Viscoelastic properties of proteoglycan subunits and aggregates in varying solution concentrations

Using a cone-on-plate mechanical spectrometer, we have measured the linear and non-linear rheological properties of cartilage proteoglycan solutions at concentrations similar to those found in situ. Solutions of bovine nasal cartilage proteoglycan subunits (22S) and aggregates (79S) were studied at concentrations ranging from 10 to 50 mg ml-1. We determined: (1) the complex viscoelastic shear modulus G (omega) under small amplitude (0.02 radians) oscillatory excitation at frequencies (omega) ranging from 1.0 to 20.0 Hz, (2) the non-linear shear rate (gamma) dependent apparent viscosity napp (gamma) in continuous shear, and (3) the non-linear shear rate dependent primary normal stress difference sigma 1 (gamma) in continuous shear. Both the apparent viscosity and normal stress difference were measured over four decades of shear rates ranging from 0.25 to 250 s-1. Analysis of the experimental results were performed using a variety of materially objective non-linear viscoelastic constitutive laws. We found that the non-linear, four-coefficient Oldroyd rate-type model was most effective for describing the measured flow characteristics of proteoglycan subunit and aggregate solutions. Values of the relaxation time lambda 1, retardation time lambda 2, zero shear viscosity no, and nonlinear viscosity parameter muo were computed for the aggregate and subunit solutions at all of the solute concentrations used. The four independent material coefficients showed marked dependence on the two different molecular conformations, i.e. aggregate or subunit, of proteoglycans in solution.     

Turing structures in an enzyme-induction system with gap junction-mediated non-linear diffusion

Two cells, each containing a reaction system modeling genetic induction, are coupled by diffusion. The substrate is moving through gap junctions, the number of which is regulated by the adjacent cells. This leads to a non-linear substrate diffusion term in the rate equations. Stability analysis reveals the conditions for the emergence of stable asymmetric solutions (dissipative structures). Due to non-linear diffusion rigid restrictions on the ratio of the two diffusion constants no longer exist. We demonstrate that substances operating as regulators of intercellular communication and participating in cellular metabolism may exhibit morphogenetic functions.     

Phytoplankton dynamics

We present a model of the phytoplankton dynamics. The distribution of the size of the phytoplankton aggregates is described by a non-linear transport equation that contains terms responsible for the growth of phytoplankton aggregates, their fragmentation and coagulation. We study asymptotic behaviour of moments of the solutions and we explain why phytoplankton tends to create large aggregates.     

A method for solution of the multi-objective inverse problems under uncertainty

We describe a method to solve multi-objective inverse problems under uncertainty. The method was tested on non-linear models of dynamic series and population dynamics, as well as on the spatiotemporal model of gene expression in terms of non-linear differential equations. We consider how to identify model parameters when experimental data contain additive noise and measurements are performed in discrete time points. We formulate the multi-objective problem of optimization under uncertainty. In addition to a criterion of least squares difference we applied a criterion which is based on the integral of trajectories of the system spatiotemporal dynamics, as well as a heuristic criterion CHAOS based on the decision tree method. The optimization problem is formulated using a fuzzy statement and is constrained by penalty functions based on the normalized membership functions of a fuzzy set of model solutions. This allows us to reconstruct the expression pattern of hairy gene in Drosophila even-skipped mutants that is in good agreement with experimental data. The reproducibility of obtained results is confirmed by solution of inverse problems using different global optimization methods with heuristic strategies.     

Effects of detrending and rescaling on correspondence analysis: solution stability and accuracy

Detrending and non-linear axis rescaling potentially improve the accuracy of gradient recovery in correspondence analyses but also reduce the stability or consistency of solutions. Variation among bootstrapped ordination solutions was compared across methods in analyses of both field and simulated data. Solution accuracy, measured with mean squared errors from Procrustes analysis, was compared using simulated data with known structure.Standard detrending-by-segments combined with non-linear rescaling entailed some cost in solution stability, but could improve the accuracy of solutions for long gradients. Without non-linear rescaling these solutions were usually less stable and less accurate. Although detrending-by-polynomials might be preferable on other grounds, it did not produce more accurate or stable solutions than detrending-by-segments.Abbreviations CA = correspondence analysis - DCA = detrended correspondence analysis - MSE = Procrustes mean squared error statistic - SD = standard deviation units of species turnover - SRV = scaled variance in species ranks     

Dynamics of prostate cancer stem cells with diffusion and organism response

We develop a systems based model for prostate cancer, as a sub-system of the organism. We accomplish this in two stages. We first start with a general ODE that includes organism response terms. Then, to account for normally observed spatial diffusion of cell populations, the ODE is extended to a PDE that includes spatial terms. Numerical solutions of the full PDE are provided, and are indicative of traveling wave fronts. This motivates the use of a well known transformation to derive a canonically related (non-linear) system of ODEs for traveling wave solutions. For biological feasibility, we show that the non-negative cone for the traveling wave system is time invariant. We also prove that the traveling waves have a unique global attractor. Biologically, the global attractor would be the limit for the avascular tumor growth. We conclude with comments on clinical implications of the model.     

