A computational framework for particle and whole cell tracking - PowerPoint PPT Presentation

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

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

Techniques based upon background removal Feng Wei Yang at BAMC in Oxford 5 April 2016 4 / 21

Directions of migration Spider plot Star plot Feng Wei Yang at BAMC in Oxford 5 April 2016 5 / 21

Individual cells Cell two Cell two Cell one Cell one Feng Wei Yang at BAMC in Oxford 5 April 2016 6 / 21

Our optimal control model The mass constrained mean curvature flow with forcing: � V V V ( x x x , t ) = ( − σ H ( x x x , t ) + η ( x x x , t ) + λ V ( t )) v v v ( x x , t ) on Γ( t ) , t ∈ (0 , T ] , x Γ(0) = Γ 0 . The phase-field approximation of the above equation - Allen-Cahn:  x , t ) − 1 x , t )) − 1 ∂ t φ ( x x x , t ) = △ φ ( x x ǫ 2 G ′ ( φ ( x x ǫ ( η ( x x x , t ) − λ ( t )) in Ω × (0 , T ] ,   ∇ φ · ν ν = 0 on ∂ Ω × (0 , T ] , ν Ω = φ 0 in Ω .  φ ( · , 0)  Feng Wei Yang at BAMC in Oxford 5 April 2016 7 / 21

Our optimal control model cont. The objective functional: � T J ( φ, η ) = 1 � x + θ � x )) 2 dx x , t ) 2 dx ( φ ( x x , T ) − φ obs ( x η ( x x x x x xdt , x 2 2 Ω 0 Ω and now we solve the minimisation problem: min η J ( φ, η ) , with J given above . Feng Wei Yang at BAMC in Oxford 5 April 2016 8 / 21

Our optimal control model cont. The adjoint equation to help computing the derivative of the objective functional: x , t ) + ǫ − 2 G ′′ ( φ ( x � ∂ t p ( x x x , t ) = −△ p ( x x x x , t )) p ( x x x , t ) in Ω × [0 , T ) , p ( x x , T ) = φ ( x x x x , T ) − φ obs ( x x x ) in Ω , and we update the control as � θη ℓ + 1 � η ℓ +1 = η ℓ − α ǫ p ℓ . Feng Wei Yang at BAMC in Oxford 5 April 2016 9 / 21

Numerical challenges Number of time steps Memory requirement (let’s consider double precision and 100 time steps) 2-D: 512 2 requires 0.4 gigabytes 3-D: 512 3 requires 215 gigabytes Feng Wei Yang at BAMC in Oxford 5 April 2016 10 / 21

Two-grid solution strategy One complete solve for the Allen-Cahn equation from t=(0,T] One time step Start the next η iteration Restrict the converged solution of ϕ Fine grid for the Allen-Cahn equation One complete solve for the adjoint equation from t=[T,0) Intermediate grid(s) Interpolate the computed η One time step Coarse grid for the adjoint equation Feng Wei Yang at BAMC in Oxford 5 April 2016 11 / 21

Cell one t=0 t=T/2 t=T Feng Wei Yang at BAMC in Oxford 5 April 2016 12 / 21

Cell one video Feng Wei Yang at BAMC in Oxford 5 April 2016 13 / 21

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

Analysis through tracking morphological changes � � { φ> 0 } 1 dx x Ω φ dx x x x − 0.8 0.1 Cell one Cell one Cell two Cell two 0.095 Desired shape for cell one − 0.81 Desired shape for cell one Desired shape for cell two Desired shape for cell two 0.09 − 0.82 0.085 − 0.83 0.08 Volume Mass − 0.84 0.075 0.07 − 0.85 0.065 − 0.86 0.06 − 0.87 0.055 − 0.88 0.05 0 5 10 15 20 25 0 5 10 15 20 25 Time (A.U.) Time (A.U.) Feng Wei Yang at BAMC in Oxford 5 April 2016 15 / 21

Real world example (2) Feng Wei Yang at BAMC in Oxford 5 April 2016 16 / 21

Euler number for topological changes We compute this Euler number for these time steps with an ”optimized“ control η : −△ φ + ∇|∇ φ | 2 · ∇ φ 1 � � � X = dx . 2 |∇ φ | 2 2 π ( a − b ) Ω( a , b ) Q. Du et al. J. Appl. Math. , 2005 Feng Wei Yang at BAMC in Oxford 5 April 2016 17 / 21

Real world example (2) Feng Wei Yang at BAMC in Oxford 5 April 2016 18 / 21

A 3-D example (a) (b) (c) (d) Feng Wei Yang at BAMC in Oxford 5 April 2016 19 / 21

A 3-D example video Feng Wei Yang at BAMC in Oxford 5 April 2016 20 / 21

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