Volume of Distribution

There are three kinds of volumes which are frequently used in the interspecies scaling.

(a) The volume of distribution of the central compartment (Vc) is used to relate plasma concentration at time zero (C0) of a drug and the amount of drug (X) in the body [26]

A small Vc (<3 L) indicates that most of the drug is in the plasma, whereas a large Vc (>7 L) indicates that the drug has concentrated in the extra vascular space.

(b) The volume of distribution at steady state (Vss) can be estimated from the following equation:

Dose x AUMC

AUC1

where MRT is mean residence time=AUMC/AUC (15)

and AUC and AUMC are area under the curve and area under the momemt curve, respectively.

The volume of distribution by area (Varea) also known as Vb can be obtained from the following equation:

Clearance

where b is elimination rate constant.

Physiological factors such as plasma protein, tissue binding, total body water and binding to erythrocytes may effect the distribution of drugs in the body. Therefore, a drug in the body can be accounted for inside plasma and outside plasma. The following equation can describe the relationship:

where Vp is plasma volume, Vt is tissue volume, and fup and fut are the fraction of unbound drug in plasma and tissue, respectively. Drugs extensively bound to plasma proteins (fup < < fut) will have small volume of distribution.

In an attempt to establish relationship between binding to plasma proteins and volume of distribution of drugs in animals and man, Swada et al. [27] investigated the relationship between the volume of distribution (Vss) and plasma protein binding of b-lactams. Swada et al. [28] also investigated the relationship between the unbound volume of distribution of tissues (V/fut) and fu (fraction unbound) of nine acidic and six basic drugs in the rat and in humans. The authors concluded that there was little difference in V/fut of basic drugs between animals and man and that volume in man from animal data was predicted with more accuracy using VJfut than using volume against fu.

Obach et al. [17] used four different methods to predict VSS, and based on their geometric mean, prediction accuracy concluded that unbound Vss can be predicted better than the total Vss.

Conceptually there should be a good correlation between body weight and volume among species and indeed this is the case. Generally the exponents of volume are around 1.0, which indicates that body weight and volume are directly proportional. However, this may not be the case for all drugs, and exponent as low as 0.58 (diazepoxide [29]) has been noted. Overall, volume of distribution can be predicted in humans from animals with reasonable accuracy. As noted by Mahmood and Balian [18], unlike clearance, volume can be predicted in humans with fair degree of accuracy using two species.

Though literature indicates that Vc, Vss, or Vb are predicted indiscriminately in humans from animals, it has been shown by Mahmood [30] that Vc can be predicted with more accuracy than Vss or Vb. In fact Vss or Vb may not be of any real significance for the first-time dosing in humans and can be estimated from human data.

Vc can play an important role in establishing the safety or toxicity for the first-time dosing in humans. Since an administered dose is always known, the predicted Vc can be used to calculate plasma concentration of a drug at time zero (C0) following intravenous administration. This initial plasma concentration may provide an index of safety or toxicity. Furthermore, Vc can also be used to predict half-life, if clearance is known (t1/2=0.693 Vc/ CL).

Elimination Half-life and Mean Residence Time

It is difficult to establish a relationship between body weight and half-life (t1/2) since half-life is not directly related to the physiological function of the body rather it is a hybrid parameter. A poor correlation between t1/2 and body weight across the species may give a poor prediction of half-life. Like clearance, the allometric exponents of half-life using body weight widely varies. In his evaluation of 18 drugs, Mahmood [30] reported that the exponents of half-life of these drugs varied from 0.066 to 0.547.

Due to the difficulty in estabishing an allometric relationship between body weight and half-life, some indirect approaches for the estimation of half-life have been suggested.

Bachmann [12], Mahmood and Balian [9], and Obach et al. [17] used the following equation to predict the half-lives of many drugs.

Though this approach has been found to be suitable for the prediction of half-life for many drugs in humans, it is also necessary that one must predict both CL and volume in humans with reasonable accuracy.

Another indirect approach to predict half-life was suggested by Mahmood [30]. In this approach, first mean residence time (MRT) was predicted and then the predicted MRT was used to predict half-life in humans using the following equation:

The results of this study indicated that MRT can be predicted in humans with fair degree of accuracy from animal data. The exponents of MRT of the studied drugs varied from -0.260 to 0.385 (Table 3). The indirect estimation of half-life using MRT was fairly close to the observed values (Table 3).

TABLE 3 Predicted vs. Observed MRT and Predicted Half-life from MRT in Humans from Animal Data

Obs.

