INTRODUCTION

TOP ABSTRACT INTRODUCTION QUALITATIVE MODEL DESCRIPTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

CEREBRAL PERFUSION PRESSURE (CPP) and CO₂ pressure have a strong impact on intracranial vessels through the action of cerebrovascular control mechanisms (7, 13, 32). Under normal circumstances, a fall in blood pressure causes a rapid dilatation of resistance vessels, whereas a rise in blood pressure causes vasoconstriction (23, 26). As a consequence, cerebral blood flow (CBF) is maintained at the preexisting level in healthy subjects in the CPP range of 50-150 mmHg [a phenomenon named "autoregulation" (11)].

Moreover, arterial PCO₂ (Pa_CO2) is a strong vasodilator of the cerebral vasculature because it is able to increase CBF by 100% during severe hypercapnia (17, 37). The role of CO₂ is mainly mediated by pH changes in the perivascular space (11, 40).

The previous mechanisms, however, not only affect cerebral hemodynamics but also may have a serious impact on intracranial pressure (ICP) via complex nonlinear relationships (7, 16). First, cerebral circulation occurs within a closed space (the skull and neuroaxis); hence, alterations in cerebral blood volume (CBV) may modify ICP through the craniospinal pressure-volume relationship. Furthermore, CBF variations modulate the cerebrospinal fluid (CSF) production rate and ICP through changes in intracranial capillary pressure.

Knowledge of the previous relationships is of paramount importance for the treatment of patients with severe head injury in neurosurgical intensive care units. Decreasing CO₂ pressure through hyperventilation, in fact, is frequently used to reduce ICP in patients at risk of intracranial hypertension in whom cerebral vasculature is reactive (13, 35). Vasoconstriction, however, may induce an excessive fall in CBF, with the potential hazard of cerebral ischemia and secondary brain damage (32). Setting a correct level of mean systemic arterial pressure (SAP) is also controversial. Although some authors (2) in recent years proposed to maintain low SAP in head-injured patients to minimize the risk of cerebral edema, a more common opinion today is that CPP should be maintained >70-80 mmHg to avoid hypoperfusion and the consequent brain ischemia (7, 15, 36, 39). The danger of ICP instability (a phenomenon often called "vasodilatory cascade") should also be considered when CPP approaches the autoregulation lower limit (38, 50).

This scenario is complicated by the observation that cerebrovascular control mechanisms may be impaired in head injury. Severe head trauma disturbs autoregulation in most patients (7, 13, 21, 35). Impairment in cerebrovascular reactivity, in turn, may be a consequence of the trauma per se, may be secondary to alterations in ICP and craniospinal storage capacity, or may occur during episodes of arterial hypotension with reduced CPP (15, 36).

Because cerebral hemodynamics adjustments in head-injured patients are complex multifactorial phenomena, they may be better understood with the use of mathematical models and computer simulation techniques. Many such models have been presented in the previous decades, with the focus on different aspects of intracranial dynamics and cerebrovascular control (8, 20, 28, 49); however, none of them focused attention on the interaction among ICP, CBV, CSF circulation, and cerebrovascular control mechanisms.

In recent years we formulated a mathematical model of the relationships among ICP, CBV, and cerebrovascular control mechanisms in acute brain damage (50, 54). The model revealed itself able to reproduce ICP and cerebral hemodynamics in different conditions of clinical interest (50-52).

The previous model, however, is computationally too heavy to be routinely used in a clinical setting. The aim of this work is to describe and validate a simplified model that can incorporate the main mechanisms involved in intracranial dynamics and permit their quantitative individual assessment with a drastic reduction in mathematical complexity.

The paper is structured as follows. First, the qualitative aspects of the model are described. The clinical and computational methods used to test the validity of the model are then presented. The results concern the reproduction of ICP and middle cerebral artery (MCA) velocity tracings during SAP and Pa_CO2 challenges. Finally, the virtues and limitations of the obtained results are critically discussed.

	QUALITATIVE MODEL DESCRIPTION

TOP ABSTRACT INTRODUCTION QUALITATIVE MODEL DESCRIPTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

The model presented in this paper is significantly simplified compared with that used in a previous recent work (54). The main simplifications are 1) that the model does not distinguish between the behavior of large and small pial arteries and 2) that the biomechanics of the pial arterial-arteriolar vascular bed are reproduced by means of a simple windkessel model (that is, the arrangement of a hydraulic resistance and a hydraulic capacity). By contrast, in previous works (50, 52, 54) the biomechanics of large and small pial arteries were described starting from the equilibrium of forces in the vessel wall, with wall elastic and active (muscular) characteristics taken into account.

