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Anesth Analg 2008; 107:1997-2008 © 2008 International Anesthesia Research Society doi: 10.1213/ane.0b013e318187b987
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A more detailed description of the different mathematical relationships included in the model is given below.
Large Intracranial Arteries
The first segment in the model (Fig. 2) 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 (Rla–resistance of large intracranial arteries). Because the impact of cerebrovascular regulation mechanisms on the basal intracranial arteries is quite small, Rla has been maintained constant throughout the simulations. An approximate estimation of blood flow velocity at an MCA (VMCA) is computed by assuming that blood flow in an MCA is about one-third of total CBF. We can thus write:
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where qla is the total CBF at the level of the basal intracranial arteries (i.e., at the level of the resistance Rla in Fig. 2) and rMCA is MCA inner radius. Finally, kV is a scaling down factor used to compare the results of the model with empiric results. This factor considers the difference in inclination between the ultrasonic probe and blood vessel, and the fact that the radius of the MCA is not the same in all subjects. The value of rMCA is a function of cerebral artery transmural pressure (SAP – ICP) through a monoexponential pressure-radius relationship taken from Hayashi et al.16 This means that the MCA behaves passively and becomes progressively more rigid when transmural pressure increases.
Pial Arterial-Arteriolar Cerebrovascular Bed
The second segment in Figure 2 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 (Rpa–resistance of pial arterioles) and compliance (Cpa). Finally, both parameters are actively regulated by cerebrovascular control mechanisms.
The present model does not explicitly incorporate the mechanism of critical closing pressure. Mathematically, CBF becomes zero when mean SAP is equal to ICP, hence, the real closing pressure would be equal to ICP. However, in the lower autoregulation range, thanks to active changes in arteriolar resistance, the curve CBF versus mean SAP is flatter and shows an intercept at about 15–16 mm Hg. This value is close to the critical closing pressure normally measured in vivo (about 20 mm Hg).17
Venous Intracranial Circulation
The intracranial venous vascular bed is described by the series arrangement of two segments. The first segment extends from the small postcapillary venules down to the large cerebral veins and includes the venous resistance (Rpv) and the intracranial venous capacity (Cvi). Because the effect of cerebrovascular control mechanisms on the venous vasculature is negligible, and large veins do not collapse even at an elevated ICP, 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 mono-exponential pressure-volume relationship for the veins. We are aware that the exact mathematical expression for venous compliance may be affected by head injury. However, changes in intracranial venous transmural pressures are always modest in our model (because of the Starling resistor mechanism).
The last segment in Figure 2 represents the terminal intracranial veins (lateral lakes and bridge veins). During intracranial hypertension, these vessels collapse or narrow at their entrance into the dural sinuses, with a mechanism similar to that of a Starling resistor. Due to this mechanism, pressure in the large cerebral veins (Pv) remains a little higher than Pic even during states of intracranial hypertension.
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 (Pc – Pic, where Pc is capillary pressure) and is reabsorbed at the dural sinuses because of a negative transmural pressure gradient (Pvs – Pic, where Pvs is venous sinus pressure). The resistances to CSF formation and CSF outflow are Rf and Ro, respectively. However, both processes are unidirectional, so we have assumed that resistances increase to infinity when the corresponding transmural pressure is reversed.
We are aware that CSF production is a more complex process than the one described in our model. In fact, CSF is produced at the choroid plexus and is regulated by active mechanisms. However, the simple choice adopted in our model allows a pressure-independent process to be simulated quite well, without the introduction of additional variables and equations (i.e., without a separate description of the choroid plexus and of active mechanisms controlling CSF). In fact, thanks to the presence of autoregulation mechanisms and of a downstream Starling resistor, capillary transmural pressure in our model is proportional to CBF. Hence, CSF production is actively controlled by the same autoregulation mechanisms which control CBF, and is almost independent of pressure changes.
