ANIMAL SCALE-UP

Robert L. Dedrick, Ph.D.

Division of Bioengineering and Physical Science, ORS, NIH

SUMMARY

Despite the influence of allometry on the development of biology and an
underlying belief that experimental systems provide useful information
about humans, disproportionate emphasis has been placed on species
differences. This would appear to derive from the culture of biology
because differences among species have often been more interesting than
similarities and because these differences can provide important
information on the development of species. We recognize that no other
animal is the same as a human in any general biological sense and that
insistence on "sameness" in a model system is illusory. I would propose
that we adopt more of an engineering-design view when we develop
experimental systems in pharmacokinetics and attempt to use data from
these systems for predictive purposes. If we do this, it is axiomatic in
biology as in engineering that the model system is [never]{.underline}
the same as the prototype. Interpretation is always required. In some
simple systems, concepts of similitude place design on a sound
theoretical basis. But in more complex situations rigorous similitude
may not be attainable. In these cases it is often possible to model
parts of a complex system and use model-dependent information in a
design process which incorporates sound theoretical principles but often
contains judgment and experience as well. This approach is illustrated
by an examination of the use of experimental *in-vitro* and *in-vivo*
data to predict pharmacokinetics of drugs in humans.

ALLOMETRIC EQUATIONS

It has been observed that many physiological processes and organ sizes
show a relatively simple power-law relationship with body weight when
these are compared among mammals. The well known allometric equation is

$P = a(\text{BW})^{m}$ (1)

where P = physiological property or anatomic size

a = empirical coefficient

BW = body weight

m = allometric exponent

Note that a is not dimensionless; its value depends on the units in
which P and BW are measured, while the exponent, m, is independent of
the system of units. Note further that if m = 1, then P is proportional
to BW. If m\<1, P increases less rapidly than BW. If m\>1, P increases
more rapidly that BW. Dividing both sides of Eq (1) by BW shows that

$\frac{P}{\text{BW}} = a(\text{BW})^{m - 1}$ (2)

Thus, if the allometric exponent is less than unity, as observed for
many measures of physiologic function such as basal oxygen consumption
and creatinine clearance, the function per unit of body weight
[decreases]{.underline} as body weight increases. If m = 0.7 for the
renal clearance of a particular drug, the clearance per unit body weight
in a 20-g mouse would expected to be \[(70,000)/(20)\]^0.3^ = 12 times
that in 70-kg human. If the volume of distribution is similar between
the two species (such as body water) and the drug is cleared only by the
kidney, then as a rough approximation pharmacokinetics would be
occurring 12-times faster in the mouse. One hr in a mouse would be
pharmacokinetically equivalent to 12 hr in a human. Such considerations
are important in the design of drug studies, because pharmacokinetic
time scales vary greatly among species.

PHYSIOLOGICAL PHARMACOKINTICS

The distribution and disposition of a drug in the body result from a
complex set of physiological processes and biochemical interactions. In
principle it is possible to describe these processes and interactions in
mathematical terms and, if sufficient data are available, to predict the
time course of drug and metabolite(s) in specific anatomic sites.

The basis of a physiological pharmacokinetic model is a flow diagram
showing the anatomic relationships among the various organs and tissues.
The accumulation of a drug within a compartment is described by an
appropriate mass-balance equation. As an illustration, we consider the
accumulation of a drug in the kidney, which is assumed both to
metabolize the drug by a saturable process and to clear it by filtration
and possibly secretion. It is further assumed that the concentration
within the compartment is uniform and equal to that of venous blood.

$V_{K}\frac{\text{dC}_{K}}{\text{dt}} = Q_{K}C_{B} - Q_{K}C_{K} - \text{CL}_{K}C_{B} - \left\lbrack \frac{v_{\text{max,}K}C_{K}}{K_{m,K} + C_{K}} \right\rbrack V_{K}$
(3)

where V = compartment volume, ml

C = drug concentration, µg/ml

t = time, min

Q = blood flow rate, ml/min

v~max~ = maximum rate of metabolism, µg/(min ml)

K = Michaelis constant, µg/ml

CL = non-metabolic clearance, ml/min

and the subscripts K and B refer to kidney and arterial blood,
respectively.

Similar equations can be written for all relevant compartments. If
parameters are chosen, the resulting set of ordinary differential
equations can be solved numerically to yield predictions of the
concentration of the drug and metabolite(s) in each of the compartments
as a function of time. Of course, the simplifying assumptions above can
be relaxed to include much more detail concerning plasma and tissue
binding, transport at the level of the blood capillary and cell
membrane, and spatial nonuniformity -- but at the cost of increasing
complexity and the requirement for more parameters.