Journal of Pharmacokinetics and Biopharmaceutics, VoL 4, No. 4, 1976

Theoretical and Computational Basis for Drug Bioavailability Determinations Using Pharmacological Data. I. General Considerations and Procedures Victor F. S m o l e n 1

Received Sept. 27, 1974--Final Mar. 2, 1976 The use of data deriving from monitoring the time variation of the intensity of pharmacological effect(s) following dosing can often present an advantageous alternative to the more conventional approach ofusing chemical or radiologieal assay of blood and/or urine level data for bioavailability evaluations of drug products: bioavailability studies can be performed with drugs where no assay exists. A relatively simplified discussion of the general theoretical principles on which the use of pharmacological data is based and a stepwise description of the approach for its routine application in bioavailability studies Ore presented. Approaches for computing rates and extents of drug bioavailability vs. time profiles on analog and digital computers are qualitatively described and quantitatively p~'esented in a subsequent report. The concept of preabsorption (gastrointestinal bioavailability) is introduced and biophasic availability of drugs to local sites ofaction is discussed.

KEY WORDS: bioavailability;pharmacologicaldata; pharmacokinetics;modeling. INTRODUCTION The determination of drug bioavailability represents an important application of pharmacokinetics. Inadequate bioavailability of a drug product is tantamount to an inadequate dose. Bioavailability studies of drug products are most commonly performed using chemical or radiological assay techniques to periodically detect levels of drug present in plasma or urine. However, sensitive assays for the detection of many drugs in body fluids do not exist. When this is the case, clinical testing of the drug products Research supported by Food and Drug Administration Contract No. 223-73-3026, Alcon Laboratories, Fort Worth, Texas, and the Purdue BiomedicalEngineeringCenter. I Interdisciplinary Drug Engineering and Assessment Laboratory (IDEAL), School of Pharmacy and Pharmacal Sciences,Purdue University,West Lafayette, Indiana 47907. 337 9 1976 Plenum Publishing Corporation, 227 West 17th Street, New York, N.Y. 10011. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission of the publisher.

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is often resorted to. HoWever, because of such factors as pathologically induced instability in the patient's physiological state, the simultaneous administration of other drugs, placebo effects, and the difficulty of accurately quantitating clinical drug response, highly variable results are often produced. In addition to raising ethical considerations about the treatment of patients with drug products of unproven quality, such testing can require a large number of subjects and involve considerable expense. An alternative method of performing bioavailability studies utilizing pharmacological data is often overlooked. Such data, when properly treated, can provide as much and even more bioavailability information as is obtainable from blood level data. For example, in addition to permitting the determination of drug bioavailability to the systemic circulation, pharmacological data permit the determination of the rates and amounts of drug penetrating to its site(s) of action (biophase) as well. Such "biophasic availability" can be quite different from systemic availability (1,2). This is particularly important for dosage forms intended to release the drug for local effects; in this case, the presence of the drug in the systemic circulation would indicate a toxic dose or a poorly designed drug dosage form. Pharmacological data have the advantage, relative to direct chemical assay data, that they can be applied to study drug bioavailability by any route of administration and can be recorded more frequently or even continuously by noninvasive methods. In contrast, blood sampling is always performed intermittently by venipuncture. The use of pharmacological data for bioavailability studies should be considered irrespective of whether an assay method for the drug exists. The decision to use either or both types of data should depend on their relative sensitivity, precision, convenience, and economy. Bioavailability is best defined operationally in terms of the access of parent drug and/or its metabolites to a sampling site in the body. For example, when sampling blood by venipuncture, bioavailability relates to the access of the drug entity or entities, which are detected by the particular assay method employed, to the venous circulation at the particular site from which the blood is withdrawn. The bioavailability could conceivably be different if arterial blood were sampled. Similarly, the use of pharmacological data, in effect, samples the drug and/or all the active drug entities at their site(s) of action (biophase) which contribute to inducing the observed drug effect(s). Just as the drug entities sampled in blood depend on the specificity and sensitivity of the assay employed, the sampled drug entities will depend on the particular drug effect chosen to be monitored and the techniques employed in its recording. Nearly all drugs elicit multiple actions. The selection of any particular drug response for bioavailability purposes can be based on considerations of the following: (a) Its direct

