A New Lumped-Parameter Model of Cerebrospinal ... - IEEE Xplore

IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 54, NO. 3, MARCH 2007

483

A New Lumped-Parameter Model of Cerebrospinal Hydrodynamics During the Cardiac Cycle in Healthy Volunteers Khalid Ambarki*, Olivier Baledent, Guy Kongolo, Robert Bouzerar, Sidy Fall, and Marc-Etienne Meyer

Abstract—Our knowledge of cerebrospinal fluid (CSF) hydrodynamics has been considerably improved with the recent introduction of phase-contrast magnetic resonance imaging (phase-contrast MRI), which can provide CSF and blood flow measurements throughout the cardiac cycle. Key temporal and amplitude parameters can be calculated at different sites to elucidate the role played by the various CSF compartments during vascular brain expansion. Most of the models reported in the literature do not take into account CSF oscillation during the cardiac cycle and its kinetic energy impact on the brain. We propose a new lumped-parameter compartmental model of CSF and blood flows in healthy subjects during the cardiac cycle. The system was divided into five submodels representing arterial blood, venous blood, ventricular CSF, cranial subarachnoid space, and spinal subarachnoid space. These submodels are connected by resistances and compliances. The model developed was used to reproduce certain functional characteristics observed in seven healthy volunteers, such as the distribution (amplitude and phase shift) of arterial, venous, and CSF flows. The results show a good agreement between measured and simulated intracranial CSF and blood flows. Index Terms—Blood flow, compliance, conductance, CSF flow, hydrodynamics, phase-contrast MRI.

I. INTRODUCTION

O

VER recent decades, a large number of mathematical models have been proposed to provide a better understanding of the intracranial dynamics. Pressure-volume (P-V) models [1]–[4] are based on the relationships observed between intracranial pressure and cerebrospinal fluid (CSF) volume during injection or subtraction of CSF in the cranium. These models incorporate the CSF mass conservation principle and secretion-absorption rates, and subsequently do not address the problem of their response to cardiac pulsations, that is do not consider the pulsatile component of the CSF motion. Some Manuscript received October 24, 2005; revised August 26, 2006. Asterisk indicates corresponding author. *K. Ambarki is with the Department of Imaging and Biophysics, Teaching Hospitals, Jules Verne University of Picardie, Amiens 80054, France (e-mail: [email protected]). O. Baledent, S. Fall, and M.-E. Meyer are with the Department of Imaging and Biophysics, Teaching Hospitals, Jules Verne University of Picardie, Amiens 80054, France (e-mail: [email protected]). G. Kongolo is with the Department of Pediatrics and Neonatal Intensive Care Unit, Teaching Hospitals, Jules Verne University of Picardie, Amiens 80054, France. R. Bouzerar is with the Condensed Matter Physics Laboratory, Jules Verne University of Picardie, Amiens 80054, France. Color versions of Figs. 2–4 and Tables II and III are available online at http:// ieeexplore.ieee.org. Digital Object Identifier 10.1109/TBME.2006.890492

Fig. 1. Cerebrospinal space model. The CSF is distributed into three spaces that communicate with each other: The ventricular space inside the brain, the cranial subarachnoid space around the brain and the spinal subarachnoid space around the spinal.

mechanical models [5], [6] regard the brain as a sponge-like material through which CSF diffusion takes place, but do not either consider that question. These models share a common attempt to describe hydrocephalus by studying the mechanical behavior of the cerebral parenchyma. However, these models are difficult to apply due to the lack of knowledge concerning brain properties such as the Young’s modulus, the Poisson’s ratio and its porosity. Another type of pulsatile model [7] describes the CSF dynamics within the ventricles excited by surrounding compartments, brain, arteries, veins and so on. Lumped-parameter pulsatile models [8]–[13] assume that the cranial cavity can be divided into a set of compartments which reflects faithfully the anatomic structure. Each of these compartments is characterized by a single pressure value and interacts with the adjacent compartments through fluid exchange (volume displacements). Based on ideas discussed above, the model presented in this paper represents the cranial cavity as a rigid box composed of three volumes: brain volume, blood volume and CSF volume. The CSF, with a viscosity close to that of water, is distributed into three spaces that communicate with each other (Fig. 1): the ventricular space inside the brain, the cranial subarachnoid space around the brain and the spinal subarachnoid space around the spine. The brain floats in CSF and the brain parenchyma and brain surface are crossed by a vascular network composed of arteries (30%) and veins (70%) [14]. The overall intracranial volume is constant. The MonroKellie doctrine [15], [16] describes blood, brain, and CSF as incompressible. If one of the three intracranial volumes increases, homeostatic mechanisms force one or both of the other volumes out of the cranial cavity to maintain a constant intracranial volume. Due to cardiac pulsations, intracranial arteries cause a cyclic cerebral blood expansion, establishing

