A LATTICE CELLULAR AUTOMATA MODEL FOR ION DIFFUSION …miura/directory/1980-1999/1999/Dai-Miura99.pdf · but the rst lattice gas cellular automata in a two-dimensional square lattice

A LATTICE CELLULAR AUTOMATA MODEL FOR ION DIFFUSIONIN THE BRAIN-CELL MICROENVIRONMENT AND

DETERMINATION OF TORTUOSITY AND VOLUME FRACTION∗

LONGXIANG DAI†‡ AND ROBERT M. MIURA†§

SIAM J. APPL. MATH. c© 1999 Society for Industrial and Applied MathematicsVol. 59, No. 6, pp. 2247–2273

Abstract. In the brain-cell microenvironment, the movement of ions is by diffusion when thereis not any electrical activity in either the cells or the externally applied electric field. In this complexmedium, the primary constraints on long-range diffusion are due to the geometrical properties of themedium, especially tortuosity and volume fraction, which are lumped parameters that incorporatelocal geometrical properties such as connectivity and pore size. In this paper, we study the effectsof these geometrical properties in mimicking the experimental situation in the brain. We build alattice cellular automata model for ion diffusion within the brain-cell microenvironment and performnumerical simulations using the corresponding lattice Boltzmann equation. In this model, particleinjection mimics extracellular ion injection from a microelectrode in experiments. As an applicationof the model, we combine the results from the simulations with porous media theory to computetortuosities and volume fractions for various regular and irregular porous media. Porous media theorypreviously had been combined with diffusion experiments in brain tissue to determine tortuosity andvolume fraction. As in the case of the diffusion experiments, porous media theory gives a goodapproximation to the numerical simulations. We conclude that the lattice Boltzmann equation canaccurately describe ion diffusion in the extracellular space of brain tissue.

Key words. brain-cell microenvironment, lattice cellular automata, lattice Boltzmann equation,ion diffusion, tortuosity, volume fraction, porous medium

AMS subject classifications. 92C20, 92C36, 9208, 76R50, 76S05, 76P05

PII. S0036139997323942

1. Introduction. The brain contains a complicated network of specialized cellscalled neurons (or nerve cells) and glia (glial cells), each of which is bounded by a thinmembrane. The membrane separates the brain into two compartments: extracellularspace (ECS), consisting of narrow spaces between cellular elements which constitutethe brain-cell microenvironment, and intracellular space (ICS). The structure of theECS resembles that of a porous medium. In the absence of electrical activity in thecells and externally applied electric fields, ions move within the ECS by diffusion.

Because the ECS and ICS are distinct compartments separated by cell mem-branes, the ionic concentration will fluctuate between the local ECS value and thelocal ICS value as one moves in a given direction. The discontinuous concentrationprofile yields derivatives that are not well defined. To avoid these discontinuities,Nicholson and his colleagues [29], [32], [33], [34], [35], [39] used volume-averaging the-ory [19], [30], [45] to study this diffusion phenomenon. They introduced phase-average

∗Received by the editors July 7, 1997; accepted for publication (in revised form) December 29,1998; published electronically October 27, 1999. This research was supported in part by NaturalSciences and Engineering Research Council of Canada grant 5-84559.

http://www.siam.org/journals/siap/59-6/32394.html†Department of Mathematics and Institute of Applied Mathematics, University of British

Columbia, Vancouver, BC, Canada V6T 1Z2 ([email protected], [email protected]).‡Mathematical Research Branch, NIDDK, NIH, BSA 350, 9190 Wisconsin Avenue, Bethesda,

MD 20814.§Department of Pharmacology and Therapeutics, University of British Columbia, Vancouver,

BC, Canada V6T 1Z3.

2247

2248 LONGXIANG DAI AND ROBERT M. MIURA

quantities in the form

〈C〉 =1

V

∫V0

Cd3x =1

V

∫V

Cd3x

and

〈C〉0 =1

V0

∫V0

Cd3x =1

V0

∫V

Cd3x,

where C is the concentration in the ECS and is zero in the ICS, V is the volume of a“suitable” averaging region, called the representative elementary volume in [5], of thebrain (a characteristic property of the porous medium [3]), V0 is the volume of theextracellular component within V , and d3x is the differential volume element. HereV remains fixed while V0 could vary with position.

When the diffusing substance is neither absorbed nor bound on the membranes,and it cannot pass through these membranes, i.e., in the case of the zero-flux condition,Nicholson and Phillips [33] use the averaging theory to derive the modified diffusionequation for the averaged substance concentration given by

D

λ2∇2〈C〉0 +

〈q〉α

=∂

∂t〈C〉0,(1)

where α is the extracellular volume fraction, λ is the tortuosity, which representsthe factor increasing the path length that a diffusing ion traverses as a result ofthe presence of cellular obstructions, D is the diffusion coefficient of the substancediffusing within the ECS, and q is a local source.

Thus, diffusion in the brain-cell microenvironment is equivalent to diffusion ina simple medium with an effective diffusion coefficient De ≡ D/λ2 and with an al-tered source term 〈q〉/α. The effective diffusion coefficient De is affected by thepore structure since it is obtained by averaging over many cells. The tortuosity wasformally defined as λ =

√D/De by Nicholson and Phillips [33]. There are other def-

initions of the tortuosity in the literature. For example, Bear [4] defines tortuosity asλ = De/D, while Schultz and Armstrong [41] use λ = αD/De. Here we use Nicholsonand Phillips’s definition because of its interpretation in terms of path length.

The ion-selective microelectrode permits the direct monitoring of ionic concentra-tion in real time at a specific point in a given tissue; thus, by fitting the experimentaldata to a solution of the modified diffusion equation (1) at two different times, theparameters λ and α can be computed [33]. For rat cerebellum, the tortuosity λ is1.55 ± 0.05 and the volume fraction α is 0.21 ± 0.02. The tortuosities and volumefractions have been determined for many other brain tissues [29], [37], [39], [43]. Arecent review has been given by Nicholson and Sykova [36].

The tortuosity incorporates the brain structure into the diffusion process. It isa lumped parameter that is directly dependent on the geometrical properties of theporous medium, e.g., connectivity (the connectedness of points in the ECS) and poresize (the width of the ECS pathways). However, the determination of the effectsof these geometrical properties on the tortuosity and volume fraction is one of theobjectives of this research. The experimental data coupled with porous media theoryis inadequate to resolve these issues because we cannot alter the existing geometryof brain tissue, nor is the structure known exactly. Therefore, we seek an alternativetheoretical model to achieve this.

In this paper, we build a theoretical model for the diffusion of a single ionicspecies in the ECS using an alternative computational method. This method mimics

DIFFUSION IN THE BRAIN 2249

the experimental situation of diffusion of ions within the brain-cell microenvironmentby a microscopic-level discrete model, i.e., the lattice cellular automata (LCA) model[14], [15], [46]. Numerical simulations will be performed using the correspondinglattice Boltzmann equation (LBE). As applications of the model, we will combine theresults from the simulations with porous media theory to compute the tortuositiesand volume fractions for various regular and irregular media and study the effects ofthe geometrical properties on the tortuosity and volume fraction.

Both the tortuosity λ and the volume fraction α are geometrical factors and somebounds on tortuosity in terms of volume fraction have been given [31], [44]. In thederivation of these parameters [33], they are treated as independent quantities, but,intuitively, it seems that some relationship between them should exist.

Using the LBE method has the following advantages: (1) the assumptions usedto derive the modified diffusion equation (1) can be satisfied rigorously by suitablychoosing the membrane boundary conditions; (2) accurate calculations in arbitrarycomplex pore-space geometries can be carried out; (3) the LCA can provide the detailsof the movements of the ions and improve our understanding of diffusion within thebrain; and 4) additional movements of ions between the ECS and ICS through cellularmembranes with the complicated membrane geometries can be taken into accountsimultaneously (see [13] to see how to incorporate membrane ionic currents).

In section 2, we present an LCA model, the LBE, for ion diffusion in a porousmedium. The choices of additional conditions, such as the membrane boundary con-dition, to match the assumptions and how we do the numerical experiments are givenin section 3. In section 4, we describe how the computations of the tortuosity andthe volume fraction are performed. In sections 5 and 6, we present how we generatetwo- and three-dimensional porous media and the corresponding numerical results.Finally, we conclude with a discussion of the method developed here and the results.

