Model compartment

Historically, the oldest compartment model proposed for portraying the nonideal mixing condition in stirred-tank bioreactors is that by Sinclair and Brown [8]. 

The inherent problem of all compartment models is the poor spatial resolution. By significantly increasing the number of compartments (up to 40000 zones) the network-of-zone (NOZ) models could overcome this problem [23-25]. However, the problem of identification and physical interpretation of the model parameters remains unchanged. The macromixing effects are reflected in the choice 

The shaded areas represent the fraction of volume elememts returning to the micromixer at any time 

In spite of the success of CFD simulations for the multiphase turbulent fluid flow in stirred-tank bioreactors (see Section 3.4), their application to coupled material balance equations in case of more complicated reaction networks is still limited by the required computing power. Even in case of successful approaches for model reduction, the number of compounds necessary for reliable portrayal of cellular dynamics in response to spatial variation of extracellular compounds may be still too large. An interesting method to overcome these numerical difficulties is the general hybrid multizonal/CFD [27-36], which gave momentum to the application of CFD modeling for bioreactors. 

The same authors also presented an example of the use of the population balance equation (PBE) (distribution of biomass m) coimected to the multi-zone/CFD model. This example is in several respects relevant for the assessment of the modeling approach. The coupling of the integro-differential equation of the population balance is a numerical challenge, which can nowadays be tackled within the environment of a CFD approach, albeit without consensus on the proper closure assumptions. Still, the computational effort for the numerical solution of the population balance embedded in the multizonal model is extensive, and it is difficult to extend this approach to multiple state variables necessary for dynamic metabolic models. This is an important argument to favor the alternative method of an agent-based Lagrange-Euler approach discussed in Section 3.5. 

The last approach to characterizing nonideal reactors is the use of compartment models. This methodology is based on the idea that a real reactor can be described as an assonbly of 

Once again, the use of tracer response experiments is a critical element in the development of a compartment model. As with the Dispersion and CIS models, the parameters of a compartment model may have to be calculated from the moments of the external-age distribution function. Moreover, the shapes of the E t) curve must he used to help choose the types and arrangement of compartments that will comprise the model. In order to conceptualize a compartment model from the measured E t) curve at the reactor outlet, it is important to inject the tracer as a sharp pulse, as close to the reactor inlet as possible. If the tracer that enters the reactor is too dispersed, the shape of the tracer curve that leaves the reactor will not reflect its behavior in sufficient detail to formulate an accurate model. 

Compartment models are multiparameter models, in that more than one parameter is required to characterize flow and mixing. The exact number of parameters depends on the number of compartments in the model. 

At this point, our understanding of CSTRs and PFRs is well developed, so that the discussion of compartment models can begin with combinations of these two elements. 

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In a two-compartment model, /3 is equivalent to k in the one-compartment model. Therefore, the terminal half-life for the elimination of a chemical compound following two-compartment model elimination can be calculated from the equation (i = 0.693/ti/i ... 

Physiologically based toxicokinetic models are nowadays used increasingly for toxicological risk assessment. These models are based on human physiology, and thus take into consideration the actual toxicokinetic processes more accurately than the one- or two-compartment models. In these models, all of the relevant information regarding absorption, distribution, biotransformarion, and elimination of a compound is utilized. The principles of physiologically based pharmaco/ toxicokinetic models are depicted in Fig. 5.41a and h. The... 

Drug elimination may not be first order at high doses due to saturation of the capacity of the elimination processes. When this occurs, a reduction in the slope of the elimination curve is observed since elimination is governed by the relationship Vmax/(Km- -[conc]), where Vmax is the maximal rate of elimination, Km is the concentration at which the process runs at half maximal speed, and [cone] is the concentration of the drug. However, once the concentration falls below saturating levels first-order kinetics prevail. Once the saturating levels of drugs fall to ones eliminated via first-order kinetics, the half time can be measured from the linear portion of the In pt versus time relationship. Most elimination processes can be estimated by a one compartment model. This compartment can... 

The limited efficacy of classical anticancer diugs can be explained in part by the compartment model of dividing (growth fraction, compartment A) and nondividing (compartment B) cells. The majority of antineoplastic diugs acts upon cycling cells and will hit, therefore, compartment A only. 

