Mixed model for simulating blood flow in thecerebral microcirculation

Farah Yasmina Houdroge∗Supervised by Sylvie Lorthois (CR CNRS), Michel Quintard (DR CNRS), Yohan Davit (CR CNRS)

Institut de Mecanique des Fluides (IMFT), GEMP groupCNRS, UPS, Universite de Toulouse

This study was aimed at creating a coupled model to simu-late blood flow in the cerebral microcirculation. Since bloodvessels in the brain are of widely varying geometries anddiameters, two different models, based on two different char-acteristic length scales, underpin our new multi-scale model.Large vessels such as arteries, veins, arterioles and venulesare represented as a graph, where the flow of blood is gov-erned by the Hagen–Poiseuille equation. On the other hand,the capillary bed can be reasonably represented as a con-tinuum and is considered homogeneous on the Darcy scale.In terms of coupling, the porous network model provides uswith a throughput that is directly injected into the homoge-neous medium at the junctions with the larger vessels. How-ever, it is found that the pressure computed following thisinjection varies with the mesh size and is not representativeof the real pressure imposed at the tip of the vessel. To ac-count for this discrepancy, a corrective model was derivedand validated based on an approach developed for under-ground reservoir simulations. The overall numerical modelwas optimised to reduce the otherwise substantial comput-ing time requirements. The numerical code and models weredeveloped with OpenFOAM.

Nomenclatureµ Blood dynamic viscosity (Pa.s)U Blood velocity (m.s−1)h Mesh size (m)K Permeability of the porous medium (m2)L Vessel length (m)P Pressure (Pa)Q Volumic flow rate (m3.s−1)q Source term (s−1)R Vessel resistance (Pa.s.m−3)req Equivalent radius (m)S Vessel cross-sectional area (m2)

∗E-mail : farah.houdroge@hotmail.fr

1 IntroductionBlood flow is essential for the activity of a living organ-

ism. Arteries supply oxygen, glucose and other nutrimentsto the tissues whereas veins carry deoxygenated blood backtowards the heart. The junction between these large vesseltypes occurs through a capillary network consisting of small-diameter capillary vessels. In the brain, blood circulation isindispensable for neuronal activity. Not only does it ensurethe exchange of substances with the surrounding tissue, butit also supplies neurons with oxygen and glucose, and regu-lates blood flow depending on levels of neuronal activity.

One of the major difficulties in modelling such systemslies in the varying complex geometries and the widely vary-ing characteristic spatial and temporal dimensions. As a mat-ter of fact, it has been observed that large vessels, such asveins, venules, arteries and arterioles, display fractal prop-erties [1], whose diameters vary between 9 and 100 µm inthe brain’s cortex, whereas small capillary vessels consist ofa meshed network, essentially homogeneous above a cut-offlength which corresponds to the typical length of these ves-sels (≈ 50µm) [2], and whose diameters vary between 4 and9 µm.

In terms of computational modelling of the entire bloodtransport network, in order to reduce simulation time costsand complexity, we avoid using a discrete approach and in-stead we describe the blood flow using two distinct models.Large vessels are considered as a graph or a network wherethe flow is of a Hagen–Poiseuille type, whereas the capillarybed is considered as a porous medium in which Darcy’s lawgoverns the flow.

Representing larger vessels as a graph for the modellingof blood flow, and particle transport and diffusion, in the vas-cular system is an approach used by many research groups,e.g., (Reichold et al. [3], Erbertseder et al. [4], Lorthois etal. [5, 6]). In the capillary bed, using a discrete approachin such a large system would result in high computationalcosts; it therefore is favourable to treat this as a continuum.Because the tissue surrounding the capillary vessels allowsthe blood to flow freely within it, we can consider the cap-illary bed as a porous medium with a fluid phase where thepores model the capillary vessels and the tissue is represented

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Fig. 1. Schematic representation of the coupled model.The black lines represent the larger vessels such as the arteries andthe veins in which the flow is governed by the Hagen–Poiseuille equa-tion, and the grey square is the homogenized capillary bed in whichthe blood flows according to Darcy’s law.

by the porous medium. Amongst the few research groupsthat have adopted this approach, an important related con-tribution is that of Erbertseder et al. [4], who have createda coupled discrete/continuum model to describe therapeutictransport in the lungs for cancer treatment.

