Math - Circle of Willis Blood Flow

8/12/2019 Math - Circle of Willis Blood Flow

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BLOOD FLOW IN THE CIRCLE OF WILLIS: MODELING AND

CALIBRATION

KRISTEN DEVAULT, PIERRE A. GREMAUD, VERA NOVAK, METTE S. OLUFSEN,

GUILLAUME VERNIERES, AND PENG ZHAO

Abstract. Modeling of blood flow in arterial networks is considered. The study concentrateson the Circle of Willis, a vital subnetwork of the cerebral vasculature. The main goal is to obtainefficient and reliable numerical to ols with predictive capabilities. The flow is assumed to obey theNavier-Stokes equations while the mechanical reactions of the arterial walls follow a viscoelasticmodel. Like many previous studies, a dimension reduction is performed through averaging. Unlikemost previous work, the resulting model is both calibrated and validated against in vivo digitaltranscranial Doppler data using ensemble Kalman filtering techniques. The results demonstrate theviability of the proposed approach.

Key words. Blood flow, viscoelastic arteries, fluid-structure interaction, Kalman filtering

1. Introduction. The brain is one of the vital organs in the body and stable

perfusion is essential to maintain its function. Cerebral circulation receives 15-20%of the cardiac output and is closely regulated to maintain perfusion in response tometabolic and physiological demands. The main cerebral distribution center for bloodflow is the Circle of Willis [14, 33], a ring-like network of collateral vessels, see Fig-ure 1.1, left1 . Blood is delivered to the brain through the two internal carotid arteriesthat contribute 80% of the blood supply, and the two vertebral arteries that join in-tracranially to form the basilar artery. Each of the internal carotid arteries branchesto form the middle and anterior cerebral arteries, which supply blood to the frontand the sides of the brain (the frontal, temporal, and parietal regions of the brain).The basilar artery bifurcates into the right and left posterior cerebral arteries, whichperfuse the back of the brain (the occipital lobe, cerebellum and the brain stem). Thering is completed by communicating arteries that connect the posterior and anteriorcerebral arteries (via posterior communicating arteries) and the two anterior cerebral

arteries (via the anterior communicating artery).

This project was initiated at and supported by the Statistical and Applied Mathematical SciencesInstitute (SAMSI), Research Triangle Park, NC 27709-4006, USA.

Department of Mathematics and Center for Research in Scientific Computation, North CarolinaState University, Raleigh, NC 27695-8205, USA ([email protected]). Partially supported by theNational Science Foundation (NSF) through grant DMS-0410561.

Department of Mathematics and Center for Research in Scientific Computation, North CarolinaState University, Raleigh, NC 27695-8205, USA ([email protected]). Partially supported by theNational Science Foundation (NSF) through grants DMS-0410561 and DMS-0616597.

Division of Gerontology, Beth Israel Deaconess Medical Center, Harvard Medical School, Boston,USA ([email protected] and [email protected]). Partially supported by theAmerican Diabetes Association through grant 1-06-CR-25 to V. Novak, by the National Institutesof Health (NIH) through grants NIH-NINDS R01 NS45745-01A2, 1R41NS053128-01A2 and NIH-NIA-P60 AG8812-11A1 RRCB and by the National Science Foundation (NSF) through grants DMS-

0616597.Department of Mathematics, North Carolina State University, Raleigh, NC 27695-8205, USA

([email protected]). Partially supported by the National Science Foundation (NSF) through grantDMS-0616597.

Statistical and Applied Mathematical Sciences Institute (SAMSI), Research Triangle Park, NC27709-4006 ([email protected]).

1Throughout the text, the standard abbreviated names for the vessels are used; ACA: ante-rior cerebral artery, MCA: middle cerebral artery, PCA: posterior cerebral artery, ACoA: anteriorcommunicating artery, PCoA: posterior communicating artery, see also Figure 3.1 and Table 3.1.

