern of V_MCA, which, after the first reduction in mean SAP, appears to be better autoregulated in the model than in the clinical tracing. With regard to the first CO₂ maneuver, executed during arterial hypotension, both ICP and V_MCA hardly increase despite the CO₂ rise, denoting a condition of maximal vasodilation. Accordingly, the dynamic relationship depicted in Fig. 8C shows the presence of an inflection point; however, ICP stabilizes above inflection, notwithstanding a continuing rise in PET_CO2, confirming that CBV cannot increase further. In fact, the sum of the responses to hypotension and to hypercapnia brings the pial vasculature to its maximal vasodilation state.

View larger version (23K): [in this window] [in a new window]   Fig. 8.   Best fitting between model simulation and clinical data in patient 6. A: measured quantities of mean SAP (top) and PETCO2 (bottom) used as inputs for model. B: time pattern of VMCA (top) and ICP (bottom) measured in patient (continuous fine line with ) and simulated with model using parameters listed in Table 3 (continuous bold line). CO2 maneuvers were performed during both hypotension and hypertension. C: dynamic relationships linking ICP to VMCA during CO2 maneuvers delimited between arrows in A. Maneuver 1 was performed during hypotension (, clinical data; continuous bold line, model results), and maneuver 2 was performed during hypertension (, clinical data; continuous bold line, model results).

The subsequent maneuver consists of a sudden elevation in arterial pressure. The initial, passive response of the cerebrovascular bed is evident, reflected in the increase of both V_MCA and ICP, and then, when the regulatory action is triggered, CBV is actively reduced and ICP diminishes with the further rise in SAP. The behavior of the system during the following CO₂ change is similar to that reported in tracing 2 of patient 3 (see Fig. 7, C and D). Given the high value of CPP, V_MCA and ICP rise together as long as hypoventilation continues, as indicated in Fig. 8, B and C.

	DISCUSSION

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In the present study, we investigated the relationships among ICP, CBV, CSF circulation, and the action of cerebrovascular regulatory mechanisms in dynamic conditions during maneuvers that alter Pa_CO2 and mean SAP. The main objective was to elucidate the complex links between ICP and the action of cerebrovascular regulatory mechanisms and to gain a quantitative understanding of how these relationships may actually affect intracranial dynamics and CBF in neurosurgical subjects.

The idea that cerebrovascular regulation and ICP changes are strictly related is not new and has been the subject of several reports in previous years. Among others, Lundberg (21) first ascribed the occurrence of ICP waves to active changes in CBV occurring within the cranial cavity. Rosner and Becker (27) formulated a qualitative model, named the vasodilatory cascade, to explain how autoregulation may induce disproportionate ICP changes. Bouma et al. (7) and Gray and Rosner (16) observed that acute arterial hypotension may induce an ICP increase, which can differ widely from one subject to the next.

In a previous, recent study (30), we used a similar but simpler model of intracranial dynamics to analyze the ICP time pattern in patients with head injury during bolus injection or withdrawal maneuvers, the so-called PVI tests. The main finding of the study was that, in more than one-half of the examined patients, ICP exhibited a "paradoxical response" to the maneuver, that is, a delayed increase after the end of the injection period and a delayed decrease after the end of the withdrawal period. The model was able to explain these responses quite well, attributing them to active CBV changes induced by the maneuver. The study in question (30) dealt only with autoregulation, however, and the examined ICP tracings lasted for a few minutes, which is a time consistent with autoregulation dynamics following bolus perturbations.

The present study significantly extends and improves the former in two major areas: 1) we included the effect of CO₂ tension on cerebral hemodynamics, and 2) we examined the response to long-lasting perturbations. As reported by various authors (8, 10, 13, 28, 35), CO₂ reactivity is well correlated with neurological condition and clinical outcome; thus studying the cerebrovascular response to a CO₂ perturbation has a great clinical importance. Moreover, the clinical tracings examined here are of between 20- and 90-min duration and therefore are representative of a much longer dynamic than those analyzed with PVI tests.

The study was divided into two distinct parts, one devoted to a sensitivity analysis of the main model parameters and another to best fitting of real clinical tracings.