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relation or correlation with the drug's sought therapeutic action. Although desirable, clinically significant drug effects need not necessarily be monitored. When a side effect of little or no clinical interest is recorded, it merely serves as an analytical tool for bioavailability studies similar to a chemical assay. (b) Intrasubject and intersubject variability in the data. Such variability in drug response can be both quantitative and qualitative in that all subjects do not always manifest all of a drug's known effects. (c) Suitability of the data for quantitative pharmacokinetic analysis. The data should allow the time variation of rates and extents of drug bioavailability to be reliably computed. When the data are insufficient or the scatter in the results is too large to justify such quantitative treatment, parameters such as peak response intensity, time of peak response, and area under the response curve can still be estimated. The relative significance of these parameters derived from pharmacological data is nearly the same as if they were obtained from blood or urine data. When sufficient data are available to construct a doseeffect curve, it can be used to convert the observed drug response intensities into their corresponding biophasic drug levels (3). The bioavailability parameters can then be estimated from the resulting biophasic drug level vs. time plots. This conversion of the data corrects for any nonlinearities which may exist in the relationship between dose and peak response intensities and areas under the response vs. time profiles ; the peaks and areas are then directly related to the amounts of drug which become bioavailable from the dosage forms being tested. (d) Convenience and ease of recording data. Continuous recording or frequent discrete data sampling by noninvasive physiological monitoring techniques should be used whenever possible. (e) Ease with which pharmacokinetically significant results can be resolved from the recorded data. Many drugs induce changes in, for example, the electrocardiogram, evoked or spontaneous electroencephalogram, phonocardiogram, plethysmogram, and electromyogram. However, the resolution and quantitation of drug-induced changes in these biological signals can require the use of complicated signal-processing techniques and sophisticated instrumentation. Whether this consideration imposes serious limitations or not depends largely on the signal analysis capability of the investigators. However, methods of bioavailability analysis which require using sophisticated techniques would not be widely applicable for routine use by most pharmaceutical scientists. Unless the usually required computerized processing is performed simultaneously with the recording of the data from the subjects, the treatment of such data can be highly time consuming. Whenever appropriate, such drug responses as pupil size, intraocular pressure (measured by applanation tonometer), temperature, and blood pressure, which can be measured directly and interpreted easily, are to be preferred.

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For practical purposes, pharmacokinetics can be described as principally concerned with the relationship between the dynamic manner in which drugs enter the body and the therapeutic and toxic responses that are elicited; i.e., drug bioavailability input ~ pharmacological response output relationships are of interest. Since drug inputs are affected by drug dosage forms, the practical concern in pharmacokinetics is with the design, development, bioavailability evaluation, and use of drug products for maximal therapeutic benefit. In order to avoid possible confusion concerning the terms "bioavailability" and "bioequivalency" of drug products, it is useful to define and categorize the various types of bioavailability that may be encountered. The generic term "bioavailability" can be interpreted to include two main categories: (I) absolute (drug input) bioavailability, (b) biophasic (site of action) bioavailability, (c) preabsorption bioavailability, and (d) in vitro predicted or correlated bioavailability; this refers to in vivo drug bioavailability input or response profiles, or some characteristic of these profiles, which is estimated or inferred from drug dissolution testing of dosage forms. In vitro results are of interest, however, only if they have been demonstrated to reflect in vivo drug product performance (2, 4, 5). The second bioavailability category is (II) comparative (drug response) bioequivalency ; this refers to the comparison of drug response outputs obtained with different drug dosage form inputs. Comparisons are made of (a) pharmacological response vs. time profiles, (b) drug and/or metabolite blood levels vs. time profiles, and (c) urinary recovery vs. time profiles for unchanged drug and/or metabolites. Absolute drug input bioavailability is determined by the dynamic manner (rates and extents) in which the drug enters the body to reach the systemic circulation or the biophase or to be released at its sites of absorption. Biophasic availability describes the rates and extents of a drug's access to its sites of action in the body. Preabsorption bioavailability vs. time profiles are in vivo analogues of in vitro drug dissolution vs. time profiles and can be computed from appropriate biological data (2.6). In all cases, in vivo bioavailability input profiles are never observed directly from the results of experiments, but must be calculated from observed drug response profiles. Therefore, mathematical relationships between the experimentally observed drug response data and the in vivo bioavailability inputs must be determined before the absolute bioavailability input profiles can be computed (2-17). When the drug input ~ response output relationships are properly established, the computed drug bioavailability input vs. time profiles are independent of whether observed blood level, urinary recovery, or pharmacological response data were used to perform the computations; this is the case provided that the dynamics of the system can at least be approximately described as linear (18).

