Joseph J. DiStefano

Identifiability of Model Parameter

Abstract The first part is concerned with whether the parameters of a model can be identified (uniquely or with several solutions) from a specified input-output experiment if perfect data are available. For linear, time-invariant models, there are several approaches available for this aspect of identifiability analysis, alternatively referred to as structural, deterministic or a priori identifiabi1ity analysis, and five such approaches are described. Only one, based on the Taylor series expansion of the observations, is directly applicable to nonlinear models or tine-varying systems. It is illustrated by analysing two nonlinear models When a model is unidentifiable from a proposed experiment, physically based constraints on the model often provide a means for computing finite bounds for the parameters (interval identifiability). This is illustrated for the class of linear, time-invariant models in which the system matrix is of compartmental form The identifiability question in the presence of real, noisy data is considered last. In the context of parameter estimation, this aspect of the analysis, often referred to in the literature as numerical or a posteriori identifiabi1ity analysis, is essentially the problem of parameter estimation accuracy, given that the parameters are known to be structurally identifiable Many of the issues discussed throughout the paper are illustrated by two examples drawn from the literature, the first a linearised model of ship steering dynamics and the second a nonlinear model for a microbial growth process

A predictive model of therapeutic monoclonal antibody dynamics and regulation by the neonatal Fc receptor (FcRn)

We constructed a novel physiologically-based pharmacokinetic (PBPK) model for predicting interactions between the neonatal Fc receptor (FcRn) and anti-carcinoembryonic antigen (CEA) monoclonal antibodies (mAbs) with varying affinity for FcRn. Our new model, an integration and extension of several previously published models, includes aspects of mAb-FcRn dynamics within intracellular compartments not represented in previous PBPK models. We added mechanistic structure that details internalization of class G immunoglobulins by endothelial cells, subsequent FcRn binding, recycling into plasma of FcRn-bound IgG and degradation of free endosomal IgG. Degradation in liver is explicitly represented along with the FcRn submodel in skin and muscle. A variable tumor mass submodel is also included, used to estimate the growth of an avascular, necrotic tumor core, providing a more realistic picture of mAb uptake by tumor. We fitted the new multiscale model to published anti-CEA mAb biodistribution data, i.e. concentration-time profiles in tumor and various healthy tissues in mice, providing new estimates of mAb-FcRn related kinetic parameters. The model was further validated by successful prediction of F(ab′)2 mAb fragment biodistribution, providing additional evidence of its potential value in optimizing intact mAb and mAb fragment dosing for clinical imaging and immunotherapy applications.

Complete parameter bounds and quasiidentifiability conditions for a class of unidentifiable linear systems

Abstract Lack of unique structural identifiability for parameters of dynamic system models is a very common situation with practical experimental schemes, particularly when studying biological systems. However, for well-structured (e.g., multicompartmental) models, it is often possible to localize unidentifiable parameters between finite limits (“interval identifiability”), using the same data base, and under certain conditions these limits nearly coincide. Two new results in this area are presented: (1) The smallest ranges on all unidentifiable rate constants and pool sizes of the most general n -compartment mammillary system are derived, in an easy-to-program algorithmic form, for the common case of input forcing and output measurements in the central pool only. From these results we see why elimination rate constants (“leaks”) are difficult to distinguish from zero, whereas exchange rate constants between pools, and pool sizes, may be bounded very tightly in certain circumstances. (2) The notion of quasiidentifiability, or sufficient identifiability for practical purposes, is introduced to quantify these circumstances. Each of the rate constants between central and peripheral pools, and all pool sizes, are quasiidentifiable if the magnitude of the ratio of the coefficient to the eigenvalue of the slowest mode is very much greater than the largest coefficient in the sum-of-exponentials response function. Also quasiidentifiability is a necessary condition for applicability of noncompartmental analysis to estimate pool sizes and residence times of mammillary systems with “leaky” noncentral pools.

