Biomedical subjects

P K Maini

Publications and source records attributed to P K Maini.

At least 19 recordsLinked to original sources

PubMed1998

Spatially varying equilibria of mechanical models: application to dermal wound contraction.

Mechanochemical models based on the Oster-Murray continuum framework have been applied to a variety of biological settings to obtain an understanding of the morphogenesis of living tissues. Wound-healing in mammalian skin is an important example, because a complex sequence of biochemical and biomechanical responses are orchestrated to close a wound by a combination of new tissue formation and wound contraction. Mechanical interactions between dermal fibroblastic cells and the collagen-rich extracellular matrix are crucial in the development of a contracted wound state. We and others have previously proposed mechanochemical models for wound repair to gain a greater understanding of both normal and abnormal healing. In the present work, the existence of spatially varying equilibria of these models is investigated by using a small-stain approximation and phase-plane techniques, with numerical simulations to confirm the analytical predictions. These results are sources of novel insight into the roles of key biological parameters in determining the mechanical properties of a contracted wound. These methods may also be relevant to other morphogenetic scenarios for which similar mechanochemical models have been proposed.

Animals↗

PubMed1998

Pattern formation in a generalized chemotactic model.

Many models have been proposed for spatial pattern formation in embryology and analyzed for the standard case of zero-flux boundary conditions. However, relatively little attention has been paid to the role of boundary conditions on the form of the final pattern. Here we investigate numerically, the effect of nonstandard boundary conditions on a model pattern generator, which we choose to be of a cell-chemotactic type. We specifically focus on the role of boundary conditions and the effects of scale and aspect ratio, and study the spatiotemporal dynamics of pattern formation. We illustrate the properties of the model by application to the spatiotemporal sequence of skeletal development.

Animals↗

PubMed1997

Hierarchically coupled ultradian oscillators generating robust circadian rhythms.

Ensembles of mutually coupled ultradian cellular oscillators have been proposed by a number of authors to explain the generation of circadian rhythms in mammals. Most mathematical models using many coupled oscillators predict that the output period should vary as the square root of the number of participating units, thus being inconsistent with the well-established experimental result that ablation of substantial parts of the suprachiasmatic nuclei (SCN), the main circadian pacemaker in mammals, does not eliminate the overt circadian functions, which show no changes in the phases or periods of the rhythms. From these observations, we have developed a theoretical model that exhibits the robustness of the circadian clock to changes in the number of cells in the SCN, and that is readily adaptable to include the successful features of other known models of circadian regulation, such as the phase response curves and light resetting of the phase.

Activity Cycles↗

PubMed1997

Role of fibroblast migration in collagen fiber formation during fetal and adult dermal wound healing.

Adult dermal wounds, in contrast to fetal wounds, heal with the formation of scar tissue. A crucial factor in determining the degree of scarring is the ratio of types I and III collagen, which regulates the diameter of the combined fibers. We developed a reaction-diffusion model which focuses on the control of collagen synthesis by different isoforms of the polypeptide transforming growth factor-beta (TGF beta). We used the model to investigate the current controversy as to whether the fibroblasts migrate into the wound from the surrounding unwounded dermis or from the underlying subcutaneous tissue. Numerical simulations of a spatially independent, temporal model led to a value of the collagen ratio consistent with that of healthy tissue for the fetus, but corresponding to scarring in the adult. We investigated the effect of topical application of TGF beta and show that addition of isoform 3 reduces scar tissue formation, in agreement with the experiment. However, numerical solutions of the reaction-diffusion system do not exhibit this sensitivity to growth factor application. Mathematically, this corresponds to the observation that behind healing wave-front solutions, a particular healed state is always selected independent of transients, even though there is a continuum of possible positive steady states. We explain this phenomenon using a caricature system of equations, which reflects the key qualitative features of the full model but has a much simpler mathematical form. Biologically, our results suggest that the migration into a wound of fibroblasts and TGF beta from the surrounding dermis alone cannot account for the essential features of the healing process, and that fibroblasts entering from the underlying subcutaneous tissue are crucial to the healing process.

Animals↗

PubMed1997

Bones, feathers, teeth and coat markings: a unified model.