The main aspects of the model are described in qualitative terms. All model equations, with considerations of numerical integration methods, are given in APPENDIX A. A biomechanical analog of the model is shown in Fig. 1.

View larger version (11K): [in this window] [in a new window]   Fig. 1.   Biomechanical analog of the mathematical model, in which resistances are represented with restrictions and compliances with bulges. Pa, systemic arterial pressure; Pla and Rla, pressure and resistance of large intracranial arteries, respectively; Ppa, Rpa, and Cpa, pressure, resistance, and compliance of pial arterioles, respectively; Pc, capillary pressure; q, tissue cerebral blood flow; Rpv, resistance of proximal cerebral veins; Cvi, intracranial venous compliance; Pv, cerebral venous pressure; Pvs and Rvs, sinus venous pressure and resistance of the terminal intracranial veins, respectively; Rf and Ro, cerebrospinal fluid (CSF) formation and CSF outflow resistance, respectively; qf and qo, CSF formation rate and CSF outflow rate, respectively; Pic and Cic, intracranial pressure and intracranial compliance.

Intracranial Hemodynamics and Hydrodynamics

Large intracranial arteries. The first segment in the model represents circulation in the basal intracranial arteries, down to and excluding the large pial arteries. The hemodynamics of this segment are described by means of a single hydraulic resistance (R_la). Because the impact of cerebrovascular regulation mechanisms on the basal intracranial arteries is quite small (14), R_la has been maintained constant throughout the simulations. An approximate estimation of blood flow velocity at an MCA (V_MCA), to be compared with the results of transcranial Doppler (TCD), is computed by assuming that blood flow in an MCA is about one-third of total CBF. We can thus write

(1)

where q_la is total CBF at the level of the basal intracranial arteries and r_MCA is MCA inner radius. Finally, k_V is a scaling factor that accounts for the slope of the Doppler probe relative to the vessel under examination and for the proportionality between average and maximum velocity in the section. In fact, the TCD technique measures maximum velocity, while the right-hand member of Eq. 1 (see Eq. A12 in the APPENDIX A) refers to mean time velocity.

Pial arterial-arteriolar cerebrovascular bed. The second segment in Fig. 1 simulates the pial arterial circulation, extending from the large pial arteries down to and including small pial arteries and intraparenchymal arterioles. For the sake of simplicity, the model does not distinguish between hemodynamics in different-sized vessels; hence, we consider only one segment, characterized by its values of hydraulic resistance (R_pa) and compliance (C_pa). As described in Cerebrovascular Regulation Mechanisms, both parameters are actively regulated by cerebrovascular control mechanisms.

Venous intracranial circulation. The intracranial venous vascular bed is described by the series arrangement of two segments. The first extends from the small postcapillary venules down to the large cerebral veins and includes the venous resistance (R_pv) and the intracranial venous capacity (C_vi). Because the impact of cerebrovascular mechanisms on the venous vasculature is negligible and large veins do not collapse even at elevated ICP (3), venous resistance has been maintained constant throughout the simulations. By contrast, venous capacity is inversely proportional to the local transmural pressure level, which implies a monoexponential pressure-volume relationship for the veins. We are aware that the exact mathematical expression for venous compliance may be affected by the head injury. However, changes in intracranial venous transmural pressure are always modest in our model (because of the Starling resistor mechanism described below); as a consequence, the exact expression for venous compliance has only a minor impact on the results.

CSF circulation. The circulation of CSF is described as a passive process. CSF is produced at the cerebral capillaries because of a positive transmural pressure gradient (P_c  P_ic, where P_c is capillary pressure and P_ic is intracranial pressure) and is reabsorbed at the dural sinuses because of a negative transmural pressure gradient (P_vs  P_ic, where P_vs is venous sinus pressure). The resistances to CSF formation and CSF outflow are R_f and R_o, respectively. However, both processes are unidirectional, so we have assumed that resistances rise to infinity when the corresponding transmural pressure is reversed.

Intracranial compliance. Because the overall intracranial volume must remain constant, any change in the content of one of the previous compartments (i.e., pial artery volume at C_pa, venous volume at C_vi, and CSF volume) must be accompanied by an opposite change in the remaining intracranial volumes with a concomitant variation in ICP. This phenomenon is described through the intracranial storage capacity (C_ic). According to Marmarou et al. (28) and Avezaat et al. (4), an expression for the craniospinal capacity can be computed by assuming that the intracranial pressure-volume relationship is approximately monoexponential in type. We then have
where k_E is the intracranial elastance coefficient as defined by Avezaat et al. (4). k_E is an index of the rigidity of the intracranial compartment; hence, it is inversely proportional to the pressure volume index (PVI) introduced by Marmarou et al. (28).