2. LCA model. The basic idea of the LCA method is to study the macroscopicevolution of particle concentrations by using a simple microscopic-level model thatwill mimic the behavior of the real system on a macroscopic scale. The idea ofusing discrete methods for modeling partial differential equations occurred very early,but the first lattice gas cellular automata in a two-dimensional square lattice wereproposed by Hardy, Pomeau, and DePazzis [21], [22] in 1972. In 1986, Frisch etal. [17], [18] proposed a lattice gas model that was based on a triangular latticestructure. The invention of these models has stimulated much research on lattice gascellular automata and their floating-point-number counterpart, LBE. These have beenapplied successfully to many fields, such as fluids in porous media [8], [20], [40], phasetransitions [1], [2], and chemical reaction-diffusion systems [11], [23], [24], [25], [26],[28]. The lattice gas cellular automata and the LBE also can be used as numericalschemes to “solve” partial differential equations such as the Navier–Stokes equation[17], Burgers’s equation [6], the wave equation [7], and reaction-diffusion equations[12], [38].

The problem of interest here is pure diffusion in a complicated geometry, so wecan use the ideas developed by Chopard and Droz [9] and by Kapral, Lawniczak, andothers [24], [25], [26], [28]. In this paper, we will study ion migrations in both two andthree dimensions; the LCA models for both can be built similarly. Here we show howto build the three-dimensional LCA model, from which the two-dimensional modelcan be obtained. For simplicity, we choose a cubic lattice L, which is easy to handleand is generally sufficient to solve diffusion problems.

We assume that all particles have unit masses, move on the cubic lattice L, and


update on the lattice. Each node of the lattice is labeled by the discrete vectorr = (x, y, z), and each particle is associated with discrete directional length vectorsci, i = 1, 2, 3, 4, 5, 6, which point in one of six possible directions on the lattice.The ci are vectors connecting the node to its nearest neighbors, where i = 1, 2, 3correspond to the directions along the positive x-, y-, z-axes, respectively; specifically,c1 = ε(1, 0, 0), c2 = ε(0, 1, 0), c3 = ε(0, 0, 1), where ε is the lattice link length. Theother three directions are defined according to the relation

ci+3 = −ci, i = 1, 2, 3.

Let S be the set of all vectors s = (s1, s2, s3, s4, s5, s6) such that si is a nonnegativeinteger. A configuration of particles at node r at time t can be described by a vectorn(r, t) = (n1, n2, n3, n4, n5, n6)(r, t) with values in the state space S. The valueni(r, t) = m ≥ 0 indicates that there are m particles at the node r at time t that willmove in the direction ci. A configuration nL(t) of the lattice L at time t is describedby the field

nL(t) = n(r, t) : r ∈ L, n ∈ S.The evolution of this LCA model is a sequential change of configuration nL(t) of

the lattice at discrete time steps t = 0, τ, 2τ, . . . . For each configuration change, ourmodel proceeds with three successive steps at each node: particle injection, particlerotation (or collision), and propagation.

Injection operator I. During the injection step, particles are placed at nodeson the lattice. We define the injection operation as

Ii : ni(r, ·) −→ ni(r, ·) + qi(r, ·),(2)

where qi represents the number of particles injected and qi/τ is the injection rate ofthis substance per unit time step τ at node r at the time t that is moving in thedirection i.

The particle injection step tries to mimic the experimental situation of injection ofions into the brain-cell microenvironment from a micropipette. For this application,a constant injection amount qi of the substance for some time period will suffice.Without loss of generality, we always choose an extracellular node near the center ofthe lattice L as the position where substances are injected.

Rotation (collision) operator R. At each node r, each incoming particle hassix possible directions along which it can leave. The operation Ri, i = 1, . . . , 6,deflects the particle entering in the direction ck into the direction ci+k, k = 1, . . . , 6.We assume the rotation operator R picks out an angle specified by one of the Ri, andall particles entering a specific node rotate by this angle. Each node on the latticeL has an independently chosen rotation. Mass is conserved at each node, but themomentum is not. The probability of a rotation Ri is pi, i = 1, . . . , 6, where the piare subject to the normalization condition

p1 + p2 + p3 + p4 + p5 + p6 = 1.

To explicitly construct the rotation operator, we let ξi(r, t), r ∈ L, t = τ, 2τ, . . . ,be independent sequences of identically distributed, independent, Bernoulli-type ran-dom variables, i.e., ξi is either 1 or 0, satisfying

ξ1 + ξ2 + ξ3 + ξ4 + ξ5 + ξ6 = 1,


where the probability that the ith variable ξi is 1 is pi. Using these variables, we thencan write the rotation operator R as

Ri : ni −→ nCi =6∑j=1

ξjni+j

= ni +6∑j=1

ξj(−ni + ni+j)

≡ ni +4Ci (n),(3)

where 4Ci (n) is the collision function. It is understood that the index is modulo 6.Propagation operator S. After particles rotate, then they propagate. In this

propagation step, each particle moves from its present node r in the direction ci toits nearest-neighbor node r + ci. The propagation can be expressed as

Si : ni(r, t) −→ ni(r + ci, t+ τ), i = 1, 2, . . . , 6;

S also is called the streaming operator.Thus each configuration change, S ·R · I, of the automata is specified by

ni(r + ci, t+ τ) = ni(r, t) + qi(r, t) + ∆i((n + q)(r, t))

=6∑j=1

ξjni+j(r, t) +6∑j=1

ξjqi+j(r, t).(4)

This is the lattice gas microdynamical equation; it is fully discrete in space, time, andconcentration. The propagation and rotation operations are intended to describe thefree streaming and collisions that occur during diffusion.

The occupancy variable ni in the lattice gas microdynamical equation is a non-negative integer and can be noisy in a computation. To avoid this, we use its averageNi = E(ni), where E denotes the average (for details, see [18]) and Ni(r, t) is anonnegative real number. The average of (4) gives

Ni(r + ci, t+ τ) =6∑j=1

pjNi+j(r, t) +6∑j=1

pjQi+j(r, t),(5)

where Qi is the average of the source qi, i.e., Qi = E(qi). Equation (5) is the LBE forour system. Note that this derivation is simpler than for the hydrodynamic case in[18] because the Bernoulli-type random variables ξi are independent of the occupationnumbers ni and the injection amount qi.

Introduce a variable density by

C(r, t) =6∑i=1

Ni(r, t).

For our application, we will assume that all particles entering a node will rotateaccording to Ri with equal probability, i.e.,

p1 = p2 = p3 = p4 = p5 = p6 = 1/6.


Thus, we can rewrite the LBE (5) as

Ni(r, t+ τ) =1

6

6∑j=1

Ni+j(r− ci, t) +1

6

6∑j=1

Qi+j(r− ci, t).

Summing over i, we have

C(r, t+ τ) =1

6

6∑i=1

C(r− ci, t) +1

6

6∑i=1

6∑j=1

Qi+j(r− ci, t).(6)

Taylor expanding C(r− ci, t), C(r, t+ τ) about (r, t), we obtain

C(r− ci, t) = C(r, t)− ci · ∇C(r, t)(7)

+1

2∇ · ((ci · ∇C)ci)(r, t) + h(C, ci)(r, t) +O(ε4),

C(r, t+ τ) = C(r, t) + τ∂C

∂t(r, t) +O(τ2),(8)

where h(C, ci) is a function determined by a third-order Taylor expansion and ε isthe lattice-link length.

We assume that τ is of order ε2 and let the diffusion coefficient be given by

D =1

6τε2.(9)

Expanding (7) and (8) to second order in ε, (6) becomes the diffusion equation

∂C

∂t(r, t) = D∇2C(r, t) + f(r, t),(10)

where the source term is defined by1

f(r, t) = limε→0

6D

ε2

6∑i=1

Qi(r, t).(11)

Derivation of the diffusion equation (10) implies that the microdynamical equation(4) and the microscopic dynamical LBE (5) describe the diffusion phenomenon withinthe pore space on the continuous macroscopic level. The LBE (5) is continuous inthe density variable, whereas it is discrete in time and space. It can be related toa finite-difference scheme for a continuous equation. In fact, (6) is related to theforward-time central-space finite-difference scheme [42] for (10).

