An approximate solution to the Lamm equation

An approximate solution to the Lamm equation subject to the initial and boundary conditions for conventional sedimentation velocity experiments is derived and compared with the approximate solution of Fujita and MacCosham. Calculations with this solution demonstrate that the half-height method of estimating sedimentation coefficients yields correct values for epsilon < 0.02.     

Solvers for the cardiac bidomain equations

The bidomain equations are widely used for the simulation of electrical activity in cardiac tissue. They are especially important for accurately modeling extracellular stimulation, as evidenced by their prediction of virtual electrode polarization before experimental verification. However, solution of the equations is computationally expensive due to the fine spatial and temporal discretization needed. This limits the size and duration of the problem which can be modeled. Regardless of the specific form into which they are cast, the computational bottleneck becomes the repeated solution of a large, linear system. The purpose of this review is to give an overview of the equations and the methods by which they have been solved. Of particular note are recent developments in multigrid methods, which have proven to be the most efficient.     

A finite volume method solution for the bidomain equations and their application to modelling cardiac ischaemia

This paper presents an implementation of the finite volume method with the aim of studying subendocardial ischaemia during the ST segment. In this implementation, based on hexahedral finite volumes, each quadrilateral sub-face is split into two triangles to improve the accuracy of the numerical integration in complex geometries and when fibre rotation is included. The numerical method is validated against previously published solutions obtained from slab and cylindrical models of the left ventricle with subendocardial ischaemia and no fibre rotation. Epicardial potential distributions are then obtained for a half-ellipsoid model of the left ventricle. In this case it is shown that for isotropic cardiac tissue the degree of subendocardial ischaemia does not affect the epicardial potential distribution, which is consistent with previous findings from analytical studies in simpler geometries. The paper also considers the behaviour of various preconditioners for solving numerically the resulting system of algebraic equations resulting from the implementation of the finite volume method. It is observed that each geometry considered has its own optimal preconditioner.     

Eigensolutions of nonlinear wave equations in one dimension

A class of nonlinear equations describing the steady propagation of a disturbance on the infinite interval in one dimensional space are shown under certain conditions to admit solution with a unique velocity of propagation. The class of equations describe both initial and final homogeneous steady states which are asymptotically stable with respect to uniform perturbations, in contrast to the Fisher equation, which does not.     

A perturbation solution of the mechanical bidomain model

This research focuses on finding analytical solutions to the mechanical bidomain model for cardiac tissue. In particular, a perturbation expansion is used to analyze the equations, with the perturbation parameter being inversely proportional to the spring constant coupling the intracellular and extracellular spaces. The results indicate that the intracellular and extracellular pressures are not equal and that the two spaces can move relative to each other. This calculation is complicated enough to illustrate the implications of the mechanical bidomain model but is nevertheless simple enough to solve analytically. One application of the calculation is to the mechanical behavior of active cardiac tissue surrounding an ischemic region.     

An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso

In this paper we present a numerical method for the bidomain model, which describes the electrical activity in the heart. The model consists of two partial differential equations (PDEs), which are coupled to systems of ordinary differential equations (ODEs) describing electrochemical reactions in the cardiac cells. Many applications require coupling these equations to a third PDE, describing the electrical fields in the torso surrounding the heart. The resulting system is challenging to solve numerically, because of its complexity and very strict resolution requirements in time and space. We propose a method based on operator splitting and a fully coupled discretization of the three PDEs. Numerical experiments show that for simple simulation cases and fine discretizations, the algorithm is second-order accurate in space and time.     

A numerical solution of the mechanical bidomain model

Introduction: The mechanical bidomain model predicts forces on integrin proteins in the membrane. It has been solved analytically for idealized examples, but a numerical algorithm is needed to address realistic problems. Methods: The bidomain equations are approximated using finite differences. An ischemic region is modeled as a circular area having no active tension, surrounded by normal tissue. Results: The membrane force is large in the ischemic border zone, but is small elsewhere. Strain is distributed widely throughout the ischemic region and surrounding tissue. Conclusion: This calculation provides a testable prediction for the mechanism of mechanotransduction and remodeling in cardiac tissue.     

An accurate two-phase approximate solution to an acute viral infection model

During an acute viral infection, virus levels rise, reach a peak and then decline. Data and numerical solutions suggest the growth and decay phases are linear on a log scale. While viral dynamic models are typically nonlinear with analytical solutions difficult to obtain, the exponential nature of the solutions suggests approximations can be found. We derive a two-phase approximate solution to the target cell limited influenza model and illustrate its accuracy using data and previously established parameter values of six patients infected with influenza A. For one patient, the fall in virus concentration from its peak was not consistent with our predictions during the decay phase and an alternate approximation is derived. We find expressions for the rate and length of initial viral growth in terms of model parameters, the extent each parameter is involved in viral peaks, and the single parameter responsible for virus decay. We discuss applications of this analysis in antiviral treatments and in investigating host and virus heterogeneities.     

