similarity principles and intrinsic geometries: contrasting approaches to interspecies scaling

November 1986 Volume 75, Number 11

Journal of Pharmaceutical

Sciences A puMicatbn of the American Pharmaceutical Association

SYMPOSIUM ARTICLES

Similarity Principles and Intrinsic Geometries: Contrasting Approaches to lnterspecies Scaling

F. EUGENE YATES’ AND PETER N. KUGLER

Received May 5, 1986, from the Crump Institute for Medical Engineering, UCLA, Los Angeles, CA 90024. September 2, 1986.

Accepted for publication

Abstract 0 We criticize standard allometric approaches on the grounds that they emphasize scaling to one variable at a time, whereas cherni- cally reactive hydrodynamic systems involved in phamacokinetic phenomena are of higher dimension. We show that attempts based on mechanical similitude to set a dosage that would be equivalent across species (for example, from mouse to humans) lead to ambiguous results. Another failing of standard allometry may be its incapability to accommodate the neoteny of Homo sapiens, even though it helped discover the phenomenon. The retarded development in our species implied by neoteny can most clearly be seen in the evidence that both our brain size and our lifespan lie well above the allometric curve for Class Mammalia for these features. In contrast to allometry, which proposes a search for scaling coefficients through invariant external measurement reference frames, we propose a search for transformations of coordinate space coefficients in an intrinsic geometry for the mammalian body plan.

Some of the familiar questions that arise in the testing of a new agent for its therapeutic potential in humans against a chosen disease include

What is the definition of a “dose”? (Is it mass or rate [masdtime], and, in either case, how is it scheduled?)’ What is the trajectory (time course) of plasma concentrations of (unbound, free) drug as a function of dose and schedule? What are the relations between the trajectory of plasma concentration and the desired and side effects (including the separate thresholds for least effective dose and for toxicity)? Is there a model (animal or mathematical) that can be used to answer the above three questions?

From a systems viewpoint, we can say that the problem is to transform the input (drug dose) applied to the system (diseased human) into the output (well human, or human in

improved state of health). This transformation is often de- composed into two separate parts,’ each with an input, a central system, and an output:

Pharmacokinetics Pharmacodynamics

drugs) body) (What the body does to (What drugs do to the

Input Drug dose Temporal pattern of plasma concentration of unbound drug

System Distribution, transport, Cells or tissues with binding, metabolism, receptors to drug (or and excretion elements otherwise responsive and processes to it)

Output Temporal pattern of Changed performance plasma concentration of in desired direction; unbound drug side effects

In this simplified scheme the possibility that some of the metabolites of a drug may be as active as the drug itself, or more so, is ignored. It need not affect the main points of this discussion.

Both the pharmacokinetic and the pharmacodynamic aspects of drug dose-response relations are functions of genet- ics, size, age, diseases, diet, and other drugs being taken concurrently. They vary across species. Yet to carry out drug development in a regulated industry we must often scale figuratively, if not literally, from mouse to human for phar- macologic and toxologic data in spite of the many uncertainties. We need principles of data extrapolation to justify transforming data from one system (species) into data about another, different system. We need to know what scaling principles can guide the designs of our experimental proto- cols. Below, we discuss two different approaches to pharmacokinetic scaling. (Pharmacodynamic scaling is more difficult; e.g., there is no known physical basis for an a priori expecta-

0022-3549/86/1 l~-lOl9$Ol.oa/o 0 1986, American Pharmaceutical Association

Journal of Pharmaceutical Sciences / 101 9 Vol. 75, No. 17, November 1986

tion that opiates would behave differently in some mammalian species than in others, at similar plasma concentrations. Here genetic, “informational” differences may override dynamic, physical similitudes that apply to transport processes.)

First Approach: Allometric Scaling Similitude and Dimensional Analysis-The excellent re-

view and commentary by Calder2 contains the empirical facts needed for this discussion as well as an introduction to similarity principles. Peters3 also provides a comprehensive set of data. Another clear account of traditional dimensional analysis and the theory of dynamically similar models, as these were developed in mechanics, has been given by McMahon and Bonner.4

The history of the use of principles of similitude in physical science began with Galileo, around 1640, but the modem formulation came from Fourier in 1822. He emphasized that the measurable quantities appearing in the mathematical representations of physical laws usually are not pure numbers; most of them have associated dimensions, such as velocity (lengthhime), momentum (mass x lengthhime), and force (mass x lengthhime’). These can be expressed in terms of a smaller set of fundamental dimensions, which he took to be mass, length, and time. That is, every mechanical magnitude has a dimension that can be expressed as a monomial in the fundamental ones-(mass)a(length)b(time)‘. (Later we shall suggest that to advance allometry it may be necessary to extend the group of fundamental quantities to embrace thermodynamic as well as classical mechanical variables.) Each dimension, fundamental or derived, has a scale of units specified by the measurement situation.

Magnitudes with different dimensions or units cannot be combined arbitrarily; they may be multiplied but they cannot be added. Fourier’s principle of dimensional homogeneity requires that any mathematical expression which purports to describe a physical system must accord with the rules for manipulating dimensional qualities. This principle severely restricts the mathematical expressions that can describe physical relationships. These ideas were further developed by Buckingham,s and recently they have been generalized by Rosen.6

Generalization of Similitude Principles-Rosene argues that the principle of dimensional homogeneity which under- lies dimensional analysis is not strong enough to cope effec- tively with systems involving many phases, such as are addressed in biological studies. To reach that conclusion, he starts with the Buckingham Il-Theorem,” which asserts that to each nonfundamental (derived) quantity xi, we can asso- ciate a corresponding dimensionless quantity ni. An equation of state, typically involving dimensional quantities, can be systematically rewritten in terms of dimensionless quantities Hi, and it will then involve a smaller number of arguments. The Principle of Similitude is then formulated: two systems governed by the same equation of state are similar if, and only if, the dimensionless numbers associated with them in this way are pairwise equal.