The two-dimensional LCA model can be built in the same way as the three-dimensional model except for dimension-related changes, e.g., a square lattice ratherthan a cubic lattice. The discrete direction-length vectors are ci = ε(cos((i− 1)π/2),sin((i− 1)π/2)), i = 1, 2, 3, 4, and the two-dimensional LBE has the form

Ni(r + ci, t+ τ) =4∑j=1

pjNi+j(r, t) +4∑j=1

pjQi+j(r, t).

1The limits in (11) and in (12) do not exist; however, with our choices of the injection amountQi in the next section, the function f is related to the Dirac delta function.


The diffusion coefficient is given by D = ε2/(4τ) and the corresponding source termis given by

f(r, t) = limε→0

4D

ε2

4∑i=1

Qi(r, t).(12)

3. Numerical procedures and membrane boundary condition. Since thelattice gas equation (4) results in noisy solutions when it is applied to diffusion withinthe ECS, instead we will use the LBE (5), which also can be split into three steps,i.e., particle injection, rotation, and propagation. During the particle injection androtation steps, a boundary condition can be implemented by choosing a suitable col-lision rule at a boundary node. Since these two operations occur locally and do notdepend on other nodes, the geometry of the membrane boundary is not important.Thus, the LBE is capable of efficiently handling complicated boundary geometries,which are difficult to incorporate when using conventional numerical methods such asthe finite-difference scheme (6).

To perform the numerical simulations, we first set up the porous medium in threedimensions. We indicate the alterations necessary for the two-dimensional compu-tations. The pore space of the porous medium can approximate any kind of braingeometry by choosing the lattice link length ε to be sufficiently small. To construct aporous medium, we use an integer variable to represent whether the node is in extra-cellular (void) or in intracellular (solid) space, or is at the membrane. The membraneof each cell always is chosen to be along the lattice links connecting the nearest mem-brane nodes, so the membrane forms a closed surface which separates the ICS fromthe ECS. Each node on the membrane is called a membrane boundary node, whereasthe ECS node with at least one link connected to a membrane boundary node is anECS boundary node.

Rothman [40] defined the porosity of a porous medium as the number of latticenodes in the pore space divided by the total number of lattice nodes. Here we needa more precise definition, so half the membrane boundary nodes will be counted asbeing in the pore space. This definition of porosity corresponds to the ratio betweenthe total volume of the pore space and the total volume of the medium. On theother hand, the volume fraction is defined as a local quantity by α = V0/V , whichis not the same as the porosity because the representative elementary volume V isnot necessarily the total volume of the medium [3]. We will call α the local volumefraction and the volume fraction of the medium will be referred to as the averagevalue of the local volume fractions over all the pore space. In this paper, the porosityand the volume fraction of a medium are treated as distinct parameters.

When we generate various porous media for our numerical experiments (details aregiven in sections 5 and 6), we always place the center of the lattice in the pore space,without loss of generality. To match Nicholson and Phillips’s experimental situation,we assume that particles are injected at a constant rate for some time period T onlyat the center node r = r0 of the lattice L, i.e., Qi(r, t) = Q/6 if r = r0, t ≤ T ;Qi(r, t) = 0 otherwise, i = 1, . . . , 6. Thus, the source term (11) becomes

f(r, t) =

limε→0 6DQ 1

ε2 if r = r0, t ≤ T,0 otherwise.

Let

d(r) =

1/ε3 if r = r0,0 otherwise.


Then,∫d(r)d3x = 1 because the volume of each basic cube associated with each

node of the lattice L is ε3. As ε → 0, the function d(r) tends to the Dirac deltafunction δ(r− r0), but the mass 6DεQ injected per unit time per unit volume tendsto 0. In the numerical simulations, ε cannot be 0; therefore, for the three-dimensionallattice, we have

f(r, t) ≈ 6DεQδ(r− r0)H(T − t),(13)

where H is the Heaviside step function.For the two-dimensional lattice, we choose Qi(r, t) = Q/4 if r = r0, t ≤ T ;

Qi(r, t) = 0 otherwise, i = 1, . . . , 4. In two dimensions, the function

d(r) =

1/ε2 if r = r0,0 otherwise

tends to the two-dimensional Dirac delta function δ(r− r0) as ε → 0. Thus, thesource term (12) becomes

f(r, t) = 4DQδ(r− r0)H(T − t).(14)

Note that (13) for three dimensions is only an approximation, whereas (14) for twodimensions is an equality. Also, the small parameter ε in (13) does not occur in (14).

To simulate the LBE (5), we need initial conditions and boundary conditions onthe lattice L. The initial condition Ni(r, 0) is set equal to zero at each node of thelattice. Since this problem involves only linear diffusion in the ECS, we need notinclude any background density of ions. We choose an absorbing boundary conditionfor the lattice L, i.e., once the particle reaches the boundary, it is absorbed and canno longer return to L. This is equivalent to setting the Ni(r, t) equal to zero.

In order to obtain the modified diffusion equation (1), it was assumed that ionscannot cross the membrane [33], i.e., there is zero flux of ions. At the ECS boundarynode r, at least one of the nodes r + ci, i = 1, . . . , 6, is a membrane node. Thus,one or more of the Nj(r + ci, t) may not be well defined. To achieve the zero-fluxcondition, we can use a bounce-back boundary condition, i.e., when a particle hits amembrane node, the particle will bounce back. In terms of the concentration variableC, this is represented for the case of one membrane node, say, r + c1, as

C(r + c1, t) = C(r, t).(15)

Thus, the concentration at r at time t+ τ is given by

C(r, t+ τ) =1

6

6∑i=2

C(r + ci, t) +1

6C(r, t).(16)

Taylor expanding (16) at node r and at time t, we obtain at the ECS boundary node

Cx(r, t) =

1

2Cxx + Cyy + Czz − Ct

D

(r, t)ε+O(ε2).(17)

(Note the factor 1/2 in front of Cxx.)Alternatively, a zero-flux condition is achieved by the reflection condition

C(r + c1, t) = C(r− c1, t),(18)


which implies that

C(r, t+ τ) =1

6

6∑i=2

C(r + ci, t) +1

6C(r− c1, t).

Again using a Taylor expansion, we obtain

Cx(r, t) =1

2

Cxx + Cyy + Czz − Ct

D

(r, t)ε+O(ε2).(19)

Thus, at the ECS boundary node r, to approximate the zero flux using either thebounce-back condition or the reflection condition to second order, we need either

Ct = D

(1

2Cxx + Cyy + Czz

)(20)

or

Ct = D(Cxx + Cyy + Czz),(21)

respectively. Here, we use the bounce-back condition for the following two reasons.First, when using the LBE, the bounce-back condition can be obtained by simplyaltering the local rotation operation (probabilities), i.e., by deflecting the particleentering in the direction ck into the direction ck+3, k = 1, . . . , 6. To achieve thereflection condition, we have to alter both the local rotation operation and the propa-gation operation so the information needed occurs at the node r and at its neighbors.Thus, the local geometry of the medium is needed, and the computer code is moredifficult to write. (This is a major advantage of the LBE over the finite-differencemethod.) Second, the bounce-back condition conserves the mass at each iteration,whereas the reflection condition does not. With the reflection condition at r + c1, anamount of mass C(r + c1, t)− C(r, t) = O(ε2) would be added at each iteration.

The possibility of a Knudson layer effect has been raised by Cornubert, D’Humieres,and Levermore [10]. However, they have shown that with a bounce-back boundarycondition with the lattice link perpendicular to the boundary, no Knudson layer isgenerated.

From (5) or (6), it is obvious that with the choices of initial condition, the bounce-back membrane boundary condition, and Qi(r, t), the solution is proportional to Qwith proportionality coefficient dependent on r and t.

4. Tortuosity and volume fraction calculations. The assumptions underwhich the numerical simulations can be performed in the last section match thoseused to derive the modified diffusion equation (1), which should describe the volumeaverage 〈C〉0 of the numerically simulated solution. We can use the experimentalnumerical results to compute the tortuosity and the volume fraction of the porousmedium, as done by Nicholson and Phillips [33].

In the LBE, the concentration C is discrete in space and the volume average 〈C〉0at a node r = (x, y, z) can be defined as the sum of the Cs over the neighboringnodes of r divided by the total number of neighboring nodes used for averaging. Forexample, we can choose M ×M ×M neighbors centered at r. Thus, if the set of thesecubic neighbor nodes is denoted by N , then the volume average of C can be writtensymbolically as

〈C〉(r, t) =1

M3

∑q∈N

C(q, t).