Intracranial Compliance
Because the overall intracranial volume must remain constant, any change in the content of one of the compartments (e.g., pial artery volume at Cpa, venous volume at Cvi, 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 (Cic). Most authors have approximated the volume-pressure curve by an exponential function.6 Therefore we used the following equation:
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where kE is the intracranial elastance coefficient, Piceq is a constant equal to the ICP at the equilibrium point, and V is the change in total volume of the craniospinal contents with regard to the equilibrium volume. The elastance (Eic) of the craniospinal system, defined as the slope of the pressure-volume curve, is the derivative of Eq. 2
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So kE determines the elastance at a given pressure and is, therefore, termed the elastance coefficient. Because Cic is the reciprocal of kE, we then have
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kE is an index of the rigidity of the intracranial compartment. Equation 4 implies that the cerebrovascular pulsatility progressively increases at high values of ICP.
Cerebrovascular Regulation Mechanisms
Cerebrovascular regulation mechanisms work by modifying Rpa and Cpa (and hence CBV) in the pial arterial-arteriolar vasculature. However, changes in these two parameters are not independent but are related through biomechanical and geometrical laws.
Two distinct control mechanisms are considered in this model, i.e., autoregulation and CO2 reactivity. The role of arterial oxygen tension and content is not included.
We assumed that autoregulation is activated by changes in CBF. Its action on the arterial-arteriolar pial vessels includes a static gain (Gaut) and first-order low-pass dynamics with the time constant 
aut. We chose to incorporate a low pass filter to simulate the typical elapse in time required for the mechanism to accomplish its action (approximately three time constants). An increase in CBF causes vasoconstriction, with a consequent decrease in pial vessel compliance and an increase in resistance, whereas a CBF decrease causes vasodilation.
CO2 reactivity also includes a static gain (GCO2), and first-order dynamics with the time constant CO2. We used the logarithm of Paco2 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 the CO2 concentration via the Henderson-Hasselbalch equation.
Finally, the two mechanisms do not superimpose linearly on pial vessels, but their interaction is characterized by significant nonlinearities. To consider experimental and clinical evidence, the model introduces 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. In fact, severe ischemia is associated with tissue acidosis, which, in turn, is buffered by the effect of HCO–3 and CO2 changes on the perivascular pH. A second nonlinearity considers that the overall regulatory action is not the sum of the two mechanisms’ actions, but is instead passed through a sigmoidal static relationship with upper and lower saturation levels. The sigmoid relationship 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.15
In this article, the existing mathematical model of intracranial dynamics15 was modified and expanded as follows:
- Changing constant inputs to pulsatile inputs (SAP, central venous pressure). These beat-by-beat variations were reproduced in order to simulate clinical reality as observed by clinicians. For instance, as the intracranial compliance decreases, the pulsatility of the ICP increases.
- Simulating the effects of various amounts of intracerebral bleeding on among others, ICP, CPP, and CBF.
- Simulating the effects of the head-up position on the cerebrovascular model.
Every change was aimed at examining the interactions among ICP, CBF, CPP, blood velocity changes, and cerebrovascular regulation in conditions of clinical relevance.
We now describe intracranial bleeding and the head-up position.
Intracranial Hemorrhage
An intracranial hemorrhage is bleeding in the intracranial vault caused by the rupture of a blood vessel (arterial or venous) within the head. Therefore the rate of bleeding may vary greatly. An intracranial hemorrhage can be caused by a traumatic brain injury or abnormalities of the blood vessels (e.g., aneurysm or angioma).
In our model, an intracranial hemorrhage is simulated by creating an increase in the volume of the interstitial portion of the intracranial compartment. So the application of the mass preservation at the intracranial storage capacity becomes:
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where Vpa and Vvi are the blood volumes of the pial arterial-arteriolar vascular bed and of the intracranial veins, respectively, qf and qo are the rates of CSF formation at cerebral capillaries and CSF outflow at the dural sinuses, respectively and hbf is the rate of blood going from the cerebral vessels to the intracranial space. The intracranial hemorrhage blood flow is also in the arterial-arteriolar compartment, as follows:
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where Ppa is the pial arterial pressure. See Figure 2 for a better understanding of the equations.
We have implemented an exponential time course, according to the typical bleeding modes (arterial or venous) encountered in the literature,18 considering that both arterial and venous bleeding usually stop due to vasoconstriction, tamponnage and/or clotting of blood.