Rosen rewrites the equation of state (eq. l a ) as a parameterized family of mappings (fg: X + Y), such that any two mappings in the family can be related by transformations of the form in eq. lb:

@(XI, . . ., x,) = 0 (la)

which depend only on the original and final values assigned to fundamental quantities. (In eq. lb, the parameter g belongs to a space, G, of the fundamental quantities, where X and Y are the appropriate subspaces of the space of derived

quantities.) For every g,g’ in G, we can construct a diagram of mappings of the form:

X 4 Y

If we make an arbitrary change (g -+ g‘) in values of fundamental arguments, there is a transformation of the derived arguments that leaves the equation of state invariant.

From the above formulation, the Buckingham II-Theorem can be generalized so that there is no reliance on specific dimensional arguments. Consider the set of triples of the form (g, x, y). Two such triples would be (g, x, y) and (g’, x’, y‘). The triples are corresponding if, and only if, the mappings shown in eq. l b hold so that we have:

This relation of correspondence is an equivalence relation on the set of triples; a complete set of invariants for this equivalence relation are the analogs of the dimensionless numbers, ni, in the Buckingham Theorem. This development goes beyond the simple scale transformations that are characteristic of traditional dimensional situations, but the Prin- ciple of Similitude is still formulated in terms of these invariants as before. Insofar as system behavior can be described by an equation of state (eq. la) , the modeling relation of Fig. 1 represents a generalized form of similitude.

A mathematical model is a formal system in which rules of inference replace the causal mechanisms of the prototype system. An inference drawn in the mathematical model is a prediction about the prototype. %sen argues that a natural system, S1 (e.g., a mouse), is a “model” of selected behaviors of a natural system prototype, Sz (e.g., a human), if and only if both systems can be described for those behaviors by a common muthemutical model. Figure 1 shows the most general case in which two natural systems, S1 and Sz, are being thought of as prototype and model, respectively. In that case, for each there must be a mathematical model (MI; Mz), related to each parent natural system (S1; Sz) by an encoding process that abstracts the physical dynamics into the world of symbolic representations, and a decoding process that oper- ates in reverse. When two systems have isomorphic mathematical models, they are analogous physical systems and one can be considered as an analog model of the other (dashed lines in Fig. 1). (Note that any given system, S1, may have many different nonisomorphic mathematical models that emphasize different aspects of its behavior and code it into completely different formal systems.)

Shortcomings of Standard Allometric Approaches-The usual allometric approach relates one biological form or process (y) to another (x) through an empirical power function:

Typically, in pharmacokinetics, x is taken to be some average value of body mass (M) of a species (expressed as body weight), and the data vary for the many species in a taxonomic class (e.g., Class Mammalia has a 107-fold size range). Even if we restrict attention to eutherian (placental) land mammals, we still have a range in weight from the 3-g shrew to the 3 x lo6-g elephant, covering six orders of magnitude.

1020 / Journal of Pharmaceutical Sciences Vol. 75, No. 11. November 1986

PHYSICAL SYSTEMS FORMAL MATHEMATICAL IN NATURE SYSTEMS

and knowing either Dh or D, from clinical or laboratory experience, solve for the other:

5, BEHAVIORS

ENCODE

I

INFERENCES

I I ISOMORPHISM I

i I I !

I I I

ENCODE S2BEHAVIOR3 INFERENCES

. B 1 I L

U U i

Figure 1-Relations among analogous natural physical systems IS,) and their mathematical models (M,). Reprinted (modified from Fig. 3) with permission from ref 6. Copyright 1983, The American Physiological Society.

(Size ranges within a species, of course, are much less. In this paper we address only interspecies allometry, not intraspe- cies allometry, to avoid the uncertainties associated with the latter.)

In Table I we show the standard allometric coefficients for six typical anatomic/physiologic mammalian variables with body mass designated as x ; coefficient a based on y = a(x)b; and coefficient b based on eq. 3 (details given in footnote of Table I):

(3)

Using the allometric relations shown in Table I, what are our options for scaling the dose of a drug from mouse (typical adult body weight of 20 g) to humans (typical adult body weight of 70 kg)? For the moment we shall suppose that the data for Homo sapiens obey the allometry of the other mammals (exceptions will be pointed out later). We could scale between species by setting up ratios of the form below,

Table CSome Aliometric Coefficients for Eutherlan Mammals’

Y

b

a Nearby Raw Data Rational Fraction

Body Volume Surface Area, cm2 Standard Metabolic

Rate, kcalld Cardiac Cycle, s Brain Mass, g Life Span, year

1 .o 1 .o 1 1045 0.64 2/3

70.5 0.73 314 0.25 0.25 1 I4

11.24 0.76 314 11.6 0.20 1 I5

‘Based on y = a(Nb. Here x is body mass (expressed as weight) in kilograms. (Units and magnitude of ”a” depend on units chosen for x.) Data from human beings do not fit exponents given for brain mass or life span-see text. (Data selected from ref 2.) blberall (ref 12) has re- examined some of the metabolic and cardiovascular flow data for mammals and estimated that the exponent for cardiac output is -0.85, and that for oxygen consumption (metabolic rate) is 0.79, rather than 3/4. The residual difference ‘is thought to be beyond experimental error. Dimensional analysis based on geometric or elastic similarity arguments predicts a rational exponent (e.g., 314); any residual must be a function of some other scaling factor. The general allometric equation would then become y = a[(NbIc. When c = 1, a “flat” geometry for event spaces is assumed; for c # 1, some other, intrinsic geometry may be posited (see ref 9 for details of this argument, and see text for distinction between curved and flat spaces and their geometries).