1


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ACoA

R ACA L ACA

R MCA L MCA

R ICA L ICA

R PCoA L PCoA

R PCA L PCA

BA

(A)

(B)

(C)

Fig. 1.1. (A) Structure of the Circle of Willis basilar artery (BA); right posterior cerebralartery (R PCA), left posterior cerebral artery (L PCA), right posterior communicating artery (RPCoA), left posterior communicating artery (L PCoA), right internal carotid artery (R ICA), leftinternal carotid artery (L ICA), right middle cerebral artery (R MCA), left middle cerebral artery(L MCA), right anterior cerebral artery (R ACA), left anterior cerebral artery (L ACA), anteriorcommunicating artery (ACoA); (B): Time of flight (TOF) magnetic resonance angiography of the

Circle of Willis; (C) Blood flow velocities measurements obtained by transcranial Doppler ultrasound(TCD) for the right anterior cerebral artery (R ACA), right middle cerebral artery (R MCA) andright posterior cerebral artery (R PCA)..

Under normal conditions, blood flow in the communicating arteries is negligible.However, if a subject has an atypical Circle of Willis, e.g., missing one of the mainarteries or communicating arteries or under pathological conditions such as completeor partial occlusion of one of the cerebral or carotid vessels, the flow can be redirectedto perfuse deprived areas [22, 23]. The borderzones are then perfused through thenetwork of communicating arterioles. The ring-like structure of the Circle of Willisis often incomplete or not fully developed. It has been found that in more than 50%of healthy brains [2, 42, 43] and in more than 80% of dysfunctional brains [51], theCircle of Willis contains at least one artery that is absent or underdeveloped. The mostcommon topological variations include missing communicating vessels, fused vessels,string-like vessels, and presence of extra vessels [3]. These topological variations mayaffect the ability to maintain flow through arteriols, which may increase the risk ofstroke and transient ischemic attack in patients with atherosclerosis [34]. Limitedtechnology exists to predict perfusion response to acute occlusion due to embolus (i.e.embolic stroke) and to chronic occlusion due to atherosclerosis (i.e. carotid or other


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large vessel stenosis), in particular for patients with an incomplete Circle of Willis.These clinical scenarios typically occur in older patients, who have a limited ability tocompensate to acute changes in blood flow and thus are at greater risk for developingan acute ischemia (stroke) or chronic hypofusion. The significance of these problems

cannot be underestimated since stroke ranks third among leading causes of deathand is the leading cause of disability in older adults [10]. Therefore patient specificmodeling is critically important to plan and predict perfusion needs in patients withsignificant carotid artery stenosis who need surgical repair.

One way to assess the state of the blood flow to the brain is to use a fluid dynamicmodel combined with subject specific anatomical information. Fluid dynamic modelshave long been used to predict blood flow dynamics in almost any section of the arterialsystem, see for instance [6, 8, 49] for classic studies and [11, 12, 25, 26, 56] for morerecent work. A number of existing fluid dynamic models have been proposed to predictblood flow in the Circle of Willis. These models include one-dimensional approaches[4, 15, 16, 36, 37, 44, 52, 53, 58], two dimensional approaches [22, 23, 39], and threedimensional approaches [5, 14, 21, 44, 45]. Due to the complexity of the underlyingproblem, vessels are usually treated as rigid in three dimensional calculations. More

complex models have however been considered, see for instance [24], but usually forgeometries significantly simpler than the Circle of Willis. On the other hand, oneand two dimensional models allow the inclusion of fluid structure effects relativelyeasily, although at the price of severely simplified fluid dynamics. As noted in [15],most of the above models are qualitative and should be taken some steps furtherto make possible patient specific studies and thereby provide powerful clinical toolswhich would greatly benefit neurosurgeons and patients.

The goal of this paper is to show that proper one-dimensional models can lead tosimple and reliable predictions of blood flow circulation in the Circle of Willis. Thepresent contribution differs from previous work in two essential aspects. First, thevessel walls are taken as viscoelastic as opposed to rigid or elastic as in most previouswork, see Section 2. While viscoelasticity of the arterial wall is by itself not new,see, e.g., [11], the model considered here includes both stress and strain relaxation2.Second, thorough comparison and calibration of the model to experimental resultsare conducted, see Section 6. The original data used here was obtained using digi-tal Doppler technology, see Figure 1.1, right, MRI imaging, and non-invasive fingerblood pressure measurements. Previous studies of cerebral blood flow have used MRImeasurements to obtain detailed patient specific geometries (e.g. [14, 45]). However,patient specific information was not used to obtain the remaining model parameters.This is done here through Ensemble Kalman filtering techniques, see Section 5, whichare used to calibrate various computational boundary conditions, see Section 3. Tothe authors knowledge, combining fluid dynamic simulations for arterial networkswith parameter identification methodology is fairly new. As such, it provides onemore step toward patient specific predictive models as set forth by Charbel et al.[15].