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Fig. 2. (A.1) CSF flow measured at the Sylvius aqueduct level. (A.2) Blood and cervical CSF flows measured at the C2–C3 level. (B.a) Blood and (B.b) cervical and T are the delay of venous and cervical CSF peak flows compared to arterial systolic CSF flows measured by phase-contrast MRI. T peak flow.

CSF pulsatile motion. The CSF pulse is dissipated by transfer of the CSF volume out of the cranial cavity (into the spinal subarachnoid space) or by compression of cerebral veins [17], [18]. The spinal CSF volume can be expanded by the elasticity of the dural sac [19], which acts as a reservoir. The CSF oscillates through the foramen magnum in response to cerebral blood flow changes, resulting in a coupling between blood volume changes and CSF volume changes. Phase-contrast magnetic resonance imaging (phase-contrast MRI) can be used to measure CSF and blood velocities [20]–[22], throughout the cardiac cycle. Image processing software was developed to analyze phase-contrast MRI data [18], [23], [24]. Phase-contrast MRI measurements show that arterial blood flow entering the cranium at a given point in time is not equal to the venous blood flow leaving the cranium measured at the same moment of the cardiac cycle [8], [18]. However, the mean blood flow rate entering the cranium via the arteries is equivalent to the mean venous blood flow throughout the cardiac cycle. Based on the available anatomical and physiological knowledge of the intracranial system, we propose a new pulsatile lumped-parameter model to describe the mechanical interactions between arterial and venous blood and CSF. We neglected secretion-absorption processes and fixed our attention on a purely “pulsatile” model to be more realistic and model in the simplest way the cardiac pulsation. Our overall objective was to mimic the main feature of the intracranial dynamics, namely the flow response of the blood and CSF to the cardiac pulsation. To achieve this goal, we compared the model-computed flows to those revealed by phase-contrast MRI data in healthy subjects.

C2–C3 subarachnoid space [Fig. 2(A2)] and cerebral aqueduct [Fig. 2(A1)]. Axial vascular flow was measured in the internal carotid and vertebral arteries and in the internal jugular veins at the C2–C3 level [Fig. 2(a.1)]. Flow images were acquired with velocity-encoded phase-contrast MRI with peripheral gating. MRI parameters were as follows: repetition time: 29 ms; echo time: 6 ms; 160 mm 120 mm rectangular field-of-view; 256 128 matrix; 5-mm thickness, 30 flip angle, and one excitation. Velocity sensitization was set at 10 cm/s for the cerebral aqueduct, 5 cm/s for cervical subarachnoid space, and 80 cm/s for vessels. For each flow series, the acquisition time was about 1.5 min depending on the heart rate. Phase-contrast MRI data were analyzed using in-house image processing software with Interactive Data Language [18]. This software automatically computes the CSF flow curve during the cardiac cycle on the basis of the measured velocity fields. Craniocaudal flows were positive and called CSF flush, whereas caudocranial flows were negative and called CSF fill. Blood and CSF flows signals were divided into 16 cardiac cycle segments each representing 6.25% of the entire cycle, where RR represents one cardiac cycle (100%). Fig. 2(b) shows an example of blood and CSF flow meaand surements during two cardiac cycles. represent two characteristic times at which are located the venous and cervical CSF flow peaks compared to the arterial systolic flow peak. These times are expressed as a percentage of the cardiac cycle. We also calculated the ventric. Maximum and ular peak of CSF flush flow minimum amplitudes of CSF and blood flows were measured on our population.

