Moment Models for kinetic chemotaxis/haptotaxis equations

Transcription

1 Moment Models for kinetic chemotaxis/haptotaxis equations An overview Gregor Corbin March 20, 2017 Contents 1 Introduction The different model scales The kinetic cell model A moment model for the state variable 3 3 Moment models for the velocity The general procedure Some examples Discrete Ordinates The P 1 model in 1D The P N model in 3D The M 1 model in 1D The Kershaw closure Partial Moments models References 12 1 Introduction 1.1 The different model scales Microscale Equations of motion for individual cells/particles x i (t), v i (t), y i (t) many small particles Mesoscale Kinetic equation for the probability density function p(t, x, v, z) t ɛ 2 t, x ɛx Macroscale Advection-Diffusion equation for the density of particles ρ(t, x) = p Figure 1: The different model scales Here we consider the mesoscale only. The derivation of macroscopic equations from the kinetic levels is done in [2, 3] in the context of glioma tumor cell populations. 1

2 1.2 The kinetic cell model We consider the following kinetic model for glioma cell populations described in [2, 3]: t p + v x p }{{} + z ( }{{ Gp) } = L(p) }{{} straight movement change of state instant change of direction λ A p }{{} death or proliferation. (1) Here, p(t, x, v, z) is a distribution function for the number of cells at time t R +, position x D x R 3, with velocity v S 2 and internal state deviation z Z R. The internal state y in each cell follows an ODE ẏ = G(y) = k + (1 y)a(t, x) k y, (2) for some constant model parameters k +, k. The time and space dependent volume fraction of ECM (see [2]) is denoted A(t, x). Equation (2) has a steady state y, where G(y ) = 0. The variable z is then defined as the deviation from the steady state z = y y. With this substitution we obtain the G(z) G(z) = α(a(t, x))z + v x β(a(t, x)). (3) from equation (1) with α = k + A + k and β = k+ A α.(c.f [3]). The linear collision operator L corresponds to a local but instant change of velocity of the cells. It is given by L(p) = k z (v, v )p(v ) k z (v, v)p(v)dv, (4) S 2 = k z (v, v )p(v )dv λ(z)p(v). (5) S 2 The collision kernel k z (v, v ) has also to be modeled. It can be interpreted as the rate at which cell with velocity v change their velocity to v. In general it depends on the internal state of cells. It satisfies λ(z) := k z (v, v)dv, (6) S 2 for all v S 2. An important assumption on the collision kernel is, that it has an equilibrium distribution. Definition (Equilibrium distribution). A distribution F eq (v) that fulfills the detailed balance is an equilibrium distribution of the kernel k z (v, v ). k z (v, v )F eq (v ) = k z (v, v)f eq (v) v, v S 2 (7) The equilibrium could in general be different for every state z. Example (A simple kernel). where Q(v) is normalized, i.e. k z (v, v ) = λ(z)q(v), (8) S 2 Q(v)dv = 1. (9) The equilibrium distribution F eq (v) is obviously Q(v), independent of the state z. In the following we will only consider kernels of the above form (8). The model (8) comes from a haptotaxis setting, where cells orient themselves according to an underlying fibre distribution Q(v). With the notation := dv (10) the collision operator is V L = λ(z) [Q(v) p p]. (11) 2