(4)

where M is given as weight in kilograms; D is dose of drug (however defined); h and m designate humans and mice, respectively; ai and bi are the parameters of the allometric eq. 2. (But the point of allometry is that, for most measures,

If we use the allometric coefficients for body volume, the doee for humans should be 3500 times that for the mouse; if we use metabolic rate, it would be 400 times greater; if we use surface area, it would be 200 times greater; if we use heart-beat interval, it would be only 8 times greater for humans than the mouse. There is an enormous range of possible doses, and this ambiguity is unsatisfactory. Either we need a theory of the allometric coefficients to guide us, or we need to use some formulation that takes into account more than one physiological variable as the basis. In prac- tice, body weight and maximum lifespan p~ ten t i a l ,~ or brain weight, are sometimes used as arguments in a modified allometric equation to get improved results; but even so, the purely empirical approach leaves us without confidence that we know what we are doing. A thoughtful exploration of the problems of interspecies scaling8 has been provided by Box- enbaum and Ronfeld,’ who offer a “theory of pharmacokinetic similarities which states that both physiological and pharmacokinetic processes are biologically interrelated and governed by a master synchronization mechanism. . . .”

Mechanical variables, such as the velocity associated with various gaits, can be well scaled by elastic similitude arguments based on firm physical principles.‘ But, when we leave the world of mechanical systems and enter the world of strongly nonlinear biochemical networks, we lose contact with theory and the allometric scaling approach is in no way fundamental. Nevertheless, it can be used to raise provoca- tive questions, such as the importance (or not) of neoteny as a factor in scaling pharmacokinetic phenomena from mice to humans.

The Problem of Neoteny-Neoteny is usually thought of as a sort of sustained juvenilization or retarded develop- m e n t a n evolutionary phenomenon. It applies particularly to H o r n sapiens. Humans reach puberty a t -60% of their final body weight, whereas most laboratory and farm ani- mals do so earlier, a t only 30% of their final weight. In this discussion we shall accept neoteny as a useful view of human development to the extent that it may be true that there is a similarity of body plans among mammals, but that the unfolding of that design need not be a t the same relative rate for every species and that the rate of unfolding does not determine the developmental sequence. The arguments for and against neoteny as a useful explanation of the higher degree of encephalization and longevity of humans compared with other primates and all other mammals have been discussed by Gould.8

The coefficients in Table I fit humans as well as any other mammalian species except for the items of brain mass and lifespan. Using our typical adult body weight and the coefficients suitable for other mammals, the mammalian allometric equation predicts a lifespan for us of only 27 years! Similarly, our brain mass should be only 275 grams instead of the actual value of -1200 g ram. (It has not gone unno- ticed that the phenomenon of neoteny, involving both an increase in relative brain size and an increase of longevity, suggests a linkage between the two through some common scaling factors. Perhaps a few common “chronogenes” govern our higher encephalization and greater longevity, both of

a. = a . b. = b..) 1 1 ’ 1 1

Journal of Pharmaceutical Sciences / 1021 Vol. 75, No. 7 7 , November 7986

which have had an unusually high rate of evolution. Howev- er, we shall not debate these ideas here.)

The strong departure of data for humans with respect to brain size or lifespan, compared with the allometric line for data from other mammals, has sometimes been called “verti- cal allometry.”* It is perhaps better regarded as evidence that the exponent c in eq. 3 (see footnote to Table I) has a value other than 1.0, when b = 1/5 (or, according to some data, 1/41 for maximum lifespan potential and b = 314 for brain mass. Substituting actual data for Homo sapiens in that equation gives values of c = 1.45 for human brain size and c = 2.54 for maximum human lifespan. For nonprimate mammals, c 2 1 for both brain and lifespan, and there is no significant residual. These departures of the coefficient c from 1.0, for humans, is a discrepancy beyond measurement error. We believe that these discrepancies (c # 1.0) may mean that a thermodynamic variable is needed for the scaling, in addition to mechanical variables, to account for invariances in the mammalian body plan.

Effects of Uncertainty in Estimates of the Exponent b- In order to know whether or not there is a significant discrepancy between the estimate of the exponent b (eq. 3, see footnote to Table I) and its expected value (on whatever grounds) as a nearby rational fraction predicted or demanded by arguments based on mechanical similitude only (such as 114,213 or 314, etc.), it is necessary to know how experimental error will affect the estimates, and to believe that the coefficient a is invariant-a strong assumption! In using eq. 2, errors in the estimate of b from the best straight line (log y = log a + b log x) fit to data from multiple species, propagate as follows when two species are later being compared as in eq. 4. Let rl be the body mass for a mou~e~(0.02 kg) and z2 be that for a human (70 kg). The range of the data is ~ 2 1 x 1 = 3500. Consider the case where b = 1, in a sample of the data from four other species, but that the “true” value was 1.04 for a universe of measurements across all Class Mammalia. The 4% error in the estimate of b would propagate so that the estimate of the dose for a human, based on that for mouse, would be (3500)“.O” - = 1.39, or 39%. In the range for all members of Class Mammalia (7 orders of magnitude of mass), uncertainty in b of 0.04 would give an overall error across the range of (107)0.04 = 1.91, or 91%.

