Functional allometric

Today, well over 100 biological parameters of mammals are known to be linearly related to body weight and highly predictable on an mterspecies basis (Davidson et al. 1986, Voisin et al. 1990, Calabrese et al. 1992). The allometric equation has traditionally been used for extrapolation of experimental data concerning physiological and biochemical functions from one mammalian species to another. In addition, the allometric equation has also been used extensively as the basis for extrapolation, or scaling, of e.g., a NOAEL derived for a chemical from studies in experimental animals to an equivalent human NOAEL, i.e., a correction for differences in body size between humans and experimental animals. 

Finally, no discussion of human pharmacokinetic predictions is complete without a consideration of allometric scaling [67-69]. In general, allometry is the examination of relationships between size and function and it has been applied to the prediction of human pharmacokinetic parameters from animal pharmacokinetic parameters for decades [70]. Allometry has been shown to work reasonably well for predicting human VD from animal VD data, probably because volumes of plasma and various tissue across species are allometrically scaleable to body weight, a notion reinforced... 

Allometric scaling (allometry) is the discipline that predicts human PK using pre-clinical data (Ritschel et al., 1992). This approach is based on empirical observations that various physiological parameters are functions of body size. The most widely used equation in allometry is a one-term power function ... 

The models were developed to simulate the physiology (e.g., blood flows and body composition) of adult rats (Table 3-6). These parameter values were then extrapolated to juvenile rats to accommodate calibration and validation data in which juvenile rats were the test organisms. The extrapolation was achieved by scaling blood flows, metabolic constants, and adipose volumes to various functions of body weight (e.g., allometric scaling). 

Power-law expressions are found at all hierarchical levels of organization from the molecular level of elementary chemical reactions to the organismal level of growth and allometric morphogenesis. This recurrence of the power law at different levels of organization is reminiscent of fractal phenomena. In the case of fractal phenomena, it has been shown that this self-similar property is intimately associated with the power-law expression [28]. The reverse is also true if a power function of time describes the observed kinetic data or a reaction rate higher than 2 is revealed, the reaction takes place in fractal physical support. 

ThuS/ if the allometric exponent is less than unity/ as observed for many measures of physiologic function/ the function per unit of body weight decreases as body weight increases. 

While the interspecies variability in metabolism precludes the possibility of a simple allometric relationship for the plasma kinetics of ARA-C, the non-metabolic clearance by the kidney does exhibit a power-law relationship with body weight. Figure 30.7 shows the kidney clearance of ARA-C and its deami-nated product ARA-U on a log-log plot as a function of body weight for mice, monkeys, dogs, and humans. The slope is 0.80, which is essentially the same as the value of 0.77 for inulin (1). 

Physiological parameters. Comprehensive sets of the physiological parameters for humans and common laboratory animals are available in the literature [5-7], When information gaps exist, necessary values can often be obtained via experimentation or through allometric extrapolation, usually based on a power function of the body weight [8] (e.g., X = aWp, where X is the parameter of interest, W is body weight, and a, (5 are constants). 

Using limited data, allometric scaling may be useful in drug discovery. We assume that, for the formula Y = aWb, the value of the power function (or slope of the line from a log vs. log plot) is drug independent, unlike the intercept o , which is drug dependent. By doing this, we can use data from a single species (rat) to successfully predict the pharmacokinetics of compound X in humans... 

It is possible to test the allometric relation of Taylor using computergenerated data. But before we do so, we note that Taylor and Woiwod [20] were able to extend the discussion from the stability of the population density in space, independent of time, to the stability of the population density in time, independent of space. Consequently, just as spatial stability, as measured by the variance, is a power function of the mean population density over a given area at all times, so too the temporal stability, as measured by the variance, is a power function of the mean population density over time at all locations. With this generalization in hand we can apply Taylor s ideas to time series. 

When developing a new therapeutic compound, relevant toxicokinetic studies are performed in different species such as mice, rabbits, dogs, and monkeys. The results of such studies lead to interspecies scaling in PK parameters based on the assumption that there are physiological and biochemical analogies between mammals, which can be expressed mathematically by allometric equations (58). The allometric approach is based on the power function, where the PK parameters Y are plotted against the body weight W of the different species ... 

For an excellent representation of the arguments in one recent and particularly fascinating instantiation of this debate, see the theme issue of Functional Ecology (18 2, 2004) dedicated to internalist and externalist understandings of allometric trends in metabolic rate. 

Darveau et al. (2002, 2003) propose that the relationship between body mass and metabolic rate reflects the contribution of multiple factors - ATP-utilization processes in parallel, supply processes in series - that each have different power functions. This hierarchical layering results in an allometric cascade that has different scaling implications for different measures of metabolism. They contend that only a multiple-factor account, and not West s (or any) single-cause account, can explain the scaling difference between basal and maximum metabolic rate (Bishop, 1999). However, the mathematical formulation of their model has been severely criticized (Banavar etal., 2003 West etal., 2003), and in any case it does not provide an account of why individual processes scale as power functions of mass or why the causal cascade results in a whole-organism metabolism that approximates the 3/4 rule (Bokma, 2004 West et al., 2003 West and Brown, 2004). 

As with the trends previously mentioned, proposals have been promulgated for internal and external constraints. At first pass, it is tempting to account for relations between life history variables almost purely on the basis of fundamental allometric constraints. Metabolic rate, lifespan, fecundity, age at maturity, and maternal investment all vary with body mass as power functions. In fact, relations are invariant between some of these variables. For example, lifespan scales with body mass by a 1/4 power, and heart rate (or the rate of ATP synthesis) scales with body mass by a — 1/4 power. The product yields an approximately constant number of metabolic events in mammal species, independent of body mass or lifespan. Age at maturity / lifespan, and annual maternal investment / lifespan (for indeterminate growers), are also invariant ratios (Chamov, 1993 Chamov et al., 2001 Steams, 1992). West and Brown (2004) point out that invariant ratios, and universal quarter-power allometric trends in general, suggest underlying physical first principles. They employ their model to explain these life history relations (Enquist et al., 1999 Niklas and Enquist, 2001 West et al., 2001). 

This is not the case for the other two formalisms commonly used in biochemistry—the Linear Formalism and the Michaelis-Menten Formalism. The Linear Formalism implies linear relationships among the constituents of a system in quasi-steady state, which is inconsistent with the wealth of experimental evidence showing that these relationships are nonlinear in most cases. The case of the Michaelis-Menten Formalism is more problematic. An arbitrary system of reactions described by rational functions of the type associated with the Michaelis-Menten Formalism has no known solution in terms of elementary mathematical functions, so it is difficult to determine whether or not this formalism is consistent with the experimentally observed data. It is possible to deduce the systemic behavior of simple specific systems involving a few rational functions and find examples in which the elements do not exhibit allometric relationships. So, in... 

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