II. MATERIAL AND METHODS

B. Structure of the Model Fig. 3 shows a schematic representation of lumped-parameter compartmental model with two incompressible liquids (blood and CSF) which do not mix. These two liquids are coupled through the elasticity of intracranial tissues (cerebral tissue, vascular tissue). In this model, cerebral tissue is considered to resemble an interface delimiting the boundaries between intracranial blood and intracranial CSF. Our approach [Fig. 3(a)] is an extension of previous electrical analogue models found in the literature [8], [13] incorporating

A. Measurement MRI was performed in seven healthy male volunteers with years using a 1.5-T machine (Signa, GE a mean age of Medical Systems, Milwaukee, WI). The subjects were supine with their neck in a neutral position. Conventional morphologic sequences were acquired in the sagittal and axial planes. CSF flow acquisition planes perpendicular to the presumed direction of flow were then selected on midsagittal sections through the

AMBARKI et al.: NEW LUMPED-PARAMETER MODEL OF CEREBROSPINAL HYDRODYNAMICS

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a pressure difference represented by compliance. The compli, ances in our model are: extracranial arterial compliance , intracranial venous intracranial arterial compliances compliances ( , , and ), extracranial venous compliance , and spinal compliance . The components of intracranial dynamics, i.e., arterial and venous blood and CSF are coupled by compliances , , , , and . Compliance represents a direct and mutual influence between the cranial subarachnoid space and the venous submodels. The asymmetry it generates between arterial and venous side visible on Fig. 3(a) by the absence of any arterial counterpart of that compliance, is only spurious since it describes a transmission from arterial pulsations to the venous vessels via the cranial subarachnoid space CSF (M4) [17]. C. Electrical Analogue of Cerebral Hydro-Dynamics Fig. 3(b) gives the electrical analogue model of cerebrospinal correspond to voltages, hydro-dynamics. The pressures correspond to electrical currents, hydraulic the flows correspond to electrical conductances and resistances compliance C correspond to capacitance. D. Model Equations In our model, the pressure differences across the eight compliances were considered as the state variables. The differential equations of the electrical analogue model of cerebrospinal hydro-dynamics [Fig. 3(b)] can be written in the matrix form which is the most general form associated with a linear dynamics

(1) where is the is the input vector, and state vector, and are arterial pressure and venous pressure, respectively. A is an 8 8 matrix and is a 2 8 matrix whose coefficients and are given in Appendix I. , venous flow , The output (2) linking the arterial flow , and spinal CSF flow to the ventricular CSF flow pressures read Fig. 3. (A) Model of cerebrospinal hydro-dynamics. (B) The electrical analogue of cerebrospinal hydro-dynamics. Gi, Ci, qi, and P , correspond to conductances, compliances, flows and pressures, respectively. The electrical model is composed of five submodels namely arterial blood (M1), venous blood (M2), ventricles (M3), cranial subarachnoid space (M4), and spinal subarachnoid space (M5) demarcated by dotted boxes.

eight compartments composed of five submodels namely arterial blood (M1), venous blood (M2), ventricles (M3), cranial subarachnoid space (M4), and spinal subarachnoid space (M5). These submodels are connected by flow conductances and compliances and demarcated by dotted boxes on Fig. 3(b). , cerebral Flow conductances are extracranial arteries , capillaries , cerebral veins , extracraarteries , cranial subarachnoid space ( and ), the nial veins cerebral aqueduct , and spinal subarachnoid space . The volume changes in two adjacent compartments result from movements of their common elastic membrane in response to

(2) is the output flow vector. where is a 4 8 matrix and is a 4 2 matrix whose explicit expresand coefficients are given in Appendix II. sions of the These last equations are physical constraints prescribed to the abovementioned fluid flows, the input pressures and the state corresponds to the sum of the flows vector. Arterial flow in the internal carotids and vertebral arteries and we do not sepcorresponds arate posterior and anterior arterial pathways. to the sum of the flows in the internal jugular veins amplified to equalize the mean arterial blood flow during the cardiac cycle [8], [18]. We used a sinusoidal input pressure with appropriate features to model in the simplest way the arterial pressure. These features are chosen to reproduce the physiological conditions, that is: diastolic blood pressure: 80 mmHg, systolic blood pressure: 120 mmHg, period 0.9 s. In this paper, we assumed that venous