Head Elevation
Elevation of the head of the bed is a standard (but controversial) neurosurgical practice for management of increased ICP, and it refers to the angle between the trunk and the horizontal plane. In some intensive care units, in a patient with increased ICP, it is a common practice to position the patient in bed with the head elevated well above the level of the heart in order to decrease an elevated ICP (however, this practice may potentially lead to inadequate CPP and CBF, which are common causes of secondary brain damage in head-injured patients). The standard position for severely head-injured patients in other intensive care units is the semi-recumbent 30 degrees head elevation. Practically, this (limited head-up position) is thought to allow reduction of ICP without compromising CBF, CPP and/or cardiac output. On the other hand, some clinicians prefer to leave patients flat, as opposed to any degree of head elevation, in order to reduce even the small risk for systemic hypotension inherent in a semi-recumbent posture. Any systemic hypotension could possibly result in a net reduction in CPP and effectively negate any benefit that might occur even if there were a decrease of ICP.
As the head elevation increases, there is a gradual decrease in SAP at the head level. This relationship is described by the following equation taken from Rosner and Coley.19
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where SAPnew is the arterial pressure at the level of the head modified by the head elevation maneuver, and x is the elevation in degrees.
The ICP similarly declines.
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Because the SAP at the head level declines faster than the ICP, there is a net decrement in CPP.18,19 Furthermore, the central venous pressure (CVP) decreases slightly as the ICP decreases19,20
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where CVPnew is the CVP after head elevation.
The elevation of the head is simulated with a rate of change of 1 degree per second, so all other variables are modified at this rate. Using the model, we hope we can study and clarify the issue of optimal head position in the care of head-injured patients.
The model has been extensively validated in previous works, both by simulating data in the physiological literature on CBF response to pressure changes and CO2 changes,15 and by simulating the time pattern of ICP and VMCA velocity during clinical maneuvers (pressure volume index tests,21 MAP manipulation and CO2 manipulation).15
The mathematical model has been implemented in two consecutive steps. In the first we used the software package Berkeley Madonna for simulation of dynamical systems
. In the second, in order to develop a stand-alone cerebrovascular simulator which could be used as a medical education tool, the model was implemented in JAVA (jdk version 1.6, Sun Microsystems, Santa Clara, CA) using the software NetBeans (version 5.5, Sun Microsystems, Santa Clara, CA). This second version includes a user-friendly interface (Fig. 3), which allows the user to interact with the mathematical model in a very simple way. The executable file (Microsoft WindowsTM, Redmond, WA) is available free of charge from the following URL: http://campus.cib.unibo.it/2446/.
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As shown in Figure 3, the screen is divided in two parts: on the left is a monitor for the visualization of three output quantities: SAP, ICP, and CBF. The monitor can be dragged to a separate (second) screen and/or widened for a better visualization as a usual Microsoft window. On the right, is a control board in which different panels can be visualized in order to choose the event or action to simulate. Using this control board, the user can simulate different events (e.g., hemodynamic reactions after changes in SAP, Paco2 head-up elevation, intracranial hemorrhage) and can modify one or more of five parameters to represent different patients. At present, the modifiable parameters are (Fig. 3) intracranial elastance coefficient, CSF outflow resistance, autoregulation gain, CO2 reactivity gain, and CO2 reactivity time constant.
Furthermore, clicking on a button placed at the top of the screen enables the user to run two different types of simulation: off-line simulation or runtime simulation. The off-line simulation is much quicker in terms of simulation time (the computational time is about 1 min to simulate 1 h in a 2 GHz personal computer, 512 MB RAM), but it requires that all aspects of the simulation are decided before running the model and all results can be seen only at the end. By contrast, in real-time simulation the output quantities are visualized in "real time" on a bedside monitor emulator (one real second per 1 s of simulation), and the user can change input quantities or parameters while the simulator is running. At the end of a real-time simulation, the user can visualize the overall simulated tracings.
RESULTS
Simulations have been performed to show how alterations or manipulations in input quantities (e.g., Pa, Paco2, head elevation) can affect clinical intracranial quantities (ICP and CBF). Among the vast numbers of potential permutations and combinations, we present and discuss some illustrative examples:
- The effect of modest hypotension in normal and head-injured patients;
- The effect of an intracranial hemorrhage, with and without previous head injury;
- The effect of head elevation on ICP and CBF in head-injured patients;
- The effect of CO2 on ICP waves.