Similar to the representative elementary volume V , the number M is a characteristicparameter of the porous medium, and it is difficult to choose an appropriate value.To avoid this difficulty, Nicholson and Phillips (see Appendix, [33]) showed that thesolution to (1) given by 〈C〉0 can be approximated by the solution u to the equation

D

λ2∇2u+

q

α= ut

in a simple medium. Thus, adding homogeneous initial and homogeneous Dirichletboundary conditions, the volume average 〈C〉0 of the numerical simulation using theLBE (5) is approximated by the solution of the system

ut =D

λ2∇2u+ f,

u(x, y, z, 0) = 0, 0 ≤ x, y, z ≤ 1,(22)

u(x, y, z, t) = 0, (x, y, z) ∈ ∂Ω,

where ∂Ω is the boundary of the unit cube Ω = (x, y, z) : 0 ≤ x, y, z ≤ 1 and

f(x, y, z, t) =Qαδ

(x− 1

2

)δ

(y − 1

2

)δ

(z − 1

2

)H(T − t).

Here Q is the source constant, which is approximately 6DεQ for the three-dimensionallattice and is 4DQ for two dimensions.

The Dirac delta function δ(x) cannot be represented by its corresponding Fourierseries expansion, and the analytical solution of the above system (22) is a generalizedsolution. The generalized solution is not convergent, for example, at x = 1/2, y = 1/2,and z = 1/2 and, consequently, is not suitable for computation of the tortuosity andvolume fraction. Therefore, we seek an asymptotic solution. The delta function δ(x)can be approximated by

ϕσ(x) =1√πσ

e−x2/σ2

(23)

as σ tends to zero. Thus, the source term f(x, y, z, t) can be approximated by

fσ(x, y, z, t) =Qαϕσ

(x− 1

2

)ϕσ

(y − 1

2

)ϕσ

(z − 1

2

)H(T − t).

The analytical series solution of the above diffusion system (22) with the sourceterm fσ(x, y, z, t) instead of f(x, y, z, t) for 0 ≤ t ≤ T is

u(x, y, z, t) =8Qλ2

Dαπ2

∞∑l=1

∞∑m=1

∞∑n=1

alaman

(l2 +m2 + n2)(24)

(1− exp

(−D(l2 +m2 + n2)π2t

λ2

))sin(lπx) sin(mπy) sin(nπz)

,

where 8Qalaman/α, l, m, n = 1, 2, . . . , are Fourier coefficients of the function fσand each an is given by

an =

∫ 1

0

ϕσ

(x− 1

2

)sin(nπx)dx(25)

= 2 sin(nπ

2

)∫ 12

0

ϕσ(x) cos(nπx)dx

= sin(nπ

2

)e−(nπσ)2/4 + sin

(nπ2

)e−1/(4σ2)p(n, σ)


for any σ and n where |p(n, σ)| ≤ 1. Note that the second term on the right is muchsmaller than the first term when σ is small.

Setting

U(x, y, z, t) =u(x, y, z, t)

8Q , β =Dπ2

λ2, γ = α

Dπ2

λ2= αβ(26)

yields

γU(x, y, z, t) =∑∞l=1

∑∞m=1

∑∞n=1

alaman sin(lπx) sin(mπy) sin(nπz)l2+m2+n2

− ∑∞l=1

∑∞m=1

∑∞n=1

alaman sin(lπx) sin(mπy) sin(nπz)l2+m2+n2 e−(l2+m2+n2)βt.

(27)If t is sufficiently large, and since an = 0 for even integer n, then using only the

first term of (25), we have the approximation

γU(x, y, z, t) ≈B0(x, y, z)−2∑l=0

2∑m=0

2∑n=0

(−1)l+m+n

(2l + 1)2 + (2m+ 1)2 + (2n+ 1)2

×e−((2l+1)2+(2m+1)2+(2n+1)2)(βt+π2σ/4)(28)

× sin((2l + 1)πx) sin((2m+ 1)πy) sin((2n+ 1)πz)

,

where

B0(x, y, z) =N∑l=0

N∑m=0

N∑n=0

(−1)l+m+ne−((2l+1)2+(2m+1)2+(2n+1)2)π2σ2/4

(2l + 1)2 + (2m+ 1)2 + (2n+ 1)2

× sin((2l + 1)πx) sin((2m+ 1)πy) sin((2n+ 1)πz)

.

Here N depends on the size of σ and on the desired accuracy of (28). If we denotethe error in using (28) compared to (27) for t = T0 by E(N,T0), then we have (seethe web page http://www.math.ubc.ca/∼miura/)

E(N,T0) ≤ E0(N,T0) + 3e−1/(4σ2)

+3

√π

3+ 8π

√(πσ/2)2 + βT0 + 16

√π((πσ/2)2 + βT0)

44 × 40√

(πσ/2)2 + βT03 e−40((πσ/2)2+βT0),(29)

where E0(N,T0) is the smaller of

3(√π)3 + 4π2σ + 4π2

√πσ2

32(N2 + 3N)π3σ3e−(N2+3N)π2σ2

(30)

and

3e−((2N+2)2+2)π2σ2/4

| sinπ(x− 12 ) sinπ(y − 1

2 ) sinπ(z − 12 )|((2N + 2)2 + 2)

.(31)

To achieve a small error (29), we need all three terms on the right side to besmall. The first term is small when either (30) or (31) is small. When σ is small, the


second term is small. To have the third term small, we need T0 to be large enoughsuch that e−40((πσ/2)2+βT0) is small. This actually means that the values of tortuosityand volume fraction can be computed accurately only after a long injection time T0

when the injected particles are fully distributed in the medium.For a given small σ and a given N , if (x, y, z) is a node such that, for example,

| sinπ(x− 12 ) sinπ(y− 1

2 ) sinπ(z− 12 )| ≥ σ, then (31) is smaller than (30). As | sinπ(x−

12 ) sinπ(y− 1

2 ) sinπ(z− 12 )| becomes larger, the error E(N,T0) becomes smaller, which

implies that E(N,T0) is smaller for those nodes that are farther away from the planesx = 1/2 or y = 1/2 or z = 1/2. In order to get a uniform error E(N,T0) for all nodes,N must be chosen larger for nodes that are close to or on the planes than for pointsthat are further away.

From the numerical simulations, the concentration C within the pore space can beobtained at each nodal point. Since the LBE simulates the diffusion process within thepore space with the zero-flux membrane boundary conditions and the zero boundarycondition for the lattice, the solution of system (22) can be approximated by C, i.e.,u ≈ C [33]. To test this assumption, we can perform simulations with a simple medium(to represent agar in the experimental situation). Using Nicholson and Phillips’sformula (equation 20, [33]), which is valid only for short times in our medium becauseof different boundary conditions, we obtain values for the diffusion coefficient. In oursimulations, we choose the value D = 1× 10−5 cm2/s, whereas from the simulationsbased on Nicholson and Phillips’s formula, D is 1.12926× 10−5 cm2/s.

With the approximation u ≈ C we can calculate U(x, y, z, T0) and U(x, y, z, T )through (26) at the nodes of L. Then we compute at each node (x, y, z) withinthe ECS by solving (28) using the Newton–Raphson method at the times T0 < T .Finally, (26) yields the local tortuosity λ and the local volume fraction α at eachnode approximately by the combination of the asymptotic solution of (22) and thenumerical simulation of the LBE (5). The tortuosity and volume fraction of the porousmedium then can be obtained by averaging λ and α over all ECS points.

For two-dimensional porous media, the tortuosity and the volume fraction can becomputed in exactly the same way as for the three-dimensional case except we usetwo-dimensional versions of (22), (24), (28) with Q = 4DQ.

As a test of our assumption that the experimental numerical solution shouldmatch the solution of (22), we performed the computations for simple agar media intwo and three dimensions for which both the tortuosity and volume fraction are 1 [33].The resulting tortuosity and volume fraction were 0.9999 and 1.0003, respectively fora two-dimensional simple medium consisting of a 100 × 100 lattice with the choicesof σ = 0.01, N = 100, Qi = 3, NT0

= T0/τ = 9500, and NT = T/τ = 10000, and1.0008 and 1.0005, respectively for a three-dimensional simple medium consisting ofa 50 × 50 × 50 lattice with σ = 0.01, N = 100, Qi = 3, NT0 = T0/τ = 4500, andNT = T/τ = 5500.