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III. RESULTS

TABLE I PARAMETERS VALUES USED IN MODEL SIMULATION

pressure is negligible. The maximum and minimum measured and simulated values of blood and CSF flows were compared, , , and as well as the delays. E. Parameters Values Blood and CSF flows were simulated using a sinusoidal pressure signal input, and we used values of the parameters very close to those available in the literature. For unavailable (or simply not measured) parameters, we carried out empirical fitting of the curves to obtain parameter values in agreement with the observed amplitudes and delays between the peaks. This is , , , and the case of the vascular conductances ( , ); the CSF conductances , and the compliances ( , , , , , , and ). It should be noticed that this procedure does not provide a rigorous estimate of the parameters, but instead allows for a qualitative order of magnitude assessment of these parameters. This step was only supposed to show our model’s ability to reproduce the amplitudes and the delays. The next step (in progress) will consist in parameters estimation based on data collected through phase-contrast MRI on healthy subjects. conductance of the cerebral aqueduct and conductance of spinal subarachnoid space were calculated by Poiseuille’s equation [25]

(3) , cereusing mean values from healthy adults [25]: , cerebral aqueduct length; bral aqueduct area; , area of subarachnoid space at C2–C3 level , spinal length [26]; , [18]; . CSF viscosity. The calculus shows compliance was derived from the literature The value of [11]. The parameters used for model simulation are summarized in Table I. The model simulation was developed using MatlabSimulink software (Mathworks®, Inc).

Table II presents the comparative maximum and minimum of measured and simulated blood and CSF flows. Table III presents the comparative measured and simulated delays of , , and . In our population, the systolic phase arterial flow undergoes marked accel, whereas eration and the maximum value is the maximum venous flow is . During the diastolic phase, arterial and venous flows decrease slowly to reach and , their minima of respectively. Systolic and diastolic peaks measured on CSF and for C2–C3 subflow were and in the arachnoid space and , , cerebral aqueduct. The values of were , , and and , respectively. The maximum and minimum simulated arterial flows were 1186 and 306 ml/min. The maximum and minimum simulated venous flows were 847.8 and 546 ml/min. Simulated cervical and ventricular CSF flows fluctuated , , , and between , respectively. The values of , , and were 8.7%, 1.1%, and 20%, respectively. Fig. 4(A.a) Measured arterial and venous blood flows Fig. 4(A.b) measured cervical CSF flow, and Fig. 4(A.c) measured ventricular CSF flow in a healthy volunteer from our population during two cardiac cycles. Fig. 4(B.a) Simulated curves for arterial and venous flows Fig. 4(B.b) cervical CSF flow, and Fig. 4(B.c) ventricular CSF flow during two cardiac cycles. IV. DISCUSSION Periodicity of filling/empting of cerebral blood vessels (arterial and venous) caused by cardiac contractions is the source of brain and CSF pulsations observed by phase-contrast MRI [17], [27]. In this study, we showed that the characteristics of cerebrospinal CSF and blood circulation during the cardiac cycle can be represented by mathematical models. The goal with our model is to propose an analysis of the amplitude and phase variations of CSF and blood flows compared to the events of cardiac contraction (systole, diastole), which is regarded as the source of excitation. Our simulations showed the model’s ability to account for the distribution of blood flow and CSF flow measured by phase-contrast MRI. The maximum and minimum simulated blood and cervical CSF flows are close to the measured values (Table II). Nevertheless, a major difference was observed for cerebral aqueduct CSF flow. The simulated aqueduct CSF flow was almost twofold higher than the corresponding measured value. Aqueductal CSF flow is due to vascular pulsations through the parenchyma via compliances and . The flow values depend on the pressure difference between ventricular CSF and cranial subarachnoid space CSF is deterand on the conductance in the cerebral aqueduct. mined by geometric properties of the cerebral aqueduct and by rheologic properties of CSF. CSF pulsations in the cranial subarachnoid space result from transmission of vascular pulsations and . The amplitude of through the brain via compliances CSF pulsations can be controlled by the values of conductances and .


487

TABLE II MAXIMUM AND MINIMUM FLOW MEASURED AND SIMULATED

TABLE III DELAY AFTER PEAK ARTERIAL FLOW MEASURED AND SIMULATED

Fig. 4. (A) Blood and CSF flows measured in a healthy volunteer: (a) Arterial and venous flows, (b) cervical CSF flow, and (c) ventricular CSF flow. T represents the time to venous peak flow compared to the arterial and vensystolic peak flow, the cervical peak of CSF flush flow T tricular peak of CSF flush flow T . (B) Blood and CSF flows simulated (a) arterial and venous flows, (b) cervical CSF flow, and (c) ventricular CSF flow. T represents the time to venous peak flow compared to the arand terial systolic peak flow, the cervical peak of CSF flush flow T ventricular peak of CSF flush flow T .