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In contrast to absolute drug input bioavailability, which must be calculated, comparative bioequivalency judgments are made directly from observed drug response data and therefore do not require constructing a pharmacokinetic model to perform the calculations. Also, in contrast, the results of such bioequivalency comparisons of drug response output profiles can be highly dependent on the type of data collected. For example, orally administered chlorpromazine is extensively metabolized in the gut wall and liver by saturable enzymatic processes prior to reaching the systemic circulation (7). The extent of the metabolism to produce therapeutically inactive sulfoxide metabolites is dependent on the rates of release of the drug from its dosage form. The bioequivalency of two drug products can be judged from the extents of systemic bioavailability as reflected by the areas under observed response vs. time profiles. For two drug formulations having different release rates, such areas will be different when measurements reflecting only the presence of unchanged chlorpromazine in the body are made using a chemical assay or a pharmacological measure such as pupilometry; measurements such as using a nonspecific assay for the phenothiazine nucleus or intraocular pressure lowering which detect both the drug a n d its metabolites will cause product formulations to appear to be bioequivalent despite the existence of likely differences in their therapeutic performance (6). The purpose of the present communication is to provide a qualitative overview of the theoretical basis and describe a systematic approach to performing bioavailability studies with the use of pharmacological data. The various methods of computing rates and extents of absolute drug bioavailability are briefly reviewed. THEORY AND DISCUSSION

Conversion of Observed Pharmacological Response Intensities into Biophasic Drug Levels When the dose-effect curve is linear, observed pharmacological response intensities can be used directly in computations just like results of a chemical assay. This has been found to be the case with pupilometric and intraocular pressure lowering responses to oral and intravenous doses of chlorpromazine in human subjects (6, 7). Generally, however, this is not the case and the conversion of pharmacological data via a dose-effect curve into biophasic drug levels is performed to simplify the construction of pharmacokinetic models and their use for the computation of bioavailability. The theoretical basis for these transformations has been described (2, 8-14). The method is predicated on two major assumptions:

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Assumption No. 1 : "The dynamics of the drug's disposition are linear." This means that dose-dependent (capacity-limited) processes are not operable in the dosing range of interest. When this assumption is valid, the time variation of the quantity of the drug, QB, in a biophase compartment following bolus intravenous dosing (or administration by any other route where the absorption of the drug is a linear process) is described by a sum of exponentials where the Ai's and mi's are constants; t is the time after dosing and D denotes the dose.

QB = D ~ A i e - m a

(1)

i=1

At any arbitrarily chosen constant time tr, QB will be directly proportional to the dose since the sum of the exponentials in equation 1, will be constant and defined by

i Ai e-m't" = fit,. = constant

(2)

i=1

Substituting flt. into equation 1 yields

QB = Dflt,.

(3)

Equation 3 merely expresses the direct relationship between biophasic drug levels and dose at a constant time after dosing. It should be apparent from equations 3 and 4 (below) that the dose is a relative biophasic drug level since fltr is constant and D is directly proportional to QB. The units of Q~ are the same as the dose ; fltr and A~ are dimensionless.