Cerebrovascular Regulation Mechanisms

Two distinct control mechanisms are considered in this work, i.e., autoregulation and CO₂ reactivity (see Fig. 2, top). The role of arterial oxygen is not included because O₂ pressure was maintained constant throughout the present trials.

View larger version (16K): [in this window] [in a new window]   Fig. 2.   Top: block diagram describing the action of cerebrovascular regulation mechanisms according to the present model. The upper branch describes autoregulation, while the lower branch indicates CO2 response. The input quantity for the CO2 mechanisms is the logarithm of arterial CO2 tension (PaCO2), i.e., PaCO2 = log10 (PaCO2/PaCO2 n). The input quantity for autoregulation is cerebral blood flow percent change (CBF). The dynamics of each mechanism are simulated by means of a gain factor (G) and a first-order low-pass filter with time constant . The gain factor of CO2 mechanisms is multiplied by the corrective factor ACO2, which depresses 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, thus producing changes in pial artery compliance and resistance, with upper and lower saturation. Bottom: frequency responses of the two low-pass filters.

As shown in the upper branch of Fig. 2, we assumed that autoregulation is activated by changes in CBF. Its action on the arterial-arteriolar pial vessels includes a static gain (G_aut) and first-order low-pass dynamics with the time constant _aut.

The lower branch in Fig. 2, top, represents CO₂ reactivity. This includes a static gain as well (G_CO2) and first-order dynamics with the time constant _CO2. However, we used the logarithm of Pa_CO2 as input to the controller. The latter choice is justified because the response of pial vessels to CO₂ is correlated quite linearly with pH changes in the perivascular space. pH, in turn, depends on the logarithm of CO₂ concentration via the Henderson-Hasselbalch equation. The minus sign in the upper branch signifies that an increase in CBF causes vasoconstriction, with a consequent decrease in pial vessel compliance and an increase in resistance. The dynamics of each mechanism are summarized through the low-pass frequency response shown in Fig. 2, bottom.

Finally, the two mechanisms do not superimpose linearly on pial vessels, but their interaction is characterized by significant nonlinearities (17, 56). To account for experimental and clinical evidences, we introduced two main nonlinearities. First, the strength of CO₂ reactivity in the model is not independent of the level of CBF but decreases significantly during severe ischemia (see Fig. 2, corrective factor A_CO2). In fact, severe ischemia is associated with tissue acidosis, which, in turn, buffers the effect of CO₂ changes on perivascular pH. A second nonlinearity considers that the overall regulatory action is not the sum of the two mechanism actions but is instead passed through a sigmoidal static relationship with upper and lower saturation levels. The sigmoid accounts for the existence of maximal limits for the vascular response, i.e., total vasodilatation and maximal vasoconstriction.

Values were given to all model parameters under normal conditions to reproduce the intracranial hydro- and hemodynamics of a healthy subject. A list of all model parameters and their values under basal conditions can be found in Table 1. Details on the main parameter values can be found in previous papers (50, 53). A deeper comment on normal and pathological values for the six parameters individually estimated is provided in the DISCUSSION.

Figure 3 shows how the model can reproduce the regulatory mechanism actions in steady-state conditions with the use of basal values for the autoregulation and CO₂ reactivity gains. Figure 3A shows the CBF percent changes vs. CPP during normocapnia, while Fig. 3B reports CBF percent changes vs. Pa_CO2 during normotension. In both cases, we can observe a satisfactory agreement between model results and physiological data.

View larger version (15K): [in this window] [in a new window]   Fig. 3.   A: percent changes in CBF vs. mean systemic arterial pressure (SAP) evaluated with the model in steady-state conditions using a normal level of CO2 arterial pressure (PaCO2 = 40 mmHg). Throughout the simulations, intracranial pressure (ICP) was set at a constant level (~9.5 mmHg). Model results (continuous line) are compared with the experimental data by Harper et al. (18) in rats (×) and MacKenzie et al. (26) in cats (). B: percent changes of CBF vs. PaCO2 evaluated with the model in steady-state conditions at a normal level of mean SAP (100 mmHg). Model results (continuous line) are compared with the experimental data by Reivich (37) in monkeys (+) and Harper and Glass (17) in dogs (*).

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION QUALITATIVE MODEL DESCRIPTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

The model was used to reproduce the time pattern of ICP and V_MCA in 13 neurosurgical patients during maneuvers that alter Pa_CO2 and mean SAP. We examined several tracings, obtained at different hours after head injury, in all patients. A total of 44 distinct tracings was thus simulated, each with a duration ranging between 12.5 and 93 min.

This study was approved by the local Ethics Committee, and informed consent of each subject's next of kin was duly obtained.