Non-linear bioconvection in a deep suspension of gyrotactic swimming micro-organisms

 The non-linear structure of deep, stochastic, gyrotactic bioconvection is explored. A linear analysis is reviewed and a weakly non-linear analysis justifies its application by revealing the supercritical nature of the bifurcation. An asymptotic expansion is used to derive systems of partial differential equations for long plume structures which vary slowly with depth. Steady state and travelling wave solutions are found for the first order system of partial differential equations and the second order system is manipulated to calculate the speed of vertically travelling pulses. Implications of the results and possibilities of experimental validation are discussed. Received: 26 May 1997 / Revised version: 10 May 1998     

Weak convergence of discrete time non-markovian processes related to selection models in population genetics

We consider discrete time stochastic processes defined by solutions to some non-linear difference equations whose coefficients are autocorrelated random sequences. It is proved that these processes converge weakly in D[0, T] to diffusion processes, under the assumption that the random sequences satisfy some mixing condition. Diffusion approximation for stochastic selection models in population genetics is discussed, as the application of this limit theorem.     

An integral method for the analysis of blood flow

The existing methods to solve the problems of pulsatile flow in the cardiovascular system are based on either linear axisymmetric equations or non-linear one-dimensional equations. The solutions thus obtained give only a mediocre comparison with measurements. In this paper, a non-linear axisymmetric theory is proposed. The starting point of the present theory is a third degree polynomial representation of the velocity profile. Integral methods are then applied to obtain the governing equations. To ascertain the accuracy of the theory proposed above, the calculations for a simple case involving pulsatile flow in a long rigid tube were performed. The results are: (a) the average velocities compare very well with exact solutions and (b) the velocity profiles for a given frequency agree very well with exact solutions for flow in small tubes, but tend to differ as tube size is increased.     

Dynamic finite element implementation of nonlinear,anisotropic hyperelastic biological membranes

We present a novel method for the implementation of hyperelastic finite strain, non-linear strain-energy functions for biological membranes in an explicit finite element environment. The technique is implemented in LS-DYNA but may also be implemented in any suitable non-linear explicit code. The constitutive equations are implemented on the foundation of a co-rotational uniformly reduced Hughes-Liu shell. This shell is based on an updated-Lagrangian formulation suitable for relating Cauchy stress to the rate-of-deformation, i.e. hypo-elasticity. To accommodate finite deformation hyper-elastic formulations, a co-rotational deformation gradient is assembled over time, resulting in a formulation suitable for pseudo-hyperelastic constitutive equations that are standard assumptions in biomechanics. Our method was validated by comparison with (1) an analytic solution to a spherically-symmetric dynamic membrane inflation problem, incorporating a Mooney-Rivlin hyperelastic equation and (2) with previously published finite element solutions to a non-linear transversely isotropic inflation problem. Finally, we implemented a transversely isotropic strain-energy function for mitral valve tissue. The method is simple and accurate and is believed to be generally useful for anyone who wishes to model biologic membranes with an experimentally driven strain-energy function.     

Stochastic hybrid delay population dynamics: well-posed models and extinction

Nonlinear differential equations have been used for decades for studying fluctuations in the populations of species, interactions of species with the environment, and competition and symbiosis between species. Over the years, the original non-linear models have been embellished with delay terms, stochastic terms and more recently discrete dynamics. In this paper, we investigate stochastic hybrid delay population dynamics (SHDPD), a very general class of population dynamics that comprises all of these phenomena. For this class of systems, we provide sufficient conditions to ensure that SHDPD have global positive, ultimately bounded solutions, a minimum requirement for a realistic, well-posed model. We then study the question of extinction and establish conditions under which an ecosystem modelled by SHDPD is doomed.     

A perturbation method for the analysis of impulse propagation in a mathematical neuron model

A perturbation method is presented for the calculation of signal propagation velocities in a mathematical neuron model. Impulse frequencies and waveforms are also determined. The method uses a non-linear conduction model containing a small parameter, such that analytical solutions are possible, yet the inherently non-linear physiological phenomena involved, can be explained. A simple two-variable membrane equation is used, but the method is generally applicable to various more complete systems.     

