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A mathematical model of intracranial dynamics aimed at clinical investigation

Dipartimento di Elettronica, Informatica e Sistemistica - Bologna

In this work, some examples of computer simulations of intracranial dynamics are presented. The simulations have been performed by using a mathematical model developed by the authors in previous years. The main physiological and biomechanical factors covered by the model are: the cerebrospinal fluid production and reabsorption processes, the cranio-spinal pressure-volume relationship, the collapsing intracranial venous vascular bed, blood velocity at the middle cerebral artery, and the regulated arterial-arteriolar pial circulation. Both autoregulation mechanisms and CO₂ reactivity affect the caliber of large and small pial arteries according to the present physiological knowledge.
The main conditions simulated with the model include: cerebral autoregulation in physiological conditions; genesis of self-sustained intracranial pressure waves (plateau waves); the sensitivity of the main quantities extracted from the Transcranial Doppler signal (systolic, diastolic, mean velocity and Gosling pulsatility index) to intracranial pressure changes; the effect of acute hypercapnia and hypocapnia on the intracranial pressure time pattern. The results obtained suggest that intracranial dynamics depend on several factors acting simultaneously, whose interactions may give rise to different responses depending on model parameters (hence on the patient's status). Worth noting is the presence of a break-point in model parameters, after which intracranial dynamics may suddenly become unstable. Instability manifest itself with disproportionate changes in intracranial pressure, cerebral blood flow, and of other hemodynamic quantities, with consequent risk of ischemia, intracranial hypertension and secondary brain damage.
We suggest that mathematical modeling and computer simulation techniques may help medical doctors in the analysis of the complex non-linear phenomena characterizing cerebral hemodynamics in health and diseases, and in the design of a more correct management of patients in neurosurgical intensive care units.
 

Understanding the relationships between cerebral hemodynamics, cerebrospinal fluid dynamics and intracranial pressure is of the greatest importance for the treatment of patients with severe brain lesions. There are several reasons which make the analysis of intracranial dynamics and cerebral hemodynamics particularly unwieldy. First, cerebral vessels are subjected to the action of strong and sophisticated regulatory mechanisms, which work in response to various perturbations to maintain cerebral blood flow (CBF) always adequate to tissue metabolism and function. It is well known that CBF remains pretty constant until cerebral perfusion pressure (CPP, that is mean systemic arterial pressure - intracranial pressure) lies in the range 50-150 mmHg. This phenomenon is usually known as 'autoregulation'. Autoregulation is probably achieved via different mechanisms (myogenic, neurogenic, metabolic and endothelium-dependent). Moreover, results of several authors underline the existence of a significant 'segmental heterogeneity' in cerebrovascular response to perfusion pressure changes (1); in fact, large pial arteries are especially sensitive to moderate pressure alterations, whereas small pial arteries and arterioles exhibit a massive vasodilation only at low CPP levels (close to the autoregulation lower limit).
Furthermore, cerebral vessels are strongly sensitive to changes in arterial concentration levels, mainly O₂ or CO₂ in the arterial blood. Hypercapnia is known to be a powerful vasodilator of cerebral vessels, being able to increase CBF more than twice of its normal value, whereas hypocapnia can significantly reduce CBF and cerebral blood volume (2). Changes in carbon dioxide levels are frequently used in the management of patients with severe brain damage, to control intracranial pressure (ICP) and set the appropriate level of CBF (3).
A third aspect deserving much attention is that cerebral hemodynamics is confined within a closed space (i.e., the skull and neuro-axis) which exhibits a limited capacity to store a volume load. This means that any increase in cerebral blood volume, caused by action of the afore-mentioned vasodilatory mechanisms, may cause a compression or dislocation of the other intracranial structures, with a consequent rise in intracranial pressure. There are two main mechanisms by which the cranio-spinal system can cope with a volume load, i.e., cranio-spinal compliance, usually reproduced via a mono-exponential pressure-volume relationship, and the cerebrospinal fluid (CSF) circulation (mainly CSF reabsorption at the dural sinuses). Both these mechanisms are of the greatest clinical importance, and their impairment may have dramatic consequences for the patients. One can expect that, if the intracranial buffering capacity is reduced, due to a decrease in compliance and/or in CSF outflow, the changes in cerebral blood volume induced by cerebral autoregulation mechanisms may have a serious impact on ICP, with the risk of intracranial hypertension, cerebral ischemia and secondary brain damage.
