Liver compartmental modeling

Stirred tank models have been widely used in pharmaceutical research. They form the basis of the compartmental models of traditional and physiological pharmacokinetics and have also been used to describe drug bioconversion in the liver [1,2], drug absorption from the gastrointestinal tract [3], and the production of recombinant proteins in continuous flow fermenters [4], In this book, a more detailed development of stirred tank models can be found in Chapter 3, in which pharmacokinetic models are discussed by Dr. James Gallo. The conceptual and mathematical simplicity of stirred tank models ensures their continued use in pharmacokinetics and in other systems of pharmaceutical interest in which spatially uniform concentrations exist or can be assumed. 

Compartmental models are normally based on a central compartment, which represents the plasma and the highly perfused tissues (Figure 8.3). Elimination (see section 2.7.1) of a drug is assumed to occur only from this compartment since the processes associated with elimination occur mainly in the plasma and the highly perfused tissues of the liver and kidney. Other compartments are connected to the central compartment as required by the nature of the investigation. 

Because there are many ways to achieve a given level of bioavailability, it makes sense to consider using a compartmental model to predict bioavailability rather than simply training a model on a set of bioavailability results. The role of metabolism tends to dominate most often and variability in drug response is greatly influenced by this. Drugs that are efficiently eliminated by the liver often have high variability in the plasma levels both within and between individuals... 

Figure 3. This kinetic model for zinc in humans was based on averaged data obtained following oral and i.v. administration of Zn to 17 patients with abnormalities of taste and smell. The compartmental model used all kinetic data from Zn activity in plasma, red blood cells, urine, liver, and thigh as well as stable zinc parameters, including dietary intake, serum, and urinary concentration. The SAAM27 computer program was used to obtain the simplest set of mathematical relationships that would satisfy the data characteristics for each measurement time in the study and remain consistent with accepted concepts of zinc metabolism. Although the short physical half-life of Zn limited the data collection period, this model allowed for analysis of the rapid phases of zinc metabolism (about 10% of total body zinc) and derivation of a number of fundamental steady state...

FIG. 6. Compartmental model proposed by Green et al. (1993) for liver and whole-body vitamin A metabolism. Compartment 11 is plasma retinol. PC, parenchymal cells NPC, nonparenchymal cells (assumed here to be perisinusoidal stellate cells) ROH, retinol RE, retinyl esters CM, chylomicrons. 

If conversion of /3-carotene to retinyl ester in the enterocyte were small, it might be a rate-limiting step that could be augmented by hepatic conversion, since the liver also contains the carotenoid-lS,lS -dioxygenase enzyme that catalyzes this process. To test if the compartmental model could predict such system behavior accurately if the liver were the only site for conversion, model parameters were altered so that only intact /3-carotene-ds was absorbed and conversion to retinyl-d4 ester in the enterocyte did not occur. Under these conditions, the compartmental model could not predict the experimental observations because fitting the first /3-carotene-dg peak limited the amount of /3-carotene-dg and retinoid-d4 which was introduced into the system, and thus underestimated either the second /3-carotene-dg peak or the retinoid-d4 peak. It was concluded therefore, that the enterocyte is an important site of conversion of /3-carotene to retinoid. Considerable evidence already exists for conversion of /3-carotene to retinoid in the intestine (Dimitrov et al, 1988 Olson, 1989 Wang et al., 1992 Sdta et aL, 1993). 

It must be realized, however, that the data used to build a particular compartmental model may not always provide sufficient statistical certainty of a given parameter s value. Because retinol-d4 and retinyl-d4 ester were not measured individually in the plasma after the subject ingested the /3-carotene-dg, we were unable to determine with statistical certainty the FTCs specifically for retinyl ester. Movement of retinyl ester from the enterocyte into the plasma was highly correlated with its removal from the plasma into the liver via chylomicron remnant. Therefore, the FTCs describing movement of retinyl ester from the enterocyte to the plasma and from the plasma to the liver could be increased simultaneously without compromising the model s prediction of the experimental observations. 

Because of the need to employ minimally invasive procedures in studies involving human subjects, compartments that can be experimentally observed are usually limited to diet, blood, urine, and feces, while compartments such as liver and EHT, etc., usually can be observed only under special circumstances. Compartmental models are ideally suited to human kinetic studies because unobservable compartments, espedally those that exchange analyte with blood, can be included in the model, and masses... 

The compartmental model of /8-carotene metabolism presented here is compatible with previously developed compartmental models of retinol metabolism (Green et aL, 1985 Lewis et aL, 1990). For example, the compartmental model of the dynamics of /S-carotene metabolism features two kinetically distinct pools of retinol in the liver, recycling of plasma retinol by liver, and irreversible loss of retinol from the plasma. These aspects of retinol metabolism (predicted by the compaitmental model) are compatible with already described aspects of retinol metabolism. 

FIG. 6. Compartmental model predicted masses and concentrations of jS-carotene in plasma, liver, and extrahepatic tissue of a healthy adult who ingested a angle 73-/imol dose of JS-carotene-dg orally. Panels labeled Past turnover liver total -carotene, Slow turnover liver total /3-carotene, and Extrahepatic total /3-carotene each include protio and deuterated... 

Questions regarding the extent of postabsorptive bioconversion of /3C to vitamin A persist. Animal data indicate the liver possesses this capability, but the relative importance of intestinal mucosa versus liver is unknown. Novotny and co-workers (1995) reported a compartmental model which predicted that both liver and intestinal mucosa were important sites for biotransformation of /SC in the human, with 43% of total conversion occurring in the liver and 56% in the intestinal mucosa. However, the model assumed a stoichiometry of 1 mol retinol per mole /SC, and the effect on the model assuming a 2 1 ratio was not discussed. 

The second broad peak between 24 and 48 hr represents labeled /3C secreted by the liver associated with very low density lipoproteins (VLDL) and very likely encompasses the period during which these VLDL particles undergo lipolysis to IDL and LDL. As this lipolytic process occurs relatively rapidly, the broad and extented nature of this second peak probably reflects recycling of labeled /8C into and out of the liver, i.e., hepatic reprocessing of VLDL or LDL particles. This phenomenon may be investigated further using modeling approaches and is reflected in the compartmental model recently proposed by Novotny et al. (1995). 

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. 

Rates of hydroquinone glucuronidation in human liver microsomes showed a two- to three-fold variation between individual liver samples they were somewhat higher than in the rat, and lower than in the mouse liver (Seaton et al., 1995). A compartmental pharmacokinetic model was derived to describe the pharmacokinetics of hydroquinone in vivo in humans, rats and mice, incorporating hydroquinone glucuronidation rates sulfation of hydroquinone was not included in this model. NAD(P)H quinone acceptor oxidoreductases protect against reactive quinones by reducing them to the hydroquinone this enzyme seems to be absent in some individuals, which will lead to loss of such protection and make them more sensitive to hydroquinone toxicity (Ross, 1996). 

As in humans ethical constraints limit sampling sites to peripheral blood vessels, absorption cannot directly be estimated. Hence, the absorption rate (ka, fe0i) as calculated from peripheral plasma concentration data by compartmental or noncom-partmental models reflects rather an oral bioavailability rate, unless any gut wall and liver first-pass elimination is ruled out [112]. 

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