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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.
    No preview · Article · Apr 2012 · Journal of Theoretical Biology
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    ABSTRACT: Tendons are composed of cells, blood vessels and extracellular matrix, intricately wo-ven together to form a vital musculoskeletal connective tissue. They act as a mechanical buffer for transmitting forces generated in muscles to bones, thus enabling movement. There is a growing need for functional imaging of tendon, for example to provide non-invasive biochemical and biomechanical insight into injured, diseased or repairing ten-don. This study focuses on ex vivo imaging using a novel combination of state of the art imaging technology, specifically ultra-high field magnetic resonance imaging (MRI) and near infrared-multiphoton laser scanning microscopy (NIR-MPLSM). We show that both imaging modalities are able to distinguish between normal and damaged tendon. MR imaging revealed macroscopic changes and evidence of tissue disruption; high sig-nal intensity corresponding to surrounding sheaths and some intra-tendinous regions. NIR-MPLSM demonstrated that local interactions between collagen and other matrix components are altered in enzyme-digested tendon. We propose a novel method for quantifying tendon disruption based on tendon crimp waveform parameterisation. Our findings suggest that MRI and NIR-MPLSM are useful technologies for extracting func-tional information from tendon. We provide insight into how biomechanical properties are related to biochemical parameters, and thus consider the future of in vivo functional MR imaging of tendon.
    Full-text · Article · Jan 2009
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    ABSTRACT: This paper presents a Particle Filter (PF) framework that stochastically detects and labels vessel bifurca-tions. The PF has previously been utilised in the segmentation of vascular structures, however has not demonstrated a consistent ability to detect and track bifurcations without the aid of user intervention. By incorporating a number of techniques into a specially designed vascular PF, including Markov Chain Monte Carlo (MCMC) rejuvenation and a spatially adaptive likelihood, we show that the simultaneous extraction of vessel centrelines, bifurcations and a hierarchical vascular representation is possible. This algorithm is shown to perform well on both synthetic and clinical data.
    Preview · Article · Jan 2008
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