Mathematic model parameter estimation

This chapter serves three purposes (a) to provide a brief overview of PBPK modeling, (b) to present a tutorial on the issues and steps involved in the development of a PBPK model, and (c) to present an application and discuss relevant issues associated with model refinement, evaluation, parameter estimation, and sensitiv-ity/uncertainty analysis. First, some basic background information is provided, and references to important resources are presented. Then the process of developing a PBPK model is discussed, and a step-by-step description of a PBPK modeling example is provided, along with a brief discussion on relevant complementary issues such as model parameter estimation and sensitivity/uncertainty analysis. The example is presented in a manner that a novice PBPK modeler can follow the model structure, mathematical equations, and the code. Relevant cross-references between the equations, parameter tables, and the actual code is presented. Though the example is implemented in Matlab (5), it does not require substantial Matlab... 

Parameter estimation is a procedure for taking the unit measurements and reducing them to a set of parameters for a physical (or, in some cases, relational) mathematical model of the unit. Statistical interpretation tempered with engineering judgment is required to arrive at realistic parameter estimates. Parameter estimation can be an integral part of fault detection and model discrimination. 

Parameter Estimation Relational and physical models require adjustable parameters to match the predicted output (e.g., distillate composition, tower profiles, and reactor conversions) to the operating specifications (e.g., distillation material and energy balance) and the unit input, feed compositions, conditions, and flows. The physical-model adjustable parameters bear a loose tie to theory with the limitations discussed in previous sections. The relational models have no tie to theory or the internal equipment processes. The purpose of this interpretation procedure is to develop estimates for these parameters. It is these parameters hnked with the model that provide a mathematical representation of the unit that can be used in fault detection, control, and design. 

As with troubleshooting, parameter estimation is not an exact science. The facade of statistical and mathematical routines coupled with sophisticated simulation models masks the underlying uncertainties in the measurements and the models. It must be understood that the resultant parameter values embody all of the uncertainties in the measurements, underlying database, and the model. The impact of these uncertainties can be minimized by exercising sound engineering judgment founded upon a famiharity with unit operation and engineering fundamentals. 

The reader is encouraged to use a two-phase, one spatial dimension, and time-dependent mathematical model to study this phenomenon. The UCKRON test problem can be used for general introduction before the particular model for the system of interest is investigated. The success of the simulation will depend strongly on the quality of physical parameters and estimated transfer coefficients for the system. 

The described experimental rig for the anionic polymerisation of dienes has been shown to behave as an ideal CSTR. The mathematical model developed allows the prediction of the MWD at future points in the reactor history, once suitable kinetic parameters have been estimated. 

The most common way in which the global carbon budget is calculated and analyzed is through simple diagrammatical or mathematical models. Diagrammatical models usually indicate sizes of reservoirs and fluxes (Figure 1). Most mathematical models use computers to simulate carbon flux between terrestrial ecosystems and the atmosphere, and between oceans and the atmosphere. Existing carbon cycle models are simple, in part, because few parameters can be estimated reliably. 

Employ a model that mathematically describes a size distribution of this type, adjust the model parameters for best fit, and estimate the missing fraction above 564 /tm after correcting the observed frequencies, continue with a correct statistical analysis. 

Topaz was used to calculate the time response of the model to step changes in the heater output values. One of the advantages of mathematical simulation over experimentation is the ease of starting the experiment from an initial steady state. The parameter estimation routines to follow require a value for the initial state of the system, and it is often difficult to hold the extruder conditions constant long enough to approach steady state and be assured that the temperature gradients within the barrel are known. The values from the Topaz simulation, were used as data for fitting a reduced order model of the dynamic system. 

Computer tools can contribute significantly to the optimization of processes. Computer data acquisition allows data to be more readily collected, and easy-to-implement control systems can also be achieved. Mathematical modeling can save personnel time, laboratory time and materials, and the tools for solving differential equations, parameter estimation, and optimization problems can be easy to use and result in great productivity gains. Optimizing the control system resulted in faster startup and consequent productivity gains in the extruder laboratory. 

Re-examine the final test conditions to ensure that, when Implemented, the test data will be sufficient to estimate all parameters In the fitted mathematical models ... 