It is crucial for the combined model that both sub-models communicate efficiently and effectively with accu-rate representation of physical quantities such as flow rates,pressures and particle concentrations. This coupling aspecttherefore requires particular attention and demands a thor-ough study; this is what the present paper mainly focuses on.

This paper is organised as follows: in section 2 we coverthe state-of-the-art modelling of blood flow in large andsmall vessels; and in section 2.4 we examine the problemsencountered with coupling the sub-models, and develop thecoupling and solution algorithm. Finally, section 3 presentssimulation results, which demonstrate the validity of the cou-pled model and opens the road to future developments.

2 Methods and modelling2.1 Modelling large vessels

In this section, inspired by Reichold et al.’s ( [3]) pa-per, we give a brief but necessary outline of the modellingof blood flow in larger vessels consisting of veins, arteries,venules and arterioles.

In this representation (Fig. 2), a node i represents a ves-sel’s bifurcation or ending and the edges i j represent the ves-sels themselves. The vessels are considered as rigid cylindri-cal bodies.

The flow in each vessel follows Hagen–Poiseuille’s law:

Qi j =∆Pi j

Ri j=

S2i j

8πµLi j∆Pi j, (1)

where i j denotes the edge connecting the node i to the nodej, Qi j the volumetric flow rate, Ri j the vessel’s resistance, Si jand Li j the vessel’s cross-sectional area and length respec-tively, µ the blood’s dynamic viscosity and ∆Pi j the pressure

Fig. 2. Schematic representation of a vascular graph. Acollection of nodes i connected by edges ij. Taken from [3].

difference across the vessel’s length.Figure 3 shows a schematic representation of this model.The black edges represent the large vessels and the outlinedsquare the capillary bed.

Fig. 3. Representation of the vascular graph model withthe corresponding boundary conditions. The black lines rep-resent the large vessels and the square the capillary bed. In redare the boundary conditions associated with the arteries and arteri-oles, and in blue the those associated with the veins and venules. Inthese vessels, the flow rates are calculated using Hagen–Poiseuille’sequation and boundary conditions on the pressure are imposed atthe extremities and bifurcations of each vessel.

2.2 Modelling small vesselsRepresenting the capillary bed as a continuum consti-

tutes an entirely different modelling approach since it re-quires defining new effective parameters associated with theupscaled capillary bed, such as the porosity and the perme-ability of the medium. In our study, to initially proceed, weassociate constant but arbitrary values to the different phys-ical parameters. Whilst ideally, relevant physically correctvalues would be used, this is not necessary for this work as itdoes not impact on the aims or key results.

The flow of blood here is taken as stationary, i.e., theinfluence of the pulsatile component of the flow is not con-sidered. From the vascular graph model we collect volumet-

2

ric flow rates which we then inject into the capillary mediumat the junction with the larger vessels (see Fig. 4). In thecapillary upscaled medium, the flow is governed by the con-tinuity equation ∇ ·U = qδi where U is the blood velocity,q a source term and δ the Dirac distribution function. Theright-hand side of this equation is obtained from the bloodflow obtained from the vascular graph calculations.

Fig. 4. Representation of the capillary bed as a contin-uum. Since the capillary bed is described as a porous medium,Darcy governs the flow in this region. The grid represents the meshof the upscaled capillary bed and the black dots represent the sourcepoints at which the flow rates obtained from the network calculationsare applied.

