Compartmental modeling dynamic systems

The first two sections of Chapter 5 give a practical introduction to dynamic models and their numerical solution. In addition to some classical methods, an efficient procedure is presented for solving systems of stiff differential equations frequently encountered in chemistry and biology. Sensitivity analysis of dynamic models and their reduction based on quasy-steady-state approximation are discussed. The second central problem of this chapter is estimating parameters in ordinary differential equations. An efficient short-cut method designed specifically for PC s is presented and applied to parameter estimation, numerical deconvolution and input determination. Application examples concern enzyme kinetics and pharmacokinetic compartmental modelling. 

The first step in constructing a compartmental model is to examine the experimental observations for clues concerning the functionality of the system. This is especially true if the experimental observations are available prior to modeling because it helps direct the construction of the model. The experimental observations were available when construction of the compartmental model of the dynamics of /3-carotene metabolism started. 

Compartmental models are lumped models. The concept of a compartmental model assumes that the system can be divided into a number of homogeneous well-mixed components called compartments. Various characteristics of the system are determined by the movement of material from one compartment to the other. Compartmental models have been u to describe blood flow distribution to various organs, populatirMi dynamics, cellular dynamics, distribution of chemical species (hormones and metabolites) in various organs, temperature distribution, etc. 

Compartmental models are a class of dynamic, that is, differential equation, models derived from mass balance considerations, which are widely used for quantitatively studying the kinetics of materials in physiologic systems. Materials can be either exogenous, such as a drug or a tracer, or endogenous, such as a substrate or a hormone, and kinetics include processes such as production, distribution, transport, utilization, and substrate-hormone control interactions. 

Almost aU the biological models are nonhnear dynamic systems, including for example saturation or threshold processes. In particular, nonlinear compartmental models. Equation 9.5, are frequently found in biomedical applications. For such models the entries of K are functions of q, most commonly fcy is a function of only few components of q, often q, or qj. Examples of fcy function of q,- or qj are the Hill and... 

Nassar et al. [20] proposed a stochastic compartmental model to simulate the concentration dynamics of suspended particles in the liquid and solid parts over the different section of flow. In their work, a deep-bed filter is considered to be an open system composed of an arbitrary number of sections or compartments distributed in the axial direction. 

In order to understand these complex metabolic interactions more fully and to maximize the information obtained in these studies, we developed a detailed kinetic model of zinc metabolism(, ). Modeling of the kinetic data obtained from measurements of biological tracers by compartmental analysis allows derivation of information related not only to the transient dynamic patterns of tracer movements through the system, but also information about the steady state patterns of native zinc. This approach provides data for absorption, absorption rates, transfer rates between compartments, zinc masses in the total body and individual compartments and minimum daily requirements. Data may be collected without disrupting the normal living patterns of the subjects and the difficulties and inconveniences of metabolic wards can be avoided. 

Most importantly, reaction rates in nanofluidic systems can be controlled both by shape and volume changes. The important interplay between chemical reactions and geometry has been conceptualized within a theoretical framework for ultra-small volumes and tested on a number of experimental systems, opening pathways to more complex, dynamically compartmentalized ultra-small volume reactors, or artificial model cells, that offer more detailed understanding of cellular kinetics and biophysical phenomena, such as macromolecular crowding. 

Models taking into consideration the metabolism and time variation of the radionuclide concentrations in man are conceptualized as compartmental systems, including dynamic processes (ICRP 1994 Jacquez 1996 Thorne 2001). 

Compartmental or media box models offer an alternative practical approach. They are derived from applying integrated forms of the CE. These involve volume and area integrals over the boxes. The volume integrals sum the mass accumulation and reaction terms while the area integrals direct the flux terms to account for the movement of chemicals between the boxes. Typically, the result is a set of linear ordinary differential equations capable of mathematically mimicking many of the key dynamic and other features of the chemodynamics in natural systems. This handbook provides the mass transport parameters needed for both model types. 

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