Implementation of cellular traction forces in agent-based...
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Implementation of cellular traction forces inagent-based models
Wilbert Gorter
Delft University of TechnologyVORtech bv
June 4th, 2020
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Content
▸ Introduction
▸ Biology of burn injuries and contraction
▸ Purely elastic model
▸ Viscoelastic model
▸ Morphoelastic model
▸ Extension to two dimensions
▸ On programming
▸ Future work
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Introduction
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Introduction
Research on burn injuries and scars
Elasticity model: Daan Smits
Agent-based model: Eline Kleimann
Special objective: obtain more understanding of burn contraction
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Biology of burn injuries andcontraction
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Structure of the skin
Skin model
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Wound contraction
The way to wound contraction:
Platelets → chemokines → (myo)fibroblasts → contraction
Plastic and elastic deformation
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Model types
Elastic models:
▸ Purely elastic model
▸ Viscoelastic model
▸ Morphoelastic model
Carried out in one and two dimensions
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Purely elastic model
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Initializing quantities and more
Location x(X , t)Langrangian location X ∶= x0Displacement u ∶= x −XVelocity v ∶= ∂x/∂tStress σ ∶= force per areaStrain ε ∶= ∂u/∂x
All quantities c can be expressed Eulerian: c(x , t)as well as Lagrangian: c(X , t)
e.g. u(x , t) = x −X (x , t) while u(X , t) = x(X , t) −X
Material derivative: DDt ∶=
∂∂t + v ∂
∂x
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Stress and strain tensor
In more dimensions, stress and strain are tensors.
σ =
⎛
⎜
⎝
σ11 σ12 σ13σ21 σ22 σ23σ31 σ32 σ33
⎞
⎟
⎠
ε =⎛
⎜
⎝
ε11 ε12 ε13ε21 ε22 ε23ε31 ε32 ε33
⎞
⎟
⎠
normal stress: σiishear stress: σij (i ≠ j)
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Normal and shear stress
Cube with normal and shearstress
Shear stress
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Hooke’s law (1D)
σ = κε = κ∂u∂x
κ is called ‘Young’s modulus’.
Hooke’s law in more dimensions (normal and shear modulus).
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Cauchy momentum equation
Newton’s second law: impulse is proportional with force
ρDvDt = ∇ ⋅ σ + f
One dimension: ρDvDt =
∂σ∂x + f
ρ: density,v : velocity,σ: stress, σ = κ∂u∂xf : internal force; caused by (myo)fibroblasts
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Numerical aspects
ρDvDt =
∂σ∂x + f
σ = κ∂u∂x
Finite Element Method with moving mesh
Euler Backward
ρMk+1vk+1 =Mkvk +∆tSk+1uk+1 +∆tfk+1
Approximate:xk+1i ≈ xki +∆t ⋅ vki (forward)xk+1i ≈ xki +∆t ⋅ vk+1i (backward)
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Results: purely elastic model
f (x , t) ∶= 5.0 ⋅
⎧⎪⎪⎨⎪⎪⎩
1 − exp(−4 ⋅ t/tf ) if 0 ≤ t < 20,
(1 − exp(−4 ⋅ t/tf )) exp(−(t − tf )) if t ≥ 20.
plot of current (red, x) againstinitial (blue, X)
plot of length against time (t)
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Viscoelastic model
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Viscosity
Suppose the blue fluid isn’t water, but honey...
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Difference between pure elasticity and viscoelasticity
Instead of
σ = κε,
we have
σ = κε + µDεDt .
µ = viscosity rate
(DεDt can also be written as ∂v∂x .)
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Numerical aspects
ρDvDt =
∂σ∂x + f
σ = κ∂u∂x + µ∂v∂x
Finite Element Method
Euler Backward
Mk+1vk+1 =Mkvk +∆tSk+1uk+1 +∆tSk+1vk+1 +∆tfk+1
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Results: comparing purely elastic and viscoelastic model
Plots of length against time (t)
Purely elastic Viscoelastic
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Morphoelastic model
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Elastic and plastic behaviour
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Elastic and plastic deformation
Deformation gradient F ∶=∂x∂X
F =∂x∂z
∂z∂X ∶= αγ
α: elastic deformationγ: plastic deformationX : initial statez : zero-stress state, equals X as long as Dγ
Dt = 0x : current state, equals z and X at t = 0
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Elastic and plastic deformation (2)
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Elastic and plastic deformation (3)
DγDt = Fg
g : growth rate; g = ξε (choice)
uz ∶= x − z
Strain evolution equation:
DεDt + (ε − 1)∂v∂x = −g
ε = new strain based on uz , i.e. ε = ∂uz∂x
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Numerical aspects
Cauchy momentum: ρDvDt =
∂σ∂x + f
Viscoelasticity: σ = κε + µ∂v∂xStrain evolution: Dε
Dt + (ε − 1)∂v∂x = −g
⇓
Sk+1wk+1= T kwk
+∆tΦk+1
wk∶= (
εεεεεεεεεk
vk) and Φk
∶= (0
fk)
Equations combined in one Euler Backward FEM system.
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Results: comparing visco- and morphoelastic model
Plots of length against time (t)
Viscoelastic Morphoelastic
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Two-dimensional models
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More-dimensional equations
Cauchy momentum: ρDvDt = ∇ ⋅ σ + f
Viscoelasticity:
σ = µ1sym (∇v) + µ2Tr (∇v) I +κ√ρ
1+η (ε +η
1−2ηTr(ε)I) .
Strain evolution:
Dε
Dt + εskw (∂v∂x) − skw (∂v∂x) ε + (Tr (ε) − 1) sym(∂v∂x) = −G .
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Numerical aspects
System to solve:
Mk+1
wk+1=Mkwk
+∆tSk+1wk+1+∆tfk+1(wk+1
)
wk∶=
⎛
⎜⎜⎜⎜⎜⎜
⎝
εεεεεεεεεk11εεεεεεεεεk12εεεεεεεεεk22vk1vk2
⎞
⎟⎟⎟⎟⎟⎟
⎠
Non-linear system: use iterative method of Picard (in each B.E.iteration)
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Results: morphoelastic model in 2D
current state (black) and initialstate (red)
plot of area against time (t)
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On programming
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Specifying the path..
Python⇓
C++⇓
GPU (Cuda)
⇒ C++ (much) faster than Python⇒ work of Eline Kleimann in C++ and on GPU
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Future work
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Future (or actually: current) work
Doing something on parameters...
Combining models (agent-based and elastic, in C++)
Triangulation
GPU-implementation?
Three-dimensions?
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Questions?