Eriksson, E. (1971). Compartment models and reservoir theory. Ann. Rev. Ecol. Syst. 2, 67-84. 

Hill, B.D. and Schaalje, G.B. (1985). A two compartment model for the dissipation of delta-methrin in soil. Journal of Agriculture and Food Chemistry 33, 1001-1006. 

Simkins S, R Mukherjee, M Alexander (1986) Two approaches to modeling kinetics of biodegradation by growing cells and application of a two-compartment model for mineralization kinetics in sewage. Appl Environ Microbiol 51 1153-1160. 

The synthetic data have been derived from a theoretical one-compartment model with the following settings of the parameters ... 

Fig. 39.9. Time courses of plasma concentration Cp in a one-compartment model for extravascular administration, with different contingencies of (a) the transfer constant of absorption k p, (b) the transfer constant of elimination kpt and (c) the volume of distribution Vp.

In the previous discussion of the one- and two-compartment models we have loaded the system with a single-dose D at time zero, and subsequently we observed its transient response until a steady state was reached. It has been shown that an analysis of the response in the central plasma compartment allows to estimate the transfer constants of the system. Once the transfer constants have been established, it is possible to study the behaviour of the model with different types of input functions. The case when the input is delivered at a constant rate during a certain time interval is of special importance. It applies when a drug is delivered by continuous intravenous infusion. We assume that an amount Z) of a drug is delivered during the time of infusion x at a constant rate (Fig. 39.10). The first part of the mass balance differential equation for this one-compartment open system, for times t between 0 and x, is given by ... 

We consider again the pharmacokinetic parameters of the one-compartment model for a single intravenous injection (eq. (39.6)). 

This model is an extension of the one-compartment model for intravenous injection (Section 39.1.1) which is now provided with a peripheral buffering compartment which exchanges with the central plasma compartment. Elimination occurs via the central compartment (Fig. 39.12). The model requires the estimation of the plasma volume of distribution and three transfer constants, namely for... 

Usually, the buffer compartment is not accessible and, consequently, the absolute amount of X cannot be determined experimentally. For this reason, we will only focus our discussion on the plasma concentration Cp. It is important to know, however, that the time course of the contents in the two compartments is the sum of two exponentials, which have the same positive hybrid transfer constants a and p. The coefficients A and B, however, depend on the particular compartment. This statement can be generalized to mammillary systems with a large number of compartments that exchange with a central compartment. The solutions for each of n compartments in a mammillary model are sums of n exponential functions, having the same n positive hybrid transfer constants, but with n different coefficients for each particular compartment. (We will return to this property of linear compartmental systems during the discussion of multi-compartment models in Section 39.1.7.)... 

A random noise with standard deviation of 0.4 pg 1 has been added to the theoretical values in order to produce a realistic example. The specifications of the model are in part the same as those used for the one-compartment models which have been discussed above. The major distinction between this model and the... 

To conclude this section on two-compartment models we note that the hybrid constants a and p in the exponential function are eigenvalues of the matrix of coefficients of the system of linear differential equations ... 

The inverse transform Xp(t) in the time domain can be obtained by means of the method of indeterminate coefficients, which was presented above in Section 39.1.6. In this case the solution is the same as the one which was derived by conventional methods in Section 39.1.2 (eq. (39.16)). The solution of the two-compartment model in the Laplace domain (eq. (39.77)) can now be used in the analysis of more complex systems, as will be shown below. 

In the previous section we found that the hybrid transfer constants of a two-compartment model are eigenvalues of the transfer constant matrix K. This can be generalized to the multi-compartment model. Hence the characteristic equation can be written by means of the determinant A ... 

This general approach for solving linear pharmacokinetic problems is referred to as the y-method. It is a generalization of the approach by means of the Laplace transform, which has been applied in the previous Section 39.1.6 to the case of a two-compartment model. 

Our pharmacokinetic data indicate that detectable PCP levels may remain in the urine for 4 to 5 weeks after the last use, similar to previous reports (Khajawall and Simpson 1983). The observed elimination kinetics were equally consistent with a one- or two-compartment model, but methodological problems with our data make... 