A fourth significant problem is that the mechanisms regulating CBF may be easily damaged, especially following head injury or subarachnoid hemorrhage. Assessing the status of cerebrovascular regulatory mechanisms in these patients is of the greatest importance, both to establish the proper treatment, and avoid ischemic insults. As a matter of fact, the same maneuver, which may have a particular impact in a patient with preserved autoregulation mechanisms and CO₂ reactivity, may produce completely different responses in patients with damaged cerebrovascular control and impaired autoregulation.
A further problem arises when one tries to assess intracranial hemodynamics with non-invasive means. Although the measurement of ICP and CBF are the most suitable to get information on cranio-spinal dynamics, these kind of measurements can be achieved only in particular conditions. The use of the Transcranial Doppler (TCD) technique, which measures blood flow velocity at the middle cerebral artery (MCA), is becoming today the most common method to analyze intracranial hemodynamics in routine clinical units thanks to its benefits of non-invasivity and continuous data acquisition (4). However, the problem of how the main quantities extrapolated from the TCD waveform are related to intracranial hemodynamics and ICP changes is still far from being satisfactorily clarified, despite the increasing number of experimental and clinical studies appeared on this subject in recent years (5,6).
Analysis of the various problems delineated above is made even more complex by the observation that the different constituents the craniospinal system interact in a non-linear way. This means that the superimposition of the effects usually does not hold, and the result of the various components acting together may be widely different from the sum of the single actions considered individually.
The complexity of the relationships among intracranial quantities, and their alterations in different patho-physiological conditions can be elucidated by using mathematical models and computer simulation techniques. These, in fact, allow the different constituents the craniospinal system to be studied simultaneously, in rigorous quantitative terms, taking their complex non-linear links and mutual dependencies into account. Moreover, the software facilities now available permit to visualize the results, and to provide input data for the models, using simple user-friendly interfaces.
In previous years, we developed various mathematical models of the intracranial dynamics and cerebrovascular control mechanisms, devoted to the investigation both of physiological (7,8) and clinical (9-12) problems. In particular, by using the more clinically oriented models, we were able to simulate correctly much of the phenomena described above.
Aim of this work is to present a general overview of the model capabilities, without entering in specific mathematical considerations. After a brief qualitative description of the physiological and biomechanical bases of the model, some example of clinical relevance are simulated and discussed. The main phenomena analyzed with the model concern: i) cerebral autoregulation in physiological conditions; ii) cerebral autoregulation in patients with poor craniospinal compensatory mechanisms, with development of ICP waves; iii) the effect of ICP changes on the pattern of blood flow velocity in the middle cerebral artery as measured with the TCD technique; iv) the effect of changing arterial CO₂ concentration (both hypercapnia and hypocapnia) on ICP. Finally, the results obtained are commented and their clinical relevance discussed. All mathematical aspects of the model, with equations and parameter numerical values can be found in previous reports (9-12).

The analysis has been performed using a mathematical model of intracranial CSF dynamics and cerebral hemodynamics previously developed by Ursino et al. (9-12). In the model, the time pattern of the main intracranial quantities (intracranial pressure, cerebral blood flow, inner radius at the level of proximal arteries and pial arterioles, blood flow velocity, intracranial venous pressure, etc...) originates from the interaction among several different compartments, each characterized by its dynamics and specific parameters. The main compartments included in the model, and the major biomechanical laws adopted to describe their behavior, are briefly summarized in the following. A complete description of mathematical equations, with assignment of parameter values, and simulation of specific patho-physiological events can be found in previous papers.
The Middle Cerebral Artery. In order to compute blood flow velocity in the middle cerebral artery, we assumed that the MCA behaves in a passive way in response to transmural pressure alterations (SAP - ICP): direct measurements demonstrated that MCA diameter changes due to haemodynamic stimuli are limited (13). The pressure-radius relationship has been reproduced by means of a mono-exponential function the parameters of which have been given to reproduce data reported in Hayashi (14). Finally, blood flow velocity in the MCA has been estimated as the ratio of blood flow to cross sectional area, and assuming that about 1/3 of total CBF passes through each MCA.
