A coupled drug kinetics-cell cycle model to analyse the response of human cells to intervention by topotecan.

A model describing the response of the growth of single human cells in the absence and presence of the anti-cancer agent topotecan (TPT) is presented. The model includes a novel coupling of both the kinetics of TPT and cell cycle responses to the agent. By linking the models in this way, rather than using separate (disjoint) approaches, it is possible to illustrate how the drug perturbs the cell cycle. The model is compared to experimental in vitro cell cycle response data (comprising single cell descriptors for molecular and behavioural events), showing good qualitative agreement for a range of TPT dose levels.

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.

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.

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.

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.

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.

Indistinguishability for a class of nonlinear compartmental models.

Indistinguishability, as applied to nonlinear compartmental models, is analyzed by means of the local state isomorphism theorem. The method of analysis involves the determination of all local, diffeomorphic transformations connecting the state variables of two models. This is then applied to two two-compartment models, in the first instance with linear eliminations, and then with the addition of eliminations with Michaelis-Menten kinetics. In the nonlinear example, the state transformation turns out to be linear or possibly affine. It is found that the nonlinear analysis could be eased by splitting the state isomorphism equations into those of the initial linear models together with extra equations due to the nonlinearities.

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.

The structural identifiability and parameter estimation of a multispecies model for the transmission of mastitis in dairy cows.

A structural identifiability analysis is performed on a mathematical model for the coupled transmission of two classes of pathogen. The pathogens, classified as major and minor, are aetiological agents of mastitis in dairy cows that interact directly and via the immunological reaction in their hosts. Parameter estimates are available from experimental data for all but four of the parameters in the model. Data from a longitudinal study of infection are used to estimate these unknown parameters. A novel approach and application of structural identifiability analysis is combined in this paper with the estimation of cross-protection parameters using epidemiological data.

The structural identifiability and parameter estimation of a multispecies model for the transmission of mastitis in dairy cows with postmilking teat disinfection.

A mathematical model for the transmission of two interacting classes of mastitis causing bacterial pathogens in a herd of dairy cows is presented and applied to a specific data set. The data were derived from a field trial of a specific measure used in the control of these pathogens, where half the individuals were subjected to the control and in the others the treatment was discontinued. The resultant mathematical model (eight non-linear simultaneous ordinary differential equations) therefore incorporates heterogeneity in the host as well as the infectious agent and consequently the effects of control are intrinsic in the model structure. A structural identifiability analysis of the model is presented demonstrating that the scope of the novel method used allows application to high order non-linear systems. The results of a simultaneous estimation of six unknown system parameters are presented. Previous work has only estimated a subset of these either simultaneously or individually. Therefore not only are new estimates provided for the parameters relating to the transmission and control of the classes of pathogens under study, but also information about the relationships between them. We exploit the close link between mathematical modelling, structural identifiability analysis, and parameter estimation to obtain biological insights into the system modelled.

Evaluation of frequency and time-frequency spectral analysis of heart rate variability as a diagnostic marker of the sleep apnoea syndrome.

The sleep apnoea/hypopnoea syndrome (SAHS) elicits a unique heart rate rhythm that may provide the basis for an effective screening tool. The study uses the receiver operator characteristic (ROC) to assess the diagnostic potential of spectral analysis of heart rate variability (HRV) using two methods, the discrete Fourier transform (DFT) and the discrete harmonic wavelet transform (DHWT). These two methods are compared over different sleep stages and spectral frequency bands. The HRV results are subsequently compared with those of the current screening method of oximetry. For both the DFT and the DHWT, the most diagnostically accurate frequency range for HRV spectral power calculations is found to be 0.019-0.036 Hz (denoted by AB2). Using AB2, 15 min sections of non-REM sleep data in 40 subjects produce ROC areas, for the DFT, DHWT and oximetry, of 0.94, 0.97 and 0.67, respectively. In REM sleep, ROC areas are 0.78, 0.79 and 0.71, respectively. In non-REM sleep, spectral analysis of HRV appears to be a significantly better indicator of the SAHS than the current screening method of oximetry, and, in REM sleep, it is comparable with oximetry. The advantage of the DHWT over the DFT is that it produces a greater time resolution and is computationally more efficient. The DHWT does not require the precondition of stationarity or interpolation of raw HRV data.

Comment on "Parameter and structural identifiability concepts and ambiguities: a critical review and analysis".

Identifiability of a model for the glucose tolerance test, considered by Cobelli and DiStefano [Am. J. Physiol. 239 (Regulatory Integrative Comp. Physiol. 8): R7-R24, 1980], is shown to depend on the shape of the external perturbation. For systems with two or more inputs applied simultaneously, it is essential in Laplace transform identifiability analysis to examine the Laplace transform(s) of the observation(s) rather than the individual transfer functions, which are not measurable separately.

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.

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.

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.

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.