3 2 A moment model for the state variable In this section we will derive the simplest possible moment model to discretize the state variable z of equation (1). This is done in the following steps: 1. define a basis b z = (b 0, b 1,...) of P (Z) 2. test (1) with all basis functions b i (z), i.e multiply by b i (z), and integrate ( t p + x (vp) + z (Gp)) b i (z)dz = L(p)b i (z)dz, i = 0, 1,... (12) Z 3. take only a finite number of basis functions b := (b 0,..., b N 1 ) to obtain a system of N equations for the moments f = (f 0, f 1,..., f N 1 ) := p(z)b z dz (13) As a basis of P (Z), we take the monomials b z = (1, z, z 2,...). We will use the simplest possible discrete moment model, with only a single basis function: How is this justified? Z Z b z = 1. (14) Assumption (Fast state dynamics). The change of state is fast, compared to the other effects. Therefore, most of the distribution p(z) is close to the steady state of G at z = 0. the solution is adequately described by its zeroth-order moment we neglect all second-order or higher moments f i, i 2 the distribution is negligibly small at the boundaries of Z: p(z min ) 0, p(z max ) 0 We expand the state dependent part of the kernel λ(z) in the basis Testing (1) by the finite basis b z = 1 yields t f 0 + x (vf 0 ) + Z λ(z) = λ i z i. (15) i=0 z (Gp)dz = i=0 λ i (Q(v) f i f i (v)) (16) where the integral over z (Gp) and the terms for i 2 vanish due to our assumptions: t f 0 + x (vf 0 ) = λ 0 (Q(v) f 0 f 0 (v)) + λ 1 (Q(v) f 1 f 1 (v)) (17) This equation for f 0 contains an additional unknown, namely the first moment f 1. We use a natural closure, meaning that we use the equation for f 1, which can be obtained by testing equation (1) by z: t f 1 + x (vf 1 ) Gpdz = λ 0 (Q(v) f 1 f 1 (v)). (18) Z Again using the assumption that most of the distribution is close to z = 0, we neglect the transport of the first moment f 1. Inserting the expression for G(z) yields: Solving this equation for f 1 gives αf 1 (v x β)f 0 = λ 0 (Q f 1 f 1 ) (19) f 1 = 1 α + λ 0 (λ 0 Q f 1 + (v x β)f 0 ). (20) Integrating equation (19) with respect to velocity gives an expression for f 1 f 1 = xβ α f 0v, (21) 3

4 where we used that Q = 1. Now we have an expression for the first-order moment f 1 that depends only on the zeroth-order moment f 0. Plugging equations (20) and (21) into the zeroth order equation (17) finally yields: t f 0 + x (vf 0 ) = λ H (Q f 0 f 0 ) + λ H (Q f 0 v f 0 v) (22) = L 0 (f 0 ) + L 1 (f 0 ) (23) where λ H := λ 0 and λ H := λ1 xβ λ 0+α. In contrast to the classical transport equation of radiation, there are now two collision terms. The first pushes the distribution f 0 towards the equilibrium Q, while the second does the same for f 0 v. In the following section we apply the method of moments again to discretize the velocity variable. 3 Moment models for the velocity 3.1 The general procedure All following considerations start from the simple moment model (23) from the previous section. We apply a similar procedure to discretize the velocity v. With the basis b for a finite-dimensional subspace of P (V ), the moments are defined as u = fb. (24) Then we take the moments of equation (23) w.r.t b t u + x v bf }{{} F = L0 (f) + L1 (f) } {{ } C (25) In general (but not always) F, C contain moments that are not components of u. We have to provide additional models for these moments. Usually, a reconstruction of the distribution ˆf(u) f is defined from the known! moments, with the condition that ˆfb = u. This is then used to compute approximations of F v b ˆf. The closed system then reads: t u + x v b ˆf = L0 ( ˆf) + L1 ( ˆf) (26) 3.2 Some examples Discrete Ordinates We choose b = (b 0,..., b N 1 ), b i = δ(v v i ) together with a linear ansatz for f: ˆf = i α iδ(v v i ). Then The coefficients α i follow from the moment constraints ˆfbi = α j b j b i The moment system becomes fb i = f(v i ) := f i = u i, (27) vb i f = v i f i, (28) j (29) = α i! = f i (30) t f i + x (v i f i ) = λ H (Q(v i ) f f i ) + λ H (Q(v i ) vf f i v i ). (31) The additional moments f, vf can be obtained from the ansatz: f ˆf = f i, (32) i vf v ˆf = v i f i. (33) i This is the discrete ordinates method(see also [7]). 4