The rest of the proposed approach is relatively standard and is based on conser-vation of mass and momentum, see Section 2. In each vessel, a system of balance lawshas to be solved. When compared to elastic models, see e.g. [4], the present systemhas an additional equation per vessel. The computational domain is linked to the rest

2In a previous study [57], we showed that for most of the larger arteries, including the carotidartery, it is not possible to accurately predict pressure as a function of area without accounting forboth of those factors.


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of the vascular system through boundary conditions as described in Section 3 wherethe conditions at vessel bifurcations are also discussed. Discretization techniques areintroduced in Section 4.

2. Derivation of the model. The following assumptions are semi-standard inone-dimensional hemodynamics and are adopted here

the blood density is constant, the blood flow is axisymmetric and no has no swirl, the vessels are tethered in their longitudinal direction, the equations are expressed in terms of variables averaged on cross-sections.

Further, the flow is assumed to obey to the incompressible Navier-Stokes equations

(tu+u u) = g, (2.1)

u= 0, (2.2)

where is the density, u is the velocity, is the stress tensor, andg is the accelerationdue to gravity. The stress tensor is = pI + 2 where p is the pressure, =1

2(u+ uT

) is the strain rate tensor, and is the dynamic viscosity. Althoughnot done here, the possible non-newtonian behavior of blood can be accounted for byletting depend on , see Conclusions.

Each vessel is assumed to be axisymmetric with a variable variable diameter.In each individual vessel, cylindrical coordinates (r,,x) are used with x being thedistance on the longitudinal axis. Further, the shape of each axisymmetric vessel isdescribed by a function R such that R(x, t) is the actual radius of the vessel at thepointx on the x-axis, at time t. Using those coordinates and the above assumptions,the velocity is u =< ur, 0, ux> and the strain rate tensor becomes

=

rur 0

1

2(rux+xur)

0 urr 01

2(rux+xur) 0 xux

.

The Navier-Stokes equations (2.1,2.2) can then be rewritten as

(tur+urrur+uxxur) = rp+r(rur) +r

urr

+x(xur)

+r rur+x rux+gr, (2.3)

(tux+urrux+uxxux) = xp+r(rux) +

rrux+x(xux)

+r xur+x xux+gx, (2.4)

1

rr(rur) +xux= 0, (2.5)

where (2.3) and (2.4) express respectively the radial and axial conservation of mo-mentum and (2.5) corresponds to the continuity equation (2.2); further,gr and gx are

respectively the radial and axial components ofg .The Kelvin model postulates (see [27])

pp0+tp=Eh

r0(s+ts), (2.6)

wheres = 1A0A (see [57]), A = R

2, and and are relaxation times.


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We introduce the following characteristic quantities

flow: q0, radius: r0, radial velocity: v0.

From these, additional characteristic quantities follow

surface area: A0 = r20, axial velocity: u0 =

q0A0

, length: x0 =r0u0

v0,

time: t0 = r0v0

=x0u0

, pressure: p0= u20, dynamic viscosity: 0.

Nondimensional quantities are then introduced in a standard way. In terms of thenondimensional variables, the Navier-Stokes equations (2.3, 2.4) and the viscoelasticconstitutive equation (2.6) take the form

rp=

v0u0

2(tur urrur uxxur)

