The determination of the parameter of Bertalanffy's growth differential equations by means of nonlinear internal regression

A short survey is given on various parameterized versions of the logistic law of growth and of Bertalanffy's growth differential equations. To examine the validity of these various growth expressions internal nonlinear regressions were performed, and the results of the calculations are presented. The body length growth of man within the embryonic development serves as examples of a growth process. The parameters in the differential equations will be adjusted to the course of the divided central differences calculated from means of measured values of this growth process.     

Stability analysis of a continuum-based constrained mixture model for vascular growth and remodeling

A stabilizing criterion is derived for equations governing vascular growth and remodeling. We start from the integral state equations of the continuum-based constrained mixture theory of vascular growth and remodeling and obtain a system of time-delayed differential equations describing vascular growth. By employing an exponential form of the constituent survival function, the delayed differential equations can be reduced to a nonlinear ODE system. We demonstrate the degeneracy of the linearized system about the homeostatic state, which is a fundamental cause of the neutral stability observations reported in prior studies. Due to this degeneracy, stability conclusions for the original nonlinear system cannot be directly inferred. To resolve this problem, a sub-system is constructed by recognizing a linear relation between two states. Subsequently, Lyapunov’s indirect method is used to connect stability properties between the linearized system and the original nonlinear system, to rigorously establish the neutral stability properties of the original system. In particular, this analysis leads to a stability criterion for vascular expansion in terms of growth and remodeling kinetic parameters, geometric quantities and material properties. Numerical simulations were conducted to evaluate the theoretical stability criterion under broader conditions, as well as study the influence of key parameters and physical factors on growth properties. The theoretical results are also compared with prior numerical and experimental findings in the literature.     

Solutions to axon equations

The solutions to a general class of axon partial differential equations proposed by FitzHugh which includes the Hodgkin-Huxley equations are studied. It is shown that solutions to the partial differential equations are exactly the solutions to a related set of integral equations. An iterative procedure for constructing the solutions based on standard methods for ordinary differential equations is given and each set of initial values is shown to lead to a unique solution. Continuous dependence of the solutions on the initial values is established and solutions with initial values in a restricted (physiological) range are shown to remain in that range for all time. The iterative procedure is not suggested as the basis for numerical integration.     

Seasonally modified forms of the revised Janoschek function

Early attempts for modifying growth functions to annual variations dating back up to 2 decades are recalled together with examples for their application showing rather different degrees of approximation. In order to secure independence from special functions, Sager (1982) has proposed a general concept for the modification of growth functions. In this way, examples were treated with the Pütter-Bertalanffy's for crustaceans and clams, with the Gompertz' for the goby, and with the Richards' for the pollack - a near relative of the cod. In continuation of these endevours, the revised Janoschek's is presented in 2 forms, namely one adapted to birth data or in an unbound variant. Special attention is given to the evaluation of the parameters used in nonlinear regressions for both forms. The new equation for bound growth opens a chance for giving real birth values even in seasonally changing growth if one deciding parameter will surmount 1 as has been the case in many applications heretofore. As an example the mussel Mytilus edulis taken from marine "farms" in the Menai-Straits of North Wales is treated with the seasonally modified Janoschek function. Although the special case cannot be realized in this case, the curves for length growth and growth increase are rather instructive compared with the basic behaviour with lacking annual variations. Approximations from 0.5 to 2.7 a show very close agreement with natural values of length as can be verified from numerical and graphical displays.     

The Routine Fitting of Kinetic Data to Models: A Mathematical Formalism for Digital Computers

A mathematical formalism is presented for use with digital computers to permit the routine fitting of data to physical and mathematical models. Given a set of data, the mathematical equations describing a model, initial conditions for an experiment, and initial estimates for the values of model parameters, the computer program automatically proceeds to obtain a least squares fit of the data by an iterative adjustment of the values of the parameters. When the experimental measures are linear combinations of functions, the linear coefficients for a least squares fit may also be calculated. The values of both the parameters of the model and the coefficients for the sum of functions may be unknown independent variables, unknown dependent variables, or known constants. In the case of dependence, only linear dependencies are provided for in routine use. The computer program includes a number of subroutines, each one of which performs a special task. This permits flexibility in choosing various types of solutions and procedures. One subroutine, for example, handles linear differential equations, another, special non-linear functions, etc. The use of analytic or numerical solutions of equations is possible.     