5 3.2.2 The P 1 model in 1D For simplicity of notation, let us first consider a one-dimensional example, the so-called slab geometry, where x R and v V = [ 1, 1]. I. this setting the kinetic equation is t f + v x f = L 0 + L 1. (34) We choose the basis as b = (1, v), moments u = (u 0, u 1 ) = fb together with a modified linear ansatz ˆf = Q(v) α i b i = Q(v)(α 0 + α 1 v). (35) Remark. This ansatz incorporates the equilibrium distribution Q(v) of the kernel. It is thus able to represent the equilibrium exactly. Testing equation (23) by b gives t u 0 + x u 1 = λ H ( Q u 0 u 0 ) + λ H ( Q u 1 u 1 ), (36) t u 1 + x u 2 = λ H ( Qv u 0 u 1 ) + λ H ( Qv u 1 u 2 ), (37) There are two equations for the three unknowns u 0, u 1, u 2. We approximate the second moment u 2 from the ansatz u 2 (u 0, u 1 ) v 2 ˆf = Q(v)(α 0 + α 1 v)v 2, (38) where the multipliers α 0, α 1 have to be chosen, such that the ansatz has the coorect moments: condition leads to a linear system of equations:! ˆfb = u. This u 0 = α 0 Q + α 1 Qv, (39) u 1 = α 0 Qv + α 1 Qv 2. (40) In order to get the second moment u 2, we have to solve this system for α 0, α 1 and plug these values into the expression for u 2. In the special case of Q(v) = 3 2 v2, we have and therefore α 0 = u 0, (41) α 1 = 5 3 u 1, (42) u 2 (u 0, u 1 ) = 3 5 u 0 (43) The P N model in 3D Now we consider the general three-dimensional case, where x R 3 and v V = S 2. Also, we include higherorder functions in the basis. The following two choices are equivalent analytically (although not numerically): 1. b = (v k ) k for all multiindices k = (k 0, k 1, k 2 ) with k N. 2. b = (Yl m ) l,m, for l N, l m l, where Yl m are the spherical harmonics. Again, we choose a linear ansatz ˆf = Q α i b i = ˆαˆαˆα b. The multipliers α follow from the linear system of moment constraints:! ˆfb = u, (44) Qb b α! = u. (45) Note that in the special case of Q 1, and with the spherical harmonics, due to orthogonality, Qb b reduces to the identity matrix and therefore α = u. In the monomial basis, we have v b n f b n+1 f = u n+1, where the subscript n denotes all components with k = n. Only the highest-order moments v b N f have to be approximated by v b N ˆf = v b N (Qα b) = Qv b N b α. 5

6 3.2.4 The M 1 model in 1D Let us go back to the one-dimensional example x R, v V = [ 1, 1] with the first-order basis b = (1, v). But this time we choose a a nonlinear ansatz ˆf = Q exp( i α i b i ) = Q exp(α 0 + α 1 v) (46) Why should we do this? For Q 1, the ansatz minimizes the Maxwell-Boltzmann entropy under the moment constraints: Recall the H-Theorem for the following linear kinetic equation η(f) = f log f f, (47) ˆf = argmin{η(g) gb = u}. (48) g t f + v x f = λ 0 ( 1 f f). 4π (49) Definition (Entropy). For a convex function η, we call H(f) := η(f)dvdx (50) an entropy of the kinetic equation. X Theorem 1 (Boltzmann s H-Theorem). Any entropy for the linear kinetic equation (49) decreases monotonically with time d H(f) 0. (51) dt Thus, with this ansatz, the discretized system also dissipates the Maxwell-Boltzmann entropy. Additionally, it is guaranteed to be positive. For a detailed derivation of entropic moment closures, we refer to [1, 8]. Caution: In the context of haptotaxis, this ansatz is not rigourosly justified. There are several problems: V It is not known yet, if ˆf = Q exp(α b) even minimizes an entropy for (23). Also the H-Theorem does not hold for arbitrary entropies any more. The Maxwell-Boltzmann entropy is physically relevant for certain particle types. How sensible this choice is for biological cells we do not know. However: The M N model still guarantees positivity of the reconstruction, as opposed to the P N model. Intuitively, it makes sense to include Q into the ansatz to be able to reconstruct the equilibrium distribution correctly. Now, let us go back to compute the M 1 model. Writing out the moment constraints ˆfb = u leads to the following nonlinear system R u (α) = [ ] [ ] Q exp(α0 + α 1 v) u0! = 0 (52) Q exp(α 0 + α 1 v)v u 1 for α 0, α 1. Typically this is solved numerically, e.g. by Newton s method. Newton s method also needs the Jacobian of this expression: JR u = R [ ] u 1 v α = Q exp(α 0 + α 1 v) v v 2. (53) Note that R u (α) = 0 does not have a solution for all u. It turns out that it has a unique solution for the realizable moments. Definition (Realizable set). For a given basis b, the set of realizable moments is given by where P (V ) is the space of all non-negative distributions on V. R b = {u R n, f P (V ) : fb = u}, (54) The realizable set contains all those moments, for which there exists a non-negative distribution f on V. Conversely, if f is non-negative, its moments lie in the realizable set. Note that this definition only depends on the basis. In this case, the realizable set is simply ( ) 0 u0 R b = (55) u 1 u 0 a con as shown in figure 2. 6