The error can be looked a t another way, for cases in which the allometric equation is used in reverse. lfsome function, y, in humans follows some isometry rule for all mammals, for exam le, y = and the data sample indicates that ye =

humans, then we would calculate a value for humans of (70)’.04 = 83, when the true value should have been (70) 70, a 19% error. I t is impractical to establish values of b for pharmacokinetic phenomena by studying numerous species of Class Mammalia from the shrew to the largest of marine mammals. In many cases, we may even want to predict the pharmacokinetic properties of humans with respect to a particular substance from studies on only one other species, such as the dog.7 However, it would be better to evaluate b from a study of five species whose body weights cover a range of four orders of magnitude, such as 0.02 kg (mouse), 0.20 kg (rat), 2.0 kg (rabbit, cat), 20 kg (dog), and 200 kg (pig). These data would bracket the point for human beings at 70 kg and give us more confidence that the estimate of b might be appropriate for humans because we would not be depending on an extrapolation of the allometric straight line. When the body mass range extends over four orders of magnitude (0.02-200 kg), an uncertainty in b of 0.04 would mean a 44% error in a dose ratio calculated across the whole range, for example, from a mouse to a pig.

The various errors pointed out above (19,39,44, or 91%), associated with the particular coefficient b, are all small

(MI’. B 4, based on a study of four species, not including

1.z

compared with the errors described earlier that arise if the wrong allometric coefficient altogether is used for scaling from mouse to humans, for which we showed that the range of possible doses was 8 to 3500. We conclude, then, that within the typical bounds of experimental errors in physiological measurements the uncertainty in b is likely to be no more than k0.02 (range = 0.04), and that this uncertainty would be of little practical consequence in deciding dose ratios across species. However, any departure greater than 0.04 from that of an expected value of b (for example, the expected value based on mechanical similitude arguments) would be a basis for believing that c # 1.0 and that a theory of the exponent c is needed to comprehend the physical basis of the scaling, even though the discrepancy might not have practical, pharmacokinetic consequences. In the history of physics, small discrepancies have sometimes had powerful effects on the development or testing of deeper theory, as in the case of the motions of the perihelion of Mercury.

Scaling Physiologic Time and Temperature-Many physiological time measures, such as cardiac cycle length or metabolic half-lives, scale in us and other mammals with a mass exponent not clearly different from 0.25, namely:

t a (5 ) except for our lifespan, as noted.

Is lifespan a manifestation of a deep “biological time” that has relevance to pharmacokinetic process scalings? (See also Boxenbaum and Ronfeld7 for discussion of this question.) Because the effects of neoteny are manifested both in brain mass and lifespan, it might be important to scale across species using brain mass (if one of the species is the human) rather than, or combined with, body mass. This has been done, and the ability to extrapolate some data from humans to laboratory mammals has been improved.

Pharmacokinetic processes consist of transports (convec- tions, diffusions) and transformations. The transports are essentially mechanical variables and it would be expected that the fundamental units would be related to length, mass, and time. But they, like chemical transformations, involve thermodynamic variables as well (such as temperature), and these might enter the scaling too. However, most mammals have approximately the same core temperature, so this variable has not been useful in the standard scalings. (Should we wish to scale among all vertebrata, instead of restricting ourselves to homeotherms, then core temperature would not be so restricted and might have to be taken into account as a fundamental dimension.)

It is amusing to view human beings as being metaphorical- ly a partially hypothermic species because of neoteny. Tem- perature is a macroscopic, thermodynamic variable express- ing the average of an energy distribution in an ensemble of microscopic interactions. The well-known Arrhenius equation is a crude model for the effect of temperature on rates of chemical reactions. A familiar form of the law (for a bimolecular reaction) is:

In k = In A - (Eact/R) (1/T) (6)

where k is a reaction rate coefficient, Eact is energy of activation, R is the gas constant, and T is absolute temperature. (The A term is taken as a constant, approximately. The [1/n term is separated out because traditionally it is the independent variable in an “Arrhenius plot.”)

Using typical mammalian enzyme values for the energy of activation and assuming that the whole constellation of physiological and biochemical processes within a mammal are typified by that averaged value, and using the model of the bimolecular reaction, we can calculate from eq. 6 that we are operating our developmental and aging processes at the

1022 / Journal of Pharmaceutical Sciences Vol. 75, No. 7 7 , November 7986

equivalent of a core temperature of 21 “C instead of the 37 “C typical of Class Mammalia overall; namely, a t this “hypothermic” temperature, our rate of living is 1/3 that of other mammals. That is a metaphorical way to express our neoteny, but of course we actually operate at a core temperature of 37 “C and our microscopic systems have the same temperature coefficients as do those of other mammals. The above pseudocalculation is meant only to say that in human beings the coordinate system may have undergone a transformation such that our 37 “C is equivalent to the 21 “C in the coordinate system of other mammals, for some few processes.

If, contrary to fact, data from human beings for all forms and processes fell on the straight line fitting data from all other mammals (using the allometric relation in its logarith- mic form) and if the mass coefficients were all rational fractions without residuals, then the mechanical allometry between humans and mammals generally would be fully established and we would need no new laws or no new geometries or reference frames to account for our pharmacokinetic behaviors compared with those of a mouse. Because all mammals have the same size of genome by weight, we presumably do not have to assume any important differences in total genetic information. Thus, we could attribute all differences solely to that fraction of the genome that is different in mouse and humans. But, for these claims to be true, there would have to be no significant residuals in the allometric data (see footnote to Table I). Kugler and Turveyg have suggested that if c # 1.0, it may mean that we should reject the assumption of mechanical similitude in a “flat” space. (Curvature of space is discussed later.)