A novel approach for estimating growth phases and parameters of bacterial population in batch culture

Using mathematical analysis, a new method has been developed for studying the growth kinetics of bacterial populations in batch culture. First, sampling data were smoothed with the spline interpolation method. Second, the instantaneous rates were derived by numerical differential techniques and finally, the derived data were fitted with the Gaussian function to obtain growth parameters. We named this the Spline-Numerical-Gaussian or SNG method. This method yielded more accurate estimates of the growth rates of bacterial populations and new parameters. It was possible to divide the growth curve into four different but continuous phases based on changes in the instantaneous rates. The four phases are the accelerating growth phase, the constant growth phase, the decelerating growth phase and the declining phase. Total DNA content was measured by flow cytometry and varied depending on the growth phase. The SNG system provides a very powerful tool for describing the kinetics of bacterial population growth. The SNG method avoids the unrealistic assumptions generally used in the traditional growth equations.     

A novel approach for estimating growth phases and parameters of bacterial population in batch culture

Traditionally, microbiologists divided bacterial growth in batch cultures into lag, exponential, station-ary and death phases[1], following the Logistic equa-tion that has been applied to the growth of human populations. The growth curves can always be ch…     

Parameter identification of the calvin photosynthesis cycle

Summary This article is concerned with the determination of kinetic parameters of the Calvin photosynthesis cycle which is described by seventeen nonlinear ordinary differential equations. It is shown that the task requires dynamic data for several sets of initial conditions. The numerical technique is based upon an algorithm for non-linear optimization and Gear's numerical integration scheme for stiff systems of differential equations. The sensitivity of the parameters to noise in the data is tested with a method adapted from Rosenbrook and Storey. A preliminary set of parameters has been obtained from a preliminary set of experimental data. The numerical methods are then tested with synthetic data derived from these parameters. The mathematical model and the results obtained in the simulation are used as an aid in designing new experiments.     

Simulation of citric acid production by rotating disk contactor

A simple model was presented to describe the time courses of citric acid production by a rotating disc contactor (RDC) using Aspergillus niger. The model is expressed by Monod-type cell growth, Luedeking-Piret-type citric acid production rate equations, and the diffusion equation for oxygen in the biofilm. The model contains five parameters which were determined by the nonlinear least squares method by fitting the numerical solution to the experimental data. In solving the equations, the cell density of the biofilm was estimated from the value of cellular mass per unit of biofilm area using an empirical equation. The experimental time courses in citric acid production period were well simulated with this model. The relation between the specific biofilm surface area and the rate of citric acid production was also explained by the simulation using the average values of five parameters of twelve runs. (c) 1997 John Wiley & Sons, Inc. Biotechnol Bioeng 56: 689-696, 1997.     

A survey of non-linear optimization techniques

Optimization means the provision of a set of numerical parameter values which will give the best fit of an equation, or series of equations, to a set of data. For simple systems this can be done by differentiating the equations with respect to each parameter in turn, setting the set of partial differential equations to zero, and solving this set of simultaneous equations (as for exwnple in linear regression). In more complicated cases, however, it may be impossible to differentiate the equations, or very difficultly soluble non-linear equations may result. Many numerical optimization techniques to overcome these difficulties have been developed in the least ten years, and this review explains the logical basis of most of them, without going into the detail of computational procedures.The methods fall naturally into two classes - direct search methods, in which only values of the function to be minimized (or maximized) are used - and gradient methods, which also use derivatives of the function. The author considers all the accepted methods in each class, although warning that gradient methods should not be used unless the analytical differentiation of the function to be minimized is possible.If the solution is constrained, that is, certain values of the parameters are regarded as impossible or certain relations between the parameter values must be obeyed, the problem is more difficult. The second part of the review considers methods which have been proposed for the solution of constrained optimization problems.     

Parameter estimation for stiff equations of biosystems using radial basis function networks

Background  

The modeling of dynamic systems requires estimating kinetic parameters from experimentally measured time-courses. Conventional global optimization methods used for parameter estimation, e.g. genetic algorithms (GA), consume enormous computational time because they require iterative numerical integrations for differential equations. When the target model is stiff, the computational time for reaching a solution increases further.     

Efficient simulations of tubulin-driven axonal growth

This work concerns efficient and reliable numerical simulations of the dynamic behaviour of a moving-boundary model for tubulin-driven axonal growth. The model is nonlinear and consists of a coupled set of a partial differential equation (PDE) and two ordinary differential equations. The PDE is defined on a computational domain with a moving boundary, which is part of the solution. Numerical simulations based on standard explicit time-stepping methods are too time consuming due to the small time steps required for numerical stability. On the other hand standard implicit schemes are too complex due to the nonlinear equations that needs to be solved in each step. Instead, we propose to use the Peaceman–Rachford splitting scheme combined with temporal and spatial scalings of the model. Simulations based on this scheme have shown to be efficient, accurate, and reliable which makes it possible to evaluate the model, e.g. its dependency on biological and physical model parameters. These evaluations show among other things that the initial axon growth is very fast, that the active transport is the dominant reason over diffusion for the growth velocity, and that the polymerization rate in the growth cone does not affect the final axon length.     

