Mathematical Medicine and Biology - recent issues

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.

Modelling HA protein-mediated interaction between an influenza virus and a healthy cell: pre-fusion membrane deformation

Vaidya, N. K., Huang, H., Takagi, S. — 2007-11-06

We present a mathematical model for pre-fusion interaction between an influenza virus and a healthy cell. Our model describes the role played by hemagglutinin (HA) protein clusters in bringing the viral membrane into close contact with the host cell membrane as a first step of the fusion process between the two membranes. The viral membrane is modelled as a lipid bilayer with bending rigidity. Using the calculus of variations, we compute the deformation of the viral membrane under the influence of HA protein clusters. Our numerical results support the hypothesis of dimple formation in the fusion site proposed in the literature. The asymmetric nature of the protein molecules due to various reasons such as tilting is the primary cause for the dimple formation. We discuss the effects of spontaneous curvature, the protein cluster radius, fusion-site size and the bending moment exerted by the protein cluster. We also examine the effects of membrane tension and the presence of a host cell on the dimple shape. Our results support previous experimental observations.

Modelling the geometric features and investigating electrical properties of dendrites in a fish thalamic neuron

Krzyzanski, W., Bell, J. — 2007-11-06

A certain pacific fish has a thalamic nerve cell with a unique dendritic geometry. Instead of the cell having a branching dendritic structure, the large cell of the corpus glomerulosum has, generally, a single dendritic stalk with a large bulbous tip. We formulate a cable-theory model that incorporates the geometry, and then, we solve the problem for a single, localized synaptic current source at the bulb. From the solution representation, we numerically examine the characteristics of the postsynaptic potential due to the presence of the bulbous tip.

Assessment of cancer immunotherapy outcome in terms of the immune response time features

2007-11-06

A cytokine-based periodic immunotherapy treatment is included in a model of tumour growth with a delay. The effects of dose schedule are studied in the case of a weak immune system and a growing tumour. We find the existence of ‘metastable’ states (that may last for tens of years) induced by the treatment and also potentially adverse effects of the dosage frequency on the stabilization of the tumour. These two effects depend on the delay between the tumour growth and the immune system response, the cytokine dose burden, and other parameters considered in the model.

Role of horizontal incidence in the occurrence and control of chaos in an eco-epidemiological system

Chatterjee, S., Kundu, K., Chattopadhyay, J. — 2007-11-06

A predator–prey model with disease in the prey population is proposed and analysed. The mode of disease transmission plays an important role in such dynamics. Keeping this factor in mind, we observe the dynamics of such a system for simple mass action incidence and standard incidence. Our observations indicate that the phenomenon of rarity or non-occurrence of chaos in our proposed model is well defined if the mode of disease transmission follows standard incidence. Moreover, using the method of Latin hypercube sampling, we show that the region of stability increases if the disease transmission follows the standard incidence law.

Non-linear modelling of breast tissue

Whiteley, J. P., Gavaghan, D. J., Chapman, S. J., Brady, J. M. — 2007-11-06

Previous approaches to modelling the large deformation of breast tissue, as occurs, e.g. in imaging using magnetic resonance imaging or mammography, include using linear elasticity and pseudo-non-linear elasticity, in which case the non-linear deformation is approximated by a series of small linear isotropic deformations, with the (constant) Young's modulus of each linear deformation an exponential function of the total non-linear strain. In this paper, these two approaches are compared to the solution of the full non-linear elastic problem for tissue with an exponential relationship between stress and strain. Having formulated each model and related the coefficients between the models, numerical simulations are performed on a block of incompressible material. These demonstrate that the simpler models may not be appropriate even in the case of modelling deformations of the human breast under gravity.

Spectral analysis of two-signed microarray expression data

Higham, D. J., Kalna, G., Vass, J. K. — 2007-05-23

We give a simple and informative derivation of a spectral algorithm for clustering and reordering complementary DNA microarray expression data. Here, expression levels of a set of genes are recorded simultaneously across a number of samples, with a positive weight reflecting up-regulation and a negative weight reflecting down-regulation. We give theoretical support for the algorithm based on a biologically justified hypothesis about the structure of the data, and illustrate its use on public domain data in the context of unsupervised tumour classification. The algorithm is derived by considering a discrete optimization problem and then relaxing to the continuous realm. We prove that in the case where the data have an inherent ‘checkerboard’ sign pattern, the algorithm will automatically reveal that pattern. Further, our derivation shows that the algorithm may be regarded as imposing a random graph model on the expression levels and then clustering from a maximum likelihood perspective. This indicates that the output will be tolerant to perturbations and will reveal ‘near-checkerboard’ patterns when these are present in the data. It is interesting to note that the checkerboard structure is revealed by the first (dominant) singular vectors—previous work on spectral methods has focussed on the case of nonnegative edge weights, where only the second and higher singular vectors are relevant. We illustrate the algorithm on real and synthetic data, and then use it in a tumour classification context on three different cancer data sets. Our results show that respecting the two-signed nature of the data (thereby distinguishing between up-regulation and down-regulation) reveals structures that cannot be gleaned from the absolute value data (where up- and down-regulation are both regarded as ‘changes’).