5. Two-dimensional porous media. Two-dimensional porous media with aporosity as small as 0.2 could not be generated randomly. Chen et al. [8] generatedporous media in two dimensions with a connected porous space, but these had volumefractions above 0.5. Here, we will generate porous media with connected ECS andporosity as small as 0.2 in the following ways.

Since the aim here is to study how the geometrical properties of the mediumaffect the tortuosity and the volume fraction, we will perform computations for someregular shapes, e.g., rectangular cell shapes that are in a periodic arrangement, and forirregular cell shapes and arrangements. A type-one medium has square cells aligned in


(a)

(b)

Fig. 1. A schematic representation of two-dimensional porous media of type two (a) and typethree (b).

two directions, and a type-two medium has rectangular cells aligned in one directionand staggered in the other direction; see Figure 1(a).

We construct a type-three medium (see Figure 1(b) for example) as follows. Togenerate the pore space with width of, say, three lattice links, we choose three adjacentnodes on the bottom boundary, i.e., along the x-axis, of the lattice L. These threenodes are advanced one step in the y-direction and kept adjacent, but they can shifteither one node to the left or one node to the right, or not shift at all. This shiftor lack of shift is determined using a random g number generator which produces asequence of numbers ηi, i = 1, 2, . . ., between 0 and 1. If either 0 ≤ ηi < 0.4 or0.6 < ηi ≤ 1, then the shift is to the left or right, respectively, and if 0.4 ≤ ηi ≤ 0.6,then there is no shift. This process continues until the three nodes exit the latticeeither through the sides or through the top. The resulting path becomes the pore


space of the medium. This procedure is then used starting on the y-axis of the latticeL. If the resulting porosity is smaller than the desired value, then is repeated thiscycle. Also, we can always place the center of the lattice in the pore space.

Figure 2 shows the concentration profiles versus time at different positions forthree porous media of different types. The solid lines are the numerical results of theLBE model, and the dashed lines are the asymptotic solution of system (22) withσ = 0.01. All these figures are obtained for 100×100 lattices, and the computationshave been carried out until time step 40000. During each time step, three unit particlesare injected on each link of the center node (50, 50) until time step 20000. The specificnode position for each curve is shown in the figure.

Figure 3 shows the tortuosity versus volume fraction for various porous media.Computations have been done with 100×100 lattices and an injection rate Qi = 3 oneach link of the central node; the numbers of the time steps NT0

and NT were setequal to 9500 and 10000, respectively.

We computed the averaged tortuosity and volume fraction based on only thosenodes that were approximately 10, 20, 30, 35, 40, and 45 lattice-link distances fromthe central node. Figures 3A and 3B are for the porous media of types one and two,respectively, where the effects of the widths of the ECS are considered. The data arefrom two sets of media with ECS widths equal to three and four lattice links.

Figure 3C shows the results for type-three media with ECS width equal to threelattice links. When we generate type-three media, we use a random-number generator.To start the random-number generator, we have to give an initial value. Differentinitial values may lead to a different sequence of numbers, ηi, i = 1, 2, . . ., andconsequently a series of different porous media. In Figure 3C, (a), (b), (c), and (d)are the tortuosities versus volume fractions for four different series of media producedby four different initial values.

Figure 4 shows plots of the porosity versus the volume fraction for the corre-sponding media used in Figure 3. The average absolute difference and maximumdifference between the porosity and the volume fraction plotted in Figure 4 for mediaof types one, two, and three used in Figure 3 are monotone increasing and range over0.032–0.039 and 0.067–0.085, respectively. The average absolute difference and themaximum difference are taken over the different media of the same type, and theaverage absolute difference is computed from

average absolute difference =1

Nm

∑i

|αi − ρi|,

where the sum is taken over media i of the same type, Nm is the total number ofmedia used for averaging, and ρi is the porosity of medium i.

6. Three-dimensional porous media. In this section, we discuss the genera-tion of three types of three-dimensional porous media and present the results.

There is a set of media studied by El-Kareh et al. [16] that we call type-onemedia. Each medium is composed of uniform rectangular parallelopiped cell shapes,which are arranged in ordered periodic arrays where the widths of the connected ECSchannels are uniformly constant (see Figure 5). Type-one (a) media have the cellsaligned in two directions and elongated in one direction; type-one (b) media have thecells aligned, staggered, and elongated in the three directions, respectively; type-one(c) media have the cells aligned in three directions; and type-one (d) media have thecells aligned in two directions and staggered in one direction.

We will generate type-two porous media in a way similar to that of Chen et al. [8].


0

50

100

150

0 10000 20000 30000 40000

Con

cent

ratio

n (C

)A (70,50)

(85,50)

(92,50)

LBE modelDIFF model

0

50

100

150

0 10000 20000 30000 40000

Con

cent

ratio

n (C

)

B(70,50)

(78,50)

(92,50)

LBE modelDIFF model

0

50

100

150

0 10000 20000 30000 40000

Con

cent

ratio

n (C

)

Time (t)

C

(50,70)

(50,78)

(90,80)

LBE modelDIFF model

Fig. 2. Concentration-time profiles at different positions for three porous media of differenttype in two dimensions. The solid lines are the numerical results of the LBE model, the dotted linesare from the modified diffusion equation. A. The medium is of type one with aligned cells with aporosity of 0.1931, tortuosity of 1.5849, and volume fraction of 0.1847. B. The medium is of typetwo with staggered cells with porosity of 0.1816, tortuosity of 1.6725, and volume fraction of 0.1785.The widths of ECS of the media in A and B are three lattice links. C. A medium of type one with thewidth of the ECS channel equal to three lattice links. The medium has porosity of 0.2228, tortuosityof 2.4027, and volume fraction of 0.2007.


1

1.2

1.4

1.6

0.2 0.4 0.6 0.8 1

Tort

uosi

ty

A

3 lattice links4 lattice links

1

1.2

1.4

1.6

0.2 0.4 0.6 0.8 1

Tort

uosi

ty

B


1

1.25

1.5

1.75

2

2.25

0.2 0.4 0.6 0.8 1

Tort

uosi

ty

Volume fraction

C

(a)(b)(c)(d)

Fig. 3. Tortuosity λ versus volume fraction α for various porous media in two dimensions. A.The media are of type one. B. The media are of type two. In A and B, one set of data is from themedia with the width of the connective ECS equal to three lattice links, and the other one is fromthe media with width equal to four lattice links. C. The media are of type three.

Initially, we set the entire medium to be void space, then we randomly distributevarious sizes of solid cubes inside. The larger solid cubes are distributed before thesmaller ones. Cubes that are distributed later can overlap those distributed previously.This procedure produces a medium with a porosity close to any desired value.

Type-three media are a mixture of type-one and type-two media. We first set up


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Poro

sity

A


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Poro

sity

B


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Poro

sity

Volume fraction

C

(a)(b)(c)(d)

Fig. 4. Porosity versus volume fraction for the corresponding media used in Figure 7. A. Themedia are of type one. B. The media are of type two. C. The media are of type three.

a medium of type one such that the medium has a porosity bigger or smaller than theporosity we want. Then we randomly put in solid or void cubic shapes such that themedium has the desired porosity.

Both type-two and type-three media may have dead ends or holes, i.e., isolatedECS regions, in them. Although such structures are not expected in either normal orpathological tissues, they may approximate tissues in some circumstances.

From the derivation, the tortuosity should not depend on the diffusion coefficient


(b)

(d)(c)

(a)

Fig. 5. A schematic representation of porous media in three dimensions of type one (a), (b),(c), and (d).

if the LBE is a good approximation of the averaged equation (1). However, in practice,we are unable to simulate thousands of cells, so the accuracy of the approximationis not guaranteed. Therefore, we computed the dependence of the tortuosity on thediffusion coefficient as a check on the accuracy of this approximation.

The dependence of tortuosities for three different three-dimensional media of dif-ferent types on diffusion coefficients is plotted in Figure 6. For the media of type one(c), the tortuosities computed for the diffusion coefficients (×105) 0.6, 0.8, 1.0, 1.2 are1.2655, 1.2654, 1.2581, 1.2682, respectively, and their biggest difference is 0.0101. Forthe media of type two, the maximum tortuosity difference is 0.0222. For the media oftype three, the tortuosity difference is less than 0.0906. In all these cases, the diffusioncoefficients have little influence on the tortuosities, so we apply the LBE to performsimulations on other media.