The arterial-arteriolar intracranial compartments. Two different compartments have been used to describe hemodynamics in the large and small pial arteries, respectively. Since, according to various recent physiological reports (1,15) the active response of proximal pial arteries and of distal arterioles are different, both as to their time pattern and the regulatory mechanisms involved, we decided to maintain a clear distinction between them. Accordingly, large pial arteries are assumed to be controlled by a pressure-dependent feedback regulatory mechanism (maybe myogenic or, more probably, neurogenic in nature) whereas distal arterioles are under the control of flow-dependent vasoactive factors. The main hemodynamic properties of each segment are summarized via two biomechanical quantities, i.e. the hydraulic resistance, which describes viscous energy losses and pressure drop, and the compliance which reproduces the non-linear pressure-volume characteristic of haematic vessels. Both quantities seriously depend on the value of vessel inner radius, hence are affected by the action of feedback autoregulation mechanisms. Action of feedback mechanisms on vessel caliber, hence on compliance and resistance, has been simulated through the Laplace law (16), which establishes the condition for the equilibrium of forces (elastic, viscous and muscular) in the vessel wall, and assuming that vascular smooth muscle tension depends on feedback regulatory factors.
The venous intracranial compartment. As for the arterial-arteriolar cerebrovascular bed, the main hemodynamic properties of intracranial veins have been described by means of the two quantities: hydraulic resistance and compliance. Furthermore, according to physiological results, we presumed that action of cerebrovascular regulatory mechanisms on intracranial veins is almost negligible, hence that cerebral veins behave passively. Venous compliance has been taken as inversely proportional to the local transmural pressure value. Venous resistance is known to be significantly affected by a passive collapse or narrowing of terminal intracranial veins (lateral lakes and bridge veins) (17). In the model, collapse of the terminal venous cerebrovascular bed has been reproduced according to the Starling resistor mechanism (16).
The cerebrospinal fluid compartment. Both CSF production at the cerebral capillaries, and CSF outflow at the dural sinuses have been described as passive mechanisms. The values of CSF production rate was computed as the ratio of the transmural pressure at the capillary level (cerebral capillary pressure minus ICP) over a resistance to CSF production. Similarly, the CSF outflow rate was computed as the ratio of transmural pressure at the dural sinuses (ICP minus sinus venous pressure) over a resistance to CSF outflow.
The elastic compartment. This embodies all the intracranial constituents not included in the previous three compartments. According to the Monro-Kellie doctrine, the overall volume in the craniospinal system must remain constant. This means that any change in one of the other volumes (either the cerebral arterial-arteriolar blood volume, cerebral venous blood volume or CSF volume) must be accompanied by a compression of intracranial structures and/or volume shifting along the spinal axis, with a consequent increase in ICP. This behavior has been described through the so-called storage capacity of the craniospinal system (analogous to the PVI previously introduced by Marmarou (18)). The storage capacity has been taken as inversely proportional to ICP. This means to have a mono-exponential pressure-volume curve, as in the works by Avezaat et al. (19) and Marmarou et al. (18). Three examples of mono-exponential pressure volume relationships are shown in Fig. 1, with reference to patients with normal, moderately reduced and severely reduced intracranial elasticity. These curves describe the effect, on ICP, of an intracranial volume change. According to the exponential profile, the intracranial compliance, which measures the capacity of storing a volume increase in the craniospinal space, is inversely proportional to ICP, through the elastance coefficient K_E. Greater the parameter K_E, more rapidly ICP increases, as a consequence of the same intracranial volume change.
Input quantities for the model are the arterial pressure at the level of major extracranial arteries, the central venous pressure and the CO₂ pressure in the arterial blood (PaCO₂). Venous return from dural sinuses to the heart has been described by means of a simple wind-kessel model, i.e., the parallel arrangement of the resistance and the compliance of the extracranial venous drainage pathways.
The model includes several parameters any of which might change from one subject to another. A basal value for each parameter was assigned according to clinical and physiological data, in order to reproduce normal intracranial hemodynamics and CSF dynamics of healthy young subjects. Details on how these values were assigned, and on their possible changes occurring in pathological conditions can be found in previous works (11,12).
