Dependent eliminating from models

The absorption, distribution, and accumulation of lead in the human body may be represented by a three-part model (6). The first part consists of red blood cells, which move the lead to the other two parts, soft tissue and bone. The blood cells and soft tissue, represented by the liver and kidney, constitute the mobile part of the lead body burden, which can fluctuate depending on the length of exposure to the pollutant. Lead accumulation over a long period of time occurs in the bones, which store up to 95% of the total body burden. However, the lead in soft tissue represents a potentially greater toxicological hazard and is the more important component of the lead body burden. Lead measured in the urine has been found to be a good index of the amount of mobile lead in the body. The majority of lead is eliminated from the body in the urine and feces, with smaller amounts removed by sweat, hair, and nails. 

Due to the second criterion, time-to-tumor models were eliminated from consideration. These models require more detailed experimental data than is generally available. Moreover, it is difficult and unproductive to interpret the distribution of time-to-tumor in the context of human exposures. In most cases, the time-to-tumor variable would be integrated over a human lifetime, thus reducing the model to a purely dose-dependent one. Therefore we restrict our attention to quantal response models that estimate lifetime risks. 

We can eliminate all the false dependent variables from the statistical model thanks to the correlation analysis. When we obtain = 1 for a process with two dependent variables (yj, y2), we have a linear dependence between these variables. Then, in this case, both variables exceed the independence required by the output process variables. Therefore, yj or can be eliminated from the list of the dependent process variables. 

Both complexes 75 and 76 promote the hydrolysis of urea in a two-step process with the same initial rates (118). Heating of 75 or 76 in acetonitrile solution produced ammonia with kinetic first-order dependence on complex concentration and an observed rate constant of (7.7 0.5) x 10 " h to yield a cyanate complex as the reaction product. It remains unclear, however, which binding mode of urea (terminal or bridging as found in 76) facilitates the ehmination reaction. Ammonia elimination from the O bound terminal substrate appears to be in accordance with quantum chemical studies on that model system (34). Although no crystals could be obtained for the cyanate-containing reaction product, an analogous complex (77) with virtually identical Vas(OCN) (as = asymmetric) vibration (at 2164cm )... 

FIGURE 3.6 Compartmental analysis for different terms of volume of distribution. (Adapted from Kwon, Y., Handbook of Essential Pharmacokinetics, Pharmacodynamics and Drug Metabolism for Industrial Scientists, Kluwer Academic/Plenum Publishers, New York, 2001. With permission.) (a) Schematic diagram of two-compartment model for compound disposition. Compound is administrated and eliminated from central compartment (compartment 1) and distributes between central compartment and peripheral compartment (compartment 2). Vj and V2 are the apparent volumes of the central and peripheral compartments, respectively. kI0 is the elimination rate constant, and k12 and k21 are the intercompartmental distribution rate constants, (b) Concentration versus time profiles of plasma (—) and peripheral tissue (—) for two-compartmental disposition after IV bolus injection. C0 is the extrapolated concentration at time zero, used for estimation of V, The time of distributional equilibrium is fss. Ydss is a volume distribution value at fss only. Vj, is the volume of distribution value at and after postdistribution equilibrium, which is influenced by relative rates of distribution and elimination, (c) Time-dependent volume of distribution for the corresponding two-compart-mental disposition. Vt is the starting distribution space and has the smallest value. Volume of distribution increases to Vdss at t,s. Volume of distribution further increases with time to Vp at and after postdistribution equilibrium. Vp is influenced by relative rates of distribution and elimination and is not a pure term for volume of distribution. 

Dose-dependent clearance and distribution was then later observed in a Phase 1 study in children with solid tumors (Sonnichsen et al., 1994). In a study in adults with ovarian cancer, Gianni et al. (1995) used a 3-compartment model with saturable intercompart-mental clearance into Compartment 2 and saturable, Michaelis-Menten elimination kinetics from the central compartment to describe the kinetics after 3 hour and 24 hour infusion. Now at this point one would typically assume that the mechanism for nonlinear elimination from the central compartment is either saturable protein binding or saturable metabolism. But the story is not that simple. Sparreboom et al. (1996a) speculated that since Cremophor EL is known to form micelles in aqueous solution, even many hours after dilution below the critical micellular concentration, and can modulate P-glycoprotein efflux, that the nonlinearity in pharmacokinetics was not due to paclitaxel, but due to the vehicle, Cremophor EL. This hypothesis was later confirmed in a study in mice (Sparreboom et al., 1996b). 