Optimal nonuniform sampling interval and test-input design for identification of physiological systems from very limited data

Optimal design of test-inputs and sampling intervals in experiments for linear system identification is treated as a nonlinear integer optimization problem. The criterion is a function of the Fisher information matrix, the inverse of which gives a lower bound for the covariance matrix of the parameter estimates. Emphasis is placed on optimum design of nonuniform data sampling intervals when experimental constraints allow only a limited number of discrete-time measurements of the output. A solution algorithm based on a steepest descent strategy is developed and applied to the design of a biologic experiment for estimating the parameters of a model of the dynamics of thyroid hormone metabolism. The effects on parameter accuracy of different model representations are demonstrated numerically, a canonical representation yielding far poorer accuracies than the original process model for nonoptimal sampling schedules, but comparable accuracies when these schedules are optimized. Several objective functions for optimization are compared. The overall results indicate that sampling schedule optimization is a very fruitful approach to maximizing expected parameter estimation accuracies when the sample size is small.

A two-tiered physiologically based model for dually labeled single-chain Fv-Fc antibody fragments

Monoclonal antibodies (mAb) are being used at an increasing rate in the treatment of cancer, with current efforts focused on developing engineered antibodies that exhibit optimal biodistribution profiles for imaging and/or radioimmunotherapy. We recently developed the single-chain Fv-Fc (scFv-Fc) mAb, which consists of a single-chain antibody Fv fragment (light-chain and heavy-chain variable domains) coupled to the IgG1 Fc region. Point mutations that attenuate binding affinity to FcRn were introduced into the Fc region of the wild-type scFv-Fc mAb, resulting in several new antibodies, each with a different half-life. Here, we describe the construction of a two-tiered physiologically based pharmacokinetic model capable of simulating the apparent biodistribution of both 111In- and 125I-labeled scFv-Fc mAbs, where 111In-labeled metabolites from degraded 111In-labeled mAbs tend to become trapped within the lysosomal compartment, whereas free 125I from degraded 125I-labeled mAbs is quickly eliminated via the urinary pathway. The different concentration-time profiles of 111In- and 125I-labeled mAbs permits estimation of the degradation capacity of each organ and elucidates the dependence of cumulative degradation in liver, muscle, and skin on FcRn affinity and tumor mass. Liver is estimated to account for ∼50% of all degraded mAb when tumor is small (∼0.1 g) and drops to about 35% when tumor mass is larger (∼0.3 g). mAb degradation in residual carcass (primarily skin and muscle) decreases from ∼45% to 16% as FcRn affinity of the three mAb variants under consideration increases. In addition, elimination of a small amount of mAb in the kidneys is shown to be required for a successful fit of model to data. [Mol Cancer Ther 2006;5(6):1550–8]

On the relationships between structural identifiability and the controllability, observability properties

It is shown that if only identifiability properties of a linear system are of interest, it is neither necessary nor sufficient to first determine whether the structure is controllable or observable.

Identification of the dynamics of thyroid hormone metabolism

We have developed a new nonlinear dynamical model, a linearized dynamical model and a minimal steady-state model of a portion of the system responsible for thyroid hormone regulation in man. Each model is quasiisomorphic in structure and serves a different purpose. A Newton-Raphson algorithm and nonlinear regression were used to fit separately the model nonlinearities; and a modified hyperconical random search scheme was used to estimate twelve identifiable combinations of parameters of the nonlinear model. Our modeling approach greatly facilitated parameter estimation. From these results, we were able to quantify completely the steady-state model, the results of which yielded eight key parameters, previously unobtainable and of great interest to the thyroidologist.

Predicting chemical impacts on vertebrate endocrine systems

Animals have evolved diverse protective mechanisms for responding to toxic chemicals of both natural and anthropogenic origin. From a governmental regulatory perspective, these protective responses complicate efforts to establish acceptable levels of chemical exposure. To explore this issue, we considered vertebrate endocrine systems as potential targets for environmental contaminants. Using the hypothalamic-pituitary-thyroid (HPT), hypothalamic-pituitary-gonad (HPG), and hypothalamic-pituitary-adrenal (HPA) axes as case examples, we identified features of these systems that allow them to accommodate and recover from chemical insults. In doing so, a distinction was made between effects on adults and those on developing organisms. This distinction was required because endocrine system disruption in early life stages may alter development of organs and organ systems, resulting in permanent changes in phenotypic expression later in life. Risk assessments of chemicals that impact highly regulated systems must consider the dynamics of these systems in relation to complex environmental exposures. A largely unanswered question is whether successful accommodation to a toxic insult exerts a fitness cost on individual animals, resulting in adverse consequences for populations. Mechanistically based mathematical models of endocrine systems provide a means for better understanding accommodation and recovery. In the short term, these models can be used to design experiments and interpret study findings. Over the long term, a set of validated models could be used to extrapolate limited in vitro and in vivo testing data to a broader range of untested chemicals, species, and exposure scenarios. With appropriate modification, Tier 2 assays developed in support of the U.S. Environmental Protection Agencys Endocrine Disruptor Screening Program could be used to assess the potential for accommodation and recovery and inform the development of mechanistically based models.