All the simulations were performed on 486 MS-DOS personal computers by using the software package SIMNON (SIMNON/PCW for Windows, version 2.01, SSPA Maritime Consulting AB, Goteborg, Sweden) for the numerical integration of differential equations.

The aim of this section is to describe the results of some model simulations of clinical interest. During these trials, we have changed the input quantities and the parameter values, in order to reproduce some important features of the intracranial dynamics, during physiological or pathological conditions. In particular, we will analyze: i) the cerebral autoregulation in physiological conditions; ii) the occurrence of intracranial instability, due to autoregulation in pathological cases; iii) the impact of cerebral perfusion pressure changes on blood flow velocity quantities measured with Transcranial Doppler Technique; iv) the impact of CO₂ reactivity on ICP.
i) cerebral autoregulation in physiological conditions - Fig. 2 shows the percentage changes of CBF and arterial-arteriolar blood volume (Va) computed with the model in steady state conditions at different values of systemic arterial pressure (SAP). Throughout these simulations we assumed that ICP is constant at its normal value (about 9.5 mmHg).
Looking at the upper panel, it can be seen that cerebrovascular autoregulation is able to maintain quite a constant CBF within a wide pressure range: in accordance with physiological literature, the lower and upper autoregulation limits are about 50 and 150 mmHg, respectively. Model results are compared with those obtained during experiments on animals reported in (15,20): the agreement is fairly good.
As represented in the lower panel of Fig. 2, the effect of cerebrovascular vasodilation and vasoconstriction induces significant active changes of blood volume in pial arterial-arteriolar vascular bed. The constancy of flow in the central autoregulation region is attained by means of quite moderate vasoconstriction and vasodilation, without excessive blood volume changes. This is the region where the response of large pial arteries dominates. As arterial hypotension becomes more pronounced, however, active vasodilation in small pial arteries rapidly rises, causing a larger increase in cerebral blood volume. It is worth noting that, at the point of maximum vasodilation (mean SAP 40 mmHg) CBF is already reduced to half its normal value. Hence, according to the classical experimental studies by Kontos et al. (1) and MacKenzie et al. (15), the lower limit of autoregulation, that is when CBF starts to decrease (about 50-60 mmHg in this model), is not reached at the point where vasodilation is exhausted, but where the active vasodilation, although present, becomes insufficient to maintain CBF. Finally, in case of very strong variations in arterial pressure, the vessels behave in a completely passive way: hence, they collapse during severe hypotension, producing a fall in Va and CBF, and are forced to dilate with massive increases in SAP, causing a progressive rise of cerebral blood flow and volume.
ii) Autoregulation and intracranial instability in pathological conditions - In patients with normal intracranial dynamics, the changes in cerebral blood volume caused by arterial-arteriolar vasodilation (lower panel of Fig. 2) are usually buffered by the intracranial compliance, and subsequently compensated by CSF outflow without causing any significant rise in ICP. By contrast, in pathological subjects, when intracranial compensatory mechanisms are impaired, autoregulation may induce significant changes in ICP. In these circumstances, intracranial dynamics can become unstable and self-sustained oscillations develop. The occurrence of ICP oscillations in neurological patients was first recognized by Lundberg, who classified several types of ICP waves (21). In the most serious pathological cases these oscillations, named A-waves or plateau waves, may reach a plateau as great as 60-70 mmHg, lasting for 20-30 minutes.
The upper panel of Fig. 3 shows an example of ICP self-sustained oscillations obtained from the model equations simulating a pathological condition. The shape, amplitude and frequency of the simulated time pattern closely resemble ICP fluctuations observed in real patients. The correspondence between clinical and theoretical context is confirmed also by the similar conditions under which this phenomenon develops: a reduced intracranial compliance (that is a high value of the parameter K_E), an impaired process of reabsorption of cerebrospinal fluid (that is a high value of CSF outflow resistance), and finally preserved autoregulation mechanisms are documented during plateau waves (see 11).