Figure 7 shows concentration profiles versus time at different positions for someporous media. All these figures were obtained for 50×50×50 lattices and computationswere carried out until time step 15000. During each time step, three unit particleswere injected on each link of the central node until time step 7500. The specific nodeposition for each curve is indicated. The dashed lines for the diffusion model are theasymptotic solutions of (22) with σ = 0.01.

Figure 8 shows plots of tortuosity versus volume fraction for various porous mediaof different types. All computations in the figure have been done with 50×50×50lattices and an injection amount Qi = 3 on each link of the central node. To computethe tortuosity λ and volume fraction α, the numbers of time steps NT0

= T0/τ andNT = T/τ were set equal to 4000 and 5500, respectively.

Both the tortuosity and the volume fraction may depend on the number of timesteps NT0 = T0/τ and NT = T/τ used in the computations. To make sure that thechoice of NT0 = T0/τ and NT = T/τ can guarantee the desired accuracy, we computedthe tortuosities for a medium of type one with porosity of 0.3000 and a medium oftype two with porosity of 0.2166. When NT − NT0

= 1000 or NT − NT0= 500 and

when NT is 5500 or 6000 or 7000, the corresponding tortuosity differences are lessthan 0.001.

Figure 9 shows plots of the porosity versus the volume fraction for the correspond-


Type threeType twoType one

D×10−5cm2/s

Tor

tuos

ityλ

1.210.80.6

2.6

2.4

2.2

2

1.8

1.6

1.4

1.2

Fig. 6. Tortuosity versus the diffusion coefficient for three different three-dimensional porousmedia of different type. The type-one medium has a porosity of 0.2166 and the ECS has a widthof three lattice links. The type-two medium has a porosity of 0.30. The type-three medium has aporosity of 0.20, which was produced from a type-one (c) medium with a porosity of 0.282, and thenrandomly inserting additional 2× 2× 2 cubes.

ing media used in Figure 8. The average absolute difference and maximum differencebetween the porosity and the volume fraction plotted in Figure 9 for the three differ-ent types of media used in Figure 8 are approximately 0.041–0.072 and 0.095–0.245,respectively. The maximum values occur for type-two media.

For Figures 6, 8, and 9, the local tortuosities and volume fractions are based onthose nodes that were approximately 10, 15, 18, 20, 23, and 25 lattice-link distancesfrom the central node.

7. Discussion and conclusions. In this study, we have mimicked the experi-ments carried out by Nicholson and his colleagues on diffusion in the brain-cell mi-croenvironment. Modeling diffusion of ions in the brain can be done more realisticallyby suitable choice of the medium and by appropriate specification of the detailed move-ments of particles in this medium. We have chosen the medium to be a lattice of nodesand specified particle movement according to the LBE, which accurately representsdiffusion. The LBE has been applied to other problems in simple media [11], [23],[24], [25], [26], [27], [28].

Here, we have shown that application of an LCA model to diffusion of particlesin the pore space of a porous medium is an effective method to study the diffusion ofextracellular ions in the brain. The numerical solution of the LBE gives the number ofparticles at each node in the ECS and is zero within the ICS. This discrete distributionrepresents the discontinuous ionic concentration values in the brain for ions that do notcross the cellular membrane. This is analogous to a porous medium consisting of porespace and solid space. The conditions imposed in the numerical simulations usingthe LBE satisfy the same assumptions as required to derive the modified diffusionequation (1).

Solving the diffusion equation with a singular function δ(x) as a source usingconventional methods would require us to approximate the delta function δ(x) aswe did in section 4 to compute the tortuosity and the volume fraction. Solving such


0

1

2

3

4

5

0 2000 4000 6000 8000 10000

Con

cent

ratio

n (C

)

A

(18, 26,18)

(13,26,16)

(5, 26,24)

LBE modelDIFF model

0

2

4

6

8

10

0 2500 5000 7500 10000

Con

cent

ratio

n (C

)

B

(18, 20,20)

(37,34,26)

(9, 17,32)

LBE modelDIFF model

0

1

2

3

4

5

6

0 2500 5000 7500 10000

Con

cent

ratio

n (C

)

Time (t)

C

(20, 20,32)

(13,24,16)

(5, 24,24)

LBE modelDIFF model

Fig. 7. Concentration versus time profiles at four different positions for three different types ofporous media in three dimensions. A. The medium is of type one (c) with porosity 0.217, tortuosity1.2968, volume fraction 0.2026. The width of the ECS channel is three lattice links. B. A mediumof type two with porosity 0.28, tortuosity 1.9601, volume fraction 0.2784. The porous medium wasgenerated by first inserting 10× 10× 10 cubes, then 8× 8× 8 cubes, . . . , finally 2× 2× 2 cubes. C.A medium of type three with porosity 0.20, tortuosity 1.7348, generated by first obtaining a mediumof type one (c) with porosity 0.249, then randomly inserting 2 × 2 × 2 cubes until the medium hasporosity 0.20.


1

1.1

1.2

1.3

1.4

0.2 0.4 0.6 0.8 1

Tort

uosi

ty

A

(a)(b)(c)(d)

1

1.4

1.8

2.2

2.6

0.2 0.4 0.6 0.8 1

Tort

uosi

ty

B

(a)(b)(c)(d)

1

1.5

2

2.5

3

0.2 0.25 0.3 0.35

Tort

uosi

ty

Volume fraction

C

(a)(b)(c)(d)

Fig. 8. Tortuosity λ versus volume fraction α for various porous media in three dimensions.(Note the different scales for tortuosities.) A. Media of type one (a), (b), (c), (d). The width of theECS channels is three lattice links. B. Media of type two. (a) is for a set of media produced by firstinserting 10× 10× 10 cubes, then 8× 8× 8 cubes, . . . , finally 2× 2× 2 cubes until the porosity is thedesired value; (b) is for media obtained by first inserting 8×8×8 cubes; (c) is for media obtained byfirst inserting 12× 12× 12 cubes, then 10× 10× 10 cubes, and so on; (d) is the plot of the functionλ = α−β/2 with β = 0.9939. C. Media of type three. (a) and (b) are for the media produced by firstsetting the medium of type one (d) with porosity 0.339 and 0.217, then inserting 2× 2× 2 cubes; (c)and (d) are for the media produced by first setting the media of type one (c) with porosity 0.333 and0.217, respectively.


0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Poro

sity

A

(a)(b)(c)(d)

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Poro

sity

B

(a)(b)(c)

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6

Por

osity

Volume fraction

C

(a)(b)(c)(d)

Fig. 9. Porosity versus volume fraction for the corresponding porous media used in Figure 3.A. The media of type one. B. The media of type two. C. The media of type three.

equations using the LBE can be achieved easily by a simple suitable choice of the Qi aswe did in section 3. In addition, the LBE is well suited to deal with arbitrary boundarygeometry and with singularities such as point sources or sinks in two dimensions andline sources or sinks in three dimensions.

We believe this study is the first application of the LBE to porous media thatincorporates particle injection, rotation (collision), and propagation as three distinctsteps in implementing the LBE. Particle injection into the medium mimics the ion-


tophoretic injection of ionized particles in brain tissue.The LBE method developed here consists of injecting particles for some time

period into the central node (taken to be in the ECS) of our lattice. During thisprocess, the particles are moved at each time step according to the rotation andpropagation operations. This produces a record of the concentration of particles ateach nodal point at each discrete time. The concentrations at two different times arecombined with the results from porous media theory, developed by Nicholson and hiscolleagues, to evaluate the local tortuosity and local volume fraction. Averaging thesequantities over the entire pore space of the lattice yields the tortuosity and volumefraction for the entire medium. Using these quantities in the porous media theorythen permits us to compare the time evolutions of the local concentration as computedfrom the LBE model and from porous media theory. We find this comparison to bevery good, and the accuracy of the comparisons is similar to that obtained betweenexperimental data and porous media theory. In addition, we can use the LBE methodto study how the geometrical properties affect the tortuosity and volume fraction,which are difficult to achieve by experiments on brain tissue.

