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Brain tumors Model Extensions Conclusions

Cell Movement in Network Tissue, Part III

Thomas Hillen

University of Alberta

fibres

movement direction

degraded fibres

directional change


Overview

• Part I:Biological backgroundDevelopment of the ModelPattern FormationThe 1-D Case

• Part II:Weak SolutionsSteady StatesParabolic LimitRelation to ODEs

• Part III:Application to Brain Tumor InvasionModel Extensions


Outline

Brain tumors

Model ExtensionsPainter JMB 2009Chauviere, H’, Preziosi, 2006, 07

Conclusions


Brain tumors

[Swanson 2000]

ut = ∇(D(x)∇u) + ρu

D(x) =

{5 x ∈ {white matter}1 x ∈ {gray matter}

However

D(x) =1

ω

∫VvvTq(x , v) dv

is the variance-covariance matrix. Swansons assumption does notreflect the fibrous structure of white matter.


White Matter

If at a point x ∈ white matter we find a dominant direction of e1,say, then we would expect

D(x) =

5 0 00 1 00 0 1


DTI - diffusion tensor imaging[ Jagersand, Murtha, Beaulieu]

Jagersand-Murtha / Patient-specific mathematical modeling of brain tumor growth beyond the visible margin

Figure 5: DIFFUSION TENSORS. An example of a DTI image, where tensors are represented by ellipsoids. Each ellipsoid is characterized by the 3 eigenvectors that characterize diffusion along (v ) and across (v ,v ). The eigenvalues , , are the diffusion rates in the corresponding directions.

1

2 3 1 2 3

Figure 6: TRACTOGRAPHY OF WHITE MATTER PATHWAYS. Diffusion tensor tractography demonstrates (a) the corpus callosum and internal capsule, (b) corticospinal tracts, and (c) optic radiations in a healthy control subject.

November 15, 2009 Competition


Tumor diffusion tensor

Water diffusion tensor Dw ∈ IR3×3:

Dw = λ1v1vT1 + λ2v2v

T2 + λ3v3v

T3

[Jbabdi, Swanson et al. 2005]:

DT = λ1(r)v1vT1 + λ2(r)v2v

T2 + λ3(r)v3v

T3 ,

where r is a scaling parameter such that

DT =

r r 11 r 11 1 1

Dw .


Jbabdi et al. 2005, Figure 1


Jbabdi et al. 2005, Figure 3


One idea to relate Dw and DT

Dw is measured to infer the orientational structure of the braintissue. Hence

q(θ) ∼ θDwθT

Using proper normalization∫q(θ)dθ = 1, we find

q(θ) =n

|Sn−1| tr(Dw )θDwθ

T


To compute the corresponding tumor diffusion tensor, we use thisq and compute the diffusion tensor of the parabolic limit:

DT = αVq

We find

DT =α

n + 2

(I +

Dw

tr(Dw )

)


Open questions

• Does our tumor diffusion tensor justify the scaling parameterr = 10 from Jbabdi?

• How does a diffusion model that uses DT compare to clinicalobservations?

• How would the solution of the full transport model look inthis case?


Fingering growth[ Jagersand, Murtha, Beaulieu; Painter, 2009]

Jagersand-Murtha / Patient-specific mathematical modeling of brain tumor growth beyond the visible margin

Figure 10: MICROSCOPIC TRANSPORT EQUATION. The top image shows an example of a finger-like tumor that cannot be modeled using a traditional diffusion equation. A better model uses a microscopic transport equation, which, as illustrated in the bottom panel [Painter 2009], can describe finger-like tumor invasion in network tissue.

November 15, 2009 Competition


Outline

Brain tumors


Conclusions


Amoeboid versus mesenchymal(P1) Amoeboid case: κ = 0, µ large, |v | large

Modelling cell migration strategies in the extracellular matrix 523

Fig. 3 Two-dimensional numerical simulations of the transport model for amoeboid migration. a1–e1Schematic showing the matrix structure; a2–e2 matrix alignment. a3–e3 Cell alignment after 3 h simulationtime. a4–e4 Macroscopic cell density after 3 h simulation time. Numerical details: space scale 1 = 100 µm,numerical simulations use a 50 ! 50 discretisation over space and a 64 (32) point discretisation for thecell (matrix) angle. A larger version of this figure, which shows more clearly the arrowheads in row 3, isavailable in the supplementary material

fibre distribution, for example if the matrix comprises of an interlocking network oforthogonally oriented fibres.

3.2.2 Numerical simulations

Equation (5) is solved numerically on a grid of size 500 ! 500 µm2, subject to eachof the matrix types represented in Fig. 3. Except where stated, an initially uniformdistribution of cells across both space and orientation is assumed, i.e. c(x, !, 0) =1/2" (and hence c(x, 0) = 1). For these and the rest of the simulations in Sects. 3and 4 we shall employ periodic boundary conditions. Parameter values are taken fromthe default values listed in Table 2 and the equations are solved up until t = 3 h,at which point solutions have evolved close to a steady-state distribution. The cell

123


Tissue orientation(P2) Isotropic versus oriented tissue (amoeboid case):

(a): initial condition(b): isotropic network at time t=60, t=120(c): diagonal oriented tissue at time t=60, t=120.