In the system illustrated in Fig. 4, Darcy’s generalisedlaw:

U =−Kµ

∇P, (2)

relates the pore blood velocity U to the pressure gradient,where K is the permeability of the medium. Coupling thecontinuity equation ∇ ·U = qδi with equation (2) gives:

∇ ·(−K

µ∇P)= qδi, (3)

which is the equation resolved in this medium. The porosityand permeability of the porous medium, the spatial positionof the vessels as well as the viscosity of the blood are con-stants.

2.3 Coupled modelIn order to study the coupling of both models and de-

velop the solution algorithm, we built two test models: aDNS (Direct Numerical Simulation) model where a cylin-der placed in the middle of a cube represents an artery inthe capillary bed. Constant pressures were imposed at the

base of the cylinder (Pcyl) and at the bottom face of the cube(Pcube < Pcyl) and the remaining faces have zero flux bound-ary conditions. The resolution of equation (3) with no righthand term gives a value of the flow rate QDNS between theartery and the bottom of the cube.

For the second model, referred to as the source model,only the capillary bed was taken into account. It is repre-sented as a cube with a regular Cartesian mesh, with the sameboundary conditions as in the DNS model imposed on eachface of the cube. The flow rate QDNS is injected in the centreof the cube where the artery supposedly spills the blood intothe capillary bed and equation (3) is resolved in this system.Both models are represented in Fig. 5.

Fig. 5. Models created for studying the coupling algo-rithm. On the left, the DNS model and on the right the sourcemodel, which are related by the value of the blood flow QDNS. Thecolours correspond to the pressure distribution obtained from thesimulations.

In the remainder of this section, all simulations wereperformed based on the following properties.

Simulation propertiesBlood viscosity µ(kg.m−1.s−1) 6∗10−3

Porous medium permeability K (m2) 10−9

Porous medium length (mm) 2Large vessel radius (mm) 0.01Large vessel length (mm) 1

Different numbers of cells in each direction were studiedfor the source point model, respecting the fact that a cell’ssize should be larger than the large vessel’s diameter. Forthe presented results, only two mesh sizes are considered:31×31×31 and 49×49×49.

As described above, large vessels are considered assource terms for the capillary network. When blood volu-metric flow rates QDNS are injected into the capillary bed,the pressure computed in the cell containing the source termdepends on the mesh size and is not representative of the realpressure at the tip of the vessel Pcyl . Figure 6 represents thepressure curves in the vicinity of the vessel (from the DNSmodel) and the source points. It depicts the mesh dependentbehaviour of the pressure, which is a numerical error that

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does not represent the reality of physical phenomena that oc-cur at this scale. Of course, any physical quantity should begrid independent.

Fig. 6. Comparison of pressure values between the DNSand the source model in the vicinity of the vessel and thesource point for different mesh sizes. In red: the curve of thepressure in the vicinity of the artery for the DNS model. In green andblue: the curves of the pressure in the vicinity of the source mesh,where the domain is divided into 313 and 493 cells, respectively.Coupling both models results in a pressure that depends on the meshsize, and that is not representative of the real pressure imposed inthe DNS. In fact, the values of the computed pressures in the sourcemodel are considerably smaller than the value of the real pressurePcyl , and they grow smaller as the mesh sizes are larger.

This problem is typically encountered in reservoir simu-lations where the mesh cell size is much larger than the radiusof the well [7]: the wells are therefore considered as sourceterms in the mesh. However, because of the peculiarities ofthe pressure in the surroundings of the wells, it is difficult todetermine which value of the pressure (or flow) must be im-posed within the cell that contains the source term. In fact,for an imposed flow, the computed pressure in the cell con-taining the source depends on the mesh and in the same way,for an imposed pressure, the flow rate depends on the meshsize. In order to correct this numerical error, it is necessaryto perform a theoretical and numerical study and to find anequation that relates the flow rate to the bottom hole pressurein the well and the computed pressure in the mesh.