Acute Arterial Hypotension
We simulated a decrease in MAP from 100 to 80 mm Hg, in a normal and head-injured patient. A head injury has been simulated by assuming an increase in the craniospinal elastance coefficient, (i.e., reduced craniospinal compliance), an increase in CSF outflow resistance and impaired autoregulation. These changes are based on those previously estimated on patients with severe head injury15 and are further justified in the Discussion section.
The results (Fig. 4) show that even moderate hypotension (with acute onset) may lead to a significant increase in ICP and a significant decrease in CBF in subjects with a stiffer intracranial compartment. In particular, as a consequence of the transient ICP increase (up to 40–50 mm Hg), CPP is reduced to 30–35 mm Hg, (i.e., below the normal autoregulation range).
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Intracranial Hemorrhage
We simulated a 10 mL hemorrhage, occurring over a period of 30 s. Two examples are illustrated (Fig. 5):
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- In a subject with normal intracranial dynamics (basal values of parameters, left panels) hemorrhage causes only moderate intracranial hypertension with minor changes in CBF;
- In a patient with severe head injury (reduction in intracranial compliance and in CSF outflow, right panels) hemorrhage causes a dramatic increase in ICP, with the appearance of a plateau wave,22–24 and a dramatic decrease in CBF. These alterations may induce secondary brain damage.
Head Elevation
This maneuver is usually performed during clinical practice in order to decrease the level of ICP in a patient with an increased ICP. Indeed, as we can see in Figure 6, this maneuver does decrease the ICP in a head-injured patient.
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However, this decreased ICP might not necessarily be beneficial as we observe that head elevation causes a further decrease of CBF (Fig. 6), with a possible detrimental effect on cerebral perfusion. While we can readily measure ICP, CBF, which is not measured readily, is actually the more important variable.
Effect of CO2 Changes
The last group of simulations illustrated here has been performed to show the different effects that CO2 manipulation can have on ICP and CBF in severely head-injured patients who exhibit intracranial instability and plateau ("Lundberg") waves.22–24 In these simulations, we eliminated the pulsatility for the sake of clarity but similar results are enabled using pulsatile outputs. In a patient with a serious reduction in CSF outflow and moderate reduction in intracranial compliance, the model predicts the presence of small amplitude ICP plateau waves (Fig. 7, continuous line). In this patient, an increase in CO2 pressure may be beneficial, causing the interruption of the ICP plateau waves and improvement in CBF (Fig. 7, dashed line). In contrast, in more seriously compromised patients, with larger decreases in craniospinal compliance, we can observe the occurrence of much larger ICP plateau waves (Fig. 8, continuous line). In this patient, CO2 pressure should be decreased to abort the waves and improve patient status (Fig. 8, dashed line). The reason is that, in the last patient, vasodilation causes an uncontrolled increase in ICP, which induces a consequential decrease in CBF. Hence, vasoconstriction is more indicated than vasodilation.
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DISCUSSION
A mathematical model is abstract and uses mathematical language to describe the behavior of a system. A cerebrovascular model consists of variables, which are abstractions of quantities of interest in the cerebrovascular system. Ideally, the mathematical model should provide an output in a visual format that is familiar to the user, in this case, the clinician.
In our case, we believe this mathematical model is valuable to the clinical community. For instance, our simulation outputs have been designed within a clinical context and the results were shown to clinicians with expertise in neurosciences. The results were deemed to be clinically accurate with applications in teaching, research, and clinical environments being envisaged.
A model is only an incomplete representation of reality and choices have to be made as to what to include and exclude from the model. For instance, this model was chosen to represent a single intracranial compartment with blood supply from a single blood vessel. The choice was deliberate, as this simplified model would be more readily understood by users.
Therefore, this simplified model was created for teaching, research and clinical purposes. For instance:
Learning and Understanding Cerebral Physiology as well as Pathology, Such as Head Injury and Intracranial Hemorrhage
Using the cerebrovascular model, the user can directly visualize the effects of a maneuver on a patient and understand, after different simulations, the mechanisms and causes of these alterations.
Simulations which the user can perform to study and understand more about the physiology and the pathology of head injury are, for example:
- The simulation of an intracranial hemorrhage.