An approximate analytical solution to the lag period of monophenolase activity of tyrosinase

Tyrosinase shows a lag period in its action on monophenols (l-tyrosine). We propose an approximate analytical solution for the lag period, which fulfils the dependences with regard to initial enzyme concentration, and initial monophenol concentration. Furthermore, from a study of the dependences of the lag period on these variables, we can determine experimentally the o-diphenol concentration in the steady state. The Michaelis constant of the o-diphenol in the presence of the monophenol can be determined from the relationship between the o-diphenol concentration in the steady state and the initial monophenol concentration, taking into consideration the experimentally calculated Michaelis constant for the monophenol substrate. Although this Michaelis constant is much lower than the Michaelis constant for diphenol in the absence of monophenol, the binding site is the same. A kinetic analysis of the action mechanism of tyrosinase explains this difference in the values of the Michaelis constants.     

On boundary stimulation and optimal boundary control of the bidomain equations

The bidomain equations with Neumann boundary stimulation and optimal control of these stimuli are investigated. First an analytical framework for boundary control is provided. Then a parallel finite element based algorithm is devised and its efficiency is demonstrated not only for the direct problem but also for the optimal control problem. The computations realize a model configuration corresponding to optimal boundary defibrillation of a reentry phenomenon by applying current density stimuli.     

Linear algebraic transformations of the bidomain equations: Implications for numerical methods

A mathematical framework is presented for the treatment of the bidomain equations used to model propagation in cardiac tissue. This framework is independent of the model used to represent membrane ionic currents and incorporates boundary conditions and other constraints. By representing the bidomain equations in the operator notation , various algebraic transformations can be expressed as , where P and Q are linear operators. The authors show how previous work fits into this framework and discuss the implications of various transformation for numerical methods of solution. Although such transformations allow many choices of independent variable, these results emphasize the fundamental importance of the transmembrane potential.     

Galerkin-Ritz procedures for approximate solutions to systems of reaction-diffusion equations

For any essentially nonlinear system of reaction-diffusion equations of the generic form ∂c_i/∂t=D_i∇²c_i+Q_i(c,x,t) supplemented with Robin type boundary conditions over the surface of a closed bounded three-dimensional region, it is demonstrated that all solutions for the concentration distributionn-tuple function c=(c ₁(x,t),...,c _n (x,t)) satisfy a differential variational condition. Approximate solutions to the reaction-diffusion intial-value boundary-value problem are obtainable by employing this variational condition in conjunction with a Galerkin-Ritz procedure. It is shown that the dynamical evolution from a prescribed initial concentrationn-tuple function to a final steady-state solution can be determined to desired accuracy by such an approximation method. The variational condition also admits a systematic Galerkin-Ritz procedure for obtaining approximate solutions to the multi-equation elliptic boundary-value problem for steady-state distributions c=−c(x). Other systems of phenomenological (non-Lagrangian) field equations can be treated by Galerkin-Ritz procedures based on analogues of the differential variational condition presented here. The method is applied to derive approximate nonconstant steady-state solutions for ann-species symbiosis model.     

Diffusion-controlled reactions of enzymes. An approximate analytic solution of Chou's model

Based on Chou's model, a criterion was derived, by which one can judge whether or not the physical picture of the critical spherical shell described by Chou for an enzyme-substrate fast reaction system can emerge. Furthermore, for those reaction systems with such a physical picture, an approximate analytical solution was presented, which can be easily handled to calculate the upper limit of the diffusion-controlled reaction and the corresponding concentration distribution of substrate molecules on the surface of the major protein outside the active site. The results thus obtained are in good agreement with those computed by Chou et al. through the approach of numerical solution. Furthermore, the physical significance of the criterion and its relation to the critical spherical shell are substantiated during the process of derivation, which is very helpful for gaining an insight into this kind of biomolecular system with surprisingly high reaction rates.     