The chief facts are that we are genetically almost identical to chimpanzees (99’% overlap of genetic material) and dynamically almost equivalent to all mammals (data for our species fall on the allometric line for Class Mammalia for a wide variety of dynamic measures) except for our retarded development and high encephalization, which suggest some warping of biological time in us. On the other hand, both mice and humans have strong circadian rhythms in many physiological processes, a result that suggests that the external earth day is reflected by an internal timing system easily entrained to that period. No neotenic effect here, nor any allometric scaling: a day is a day for all mammals. The situation looks murky when we consider scaling of the duration of exposure to a drug using an animal model of humans for the test. The protocol design might then have to take into account the neotenic aspects of human development. Using the (non-neotenic) allometric exponent of 0.25 to scale mammalian physiological time, we can calculate that one mouse day equals 9 human days. Using the circadian rhythm as an endogenous clock, one mouse day equals 1 human day. Taking neoteny into account by using actual lifespans (-3.5 years for the mouse and -75 years for the human), we conclude that 1 mouse day equals -21 human days. Which of the three estimates is right?

Suppose we want to design a protocol to test whether or not a new antibiotic of the tetracycline class avoids the risk of photosensitivity often associated with tetracycline administration to human beings for 1 week. How long should the mouse be exposed to the tetracycline to act as a model for this process in the human: 1 week (circadian rhythm scaling) 18 hours (allometric physiological time scaling), or 8 hours (neotenic time scaling)? Most toxicologists would probably choose the week, because it is not inconvenient and would be playing it safe; if the drug does cause photosensitivity, the effect would be likely to be found in a week even though it might be missed at 8 hours. (Of course, in this discussion of time scaling, we are assuming that the plasma levels of the drug were, and should have been, pharmacodynamically equivalent between the mouse and the human.)

The Problem of Using Too Few Dimensions-We become exasperated after pursuing such elusive arguments based on mechanical similarity principles, sometimes combined with the hidden assumption that a product, dose x time, is the governing quantity-so that strength-duration relationships are hyperbolic: twice the dose for half the time is equivalent to the regular dose for the regular time. Thresholds and other strong nonlinearities should caution us against making that assumption casually.

The difficulties encountered in standard approaches to pharmacokinetic scaling across species may arise from using too few dimensions. Scaling by body mass alone or by brain mass alone, or even by a combination of the two, may not always be sufficiently accurate for hydrodynamic, chemically reactive systems such as constitute the processes of distribution, binding, metabolism, transport, and excretion of drugs. The integrated aspect of organisms suggested to Adolph10 that perhaps none of the properties of an organism is entirely uncorrelated with others; properties are demonstrably caus- ally interlinked, o h n through multiple pathways, as well as being correlated according to measurement. In the same spirit, Gould and Lewontinll have insisted that no adapta- tion and no selection during biological evolution operate directly on parts, because the form of a part is a correlated consequence of selection directed a t the phenotype overall, or even the species. We must face the fact that organisms are integrated wholes, not always usefully decomposable into independent and separately optimized parts. “he allometric approach seems to take one part and one relation a t a time, focusing on the scaling of specific organs, functions, or requirements. Calder2 comments wryly: “Attempts to put the allometric animal together again have been the rare exceptions. . . . It is difficult or even impossible to reassemble anything when a number of pieces are missing . . .” More pieces are being discovered, but the integration is not explained.

When only physical or morphological variables are involved, there are three approaches to scaling: (a) the geometric approach in which linear dimensions must scale as (b) the elastic similarity approach which predicts that linear dimensions will scale as (M)’l4; and (c) a dynamic similarity approach which could scale either way. The empirical facts are that most of the relevant scalin fits the 114 exponent.

been reviewed by Calder.2 When a wider range of variables is considered, the exponents on body mass of 1,3/4,2/3,1/3, and 114 are all reported frequently. Of these the scaling of standard metabolic rates according to (M)”’ has provoked perhaps the most discussion.

In one line of reasoning, the 314 exponent is the result of energy turnover proportional to body mass (exponent 1) per unit of physiological time (exponent 1/4), giving us a combined mass exponent of 1 - 1/4 = 314. Then we need only a theory of the 114 exponent, because the 1.0 exponent is given by the assumption that body densities among mammals do not depart much from 1.0. (But see Iberall12 for evidence that the exponent is not 314.)

This kind of speculation can go on indefinitely in the absence of a more general physical basis for scaling than is provided by purely mechanical considerations. Meanwhile, many pharmacokineticists assume that because metabolism scales according to the exponent b = 3/4, drug dosages should be scaled accordingly, assuming that drug metabolism and/or excretion follows the same rule as does energy metabolism. That assumption works very well in many cases. Neverthe- less, we propose below that the phenomenon of neoteny means that human beings operate in a biochemical and physiological coordinate system that is, in some respect, “slower” than that of other mammals.

Some, however, seem close to an (M) Y/3 rule, and these have


“Biological” time is changing in relation to external clock time during all mammalian development and aging. For example, oxygen consumption per unit mass starts very high and falls rapidly after birth, and then falls more slowly. The probability of dying over a 24-hour period increases monoton- ically aRer about age 30 in humans. Biological time can be defined in many different ways; for example, it can be scaled to some aspect of caloric throughput or to the probability of dying. In any form we have seen, its putative relation to clock time is nonlinear. Thus, in biology we are always confronted with changing rates of processes in chemical fields. We believe that this situation is similar to those addressed by general relativity, which is concerned with accelerated mechanical fields. It is the measurement issues, not the potentials, that seem common between the two cases.

When c # 1 in eq. 3, the operation of nonmechanical, thermodynamic coefficients, such as a diffusion coefficient, may be implied. Somebody has to pay the energetic bill, as it were, for the residual. To understand that, a deeper physical view of similitude may be helpful.

