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A computational framework for particle and whole cell tracking applied to a real biological dataset Feng Wei Yang F.W.Yang@sussex.ac.uk 5 April 2016 Feng Wei Yang at BAMC in Oxford 5 April 2016 1 / 21 Objectives The human fibrosarcoma cell

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A computational framework for particle and whole cell tracking applied to a real biological dataset

Feng Wei Yang

F.W.Yang@sussex.ac.uk

5 April 2016

Feng Wei Yang at BAMC in Oxford 5 April 2016 1 / 21

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Objectives

The human fibrosarcoma cell line HT-1080 obtained from DSMZ, Germany Feng Wei Yang at BAMC in Oxford 5 April 2016 2 / 21

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Outline

F. Yang, C. Venkataraman, V. Styles, V. Kuttenberger, E. Horn, Z. von Guttenberg, A. Madzvamuse, Journal of Biomechanics,

Accepted for publication, http://dx.doi.org/10.1016/j.jbiomech.2016.02.008, 2016.

Identification from phase contrast microscopy Particle tracking Whole cell tracking for morphological changes

Optimal control of phase field formulations of geometric evolution laws Efficient solver Applications

Feng Wei Yang at BAMC in Oxford 5 April 2016 3 / 21

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Techniques based upon background removal

Feng Wei Yang at BAMC in Oxford 5 April 2016 4 / 21

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Directions of migration

Spider plot Star plot

Feng Wei Yang at BAMC in Oxford 5 April 2016 5 / 21

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Individual cells

Cell two Cell one Cell one Cell two

Feng Wei Yang at BAMC in Oxford 5 April 2016 6 / 21

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Our optimal control model

The mass constrained mean curvature flow with forcing:

V V (x x x, t) = (−σH(x x x, t) + η(x x x, t) + λV (t))v v v(x x x, t) on Γ(t), t ∈ (0, T], Γ(0) = Γ0.

The phase-field approximation of the above equation - Allen-Cahn:

     ∂tφ(x x x, t) = △φ(x x x, t) − 1

ǫ2 G ′(φ(x

x x, t)) − 1

ǫ(η(x

x x, t) − λ(t)) in Ω × (0, T], ∇φ · ν ν νΩ = 0 on ∂Ω × (0, T], φ(·, 0) = φ0 in Ω.

Feng Wei Yang at BAMC in Oxford 5 April 2016 7 / 21

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Our optimal control model cont.

The objective functional:

J(φ, η) = 1 2

(φ(x x x, T) − φobs(x x x))2 dx x x + θ 2 T

η(x x x, t)2dx x xdt,

and now we solve the minimisation problem:

minηJ(φ, η), with J given above.

Feng Wei Yang at BAMC in Oxford 5 April 2016 8 / 21

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Our optimal control model cont.

The adjoint equation to help computing the derivative of the objective functional:

∂tp(x

x x, t) = −△p(x x x, t) + ǫ−2G ′′ (φ (x x x, t))p(x x x, t) in Ω × [0, T), p(x x x, T) = φ(x x x, T) − φobs(x x x) in Ω,

and we update the control as

ηℓ+1 = ηℓ − α

θηℓ + 1

ǫ pℓ

Feng Wei Yang at BAMC in Oxford 5 April 2016 9 / 21

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Numerical challenges

Number of time steps Memory requirement (let’s consider double precision and 100 time steps)

2-D: 5122 requires 0.4 gigabytes 3-D: 5123 requires 215 gigabytes

Feng Wei Yang at BAMC in Oxford 5 April 2016 10 / 21

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Two-grid solution strategy

One time step One complete solve for the Allen-Cahn equation from t=(0,T] Intermediate grid(s) Restrict the converged solution

f ϕ

Fine grid for the Allen-Cahn equation Coarse grid for the adjoint equation One time step One complete solve for the adjoint equation from t=[T,0) Interpolate the computed η Start the next η iteration

Feng Wei Yang at BAMC in Oxford 5 April 2016 11 / 21

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Cell one

t=0 t=T/2 t=T

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Cell one video

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Cell two video

Feng Wei Yang at BAMC in Oxford 5 April 2016 14 / 21

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Analysis through tracking morphological changes

Ω φdx

x x

5 10 15 20 25 −0.88 −0.87 −0.86 −0.85 −0.84 −0.83 −0.82 −0.81 −0.8 Time (A.U.) Mass Cell one Cell two Desired shape for cell one Desired shape for cell two

{φ>0} 1dx

x x

5 10 15 20 25 0.05 0.055 0.06 0.065 0.07 0.075 0.08 0.085 0.09 0.095 0.1 Time (A.U.) Volume Cell one Cell two Desired shape for cell one Desired shape for cell two

Feng Wei Yang at BAMC in Oxford 5 April 2016 15 / 21

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Real world example (2)

Feng Wei Yang at BAMC in Oxford 5 April 2016 16 / 21

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Euler number for topological changes

We compute this Euler number for these time steps with an ”optimized“ control η: X = 1 2π(a − b)

Ω(a,b)
−△φ + ∇|∇φ|2 · ∇φ

2|∇φ|2

dx.
Q. Du et al. J. Appl. Math., 2005

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Real world example (2)

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A 3-D example

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

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A 3-D example video

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The end

F.W. Yang, C.E. Goodyer, M.E. Hubbard and P.K. Jimack “An Optimally Efficient Technique for the Solution of Systems of Nonlinear Parabolic Partial Differential Equations” AiES in review, 2015

F. Yang, C. Venkataraman, V. Styles and A. Madzvamuse

“A Robust and Efficient Adaptive Multigrid Solver for the Optimal Control of Phase Field Formulations of Geometric Evolution Laws” CiCP in review, 2015

F. Yang, C. Venkataraman, V. Styles, V.Kuttenberger, E. Horn, Z.von Guttenberg and A. Madzvamuse

“A Computational Framework for Particle and Whole Cell Tracking Applied to a Real Biological Dataset” JBM Accepted for publication, http://dx.doi.org/10.1016/j.jbiomech.2016.02.008, 2016. Feng Wei Yang at BAMC in Oxford 5 April 2016 21 / 21