This problem, also known as the well model, was intro-duced by Peaceman in 1978 [8] in the context of a theoreticalstudy on the relationship between the pressure of the mesh,the well pressure and the flow rate, in the case of a single-phase flow within a square grid using a two-dimensionalfinite-difference method. Peaceman defined an equivalentradius that represents the distance at which the analytical ex-pression for the pressure, that corresponds to a flow around awell placed in an infinite medium, is equal to the pressure inthe source mesh. Since then, the problem has been extendedto take into account various more complex cases (non-regularmeshes, non-centered meshes, ...) [9–11].

Although similar, taking account of the wells inpetroleum engineering is different because of the geometry

of our problem. In the petroleum context, the geometry istwo dimensional 1, whereas our case is extended to three di-mensions.

2.4 Corrective modelTo overcome this numerical approximation problem, an

approach is to follow Peaceman’s early works [8,10,11] andEwing et al.’s [9] developments by finding a value of anequivalent radius, defined as the distance at which the com-puted pressure in the source mesh is equal to the real pressureat the tip of the vessel. To do this, we compared the analyt-ical solution of the pressure obtained for the DNS case withthe analytical solution of the computed pressure in the sourcemodel case, and established an equation relating the pressurevalues in both cases with the volumetric flow rate.

2.4.1 DNS: analytical solution of the pressureWe considered a spherical source placed in the middle

of a three-dimensional infinite porous medium. Since we areapproximating the end of the vessel as a spherical source andnot a circular one related to the vessel’s cylindrical geometry,we should keep in mind that the flow rate should be the samebetween both geometries. This condition is satisfied if theradius of the spherical source is half of that of the cylinder.In the vicinity of the source, considering the rotational sym-metry of the solution, equation (3) with no right hand termgives:

1r2

∂

∂r

(r2Ur

)= 0⇔ ∂

∂r

(r2Ur

)= 0⇔ r2Ur =C

⇒Ur =Cr2 , (4)

and as Q represents the volumetric flow rate through a sphereSe centred at the origin, the following relation can be de-duced:

Q = −∮

Se

U.nds

= −∫ 2π

φ=0dφ

∫π

θ=0

Cr2 r2 sinθdθ

= −4πC. (5)

therefore C and Ur can be rewritten as:

C =− Q4π⇒Ur =−

Q4π

1r2 . (6)

1Oil reservoirs are pipes through which the fluid flows, their geometrycan thus be reduced to a two-dimensional geometry by considering a trans-lational symmetry relative to their main axis.

4

The resolution of equation (2) thus gives:

−∇P =µK

U⇒ ∂P∂r

=µK

Q4π

1r2 , (7)

and integrating equation (7) from rvessel/2 to r gives the fol-lowing analytical expression for the pressure in the vicinityof the vessel:

P(r) = P(rvessel)−µQ

4πK

(1r− 2

rvessel

). (8)

2.4.2 Source model: analytical solution of the computedpressure

Fig. 7. Seven-point, three-dimensional, cubic numericalscheme. The numbers are references to the cells. (a) View of thecomplete numerical scheme, (b) front view and (c) side view.

We consider the problem described by (3) applied to aseven-point numerical scheme (see Fig. 7) in the case whenthe source is located at the centre of a cubic grid. Integratingequation (3) over the volume V of the mesh containing thesource term gives:

∫V

∇ ·(−K

µ∇P)

dV =−∫

Vqδ0 dV, (9)

and using the Green-Ostrogradski’s formula, we get:

∫V

∇ ·(−K

µ∇P)

dV =−Kµ

∮∂V

∇P · dS, (10)

where dV is the volume element and dS the vector surfaceelement. Knowing that the source term q is constant in mesh0 containing the source and is zero for the rest of the domain,equation (9) becomes:

−Kµ

∮∂V

∇P ·dS =−Q. (11)

Eventually, through integration by parts and discretising us-ing the finite-volume method ( [12]), we have:

− ∑ε0i

i∈[1,6]

Kµ

Pi−P0

hh2 =−Q, (12)

where i is the cell index, h the mesh size and ε0i the facebetween cells 0 and i. Since P is symmetrical in the vicinityof the source (i.e., Pi = Pj for i, j ∈ [1,6]) we have:

