Compartment concept

Although the effect compartment concept achieved wide popularity in the PK/PD modeling field, it is not suitable for a fairly large number of different pharmacological effects, and it does not help to elucidate some underlined mechanisms of pharmacological effects. 

The compartment concept is utilized in pharmacokinetics when it is necessary to describe the plasma concentration versus time data adequately and accurately, which, in turn, permits... 

The subject of mathematical models and the appropriate flux equations was introduced in the previous section. Although a few examples were given, a general approach for accommodating individual flux equations of various types into the overall accounting for the combined processes was not explicitly developed. The objective of this section is to introduce and develop the interface compartment concept as a consistent approach of combined the various flux expressions as they apply to differential equations, and compartmental-box models. As was done in the previous section the development will include the application of the concept to several realistic interface chemical transport scenarios. 

In order to be realistic, environmental models must contain multiple phases, and are often referred to as multiphasic, or multimedia compartment models. Being so requires interfaces that separate the phases and media. These interfaces can be real or idealized (i.e., imaginary). Two-dimensional interface planes are assumed to exist between the air-water, water-sediment, and soil-air phases or media. In reality, chemical transport across the air-water interphase plane involves a true phase change. The watery interface plane at the water-bed sediment junction and airy interface plane at the soil-air junction are only separated by imaginary planer surfaces. Nevertheless, due to the dramatic changes that typically occur within the fluid and the associated media fluid dynamics on either interface side, different transport processes occur on opposite sides, typically. Therefore it is also practical to define an interfacial compartment for such imaginary and idealized interface situations. The interfacial compartment concept for multimedia, interphase chemical transport is based on the following ideas ... 

The essence of the interface compartment concept has existed in the chemical engineering literature for decades (see Cussler, 1997 Bird et al., 2002). The idea originated and evolved through the efforts of Nernst, Whitman, and Lewis over the 20-year time-period 1904-1924. One current chemical engineering application is for gas-to-liquid mass transfer where it appears as a step in the derivation of the overall MTC, relating it to the individual phase MTCs (resistance-in-series concept). Another use is estimating concentrations at the gas-liquid interface. 

Applying the interface compartment concept is a two-step procedure. First, write a flux balance to sum the rate of all processes moving chemical through the interface. The approach is to write a balance of those entering being equal to those departing. In this case, it is... 

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. 

Scenario 2. Chemical volatilization from water follows. Processes AWl and AW7 are listed in Table 4.1. The chemical pathway is from solution in water with turbulent transport to the interface where vaporization occurs followed by turbulent transport away and into the bulk air beyond. The first step in applying the AW interface compartment concept is to sum the entering and departing fluxes. This is... 

The above alternative scenario development for liquid-to-gas phase chemical transport has a long history of successful application in the area of chemical separation in the chemical process industries (Bird et al., 2002). It gives support to the assumptions behind the interface compartment concept. However, the resistance-in-series algorithm shown in Equation 4.16 will work correctly only if the flux equations are of the diffusive concentration-difference-type. 

Application of the interface compartment concept follows. The fluxes of chemical to and from the interface by the respective rate equations are as follows ... 

Scenario 4. A modification of Scenario 3 follows. This example demonstrates the use of the interface compartment concept for obtaining the overall flux expression connecting adjoining media compartments when a combination of advective and... 

A final note. The interface compartment concept has application for the next generation of multimedia compartment box models and other models as well. Presently several of these models use versions of Ohm s law as noted above as well as other procedures. The numerical results produced appear to be reasonable and provide good approximations apparently. However, without the additional mass balance provided by applying the interface compartment concept, the advective transport processes fail to impact the magnitude of the interface concentfation. This influences the flux and finally the media mass concentrations. Comparative model studies using the present-day approaches of combining interface fluxes and the IC model approach need to be performed and the results evaluated. Such studies may aid the development of a more realistic and appropriate approaches for connecting chemical flux between multimedia environmental compartments. 

Use the interface diagram and apply the interface compartment concept. The flux n is equal to the sum of individual processes moving A to the plane... 

Because disturbances in fluid balance are routinely encountered in clinical medicine, it is essential to have a thorough understanding of body fluid compartments and the therapeutic use of fluids. Similarly, disturbances in serum sodium, potassium, calcium, phosphorus, and magnesium are ubiquitous and must be mastered by all clinicians. Dysregulation of fluid and/or electrolyte status has serious implications regarding the concepts of drug absorption, volumes of distribution, and toxicity. Similarly, many medications can disrupt fluid and/or electrolyte balance as an unintended consequence. 

A discussion of all the reasons for this phenomenon is beyond the scope of this chapter, but a simple example will illustrate the concept. Highly lipid-soluble drugs, such as pentobarbital, are preferentially distributed into adipose tissue. The result is that plasma concentrations are extremely low after distribution is complete. When the apparent volumes of distribution are calculated, they are frequently found to exceed total body volume, occasionally by a factor of 2 or more. This would be impossible if the concentration in the entire body compartment were equal to the plasma concentration. Thus, Vd is an empirically fabricated number relating the... 

Compartmental soil modeling is a new concept and can apply to both modules. For the solute fate module, for example, it consists of the application of the law of pollutant mass conservation to a representative user specified soil element. The mass conservation principle is applied over a specific time step, either to the entire soil matrix or to the subelements of the matrix such as the soil-solids, the soil-moisture and the soil-air. These phases can be assumed in equilibrium at all times thus once the concentration in one phase is known, the concentration in the other phases can be calculated. Single or multiple soil compartments can be considered whereas phases and subcompartments can be interrelated (Figure 2) with transport, transformation and interactive equations. 

Stochastic or probabilistic techniques can be applied to either the moisture module, or the solution of equation (3) — or for example the models of Schwartz Crowe (13) and Tang et al. (16), or can lead to new conceptual model developments as for example the work of Jury (17). Stochastic or probabilistic modeling is mainly aimed at describing breakthrough times of overall concentration threshold levels, rather than individual processes or concentrations in individual soil compartments. Coefficients or response functions and these models have to be calibrated to field data since major processes are studied via a black-box or response function approach and not individually. Other modeling concepts may be related to soil models for solid waste sites and specialized pollutant leachate issues (18). 

Summary. In summary, when modeling with the fugacity concept, all equilibria can be treated by Z values (one for each compartment) and all reaction, advection and transport processes can be treated by D values. The only other quantities requiring definition are compartment volumes and emission rates or initial concentrations. A major advantage is that since all D quantities are in equivalent units they can be compared directly and the dominant processes identified. By converting diverse processes such as volatilization, sediment deposition, fish uptake and stream flow into identical units, their relative importance can be established directly and easily. Further, algebraic manipulation... 

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