The oscillatory nature of the phenomenon is represented also by the closed trajectories plotted in the lower panel of Fig. 3, which represents ICP vs. arterial-arteriolar blood volume during plateau wave generation. The chain of events occurring during a plateau wave can be summarized as follows: i) owing to obstruction of CSF outflow, ICP slowly increases, leading to a reduction in CPP and CBF. The latter trigger cerebral autoregulation which causes active vasodilation, hence an increase in cerebral blood volume (CBV). ii) As a consequence of the reduced storage capacity, the rise in CBV provokes a significant ICP increase, and thus CPP diminishes further. This chain of events, called "vasodilatory cascade" by Rosner (22), continues until vasodilation is maximal and CPP falls below the autoregulation lower limit. iii) When cerebral vessels are maximally dilated, ICP slowly decreases thanks to CSF reabsorption. The slow ICP reduction continues until CPP overcomes the autoregulation lower limit. iv) An opposite instability phase, sustained by active vasoconstriction, develops, leading to sudden cerebral vessels constriction and ICP reduction. Thus ICP returns to its starting point and a new cycle begins again.
Effect of ICP changes on blood velocity at the MCA -
Fig. 4 shows the pattern of systolic, mean and diastolic blood velocity at the level of MCA (left panel) and the Gosling pulsatility index (PI) computed with the model at different levels of ICP. In this case, simulations have been performed in pulsatile regime, that is, giving arterial pressure in the model an oscillatory time pattern which reproduces the normal sphygmic wave. Different levels of intracranial hypertension in the model have been attained by progressively increasing the CSF outflow resistance, that is simulating a progressive obstruction in CSF circulation. The Gosling pulsatility index was computed as follows:

where v_sys, v_mean e v_dia denote systolic, mean and diastolic blood flow velocity, respectively.
Looking at the lower panel of Fig. 4, one can see that PI increases quite linearly with ICP until ICP reaches a break-point located at about 50-60 mmHg; then a rapid rise occurs. The reason is that, when the regulatory mechanisms are adequate, mean and diastolic velocities do not change substantially in the central autoregulation range, while systolic velocity increases due to the increase in vessel compliance caused by vasodilation. Hence, in the central autoregulation region PI grows mainly via an increase in its numerator. By contrast, beyond the autoregulation lower limit, mean velocity starts to fall rapidly, and consequently PI rears dramatically owing to a reduction in the denominator. The same patterns of PI and velocities vs. ICP have been observed in several clinical and experimental studies (5,6).
Effect of acute hypocapnia and hypercapnia on ICP - Fig. 5 shows the effect on ICP of a reduction in PaCO² from 40 to 20 mmHg (hypocapnia, left panel) and of an increase in PaCO₂ from 40 to 60 mmHg (hypercapnia, right panel). Simulations have been performed for three different values of the parameter K_E (the same used in Fig. 1) and supposing a partially obstruction of CSF outflow (hence, a moderate initial intracranial hypertension). Hypocapnia causes vasoconstriction: the consequent CBV reduction leads to an abrupt fall in ICP, followed by a slow return to a new equilibrium condition, through the mechanism of CSF production. The new level of ICP is lower than the initial one, since hypocapnia in the model causes a decrease in CBF with a consequent decrease in the CSF production rate. High values of elastance coefficient induce a more sudden, massive ICP decrease, with a more rapid dynamics and a prompt achievement of the same final pressure level.
On the other hand, hypercapnia induces an active vasodilation which brings about an ICP rise. Later, ICP diminishes through the process of CSF reabsorption. A growth of K_E parameter means poor storage capacity of the craniospinal system: thus, CBV alterations can hardly be compensated, and the acute ICP increase becomes dramatic. In the extreme case, the pattern of ICP reproduces that of a single plateau wave. However, conditions leading to intracranial instability are not reached and the wave still remains isolated. The new, common baseline is higher than the starting one, since the increase in CBF causes a parallel increase in CSF production rate.

The few examples shown in this work aim at illustrating some possible applications of the present model, and the potential clinical benefits of a computer simulation approach for the analysis of patients in neurosurgical units. The main value of a modeling study resides in the complexity of the relationships among intracranial quantities, in the modifications that these relationships exhibit in various pathological conditions, and in the consequent difficulties in understanding measurement results, and in making predictions on the patient's management on the basis of simple qualitative reasoning. These difficulties are made even greater by the increasing number of quantities which are simultaneously monitored in modern Intensive Care Units. All the data now available risk being insufficiently exploited or misunderstood, if the mutual relationships among them are not correctly taken into account.