7 Figure 2: The realizable set R b for b = (1, v) The Kershaw closure In the moment models that we have seen so far, the main desirable properties were representation of special distributions f (e.g. the equilibrium point) a closure that produces realizable higher-order moments The M N model naturally generates higher-order moments that are realizable. But this comes at the expense of having to solve a nonlinear system of equations for the multipliers α first. The Kershaw closure achieves these properties with a simpler ansatz that is therefore cheaper to compute. The reconstruction ˆf is done by interpolation between certain distributions f, that should be represented. To keep the higher-order moments realizable, explicit information on the higher-order realizable set is needed. For more theory on realizability and Kersahw closures, see [6, 9, 10] First-order Kershaw closure in 1D The first-order monomial basis in 1D is b = (1, v) with moments. u = (u 0, u 1 ). The Kershaw closure needs the realizable set R b + for the higher-order basis b+ = (1, v, v 2 ). The higher-order realizable set is 0 u 0 R b + u 1 u 0, (56) u 2 1 u 0 u 2 which is shown in figure 3. We get a realizable closure for u 2 by interpolation between the upper and lower bound on u 2 : u 2,K u 0 = ξ 1 + (1 ξ) ( u1 u 0 ) 2 (57) How to choose the interpolation coefficient ξ? At the equilibrium, we have u 0 = Q = 1 and u 1 = Qv = 0. Since we want to reproduce moments of the equilibrium, we need Qv 2! = u 2,K (58) Qv 2 ( ) 2 u1 = ξ + (1 ξ) (59) u 0 Qv 2 = ξ + (1 ξ) Qv 2 (60) Qv 2 Qv 2 ξ = 1 Qv 2 (61) If Q is symmetric around 0, then Qv = 0 and ξ = Qv 2. For example, using Q 1 yields ξ =

