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    ABSTRACT: Biological systems are typically modelled by nonlinear differential equations. In an effort to produce high fidelity representations of the underlying phenomena, these models are usually of high dimension and involve multiple temporal and spatial scales. However, this complexity and associated stiffness makes numerical simulation difficult and mathematical analysis impossible. In order to understand the functionality of these systems, these models are usually approximated by lower dimensional descriptions. These can be analysed and simulated more easily, and the reduced description also simplifies the parameter space of the model. This model reduction inevitably introduces error: the accuracy of the conclusions one makes about the system, based on reduced models, depends heavily on the error introduced in the reduction process. In this paper we propose a method to calculate the error associated with a model reduction algorithm, using ideas from dynamical systems. We first define an error system, whose output is the error between observables of the original and reduced systems. We then use convex optimisation techniques in order to find approximations to the error as a function of the initial conditions. In particular, we use the Sum of Squares decomposition of polynomials in order to compute an upper bound on the worst-case error between the original and reduced systems. We give biological examples to illustrate the theory, which leads us to a discussion about how these techniques can be used to model-reduce large, structured models typical of systems biology.
    Journal of Theoretical Biology 04/2012; 304:172-82.
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    ABSTRACT: Cardiac computational models of electrical conduction, mechanical activation, hemodynamics and metabolism require detailed information about the structural arrangement of functionally heterogeneous cardiac cell types. However, current state-of-the-art models lack anatomically accurate cell type localization, which limits their utility. Histological sections combine unique resolution with discrimination of tissues and anatomical structures, but they suffer from alignment and deformation problems. On the other hand, MRI datasets preserve the correct geometry, but provide less micro structural detail. This paper presents a method for aligning MRI and histological datasets to obtain a highly detailed, geometrically correct anatomical description of the heart. An iterative process is used to correct the various 2D and 3D, rigid and non-rigid transforms, introduced in the histology preparation and acquisition. Validation is performed by calculating distances between anatomical landmarks in both datasets, and by quantifying tissue overlap. Results illustrate the suitability of the proposed algorithm to produce detailed, accurate cardiac models.
    Proceedings of the 2007 IEEE International Symposium on Biomedical Imaging: From Nano to Macro, Washington, DC, USA, April 12-16, 2007; 01/2007
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