Patients

We performed CO₂ challenges at different levels of SAP. The studies were conducted as follows. After 30 min in a steady basal condition, patients were gradually hyperventilated by increasing respiratory frequency at constant tidal volume until V_MCA no longer fell. During hyperventilation, jugular Sv_O2 was not allowed to drop below 50%. A short period of hypoventilation was then allowed by decreasing the respiratory rate. Hypoventilation was allowed up to a PET_CO2 of ~40 mmHg or less if the ICP increased above 35 mmHg.

Ventilation was then adjusted to achieve the same PET_CO2 obtained at the basal state. In a second session we kept PET_CO2 constant and modulated norepinephrine perfusion to generate variations in SAP. When a new steady level of SAP was achieved, a second CO₂ challenge was performed. Quantitative details on the maneuvers (duration, PET_CO2, and mean SAP ranges) can be found in Table 3. Changes in PET_CO2 always took a few minutes to be completed. Hence, the rate of PET_CO2 variation per unit time ranged between 2 and 5 mmHg/min. Arterial pressure changes exhibited wider dynamics, ranging approximately between ±4 mmHg/min for the slowest maneuvers and ±36 mmHg/min for the fastest ones.

Model Identification

As explained in classic identification textbooks (5), individual values to the parameters can be assigned a posteriori through a best fitting procedure. This consists of minimizing a suitable criterion function of the difference between in vivo data (in our case, ICP and V_MCA) and the corresponding model outputs, assuming that the model is stimulated by the same input perturbations used in the real trials. The general process of model identification is schematized in Fig. 4.

View larger version (14K): [in this window] [in a new window]   Fig. 4.   Block diagram describing the automatic procedure for estimation of model parameters. F(), cost function; Pic(s) and VMCA(s), clinical data for Pic and middle cerebral artery velocity; Pic(m) and VMCA(m), model output for Pic and VMCA.

Unfortunately, the model contains too many parameters for them all to be individually estimated. Hence, when performing the best fitting procedure, we decided to estimate only the parameters that have the greater physiological and clinical meaning, i.e., those that summarize intracranial elasticity, CSF circulation, and cerebrovascular regulation. Hence, the parameters chosen are the CSF outflow resistance (R_o), the intracranial elastance coefficient (k_E), the gains of cerebral autoregulation (G_aut) and CO₂ reactivity (G_CO2-), the time constant of the CO₂ response (_CO2-), and the position of the sigmoidal regulation curve. To this end, we individually estimated a "set point" for the CO₂ regulation (Pa_{CO2 n}) (see Eq. A15 in APPENDIX A). Changing the latter parameter is equivalent to shifting the normal working point along the sigmoidal regulation curve, that is, modifying the position of the upper and lower autoregulation limits. The latter choice is justified by the observation that the position of upper and lower limits for cerebrovascular regulation is affected by the patient's metabolic need and by adaptation to gas level in the blood (32).

It is worth noting that the CSF production resistance (R_f) was not included in the previous list because, according to our earlier experience (52), this parameter is inversely correlated with CSF outflow resistance; hence, it cannot be estimated independently. Moreover, the time constant of autoregulation (_aut) was not individually estimated because it is significantly smaller than the other time constants in the model and, hence, has only a minimal impact on the fitting process.

Automatic estimation of the aforementioned parameters was achieved by minimizing a least-square criterion function. To assess the accuracy of parameter estimates, we also evaluated the coefficient of variation (the percent standard deviation of the estimates) by using classic statistical techniques (5). A mathematical description of the criterion function and the equations for the assessment of the coefficient of variation are reported in APPENDIX B.

When the minimization procedure was performed, the initial values of model ICP and V_MCA were assigned on the basis of clinical data. In particular, the scaling factor k_V in Eq. 1 was computed separately in each trial to ensure that model V_MCA at the beginning of the simulation is the same as the first velocity data point.

The entire software program for mathematical computations was written using the scientific programming language Fortran 77 (AbSoft Fortran 77 for Windows) running on a Pentium-based personal computer. A user-friendly interface was also written in Visual Basic (Microsoft Visual Basic 5.0); the latter program permits handling of the parameter estimation procedure, visualization of parameter values, and graphic plotting in a simple and straightforward way.