Sensitivity Analysis

Taken together, all these results emphasize that an identical maneuver may have different effects on the time patterns of intracranial quantities, depending not only on the specific individual parameters but also on the result of previous maneuvers or perturbations that may have affected the pial vasculature.

In this study we used the dynamic curve linking ICP to V_MCA obtained during Pa_CO2 changes to summarize the status of cerebral hemodynamics and its relationship to ICP dynamics. The advantage of using this curve is that its modifications can be correlated easily with changes in certain important intracranial parameters. In particular, the sensitivity analysis suggests that the relationship is quite linear in the central regulatory range and that its slope is strongly affected by the intracranial elastance coefficient. Moreover, a decrease in CSF outflow may induce an upward shift of the entire relationship. When the pial arteriolar vascular bed approaches maximal vasodilation, the ICP-V_MCA relationship exhibits an inflection point: ICP rises, whereas velocity remains constant or even decreases. In this condition, CBV changes are large and may trigger a vasodilatory cascade, leading to disproportionate ICP changes, whereas velocity and CBF cannot be further improved. The suggestion therefore is that the patient's intracranial dynamics should be kept far from this inflection point to avoid possible ischemia and intracranial instability. In particular, our simulations confirm the result of a previous study (14), according to which CPP should always be maintained >70-80 mmHg to keep the cerebral vasculature at a comfortable distance from its critical zone.

Fitting of Model Results to In Vivo Tracings

All these requisites have been addressed quite satisfactorily in the study. In particular, the clinical tracings show precisely the chain of events anticipated by the model at different arterial pressure levels. Increasing PET_CO2 from hypocapnia to normocapnia evokes an increase in ICP and a parallel increase in V_MCA, whereas hyperventilation decreases ICP and leads to a fall in velocity. During arterial hypotension, the ICP increase may become dramatic. Accordingly, the pattern of ICP versus V_MCA is significantly affected by the level of mean SAP; falling pressure generally results in a shift of the curve to the left and in the appearance of an inflection point, after which velocity falls (Figs. 7 and 8).

In all the cases examined, the in vivo time pattern of ICP and V_MCA could be reproduced quite well with the model by adjusting a few parameters that characterize intracranial elasticity, CSF outflow, and cerebrovascular reactivity (both autoregulation and CO₂ response). In most cases, moreover, the values of estimated parameters agree with those expected on the basis of the present pathophysiological knowledge.

In most of the tracings examined, the value of CSF outflow resistance was significantly increased with respect to normal. This finding suggests that an impairment in CSF circulation is probably one of the main causes of intracranial hypertension in neurosurgical patients. The values found in this study (ranging between 9.83 and 92.2 mmHg · min · ml¹) are almost similar to those measured by various authors in patients with hydrocephalus, subarachnoid hemorrhage, or severe brain injury [52-100 mmHg · min · ml¹ (18), 11.5-85 mmHg · min · ml¹ (20), 6.66-111.11 mmHg · min · ml¹ (6), or 29-100 mmHg · min · ml¹ (15)].

The estimated values of the intracranial elastance coefficient are likewise in substantial agreement with those reported in the clinical literature. The values reported in Table 3 encompass a broad range of pathophysiological levels, including cases with very low intracranial elastance coefficient, i.e., good intracranial storage capacity (k_E = 0.03 ml¹ in patient 6); cases with a mild increase in k_E; and cases with greatly increased elastance, denoting a strong pathological reduction in craniospinal capacity (k_E = 0.28 ml¹ in patient 5). According to various authors (4, 12), typical values of k_E range between 0.05 and 0.15 ml¹, albeit values as high as 0.3-0.4 ml¹ are not infrequent in neurosurgical subjects.