Summary of Our Criticisms of the Standard Allometric Approach-The allometric approach has been powerful for discovering some invariances that hold across a very wide range of size (body mass) data for mammals, all viewed from the same external frame of reference for measurements. For mechanical variables, we can use physical principles of geometric or elastic similitude to rationalize many of the coefficients in the allometric equations. In some cases, however, these are purely empirical coefficients. The principal problem is lack of a general theory for interspecies scaling of the features of biochemical networks. But there is another shortcoming: the allometric approach attracts too much attention to a scaling based on the single variable, size. It seems much more likely to us that in a complex system such as the mammalian organism, both mechanical and thermodynamic quantities will be fundamental, and that to locate a “world point” in state or event space will require use of a reference frame with multiple coordinate axes (dimensions), one for each independent, fundamental quantity, going beyond the three spatial coordinate axes.

Conventional allometry assumes that “scaling” is a one- dimensional group of transformations, applied to a one- dimensional space of “kinds of mammal” (parameterized by, for example, mass). It can be seen as either (a) an operation that, given a mammal type with a particular mass and therefore a particular lifespan, gives another type with (e.g.1 twice the mass or (b ) an operation that transforms the whole space, as a change of scale transforms the animal. But these two transformations, of a mammal and of mammal space, are quite different things. The failure of data from humans to fit a one-dimensional allometry curve shows that the event space (where “event” means a cluster of measureemass, heart rate, etc.-yielded by one specimen) is not one-dimen- swnal. The questions then become (a ) What is the dimension of allometric space? (b ) What transformation group, of what dimension, generalizes one-dimensional scaling? and (c) Does it have metrical features that relate it neatly to some geometric features of allometric space (i.e., can we define distance, curvature, etc.)?

We turn now to a brief physical-relativistic treatment of reference frames, coordinates, and geometries (see Appen- dix), in the hope of finding a clue about how to deepen the approach to interspecies scaling in pharmacokinetics. The basic ideas of relativistic physics, world points and lines, geodesics, space curvature, et cetera, that we shall draw upon, have been beautifully explained in simple but authori- tative style (without mathematics) by Martin Gardner.13

Second Approach: Physical Laws and Intrinsic Geometries

Background-Physics addresses two chief questions: (a) What is the universe made of? and (6) What are the rules that govern motion and change within it? Many physicists make the Assumption of Simplicity that supposes that Bur- face complexity arises from deep simplicity, “that there is a simple set of laws of nature, of which all our complicated present physical and chemical laws are just mathematical con~equences.”~~ Physics is the science of lawhl motion, change, trends, and happenings. But in addition to trends, happenings, and (approximate) laws, empirical physics de- pends on a privileged act-that of measurement, in which a macroscopic device maps from the physical world of dynamics to the world of symbolic representationsl5-interpreted by human conscwwness. (Measurements can be thought of more generally-i.e., a very small system can be imagined to measure microscopic events within itself, as in cellular biology, or a robot can make a macroscopic measurement never reported to, or read by, humans. But, then we have the ancient conundrum in a new guise: is there a sound in the forest when a tree falls, but no one hears it?)

Insofar as an object of interest occupies space-time, and geometry is used to describe the positions and motions of the object in space-time, certain assumptions regarding the mathematical description of the structure of spacetime must be made before a convenient physical reference fmme can be chosen to anchor observations (measurements). Clas- sically, measurement is the process of locating (specifying the coordinates of) a “world point” in some frame of reference and quantifying the coordinate system according to some convention of units. A reference frame can be thought of crudely as a “container,” embedded in a geometric universe, whose features are used to fix a measurement operation. A distance, for example, has to be measured from somewhere to somewhere. A reference frame provides the “from.” It may have more, fewer, or as many dimensions as the nominated geometry. It need not have sides a t right angles to each other. It can be circular, spherical, cylindrical, skewed. . . . Events take place inside the container and are specified by distances or angles changing with respect to nominated reference points in or on the reference container. There are no absolute reference frames in modem physics: an observer is free to nominate himself as “center” of the Universe if he wishes, though that choice may lead to complicated forms of physical laws as expressed mathematically.

Two Theories of Measurement-criteria for choosing a geometric scheme have traditionally emerged from either of two hypotheses. The first holds that the geometry should be independent of events being measured. In it, reference frames can be arbitrarily chosen for convenience. This assumption is fundamental to extrinsic (classical) theories of measurement. (For example, the space in which an event occurs is oRen assumed to be a “flat” (zero curvature) three- space with the coordinate hypersurfaces intersecting orthog- onally.) When such an extrinsic view is taken, physical laws will be invariant from one frame to another, provided that the frames of reference do not accelerate relative to other such rigid frames in which events are occurring and being compared. Subluminal velocities and lengths may be different in the different frames, however, even when these are in rectilinear, uniform motion with respect to each other and laws are invariant across frames (as in the case of Special Relativity, which assumes a flat four-dimensional space time).

The second hypothesis assumes that the appropriate geom-


etry is not independent of events. Instead, the geometric structure of events and the geometric structure of space-time itself are linked. It may then happen that the description of the geometry increases in complexity (compared with Euclid- ean geometry), but the physical concepts become simpler. This search for natural geometries intrinsic to the physical phenomena being observed constitutes the intrinsic approach to measurement.

Intrinsic Geometr ieeThe use of intrinsic geometries makes it possible to describe, by invariant laws, events in all (even accelerated) physical frames of reference (the principle of general covariance) and to geometrize many physical phenomena (the principle of geometrization of natural phenomena). Einstein’s theory of general relativity provided the first successful utilization of the principle of geometrization of natural phenomena. Although the theory of general relativity focuses on the motions of large bodies of matter in gravitational fields, we here propose that the concept of intrinsic geometries applies to any set of physical observables governed by field potentials.

