Phase differences in reaction-diffusion-advection systems and applications to morphogenesis

The authors study the effect of advection on reaction-diffusion patterns. It is shown that the addition of advection to a two-variable reaction–diffusion system with periodic boundary conditions results in the appearance of a phase difference between the patterns of the two variables which depends on the difference between the advection coefficients. The spatial patterns move like a travelling wave with a fixed velocity which depends on the sum of the advection coefficients. By a suitable choice of advection coefficients, the solution can be made stationary in time. In the presence of advection a continuous change in the diffusion coefficients can result in two Turing-type bifurcations as the diffusion ratio is varied, and such a bifurcation can occur even when the inhibitor species does not diffuse. It is also shown that the initial mode of bifurcation for a given domain size depends on both the advection and diffusion coefficients. These phenomena are demonstrated in the numerical solution of a particular reaction–diffusion system, and finally a possible application of the results to pattern formation in Drosophila larvae is discussed

Mathematical modelling of anisotropy in fibrous connective tissue

We present two modelling frameworks for studying dynamic anistropy in connective tissue, motivated by the problem of fibre alignment in wound healing. The first model is a system of partial differential equations operating on a macroscopic scale. We show that a model consisting of a single extracellular matrix material aligned by fibroblasts via flux and stress exhibits behaviour that is incompatible with experimental observations. We extend the model to two matrix types and show that the results of this extended model are robust and consistent with experiment. The second model represents cells as discrete objects in a continuum of ECM. We show that this model predicts patterns of alignment on macroscopic length scales that are lost in a continuum model of the cell population

Modelling of Tirapazamine effects on solid tumour morphology

Bioreductive drugs are in clinical practice to exploit the resistance from tumour microenvironments especially in the hypoxic region of tumour. We pre-sented a tumour treatment model to capture the pharmacology of one of the most prominent bioreductive drugs, Tirapazamine (TPZ) which is in clinical trials I and II. We calculated solid tumour mass in our previous work and then integrated that model with TPZ infusion. We calculated TPZ cytotoxicity, concentration, penetra-tion with increasing distance from blood vessel and offered resistance from micro-environments for drug penetration inside the tumour while considering each cell as an individual entity. The impact of these factors on tumour morphology is also showed to see the drug behaviour inside animals/humans tumours. We maintained the heterogeneity factors in presented model as observed in real tumour mass es-pecially in terms of cells proliferation, cell movement, extracellular matrix (ECM) interaction, and the gradients of partial oxygen pressure (pO2) inside tumour cells during the whole growth and treatment activity. The results suggest that TPZ high concentration in combination with chemotherapy should be given to get maximum abnormal cell killing. This model can be a good choice for oncologists and re-searchers to explore more about TPZ action inside solid tumour

Pattern formation of reaction-diffusion system having self-determined flow in the amoeboid organism of Physarum plasmodium

The amoeboid organism, the plasmodium of Physarum polycephalum, behaves on the basis of spatio-temporal pattern formation by local contraction-oscillators. This biological system can be regarded as a reaction-diffusion system which has spatial interaction by active flow of protoplasmic sol in the cell. Paying attention to the physiological evidence that the flow is determined by contraction pattern in the plasmodium, a reaction-diffusion system having self-determined flow arises. Such a coupling of reaction-diffusion-advection is a characteristic of the biological system, and is expected to relate with control mechanism of amoeboid behaviours. Hence, we have studied effects of the self-determined flow on pattern formation of simple reaction-diffusion systems. By weakly nonlinear analysis near a trivial solution, the envelope dynamics follows the complex Ginzburg-Landau type equation just after bifurcation occurs at finite wave number. The flow term affects the nonlinear term of the equation through the critical wave number squared. Contrary to this, wave number isn't explicitly effective with lack of flow or constant flow. Thus, spatial size of pattern is especially important for regulating pattern formation in the plasmodium. On the other hand, the flow term is negligible in the vicinity of bifurcation at infinitely small wave number, and therefore the pattern formation by simple reaction-diffusion will also hold. A physiological role of pattern formation as above is discussed.Comment: REVTeX, one column, 7 pages, no figur

Cell-scale degradation of peritumoural extracellular matrix fibre network and its role within tissue-scale cancer invasion

Local cancer invasion of tissue is a complex, multiscale process which plays an essential role in tumour progression. Occurring over many different temporal and spatial scales, the first stage of invasion is the secretion of matrix degrading enzymes (MDEs) by the cancer cells that consequently degrade the surrounding extracellular matrix (ECM). This process is vital for creating space in which the cancer cells can progress and it is driven by the activities of specific matrix metalloproteinases (MMPs). In this paper, we consider the key role of two MMPs by developing further the novel two-part multiscale model introduced in [33] to better relate at micro-scale the two micro-scale activities that were considered there, namely, the micro-dynamics concerning the continuous rearrangement of the naturally oriented ECM fibres within the bulk of the tumour and MDEs proteolytic micro-dynamics that take place in an appropriate cell-scale neighbourhood of the tumour boundary. Focussing primarily on the activities of the membrane-tethered MT1-MMP and the soluble MMP-2 with the fibrous ECM phase, in this work we investigate the MT1-MMP/MMP-2 cascade and its overall effect on tumour progression. To that end, we will propose a new multiscale modelling framework by considering the degradation of the ECM fibres not only to take place at macro-scale in the bulk of the tumour but also explicitly in the micro-scale neighbourhood of the tumour interface as a consequence of the interactions with molecular fluxes of MDEs that exercise their spatial dynamics at the invasive edge of the tumour

Malignant and morphogenetic waves

SIGLEAvailable from British Library Document Supply Centre-DSC:D192323 / BLDSC - British Library Document Supply CentreGBUnited Kingdo