D

=

QB/flt~

=

f(l)

(4)

The dose, or relative biophasic drug level, is denoted in equation 4 as f(l) to emphasize its functional relationship to the pharmacological response intensity, I ; the I is usually expressed as a drug-induced physiological change, AR, relative to its predrug level, Ro, thus [ARI -

(5)

Ro

As described below, the functional relationship between QB/fl,r and I, i.e.,f(/), is defined by plotting the intensity of response observed at an arbitrary reference time, tr, as a function of dose. For practical reasons, t r is chosen as the time of maximum response. When the maximum response intensity occurs at time zero, following bolus intravenous dosing, then

fit~= i A i = 1 i=1

and

D= Qs=f(l)

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This is the case when the exchange of drug between plasma and biophase (site of action) is so rapid so that these tissues may always be regarded in equilibrium and therefore can mathematically be treated as the same compartment. Most commonly, drug effects manifest their maxima some time after the dose is given. In such cases,

~A i =

0

tmax > 0

and

]~tr < 1

i=1

Response vs. time profiles following bolus intravenous dosing, are exemplified n 1 Ai = 1 and Z~= 1 Ai = 0. in Fig. 1 for the cases o f Zi= Assumption No. 2: "The intensity of pharmacological response is a single-valued function of biophasic drug levels." This assumption is implicit in all drug receptor site interaction models. It may be invalid in cases where a tolerance to the drug rapidly develops or for some indirect effects. In the latter case, the validity of the assumption will depend on the manner in which the response to the drug is defined. For any particular response to a drug, the validity of both assumptions 1 and 2 is often apparent by mere inspection of the drug response data, although several specific rigorous tests have been developed (3,8,9). When assumption No. 2 is valid, only one value ofa biophasic or relative biophasic drug level, shown above to equal the dose, corresponds to any given intensity of response. Therefore, the dose-effect curve must define this single-valued relationship; it is then only necessary to relabel the dose axis as Q~/fltr or f(I) and drop the tt subscript on the I,~ axis and relabel it simply as I in order to obtain a calibration curve which can be used to convert I values, observed at any time following dosing by any route, into corresponding f(I) values.

Standardized Experimental and Computational Procedure for Performing Bioavailability Analysis from Pharmacological Data The most commonly applicable procedure for performing bioavailability studies has been standardized. The procedure involves several steps: (a) Administration of several bolus intravenous doses and monitoring the time variation of pharmacological response intensities. When it is not practical to administer bolus doses, then slow intravenous infusions given at various rates over a constant period of time may be substituted. Other routes such as oral and intramuscular may be used, but, as described below, this necessitates a redefinition of the bioavailability results ultimately obtained (2). In all cases, the maximum response intensities resulting from each dose should occur at the same time, tm,x. If this is clearly not the case, then assumption No. 1 does not hold and an alternative approach must be implemented.

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(b) Construction of a dose-effect curve by plotting the maximum observed response intensities as a function of dose. At this stage, a f(I) ratio test may be performed to verify the validity of assumption No. 2 (8,9). (c) Using the dose-effect curve to convert each 1 vs. t response profile resulting from each dose into a corresponding f(I) vs. t profile. (d) Dose-normalizing the values of f(1) by dividing by the dose and plotting the f(I)/D values on the same set of coordinates. All the points should scatter around the same curve irrespective of dose. If this is not so, then a violation of assumption No. 1 is indicated. (e) Fitting a sum of exponentials equation to the f(I)/D vs. t curve. Various weighted least-squares curvilinear regression computer programs such as BMDX 85, Nonlin-R, SPSS 21, SAAM, Multifit, and PLTEST (17) can be used for this purpose. Recently two new computerized fitting procedures have been developed. These include a "pulse-testing" program (15-17) which performs a frequency response analysis of the data and provides magnitude and phase angle vs. frequency plots (Bod6 diagrams) from which the pharmacokinetic model is obtained. The second new approach, called the "method of moments" (15,16), can be used either to fit the curve or to adjust the values of equation parameters obtained from the "pulse-testing" program. A Fourier transform method for analysis of tracer data which is related to the "pulse-testing" approach has also been reported (19). This method is similar to the pulse-testing program in that it utilizes the frequency domain. However, it appears preferable in that model parameters can more accurately be read from the peaks of spectral plots rather than from breakpoints in Bod6 diagrams. In addition to being more economical in terms of computer time, frequency domain methods of curve fitting are advantageous relative to time domain methods involving exponential curve peeling, in that all the data points contribute to the estimates of each model parameter. Considering that values of initial estimates of equation parameters can appreciably influence final least-squares values obtained from curvilinear regression programs, an improved approach could utilize a frequency domain technique to determine the number of exponentials and initial estimates of equation parameters and, if necessary, improve the estimates to any degree desired by using a time domain, weighted least-squares, iterative convergence algorithm as employed in the Multifit program. (f) Developing a mathematical model describing the relationship between relative biophasic drug levels, f(1), and systemic bioavailability rates, A(t), and extents, A(t). Classically, such relationships have been developed in terms of compartment models; the Wagner-Nelson (20) and Loo-Riegelman (21) equations exemplify this approach for one- and two-compartment models, respectively. Such models are, however, generally nonunique and fictitious representations of the system. The constraints imposed in fitting compartment models can necessitate a distortion of the experimental data (9). When the only intent