We found that the tortuosity of a medium is affected by its geometry. The dif-ferences in the tortuosities among those regular media of approximately the samevolume fraction are not sufficiently small to be neglected. The difference between thetortuosities of a regular medium and an irregular medium and among the irregularmedia can be large. Other researchers (e.g., [16]) have concluded that the cell shapeand the arrangement in regular media with volume fractions in a particular rangehave little effect on the effective diffusion coefficient and, hence, on the tortuosity.

Irregular shapes should increase the tortuosity because a particle diffusing throughsuch a medium zigzags along a pathway longer than in a medium with regular shapes.Indeed, this is evident by comparing the tortuosities of three-dimensional type-onemedia with those of type two (Figure 8A, B). A medium of type one consisting ofregular shapes with porosity around 0.2 has a tortuosity less than 1.4. In contrast, amedium of type two with a porosity around 0.2 has a tortuosity as large as 2.4 whenit is generated randomly. The resulting cell shapes are irregular, and the widths ofthe ECS channels are not constant.

The effect of geometry on the tortuosity also is evident with two-dimensionalmedia (Figure 3). A medium of type one or two with a volume fraction of 0.20 hasa tortuosity less than 1.57, whereas a medium of type three has a tortuosity around2.0. That the staggered arrangement also will increase the tortuosity is evident bycomparing Figure 3A with Figure 3B. Increasing the widths of the connected ECSchannels decreases the tortuosity as can be seen in Figure 3A and Figure 3B.

Even with uniform cell size and a periodic arrangement, the effects of cell ar-rangement and width of the ECS channel are noticeable. A medium with a staggeredarrangement should have a larger tortuosity than one with an aligned arrangement.Thus, three-dimensional type-one (b) media should have the highest tortuosities, and(c) should have the lowest tortuosities, as confirmed in Figure 8A.

The tortuosity of the brain is about 1.6 as determined from experimental dataand porous media theory [33], [35]. The ECS of the brain is connected, but the porespace of three-dimensional media of types two and three may have regions that areisolated from the rest of the pore space. Such types of media are not similar to thebrain and, as expected, the calculated tortuosities using the LBE with these mediaare much larger (Figure 8). For example, three-dimensional media of type three (a)were produced by constructing a medium of type one (d) with a porosity of 0.339 and


then inserting additional solid cubes until the medium had the desired porosity. Wecan see from Figure 8C that as we put in more solid cubes, more ECS channels areblocked, so the corresponding tortuosities become larger. Constructing a medium inthis way with a porosity 0.20 can result in a tortuosity as large as 10.

In this paper, the porosity is defined as the proportion of the tissue that the ECSis composed of, whereas the volume fraction is computed by fitting the experimentalnumerical data with the modified diffusion equation (1), so porosity and volume frac-tion are not necessarily identical. This is especially true for those media with irregularcell shapes and arrangements. Sometimes the difference between them can be large.

The three-dimensional media of types two and three are highly irregular. Forthese media, it is possible to have a large “hole,” i.e., totally isolated regions from theremaining ECS, or a dead end. This means that for holes, these nodes are actuallypart of the ICS when the local volume fraction is computed, and the computed volumefraction could be thought of as the ratio between the volume of the major connectedpart of the ECS and the total volume of the medium. If there are many unconnectedextracellular regions, then the volume fraction and porosity will differ greatly (Fig-ure 9). The average absolute difference between the porosity and the volume fractionfor three-dimensional media of type two is 0.0717 . Also large fluctuations of thedifference between porosity and volume fraction around the average absolute value(Figure 9B, C) can occur; one extreme case is a medium of type two with this dif-ference as large as 0.2450. The average absolute difference over the type-three mediain Figure 9C is 0.0541. Similar to the tortuosity, as we block more of the ECS chan-nels, the difference between the porosity and the volume fraction becomes larger. Forexample, a medium that was generated by first setting the medium to be type one(c) with porosity 0.333, then randomly putting in additional 2×2×2 cubic cells, hasa porosity 0.2, but has a volume fraction 0.08. The corresponding tortuosity is 2.75.

The two-dimensional media of types one, two, and three have neither holes nordead regions. The computed average absolute differences between the porosity andvolume fraction are even smaller for most of these media and have small fluctuations(Figure 4). The two-dimensional media of types one, two, and three were generatedby aligning the rectangular cells in two directions, by aligning the cells and staggeringthem in each direction, and by using our algorithm for the random generation ofthe pore space, respectively. The average absolute differences between the porosityand the volume fraction for these three types of media are 0.0319, 0.0369, 0.0390,respectively. In spite of the small differences between these three values, an importantresult is that as the medium becomes more irregular, the difference between theporosity and the volume fraction becomes larger.

The tortuosity versus volume fraction in two dimensions and three dimensionsis shown in Figures 3 and 8, respectively. Our results show that as the volumefraction decreases, the tortuosity increases in general. However, the rate of changeof the tortuosity with respect to the volume fraction is not the same for all types ofmedia. For two-dimensional type-three media, the rate of change of tortuosity withrespect to the volume fraction is almost a constant, i.e., the relationship between thetortuosity and the volume fraction can almost be fitted by a straight line. However,this relationship is not true for other types of media.

For two-dimensional type-one and type-two media and three-dimensional type-onemedia, as the volume fraction decreases from 1.0 to 0.7, the corresponding tortuositychanges dramatically from 1.0 to 1.3. As the volume fraction decreases further to0.4, the rate of change of tortuosity with respect to the volume fraction becomes


relatively smaller. Finally, as the volume decreases from 0.4 to 0.2, the tortuositydoes not change significantly. Thus, for those regular media, the relationship betweenthe tortuosity and volume fraction can be fitted to a function that is concave down.

For three-dimensional media of types two and three, as the volume fraction de-creases from 1.0, the tortuosity gradually increases. As the volume fraction changesfrom 0.4 to 0.2, the tortuosity increases rapidly. It seems that the tortuosity will tendto infinity as the volume fraction tends to zero.

As we know, the volume fraction can change from 0.2 to 0.4 during ischemia [37],hypoxia [39], and postnatal development [29], but tortuosity does not change signifi-cantly. However, during postnatal development after X-irradiation at postnatal daystwo to five [43], the volume fraction changes from 0.2 to 0.48 and the tortuosity de-creases significantly. We cannot reproduce these results using only three-dimensionaltype-one, i.e., regular, media, but we can reproduce them using media of types twoand three.

With fixed diffusion coefficients, tortuosity and the volume fraction have stronginfluences on the time course and amplitude of the extracellular concentration ofparticles; this is evident for two- and three-dimensional media (see [2] and Figure 7,respectively). At an ECS node, a large tortuosity implies a small rate of change ofconcentration (Figures 2C and 7), whereas a small tortuosity leads to a rapidly risingor decaying concentration profile (Figures 2A and 7). On the other hand, if the rateof change of the concentration is large, the computed tortuosity will be closer to 1.

In this paper, we have built an LCA model for diffusion within the ECS of thebrain and performed numerical simulations. As an application of the model, we havecomputed tortuosities and volume fractions for various two- and three-dimensionalmedia, and we have studied how the geometrical properties of the ECS such as con-nectivity and pore size affect the tortuosity and the volume fraction.

The model is suitable for the diffusion of ions such as tetramethylammoniumand tetraethylammonium, but it cannot be applied to the movement of, for example,potassium and sodium, because these ions can cross the cell membranes. For suchions, the zero-flux membrane boundary condition is no longer valid. Modeling themovement of potassium and sodium will be developed in our upcoming papers.

We speculate that this modeling may partially replace the need for some typesof diffusion experiments in brain tissue. Furthermore, the modeling carried out herecan be applied to diffusion in a true porous medium, e.g., to geophysical applicationswhere tracer chemicals are injected into ground water and then undergo diffusion.

REFERENCES

[1] C. Appert and S. Zaleski, A lattice gas with a liquid-gas transition, Phys. Rev. Lett., 64(1990), pp. 1–4.

[2] C. Appert, D. H. Rothman, and S. Zaleski, A liquid-gas model on a lattice, Phys. D, 47(1990), pp. 85–96.