524 K. J. Painter

Fig. 4 Simulations illustrating isotropic/anisotropic spread of the cell population under matrix types (E1)and (E2). a Initial macroscopic cell density (note that cells are uniformly distributed in orientation).b Macroscopic cell density after 60 and 120 min for cells migrating in matrix type (E1). c Macrosco-pic cell density after 60 and 120 min for cells migrating in matrix type (E2). Parameters/discretisationdetails as in Fig. 3

density distributions are plotted in the bottom two rows of Fig. 3. Note that for thecell alignment here we do not employ the measures of cell alignment derived fromEq. (3): the bidirectional nature of the fibres results in symmetric cell distributionswith approximately equal densities of cells migrating in opposite directions. Thus thevector (3) gives !"c # 0 and hence limited information on the local cell alignment.For this figure, we therefore instead calculate the arguments of the local maxima inthe cell distribution and plot these using the double arrowheads in Fig. 3, row 3.

With the random matrix (E1) there is no discernible change in the cell distributionand they remain (subject to the small random perturbation of the matrix distribu-tion) uniformly distributed in space and orientation (Fig. 3a3, a4). Cell movement for(E1) is essentially an unbiased random walk, resulting in an isotropic diffusion pro-cess at a macroscopic level: numerical simulations for a population of cells initiallyconcentrated in the centre of domain demonstrate uniform radial spread (Fig. 4b). Fora globally-oriented cell matrix (E2) the impact of contact guidance is demonstratedthrough the alignment of cells to the underlying matrix (Fig. 3b3). With no spatialvariation of the matrix organisation, macroscopic cell densities c remain uniform(Fig. 3 (b4)). Contact alignment here results in an anisotropic diffusion process at themacroscopic level: radial spread of a population of cells shows enhanced movementalong the direction of fibre alignment, Fig. 4c.

In Fig. 3c and d data is plotted for two mixed matrix types. In Fig 3c, a strip oforiented matrix (type (E2)) is positioned within a non-oriented matrix (type (E1)), withfibres in the strip predominantly aligned parallel to it. Cells demonstrate a high degreeof orientation when positioned on the strip, as can be predicted from the above results.Yet a difference now emerges in the macroscopic cell density c, which demonstratesincreased cellular density on the strip (c4). Intuitively, this arises through cells migra-ting into the aligned region and becoming “trapped”. The significance of the ECMarrangement is appreciated by considering the alternative matrix in which fibres in thestrip are predominantly aligned orthogonal to it, Fig. 3d. Cells that migrate onto thestrip are quickly transported across it, resulting in a lowering of the macroscopic celldensity, c, on the strip (d4). These two simulations indicate the capacity of an orga-nised ECM to dictate cell patterning: the spatial organisation of the cells is stronglyinfluenced by the underlying ECM. A further test of this capacity is shown using the

123


Contact guidance

(P3) Random movement versus contact guidance:

Replace:

q(t, x , θ)

ω7→ α

|V |+ (1− α)

q(t, x , θ)

ω

α ∈ [0, 1].

Result: For α→ 0 the network formation is lost.


ECM production

(P4) ECM degradation and ECM production:

qt = κ( Π︸︷︷︸production

− A︸︷︷︸degradation

)pq

Same model as before.Hillen: q = fibre orientational distributionPainter: q = fibre density.


Protease distribution

(P5) Focussed protease release versus unfocussed protease release.Then the mean projection operator

Π(t, x , θ) =1

p(t, x)

∫V|θ · v |p(t, x , v)dv ,

is replaced with

Π(t, x , θ) =1

p(t, x)

∫V

(ψ − (1− ψ)|θ · v |)p(t, x , v)dv ,

ψ ∈ [0, 1].

Result: As long as ψ 6= 1 the system behaves qualitatively similar.


Chauviere, H’, Preziosi 2006, 07

Including cell-cell interaction and chemotaxis:

pt + v · ∇xp +∇v (f (c)p) = JECM + Jcell

f (c): chemotactic forceJECM: cell-matrix interactionsJcell: cell-cell interactions.


Physical quantities (moments)

• cell density

ρ(t, x) =

∫Vp(t, x , v)dv

• mean velocity U(t, x)

ρU =

∫Vp(t, x , v)vdv

• internal energy

E =

∫V

1

2p(t, x , v)|v − U(t, x)|2dx


[Chauviere, H’, Preziosi, 2006, 2007]• Derive mass, momentum and energy balance equations.• Derive biologically realistic momentum and energy dissipation

terms.• Moment closure or scaling to obtain drift-diffusion limit.• Numerical comparison of moment closure and full solution in

steady state:

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.25 0.5 0.75 1x

0

0.25

0.5

0.75

1

y

0

0.2

0.4

0.6

0.8

1

1.2


Outline

Brain tumors


Conclusions


Summary

• We derived a model for mesenchymal motion.

• The model is generalizable to many other situations (includingECM production, amoeboid-mesenchymal motion,chemotaxis, cell-cell interactions, etc.)

• We expect that model properties are typical for this class ofmodels

• We build solution theory for measure valued solutions, whichenabled us to classify steady states

• We argued that the parabolic limit might miss importantaspects (such as fingering).

• Applications to brain tumors are promising

• Other applications might arise


Applications of the M5 model

• Movement of metastases

• Vasculature formation

• Angiogenesis

• Wound healing

• Ecological: Path generation in a given habitat

• Micro: Protein movement inside a cell


Open Mathematical Problems

• Stability of patterns?

• Orientation driven instabilities?

• Geometric analysis of steady state networks?

• Boundary conditions?

• Efficient numerics ?


Special Thanks To:

K. Painter, L. Preziosi, A. Chauviere


And To:

Z.A. Wang, P. Hinow, M. Li

and NSERC, MITACS

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