Pred.

Obs.

T1/2

Drugs

Exponent

R

MRT

MRT

T1/2

from MRT

Tolcapone

0.283

0.875

1.20

1.70

1.30

1.20

Lamifiban

0.385

0.915

2.50

4.10

2.10

2.90

Interferon

0.231

0.776

2.40

2.40

5.10

1.63

AZT

0.107

0.827

0.90

1.14

1.10

0.80

Erythromycin

0.137

0.883

2.10

3.20

2.20

2.20

Oleandomycin

0.123

0.851

1.40

3.10

1.00

2.10

Amascrine

0.381

0.975

6.00

8.20

4.70

5.70

CI-921

0.228

0.682

2.00

3.80

2.60

2.60

Sematilide

0.186

0.946

3.10

3.60

2.70

2.50

Sch-34343

0.317

0.963

0.73

0.83

1.00

0.60

Ciprofloxacin

0.037

0.132

4.80

2.95

4.30

2.05

Cyclosporine

-0.260

0.337

5.50

2.90

6.20

2.00

Cefotetan

0.339

0.906

4.40

2.10

3.40

1.50

Cefmetazole

0.220

0.886

1.28

0.87

0.90

0.60

Cefoperazone

0.351

0.853

2.30

2.50

2.35

1.80

Moxalactam

0.289

0.999

2.40

1.70

1.30

1.70

Cefpiramide

0.373

0.917

6.70

4.50

5.00

3.10

Cefazoline

0.241

0.914

2.00

1.75

1.50

1.20

Though Eqs (18) and (19) are only true for one compartment model, both these equations may also be used in a multicompartment system for prediction purposes.

Species-Invariant Time Methods

In chronological time there is an inverse relationship between the size of the animal and the heart beat and respiratory rates, in other words, as the size of the animals increases their heart beat and respiratory rates decrease. On the other hand, on a physiological time scale, regardless of their size all mammals have the same number of heart beats and breaths in their lifetime. Therefore, one may define physiological time as the time required to complete a species-independent physiological event. Thus in smaller animals the physiological processes are faster and the life span is shorter.

Chronological time, also known as species-invariant time, can be transformed into physiological time. Dedrick et al. [31] were first to use the concept of species-invariant time when they used the pharmacokinetic parameters of methotrexate in five mammalian species following intravenous administration as an example. The chronological time was transformed into physiological time using the following equations:

Y a is concentrati┬░n where W is the body weight.

By transforming the chronological time to physiological time, Dedrick and co-workers demonstrated that the plasma concentrations of methotrexate were superimposable in all species. They termed this transformation as equivalent time.

Later, Boxenbaum [20] introduced two new units of pharmacokinetic time, kallynochrons, and apolysichrons. Kallynochrons and apolysichrons are transformed time units in elementry Dedrick plot and complex Dedrick plot, respectively.

Kallynochrons (elementry Dedrick plot):

^ concentration (22)

where b is the exponent of clearance.

Apolysichrons (complex Dedrick plot):

Y . concentration ^^

where b and c are the exponents of clearance and volume, respectively.

Dienetichrons

Boxenbaum [20] introduced a new time unit known as dienetichrons by incorporating the concept of MLP in physiological time. The transformation of chronological time to dienetichrons can be obtained by dividing the X-axis or time by MLP. For example, for elementry Dedrick plot, X-axis or time was normalized as follows: time 1 MLP X W1

Though some investigators [32-34] have used the concept of species-invariant time in their allometric analysis, a direct comparison of allometric approaches with species-invariant time has not been systematically evaluated.

In a study, Mahmood and Yuan [35] compared the empirical allometric approaches with species-invariant time methods using equivalent time, kallynochron, apolysichron, and dienetichrons. Clearance, volume of distribution, and elimination half-life of three drugs (ethosuximide, cyclosporine, and ciprofloxacin) were compared using allometric approach and species invariant time methods. Overall, the species invariant time method did not provide any improvement over conventional allometric approach. Especially, the equivalent time approach did not predict plasma concentrations or pharmacokinetic parameters as accurately as elementry or complex Dedrick plots. This may be due to the fact that equivalent time approach uses a fixed exponent of 0.25 for elimination half-life. It should be noted, however, that the exponent of half-life of drugs is not always 0.25 [30]. The exponents of half-life for ethosuximide, cyclosporine, and ciprofloxacin in this study were 0.47, -0.24, and 0.04, respectively.