Isolated periodic solutions of the Hodgkin-Huxley equations

Periodic solutions of the current clamped Hodgkin-Huxley equations (Hodgkin & Huxley, 1952 J. Physiol. 117, 500) that arise by degenerate Hopf bifurcation were studied recently by Labouriau (1985 SIAM J. Math. Anal. 16, 1121, 1987 Degenerate Hopf Bifurcation and Nerve Impulse (Part II), in press). Two parameters, temperature T and sodium conductance gNa were varied from the original values obtained by Hodgkin & Huxley. Labouriau's work proved the existence of small amplitude periodic solution branches that do not connect locally to the stationary solution branch, and had not been previously computed. In this paper we compute these solution branches globally. We find families of isolas of periodic solutions (i.e. branches not connected to the stationary branch). For values of gNa in the range measured by Hodgkin & Huxley, and for physically reasonable temperatures, there are isolas containing orbitally asymptotically stable solutions. The presence of isolas of periodic solutions suggests that in certain current space clamped membrane experiments, action potentials could be observed even though the stationary state is stable for all current stimuli. Once produced, such action potentials will disappear suddenly if the current stimulus is either increased or decreased past certain values. Under some conditions, "jumping" between action potentials of different amplitudes might be observed.     

Extending the limits to enzymatic catalysis: diffusion of ribonuclease A in one dimension

Bovine pancreatic ribonuclease A (RNase A) is a distributive endoribonuclease that catalyzes the cleavage of the P-O5' bond of RNA on the 3' side of pyrimidine residues. Here, RNase A is shown to cleave the P-O5' bond of a pyrimidine ribonucleotide faster when the substrate is embedded within a longer tract of poly(adenylic acid) [poly(A)] or poly(deoxyadenylic acid) [poly(dA)]. These data indicate that a ribonuclease can diffuse in one dimension along a single-stranded nucleic acid. This facilitated diffusion is mediated by Coulombic interactions, as the extent is diminished by the addition of NaCl. RNase A is more effective at cleaving a pyrimidine ribonucleotide embedded within a poly(dA) tract than within a poly(deoxycytidylic acid) [poly(dC)] tract. T45G RNase A, which catalyzes the processive cleavage of poly(A) but the distributive cleavage of poly(cytidylic acid) [poly(C)], has the same preference. Apparently, processive catalysis by the T45G enzyme arises from the expanded substrate specificity of the variant superimposed upon an intrinsic ability to diffuse along poly(A). Homologous ribonucleases with cytotoxic activity may rely on facilitated diffusion along poly(A) tails for efficient degradation of the essential information encoded by cellular mRNA.     

An approximate solution for a transient two-phase stirred tank bioreactor with nonlinear kinetics

The derivation of an approximate solution method for models of a continuous stirred tank bioreactor where the reaction takes place in pellets suspended in a well-mixed fluid is presented. It is assumed that the reaction follows a Michaelis-Menten-type kinetics. Analytic solution of the differential equations is obtained by expanding the reaction rate expression at pellet surface concentration using Taylor series. The concept of a pellet's dead zone is incorporated; improving the predictions and avoiding negative values of the reagent concentration. The results include the concentration expressions obtained for (a) the steady state, (b) the transient case, imposing the quasi-steady-state assumption for the pellet equation, and (c) the complete solution of the approximate transient problem. The convenience of the approximate method is assessed by comparison of the predictions with the ones obtained from the numerical solution of the original problem. The differences are in general quite acceptable.     

Spring-mass running: simple approximate solution and application to gait stability

The planar spring-mass model is frequently used to describe bouncing gaits (running, hopping, trotting, galloping) in animal and human locomotion and robotics. Although this model represents a rather simple mechanical system, an analytical solution predicting the center of mass trajectory during stance remains open. We derive an approximate solution in elementary functions assuming a small angular sweep and a small spring compression during stance. The predictive power and quality of this solution is investigated for model parameters relevant to human locomotion. The analysis shows that (i), for spring compressions of up to 20% (angle of attack > or = 60 degree, angular sweep < or = 60 degree) the approximate solution describes the stance dynamics of the center of mass within a 1% tolerance of spring compression and 0.6 degree tolerance of angular motion compared to numerical calculations, and (ii), despite its relative simplicity, the approximate solution accurately predicts stable locomotion well extending into the physiologically reasonable parameter domain. (iii) Furthermore, in a particular case, an explicit parametric dependency required for gait stability can be revealed extending an earlier, empirically found relationship. It is suggested that this approximation of the planar spring-mass dynamics may serve as an analytical tool for application in robotics and further research on legged locomotion.     

Periodic response to periodic forcing of the Droop equations for phytoplankton growth

The dynamics of a phytoplankton population growing in a chemostat under a periodic supply of nutrients is investigated with the model proposed by Droop. This model differs from the well-known Monod equations by incorporating nutrient storage by the cells. In spite of its nonlinearity and the time delays introduced by an internal nutrient pool, the model predicts a simple response to a periodic nutrient supply. The population is shown to oscillate with the same frequency as the forcing. To prove the existence of a periodic solution local and global bifurcation results are used. This work establishes a basis on which to evaluate experimental data against the model as a representation of the nutrient-phytoplankton interaction when nutrients fluctuate.