	RESULTS

TOP ABSTRACT INTRODUCTION QUALITATIVE MODEL DESCRIPTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

The parameter values estimated in 44 tracings on 13 patients are reported in Table 4, together with the coefficient of variation of the estimates. To facilitate the analysis, all parameters in Table 4 (with the exception of Pa_{CO2 n}) were normalized to the values in Table 1. The accuracy of the best fitting is summarized in two other columns of Table 4, in which the SD of the residuals between the predicted and the actual measurements (ICP and V_MCA) are presented. A column scatter graph showing these values in all 44 tracings is presented in Fig. 5. Figure 5 reveals that, in most cases, the SD of the residuals is of the same order as measurement accuracy (~1-2 mmHg for ICP, 4-5 cm/s for Doppler velocity), indicating that the model's reproduction of clinical tracings is satisfactory. As for ICP, only four tracings exhibit residuals >3 mmHg, denoting insufficient fitting, but three of these tracings refer to the same patient (patient Der). Velocity exhibits nine tracings in which residuals are >6 cm/s, suggesting poor fitting. However, six of these are concerned with only two patients (patients All and Gra). These patients had high basal values of velocity (>100 cm/s), which might suggest the existence of either vasospasm or hyperemia and could explain the poor velocity fitting.

View larger version (11K): [in this window] [in a new window]   Fig. 5.   Column scatter graph showing the standard deviations of the residuals (i.e., the last two columns in Table 4) for all 44 tracings examined. Horizontal lines represent the mean value of all residuals.

By looking at Table 4, one can observe that the estimated values of the parameter Pa_{CO2 n} are significantly lower than normal. This result may reflect adaptation to the low level of PET_CO2 set before the maneuvers. To check this aspect, Fig. 6 shows the correlation between the basal PET_CO2 and the parameter Pa_{CO2 n} for all 44 tracings. The correlation is high, apart from a few diverging points. However, three of these points refer to tracings (tracings Der4, Pra5, and Tho5) in which PCO₂ was maintained almost constant (see Table 3); hence, estimation of CO₂ reactivity parameters is scarcely significant. If these three points are excluded, correlation is as high as r² = 0.86. 

View larger version (13K): [in this window] [in a new window]   Fig. 6.   Correlation between the end-tidal PCO2 (PETCO2), set prior to the maneuvers, and the estimated parameter PaCO2 n for all 44 tracings. Three of these points, referring to tracings Der4, Pra5, and Tho5 () have not been used in the computation of the correlation coefficient because the corresponding runs were performed at almost constant PCO2 (see Table 3). The regression line, computed using all the other 41 points (*), has a slope of 1.13 ± 0.07, the y-axis intercept is 1.12 ± 1.75, and r2 = 0.87.

Examples of the results of the best fitting procedure, obtained in two different patients, are shown in Figs. 7 and 8. The time pattern of the input quantities (i.e., mean SAP and Pa_CO2) are shown in Figs. 7 and 8, top, while a comparison between model outputs (ICP and V_MCA) and the corresponding clinical data is shown in Figs. 7 and 8, bottom.

View larger version (32K): [in this window] [in a new window]   Fig. 7.   Results of the estimation procedures performed on 4 different tracings measured a few hours apart in patient Bab. For each tracing, the upper panel represents the manipulated quantities for mean SAP (top) and PaCO2 (bottom) used as input to the model, while the lower panel shows a comparison between model and in vivo VMCA (top) and ICP (bottom). Open circles connected with a thin line represent in vivo data, while thick lines are model simulation results obtained with the estimation algorithm. For each tracing, parameters were estimated again. The resulting parameter estimates are reported in Table 4.

View larger version (30K): [in this window] [in a new window]   Fig. 8.   Results of the estimation procedures performed on 2 different tracings measured a few hours apart in patient Dug. Symbols are defined as in Fig. 7. For each tracing, parameters were estimated again. The resulting parameter estimates are reported in Table 4.

Figure 7 shows four different tracings obtained on the same day from patient Bab. The first two tracings contain only a CO₂ challenge, while arterial pressure was maintained constant. The third and fourth tracings comprise both SAP and Pa_CO2 maneuvers. It is clear from the first two tracings that a mild increase in CO₂ pressure, performed by starting from a baseline hypocapnic level (~20 mmHg), gave rise to a velocity increase and a significant ICP rise. Finally, even a modest decrease in Pa_CO2 was able to interrupt the uncontrolled increase in ICP, leading to rapid restoration of the baseline levels. The model is able to explain this chain of events with reliable values of its parameters (Table 4), ascribing the ICP time pattern to changes in cerebral blood volume.

The third and fourth tracings in Fig. 7 differ from the previous tracings because they comprise a hypertensive maneuver as well, and the CO₂ challenge occurs at a higher level of CPP. In the third tracing, a rapid increase in mean SAP causes a change in V_MCA, so the patient is classified as having weak autoregulation (normalized G_aut = 0.65, see Table 4). An index commonly used to estimate autoregulation in the clinical literature is the static autoregulation index (sARI) (30, 48), defined as the percentage changes in cerebrovascular resistance (CVR) divided by the percent changes in CPP. Values of sARI in normal subjects are >0.85 (see DISCUSSION). However, if this index is computed from the two arterial pressure maneuvers in the third clinical tracings of Fig. 5, one obtains the values sARI = 0.47 and sARI = 0.39, respectively, in accordance with model estimation of weak autoregulation.

