Directional cell migration in dense three-dimensional (3D) environments critically depends upon shape adaptation and is impeded depending on the size and rigidity of the nucleus. Accordingly, the nucleus is primarily understood as a physical obstacle; however, its pro-migratory functions by stepwise deformation and reshaping remain unclear. Using atomic force spectroscopy, time-lapse fluorescence microscopy and shape change analysis tools, we determined the nuclear size, deformability, morphology and shape change of HT1080 fibrosarcoma cells expressing the Fucci cell cycle indicator or being pre-treated with chromatin-decondensating agent TSA. We show oscillating peak accelerations during migration through 3D collagen matrices and microdevices that occur during shape reversion of deformed nuclei (recoil), and increase with confinement. During G1 cell-cycle phase, nucleus stiffness was increased and yielded further increased speed fluctuations together with sustained cell migration rates in confinement when compared to interphase populations or to periods of intrinsic nuclear softening in the S/G2 cell-cycle phase. Likewise, nuclear softening by pharmacological chromatin decondensation or after lamin A/C depletion reduced peak oscillations in confinement. In conclusion, deformation and recoil of the stiff nucleus contributes to saltatory locomotion in dense tissues. This article is part of a discussion meeting issue ‘Forces in cancer: interdisciplinary approaches in tumour mechanobiology’.
We employ adaptive mesh refinement, implicit time stepping, a nonlinear multigrid solver and parallel computation, to solve a multi-scale, time dependent, three dimensional, nonlinear set of coupled partial differential equations for three scalar field variables. The mathematical model represents the non-isothermal solidification of a metal alloy into a melt substantially cooled below its freezing point at the microscale. Underlying physical molecular forces are captured at this scale by a specification of the energy field. The time rate of change of the temperature, alloy concentration and an order parameter to govern the state of the material (liquid or solid) is controlled by the diffusion parameters and variational derivatives of the energy functional. The physical problem is important to material scientists for the development of solid metal alloys and, hitherto, this fully coupled thermal problem has not been simulated in three dimensions, due to its computationally demanding nature. By bringing together state of the art numerical techniques this problem is now shown here to be tractable at appropriate resolution with relatively moderate computational resources. Figure 1: Snapshot of the solid-liquid interface for a typical dendrite. This image was obtained from a simulation with Le = 40, ∆ = 0.525 and ∆x = 0.78.The computational techniques we employ are: use of very fine meshing in the region around the moving boundary where phase field and solute field resolution is critical, and coarse meshing away from the boundary where only the slowly changing temperature field requires resolution; implicit time stepping to allow much larger time steps than would otherwise be possible; nonlinear smoothing in conjunction with a nonlinear multi-grid solver; and parallel processing with up to 1024 cores as the simulation progresses. The combination of all of these techniques allows an almost optimal solution process to be developed, in which the number of degrees of freedom is evolved with the dendrite, to maintain the required resolution as the interface grows, and the solution time at each time step is approximately proportional to the number of degrees of freedom. Furthermore, the use of a parallel implementation ensures that sufficient primary memory is available to support a mesh resolution which is fully converged whilst maintaining a tractable solution time.The particular phase field model we employ is an extension of [3], and is based on the three dimensional thermalphase field model of [4] and two dimensional thermal-solutal phase field model of [5]. One feature of the physical problem is that it is purely dissipative, or entropy increasing, as all natural relaxational phenomena are. The resulting PDEs are of Allen-Cahn [6] and Carn-Hilliard type [7]. That is to say, the model involves time derivatives of the three fields coupled to forms involving variational derivatives of some functional -typically the free energy functional. As the dendrite grows the free energy reduces monotonically with time but never achi...
We propose and investigate a novel solution strategy to efficiently and accurately compute approximate solutions to semilinear optimal control problems, focusing on the optimal control of phase field formulations of geometric evolution laws. The optimal control of geometric evolution laws arises in a number of applications in fields including material science, image processing, tumour growth and cell motility. Despite this, many open problems remain in the analysis and approximation of such problems. In the current work we focus on a phase field formulation of the optimal control problem, hence exploiting the well developed mathematical theory for the optimal control of semilinear parabolic partial differential equations. Approximation of the resulting optimal control problem is computationally challenging, requiring massive amounts of computational time and memory storage. The main focus of this work is to propose, derive, implement and test an efficient solution method for such problems. The solver for the discretised partial differential equations is based upon a geometric multigrid method incorporating advanced techniques to deal with the nonlinearities in the problem and utilising adaptive mesh refinement. An in-house two-grid solution strategy for the forward and adjoint problems, that significantly reduces memory requirements and CPU time, is proposed and investigated computationally. Furthermore, parallelisation as well as an adaptivestep gradient update for the control are employed to further improve efficiency. Along with a detailed description of our proposed solution method together with its implementation we present a number of computational results that demonstrate and evaluate our algorithms with respect to accuracy and efficiency. A highlight of the present work is simulation results on the optimal control of phase field formulations of geometric evolution laws in 3-D which would be computationally infeasible without the solution strategies proposed in the present work.
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