CSF pulsatile motion ensures a constant intracranial volume throughout the cardiac cycle [15], [16]. This constraint implies that the total volume entering the cranial cavity (CSF and arterial blood) must be the same as the total volume leaving the cranial cavity (CSF and venous blood). There is, therefore, an interdependent relationship between changes of intracranial blood volume and changes of intracranial CSF volume that fills the space delimited by the rigid skull. Spinal CSF (M5) pulsations result from intracranial CSF motions, i.e., from ventricular CSF (M3) and cranial subarachnoid space CSF (M4).

The flows measurements provided by phase-contrast MRI in healthy subjects demonstrate a physiological delay between arterial blood flow and venous blood flow and a significant delay between the arterial blood flow and the CSF flow. The flow results observed in our population are in agreement with those reported by [18] and [28]–[30], but do not comply with Egnor’s theory [13] according to which peak arterial blood flow, venous blood flow and CSF flow coincide in healthy subjects. The value of the phase shift between arterial blood flow and CSF flow is a significant characteristic of the mechanical properties of brain tissue [30]. The systolic times calculated on simulated curves (Table III) are globally lower than the measured times due to the empirical values for the parameters (compliances and conductances) used in the model during simulation. Our model demonstrated that the normal mechanical state of the cerebrospinal hemodynamic and hydrodynamic can be described by a forced regime value of the state vector . This state depends on the values of pressures – . These pressures control the distribution of intracranial volumes between blood and CSF. This distribution is governed by the biomechanical properties of the intracranial system. In our study, the regulation of the intracranial pressure controlled by the heart rate and cardiac output variation is ignored. The proposed model takes into account the traditional considerations of anatomy, physiology and functional of cerebrospinal hydro-dynamics. Our model comprises a large number of parameters, which makes it more difficult to estimate unknown parameters. An approximate curve (sinusoidal) of blood pressure was used as input signal due to the absence of measured curves, which would require invasive techniques, involving the use of sensors which are not compatible with MRI safety rules. A sine wave function was used as excitation to facilitate interpretation of the curves recorded at the output of the model, as sinusoidal wave analysis techniques are easy to implement for these types of signals and do not require the use of sophisticated signal processing tools. However, measured blood pressure signals should be used for final validation of our model. Marmarou et al. [2] demonstrated the existence of an exponential relationship between intracranial pressure and intracranial volume changes. In our study, we assumed that cerebral deformations due to arterial pulsations during the cardiac cycle were negligible. With this approximation, the pressure-volume relationship of the intracranial system can be considered to be linear around the resting state. This assumption of linearity justifies the use of constant values for compliance and resistance in the simulation model. The majority of models [8]–[13] do not assess specific aspects of intracranial biomechanics relative

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to interactions between arterial blood, CSF and venous blood. One of the original features of our model is that the influence of ventricular CSF, intracranial subarachnoid CSF, and subarachnoid spinal CSF are analyzed separately during cardiac cycle. The model described by Takemae et al. [10] simulates the intracranial pressure wave as a function of cerebral vascular volume. The compartmental model described by Sorek [12] estimated the compliance and resistance values. Ursino’s model [11] simulated the impact of injection of successive small volumes of CSF on intracranial pressure, capillary pressure, and venous pressure. Recent studies by Linninger et al. [31] are based on animal intracranial pressure measurements and ventricular CSF phase-contrast MRI data. Pressure measurements demonstrated similar mean pressures in brain tissue, ventricular CSF (M3), and cranial subarachnoid CSF (M4). In their model, they proposed that the choroid plexus is the main driver of ventricular CSF motion. In our description, CSF (ventricular, cranial subarachnoid space, and spinal subarachnoid space) pulsatile motion results from an instantaneous arterial and venous blood flow imbalance during the cardiac cycle.