+ v0u0

0 r0u0 2 rrur+r

urr +xrux

+

v0u0

30

r0u0xxur

r0g

u20er k, (2.7)

tux+ urrux+uxxux= xp+ 0 r0v0

rrux+

1

rrux

+ v0u0

0 r0u0

(2 xxux+rxur) g

u0v0ex k, (2.8)

tp 1

2

Eh

r0p0A3/2tA=

t0

(1p) + t0

Eh

r0p0(1A1/2), (2.9)

whereer and ex are the unit vectors associated with the coordinate directions r andx respectively while k is the unit vertical vector. The continuity equation (2.5) isleft unchanged by non-dimensionalization since v0r0 =

u0x0

. The axial velocity u0 being

assumed much larger than the radial velocity v0, i.e.,

v0


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Integration by parts of the continuity equation (2.5) over a cross-section together withthe boundary condition (2.12) gives

2R R+x2 R

0

uxr dr= 0,or, equivalently

tA+xQ= 0, (2.13)

where Q = 2R0

uxr dr is the dimensionless flux. Integrating the r-momentumequation (2.10) over a cross-section leads to R

0

rp r dr= 0 p(R,x,t) = 1

R

R0

p(r,x,t)dr P(x, t).

The pressure p is additionally assumed to be independent3 of r, i.e., p = P. Thex-momentum equation (2.11) is now integrated, yielding, together with (2.12)

tQ+x

2

R0

u2xr dr

+A xP=

1

RR rux(R,x,t)

ex k

F A, (2.14)

where the following nondimensional parameters have been introduced

Reynolds number R =r0v020

,

Froude numberF=u0v0

gr0.

To close the model, an additional assumption is needed to relate ux to the averagedquantities A, Q and P in terms of which the entire problem will be expressed. LetU = Q/A be the average axial velocity. The axial velocity ux is sought with the

following profile

ux(r,x,t) =+ 2

U(x, t)

1

r

R(x, t)

, (2.15)

In (2.15), determines the profile (for instance, = 2 corresponds to the classicalPoiseuille profile), see Figure 2.1, while the factor +2 ensures that the average ofuxis indeed U. The parameter is taken as constant = 2 in each vessel in the presentstudy.

The x-momentum equation (2.14) can now be re-expressed in terms of the aver-aged variables

tQ++ 2

+ 1x

Q2

A +A xP = + 2

R

Q

A

ex k

F A. (2.16)

The system is closed by averaging the Kelvin relation (2.9), which just amountsto replacingp by P. Using the continuity equation (2.13), one finds

tP+

1

M2A3/2xQ=

1

W(1 P) +

2

WM2(1A1/2), (2.17)

3This is automatically satisfied ifg = 0 and/or ifer ez = 0, i.e., for a vertical vessel.


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6

10

4 2 3 5

13

9

12

7

8 11

1

15 16

14

Fig. 3.1. Topology of the Circle of Willis and boundary conditions and numbering convention,see also Table 3.1.

1 < 0, 3 > 0, the solutions are smooth.

As a result of the first observation, the resolution of the equations in each vesselrequires that one scalar condition has to be enforced at each end of that vessel toensure well-posedness. The second observation is used below to significantly simplifythe treatment of junction conditions.

For three of the vessels (basilar artery as well as left and right internal carotidarteries), inflow conditions are imposed whereby the velocity is prescribed and cor-responds to experimental measurements, see Section 5. In other words, since thevelocity is related to the unknowns through U = Q/A, the following conditions willbe imposed at the end of the corresponding vessels and at all time

Qves = UvesAves, ves {Basilar, L. Carotid, R. Carotid}, (3.1)

with the obvious naming convention, see again Figure 3.1 and Table 3.1. The velocitiesUves in (3.1) are experimentally determined time dependent functions and the surfaceareas are computed from the average radii from Table 3.1. The remaining conditionsare outflow boundary conditions. Those conditions have to mimic the effects of therest of the vascular system on the Circle of Willis. While the issue is delicate anddeserves further research, simple ad hoc conditions can be used. In the present work,

two such types of conditions are considered. Pure resistance boundary conditionshave the form (see for instance [53])

Pves = RvesQves,ves {L. PCA 2, R. PCA 2, L. MCA, R. MCA, L. ACA 2, R. ACA 2} .