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of such modeling is to relate drug bioavailability inputs to drug level vs. time response outputs, it is more direct and useful to use transfer functions (2,6,8,18). For present purposes, the transfer function, G(s), can be defined as the Laplace transform of an observed f(I) vs. time response to a unit intravenous bolus dose of drug. It is obtained from the computerized fit to the f(I)/D vs. time curve described in step (e) above. G(s) will be given by equation 6, where G(t) is known as the "weighting function" and L denotes the Laplace operator.

G(s) = LG(t)= L ~. A ie-m''= ~ Ai/(m , + s) i=l

(6)

i=1

As block-diagrammed in Fig. 1, the weighting function is defined by the

f(I)/D vs. time curve obtained by transformation of the I vs. t profiles observed following bolus intravenous dosing. (g) Verification of the transfer function model. This is obtained by comparing experimentally known drug bioavailability inputs, for example as can be obtained by a programmed intravenous infusion, with values computed from the results of monitoring the pharmacological response intensities during and after the intravenous administration of the drug. The observed time variation of drug-induced response intensity is converted into a corresponding biophasic drug level vs. time profile using the dose-effect curve and then used to compute the bioavailability input. The relationship between a constant rate of infusion intravenous input and response output is shown at the bottom of Fig. 1. Various methods have been developed to compute cumulative amounts of drug absorbed vs. time profiles, A(t), from the biophasic drug level vs. time profiles. The methods involve deconvolution or inverse Laplace transformation. This operation of obtaining cumulative amounts of drug absorbed by deconvolution of biophasic drug levels, with the weighting function, can be defined by equation 7, where r is a dummy variable used to perform the integrations in the limit of 0 to t.

A(t) =

[G(~)]-1Q~(t - ~) dz dz

(7)

The function G(r)- 1 in equation 7 is not simply the reciprocal of the weighting function ; it is defined by equation 8, where L - 1 symbolizes inverse Laplace transformation.

t

~

D ~ *A(t) ~

Q~k

t-~ ~---~

[~

t

>

A(t)

t

t

with G(t)

Deconvolve

I

Dose-EffeCtcurve

TRANSDUCTION

--

i jI(t)=f(QB(O)

DEC

I Level ] Output

QB (t)~

---~I d . . . . r f~I)

~ [ II ~ iDrug e

"

MODEL V E R I F I C A T I O N

G(S) =i_~l mi ~ s G(s) = L S(t)