−6Khµ

(P1−P0) =−Q

⇒ P0−P1 =−µQ

6Kh. (13)

If we consider that the analytical expression of the pressure(8) is a good approximation in cell 1, we get:

P1 = Pvessel−µQ

4πK

(1r1− 2

rvessel

), (14)

where r1 = h. Adding (14) and (13) gives:

P0 = Pvessel−µQ

4πK

(1h− 2

rvessel+

2π

3h

). (15)

This relation shows how the pressure computed in the sourcemesh P0 is not equal to the pressure at the tip of the vesselPvessel . The idea is therefore to find a value of an equivalentradius req that satisfies: P(req)=P0. In other words, equation(8):

P(req) = Pvessel−µQ

4πK

(1

req− 2

rvessel

),

should be equivalent to (15):

P0 = Pvessel−µQ

4πK

(1h− 2

rvessel+

2π

3h

),

which gives:(1

req− 2

rvessel

)=

(1h− 2

rvessel+

2π

3h

). (16)

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We thus find req =3

3+2πh≈ 0.32h.

3 Results and Discussion3.1 Numerical justification of the equivalent radius req

In order to verify this analytical corrective model, wealso tried to find the appropriate value of req numerically.Considering that the source is in cell 0 and that for r = rnum

eqwe have P0 = P(rnum

eq ), we get:

P(r(k,l,m)) = P0−µQ

4πK

(1

r(k,l,m)− 1

rnumeq

). (17)

For a regular Cartesian mesh, in any given direction (k, l,m),r(k,l,m) = nh where n is a positive real number. Using thisform allows (17) to be written:

P(r(k,l,m)) = P0−µQ

4πK

(1

nh− 1

rnumeq

)

= P0−µQ

4πKh

(1n− h

rnumeq

)

= P0−A

(1n− h

rnumeq

), (18)

where A =µQ

4πKh. We can write A−1

(P(r(k,l,m))−P0

)as a

linear function of 1/n:

A−1 (P(r(k,l,m))−P0)=

1n− h

rnumeq

=an+b. (19)

Figure 8 shows A−1(P(r(k,l,m))−P0

)as a function of 1/n for

different mesh sizes. The interpolation of these results givesa linear function with a slope a equal to 1.09 and a y-interceptb of -3.19. Hence we have the approximation:

− hrnum

eq=−3.19⇒ rnum

eq =h

3.19≈ 0.31h , (20)

which is very close to the value of req found analytically.

3.2 Verification of the corrective modelA final study was undertaken to justify this corrective

model. For different pressure values Pvessel imposed in theDNS model and their corresponding volumetric flow ratesQDNS, we used the value of P0 computed with the source

Fig. 8. A−1(P(r(k,l,m))−P0

)as a function of 1/n for dif-

ferent mesh sizes. The fitted curve in blue dots is a linear functionwhose y-intercept gives the value of the equivalent radius req.

Fig. 9. Graph of the predicted pressure as a function ofthe real vessel pressure. The dashed line, corresponding to theidentity function, was obtained by interpolating the data and verifiesthe validity and reliability of our model.

model for several mesh sizes and were able to predict Pvesselwith the following equation:

Pvessel = P0−µQ

4πK

(2

rvessel− 1

0.3h

), (21)

inspired by the analytical expression of the pressure (8) forr = req and P(req) = P0.

Figure 9 shows the graph of the predicted pressure as afunction of the real vessel pressure. The resulting functionis the identity function. Not only do these results show howthe predicted pressure becomes independent of the mesh sizeand representative of the real pressure at the tip of the vessel,they also verify the validity and reliability of our model.

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Fig. 10. Approximation error as a function of the distancebetween two sources. For a distance below 400µm, the errorbecomes greater than 1% regardless of the mesh size.

3.3 Additional studies3.3.1 Interaction between two sources

We further studied how the interaction between twodifferent sources can influence the results of the correctivemodel. To do this, for different mesh sizes, two sources wereplaced at a distance of 1mm from each other which was grad-ually reduced until both sources were located at the samespot.