Normalization of clearance by MLP provided substantial improvement in the prediction of clearance for cyclosporine and ciprofloxacin (according to "rule of exponents" as the exponent of simple allometry was greater than 0.7). The incorporation of MLP in the species invariant time method substantially underpredicted the clearance and overpredicted the half-life by more than 20-fold. It was noted by the authors that this inaccurate prediction of clearance and half-life was mainly due to the prolonged sampling times in humans following the normalization of MLP. This increased the AUC and prolonged the half-life of cyclosporin and ciprofloxacin.

The findings of this study were based on the limited number of drugs (n=3). Overall, the results of this study indicated that both simple allometry and species invariant time methods would give almost similar results. Species invariant time method may be helpful in gaining some insight about plasma concentrations of a drug but the accuracy of this method in predicting plasma concentrations in man may not be reliable.

Prediction of Pharmacokinetic Parameters Using Pharmacokinetic Constants

Besides Species invariant time method, pharmacokinetic constants have been also used by some investigators to predict plasma concentrations in humans from animals.

The following equation represents a two-compartment model following intravenous administration.

where A and B are the intercepts on Y-axis of plasma concentration vs. time plot and a and b are the rate constants for the distribution and the elimination phases, respectively.

Equation [27] can be used to predict plasma concentrations as well as pharmacokinetic parameters (using predicted concentrations) in humans from animal data. Swabb and Bonner [36] and Mordenti [37] predicted plasma concentrations of aztreonam (one compartment model) and ceftizoxime (two compartment model) in humans from animal data using pharmacokinetic constants. Though Swabb and Bonner and Mordenti successfully used pharmacokinetic constants approach for the prediction of plasma concentrations and pharmacokinetic parameters, the suitability of this approach for the prediction of pharmacokinetic parameters in humans from animal data has not been thoroughly investigated.

Mahmood [38] compared the predicted pharmacokinetic parameters of six drugs using either pharmacokinetic constants or conventional allometric approach. No trend in correlation between body weight and A, B, or a was found. For some drugs a good correlation between body weight and these parameters was obtained whereas a poor correlation was observed for other drugs. Though the predicted values of A and B were occasionally close to the observed values, the predicted a values were many folds higher or lower than the observed values which had substantial effect on the predicted plasma concentrations. Overall, the use of pharmacokinetic constants to predict pharmacokinetic parameters in humans from animal data did not provide any improvement over conventional allometric approach. Like species invariant time method, pharmacokinetic constant approach may provide some information about plasma concentrations of a drug but the accuracy of the method for the prediction of plasma concentrations in man may be questionable.

Absorption and Absolute Bioavailability

Prediction of absolute bioavailability in humans from animals due to the differences in the anatomical and physiological features of the gastrointestinal tract, dietry habits, blood flow through the gut and the liver, and the enzymatic activity of the metabolizing enzymes, is a complex task. Some animal models may provide a rough estimate of absolute bioavailability in humans and such rough estimates can also be of significant importance to identify problems of absorption and intestinal and hepatic metabolism in man.

Conceptually it is difficult to justify an allometric relationship between body weight and absolute bioavailability. Mahmood [39] using direct (body weight vs. absolute bioavailability) and several indirect approaches attempted to predict absolute bioavailability in humans from animal data. Five different methods were used to predict absolute bioavailability in humans:

i. body weight vs. absolute bioavailability (allometric approach)

where Q is hepatic blood flow (1500 mL/min). Methods II-V are indirect approaches. Fifteen drugs were used in this analysis. In Table 4 the ii. F=CL(IV)/CL(oral)

Drugs

Obs F (%)

Predicted F (%)

Range

Mid point

Method I

Method II

Method III

Method IV

Method V

Actisomide

30-43

135

76

65

53

68

53-76

64

Amlodipine

63

86

86

78

75

80

75-86

80

Candoxatrilat

32

15

56

87

76

81

15-87

51

CI-1007

4

12

4

31

NA

5

5-31

18

Dofetalide

83

66

72

81

74

79

66-81

74

Faradifiban

24

5

15

94

60

71

5-94

50

Meloxicam

95

89

89

99

99

99

89-99

94

Metoprolol

38

53

53

45

NA

50

45-53

49

Morphine

24

9

16

32

NA

19

9-32

20

Nicardipine

7

9

9

47

NA

15

9-15

12

Recainam

67

132

130

69

76

81

69-81

75

Remikiren

0.3

0.3

0.3

41*

NA

0.7

0.3-0.7*

0.50

Sildenafil

35

70

67

65

48

66

48-70

59

Tacrolimus

25

8

5

NA

NA

3

3-8

5

Troglitazone

43-53

53

23

88

47

15

23-88

55

% MAE

59

46

58

NC

49

33

NC=Not calculated because there were only nine drugs available for this method. NA=Not available. Oral clearance was greater than the liver blood flow (1,500mL/min).