Azacitidine, a cytidine analog, causes hypomethylation of DNA, which normalizes the function of genes that control cell differentiation to promote normal cell maturation. The suspension is administered as a subcutaneous injection daily for 7 days for the treatment of myelodysplastic syndrome, a preleukemia disease. The pharmacokinetics of azacitidine are best described by a two-compartment model, with a terminal half life of 3.4 to 6.2 hours, whereas peak concentrations are achieved 30 minutes after a subcutaneous injection.7 Azacitidine has been shown to be clinically active in the treatment of myelodysplastic syndromes. The side effects include myelosuppression, renal tubular acidosis, renal dysfunction, and injection-site reactions. 

Vinblastine is another vesicant vinca alkaloid that causes myelo-suppression and less neurotoxicity than vincristine. The pharmacokinetics of vinblastine are best described by a three-compartment model, with an a half-life of 25 minutes, a 3 half-life of 53 minutes, and a terminal half-life of 19 to 25 hours.12 Vinblastine has shown activity in the treatment of bladder, breast, and kidney cancer, as well as some lymphomas. The doses of vinblastine tend to be higher on a milligram per meter squared basis than vincristine. Nausea and vomiting are minimal with vinblastine. Other side effects include mild alopecia, rash, photosensitivity, and stomatitis. 

The vesicant vinorelbine is structurally similar to vincristine and may cause many of the same side effects as vincristine. While this vesicant is administered intravenously over 6 to 10 minutes, patients should be counseled about neuropathy, ileus, and myelosuppression. The pharmacokinetics of vinorelbine are best described by a three-compartment model, with an a half-life of 2 to 6 minutes, a 3 half-life of 1.9 hours, and a y half-life of 40 hours. Vinorelbine has shown efficacy in the treatment of breast cancer and non-small cell lung cancer. Additional side effects include myelosuppression, paresthesias, and mild nausea and vomiting. 

Docetaxel, another taxane, binds to tubulin to promote microtubule assembly. The pharmacokinetics of docetaxel are best described by a three-compartment model, with an a half-life of 0.08 hours, a 3 half-life of 1.6 to 1.8 hours, and a terminal half-life of 65 to 73 hours.14 Docetaxel has activity in the treatment of breast, non-small cell lung, prostate, bladder, esophageal, stomach, ovary, and head and neck cancers. Dexamethasone, 8 mg twice daily for 3 days starting the day before treatment, is used to prevent the fluid retention syndrome associated with docetaxel and possible hypersensitivity reactions. The fluid... 

Etoposide causes multiple DNA double-strand breaks by inhibiting topoisomerase II. The pharmacokinetics of etoposide are described by a two-compartment model, with an a half-life of 0.5 to 1 hour and a (5 half-life of 3.4 to 8.3 hours. Approximately 30% of the dose is excreted unchanged by the kidney.16 Etoposide has shown activity in the treatment of several types of lymphoma, testicular and lung cancer, retinoblastoma, and carcinoma of unknown primary. The intravenous preparation has limited stability, so final concentrations should be 0.4 mg/mL. Intravenous administration needs to be slow to prevent hypotension. Oral bioavailability is approximately 50%, so oral dosages are approximate two times those of intravenous doses however, relatively low oral daily dosages are used for 1 to 2 weeks. Side effects include mucositis, myelosuppression, alopecia, phlebitis, hypersensitivity reactions, and secondary leukemias. 

Teniposide, a topoisomerase II inhibitor, is administered as an infusion over 30 to 60 minutes to prevent hypotension. The pharmacokinetics are described by a three-compartment model, with an a half-life of 0.75 hours, a (5 half-life of 4 hours, and a terminal half-life of 20 hours. Considerable variability in clearance of teniposide in children has been reported.17 Teniposide has shown activity in the treatment of acute lymphocytic leukemia, neuroblastoma, and non-Hodgkin s lymphoma. Side effects include myelosuppression, nausea, vomiting, mucositis, and venous irritation. Hypersensitivity reactions may be life-threatening. 

Topotecan inhibits topoisomerase I to cause single-strand breaks in DNA. The pharmacokinetics of topotecan can be described by a two-compartment model, with a terminal half-life of 80 to 180 minutes, with renal clearance accounting for approximately 70% of the clearance.19 Topotecan has shown clinical activity in the treatment of ovarian and lung cancer, myelodysplastic syndromes, and acute myelogenous leukemia. The intravenous infusion may be daily for 5 days or once weekly. Side effects include myelosuppression, mucositis, and diarrhea. 

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