A survey of the mathematical models for typical chemical reactors and reactions shows that several hydrodynamic and transfer coefficients (model parameters) must be known to simulate reactor behaviour. These model parameters are listed in Table 5.4-6 (see also Table 5.4-1 in Section 5.4.1). Regions of interfacial surface area for various gas-liquid reactors are shown in Fig. 5.4-15. Many correlations for transfer coefficients have been published in the literature (see the list of books and review papers at the beginning of this section). The coefficients can be evaluated from those correlations within an average accuracy of about 25%. This is usually sufficient for modelling of chemical reactors. Mathematical models of reactors arc often more sensitive to kinetic parameters. Experimental methods and procedures for parameters estimation are discussed in the subsequent section. 

Parameter estimation is one of the steps involved in the formulation and validation of a mathematical model that describes a process of interest. Parameter estimation refers to the process of obtaining values of the parameters from the matching of the model-based calculated values to the set of measurements (data). This is the classic parameter estimation or model fitting problem and it should be distinguished from the identification problem. The latter involves the development of a model from input/output data only. This case arises when there is no a priori information about the form of the model i.e. it is a black box. 

Parameter estimation and identification are an essential step in the development of mathematical models that describe the behavior of physical processes (Seinfeld and Lapidus, 1974 Aris, 1994). The reader is strongly advised to consult the above references for discussions on what is a model, types of models, model formulation and evaluation. The paper by Plackett that presents the history on the discovery of the least squares method is also recommended (Plackett, 1972). 

The formulation of the parameter estimation problem is equally important to the actual solution of the problem (i.e., the determination of the unknown parameters). In the formulation of the parameter estimation problem we must answer two questions (a) what type of mathematical model do we have and (b) what type of objective function should we minimize In this chapter we address both these questions. Although the primary focus of this book is the treatment of mathematical models that are nonlinear with respect to the parameters nonlinear regression) consideration to linear models linear regression) will also be given. 

Let us first concentrate on dynamic systems described by a set of ordinary differential equations (ODEs). In certain occasions the governing ordinary differential equations can be solved analytically and as far as parameter estimation is concerned, the problem is described by a set of algebraic equations. If however, the ODEs cannot be solved analytically, the mathematical model is more complex. In general, the model equations can be written in the form... 

In parameter estimation we are occasionally faced with an additional complication. Besides the minimization of the objective function (a weighted sum of errors) the mathematical model of the physical process includes a set of constrains that must also be satisfied. In general these are either equality or inequality constraints. In order to avoid unnecessary complications in the presentation of the material, constrained parameter estimation is presented exclusively in Chapter 9. 

In this chapter we are focusing on a particular technique, the Gauss-Newton method, for the estimation of the unknown parameters that appear in a model described by a set of algebraic equations. Namely, it is assumed that both the structure of the mathematical model and the objective function to be minimized are known. In mathematical terms, we are given the model... 

If matrix A is ill-conditioned at the optimum (i.e., at k=k ), there is not much we can do. We are faced with a truly ill-conditioned problem and the estimated parameters will have highly questionable values with unacceptably large estimated variances. Probably, the most productive thing to do is to reexamine the structure and dependencies of the mathematical model and try to reformulate a better posed problem. Sequential experimental design techniques can also aid us in... 

Most of the constrained parameter estimation problems belong to this case. Based on scientific considerations, we arrive quite often at constraints that the parameters of the mathematical model should satisfy. Most of the time these are of the form,... 

This is rarely the case in engineering. Most of the time we do have some form of a mathematical model (simple or complex) that has several unknown parameters that we wish to estimate. In these cases the above designs are very straightforward to implement however, the information may be inadequate if the mathematical model is nonlinear and comprised of several unknown parameters. In such cases, multilevel factorial designs (for example, 3k or 4k designs) may be more appropriate. 

The mathematical model for a hydrocarbon reservoir consists of a number of partial differential equations (PDEs) as well as algebraic equations. The number of equations depends on the scope/capabilities of the model. The set of PDEs is often reduced to a set of ODES by grid discretization. The estimation of the reservoir parameters of each grid cell is the essence ofhistory matching. 

Our approach to determine the properties of heterogeneous media utilizes mathematical models of the measurement process and, as appropriate, the flow process itself. To determine the desired properties, we solve an associated system and parameter identification problem (also termed an inverse problem) to estimate the properties from the measured data. 

We also use a linearized covariance analysis [34, 36] to evaluate the accuracy of estimates and take the measurement errors to be normally distributed with a zero mean and covariance matrix Assuming that the mathematical model is correct and that our selected partitions can represent the true multiphase flow functions, the mean of the error in the estimates is zero and the parameter covariance matrix of the errors in the parameter estimates is ... 

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