*Method III not included in the analysis. Method I=body weight vs. absolute bioavailability; Method II: F-CL(IV)/CL(oral); Method III: a F=1-(CL(IV)/Q); Method V: F=Q/(Q+CL(oral)). j

Reproduced with kind permission of the copyright holder, Drug Metabolism and Drug Interactions (Ref. [39]). 0

o correlation coefficient between body weight and absolute bioavailability, exponents of allometric equation, and the predicted absolute bioavailability in humans from animals have been shown. Though for some drugs a good correlation between body weight and absolute bioavailability has been obtained, there is uncertainty in the prediction of absolute bioavailability in humans from animals. Overall, the results of the study indicated that all the five approaches predict absolute bioavailability with different degrees of accuracy and are unreliable for the accurate prediction of absolute bioavailability in humans from animals. Despite uncertainty in the prediction of absolute bioavailability in humans, the approach may provide a rough estimate of absolute bioavailability.

Sietsema [40] plotted absolute bioavailability in man against those in rodents, dogs, and monkeys. The correlation coefficient (r2) for absolute bioavailability between man and rodent, man and dog, and man and primates was 0.4, 0.3, and 0.2, respectively. This poor correlation indicates that absolute bioavailability data in animals may be of moderate use for the prediction of absolute bioavailability in humans.

In recent years, attempts have also been made to correlate fraction of oral dose between rat and humans [41]. For the prediction of absorption in humans, methods such as intestinal permeability in rats [42, 43], jejunal permeability in humans [42, 43], and caco-2 cell permeability [44] have been proposed.

Prediction of Maximum Tolerated Dose (MTD)

In phase I clinical trials, not only the selection of the first dose to be administered to the patients is a challenge but also the issue of dose escalation is a complex task. A conservative low-dose approach will result in subtherapcutic or ineffective dose. On the other hand an aggressive dose escalation may result in producing toxicity. Certain class of drugs, for example anticancer drugs, are so toxic that for ethical reasons they can not be given to healthy subjects. Therefore, predicting MTD in humans from animal data may prove to be highly beneficial. For anticancer agents, generally 1/10 of the LD10 in mice or 1/3 of the toxic dose level (TDL) in the dog in mg/m2 is used as the starting dose in phase I clinical trials [45]. Goldsmith et al. [46] reported that the use of 1/3 of the TDL would have produced significant toxicity in the patients for 5 out of 30 drugs. The authors further concluded that for a safe starting dose in phase I clinical trials, not only toxicology data from dog and monkey, but also data from rat, mice, and tumor-bearing mice should be included. Similary Homan [47] concluded that there was a 5.9% probability of exceeding the human maximum tolerated dose (MTD) if the starting dose in clinical trials were 1/ 3 of the TDL of large animal species (dog or monkeys). Rozencwig et al. [45]

concluded that 1/6 LD10 in the mouse and 1/3 toxic dose low in the dog corresponds to an acceptable dose in humans provided both preclinical and clinical data are obtained under identical schedule and compared on a mg/ m2 basis. Mice and dogs may provide different informations for a given drug but combining data from both species can be helpful in determining the starting dose in humans for phase I clinical trials [45].

Mahmood [48] using 25 anticancer drugs examined whether or not MTD can be predicted from animals to humans. The predictive performance of two different approaches of allometry for the prediction of MTD was compared in humans from animal data. The two approaches to predict MTD in humans were: (i) the use of a fixed exponent of 0.75 and the LD10 in mice; and (ii) the use of LD10 (in case of mice) or MTD data from at least three animal species (interspecies scaling). The results of the study indicated that MTD can be predicted more accurately using interspecies scaling than using a fixed exponent of 0.75. Like clearance, it was noted that incorporation of mean life-span potential (MLP) can also be used to improve the prediction of MTD for some drugs. One-third of the predicted MTD from interspecies scaling can be used as a starting dose in humans. This approach may save time and avoid many unnecessary steps to attain MTD in humans.