A first necessary criterion to be met by any model for pattern formation is that it must be able to reproduce the patterns it purports to model. The above examples show how simple ideas from self-organisation can produce spatial patterns of varying complexity that are consistent with those observed experimentally. Second, the model must be consistent with the results of experimental manipulation. Third, the model must make experimentally testable predictions. In this way, a mathematical model can help to elucidate the underlying biochemical and biophysical mechanisms of pattern formation. I have tried to illustrate these ideas with the above examples. These examples also show the breadth of patterning phenomena that can be captured by these models. In animal coat markings, the patterns are laid down simultaneously. In limb development, the skeletal elements are laid down sequentially. The application to feather germ formation illustrates complex sequential regular pattern formation, while the application to tooth primordium formation illustrates sequential irregular pattern formation. Many other patterning phenomena have been studied (see, for example, ref. 7). Pattern formation is one of the central issues in developmental biology and intense interdisciplinary research involving experimentalists and theoreticians is beginning to help us understand how this phenomenon occurs. A detailed understanding of normal development is a necessary first step to the understanding of abnormal development and, hopefully, will help medical science combat developmental defects.

Alligators and Crocodiles↗

PubMed1997

A mathematical model for the capillary endothelial cell-extracellular matrix interactions in wound-healing angiogenesis.

Angiogenesis, the process by which new blood capillaries grow into a tissue from surrounding parent vessels, is a key event in dermal wound healing, malignant-tumour growth, and other pathologic conditions. In wound healing, new capillaries deliver vital metabolites such as amino acids and oxygen to the cells in the wound which are involved in a complex sequence of repair processes. The key cellular constituents of these new capillaries are endothelial cells: their interactions with soluble biochemical and insoluble extracellular matrix (ECM) proteins have been well documented recently, although the biological mechanisms underlying wound-healing angiogenesis are incompletely understood. Considerable recent research, including some continuum mathematical models, have focused on the interactions between endothelial cells and soluble regulators (such as growth factors). In this work, a similar modelling framework is used to investigate the roles of the insoluble ECM substrate, of which collagen is the predominant macromolecular protein. Our model consists of a partial differential equation for the endothelial-cell density (as a function of position and time) coupled to an ordinary differential equation for the ECM density. The ECM is assumed to regulate cell movement (both random and directed) and proliferation, whereas the cells synthesize and degrade the ECM. Analysis and numerical solutions of these equations highlights the roles of these processes in wound-healing angiogenesis. A nonstandard approximation analysis yields insight into the travelling-wave structure of the system. The model is extended to two spatial dimensions (parallel and perpendicular to the plane of the skin), for which numerical simulations are presented. The model predicts that ECM-mediated random motility and cell proliferation are key processes which drive angiogenesis and that the details of the functional dependence of these processes on the ECM density, together with the rate of ECM remodelling, determine the qualitative nature of the angiogenic response. These predictions are experimentally testable, and they may lead towards a greater understanding of the biological mechanisms involved in wound-healing angiogenesis.

Animals↗

PubMed1996

Pattern formation by lateral inhibition with feedback: a mathematical model of delta-notch intercellular signalling.

In many developing tissues, adjacent cells diverge in character so as to create a fine-grained pattern of cells in contrasting states of differentiation. It has been proposed that such patterns can be generated through lateral inhibition--a type of cell-cell interaction whereby a cell that adopts a particular fate inhibits its immediate neighbors from doing likewise. Lateral inhibition is well documented in flies, worms and vertebrates. In all of these organisms, the transmembrane proteins Notch and Delta (or their homologues) have been identified as mediators of the interaction--Notch as receptor, Delta as its ligand on adjacent cells. However, it is not clear under precisely what conditions the Delta-Notch mechanism of lateral inhibition can generate the observed types of pattern, or indeed whether this mechanism is capable of generating such patterns by itself. Here we construct and analyse a simple and general mathematical model of such contact-mediated lateral inhibition. In accordance with experimental data, the model postulates that receipt of inhibition (i.e. activation of Notch) diminished the ability to deliver inhibition (i.e. to produce active Delta). This gives rise to a feedback loop that can amplify differences between adjacent cells. We investigate the pattern-forming potential and temporal behaviour of this model both analytically and through numerical simulation. Inhomogeneities are self-amplifying and develop without need of any other machinery, provided the feedback is sufficiently strong. For a wide range of initial and boundary conditions, the model generates fine-grained patterns similar to those observed in living systems.

Animals↗

PubMed1996

A mathematical model for collagen fibre formation during foetal and adult dermal wound healing.

Adult dermal wounds, in contrast to foetal wounds, heal with the formation of scar tissue. A crucial factor in determining the nature of the healed tissue is the ratio of collagen 1 to collagen 3, which regulates the diameter of collagen fibres. We develop a mathematical model which focuses on the stimulus for collagen synthesis due to the secretion of the different isoforms of the regulatory chemical transforming growth factor beta. Numerical simulations of the model lead to a value of this ratio consistent with that of healthy tissue for the foetus but corresponding to scarring in adult wound healing. We investigate the effect of topical application of TGF beta isoforms during healing and determine the key parameters which control the difference between adult and foetal repair.

