Compartmental model mass transfer

Modeled rates (curve d. Fig. 5) agreed with those determined experimentally. We suggest that the interaction of mass transfer and metabolism consisting of two or several bioreactions in compartmentalized media is an interesting phenomenon, which has not been fully researched to date. 

The mass transfer coefficients may also be expressed in units of time-1 by multiplying by the appropriate compartmental volume term. Irreversible drug elimination from the tissue requires the addition of an expression to the differential equation that represents the subcompartment in which elimination occurs. For instance, hepatic drug elimination would be described by a linear or nonlinear expression added to the intracellular liver compartment mass balance equation since this compartment represents the hepatocytes. Formal elimination terms are given below for the simplified tissue models. 

Let Xj(f) be the mass of a drug in the ith compartment. The notation for input, loss, and transfers is summarized in Figure 8.3. Because this notation describes the compartment in full generality, it is a little different from that used in earlier chapters. This difference is necessary to understand how one passes to the linear compartmental model. In Figure 8.3, the rate constants describe mathematically the mass transfer of material among compartments interacting with the ith compartment (Fji is the transfer of material from compartment i to compartment j, F j is the transfer of material from compartment j to compartment i), the new input F q (this corresponds to Xq in Chapter 4), and loss to the environment Fqi from compartment i. The mathematical expression describing the rate of... 

The mass transfer mechanisms operative in substrate conversion are essentially those described by Waterland et al47 in their model of the compartmentalized enzyme membrane reactor. Since kinetic parameters cannot be assumed equal to those of native enzymes, a kinetic analysis has to be performed in order to characterize enzyme behavior after the immobilization procedure. 

We began modeling under the assumption that the introduction of the tracer (mass) into the system did not affect the mechanisms present for metabolism of the tracee. The compartmental model was compatible with the assumption that non-steady-state mechanisms for metabolism of /3-carotene were not induced by the tracer because the model prediction of the tracer state, the tracee state, and the steady state could be achieved using the same set of fractional transfer coefficients (FTCs). The appropriateness of this assumption is discussed again under Statistical Considerations. FTC is the fraction of analyte in a donor compartment that is transferred to a recipient compartment per unit of time, in this case per day. 

Kinetic studies were made of Apo C in normal subjects. It was impossible to build a complex compartmental model quantifying the exchanges between lipoproteins, since quick passive exchanges were not experimentally separable from exchanges involving mass transfer appearing with the entry of chylomicron or native VLDL into circulation. The kinetics of individual lipoproteins for Apo C were thus illusory. 

A central biokinetic parameter governing the operation of this compartmental model is the compartmental transfer time. These times cover Pb movement among the various compartments set forth in Figure 9.3. Times are based on plasma Pb. At steady state, the ratios of Pb masses in tissue compartments to plasma Pb masses are equivalent to the ratios of transfer times from tissues to plasma and ECF, and from plasma/ECF to tissues. Transfers are also assumed to be from the central to tissue compartments by a first-order kinetic process (White et al., 1998). 

The fcy s are called fractional transfer coefficients. Equation 9.3 describes the generic nonlinear compartmental model. If the fcy s do not depend on the compartmental masses qfs, the model becomes... 

Mass transport also plays a major role in several other important disciplines. Environmental processes are dominated by the twin topics of mass transfer and phase equilibria, and here again an early and separate introduction to these subject areas can be immensely beneficial. This text provides detailed treatments of both phase equilibria and compartmental models, which are all-pervasive in the environmental sciences. Transport, where it occurs, is almost always based on Fickian diffusion and film theory. The same topics are also dominant in the biological sciences and in biomedical engineering, and the text makes a conscious effort to draw on examples from these disciplines and to highlight the idiosyncrasies of biological processes. 

Chapter 6 deals with phase equilibria, which are mainly composed of topics not generally covered in conventional thermodynamics courses. These equilibria are used in Chapter 7 to analyze compartmental models and staged processes. Included in this chapter is a unique treatment of percolation processes, which should appeal to environmental and chemical engineers. Chapter 8 takes up the topic of modeling continuous-contact operations, among which the application to membrane processes is given particular prominence. Finally, in Chapter 9 we conclude the text with a brief survey of simultaneous mass and heat transfer. 

The diffusive gradient is integrated over a specified film or layer thickness to accommodate the compartmental box models the resulting flux equations contain a concentration difference (see Section 4.3). What follows are some chemodynamic mass transfer scenarios using concentration difference flux equations for demonstrating the interface compartment concept used with compartmental-box models. 

The discussion above provides a brief qualitative introduction to the transport and fate of chemicals in the environment. The goal of most fate chemists and engineers is to translate this qualitative picture into a conceptual model and ultimately into a quantitative description that can be used to predict or reconstruct the fate of a chemical in the environment (Figure 27.1). This quantitative description usually takes the form of a mass balance model. The idea is to compartmentalize the environment into defined units (control volumes) and to write a mathematical expression for the mass balance within the compartment. As with pharmacokinetic models, transfer between compartments can be included as the complexity of the model increases. There is a great deal of subjectivity to assembling a mass balance model. However, each decision to include or exclude a process or compartment is based on one or more assumptions—most of which can be tested at some level. Over time the applicability of various assumptions for particular chemicals and environmental conditions become known and model standardization becomes possible. 

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. 

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