(3.2)

Alternatively, boundary conditions based on the three-parameter Windkessel modelcan be used; this model includes two resistors and one capacitor, see for instance


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Vessel # Name Radius Length(cm) (cm)

1 BA .150 .8252 R. PCA 1 .112 .333

3 L. PCA 1 .112

.333

4 R. PCA 2 .110 .7565 L. PCA 2 .110 .7566 R. PCoA .0986 1.00

7 L. PCoA .0986 1.00

8 R. ICA .210 4.819 L. ICA .210 4.81

10 R. MCA .134 2.1111 L. MCA .134 2.1112 R. ACA 1 .170 1.0713 L. ACA 1 .100 1.0714 ACoA .100 .20015 R. ACA 2 .115 2.30

16 L. ACA 2 .115 2.30

Table 3.1

Geometry data used in the calculations, see also Figure 3.1. The values were missing fromthe data and had to be estimated.

[4, 47, 48, 50, 52]. This corresponds to

RsvestQ+Rsves+R

pves

RpvesCvesQ= tP+

1

RpvesCvesP, (3.3)

where, for each vessel ves in the same list as in (3.2), Rpves and Rsves are resistance

parameters andCves is a compliance parameter.Junction conditions link vessels to their neighbors. The mathematical derivation

of proper junction conditions for systems of conservation laws is non trivial; it is infact an active field of research, see for instance [9, 17, 18, 38]. The present system ofequations (2.18) is not in conservation form which further complicates the problem.However, as mentioned at the beginning of the section, only smooth solutions areexpected (and observed) and thorny questions of selection principles [28, 29, 30] canbe avoided.

Consider a junction J at which NJ vessels intersect. Continuity of the pressureand conservation of the flow are imposed

P1= P2 = =PNJ,NJi=1

Qi = 0, (3.4)

where the flux Qi is counted positive if flowing towards J.

4. Numerical analysis. The equations are discretized in space using Chebyshevcollocation methods [13]. Such methods deliver high accuracy with a low numberof nodes for smooth solutions (which are expected here). Working in the standard


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[1, 1] interval to simplify the notation, Chebyshev collocation is considered at theusual Chebyshev-Gauss-Lobatto nodes

xj = cos j

N 1 , j = 0, . . . , N 1,

where N stands for the number of nodes. Ifv is any of the above unknowns to bedetermined forx [1, 1] andt >0, we seek an approximation of it of the form

vN(x, t) =

N1i=0

Vi(t)i(x), (4.1)

where{i}N1i=0 are the Lagrange interpolation polynomials at the Chebyshev-Gauss-

Lobatto nodes on [1, 1], i.e., i(xj) =ij, i, j = 0, . . . , N 1. Interpolation on theabove nodes of a functionv = v(x, t) simply takes the form

INv(x, t) =N1

j=0

v(xj, t)j(x).

By definition, the Chebyshev collocation derivative of v with respect to x at thosenodes is then

x(INv)(xl, t) =

N1j=0

v(xj , t)j(xl) =

N1j=0

Dljv(xj , t),

withDlj = j(xl). The collocation derivative at the nodes can be obtained throughmatrix multiplication.

We introduce the numerical method on a simple advection equation for ease ofexposition

tu+a xu= 0, (4.2)u(1, t) = g(t), (4.3)

where a > 0 and g is a given function describing the inflow boundary condition.Spatial semi-discretization using the above principles and notation leads to

tuN+ a xuN= 0.

The latter relation is enforced at the internal nodes and an extra condition is imposedto ensure the verification of the boundary condition (4.3). Typically, that conditionis simply4

uN(xN1, t) = g(t).

The above method can be applied in a straightforward way to (2.18). Each of thevariables A, Q andPis discretized according to (4.1), leading to the new unknowns

4It has been observed that such a condition may lead to both theoretical and practical stabilityproblems (for instance, the structure of the derivative matrix D is essentially altered [31]). Toalleviate those problem, weak implementation of the boundary conditions through a penalty methodhas been proposed, see [19, 31, 35]. This type of method has been tested here and was not found tobe necessary.


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AN, QN and PN. As said in Section 3, the system (2.18) requires two boundaryconditions, one at each end of the vessel. With respect to a given junction, this cor-responds to a boundary condition for each vessel involved. For illustration purposes,consider a standard vessel bifurcation with one parent vessel and two daughter ves-

sels. Since there are three vessels related to this junction, we will need three boundaryconditions. As stated previously, these take the form of (3.4). Thus, is this case, wewill have one flow condition and two pressure conditions. This is consistent with thenumber of conditions needed based on a study of the characteristics of the system.