Events, measurements, constants, laws, and coordinates are at the core of many physical theories. There is a restric- tion, though; laws must be invariant over transformations of coordinates. In the development of a physical theory for biology one might begin by assuming a law and looking for a geometry in which it can be expressed most simply. Or one can begin by positing a geometry and then looking a t world points and lines for happenings and trying to discover the law. The latter route to discovery is exhibited in the changing views of space, time, and mechanical motions from Galileo and Newton through Einstein’s Special Theory of Relativity. However, the General Theory differed: it concerns a law about the interdependence of geometry and happenings and does not posit a geometry. Newton explicitly posited a geometry; so does Special Relativity. General relativity, in great contrast, makes the curvature of space-time a physical variable. General relativity dissolved the distinction between inertial and noninertial frames of reference and the problem of inertial mass disappeared-a major achievement in similitude!

The Hypothesis-The ideas presented below follow those of Kugler and Turvey.9 We believe that in biological allometry we have the problem of finding an intrinsic geometry for (as an example) the mammalian body plan, that includes nonequilibrium thermodynamics, with both its locally changing flows within an organism, and different process rates in different species for the same chemical transformations. The intrinsic geometry should expose the invariances across species for the mammalian body plan, if such invariance exists. (If not, allometry must remain empirical.) We suggest that different species represent either (a) different coordinate transformations between reference frames in an intrinsic geometry common to all mammals or (b ) different curvatures of event space. In either case, the laws of pharmacokinetics, if these are to scale across species, must be invariant with respect to coordinate transformations. To develop these ideas, we must draw further upon the relativistic principles in physics.13.16

Relativity and Intrinsic Geometry: Summary-The general laws of classical mechanics pertain to accelerations, and these laws are the same for all reference (inertial) systems in rectilinear, uniform motion. This is not true of the distances and the velocities which can be mapped from one reference system to another through transformation equations that keep the laws invariant. These are the Galilean transformations in classical mechanics.

The Special Theory expresses the Principle of Relativity as follows.16 There are an infinite number of systems of reference (inertial systems) moving uniformly and rectilinearly

with respect to each other, in which all physical laws assume the simplest form. (That was the form originally derived for classical absolute space or the stationary ether.) To that principle was added the principle of the constancy of the velocity of light: in all inertial systems the velocity of light has the same value when measured with rods and clocks of the same kind. By means of the Lorentz transformation it was possible to show that even though neither distance nor time was constant from one coordinate system to another (e.g., time dilation appeared), the velocity of light and the laws of the motions were invariant across the transformations. The results of measurements were connected with each other by Lorentz transformations.

The essence of Einstein’s new kinematics consisted of the inseparability of space and time. Minkowski developed the kinematics as four-dimensional space-time geometry, and physics was then no longer dealing with ordinary Euclidean geometry. The final step in the unfoldmg of the Special Theory was to create a new kinetics in which the mass of one and the same body is itself a relative quantity. Unfortunately, however, the problem still remained that if physical laws were referred to accelerated coordinate systems, the laws became different.

The General Theory introduced the principle of equivalence that asserted that certain physical concepts cannot be distin- guished (in this case, gravitational mass and inertial mass). Furthermore, the four-dimensional world and its laws had to be described without assuming a specific geometry (e.g., that of Euclid) a priori.

Gauss, Reimann, and Helmholtz had already doubted the claim of Kant that the theorems of Euclidean geometry were exact and a priori, eternal truths. To preserve the invariance of physical laws in transforming from one coordinate system to another when these were accelerated with respect to each other, one had to abandon Euclidean geometry itself and set up a more general doctrine of space. Einstein chose a nonEu- clidean geometry with which he could generalize the law of inertia by introducing Gaussian coordinates into the four- dimensional space-time world. Finally, it was shown that the physical laws were invariant with respect to arbitrary transformations of the Gaussian coordinates.

Toward a New, Physical Allometry for Pharmacokinet- ics-In our view, a multidimensional coordinate system of fundamental mechanical and thermodynamic quantities, embedded in some curved, intrinsic geometry, offers the best theoretical model to eliminate some of the uncertainty from allometric descriptions and to enhance their explanatory power. The beautiful, empirical geometric transformations used by D’Arcy Thompson17 to invent intermediate forms of the pelvises of birds from Archaeopteryx to Apatornis pointed the way; it falls to his successors to find the theory. So far, we have failed. ’ 8

To build the new allometry we believe it will be helpful to work with the three elements we have discussed above: (a ) Rosen’s generalized similitude principles that define modeling relations between systems, ( b ) Einstein’s principles of general covariance and of geometrization of natural phenomena (the search for a curved intrinsic geometry), and (c) irreversible thermodynamics as a basis for time asymme- try.19

In the mechanics of Galileo and Newton, in the relativistic physics of Einstein, and in the domain of quantum mechanics, time is reversible and symmetric. Nature follows the same laws whether running “forward or “backward” in time. But at supra-atomic scales including those within living systems, in our corner of the universe, all natural processes are irreversible and time is asymmetric. To bring the irreversibility of pharmacokinetic phenomena into allometric scalings we encourage the further use of transport and


reaction rate coefficients in the modeling relations of Fig. 1. Transport coefficients address the macroscopic irreversibility of flows of mass (density differences), momentum (pressure differences), charge (voltage differences), or energy (temperature differences). In addition to the fundamental thermodynamic irreversibilities associated with diffusive transports, there are irreversibilities associated with chemical transformations. Many enzymatically catalyzed chemical reactions that may be thermodynamically reversible in principle (i.e., the rate constants for the forward and reverse reactions are both positive) are kinetically irreversible because the reverse rate constant, though positive, is so small as to be negligible.