Prediction of Inhalational Anesthetic Potency Minimum Alveolar Concentration (MAC)

Interspecies scaling is frequently performed to predict pharmacokinetic parameters from animals to man and a fair amount of research has been successfully conducted to correlate body weight with the pharmacokinetic parameter(s) of interest [5, 49, 50]. However, very little information is available for the prediction of pharmacodynamic parameters from animals to man. Travis and Bowers [51] applied the principles of allometry to the minimum alveolar concentration of several inhalational anesthetics. The authors found that not only there was a poor correlation between body weight of animals and the MAC but the slope of the allometry was statistically not different from zero. Lack of correlation between body weight and MAC and a slope nearly zero made it almost impossible to predict MAC in humans.

MAC is defined as the minimum concentration of inhalational anesthetic agent in the alveolus at steady state which will inhibit a muscular response to stimulus in 50% of patient population and is expressed as volume percent required at one atmosphere [52]. Thus MAC represents EC 5o on a conventional quantal dose response curve.

Using a correction factor, Mahmood [53] attempted to predict MAC from animals to humans. The MAC values of 10 anesthetics were obtained from the literature. At least three animal species (excluding humans) were used in the scaling. Interspecies scaling of MAC was performed using the following two methods:

i. Using traditional allometric approach, the MAC of each drug was plotted against the body weight of the species on a log-log scale and from the resultant equation MAC was predicted in humans;

ii. MAC in each species was multiplied by a "correction factor" obtained by adjusting the lung weight of the species based on per kg body weight. The product of correction factor and the MAC was then plotted against body weight on a log-log scale.

Using the simple allometric approach, the correlation between body weight of the species and the MAC was found to be poor. The exponents of the simple allometry varied from -0.026 to 0.105. The mean of the exponents of all 10 drugs was 0.027 which was statistically not different from zero. The error of predicted values ranged from 28-134%. The predicted MAC in humans was overestimated at least by 50% for six drugs.

On the other hand, incorporation of "correction factor" substantially improved the correlation between body weight and the MAC. The exponents of the allometry varied from 0.078 to 0.218. The mean of the exponents of all 10 drugs was 0.127 which was statistically different from zero. The error of predicted values ranged from 2-92%. The predicted MAC in humans was overestimated by 50% for only two drugs.

It is difficult to visualize that there will be a correlation between body weight and MAC, since a change in a pharmacodynamic parameter may not simply be a function of change in body weight. The concept of a "correction factor" for anesthetic gases and vapors is based on the fact that these anesthtics are administered to patients at appropriate inspired concentrations. Depth of anesthesia is determined by the concentration of anesthetic agent in the brain. The rate at which an effective brain concentration can be acheived depends on the rates of transfer of inhaled anesthesia from the lung to the blood and from blood to the tissues, the solubility of the anesthetic from the lungs to the arterial blood, its concentration in the inspired air, pulmonary ventilation rate, pulmonary blood flow (change in cardiac output), and the partial pressure of the anesthetic between arterial and mixed venous blood. Considering all the abovementioned factors in order to achieve an adequate concentration of an inhaled anesthetic, it appears lung plays a vital role, as the lung is the site of drug delivery. Therefore, taking into account that the role of lung in maintaining an adequate concentration of a given anesthetic in the brain is vital and since the size of the lung and the body weight varies from species to species, normalization of the lung weight based on the body weight was found to be suitable for the improved prediction of MAC in humans from animals. Though data for inhaled drugs other than anesthetics were not evaluated, the findings of this study may be extended to other inhaled drugs. The concept of a correction factor for the prediction of a parameter of interest (especially for a pharmacodynamic parameter) for inhaled drugs other than anesthetics should be examined.

CONCLUSION

The allometric scaling of pharmacokinetic parameters can be useful to select a safe and tolerable dose for the first-time administration to humans. Thus scaling can provide a rational basis for the selection of a first dose in humans. Therefore, in recent years, interspecies scaling of pharmacokinetic parameters has drawn enormous attention. Over the years many approaches have been suggested to improve the predictive performance of allometric scaling. Though not perfect, these approaches are of considerable importance to understand and refine the concept of allometric scaling.

There may be anatomical similarities among species but there are external factors which will affect the allometric scaling. Experimental design, species, analytical errors, and physico-chemical properties of drugs such as renal secretion or biliary excretion may have impact on allometric extrapolation.

Despite the fact that allometry is empirical and occasionally fails to perform adequately, further investigation should be conducted to find the underlying reasons for failure.

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