Non-linear global optimization via parameterization and inverse function approximation: an artificial neural networks approach

In this article, a novel technique for non-linear global optimization is presented. The main goal is to find the optimal global solution of non-linear problems avoiding sub-optimal local solutions or inflection points. The proposed technique is based on a two steps concept: properly keep decreasing the value of the objective function, and calculating the corresponding independent variables by approximating its inverse function. The decreasing process can continue even after reaching local minima and, in general, the algorithm stops when converging to solutions near the global minimum. The implementation of the proposed technique by conventional numerical methods may require a considerable computational effort on the approximation of the inverse function. Thus, here a novel Artificial Neural Network (ANN) approach is implemented to reduce the computational requirements of the proposed optimization technique. This approach is successfully tested on some highly non-linear functions possessing several local minima. The results obtained demonstrate that the proposed approach compares favorably over some current conventional numerical (Matlab functions) methods, and other non-conventional (Evolutionary Algorithms, Simulated Annealing) optimization methods.     

Two-dimensional standing gradient osmotic flow: a generalization of the “isotonic convection approximation”

A linear two-dimensional model of a flow system for solute and fluid transport in intercellular spaces has been obtained by using the so called isotonic convection approximation, already employed in the one-dimensional case (Segel (1970)). This is equivalent to ignoring the convective components in the relevant differential equation. The solutions found by means of the eigenfunctions of a Sturm-Liouville system have been compared with the numerical solutions of the general non-linear model, in which also the convective terms are present.The results of the linear model agree fairly well with those of the non-linear one, in the range of interest of parameters. This fact shows that the ignorance of the profile of fluid velocity does not introduce significant errors in the evaluation of solute and fluid fluxes.The differences of results between the one-dimensional and the twodimensional, the linear and non-linear models, give a way to evaluate the relative effects of the diffusive and convective terms in the differential equation and to estimate the errors introduced by the one-dimensional approximation.     

Dynamic finite element modeling of poroviscoelastic soft tissue

Clinical evidences relative to biomechanical factors have demonstrated their important contribution to the behaviour of soft tissues. Finite element (FE) analysis is used to study the mechanical behaviour of soft tissue because it can provide numerical solutions to problems that are intractable to analytic solutions. This study focuses on the development of a FE model of a poroelastic biological tissue, which incorporates the viscoelastic material behaviour, finite deformation and inertial effect. The FE formulation is based on the weak form derived from the governing equation, and Newmark-beta method as well as Newton's method is incorporated into the implicit non-linear solutions. One-dimensional analytical solutions were used to verify the theoretical formulation and the numerical implementation of the proposed model. This study was further extended to analyze two-dimensional biomechanical models and the results clearly demonstrate the importance of including finite deformation, viscoelasticity and inertial effects.     

Mathematical modelling of tissue-engineered angiogenesis

We present a mathematical model for the vascularisation of a porous scaffold following implantation in vivo. The model is given as a set of coupled non-linear ordinary differential equations (ODEs) which describe the evolution in time of the amounts of the different tissue constituents inside the scaffold. Bifurcation analyses reveal how the extent of scaffold vascularisation changes as a function of the parameter values. For example, it is shown how the loss of seeded cells arising from slow infiltration of vascular tissue can be overcome using a prevascularisation strategy consisting of seeding the scaffold with vascular cells. Using certain assumptions it is shown how the system can be simplified to one which is partially tractable and for which some analysis is given. Limited comparison is also given of the model solutions with experimental data from the chick chorioallantoic membrane (CAM) assay.     

A non-linear analysis for spatial structure in a reaction-diffusion model

A non-linear stability analysis using a multi-scale perturbation procedure is carried out on the practical Thomas reaction-diffusion mechanism which exhibits bifurcation to non-uniform states. The analytical results compare favourably with the numerical solutions. The sequential patterns generated by this model by variations in a parameter related to the reaction-diffusion domain indicate its capacity to represent certain key morphogenetic features required in a recent model by Kauffman for pattern formation in theDrosophila embryo.     

Mean-square stability of a stochastic model for bacteriophage infection with time delays

We consider the stability properties of the positive equilibrium of a stochastic model for bacteriophage infection with discrete time delay. Conditions for mean-square stability of the trivial solution of the linearized system around the equilibrium are given by the construction of suitable Lyapunov functionals. The numerical simulations of the strong solutions of the arising stochastic delay differential system suggest that, even for the original non-linear model, the longer the incubation time the more the phage and bacteria populations can coexist on a stable equilibrium in a noisy environment for very long time.     

Characterization of non-linear behaviour of an animal membrane using electro-kinetic studies

Electro-kinetic studies of water, aqueous solutions of urea, thiourea, glucose, creatinine and urine solution across urinary bladder membranes of goat have been used to explain non-linear behaviour. It has been found that fluxes are non-linearly related with forces. The structure of the membrane is a decisive factor when the range of driving forces is not too high. Stability of the steady state has also been examined and has been found to remain quite stable even in non-linear regions. The results have been examined using the methodology of non-equilibrium thermodynamics.