G(t) =.~A.e-mi t l niA. •

TRANSFER

"

tr --~OR

t

t

I

v

t

/

harmacological~

t

i__~iAi= 0

Bio~hase~ Plasma

- - ! Response i~t,~ output

~ A =,i

Biophase = ~lasma

Fig. 1. Drug input ~ pharmacological response output relationships. The upper curves illustrate the process of determining a weighting function (in the time domain, t) or transfer function model (in the complex frequency domain, s) as defined by the relative dose (D) normalized biophasic drug level [Qn(t)/D] vs. t response curve to an impulse (bolus intravenous dose) input. The Qn(t) values are obtained by conversion of observed pharmacological response intensities (I) via the use of the dose--effect curve (DEC). The weighting function model is verified for use in bioavailability computatations by comparisons of experimentally known (e.g., constant-rate intravenous infusion) inputs with the inputs computed from the observed I vs. t output curves, which are converted to Q,(t) and deconvolved with G(t). The DEC is constructed as a plot of I (observed at a time constant, tr, after dosing) vs. dose.

A(t)

Constant Rate i.v. Infusion

t

OR

Curve

Dose-Effect

DETERMINATION

Weighting Function

DRUG

(Impulse) Input (Integral) Input

A(t)~rug Bi .... Inputilability~/ -----~

i(t)

Bolus i.w. Dose

Differential Cumulative

MODEL

I NPUT --~ TRANSFERE NC E - ~ T R A N S D U C T I O N - - ~ OU T PUT

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The source of equation 8 is seen from the alternative expression for A(t) given by equation 9. A(t) = L- 1

QB(s) sG(s)

(9)

Deconvolution can be performed using either digital or analog computers (18-28). Each method has some advantages and disadvantages depending on the characteristics of the data. Most recently, a generalized digital computer deconvolution program has been developed (15-16,19). (h) Application of the verified pharmacokinetic model to the computation of extents and rates of bioavailability elicited by the drug dosage forms being evaluated. The application of steps (a) through (g) is exemplified by results presented in other reports (1-3,6,8,9,12-16). The procedure is here exemplified by the results for the miotic response to chlorpromazine in rabbits shown in Figs. 2-5. The computer-plotted, computationally predicted drug input profile shown in Fig. 5 was obtained using a Hewlett-Packard 5451B Fourier Analyzer/ 2100S computer system which performs the deconvolution by an entirely numerical method in the frequency domain. This obviates the fitting of curves and making any assumptions whatever, except for linearity, regarding a pharmacokinetic model for the system. The results obtained by this approach are identical to those obtained with the use of an explicitly defined transfer function to describe the system dynamics (6).

-0• 3000" ~" 20013 z~ W i-Z 1000 m 0

~ 4,omg/kgiv

60 120 180 240 300 360 TIME(min}

Fig. 2. Time variations of the averageintensity of miotic response to bolus intravenous doses of chlorpromazine administered to three or four rabbits. These curves were transduced into relative biophasic drug levels (Qn) using the dose-effect curve shown in Fig. 4, and dosenormalized by dividing each resulting QB value by its correspondingdose, D. A plot of QB/Dvs. time defines the weighting function (curve D in Fig. 5) for the system.

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• 02000. . ~u

1000-

60

120

160

240

300

360

TIME (rntn)

Fig, 3. Time variation of the intensity of miotic response (/) observed in rabbits during the slow intravenous infusion of chlorpromazine at the rate of 0.50 mg/kg/min. Each I value is transduced into relative biophasic drug levels (QB) using the dose-effect curve shown in Fig, 4. The resulting QB vs. time curve (curve A in Fig. 5) is deconvolved against the weighting function (curve D in Fig. 5). The integrated result provides the cumulative drug input profile shown in Fig. 5 as c u r v e C.

3000

~'02000

~1000 z

,io

31o

,io

DOSE ( m g / k g , i v ) or f(I)

Fig. 4. Intravenous dose-effect curve (DEC) for chlorpromazine-induced miotic activity in rabbits. The average values of observed maximum response intensities are plotted with their standard deviations. The form of the DEC defines the transduction function for the system which is used to convert observed intensities of miotic response into their corresponding relative biophasic (siteof-action) drug levels.

