Mathematical Medicine and Biology - current issue

Single-equation models for the tear film in a blink cycle: realistic lid motion

2008-03-11

We consider model problems for the tear film over multiple blink cycles that utilize a single equation for the tear film; the single non-linear partial differential equation that governs the film thickness arises from lubrication theory. The two models that we consider arise from considering the absence of naturally occurring surfactant and the case when the surfactant is strongly affecting the surface tension. The film is considered on a time-varying domain length with specified film thickness and volume flux at each end; only one end of the domain is moving, which is analogous to the upper eyelid moving with each blink. Realistic lid motion from observed blinks is included in the model with end fluxes specified to more closely match the blink cycle than those previously reported. Numerical computations show quantitative agreement with in vivo tear film thickness measurements under partial blink conditions. A transition between periodic and non-periodic solutions has been estimated as a function of closure fraction and this may be a criterion for what is effectively a full blink according to fluid dynamics.

Shock formation and non-linear dispersion in a microvascular capillary network

Pop, S. R., Richardson, G., Waters, S. L., Jensen, O. E. — 2008-03-11

Temporal and spatial fluctuations are a common feature of blood flow in microvascular networks. Among many possible causes, previous authors have suggested that the non-linear rheological properties of capillary blood flow (notably the Fåhræus effect, the Fåhræus–Lindqvist effect and the phase-separation effect at bifurcations) may be sufficient to generate temporal fluctuations even in very simple networks. We have simulated blood flow driven by a fixed pressure drop through a simple arcade network using coupled hyperbolic partial differential equations (PDEs) that incorporate well-established empirical descriptions of these rheological effects, accounting in particular for spatially varying haematocrit distributions; we solved the PDE system using a characteristic-based method. Our computations indicate that, under physiologically realistic conditions, there is a unique steady flow in an arcade network which is linearly stable and that plasma skimming suppresses the oscillatory decay of perturbations. In addition, we find that non-linear perturbations to haematocrit distributions can develop shocks via the Fåhræus effect, providing a novel mechanism for non-linear dispersion in microvascular networks.

Bayesian support is larger than bootstrap support in phylogenetic inference: a mathematical argument

Britton, T., Svennblad, B., Erixon, P., Oxelman, B. — 2008-03-11

In phylogenetic inference, the support of an estimated phylogenetic tree topology and its interior branches is usually measured either with non-parametric bootstrap support (BS) values or with Bayesian posterior probabilities (BPPs). Extensive empirical evidence indicates that BPP values are systematically larger than BS when measured on the same data set, but there are no theoretical results supporting such a systematic difference. In the present note, we give a heuristic mathematical argument supporting the empirically observed phenomenon. The argument uses properties of the marginal and profile likelihoods of the normal distribution. The heuristic arguments are supported in a simulation study evaluating different steps in the argument.

The ant colony algorithm for feature selection in high-dimension gene expression data for disease classification

Robbins, K. R., Zhang, W., Bertrand, J. K., Rekaya, R. — 2008-03-11

The use of gene expression data to diagnose complex diseases represents an exciting area of medicine; however, such data sets are often noisy, requiring the selection of feature subsets to obtain maximum classification accuracy. Due to the high dimensions of many expression data sets, filter-based methods are commonly used, but often yield inconsistent results. Optimization algorithms can outperform filter methods, but often require preselection of features to achieve good results. To address the problems of many commonly used feature selection methods, the ant colony algorithm (ACA) is proposed for use on data sets with large numbers of features. The ACA is an optimization algorithm capable of incorporating prior information, allowing it to search the sample space more efficiently than other optimization methods. When applied to several high-dimensional data sets, the ACA was able to identify small subsets of highly predictive and biologically relevant genes without the need for extensive preselection of features. Using the selected genes to train a latent variable model yielded substantial increases in prediction accuracy when compared to several rank-based methods and results obtained in previous studies. The superiority of the ACA algorithm was validated through simulation.