Why is the distribution of HTLV-I carriers geographically biased? An answer through a mathematical epidemic model

Eshima, N., Tabata, M., Okada, T. — 2007-05-23

Human T-cell leukemia virus type I (HTLV-I) is a retrovirus that causes adult T-cell leukemia, and the distribution of HTLV-I carriers is endemically biased, e.g. in Japan the density of carriers is high in Kyushu and Okinawa. In order to consider population dynamics over long times taking account of an increase and a decrease of a population we propose a continuous-time HTLV-I model. The model describes population dynamics of carrier numbers and that of carrier proportions, and theoretical results about an increase and a decrease of carriers are obtained. The present approach derives an explanation of the biased distribution of carriers in Japan.

Tissue growth in a rotating bioreactor. Part II: fluid flow and nutrient transport problems

Cummings, L. J., Waters, S. L. — 2007-05-23

Fluid flow and nutrient transport around a growing tissue construct within a cylindrical bioreactor of circular cross-section are considered. The bioreactor is filled with nutrient-rich culture medium, and the growing tissue construct is modelled as a cylindrical obstacle, also of circular cross-section, at a given (moving) position within the nutrient solution. The bioreactor rotates about its cylindrical axis, and its axial length is small relative to its radius (the high-aspect ratio vessel bioreactor). This small-aspect ratio means that a simple idealized model may be considered, in which (leading order) quantities are averaged across the axial direction. The leading-order fluid flow is then of Hele–Shaw type, and may be solved for explicitly. The trajectory of the tissue construct within the rotating bioreactor is determined by analysis of the various forces acting on it. Several different modes of motion are found to be possible, depending on the experimental conditions, and examples of each type of motion are presented. Additionally, we solve the problem for the nutrient transport around the tissue construct in the special case in which the construct remains fixed in the laboratory frame, and (as the cells proliferate in response to the nutrient available locally) deduce growth rates for the construct. Finally, we discuss our results in the light of possible experimental bioreactor set-ups. We note the present model's limitations, and consider how our work could be extended and improved to inform experimental protocols in future.

Mathematical analysis of a free-boundary model for lung branching morphogenesis

Hartmann, D., Miura, T. — 2007-05-23

Lung branching morphogenesis has been widely studied in the field of developmental biology. Lung airway trees consist of relatively regular-sized distal branches, but how this regular branched pattern is formed is not well understood. In the present study, we undertake a detailed mathematical analysis of the model proposed in (Hartmann & Miura (2006), which numerically captures branching morphogenesis of the simplest possible experimental system in vitro. We investigate analytically the stability of 1D travelling waves with respect to periodic perturbations in two dimensions. This linear stability analysis leads to the so-called dispersion relations, predicting that a certain representative length dominates in this model. As the analytical analysis is restricted to travelling waves, we generalize the linear analysis to any 1D solution by numerical simulations. Both results predict how the representative lengths will change by experimentally changing specific parameters. Finally, we discuss the importance of the analytical results from a biological point of view and propose an experimental scheme for a quantitative comparison between experiments and theory.

Parameter estimation of a respiratory control model from noninvasive carbon dioxide measurements during sleep

Aittokallio, T., Gyllenberg, M., Polo, O., Virkki, A. — 2007-05-23

A new method for estimating the parameters of a human gas exchange model is presented. Sensitivity analysis is used both to inspect the relative importance of the model parameters and to speed up the par-ameter estimation process. Multistart optimization is used to compensate for the effects of partial and noisy measurements. The validity of the method is first investigated with a test problem for which par-ameter identifiability is shown. The method is then applied to the estimation of sleep-related changes in the respiratory control system from the end-tidal and transcutaneous carbon dioxide measurements on human subjects. The results show that it is possible to gain insight into the behaviour of the rather complex physiological system using only a few noninvasive measurements and tractable computations.