Theoretical and Computational Basis for Drug Bioavailability Determinations. I

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t~ w

Z

3.0

/

~

~

C

2.0

_u

0



>

i-

9"-. o

60

120 TIME

180

240

300

(minutes) u

Fig. 5. Comparison of the experimentally known cumulative amount of chlorpromazine infused at a zero-order rate of 0.5 mg/kg/min to rabbits as a function of time (cui:ve B) with the amounts computed to have been administered (curve C) from the results of monitoring the time course of the intensity of observed miotic response. The computation was accomplished by numerical deconvolution performed numerically in the frequency- domain using a Hewlett-Packard 5451B/2100S Fourier Analyzer microprogrammable computer system. The weighting function (curve D), which represents dose-normalized relative biophasic drug levels resulting from bolus intravenous dosing (impulse response), was deconvolved against the relative biophasic drug level vs. time response (curve A) corresponding to the slow, zero-order, intravenous infusion (step response) and integrated. Prior to deconvolution, curves A and D were numerically smoothed once employing a computer program using the three-point smoothing algorithm DATA (N) + [DATA (N - 1) + DATA (N + 1)]/2 2 The close similarity between the experimentally known and computed drug input profiles (curves B and C, respectively) demonstrates the verity of the approach. The curves were constructed with a Hewlett-Packard 7210 digital plotter. The range of values for the weighting function (curve D) is 0-1.0. DATA (N)=

Gastrointestinal B i o a v a i l a b i l i t y

G a s t r o i n t e s t i n a l b i o a v a i l a b i l i t y (GIB) is here defined as " t h e rates a n d extent of d r u g release f r o m an o r a l d o s a g e f o r m into the g a s t r o i n t e s t i n a l contents relative to i n s t a n t a n e o u s release of the d r u g which is a s s u m e d to occur f r o m a dose of the d r u g a d m i n i s t e r e d in an a q u e o u s o r o t h e r s o l u t i o n vehicle f r o m which it d o e s n o t p r e c i p i t a t e when d i l u t e d with g a s t r o i n t e s t i n a l fluids." T h e s o l u t i o n d o s a g e f o r m is t a k e n to p r o v i d e a reference d r u g i n p u t a n a l o g o u s to a b o l u s i n t r a v e n o u s dose. D a t a deriving f r o m the o b s e r v e d p h a r m a c o l o g i c a l (or b l o o d level) d r u g r e s p o n s e to the s o l u t i o n d o s e p r o v i d e the w e i g h t i n g function for the system. T h e release o f d r u g from solid oral d o s a g e forms into the g a s t r o i n t e s t i n a l fluids can be c h a r a c t e r i z e d , relative to the reference s o l u t i o n d r u g input, by the results of d e c o n v o l v i n g the weighting

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function with the observed response to the solid dosage forms. Confidence in the fidelity of the weighting function and the validity of the procedure may first be derived from deconvolving the weighting function against the observed response to giving the same dose of the solution in divided quantities at equally spaced intervals. The cumulative amount of drug released into the gastrointestinal contents, i.e., the GIB profile, for this manner of dosing appears as a staircase; confidence in the technique may be obtained by determining how closely points on the computed GIB profile fall on the staircase (6), It is important to emphasize that GIB computed as described above will not represent drug absorption into the systemic circulation even though this must be occurring to some extent in order to observe a pharmacological response vs. time profile or blood levels of the drug. The combination of systemic drug bioavailability and GIB can be useful in determining the cause of poor systemic bioavailability; it can shed light On whether the cause of poor systemic bioavailability is poor absorption across the gastrointestinal barrier or an inadequate release of the drug into the gastrointestinal fluids. The determination of GIB for solid dosage forms, in effect, serves as an in vivo drug dissolution test. This is the case provided that all kinetic processes operative in the absorption and presystemic metabolism of the drug are linear; these processes are therefore independent of the amounts and rates at which the drug is released into the gastrointestinal contents. When this is not the case, various seemingly anomalous results may be observed. For example, an instability of the drug in gastrointestinal fluids could manifest as a greater GIB for a solid dosage form than for a solution. A capacitylimited presystemic metabolism of the drug would cause the GIB to decrease with decreasing dose and rates of release into the gastrointestinal tract. This was observed for chlorpromazine (6) when pupil response data, which are reflective of unchanged drug, were used to compute GIB. However, intraocular pressure response data, which are sensitive to both unchanged drug and metabolites, provided GIB results which were essentially independent of the rates of drug release. The use of data for these two different pharmacological responses is analogous to using a blood assay specific for the parent drug or a nonspecific measure such as total radioactivity. GIB loses its utility when nonlinear presystemic kinetic processes are operative. The GIB manifestations attributable to such nonlinear processes are also evidenced in systemic bioavailability.