8 Figure 3: The realizable set R b for b = (1, v, v 2 ). The Kershaw model interpolates the second moment u ( ) 2 2 between the upper boundary u2 u 0 = 1 and the lower boundary u2 u u 0 = 1 u 0 of the realizable set First-order Kershaw closure in 2D The velocity space is V = S 2. The first-order monomial basis is b = (1, v) = (1, v 0, v 1 ), with moments u = (ρ, q 0, q 1 ). We need second-order moments P := v vf to close the system. Define the second order basis b + = (1, v, v v) = (1, v 0, v 1, v0, 2 v 0 v 1, v1) 2 and the corresponding moments u + = (ρ, q, P ) = (ρ, q 0, q 1, P 00, P 01, P 11 ). We want to construct P such that u + R b +. Additionally,the model should also incorporate our equilibrium Q. We calculate P for two extreme cases explicitly. 1. For the equilibrium solution f eq = ρq, we have q = 0. The second moments of this distribution are P eq = ρ v vq =: ρd F. 2. In the so-called free-streaming case the distribution f δ = ρδ(v q q q = ρ. The second moments are P δ = ρ v vδ(v q q ) = ρ q q q. 2 ) is a beam in direction q q. Therefore Then we interpolate between those two cases, s.t. realizability is preserved. Figure 4 shows the projection of R b + onto q0 ρ, q1 ρ and the interpolation procedure. The ansatz for P becomes ( P (ρ, q) = ρ α(ρ, q)d F + (1 α(ρ, q)) q q ) q 2 (62) Now, we have to choose α(ρ, q) s.t. u + is realizable. For example α = 1 q 2 ρ two extreme cases, this ansatz gives the correct second moments. achieves this. Note that for the Discussion The Kershaw closure lets us compute higher-order moments explicitly from the known moments, while preserving realizability and reconstructing moments of special distributions exactly. This means it is much faster to compute than the M N model. However, the generalization beyond a first-order model is hard, because the realizable set has to be known exactly. Already the second-order set is five-dimensional. For a second-order Kershaw closure, one needs to compute the third-order realizability domain in nine dimensions. Also, point-wise directional information on the reconstruction ˆf is needed for a kinetic upwind scheme Partial Moments models A simple first-order finite-volume method with forward Euler time discretization in 1D is [ ] u n+1 i = u n i + t vĝ i+ 1 b vĝ 2 i 1 b +... (63) 2 A kinetic scheme uses upwind point-wise for velocity in the numerical flux ĝ: ĝ i+ 1 2 = { ˆf(xi ) v > 0 ˆf(x i+1 ) v < 0. (64) 8

9 Figure 4: Kershaw closure on the realizable set R b for b = (1, v x, v y ): Shown are the normalized moment components qx ρ, qy ρ This cannot be done in the Kershaw model, since we do not have ˆf, but only the moments. Idea: Define a non-overlapping decomposition of V into patches. On each patch, use a low-order moment model. Advantage: Closure problem decouples. We can use a kinetic scheme, if patches do not cross quadrant boundaries. We can add more degrees of freedom relatively easy. [4, 5] 9

10 (a) flux at the right face (b) upwind patches at the right face (c) flux at the top face (d) upwind patches at the top face Figure 5: Upwind scheme for quarter-moments: For the right face of cell i, j, moments associated with positive velocity in x-direction(u ++, u + ) are taken from the left cell i, j. Moments associated with negative velocity in x-direction are taken from the right cell i + 1, j. Analogously in y-direction. 10

11 Figure 6: The unit sphere with quadrants 11

12 References [1] B. Dubroca and J. L. Feugeas. Entropic moment closure hierarchy for the radiative transfer equation. C. R. Acad. Sci. Paris Ser. I, pages , [2] C. Engwer, T. Hillen, M. Knappitsch, and C. Surulescu. Glioma follow white matter tracts: a multiscale dti-based model. Mathematical Biology, page , [3] Christian Engwer, Alexander Hunt, and Christina Surulescu. Effective equations for anisotropic glioma spread with proliferation: a multiscale approach and comparisons with previous settings. Mathematical Medicine and Biology, 33(4): , [4] M. Frank. Partial Moment Models for Radiative Transfer. Shaker-Verlag, Aachen, [5] Martin Frank, Bruno Dubroca, and Axel Klar. Partial moment entropy approximation to radiative heat transfer. Journal of Computational Physics, 218(1):1 18, [6] DS Kershaw. Flux limiting nature s own way. Lawrence Livermore National Laboratory, UCRL-78378, [7] E. W. Larsen and J.E.Morel. Advances in discrete-ordinates methodology. Nuclear computational Science, pages 1 84, [8] D. Levermore. Moment closure hierarchies for kinetic theories. Journal of Statistical Physics, [9] Philipp Monreal and Martin Frank. Moment realizability and kershaw closures in radiative transfer. Technical report, Lehr-und Forschungsgebiet Simulation in der Kerntechnik, [10] Florian Schneider. Kershaw closures for linear transport equations in slab geometry i: model derivation. Journal of Computational Physics, 322: ,