Another interesting aspect emerging from the analysis of Table 3 is that, in most cases, the autoregulation gains are lower than the basal values set previously (32). More exactly, although in a few tracings the estimated autoregulation gains indicate that autoregulation is reasonably well maintained, in other tracings autoregulation at the small pial artery level is close to zero (tracing 1 in patients 4 and 5) or autoregulation is significantly attenuated in both the proximal vessels and arterioles (tracing 1 in patient 3). This result confirms the observation of a former study (30), in which the autoregulation gain was estimated during PVI tests in 20 patients with head injury, and suggests that poor autoregulation is probably a frequent occurrence in severe head injury (9, 10, 13, 24).

The values of gain in CO₂ reactivity exhibit the highest variability among the estimated parameters. This wide variability in CO₂ gain may depend on both an intrinsic variability among subjects and a model limitation. In particular, in Table 3 we can observe the existence of a positive correlation between the values of the CO₂ gain and the mean arterial pressure level set during the maneuver. This result emphasizes that hypotension attenuates the CBF response to CO₂, as observed by other authors (3, 19). When building the model, we attempted to take this phenomenon into account, assuming that severe ischemia almost completely inhibits CO₂ response in large and small pial vessels [see Figs. 4 and 9 in our previous, related paper (32)]. The data in Table 3 suggest that this aspect of the model must be refined in future studies and that a more direct relationship between arterial pressure and CO₂ gains should be devised.

A further important parameter in the model, for which the value has to be individually estimated to reach a satisfactory fitting, is the initial working point on the smooth muscle sigmoidal characteristic. To move the working point (which is equivalent to shifting the sigmoidal relationship left or right), we individually estimated the parameter Pa_CO2 n. The basal value for this parameter was set at 40 mmHg in the previous paper (32), with the assumption that, in normocapnia, smooth muscle works at the central point of the sigmoidal relationship. Looking at the corresponding column in Table 3, however, we can see that, in all the tracings examined, the estimated value of Pa_CO2 n is significantly lower, ranging between 19 and 38 mmHg. This result suggests that only at reduced Pa_CO2 does smooth muscle work at the central point of the sigmoidal relationship, where the regulatory capacity is maximal. There are various possible justifications for this finding. First, the patients are ordinarily maintained in a status of moderate hypocapnia to limit intracranial hypertension. Hence, the value of Pa_CO2 n found by the algorithm may simply reflect adaptation to the low CO₂ set by artificial ventilation (22). In support of this idea, we may observe that the value of Pa_CO2 n estimated by the algorithm is always close to the initial PET_CO2 pressure set in the patient by artificial ventilation. An alternative explanation can be found in the works by Aaslid et al. (1) and Newell et al. (23), who measured velocity using the TCD technique during transient decreases in SAP and observed that the autoregulation strength in both healthy volunteers and patients with head injury increases when Pa_CO2 is lowered from 37 to 28 mmHg. This result suggests that autoregulation works better during hypocapnia (24).

Naturally, despite the encouraging results obtained, this study also raises important questions that merit future study. First, the model shows some difficulties in fitting very long tracings, involving multiple maneuvers that exhibit simultaneous changes in SAP and Pa_CO2 (see Fig. 8). Of course, this difficulty may simply be a consequence of the nonstationary nature of long-lasting signals, i.e., the set of parameters might change during an observation period as long as 1-2 h. During such a long period, moreover, patient behavior may be affected by other long-term mechanisms (such as changes in osmotic pressure or in metabolism) not included in the present model. Finally, as mentioned above, we also feel that the nonlinear relationship between CO₂ reactivity and mean SAP deserves future improvements to achieve a better fitting in respect of multiple maneuvers.

Finally, one possible objection is that the model appears somewhat complex, with an excessive number of parameters. Indeed, the aim of this work is to provide a fairly comprehensive overview of the main factors involved in cerebrovascular control and intracranial dynamics, without dwelling on the problem of structural identifiability and model reduction. In future studies, it may become necessary to simplify the model and to group different mechanisms into simplified equations if the emphasis is to be on clinical practice rather than on physiological investigation. With this in mind, we recently developed a drastically simplified model of ICP dynamics designed especially to favor parameter identification at the patient's bedside (31, 33). The inclusion of CO₂ reactivity in that model, following the indications of the present study, could be of future clinical value in assisting the quantitative diagnosis and treatment of neurosurgical patients.