By contrast, the increase of mean SAP in the fourth tracing in Fig. 7 does not evoke a clear velocity change, whereas it causes a moderate decrease in ICP, ascribed to active blood volume reduction. Hence, in this case the estimation procedure provides a high value for the normalized G_aut (1.50, see Table 4). Accordingly, sARI computed from the clinical tracing is as high as 0.89, which belongs within the range of normality.

An important aspect arising from the tracings in Fig. 7 is that the percent changes in TCD velocity per mmHg of CO₂ change (V_MCA%/Pa_CO2, which is the index commonly used in the clinical literature to estimate CO₂ reactivity) depend strongly on the level of CPP. Hence, this empirical index is not representative of the "true" CO₂ reactivity. By contrast, the gain G_CO2 in the model is quite independent of the CPP level. In fact, if the empirical index is computed in the first tracing, one obtains the value 1.84%/mmHg, while CPP decreases from 57 to 37 mmHg, i.e., below the autoregulation range. In the second tracing in Fig. 7, the same index is as low as +0.28%/mmHg, suggesting exhausted cerebrovascular regulation reserve, while CPP decreased during the maneuver to 34 mmHg. However, in the third tracing, the index is as high as +3.97%/mmHg. In fact, mean SAP was increased before the maneuver, and CPP always remained >66 mmHg, i.e., the overall maneuver was performed within the autoregulation range. Finally, in the fourth tracing in Fig. 7, we computed a value of V_MCA%/Pa_CO2 = 2.94%/mmHg, while CPP decreased from 84 to 61 mmHg. We can thus conclude that, when the CO₂ challenge is performed at a higher CPP level (in the range of 60-80 mmHg), the V_MCA change per mmHg of Pa_CO2 change is significantly higher compared with the case of low CPP (35-60 mmHg). Hence, the compensatory response to CO₂ is depressed by hypotension. It is worth noting that the model can explain this behavior, ascribing it to a nonlinear interaction between CO₂ reactivity and autoregulation (Fig. 2).

In some patients we performed several simultaneous CO₂ and SAP challenges in a single tracing, and we monitored the consequent ICP and V_MCA changes over a long time period (40-90 min). Two tracings of this kind, monitored in patient Dug, are shown in Fig. 8. It is worth noting the capacity of the model to reproduce the consequences of several consecutive maneuvers at different SAP and Pa_CO2 levels using just a single set of parameters. The dependence of the CO₂ reactivity index V_MCA%/Pa_CO2 on CPP is evident by comparing the results of the two consecutive CO₂ challenges in the first tracing in Fig. 8. In the first maneuver, performed at a CPP always >72 mmHg (i.e., within the autoregulation range), we computed a V_MCA%/Pa_CO2 = 4.11%/mmHg, which is quite normal. By contrast, the second CO₂ maneuver provides an index V_MCA%/Pa_CO2 as low as 2.13%/mmHg, while CPP decreased from 57.4 to 50 mmHg, approaching the autoregulation lower limit. Accordingly, the model estimates that G_CO2- is quite normal in this patient (normalized value 0.93) and ascribes the apparent reduction in CO₂ reactivity of the second maneuver to the attainment of maximal vasodilation at low CPP. In tracing Dug2, the CO₂ maneuver is performed at normal CPP (always >74 mmHg), the index V_MCA%/Pa_CO2 = 4.55%/mmHg, and estimated CO₂ reactivity in the model is rather normal.

Finally, the sARI computed during the arterial pressure maneuvers in the two tracings of Fig. 8 is 0.42 and 0.69, respectively, denoting a weak autoregulation. Accordingly, the model provides a G_aut lower than normal (normalized values close to 0.6).

	DISCUSSION

TOP ABSTRACT INTRODUCTION QUALITATIVE MODEL DESCRIPTION MATERIALS AND METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

The present work was designed with two main purposes: to find a reliable quantitative description of the main hemodynamic changes occurring in head-injured patients during SAP and CO₂ pressure challenges, and to verify the possibility of characterizing intracranial dynamics in each patient through a limited number of parameters. Before the significance and limitations of the obtained results are examined, it is important to provide a critical analysis of both the measurement technique employed and the quality of clinical data available.