This results in the following semi-discretized system

d

dtU+ B(I3 D)U= G + F

U,

d

dtU

,

whereU= [A(x0, t), . . . A(xN1, t), Q(x0, t), . . . Q(xN1, t), P(x0, t), . . . P (xN1, t)]T,

G is the vector obtained in a natural way from GN (discretization ofG), I3 is the3 3 identity matrix, and is the Kronecker product. Finally, all the contributionsfrom the boundary conditions have been lumped into F and the matrix B is definedas

B11,0 B12,0 B13,0B11,1 B12,1 B13,1

. . . . . . . . .

B11,N1 B12,N1 B13,N1

B21,0 B22,0 B23,0B21,1 B22,1 B23,1

. . . . . . . . .

B21,N1 B22,N1 B23,N1B31,0 B32,0 B33,0

B31,1 B32,1 B33,1. . . . . . . . .

B31,N1 B32,N1 B33,N1

,

with Bij,k = BN,ij(xk), where BNis the matrix corresponding to the discretizationof the matrix B in (2.18).

Two different methods have been considered for temporal discretization: a third

order explicit TVD Runge-Kutta method [32, 54, 55] as well as a simple BackwardEuler method. In the first case, the stability of the above numerical approach appliedto (4.2,4.3) (withg 0) was analyzed in [40]. Their stability result (see Theorem 4.2)is adapted to an empirical stability condition for the present case. More precisely,the size of the n-th time step tn is adapted during the calculations and taken as

tn = C

(N 1)2,

where is the maximum over all spatial nodes of the spectral radius of the matrixBN at the current time and C is a constant. However, for the problems at hand,it was observed that Backward Euler with a limited number of Newton steps asnonlinear solver was overall faster and lead to results quantitatively comparable tomore elaborate TVD solvers. The use of an implicit solver allows us to implementthe boundary conditions directly on the primary variables without having to switchto the characteristic variables. Thus, the boundary conditions are implemented bysimply removing the appropriate differential equation corresponding to the boundarynode and replacing it with an equation for the boundary condition. The results shownbelow were obtained using Backward Euler. After appropriate numerical convergencestudy, it was determined that as little as four collocation nodes per vessel can be used.


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0 5 10 15 20 25 30 35

0

1

2

3Right Internal Carotid Artery

ECG

0 5 10 15 20 25 30 3560

80

100

120

p[mmHg]

0 5 10 15 20 25 30 35

20

40

v[cm/sec]

time [sec]

Fig. 5.1. Typical raw data file (here the right Carotid artery).

5. Data analysis. Data analyzed in this study stem from one subject and in-clude: velocity measurements obtained using digital transcranial Doppler technology5

at locations approved by the Institutional Review Board at the Beth Israel DeaconessMedical Center. These correspond to our three inflow locations (nodes in Basilar,Left and Right Carotid arteries) and six outflow locations (nodes in Left and RightACA, MCA and PCA). Blood pressure measurements obtained using a continuousnoninvasive finger arterial blood pressure monitor in supine position6 that reliablytracks intra-arterial blood pressure when controlled for finger position and temper-ature [46]. Geometric measurements of vessel lengths and areas are derived froma magnetic resonance angiogram7. Typical velocity and pressure measurements areshown in Figure 5.1. Finally, respiration and CO2 were measured from a mask usingan infrared end-tidal volume monitor (Datex, Ohmeda, Madison, WI). Electrocardio-gram, cerebral blood flow velocities, and CO2 were continuously recorded at 500 Hzusing Labview6.0 NIDQ (National instruments, Austin, TX).

The inflow velocity data is used to drive the system while the outflow velocity andthe pressure data are only used a posteriorito validate the results. Geometric areadata are used to specify the model domain and to determine inflow into the modelprovided the measured velocity.

5PMD 150, Terumo Cardiovascular Systems and Spencer Technologies Inc, Ann Arbor, MI andSeattle,VA USA.