We rewrite this last relation so to obtain an expression of depending on , the states and the inputs and

(8) with (9) and (10) The application of Kirchhoff ’s voltage law yields

V. CONCLUSION We propose a new lumped-compartment model of cerebrospinal hydro-dynamics. The results show a good agreement between measured and simulated values of blood and CSF flow distribution. The major contribution of our model in comparison with the literature is that it reflects the instantaneous volume change of individual cerebral compartments during the cardiac cycle. Our model highlights the fact that CSF flow results from a global relationship between conductance, compliance and pressure of the cerebral compartments. The present mathematical model may constitute a first step towards an optimal description of normal human hydrodynamic phenomena related to the influence of blood transmitted by cardiac contractions. Definition of the values of the various parameters of the model will provide a better understanding of intracranial hydrodynamics.

(11) Finally, by substituting (9)–(11) into (8) gives states (6)

(12) State 1 is obtained by substituting state 1

in (11). Therefore, we get

(13) The majority of the others states are obtained with (12) and (13). State 8 is calculated by substituting (9), (10), (12), and (13) in (4)

APPENDIX I We describe the electrical model [see Fig. 3(b)] in terms of across capacistate-space equations. Choosing the voltages tors as the states, we have eight states with the corresponding , and to eight flowing through currents the capacitors. The inputs of the model are arterial pressure and venous pressure . Applying the Kirchhoff ’s laws to our circuit, we obtain the following relationships:

(14) The application of Kirchhoff ’s voltage law yields

(15) (4) (5)

The general expression of state 5 is

(6) By introducing (5) and (6) in (4), we get

(16) Node e leads to

(7)

(17)


We calculate state 3 by substituting (16) in (17)

The

(18)

Node b gives

(19)

State 2 is derived from (19) by replacing tively, from (9), (13), and (18)

,

, and

, respec-

(20)

in the node f we have (21) and (22) and

(23)

489

equations

are

rearranged

in matrix form where is the state represents the input vector; is vector; an 8 8 matrix, and is a 2 8 matrix. The coefficient of matrix read

a11 = C1 0G1 0 G(G4++GG9+)GG2 1 2 9 4 1 (G 4 + G 9 )G 2 a12 = C G + G + G 1 2 9 4 1 (G 4 + G 9 )G 2 a13 = 0 C G + G + G 1 2 9 4 (G4 + G9 )(G2 + G9 ) 1 a14 = 0 C 0G9 0 G + G + G 1 2 9 4 a15 = 0 a16 = 0 a17 = C1 G9 0 G(G4++GG9+)GG4 1 2 9 4 (G 4 + G 9 )G 4 1 a18 = C G9 0 G + G + G 1 2 9 4 1 (G 4 + G 9 )G 2 a21 = C G + G + G 2 2 9 4 1 a22 = C 0G3 0G7 0G6 0 G(G4++GG9+)GG4 2 2 9 4 1 G 2 (G 4 + G 9 ) a23 = C G3 + G7 + G6 0 G + G + G 2 2 9 4 1 (G4 + G9 )(G2 + G9 ) a24 = C G9 0 G + G + G 2 2 9 4 1 G 6 a25 = C (G7 + G6 ) a26 = 0 C 2 2 G 1 9 (G 4 + G 9 ) a27 = C 0G9 + G + G + G 2 2 9 4 1 G 9 (G 4 + G 9 ) a18 = C 0G9 + G + G + G 2

2

9

a31 = 0 a32 = C (G7 + G6 ) 3

we replace (22) and (23) in (21) (24) State 4 is obtained from (20) and (24)

(25)

Node c gives and

with (node d) Finally, we have (26)

, , , and tively, so

are given by (8), (10), (16), and (25), respec(27)

4

1

a33 = 0 C1 (G7 + G6 ) a34 = 0 3 1 6 a35 = 0 C (G7 + G6 ) a36 = G C3 3 a37 = 0 a38 = 0 a41 = 0 C1 G(G4++GG9+)GG2 4 2 9 4 1 a42 = C G3 + G7 + G6 0 G(G4++GG9+)GG4 4 2 9 4 1 (G 4 + G 9 )G 2 a43 = C 0G4 0G7 0G6 0G8 0 G + G + G 4 2 9 4 1 (G4 + G9 )(G2 + G9 ) a44 = C 0G9 + G + G + G 4 2 9 4 1 G 6 a45 = 0 C (G7 + G6 ) a46 = C 4 4 1 (G 4 + G 9 )G 4 a47 = C G9 + G8 0 G + G + G 4 2 9 4 1 (G 4 + G 9 )G 4 a48 = C G9 0 G + G + G 4 2 9 4 G 6 6 a51 = 0 a52 = 0 C a53 = G C5 a54 = 0 5 G6 6 a55 = G C a56 = 0 C a57 = 0 a58 = 0 5