Figure 10 represents the approximation error dependingon the distance separating the two sources. The error wascalculated by comparing the pressure values obtained fromthe corrective model with the actual values imposed in theDNS models. We observe that for a distance below 400µm,the error is greater than 1% regardless of the mesh size. Thisinteraction between sources requires a more detailed study toquantify how it influences the pressures in order to adjust thecorrective model.

3.3.2 Code optimisationFigure 11 illustrates the algorithm employed for the cal-

culations. The inputs are the pressure values at the ends orat the bifurcations of the large blood vessels, which servefor the network computations. By applying the Hagen–Poiseuille equation, flow rates are collected and then injectedinto the capillary bed at the junctions with the larger vessels.The pressure distribution in the medium is then calculatedusing equation (3). After applying the corrective model ineach source cell, new pressure values are deduced and thesystem iterates until convergence of the solution.

However, because of the large mesh sizes used, thesimulations can run for many hours, even for the mostsimplified systems, e.g., with only one artery and one vein.For practical simulations, the code needed to be moreefficient and less cpu-time intensive, especially since realmodels would be much more complex than our simple testmodels.

Fig. 11. Algorithm for the calculations

Platform-dependent efficiency improvements

The first improvement sought was a platform-dependentone. The open source CFD toolbox OpenFOAM that wasused for this work includes many linear system solvers for it-eratively solving each matrix equation generated by the maincode. The first one that we used was the PCG linear solver(Preconditioned Conjugate Gradient), which solves linearsystems using the preconditioned conjugate gradient method.Although effective, it requires a large number of iterationsthat make the simulations very costly in terms of computertime. To address this situation, we experimented with adifferent linear solver called GAMG (Geometric-AlgebraicMulti-Grid). The basic idea of this method is to define acoarse mesh from the originally defined mesh. Once the so-lution is obtained on the coarse mesh, it is projected ontoa finer mesh and thus serves as a starting guess for the finegrid solution. A sequence of coarse grids can be consideredto assist in obtaining the solution on the desired fine mesh.The coarse meshes are further employed to quickly reducethe residual error on finer mesh through a series of extrapo-lations and interpolations during the iterative cycling.

To test the efficiency of this method, we built a testmodel (test A) where two arteries and one vein are repre-sented in the capillary bed. The simulation took 2 hoursand 40 minutes using the PCG solver but only 20 minutesusing the GAMG solver, i.e., a factor of 8 reduction in thecomputing time.

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Platform-independent efficiency improvementsRunning the simulations with initial solutions that are

far from the final solutions can also cause an excessive num-ber of iterations (thousands) before obtaining convergence.In order to reduce this number, the ideal situation would beto start from an initialised system that approximates the finalstate of the system.

Lorthois et al. [6] showed that a system similar to ours,where an artery and a vein described by a vascular graphmodel are placed in a capillary homogenised medium, canbe simplified into a unidimensional system by applying anelectrical network analogy of equivalent resistances in se-ries subject to constant pressure drops. The resistances ei-ther represent the arteriolar part of the network or the capil-lary and venular parts of the network, and are deduced fromPoiseuille’s and Darcy’s laws respectively. We used this de-scription and generalised it to take into account several largevessels by including a parallel circuit as well. This allowedus to approximate an averaged value of the pressure to beused as an initial value in the capillary domain as well asaveraged flow rates to be injected into the source meshes.

One can argue that this simplification of a three-dimensional system to a uni-dimensional one is very unre-alistic; in fact, the averaged values of the pressure and flowrates are still very different from the final solutions. How-ever, they still proved to be satisfactory initial conditions asthey considerably reduced simulation times. For instance,the simulation time for test A was reduced to 6 minutes whenboth the linear solver and the initial values of the pressureand the flow rates were optimised.