Envoi-To introduce the contribution from Einsteinian physics into a modeling relation for pharmacokinetics, we propose that it is necessary to connect or interrelate the mapping operations, f,(.), ,(.I, and v~,~,(-), described by

reference frames embedded in an intrinsic geometry. Thus, the full program for establishing a physical basis for allometry would consist of (a) identifying derived quantities that, empirically, permit superposition of pharmacokinetic data from different species (as done by Boxenbaum and Ronfeld?; (b) being sure that thermodynamic terms (such as transport coefficients) and mechanical terms are both included; (c) then incorporating these into a stable parameterized family of mappings describing an equation of state; and, finally, (4 showing that the mappings establish an equivalence relation based upon coordinate transformations of reference frames in an intrinsic geometry for a curved event space.

Ours is an ambitious program that will be difficult and maybe impossible to accomplish. In this paper we can only point to a new direction in which we think theoretical progress can be made that might ultimately provide a fum, scientific foundation for the descriptive methods of standard experimental allometry .

Of course the impatient reader would like to see a “worked problem” at this point. We are planning such studies now, using the approach of finding average amplitudes and inter- vals between fluctuations in time-series data on drug and hormone levels (and blood pressures), to define two kinds of parameters, T~ and &, analogous to characteristic mean-free- paths and relaxation times for pharmacokinetic processes within various species. This approach can expose invariants if a law is operating in different coordinate systems.@

Rosen (eq. lb), through trans 9 ormations on the coordinates of

Appendix

The technical meanings, in mechanics, of the terms Rdimension,” “unit,” “coordinate,” “geometry,” “reference frame,” law,” “theory,” and “principle” have been eet forth by Lindsay and MargenauP We use their definitions but caution that in Ramplee of the secondary literature about physics some of these terms may be confused.

The geometries of Riemann (a locally spherical geometry in which the space constant, or krvature,” is greater than zero), of Euclid (“curvature” is zero, i.e., a “flat” geometry), and of Lobatchevsky- Bolyai (“curvature” is negative) are the only types of apace that satisfy the condition offree mobility, that is, the possibility of moving physical objects from place to place and rotating them arbitrarily without altering their dimensions. This condition is the fundamental requirement for measurements within a chosen reference frame that is itself embedded in, or expressive of, a geometry. In the two non- Euclidean geometries mentioned above, magnification or diminution of a geometrical figure d w s alter its shape. Therefore, within one reference frame with curved space it is impoeeible to have exactly similar figures of different sizes.

We think that allometric coefficients that are not the rational fractions predicted by mechanical scaling principles suggest that the mammalian body (event) space is curved. However, we have

difficulty in telling from the available data whether we are seeing manifestations of an intrinsic, n o d a t geometry, characteristic of the event space of Clam Mammalia overall in which curvature is a speciesdependent variable, or just manifestations of transformations of coordinate systems between different reference frames (species) in a flat geometry.

We believe these issues reduce to three stmtegic questions that bear on the mapping of pharmacokinetic phenomena across species (e.g., from m o w to humans) that poseibly possess different intrinsic geometries or different laws or different reference frames: (a) Should we begin by aseuming that mammalian body plans all have the aame intrinsic geometry and then seek different pharmacokinetic laws in different species? (b) Should we begin by assuming a pharmacokinetic law that is invariant across species and then seek coordinate transformations that distinguish species as different references frames? (c) Or should we look a t species as having different curvatures of their event space?

Kugler and T w e y B have suggested that an allometric plot can be used to measure (or map) an intrinsic geometry if the allometric relation actually defines a physical law or universal constant. The metrics for the coordinatea are defined by the deviations of a and c (aee eq. 3) from their canonical magnitudes. The physical signifi- cance of these indices reflects variations in the size and layout of potentials (elastic, gravitational, etc.). This conclusion follows directly from the invariance postulate of natural law: natural laws (or universal constants) constitute the invariant (“objective”) frame of reference from which all measurementa are to be evaluated. This is the measurement strategy underlying both Special and General Relativity: (a) in Special Relativity, the universal constant defining the speed of light defines the objective frame of reference to which all measurements are referred; and (b) in General Relativity, the Conservation of Energy-Matter law defines the objective frame of reference for measurement. There is no absolute (inertial) reference frame for measurement in either theory. The other absolutes (velocity of light in a vacuum and the summational invariant of m a s s energy in any event) substitute for such a reference frame, to anchor measurements. (See Gardner’s for details.)

A new report21 shows clearly that dimensional analysis, correctly applied, not only fails to determine the mass scaling exponent in standard allometry (contrary to Heusner21) but also fails to constrain the relationship to a power law a t all. Again, we must conclude that a more fundamental physical approach is necessary.

References and Notes

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15. Pattee, H. H. In “Complexit Language, and Life: Mathemati- cal Approaches”; Casti, J. f.; Karlqvist, A., El.; Springer- Verlag: New York, 1985; pp 268-281.

16. Born, M. “Einstein’s Theory of Relativity”, revised ed. (Engl. transl.); Dover Publications: New York, 1962; p 232.

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19. Morris, R. “Time’s Arrows: Scientific Attitudes Toward Time”; Simon and Schuster: New York, 1984.

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21. Butler, J. P.; Feldman, H. A,; Fredberg, J. J. Am. J . Physiol.1

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246, R.8394845.

Acknowledgments We are very indebted to Donald Walter and Timothy Poston for

he1 in us clarify our thoughts and the text. They are not respon- sib& kr our opinions or any residual cloudiness of thought or expression.


similarity principles and intrinsic geometries: contrasting approaches to interspecies scaling

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