Animals↗

PubMed1996

A mathematical model for fibro-proliferative wound healing disorders.

The normal process of dermal wound healing fails in some cases, due to fibro-proliferative disorders such as keloid and hypertrophic scars. These types of abnormal healing may be regarded as pathologically excessive responses to wounding in terms of fibroblastic cell profiles and their inflammatory growth-factor mediators. Biologically, these conditions are poorly understood and current medical treatments are thus unreliable. In this paper, the authors apply an existing deterministic mathematical model for fibroplasia and wound contraction in adult mammalian dermis (Olsen et al., J. theor. Biol. 177, 113-128, 1995) to investigate key clinical problems concerning these healing disorders. A caricature model is proposed which retains the fundamental cellular and chemical components of the full model, in order to analyse the spatiotemporal dynamics of the initiation, progression, cessation and regression of fibro-contractive diseases in relation to normal healing. This model accounts for fibroblastic cell migration, proliferation and death and growth-factor diffusion, production by cells and tissue removal/decay. Explicit results are obtained in terms of the model processes and parameters. The rate of cellular production of the chemical is shown to be critical to the development of a stable pathological state. Further, cessation and/or regression of the disease depend on appropriate spatiotemporally varying forms for this production rate, which can be understood in terms of the bistability of the normal dermal and pathological steady states-a central property of the model, which is evident from stability and bifurcation analyses. The work predicts novel, biologically realistic and testable pathogenic and control mechanisms, the understanding of which will lead toward more effective strategies for clinical therapy of fibro-proliferative disorders.

Adult↗

PubMed1996

Simple versus sophisticated models of breath alcohol exhalation profiles.

For medicolegal purposes, breath alcohol content is typically determined from an end-expiratory sample. Measurements obtained by this method necessarily underestimate the alveolar breath alcohol content, and therefore underestimate the blood alcohol content. We suggest and analyse an improved paradigm which uses the entire time-series of breath alcohol measurements during exhalation, not simply the last recorded value. We present two mathematical models for the exhaling lung, and discuss the implications of each for more accurate and therefore more reliable breath alcohol measurement.

Adult↗

PubMed1995

A mechanochemical model for adult dermal wound contraction and the permanence of the contracted tissue displacement profile.

The healing of adult mammalian skin wounds involves a complex sequence of spatially and temporally coordinated processes. Wound contraction, by reducing the size of the injury, is an intrinsic component of full-thickness excisional dermal wound healing. The underlying biomechanics of wound contraction, however, are not fully understood, and little is known about the pathogenesis of severe medical conditions known as fibrocontractive diseases. The aim of this work is to investigate a deterministic mathematical model in order to obtain insight into the mechanistic relationships between wound contraction and associated normal and pathological healing processes. The model describes the essential roles of fibroblast and myofibroblast cells, a chemical growth factor and the extracellular matrix which includes type I collagen. The model results are qualitatively consistent with the biology of fibroplasia and wound contraction. It is shown that a contracted state evolves during a (long) transient phase of healing known as "proliferation", while collagen kinetics are fundamental to the considerably longer "remodelling" phase. Some quantitative results, notably on the evolution of wound contraction, compare favourably with experimental data. Application of the model to adult human dermal wound healing in vivo, with a greater understanding of the underlying biological mechanisms involved, may suggest strategies for controlling contraction and fibrocontractive diseases.

Adult↗

PubMed1995

Dictyostelium discoideum: cellular self-organization in an excitable biological medium.

The dynamics which govern the establishment of pattern and form in multicellular organisms remain a key problem of developmental biology. We study this question in the case of morphogenesis during aggregation of the slime mould Dictyostelium discoideum. Here detailed experimental information allows the formulation of a mechanistic model in which the central element is the coupling of the previously much-studied intracellular cyclic AMP signalling with the chemotactic cell response in cyclic AMP gradients. Numerical simulations of the model show quantitatively how signal relay, chemotactic movement and adaptation orchestrate the collective modes of cell signalling and migration in the aggregating cell layer. The interaction of chemotaxis with the cyclic AMP excitation waves causes the initially homogeneous cell layer to become unstable towards the formation of a branching cell stream pattern with close cell-cell contacts as observed in situ. The evolving cell morphology in turn leads to a pattern of non-homogeneous excitability of the medium and thus feeds back into the cAMP dynamics. This feedback can explain the decrease in signalling period and propagation speed with time, as well as observations on the structure of the spiral wave core in this self-organized excitable medium.