4 ConclusionThis type of modelling of blood circulation is widely

used by a number of scientific research groups e.g., [1, 3, 4].However, this study allowed these models not only to be ex-tended to a three-dimensional geometry, but also to rethinkthe coupling between the two different models used to de-scribe the flow in the large and in the small vessels. As shownin this present paper, it is not sufficient to naively couple bothmodels without taking into account the mesh-dependent be-haviour of the pressure in the vicinity of the source points.It is therefore necessary to apply a corrective model, whichpredicts the real pressure at the tip of the vessel based on thecomputed pressure in the cell in which the volumetric flowrate is injected.

This methodology can be extended to take into accountmass transfer between the blood and the surrounding tissuefor the study of particle transport and diffusion in the vas-cular system. In this case, a homogenisation of the masscontinuity equation requires an assessment of effective pa-rameters such as the effective diffusion tensor, while takinginto account real anatomical data and a capillary density thatvaries with the depth in the cortex. Finally, potential exten-sions to the model include incorporation of non-Newtonianaspects of blood or incorporating elastics vessels whose di-ameters vary for determining the effect of arterial occlusionsor expansions.

AcknowledgementsThe author acknowledges the support and ongoing su-

pervision provided by Sylvie Lorthois, Michel Quintard andYohan Davit as well as the help and contribution brought byCyprien Soulaine.

References[1] Lorthois, S., and Cassot, F., 2010. “Fractal analysis

of vascular networks : insights from morphogenesis”.Journal of Theoretical Biology.

[2] Billanou, I., 2010. “Modelisation experimentale ettheorique pour la quantification du debit sanguin partomographie a emission de positrons”. PhD thesis, Uni-versite de Toulouse.

[3] Reichold, J., Stampanoni, M., Keller, A., Buck, A.,Jenny, P., and Weber, B., 2009. “Vascular graph modelto simulate the cerebral blood flow in realistic vascu-lar networks”. Journal of Cerebral Blood Flow andMetabolism.

[4] Erbertseder, K., Reichold, J., Flemisch, B., Jenny, P.,and Helmig, R., 2012. “A coupled descrete/continuummodel for describing cancer-therapeutic transport in thelung”. PLoS ONE.

[5] Lorthois, S., Cassot, F., and Lauwers, F., 2010. “Sim-ulation study of brain blood flow regulation by intra-cortical arterioles in an anatomically accurate large hu-man vascular network: Part i: Methodology and base-line flow”. Neuroimage.

[6] Lorthois, S., Cassot, F., and Lauwers, F., 2010. “Sim-ulation study of brain blood flow regulation by intra-cortical arterioles in an anatomically accurate large hu-man vascular network. part ii: Flow variations inducedby global or localized modifications of arteriolar diam-eters”. NeuroImage, 54(2840-2853).

[7] Ding, D.-Y., 2004. Modlisation des coulements en mi-lieu poreux au voisinage des puits. Tech. rep., Avril.

[8] Peaceman, D., 1978. “Interpretation of well-blockpressure in numerical reservoir simulation”. Society ofPetroleum Engineers Journal, 18, pp. 183–194.

[9] Ewing, R., Lazarov, R., Lyons, S., Papavassiliou, D.,Pasciak, J., and Qin, G., 1998. “Numerical well modelfor non-darcy flow through isotropic porous media”.Computational Geosciences, 3, pp. 185–204.

[10] Peaceman, D., 1983. “Interpretation of well-blockpressure in numerical reservoir simulation with non-square grid blocks and anisotropic permeability”. Soci-ety of Petroleum Engineers Journal, 23, pp. 531–543.

[11] Peaceman, D., ed., 1978. Interpretation of well-blockpressure in numerical reservoir simulation - Part 3 :Some additional well geometries, 62nd Annual Tech-nical Conference and Exhibition of the Society ofPetroleum Engineers.

[12] Jasak, H., 1996. “Error analysis and estimation for thefinite volume method with applications to fluid flows”.PhD thesis, Imperial College of Science, Technologyand Medicine, Juin.

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