- The creation of a patient with a stiff (low compliance) intracranial compartment (typical of head-injured patients).
- The decrease of autoregulation.
- Simulations of the blockage of the CSF outflow path.
- Changes in Paco2 and Pa.
- The creation of a patient with a stiff (low compliance) intracranial compartment (typical of head-injured patients).
Generating Research Questions; for Instance: What is the Optimal Head-Up Position (In Terms of "Best" CBF) in a Given Patient?
The model gives the user the capability to analyze the magnitude and time course of changes in the CBF and in other variables which are not typically monitored during clinical practice. In this way, the user is able to understand whether a maneuver applied to the patient is beneficial; hence, the model can be used to teach how to make clinical decisions regarding CBF, while using ICP measurements.
Our model shows that sometimes the ICP does not reflect the status of brain perfusion, even though clinicians usually consider ICP to make decisions (the ICP is typically used, because it is one of the few variables that we can readily monitor during clinical practice). For instance, after acute arterial hypotension, ICP increases to a value higher than the normal (very high in the head-injured patient), therefore, it seems reasonable to perform head elevation in order to decrease ICP in order to increase CPP to a more normal level. Unfortunately, after head elevation, CPP decreases (due to a decrease in SAP) and it is more difficult for autoregulation to reach the CBF baseline (indeed CBF may decrease in head-injured patients). Using the simulation program, trainees can gain a "feel" for these relationships. Such training is not typically available in clinical practice.
The simulations of ICP plateau (Lundberg) waves are also of great interest, as an example of the difficulty in the choice of the correct balance between vasodilation and vasoconstriction (obtained by high and low Paco2 manipulations). ICP waves are a phenomenon of vascular instability,22–24 induced by a positive feedback between vascular vasodilation and ICP changes. Vasodilation causes an increase in ICP, that may become especially evident in cases of reduced intracranial storage capacity. The ICP increase, in turn, decreases CPP, with the effect of inducing further vasodilation. In the case illustrated in Figure 7, ICP waves are still of a moderate magnitude, hence vasodilation (induced by hypercapnia) has a beneficial effect, increasing CBF and causing the interruption of the plateau waves. In contrast, in the more severe case illustrated in Figure 8, plateau waves are fully developed, caused by intermittent maximal vasodilation and vasoconstriction. In this condition, the model demonstrates that a vasoconstrictive stimulus (induced by hypocapnia) is necessary to stop the feedback, aborting the waves. These examples highlight the complexity of the multifactorial phenomena involved in intracranial dynamics; the same maneuver may have a beneficial or detrimental effect depending on the patient status. Practice with a simulation program may help to build an intuitive feel, which otherwise can only be gained after many years of clinical experience.
Performing Clinical Optimization (e.g., by Entering Patient-Specific Parameters, Calculate Optimal Combinations, and Validate on the Patient)
The model gives us the capability to simulate a specific individual patient and different states of this patient, by changing the parameters of the model. Therefore, if we are able to measure and/or estimate some of the parameters in a real patient, we can put these values in the model and use the model to simulate the individual’s behavior (in terms of ICP and CBF). To this end, an evaluation phase is needed, during which the model and real ICP and VMCA tracings are compared, and a best fitting (optimization) "search" is performed to find the parameter set which simulates the patient’s dynamics as closely as possible (Ursino et al.15 for example). The model, with parameters individually assigned, can then be used to discover the optimal combination of maneuvers able to maintain adequate levels of CBF and ICP, reducing the risk of secondary brain damage.
There are substantially two methods for assigning parameters:
- By using paradigmatic values taken from the clinical literature, in order to simulate a prototypical patient of a given class;
- By using a best fitting procedure for parameter estimation, based on individual data, collected during clinical trials. An example of the latter method was described in previous papers.15,21
A possible objection may be that our equations contain too many parameters, which cannot all be carefully estimated on the basis of optimization procedures. Although this objection is certainly correct, one must consider that not all parameters need to be individually assigned, but only a subset of them, depending on two main criteria. First, the parameter value must exhibit significant changes as a consequence of pathological alterations; second, these changes must have a significant impact on model outputs. The second point may be decided on the basis of a sensitivity analysis, while the first requires a priori clinical physiological knowledge. All other parameters may be maintained at their basal values.