Extensions, Validation, and Clinical Applications of a Feedback Control System Simulator of the Hypothalamo-Pituitary-Thyroid Axis

BACKGROUND We upgraded our recent feedback control system (FBCS) simulation model of human thyroid hormone (TH) regulation to include explicit representation of hypothalamic and pituitary dynamics, and updated TH distribution and elimination (D&E) parameters. This new model greatly expands the range of clinical and basic science scenarios explorable by computer simulation. METHODS We quantified the model from pharmacokinetic (PK) and physiological human data and validated it comparatively against several independent clinical data sets. We then explored three contemporary clinical issues with the new model: combined triiodothyronine (T(3))/thyroxine (T(4)) versus T(4)-only treatment, parenteral levothyroxine (L-T(4)) administration, and central hypothyroidism. RESULTS Combined T(3)/T(4) therapy--In thyroidectomized patients, the L-T(4)-only replacement doses needed to normalize plasma T(3) or average tissue T(3) were 145 microg L-T(4)/day or 165 microg L-T(4)/day, respectively. The combined T(4) + T(3) dosing needed to normalize both plasma and tissue T(3) levels was 105 microg L-T(4) + 9 microg T(3) per day. For all three regimens, simulated mean steady-state plasma thyroid-stimulating hormone (TSH), T(3), and T(4) was within normal ranges (TSH: 0.5-5 mU/L; T(4): 5-12 microg/dL; T(3): 0.8-1.9 ng/mL). Parenteral T(4) administration--800 microg weekly or 400 microg twice weekly normalized average tissue T(3) levels both for subcutaneous (SC) and intramuscular (IM) routes of administration. TSH, T(3), and T(4) levels were maintained within normal ranges for all four of these dosing schemes (1x vs. 2x weekly, SC vs. IM). Central hypothyroidism--We simulated steady-state plasma T(3), T(4), and TSH concentrations in response to varying degrees of central hypothyroidism, reducing TSH secretion from 50% down to 0.1% of normal. Surprisingly, TSH, T(3), and T(4) plasma concentrations remained within normal ranges for TSH secretion as low as 25% of normal. CONCLUSIONS Combined T(3)/T(4) treatment--Simulated standard L-T(4)-only therapy was sufficient to renormalize average tissue T(3) levels and maintain normal TSH, T(3), and T(4) plasma levels, supporting adequacy of standard L-T(4)-only treatment. Parenteral T(4) administration-TSH, T(3), and T(4) levels were maintained within normal ranges for all four of these dosing schemes (1x vs. 2x weekly, SC vs. IM), supporting these therapeutic alternatives for patients with compromised L-T(4) gut absorption. Central hypothyroidism--These results highlight how highly nonlinear feedback in the hypothalamic-pituitary-thyroid axis acts to maintain normal hormone levels, even with severely reduced TSH secretion.

An algorithm for the identifiable parameter combinations of the general mammillary compartmental model

Abstract Mammillary compartmental models are not uniquely identifiable except in special cases where parameter subsets of rate constants have known values. However, even for the most general n -pool mammillary model, an intrinsic set of parameter combinations is identifiable. An explicit algorithm for computing the uniquely identifiable sums and products of unidentifiable parameters for this class of models is developed for the most practical experimental conditions, input into and output from the central pool. Also, all steady state unidirectional mass fluxes from noncentral pools to the central pool (products of pool masses Q i and fractional transport rates k 1 i ) are shown to be always uniquely identifiable, even when the Q i and k 1 i are not, and the lower bounds are the same and the upper bounds are the same for all mass fluxes to the invironment.