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K R Godfrey

Publications and source records attributed to K R Godfrey.

18 recordsLinked to original sources
PubMed1992

Structural identifiability of the parameters of a nonlinear batch reactor model.

The similarity transformation approach is used to analyze the structural identifiability of the parameters of a nonlinear model of microbial growth in a batch reactor in which only the concentration of microorganisms is measured. It is found that some of the model parameters are unidentifiable from this experiment, thus providing the first example of a real-life nonlinear model that turns out not to be globally identifiable. If it is possible to measure the initial concentration of growth-limiting substrate as well, all model parameters are globally identifiable.

Bacteria↗
PubMed1991

Optimal tumor targeting by antibodies: development of a mathematical model.

A mathematical model has been developed to optimize tumor targeting with labeled antibodies. The model is compartmental and nonlinear, incorporating saturable binding. Published parameter values have been used in the model, and the resulting stiff differential equations have been solved using FACSIMILE, a computer package that can simulate very stiff differential systems. Results show that successful tumor targeting depends on an optimal combination of antibody dose, affinity, and molecular size. The model has allowed an assessment to be made of the complicated and interrelated dynamic relationships that these factors have on tumor targeting. It has also offered an explanation for previously unsatisfactory results from tumor targeting with labeled antibodies. The structural identifiability of the model parameters is also analyzed and it is shown that, with the prior knowledge of some parameters which is likely in practice, the remaining model parameters are uniquely identifiable.

Algorithms↗
PubMed1990

Global identifiability of the parameters of nonlinear systems with specified inputs: a comparison of methods.

The two methods available for analyzing the global structural identifiability of the parameters of a nonlinear system with a specified input function, the Taylor series approach and the similarity transformation approach, are compared and contrasted through application to three examples. It is shown that, as for linear systems, it is very difficult to predict which of the available methods will result in the least effort for a particular example. The role of modern symbolic manipulation packages in the analysis is assessed. The third example proves intractable using the similarity transformation approach as originally formulated, but the analysis is completed using a reformulation that exploits the polynominal form of the system equations in the example.

Biometry↗
PubMed1990

Theoretical considerations for improving tumour targeting.

To determine the relative importance of factors influencing tumour uptake of antibodies, we used a mathematical model to simulate intravenous injection of substances of varying molecular sizes and tumour-binding affinities at several dose levels. The FACSIMILE program was used to simulate the time course of tumour uptake of the tumour-binding substance by calculating the instantaneous tumour content (TC) and tumour:background uptake ratios (UR). Relative total doses to tumour and normal tissue were calculated by integration of TC/time curves. The model was used to make theoretical predictions on the effects of altering different parameters. The size of the injected dose in relation to the number of tumour receptors was crucial:if too low, uptake could not be improved by manipulating other variables, and if too high, the UR for large binding molecules was reduced. Using the standard scanning dose of labelled antibody, absolute numbers of labelled molecules binding to tumour could be increased by injection of a large excess of unlabelled molecules. Given an adequate dose, peak tumour content increased with increasing affinity up to receptor saturation. The peak uptake ratio rose progressively with affinity for a small ligand, but reached a relatively low plateau for antibody due to constant high background levels. At low doses such as those currently administered for diagnostic scanning with antibody, no effect of increasing affinity was predicted.

Antibodies, Monoclonal↗
PubMed1989

Effect of dose, molecular size, affinity, and protein binding on tumor uptake of antibody or ligand: a biomathematical model.

A mathematical model has been developed to determine the best approach to improving tumor targeting with antibody. The amount of antibody in the tumor (tumor content) and the tumor:normal tissue antibody concentration ratio (uptake ratio) were calculated over 12 days from injection, using the computer program FACSIMILE to solve the stiff nonlinear differential equations describing the system. Results indicate that success requires an optimal combination of dose, size, and binding affinity of antibody. Increasing the dose to 100 times that presently used for scanning increased both the percentage of injected antibody in the tumor and the uptake ratio by up to 2 orders of magnitude to maximal values determined by affinity. This result could be achieved by coinjecting unlabeled antibody. Increasing affinity from Keq = 10(9) to 10(13)M-1 increased the uptake ratio from 5 to 100 for whole antibody and to 550 for a small ligand, at the calculated optimal dose, but had no effect at the current scanning dose. With decreasing molecular size at average affinity, the same maximum tumor content and uptake ratio were achieved but progressively earlier. At high affinity there was a substantial advantage for a small ligand compared with whole antibody in terms of uptake ratio (550 versus 100) and tumor:normal tissue integral dose ratio (330 versus 60). The uptake of a small ligand was not increased by binding to plasma protein but with increasing time the tumor content was higher than without protein binding.

Animals↗
PubMed1989

The problem of model indistinguishability in pharmacokinetics.

The problem of model indistinguishability is introduced in the context of linear compartmental models in pharmacokinetics. The two most widely used methods of analyzing model indistinguishability are described. It is shown that as the number of compartments increases, one approach, based on the Laplace transforms of the observations, although conceptually simple, can result in very large numbers of candidate models to be examined for indistinguishability, while the other approach, based on similarity transformations, although systematic, often results in very difficult algebraic expressions. These problems can be eased by the use of some simple geometrical rules, used at the outset of an indistinguishability analysis. The approach is illustrated by application of two 2-compartment drug, 2-compartment metabolite models.