Following these ideas, in previous work15 we have shown that a change in the value of only five parameters with a specific physiological meaning (CSF outflow resistance, intracranial elastance coefficient, gain of autoregulation, gain and time constant of CO2 reactivity) is sufficient to simulate ICP and VMCA in head-injured patients after clinical maneuvers. Hence, in preparing the present software package, we enabled these parameters to be assigned via a user-friendly interface to mimic pathological subjects.
Finally, it is worthwhile to discuss some aspects of the model which may be the subject of future improvements, modifications and extensions, and those aspects which are still debated in neurophysiology.
In the present article we assumed that head elevation decreases both SAP and ICP, as well as CVP, with a greater decrease in SAP than in ICP and CVP. These data were taken from the clinical literature.19 As a consequence, CPP decreases during head elevation. However, the way gravity affects blood supply to the brain is the subject of much controversy in the physiological literature and this problem is far from being fully assessed. Two alternative ideas can be found: the siphon concept25 and the vascular waterfall.26 The assumption used in the present work (SAP decreases more than CVP) may be considered as the result of an incomplete siphon or, alternatively, as suggested by Hicks and Badeer,25 as the result of a siphon mechanism operating in a partially collapsed venous limb, with an increase in the viscous losses in the descending pathway.
The way we simulated head injury also deserves discussion. In the present simulations, we assumed that head injury reduces craniospinal compliance and increases the CSF outflow resistance. These changes are documented in the literature,21 and agree with parameter values estimated with our model on individual head-injured patients.15 From a clinical/physiological point of view, the first mechanism after traumatic brain injury is probably brain swelling, which is essential in the etiology of intracranial hypertension in trauma. Brain swelling might be initially simulated as an increase in intracranial volume, thus inducing intracranial hypertension and reducing compliance. However, after the initial volume increase, one may expect secondary changes in intracranial dynamics; the latter may affect CSF circulation and overwhelm the redistribution mechanism (hence, may be simulated with an increase in CSF outflow resistance).
A limitation of this model is that it includes only a single path for CBF from the arterial to the venous circulation. Hence, the model cannot be used to simulate conditions characterized by nonhomogeneity in brain hemodynamics, such as those due to a stenosis or occlusion of a major intracranial artery or metabolism increase in a specific region. To overcome these limitations, a more sophisticated model, including a separate description of brain hemodynamics in different brain regions and the anastomotic role of the circle of Willis, is under construction and validation.
A further important future development may consist of a description of oxygen transport to individual brain regions and local vasodilatory adjustments in the microcirculation. This model extension may be of high value for the analysis of data in functional neuroimaging techniques based on hemoglobin oxygenation.
CONCLUSION
Model simulations clearly suggest that intracranial dynamics is a complex multifactorial phenomenon, and that there is no single quantity which is able to summarize the overall patient status and/or use as a therapeutic guideline. For instance, a maneuver which reduces ICP can induce deleterious decreases in CBF in some conditions, whereas, in other conditions, reduction in ICP is essential to improve the status of perfusion. In particular, one cannot use a single variable to express the health or perfusion of the brain. Only an analysis of multiple quantities together can provide a complete description of the patient’s status. In this context, mathematical models may play a fundamental role, to summarize the complex relationships among intracranial quantities in a coherent and comprehensive scenario as an aid to improve teaching, research and clinical care.
Given the simple, user-friendly graphic user interface and the realism given by the bedside monitor emulator, we expect that this model and software could be used in the clinical research field as well as in medical education applications.
Footnotes
*A detailed description of model mathematical equations, including normal values of their parameters, has been reported.15 In this present paper, only the major qualitative aspects of the model are presented. The Masters thesis "A mathematical cerebrovascular model for clinical and teaching purposes" of M. Giannessi under the supervision of M. Ursino and W.B. Murray, discusses the complete model and its potentialities in complete detail. 
Version 8.0, Macey R, Oster G, Zahnley T., University of CA, Department of Molecular and Cellular Biology, Berkeley, CA.
Accepted for publication July 10, 2008.
None of the authors have any conflicts of interest related to the material published here.
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