A first important problem concerns the use of the TCD technique to achieve dynamic information on CBF percent changes. This choice is acceptable because previous studies in head-injured patients demonstrated the existence of a good proportionality, with a slope close to 1, between percent variations in TCD velocity and CBF estimated with the Fick principle. However, in patients with elevated V_MCA >100 cm/s, the slope was not strictly close to 1, and TCD variations underestimated CBF variations (47). This result is probably due to the fact that hyperemia or vasospasm is associated with modifications in velocity profile and/or interhemispheric heterogeneity of vascular reactivity. This problem may have caused some of the high differences (>6-7 cm/s) between model predictions and clinical data observed in a few patients (especially patients Gra and All) who had elevated blood flow velocity.

A second problem is that we measured end-tidal CO₂ during the clinical trials, whereas arterial CO₂ is the true stimulus for cerebrovascular regulation. We are aware that in some head-injured patients the gradient between end-tidal and arterial PCO₂ can be increased as a consequence of neurogenic pulmonary edema, which increases ventilation perfusion inhomogeneity. However, none of the patients analyzed exhibited symptoms of pulmonary edema or had pulmonary disease.

A final problem is that SAP was modified with norepinephrine. In the present simulations we assumed that infusion of norepinephrine does not modify the diameter of the insonated vessel or change cerebrovascular reactivity. This assertion is speculative, but it is supported by data from Olesen (33). This author observed that intracarotid application of adrenalin or norepinephrine does not affect regional cerebral blood flow in humans, and these drugs do not alter the diameter of large intracranial arteries, as shown by angiography. This result can explain why similar V_MCA variations were observed during autoregulatory tests realized by means of pharmacological infusion of norepinephrine and by means of nonpharmacological methods (44, 48).

In most tracings examined, the model was able to show rather good reproduction of the observed ICP and V_MCA time patterns, with a standard error of the residuals of the same order as measurement errors. Moreover, the capacity of the present model to reproduce ICP and velocity tracings is equivalent to that of the previous, more complex model (25) but with much shorter computation time. However, it is important to emphasize that reproduction of real tracings was not achieved a priori, based on knowledge of a patient's condition, but only a posteriori through a best fitting procedure. In other words, the present simplified model needs to be used together with a fitting procedure to assign some parameter values.

Among the estimated parameters, two are of particular interest because they contribute to intracranial hypertension and clinical status in patients with severe head injury; i.e., intracranial elastance and CSF outflow resistance. Clinicians are strongly interested in a quantitative knowledge of these parameters because their alterations require therapeutic intervention before the occurrence of secondary brain damage. However, the methods presently available to estimate these parameters often provide contradictory results (24, 27) and are not without criticism (15, 16). As a consequence, no gold standard method to derive these parameters is yet at hand.

Furthermore, knowledge of the autoregulation and CO₂ reactivity gains could also be important in clinical practice to choose the correct management and to distinguish between the patients who need sustained elevated CPP and those who need hyperventilation (9, 39). Estimation of the strength of regulation mechanisms, however, as obtained by classic indexes (such as CBF percent change or resistance changes per mmHg), is misleading because, as in the example of Figs. 7 and 8, this may depend on CPP, ICP changes, and the working point along the regulation characteristic.

In the present study, parameters have been estimated by using a classic identification schema (Fig. 4), i.e., by minimizing the square of residuals between clinical data and model results. A crucial problem is whether the obtained parameter values have a reliable meaning or are merely the result of curve fitting. This problem will be critically examined below for each estimated parameter.

CSF Outflow Resistance

Values of R_o, measured in patients with severe head injury with stationary withdrawal (24) show a significant increase compared with normal values, ranging between 21 and 85 mmHg · ml¹ · min (normalized range 2.4-9.7). By contrast, lower values of R_o were obtained by Marmarou et al. (27) in the severe head injury with the bolus technique (3.15-36.62 mmHg · ml¹ · min; normalized range 0.4-4.17).

The values of R_o estimated in the present study show a good agreement with those measured by Kosteljanetz (24). In 12 of 13 patients, R_o appears significantly increased compared with the basal value (normalized range 2.0-9.0). This result supports the idea that impairment in CSF outflow may represent a major contribution to intracranial hypertension in severe head injury.

Intracranial Elastance Coefficient

1

1

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Normalized values of k_E estimated in this study range between 0.7 and 2.9, i.e., we observed both patients with normal elastance and patients with a significant increase in k_E, suggesting a reduction in the intracranial storage capacity. On average, the elastance coefficient is increased in head injury compared with normality. This result agrees with the observation by Kosteljanetz (24) and our previous observation with the bolus technique (55).

Autoregulation Gain

Autoregulation is known to be significantly impaired by many cerebral diseases, including head injury. Results from Bouma and Muizelaar (6) and Matta et al. (30) suggest that patients with head injury can be discriminated in three classes: normal autoregulation (sARI > 0.85), moderately impaired autoregulation (0.85 > sARI > 0.5), and severely impaired autoregulation (0.5 > sARI). According to our simulations, the values of G_aut parameters (normalized) that discriminate between these three classes are G_aut > 1.0 (normal), 0.5 < G_aut < 1 (moderately impaired), and G_aut < 0.5 (strongly impaired).