[3] J. Bear, Dynamics of Fluids in Porous Media, American Elsevier, New York, 1972.[4] J. Bear, Hydrodynamic dispersion, in Flow Through Porous Media, R. J. M. de Weist, ed.,

Academic Press, New York, 1969, pp. 109–199.[5] J. Bear and Y. Bachmat, Introduction to Modeling of Transport Phenomena in Porous

Media, Kluwer Academic Publishers, Dordrecht, 1990.[6] B. M. Boghosian and C. D. Levermore, A cellular automaton for Burgers’ equation, Com-

plex Systems, 1 (1987), pp. 17–30.[7] H. Chen, S. Chen, G. D. Doolen, and Y. C. Lee, Simple lattice gas model for waves,

Complex Systems, 2 (1988), pp. 259–267.[8] S. Chen, K. Diemer, G. D. Doolen, K. Eggert, C. Fu, S. Gutman, and B. J. Travis,


Lattice gas automata for flow through porous media, Phys. D, 47 (1991), pp. 72–84.[9] B. Chopard and M. Droz, Cellular automata model for the diffusion equation,

J. Statist. Phys., 64 (1991), pp. 859–892.[10] R. Cornubert, D. D’Humieres, and D. Levermore, A Knudson layer theory for lattice

gases, Phys. D, 47 (1991), pp. 241–259.[11] D. Dab, A. Lawniczak, J. Boon, and R. Kapral, Cellular-automaton model for reactive

systems, Phys. Rev. Lett., 64 (1990), pp. 2462–2465.[12] D. Dab, J. Boon, and Y. X. Li, Lattice-gas for coupled reaction-diffusion equations,

Phys. Rev. Lett., 66 (1991), pp. 2535–2538.[13] L. Dai and R. M. Miura, Lattice Boltzmann equation model for potassium movement within

the brain-cell microenviroment, in preparation.[14] G. D. Doolen, ed., Lattice Gas Methods: Theory, Application, and Hardware, MIT Press,

Cambridge, MA, 1991.[15] G. D. Doolen, U. Frisch, B. Hasslacher, S. Orszag, and S. Wolfram, eds., Lattice Gas

Methods for Partial Differential Equations, Addison–Wesley, Redwood City, CA, 1990.[16] A. W. El-Kareh, S. L. Braunstein, and T. W. Secomb, Effect of cell arrangement and

interstitial volume fraction on the effective diffusivity of monoclonal antibodies in tissue,Biophys. J., 64 (1993), pp. 1638–1646.

[17] U. Frisch, B. Hasslacher, and Y. Pomeau, Lattice-gas automata for the Navier–Stokesequation, Phys. Rev. Lett., 56 (1986), pp. 1505–1508.

[18] U. Frisch, D. D’Humieres, B. Hasslacher, P. Lallemand, Y. Pomeau, and J. Rivet,Lattice gas hydrodynamics in two and three dimensions, Complex Systems, 1 (1987), pp.649–707.

[19] W. G. Gray and P. V. Y. Lee, On the theorems for local volume averaging of multiphasesystems, Internat. J. Multiphase Flow, 3 (1977), pp. 333–340.

[20] A. K. Gunstensen and D. H. Rothman, Lattice-Boltzmann studies of immiscible two-phaseflow through porous media, J. Geophys. Research, 98 (1993), pp. 6431–6441.

[21] J. Hardy, Y. Pomeau, and O. de Pazzis, Time evolution of a two-dimensional classicallattice system, Phys. Rev. Lett., 31 (1973), pp. 276–279.

[22] J. Hardy, Y. Pomeau, and O. de Pazzis, Molecular dynamics of a classical lattice gas:Transport properties and time correlation functions, Phys. Rev. A, 13 (1976), pp. 1949–1961.

[23] B. Hasslacher, R. Kapral, and A. Lawniczak, Molecular Turing structures in the bio-chemistry of the cell, Chaos, 3 (1993), pp. 7–13.

[24] R. Kapral, Discrete models for chemical reacting systems, J. Math. Chem., 6 (1991), pp.113–163.

[25] R. Kapral, A. Lawniczak, and P. Masiar, Oscillations and waves in a reactive lattice-gasautomaton, Phys. Rev. Lett., 66 (1991), pp. 2539–2542.

[26] R. Kapral, A. Lawniczak, and P. Masiar, Reactive dynamics in a multispecies lattice-gasautomaton, J. Chem. Phys., 96 (1992), pp. 2762–2776.

[27] D. Krizaj, M. E. Rice, R. A. Wardle, and C. Nicholson, Water compartmentalizationand extracellular tortuosity after osmotic changes in cerebellum of Trachemys scripta,J. Physiol., 492 (1996), pp. 887–896.

[28] A. Lawniczak, D. Dab, R. Kapral, and J. Boon, Reactive lattice gas automata, PhysicaD, 47 (1991), pp. 132–158.

[29] A. Lehmenkuhler, E. Sykova, J. Svoboda, K. Zilles, and C. Nicholson, Extracellu-lar space parameters in the rat neocortex and subcortical white matter during postnataldevelopment determined by diffusion analysis, Neuroscience, 55 (1993), pp. 339–351.

[30] F. K. Lehner, On the validity of Fick’s law for transient diffusion through a porous medium,Chem. Engrg. Sci., 34 (1979), pp. 821–825.

[31] G. W. Milton, Bounds on the complex dielectric constant of a composite material, Appl.Phys. Lett., 37 (1980), pp. 300–302.

[32] C. Nicholson, Diffusion from an injected volume of a substance in brain tissue with arbitraryvolume fraction and tortuosity, Brain Research, 333 (1985), pp. 325–329.

[33] C. Nicholson and J. M. Phillips, Ion diffusion modified by tortuosity and volume fraction inthe extracellular microenvironment of the rat cerebellum, J. Physiol. (Lond), 321 (1981),pp. 225–257.

[34] C. Nicholson and M. E. Rice, The migration of substances in the neuronal microenviron-ment, in Ann. New York Acad. Sci., 481, New York Acad. Sci., New York, 1986, pp.55–66.

[35] C. Nicholson and M. E. Rice, Diffusion of ions and transmitters in the brain cell microen-vironment, in Volume Transmission in the Brain, K. Fuxe and L. F. Agnati, eds., Raven


Press, New York, 1991, pp. 279–294.[36] C. Nicholson and E. Sykova, Extracellular space structure revealed by diffusion analysis,

Trends in Neurosci., 21 (1998), pp. 207–215.[37] M. Perez-Pinzon, L. Tao, and C. Nicholson, Extracellular potassium, volume fraction,

and tortuosity in rat hippocampal CA1, CA3, and cortical slices during ischemia, J. Neu-rophysiol., 74 (1995), pp. 565–573.

[38] S. Ponce-Dawson, S. Chen, and G. D. Doolen, Lattice Boltzmann computations forreaction-diffusion equations, J. Chem. Phys., 98 (1993), pp. 1514–1523.

[39] M. E. Rice and C. Nicholson, Diffusion characteristics and extracellular volume fractionduring normoxia and hypoxia in slices of rat neostriatum, J. Neurophysiol., 65 (1991),pp. 264–272.

[40] D. H. Rothman, Cellular-automaton fluids: A model for flow in porous media, Geophys., 53(1988), pp. 509–518.

[41] J. S. Schultz and W. Armstrong, Permeability of interstitial space of muscle to solutes ofdifferent molecular weights, J. Pharm. Sci., 67 (1978), pp. 696–700.

[42] J. C. Strikwerda, Finite Difference Schemes and Partial Differential Equations, Wadsworth& Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1989.

[43] E. Sykova, J. Svoboda, Z. Simonova, A. Lehmenkuhler, and H. Lassmann, X-irradiation-induced changes in the diffusion parameters of the developing rat brain, Neurosci., 70(1996), pp. 597–612.

[44] S. Torquato, Random heterogeneous media: Microstructure and improved bounds on effec-tive properties, Appl. Mech. Rev., 44 (1991), pp. 37–76.

[45] S. Whitaker, Advances in theory of fluid motion in porous media, Ind. Engrg. Chem., 61(1969), pp. 14–28.

[46] S. Wolfram, Cellular automaton fluids 1: Basic theory, J. Statist. Phys., 45 (1986), pp.471-526.

Documents

A LATTICE CELLULAR AUTOMATA MODEL FOR ION DIFFUSION …miura/directory/1980-1999/1999/Dai-Miura99.pdf · but the rst lattice gas cellular automata in a two-dimensional square lattice