Animals↗

PubMed1995

Spatio-temporal patterns in a mechanical model for mesenchymal morphogenesis.

We present an in-depth study of spatio-temporal patterns in a simplified version of a mechanical model for pattern formation in mesenchymal morphogenesis. We briefly motivate the derivation of the model and show how to choose realistic boundary conditions to make the system well-posed. We firstly consider one-dimensional patterns and carry out a nonlinear perturbation analysis for the case where the uniform steady state is linearly unstable to a single mode. In two-dimensions, we show that if the displacement field in the model is represented as a sum of orthogonal parts, then the model can be decomposed into two sub-models, only one of which is capable of generating pattern. We thus focus on this particular sub-model. We present a nonlinear analysis of spatio-temporal patterns exhibited by the sub-model on a square domain and discuss mode interaction. Our analysis show that when a two-dimensional mode number admits two or more degenerate mode pairs, the solution of the full nonlinear system of partial differential equations is a mixed mode solution in which all the degenerate mode pairs are represented in a frequency locked oscillation.

Animals↗

PubMed1995

Rhythmic firing patterns in suprachiasmatic nucleus (SCN): the rôle of circuit interactions.

The suprachiasmatic nucleus (SCN) is believed to contain the main generator of circadian rhythmicity in mammals. In order to obtain further functional details of this, electrophysiological extracellular measurements in vitro were made. By means of an interspike interval distribution analysis, it is shown that there is a novel kind of neuronal firing pattern: the harmonic pattern. From these observations, we have developed a theoretical model based on possible filtering processes occurring during synaptic transmission. The model suffices to infer that regular ultradian oscillators could be an emergent property of circuit interactions of cells in the suprachiasmatic nucleus.

Action Potentials↗

PubMed1994

Travelling waves in a tissue interaction model for skin pattern formation.

Tissue interaction plays a major role in many morphogenetic processes, particularly those associated with skin organ primordia. We examine travelling wave solutions in a tissue interaction model for skin pattern formation which is firmly based on the known biology. From a phase space analysis we conjecture the existence of travelling waves with specific wave speeds. Subsequently, analytical approximations to the wave profiles are derived using perturbation methods. We then show numerically that such travelling wave solutions do exist and that they are in good agreement with out analytical results. Finally, the biological implications of our analysis are discussed.

Animals↗

PubMed1994

Nonlinear diffusion of a growth inhibitory factor in multicell spheroids.

A mathematical model is presented for the production of a growth inhibitory factor (GIF) within a multicell spheroid. The model is based on the assumption that the GIF diffuses within the spheroid in a nonlinear spatially dependent manner. This is in contrast with previous models, in which the nonlinearity was assumed in the production term. The results of the new model are compared with those of previous models and with experimental data.

Animals↗

PubMed1994

Mathematical modeling of corneal epithelial wound healing.

We propose a reaction-diffusion model of the mechanisms involved in the healing of corneal surface wounds. The model focuses on the stimulus for increased mitotic and migratory activity, specifically the role of epidermal growth factor. Analysis of the model equations elucidates the interaction and roles of the model parameters in determining the speed of healing and the shape of the traveling wave solutions which correspond to the migration of cells into the wound during the initial phase of healing. We determine an analytic approximation for the speed of traveling wave solutions of the model in terms of the parameters and verify the results numerically. By comparing the predicted speed with experimentally measured healing rates, we conclude that serum-derived factors can alone account for the overall features of the healing process, but that the supply of growth factors by the tear film in the absence of serum-derived factors is not sufficient to give the observed healing rate. Numerical solutions of the model equations also confirm the importance of both migration and mitosis for effective would healing. By modifying the model we obtain an analytic prediction for the healing rate of corneal surface wounds when epidermal growth factor is applied topically to the wound.

Animals↗

PubMed1993

Suicide substrate reaction--diffusion equations: varying the source.

The suicide substrate reaction is a model for certain enzyme-inhibiting drugs. This reaction system is examined assuming that the substrate diffuses freely while the enzyme remains fixed. Two sets of initial and boundary conditions are examined: one modelling an instantaneous point source, akin to an injection of substrate, the other, a continuous point source, akin to a continuing influx, or intravenous drip, of substrate. The quasi-steady-state assumption is applied to obtain analytical solutions for a limited parameter space. Finally, further applications of numerical and analytical experimentation on pharmaceutical mechanisms are described.

Enzyme Inhibitors↗