5

490


1

a61

=

a62

=

0

a63

=

0

a64

=

0

a65

=0

a68

=

a71

=

a72

=

a73

=

C6

G2

G2 G4 + G9 + G4

G2

G2 G4 + G9 + G4

1 C6

1

G 4 (G 2

C6

=0

a66

G2

G2 G4 + G9 + G4

G

1 1

a75

=

a77

=

a78

=

a81

=

a82

=

a83

=

a84

=

a85

=0

G9

C7 G7

0

a76

C7

0

1

0

1 C8

1

a88

=

b11

=

b61

=0

C8 G1 C1

G9

+

G2

G2

G2 G9 + G9 + G4 G2 G9 + G9 + G4

G2 G9

0

0 0

G9

+ G9 ) + G9 + G4 2

+

G9

+ G9 + G4 2 G9 G9 + G2 + G9 + G4

b21 b71

G 9 (G 2 G2

=0

a86

REFERENCES

=0

1

1

G2 G4 + G9 + G4

+ G9 )

G2 G9 + G9 + G4

C8

C8

G2

G2

1 C8

The authors would like to thank Dr. R. Bouzerar and F. Djaidja for their help in writing this paper.

2 G9 + G9 + G4 2 G9 G9 + G2 + G9 + G4

1 C8

0

+ G9 + G4

G2

G

C7

G9 G4 + G9 + G4

+ G9 G2 ) + G9 + G4

G2

G 9 (G 4

0 80

1 C7

G2

ACKNOWLEDGMENT

G 4 (G 4

+ G8 +

G5

C7

C6

G2 G4 + G9 + G4

0 50

1 C7

1

=

a67

G2

1 C7

+ G9 ) + G5 + G9 + G4

G2

1

=

=

1 C6

C6

a74

a87

G2 G4 + G 9 + G4

G2

=0

=0

G2

b31 b81

=0

b41

=0

b51

=0

= 0:

For matrix

APPENDIX II The outputs of model are arterial flow ; venous flow ; ; cervical CSF flow . These flows ventricular CSF flow are given by (9), (10), and (18); and (21), respectively, and where are rearranged in matrix form is the output flow vector; is a 4 8 matrix; is a 4 2 matrix and theirs coefficients are

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Olivier Balédent received the Ph.D. degree from the University of Picardie, Amiens, France. He is Associate Professor in the Department of Imaging and Biophysics, Teaching Hospitals, Amiens. His research interests are intracranial dynamics, hydrocephalus, phase-contrast magnetic resonance imaging, positron emission tomography imaging, and image processing software.

Khalid Ambarki is working towards the Ph.D. degree the University of Picardie, Amiens, France. His research interests are mathematical models for intracranial dynamics, bioengineering, and phase-contrast magnetic resonance imaging.

Marc-Etienne Meyer received the M.D. degree from the University of Strasbourg I, Strasbourg, France, in 1989, and the Ph.D. degree from the LGME of CNRS, INSERM-U.184 at the University of Strasbourg I in 1990. He is Professor in the Department of Biophysics and Nuclear Medicine, Teaching Hospitals, Amiens. His research interests are intracranial dynamics, functional magnetic resonance imaging, and positron emission tomography dynamic imaging.

Guy Kongolo received the M.D. degree in 1988, and the Ph.D. degree in biomedical engineering in 2003 from the University of Picardie Jules Verne, Amiens, France. He is currently working as Pediatrician at the Pediatric and Neonatal Intensive Care Unit of the Hospital of Amiens. His main research areas are the modeling of physiological systems and the biomedical signals analysis. He also is interested in techniques allowing the assessment of respiratory mechanics and cardiovascular hemodynamics in critically ill patients.

Robert Bouzerar received the Ph.D. degree from the University of Lille, France. He is Associate Professor in the Condensed Matter Physics Laboratory, University of Picardie, Amiens, France. His research interests are mathematical models for intracranial dynamics, theory of hydrocephalus, biophysics, semi-conductors, and electronic and transport properties in disordered semiconductors.

Sidy Fall is working towards the Ph.D. degree at the University of Picardie, Amiens, France. His research interests are functional magnetic, bioengineering, and mathematical models.