6Ohmeda, Monitoring Systems, Englewood.7More precisely, intracranial vessels were visualized using 3D-MR angiography (time of flight,

TOF): TE/TR=3.9/38ms, flip angle of 25 degrees, 2mm slice thickness, -1 mm skip, 20cm 20cmFOV, 384224 matrix size, pixel size 0.39x0.39 mm at the GE VHI 3 Tesla scanner at the Center forAdvanced Magnetic Resonance Imaging at the Beth Israel Deaconess Medical Center. The radiusand length of the vessels were measured by the software Medical Image Processing, Analysis,andVisualization (MIPAV), Biomedical Imaging Research Services Section, NIH, USA. The scale foran image can be defined to achieve accurate measurements with resolution up to one pixel size (0.39mm x0.39 mm).


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1 1.2 1.4 1.6 1.8 250

100

150

200

LMCA

time (s)

pressure(mmHg)

Fig. 5.3. Comparison of model blood pressures to data before and after running the EnKF; blue

line: data,: model-original resistance parameters, o: model-EnKF optimized resistance parame-ters.

the model results obtained using the windkessel boundary condition. In this case, theoriginal parameters were chosen in a more intelligent way and therefore the switch tothe EnKF optimized parameters provides less of an improvement.

It is also important to consider the associated pressure data. Since the bloodpressure was measured in the finger and not in the brain, the model results are notexpected to match the data, but they should be in roughly the same range. Figure 5.3shows a comparison of the pressures from the model with the pressures from the datain the LMCA. As expected, the waveforms are not the same, but they are similar.

6. Results. Validation of many blood flow models is limited by the lack of avail-able data, and is therefore usually qualitative in nature. Access to clinical data allowsthe present approach to be validated in a quantitative manner.

% within % within( , +) ( 2, + 2)

RPCA 66 90LPCA 48 100RMCA 16 100LMCA 54 100RACA 32 98LACA 40 84

Table 6.1Percentage of time the model mean is within one or two standard deviations () of the datamean () in each of the six outflow vessels.

Since the cardiac cycle varies over time, even in a single subject, a given setof outflow data is not expected to be matched exactly using a given set of inflowdata collected at a different time. Instead, all of the available data is processed


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1 1.2 1.4 1.6 1.8 20

20

40

60

80

100

120

RACA

time (s)

velocity(cm/s2)

1 1.2 1.4 1.6 1.8 220

30

40

50

60

LACA

time (s)

velocity(cm/s

2)

1 1.2 1.4 1.6 1.8 20

20

40

60

80

100

RMCA

time (s)

velocity(cm/s2)

1 1.2 1.4 1.6 1.8 20

20

40

60

80

100

LMCA

time (s)

velocity(cm/s2)

1 1.2 1.4 1.6 1.8 220

30

40

50

60

70RPCA

time (s)

velocity(cm/s2)

1 1.2 1.4 1.6 1.8 210

20

30

40

50

60LPCA

time (s)

velocity(cm/s

2)

Fig. 6.1. Mean outflow velocities resulting from running the stochastic version of the modelover 20 realizations; blue line: , green line: , dashed line: 2,: mean predicted outflow.

and a mean velocity profile is calculated for each inflow and outflow vessel, alongwith the associated variances. The available number of measurements, i.e., periods,per vessel varies between 20 and 200. The simulation is then run with 20 differentstochastically perturbed inflow velocity profiles. The inflow conditions are determinedby stochastically perturbing the mean change in velocity at each time step to avoidcreating artificial roughness in the wave form; the perturbations are drawn from anormal distribution based on the data. The mean predicted velocity in each outflowvessel is then compared to the corresponding mean velocity profile from the data, seeFigure 6.1. The breakdown of how well the model results match the data is shown inTable 6.1.

As is evident from both the figures and the table, the model is predicting thevelocities at each of the six outflow points consistently.

Figure 6.2 shows the results of running the deterministic model (where the inflowsare taken from the data, not from perturbations of the means) in the LMCA over anumber of cardiac cycles.

7. Conclusions. The proposed model and implementation agree remarkablywell with the data in spite of their simplicity. A comparison with results from 1.5Dmodels such as those proposed in [11, 12] would be interesting. The present work is


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