Numerical Simulation for the Unsteady MHD Flow and Heat Transfer of Couple Stress Fluid over a Rotating Disk

The present work is devoted to study the numerical simulation for unsteady MHD flow and heat transfer of a couple stress fluid over a rotating disk. A similarity transformation is employed to reduce the time dependent system of nonlinear partial differential equations (PDEs) to ordinary differential equations (ODEs). The Runge-Kutta method and shooting technique are employed for finding the numerical solution of the governing system. The influences of governing parameters viz. unsteadiness parameter, couple stress and various physical parameters on velocity, temperature and pressure profiles are analyzed graphically and discussed in detail.     

The role of aggregation kinetics in the sedimentation of erythtocytes

The well-known S-shaped settling curves are obtained as solutions of an autonomic dynamical system deduced mathematically from the generalized Stokes formula, the blood volume conservation law, and the Smoluchowski theory of particle coagulation. Numerical computations and parametric analysis of the deduced two nonlinear differential equations for the plasma zone thickness and aggregate size are given. It is shown that the model presented makes it possible, on the basis of experimentally recorded sedimentation curves and aggregate size growth, to identify quantitatively the values of the essential physical parameters of the coupled processes of erythrocyte aggregation and sedimentation. This method of identification could be used as a diagnostic test in hematological laboratories.     

Bend propagation in flagella. I. Derivation of equations of motion and their simulation.

A set of nonlinear differential equations describing flagellar motion in an external viscous medium is derived. Because of the local nature of these equations and the use of a Crank-Nicolson-type forward time step, which is stable for large deltat, numerical solution of these equations on a digital computer is relatively fast. Stable bend initiation and propagation, without internal viscous resistance, is demonstrated for a flagellum containing a linear elastic bending resistance and an elastic shear resistance that depends on sliding. The elastic shear resistance is derived from a plausible structural model of the radial link system. The active shear force for the dynein system is specified by a history-dependent functional of curvature characterized by the parameters m0, a proportionality constant between the maximum active shear moment and curvature, and tau, a relaxation time which essentially determines the delay between curvature and active moment.     

Improved pseudoanalytical solution for steady-state biofilm kinetics

Simple algebraic expressions for the flux of substrate into a steady-state biofilm are developed. This pseudoanalytical solution, which eliminates the need for repetitiously solving numerically a set of nonlinear differential equations, is based on an analysis of the numerical results from the numerical solution of the differential equations. The critical advantage of this new pseudoanalytical solution is that it is highly accurate for the entire range of substrate concentrations and kinetic parameters. The article also illustrates that previous pseudoanalytical solutions for steady-state biofilm kinetics are seriously inaccurate for certain ranges of substrate concentration and kinetic parameters.     

Curve fitting of germination data using the Richards function

Abstract. The fitting of the generalized Richards function to germination data by using two nested iterative and least squares regression procedures to estimate the four parameters (all of which can be associated with features of biological growth) is demonstrated. The program also involves a procedure of parallel curve analysis which makes comparisons between two curves by examining the whole process represented by the curve and not just a point or portion thereof. Excellent agreement between observed and expected values was obtained by analyzing data which defined patterns of germination exhibiting a range of rates and final percentages. The program also calculates a number of derived quantities including maximum daily rate of germination and time to 50% of final germination.     

Mathematical modeling of citric acid production by repeated batch culture

A mathematical model has been created for the process of citric acid biosynthesis by yeast (mutant strain Yarrowia lipolytica) cultivated by the repeated batch (RB) method on ethanol under conditions of nitrogen limitation. The model accounts for cell growth as a function of nitrogen concentration in the culture liquid; nitrogen uptake by growing cells; citric acid production; pH control in the fermentor by means of NaOH addition; and changes in system volume. The model represents a system of five nonlinear differential equations. Experimental measurements of cell concentration, citric acid concentration, and cultivation broth volume were used with the least squares method to determine the values of eight model parameters. The parameter values obtained were consistent with literature data and general concepts of cell growth and citric acid biosynthesis. The model has been used to predict optimum RB culture conditions.