An example of the difficulties encountered in the management of neurosurgical patients is shown by the phenomenon of ICP instability, stressed in the simulations of Fig. 3. Even though the occurrence of self sustained ICP plateau waves manifest itself only in patients with serious pathologies and severe intracranial hypertension, a similar 'vasodilatory cascade' may induce transient acute alterations in ICP even in patients who normally do not exhibit ICP oscillations. The risk of acute hypertension, induced by a transient CBV increase, is especially relevant when cerebral perfusion pressure approaches the lower autoregulation limit, as shown in the lower panel in Fig. 2. In a previous recent work (23) we were able to show by mathematical simulations that a moderate arterial pressure decrease, which is well tolerated in a patient with normal mean SAP and good intracranial compliance, may induce a dramatic acute ICP rise in a patient with low mean SAP and poor elasticity of the craniospinal compartment. The acute ICP increase, in turn, may contribute to cerebral ischemia and secondary brain damage. A similar phenomenon is shown during hypercapnia in Fig. 5 (right panel). In this case, too, in patients with abnormally reduced intracranial compliance, cerebral vasodilation may provoke a dramatic ICP rise similar to that of an isolated plateau wave. The existence of abrupt ICP rises following acute perturbations of cerebral hemodynamics, such as arterial hypotension and hypercapnia, is well documented in the recent clinical literature (24).
The capacity of the model to simulate the ICP response to various vasodilation and vasoconstrictory stimuli is of clinical value for two reasons. First, the model may be used to estimate the main parameters of intracranial dynamics (mainly compliance and CSF outflow resistance) and to assess the status of cerebrovascular regulation mechanisms starting from measurement of ICP and/or blood velocity time patterns in response to modest intracranial perturbations. For instance, we suggested in previous works that the status of cerebral autoregulation may be estimated starting from the so-called PVI tests, which consist in the injection or in the withdrawal of small amount of fluid into the cranial cavity (12). Similarly, a best fitting between model and clinical data during CO₂ alteration maneuvers may be put to use to evaluate the status of CO₂ reactivity (25), which is often related to patient's outcome.
Second, in perspective the modeling approach might be useful to improve the management of neurosurgical patients in intensive care units. For instance, CO₂ manipulation or SAP alterations are frequently used today as means to control ICP, match CBF to metabolic requirement or reduce risk of cerebral edema. However, controversial hypotheses have been formulated in the clinical literature on how these maneuvers should be carried out to avoid any danger to the patient (26). The present simulation results emphasize that the same maneuver may have completely different effects on ICP and CBF, depending on the status of cerebrovascular regulation mechanisms, craniospinal compliance and CSF circulation. Moreover, due to the non-linear nature of the relationships characterizing intracranial dynamics, the response to many perturbations often exhibit a sudden break-point, after which a dramatic disproportionate change in intracranial quantities may occur. Analysis of the pulsatility index, for instance (Fig. 4), suggests that a disproportionate increase in the PI above 2 is indicative of a serious reduction of CPP below the lower autoregulation limit, with risk of brain tamponade and ischemia. As shown in the right panel of Fig. 5, the vasodilatory cascade may be silent and scarcely influential on ICP until craniospinal compliance is reduced below a critical level; after which it can provoke an uncontrollable instability phenomenon leading to massive vasodilation and severe intracranial hypertension. Of course, the possibility of early prediction on the occurrence of such a break-point may be of great clinical usefulness.
In conclusion, the aim of this short review paper was to summarize some of the main results which can be achieved from application of mathematical models and computer simulation techniques to the analysis of intracranial dynamics. The emerging message is that ICP and CBF dynamics depends on several concomitant factors mutually related. Consequently, the pattern of the main monitored quantities cannot be understood in simple qualitative terms, without considering its relation with the simultaneous patterns of the other quantities as a part of a unique strictly interconnected system. In this context, the synthetic aptitude of mathematical models may help clinicians to reach a more comprehensive understanding of the richness and the complexity of intracranial dynamics as a whole, both in health and disease, and to design new method for patient's diagnosis and treatment.

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