Models, Biological↗
PubMed1989

Similarity transformation approach to identifiability analysis of nonlinear compartmental models.

Through use of the local state isomorphism theorem instead of the algebraic equivalence theorem of linear systems theory, the similarity transformation approach is extended to nonlinear models, resulting in finitely verifiable sufficient and necessary conditions for global and local identifiability. The approach requires testing of certain controllability and observability conditions, but in many practical examples these conditions prove very easy to verify. In principle the method also involves nonlinear state variable transformations, but in all of the examples presented in the paper the transformations turn out to be linear. The method is applied to an unidentifiable nonlinear model and a locally identifiable nonlinear model, and these are the first nonlinear models other than bilinear models where the reason for lack of global identifiability is nontrivial. The method is also applied to two models with Michaelis-Menten elimination kinetics, both of considerable importance in pharmacokinetics, and for both of which the complicated nature of the algebraic equations arising from the Taylor series approach has hitherto defeated attempts to establish identifiability results for specific input functions.

Kinetics↗
PubMed1989

Parameter space boundaries for unidentifiable compartmental models.

Methods for dealing with unidentifiable compartmental models are first reviewed, emphasizing the parameter interval analysis and exhaustive modeling approaches. More general methods are presented for generating the set of all nonnegative parameter solutions that localize the parameters within bounded regions of parameter space and extend previously published parameter bounding strategies. Each point of these regions is an equivalent solution of the parameter identification problem. If a point on the boundary is selected, at least one of the parameters vanishes and an equivalent submodel is obtained. This property shows the close relationship between the exhaustive modeling and parameter interval analysis approaches.

Mathematics↗
PubMed1989

A methodology for compartmental model indistinguishability.

The problem of constructing all minimal compartmental models that are indistinguishable through input-output knowledge alone from some given model is examined. The main tool in this analysis is a set of geometric properties that can be deduced from input-output knowledge and hence must be equally true in any two indistinguishable models. These properties, together with preservation of the form of the model's transfer function(s), provide an effective means for producing a set of candidate models for indistinguishability.

Mathematics↗
PubMed1988

Numerical deconvolution using system identification methods.

A deconvolution method is presented for use in pharmacokinetic applications involving continuous models and small samples of discrete observations. The method is based on the continuous-time counterpart of discrete-time least squares system identification, well established in control engineering. The same technique, requiring only the solution of a linear regression problem, is used both in system identification and input identification steps. The deconvolution requires no a priori information, since the proposed procedure performs system identification (including optimal selection of model order), selects the form of the input function and calculates its parametric representation and its values at specified time points.

Models, Biological↗
PubMed1985

Regional specific mean expiratory gas flow from 81mKr equilibrium inhalation data.

A new method of analysing the data available from routine 81m Kr equilibrium inhalation investigations has been developed. The data for analysis are acquired from a gamma camera in the form of a sequential series of images from which multiple breath activity-time curves are generated for eight regions in the lung. The method is based on a description of the behaviour of the radioactive gas in the lung using a mathematical model. Values of specific mean expiratory gas flow, that is mean expiratory gas flow per unit lung volume, are calculated from the application of the model to the expiratory phase only of a single breath activity-time curve which is generated from the multiple breath activity-time curve using post-acquisition gating. This method overcomes the problem of non-uniform inspiratory concentration of tracer gas experienced in previously reported techniques of analysing inhalation data obtained using poorly soluble radioactive gases. The model is shown, in simulation studies, to be an adequate description of the behaviour of radioactive gas in the lung and the analysis technique is shown, in clinical studies, to be both reproducible and sensitive to disease state.

Humans↗
PubMed1984

The deterministic identifiability of nonlinear pharmacokinetic models.

This paper deals with the deterministic identifiability of nonlinear pharmacokinetic models, namely, whether the model parameters can be identified with perfect data. It is shown that the most familiar method for analyzing the deterministic identifiability of linear models, in which the Laplace transform of the observation is examined, does not work for nonlinear models. An alternative method, in which the observation is expanded as a Taylor series about t = 0, is described and is illustrated with some examples of nonlinear models familiar in the pharmacokinetics literature, in which an elimination rate is assumed capacity limited, with Michaelis-Menten kinetics.

Animals↗
PubMed1984

On the identification of Michaelis-Menten elimination parameters from a single dose-response curve.

This paper deals with aspects of the numerical identifiability of parameters of a model with a capacity-limited elimination rate using a single dose-response curve, namely, the prospects of being able to identify model parameters with any meaning from real data. The concept of linear bounds, first proposed by Tong and Metzler, is described and it is shown that if the Michaelis-Menten constant Km is greater than all the measured concentration values, approximation by a linear model is appropriate. At the other end of the scale, if Km is small compared with measured concentration values, the nonlinear response approximates to a zero-order curve.

Animals↗