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Understanding the pathophysiological processes of osteoarthritis (OA) require adequate model systems. Although different in vitro or in vivo models have been described, further comprehensive approaches are needed to study specific parts of the disease. This study aimed to combine in vitro and in silico modeling to describe cellular and matrix-related changes during the early phase of OA. We developed an in vitro OA model based on scaffold-free cartilage-like constructs (SFCCs), which was mathematically modeled using a partial differential equation (PDE) system to resemble the processes during the onset of OA. SFCCs were produced from mesenchymal stromal cells and analyzed weekly by histology and qPCR to characterize the cellular and matrix-related composition. To simulate the early phase of OA, SFCCs were treated with interleukin-1β (IL-1β), tumor necrosis factor α (TNFα) and examined after 3 weeks or cultivated another 3 weeks without inflammatory cytokines to validate the regeneration potential. Mathematical modeling was performed in parallel to the in vitro experiments. SFCCs expressed cartilage-specific markers, and after stimulation an increased expression of inflammatory markers, matrix degrading enzymes, a loss of collagen II (Col-2) and a reduced cell density was observed which could be partially reversed by retraction of stimulation. Based on the PDEs, the distribution processes within the SFCCs, including those of IL-1β, Col-2 degradation and cell number reduction was simulated. By combining in vitro and in silico methods, we aimed to develop a valid, efficient alternative approach to examine and predict disease progression and new therapeutic strategies.
Understanding the pathophysiological processes of cartilage degradation requires adequate model systems to develop therapeutic strategies towards osteoarthritis (OA). Although different in vitro or in vivo models have been described, further comprehensive approaches are needed to study specific disease aspects. This study aimed to combine in vitro and in silico modeling based on a tissue-engineering approach using mesenchymal condensation to mimic cytokine-induced cellular and matrix-related changes during cartilage degradation. Thus, scaffold-free cartilage-like constructs (SFCCs) were produced based on self-organization of mesenchymal stromal cells (mesenchymal condensation) and i) characterized regarding their cellular and matrix composition or secondly ii) treated with interleukin-1β (IL-1β) and tumor necrosis factor α (TNFα) for 3 weeks to simulate OA-related matrix degradation. In addition, an existing mathematical model based on partial differential equations was optimized and transferred to the underlying settings to simulate distribution of IL-1β, type II collagen degradation and cell number reduction. By combining in vitro and in silico methods, we aim to develop a valid, efficient alternative approach to examine and predict disease progression and effects of new therapeutics.
In this paper we present a self--adaptive finite element method to solve flame propagation problems in 3D. An implicit time integrator of Rosenbrock type is coupled with a multilevel approach in space. The proposed method is applied to an unsteady thermo--diffusive combustion model to demonstrate its potential for the solution of complicated problems.
An integrated time--space adaptive finite element method for solving mixed systems of nonlinear parabolic, elliptic, and differential algebraic equations is presented. The approach is independent of the spatial dimension. For the discretization in time we use singly diagonally linearly implicit Runge--Kutta methods of Rosenbrock type. Local time errors for the step size control are defined by an embedded strategy. A multilevel finite element Galerkin method is subsequently applied for the discretization in space. A posteriori estimates of local spatial discretization errors are obtained solving local problems with higher order approximation. Superconvergence arguments allow to simplify the required computations. Two different strategies to obtain the start grid of the multilevel process are compared. The devised method is applied to a solid--solid combustion problem.
We present an integrated time--space adaptive finite element method for solving systems of twodimensional nonlinear parabolic systems in complex geometry. The partial differential system is first discretized in time using a singly linearly implicit Runge--Kutta method of order three. Local time errors for the step size control are defined by an embedding strategy. These errors are used to propose a new time step by a PI controller algorithm. A multilevel finite element method with piecewise linear functions on unstructured triangular meshes is subsequently applied for the discretization in space. The local error estimate of the finite element solution steering the adaptive mesh refinement is obtained solving local problems with quadratic trial functions located essentially at the edges of the triangulation. This two--fold adaptivity successfully ensures an a priori prescribed tolerance of the solution. The devised method is applied to laminar gaseous combustion and to solid--solid alloying reactions. We demonstrate that for such demanding applications the employed error estimation and adaption strategies generate an efficient and versatile algorithm.
Selfadaptive Finite Element Computations with Smooth Time Controller and Anisotropic Refinement
(1996)
We present Multilevel Finite Element computations for twodimensional reaction-diffusion systems modelling laminar flames. These systems are prototypes for extreme stiffness in time and space. The first of these two rather general features is accounted for by an improved control mechanism for the time step. The second one is reflected through very thin travelling reaction fronts for which we propose an anisotropic discretization by local directional refinement.
Dynamical simulation of industrially relevant processes strongly advises the use of algorithms, which are {\em adaptive} both in time and in space discretization. The paper presents two alternatives: (a) a fully adaptive method of lines approach, which is based on finite difference methods and essentially applicable to 1D problems; (b) a fully adaptive Rothe method, which is based on a fast multilevel finite element method and applicable to 1D up to 3D.
We present an adaptive Rothe method for two--dimensional problems combining an embedded Runge--Kutta scheme in time and a multilevel finite element discretization in space. The spatial discretization error is controlled by a posteriori error estimates based on interpolation techniques. A computational example for a thermodiffusive flame propagation model illustrates the high accuracy that is possible with the proposed method.
Large chemical computations show the need for full adaptivity supporting the development of robust and highly efficient programs. For solutions possessing sharp moving spatial transitions, as travelling wavefronts or emerging boundary and internal layers, an automatic adjustment of both the space and the time stepsize is generally accepted to be more successful in efficient resolving critical regions of high spatial and temporal activity. In contrast to the widespread discretization sequence first space then time the reversed sequence first time then space is employed. Full adaptivity of the proposed algorithm is realized by combining embedded time discretization and multilevel finite element space discretization. In this paper the algorithm is described for one--dimensional problems. The numerical results show the significantly new perspectives opened by this approach.
A Finite Element Method Adaptive in Space and Time for Nonlinear Reaction-Diffusion- Systems.
(1992)
Large scale combustion simulations show the need for adaptive methods. First, to save computation time and mainly to resolve local and instationary phenomena. In contrast to the widespread method of lines, we look at the reaction- diffusion equations as an abstract Cauchy problem in an appropriate Hilbert space. This means, we first discretize in time, assuming the space problems solved up to a prescribed tolerance. So, we are able to control the space and time error separately in an adaptive approach. The time discretization is done by several adaptive Runge-Kutta methods whereas for the space discretization a finite element method is used. The different behaviour of the proposed approaches are demonstrated on many fundamental examples from ecology, flame propagation, electrodynamics and combustion theory. {\bf Keywords:} initial boundary value problem, Rothe- method, adaptive Runge-Kutta method, finite elements, mesh refinement. {\bf AMS CLASSIFICATION:} 65J15, 65M30, 65M50.