An efficient and robust numerical algorithm for estimating parameters in Turing systems

Garvie, Marcus R.; Maini, Philip K.; Trenchea, Catalin

doi:10.1016/j.jcp.2010.05.040

Cited by 40 publications

(22 citation statements)

References 43 publications

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“…The reaction kinetics (5.2) are known as the activator-depleted substrate model [Lefever and Prigogine, 1968]. They have been used widely in the modelling of biological pattern formation phenomena and there has been progress in the analysis of RDSs on fixed and growing domains equipped with the kinetics (5.2) at the continuous level [Kolokolnikov et al, 2008;Venkataraman, 2011; and in the analysis of numerical methods to approximate the equations [Garvie et al, 2010;.…”

Section: Numerical Experiments With Artificial Datamentioning

confidence: 99%

“…Parameter identification for semilinear reaction-diffusion systems has received some attention in the literature Friedman and Reitich, 1992;Garvie et al, 2010;Jiang and Zhang, 2000] and there have also been some studies focussed on optimal control and related inverse problems in the context of geometric evolution equations and free boundary problems [Deckelnick et al, 2009[Deckelnick et al, , 2011Haußer et al, 2010Haußer et al, , 2012. There has also been some work on parameter estimation in models of cell motility.…”

Section: Introductionmentioning

confidence: 99%

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Parameter identification problems in the modelling of cell motility

et al. 2014

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ABSTRACT. We present a novel parameter identification algorithm for the estimation of parameters in models of cell motility using imaging data of migrating cells. Two alternative formulations of the objective functional that measures the difference between the computed and observed data are proposed and the parameter identification problem is formulated as a minimisation problem of nonlinear least squares type. A Levenberg-Marquardt based optimisation method is applied to the solution of the minimisation problem and the details of the implementation are discussed. A number of numerical experiments are presented which illustrate the robustness of the algorithm to parameter identification in the presence of large deformations and noisy data and parameter identification in three dimensional models of cell motility. An application to experimental data is also presented in which we seek to identify parameters in a model for the monopolar growth of fission yeast cells using experimental imaging data.

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Section: Numerical Experiments With Artificial Datamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Parameter identification problems in the modelling of cell motility

et al. 2014

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“…In some bio-medical applications the parameters could also be spatially dependent functions, like in [38,39] where the birth rate of proliferative cells in colon cancer is estimated. Recent studies on parameter estimation in the context of biological pattern formation ( [40], [41], [42]) have concentrated on development of the methodology, including demonstration of well-posedness and efficiency of solution methods. Here we apply parameter estimation methods using gradient descent algorithms in order to analyse the behaviour of our model of electrochemical pattern formation in different parameter regimes.…”

Section: Introductionmentioning

confidence: 99%

Parameter estimation for a morphochemical reaction-diffusion model of electrochemical pattern formation

Sgura

Lawless

Bozzini

2018

Inverse Problems in Science and Engineering

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Article Accepted VersionSgura, I., Lawless, A. S. and Bozzini, B. (2019) Parameter estimation for a morphochemical reactiondiffusion model of electrochemical pattern formation. ABSTRACTThe process of electrodeposition can be described in terms of a reaction-diffusion PDE system that models the dynamics of the morphology profile and the chemical composition. Here we fit such a model to the different patterns present in a range of electrodeposited and electrochemically modified alloys using PDE constrained optimization. Experiments with simulated data show how the parameter space of the model can be divided into zones corresponding to the different physical patterns by examining the structure of an appropriate cost function. We then use real data to demonstrate how numerical optimization of the cost function can allow the model to fit the rich variety of patterns arising in experiments. The computational technique developed provides a potential tool for tuning experimental parameters to produce desired patterns.

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“…It has been found that one key application of such optimal control formulations is to find solutions to pattern formation problems [10,11], and so it is natural to explore this particular application here.…”

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confidence: 99%

“…PDEconstrained optimization problems of similar form have been considered in the literature, such as in [10,11].…”

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confidence: 99%

Fast solvers for optimal control problems from pattern formation

Stoll

Pearson

Maini

2016

Journal of Computational Physics

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Abstract. The modelling of pattern formation in biological systems using various models of reaction-diffusion type has been an active research topic for many years. We here look at a parameter identification (or PDE-constrained optimization) problem where the Schnakenberg and Gierer-Meinhardt equations, two well-known pattern formation models, form the constraints to an objective function. Our main focus is on the efficient solution of the associated nonlinear programming problems via a Lagrange-Newton scheme. In particular we focus on the fast and robust solution of the resulting large linear systems, which are of saddle point form. We illustrate this by considering several two-and three-dimensional setups for both models. Additionally, we discuss an image-driven formulation that allows us to identify parameters of the model to match an observed quantity obtained from an image.Key words. PDE-constrained optimization, reaction-diffusion, pattern formation, Newton iteration, preconditioning, Schur complement.AMS subject classifications. 65F08, 65F10, 65F50, 92-08, 93C201. Introduction. One of the fundamental problems in developmental biology is to understand how spatial patterns, such as pigmentation patterns, skeletal structures, and so on, arise. In 1952, Alan Turing [36] proposed his theory of pattern formation in which he hypothesized that a system of chemicals, reacting and diffusing, could be driven unstable by diffusion, leading to spatial patterns (solutions which are steady in time but vary in space). He proposed that these chemical patterns, which he termed morphogen patterns, set up pre-patterns which would then be interpreted by cells in a concentration-dependent manner, leading to the patterns that we see.These models have been applied to a very wide range of areas (see, for example, Murray [23]) and have been shown to exist in chemistry [5,26]. While their applicability to biology remains controversial, there are many examples which suggest that Turing systems may be underlying key patterning processes (see [1,7,34] for the most recent examples). Two important models which embody the essence of the original Turing model are the Gierer-Meinhardt [13] and Schnakenberg models [33] and it is upon these models which we focus.

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An efficient and robust numerical algorithm for estimating parameters in Turing systems

Cited by 40 publications

References 43 publications

Parameter identification problems in the modelling of cell motility

Parameter identification problems in the modelling of cell motility

Parameter estimation for a morphochemical reaction-diffusion model of electrochemical pattern formation

Fast solvers for optimal control problems from pattern formation

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