The values of G_aut estimated in this study range between 0.2 (impaired autoregulation) and 1.5 (normal autoregulation) in accordance with the literature.

CO₂ Reactivity

CO₂ reactivity may decrease below 2 CBF%/mmHg in some patients with severe head injury, although in other patients CO₂ reactivity is preserved despite the trauma. Data in Tenjin et al. (46) suggest that the value 2 CBF%/mmHg may be considered as a threshold to discriminate between patients with preserved and impaired CO₂ reactivity. According to our simulations, this threshold corresponds to a value of normalized G_CO2 approximately as low as 0.66.

In this work we estimated values of normalized G_CO2 ranging between 0.21 and 2.32 (average 0.85). This range agrees with values reported in the clinical literature and confirms the existence of patients with preserved CO₂ reactivity as well as patients with almost absent CO₂ response.

The parameter _CO2, estimated in the present trials, ranges between normal and threefold normal values (i.e., 2 min). These high values might depend on the impairment of some feedback mechanisms in head-injured patients or may reflect a high time delay between changes in PET_CO2 and changes in pH at cerebral arterioles.

Finally, the set point Pa_{CO2 n} is significantly lower than normal in most patients. We think that these low values should not be ascribed to the traumatic pathology but, rather, that they represent adaptation to the low PCO₂ level set in the patients before the maneuvers, probably due to a reset of CSF pH to normal values after prolonged hyperventilation (41). This assumption is strongly supported by the high correlation between Pa_{CO2 n} and PET_CO2 shown in Fig. 6.

The previous analysis confirms that parameter estimates belong within the range reported in the clinical literature for patients with severe head injury. This observation suggests the possibility that the present method may be of clinical value to assess pathological alterations in intracranial dynamics. An important problem, however, concerns the accuracy by which parameters can be estimated from the clinical tracings and the uniqueness of the obtained solutions. Contradictory conclusions can be drawn on the basis of this major point. In favor of the accuracy of the estimates, we can observe that, in several cases, parameter estimation achieved in the same patient using tracings recorded a few hours apart furnished very similar results (compare tracings All4 vs. All5, Bab1 vs. Bab2, Dbr1 vs. Dbr2, Dug1 vs. Dug2, Hel1 vs. Hel2 and Hel3, and Tho1 vs. Tho2). The repeatability of parameter estimates is especially satisfactory when the clinical maneuvers comprise both CO₂ and SAP challenges performed at different times in the same tracing (as in Fig. 8). In a few other cases, however, parameters exhibited large fluctuations when passing from one trial to the next. This intrinsic variability might depend on an effective change in a patient's intracranial status but might also be caused by insufficient accuracy in parameter estimates. At present we have not enough elements to discriminate between these two possibilities, but a few arguments can be developed.

First, we can observe that, in most cases, the variability in the estimates of the CSF outflow resistance within the same patient is small enough to permit discrimination among patients with normal CSF outflow (normalized R_o 1-2), moderate elevation in R_o (~2.5-5), or large elevation in R_o (~6-9). Similarly, values of k_E maintain certain coherence within the same patient. In most cases, estimation of k_E remains rather constant or decreases during the treatment, suggesting an improvement in intracranial elasticity. The remarkable case of patient Rvm will be examined below.

By contrast, the autoregulation gain, as well as the gain and time constant of CO₂ reactivity, exhibits a wide dispersion, which is evident not only from one patient to the next but also within the same patient. We think that the reason for this variability may be the significant correlation existing among these parameters, which often precludes the possibility of their accurate unequivocal estimation. To attenuate this problem, it would be important to use tracings in which SAP and CO₂ challenges are performed separately (such as tracings in Fig. 8); multiple information, in fact, may provide more accurate estimation of correlated parameters.

A final important issue is that the estimated parameter values should have some relationship with the patient's clinical status to effectively support the clinical practice. Naturally, clinical validation of the model is well beyond the limits of the present work and should be the subjects of future, more clinically oriented studies. However, patient Rvm represents an exemplary case. This patient developed a secondary compressive chronic subdural hematoma during the recovery. Surgical evacuation was performed at the 16th day after trauma (i.e., between tracings Rvm2 and Rvm3). It is interesting to observe that the estimation procedure predicts an increase in R_o and a strong increase in k_E (indicating reduced intracranial elasticity) at days 15 and 16 before the surgical procedure. However, these parameters returned rather close to normal after surgical evacuation (Table 4).