Bauer and Wu have applied a method involving the phase shifts A,. But they find that their results for the reaction of Eq. (162) are approximately the same whether the phase shifts are eliminated from or included in their computation. Setting the A, equal to zero is tantamount to a Bom approximation. Therefore, the negligible dependence of the results on A, supports in a back-handed way the prior work of Golden and Peiser. This support, however, is tenuous because of the differences in the models of the two approaches. 

An extension of indirect response models are precursor pool-dependent indirect response models that include the lihera-tion of an endogenous compound from a storage pool. These models possess the unique ability to characterize hoth tolerance and rebound phenomena [90]. Such a model was, for example, used to describe the effect of interferon- Sla on neopterin, an endogenous marker for cell-mediated immunity, in humans and monkeys (Fig. 8) [91, 92]. The primary elimination mechanism of interferon- 8 la was modeled as receptor-mediated endo-cytosis, and the pharmacodynamic model was driven by the amount of internalized drug-receptor complex DR ... 

The structural information contained in the ARRs, i.e. the information on which ARR depends on which component parameters can be obtained directly by inspection of causal paths in a diagnostic bond graph [1]. There is no need to derive equations and to eliminate unknowns in order to set up a mode-dependent FSM. To that end, causal paths from model inputs to inputs of sensor elements are considered. Elements that are traversed on these causal paths contribute to the ARR of a residual related to a sensor element. An output of a source or an element that is followed directly or indirectly by switches on the causal path to a sensor element provides an entry in the FSM equal to the product of the switch states. 

The daily elimination in untreated people is estimated at 12 ixg [12] including 2.9 xg excreted with the urine [27]. Ingested bismuth from therapeutics is mainly eliminated with feces as sulfide. Within 5 days 10-20% is excreted. But elimination is not finished after 10 days. Overall 99% may be eliminated in this way [3,12,17,29]. Absorbed bismuth is mainly excreted by urine the biliary/fecal elimination route is only about half of the urinary one [3,6,62-64]. Half-lives in blood after a single dose or during a treatment depend on the kind of the Bi compound, the amounts ingested, and the blood levels. Elimination from blood of bismuth subcitrate is biphasic [17,28,65]. In cases of encephalopathy with remarkably high urine and blood levels (2000 and 1500 p-g/liter, respectively), half-lives were calculated for urine (4.5 days) and blood (5.2 days). Liquor levels decreased more slowly with a half-life of 15.9 days [53]. Elimination kinetics is also described as a three-compartment model with half-lives of 3.5 min, 0.25 hr, and 3.2 hr [6]. Biological half-times in humans are reported for the whole body 5 days, the kidney 6 days, and the liver 15 days (cited in [3]). 

Overall, the model is constructed with 15 peripheral body compartments, and elimination of Pb is structured within three pools. Diffusible plasma Pb comprises the central compartment, with bound and free fractions, and most of the Pb is in bound form, in an approximate 5 1 ratio. Movement of Pb from plasma to tissues follows first-order kinetics, and transfer constants are age- and PbB-dependent. Dependency on PbB is built into this model as nonlinearity above a threshold level, analogous to the other systems, and accounts for the increased relative fraction of Pb in plasma versus erythrocytes noted in published data (Bergdahl et al., 1997 Manton and Cook, 1984 Manton et al., 2001). Concentrations of Pb in the blood compartment are simulated on the basis of age-dependent blood volume. With other tissues, only Pb masses are calculated. The temporal resolution of the Leggett model can be as short as 1 day, similar to the O Flaherty model. Age dependency in the model is reflected in six age groups infants children adolescents and three subgroups of adults—young, middle-aged, and older adults. 

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