Biophasic Availability of Topically Administered Drugs Drugs used in ophthalmic and dermatological preparations, inhalations, implantations into the vagina or uterus, etc., are usually intended for local

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effects. The biophasic availability of a drug to its local sites of action can be determined using pharmacological data in a manner similar to the study of systemic bioavailability or GIB. The biophasic availability will reflect the permeability of the drug across an absorbing membrane at the site of administration, drug release from dosage forms, or both depending on the mode of drug administration used to obtain the weighting function. For example, in ophthalmic studies the weighting function can be obtained from the results of monitoring pupil size following the injection of the drug through the cornea into the anterior chamber. The use of the weighting function to compute the biophasic availability of drugs administered as drops placed into the conjunctival sac provides results which characterize the passage of the drug across the cornea (10). Alternatively, the weighting function could be defined by a dose-normalized relative biophasic drug level vs. time curve obtained from the results of administering eye drops. Then, provided that any loss of the drug due to the administered solution running out of the eye can be neglected, bioavailability calculations for solid ophthalmic dosage forms utilizing this weighting function could be interpreted as describing the release of the drug from the solid dosage form into the lacrimal fluid bathing the cornea and other drug-absorbing ocular tissues. By dosing only one eye and making measurements in both eyes, both the simultaneous systemic and biophasic drug availability can be determined (10). Similar considerations apply as well to other routes of administration. The drug input ~ output response scheme shown in Fig. 1 is valid in all cases.

SUMMARY When necessary conditions are satisfied, pharmacological data can provide the same and even more drug bioavailability information than data deriving from chemical assay of drug levels in body fluids. This can be implemented to study drug absorption by any route of administration. In contrast to blood sampling, pharmacological sampling can generally be done more frequently to obtain many more or, in some cases, even a continuity of data points ; complicated and time-consuming analytical procedures can be avoided. Pharmacological data can also be obtained by noninvasive methods. Since most drugs elicit more than one effect, two or more responses can be simultaneously recorded with very little additional effort. The results of such multiple recordings are useful for corroborating bioavailability differences which might only be suspected from the results of a single response, as well as for providing information concerning the relative potencies of active metabolites of the drug (5,6,28). Except for a relatively simple transformation of the observed data via a dose-effect curve, wl~ich may not always

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be necessary, the pharmacokinetic analysis of pharmacological data is nearly identical to that of assay data. The use of pharmacological data should be considered as an alternative or in addition to performing chemical bioavailability studies irrespective of whether a sensitive chemical assay for the drug exists. A generally applicable procedural and computational scheme is presented to determine the absolute (drug input) bioavailability, which is categorized as systemic, biophasic and preabsorption bioavailability and distinguished from bioequivalency. A second report (29) in this series describes the application of the drug input ~ output response scheme to accomplish a predictive conversion of observed in vitro drug dissolution vs. time profiles. Equations are derived and computational methods for bioavailability determinations are presented (29).

REFERENCES 1. V. F. Smolen and W. A. Weigland. Drug bioavailability and pharmacokinetic analysis from pharmacological data. J. Pharmacokin. Biopharm. 1:329-336 (1973). 2. V. F. Smolen, P. B. Kuehn, and E. J. Williams. Idealized approach to the optimal design, development, and evaluation of drug delivery systems II : Drug bioavailability inputs and in-vitro drug release testing. Drug. Dev. Comm. 1:231-258 (1974/1975). 3. V. F. Smolen, R. D. Barile, and T. G. Theophanous. The relationship between dose, effect, time, and biophasic drug levels. J. Pharm. Sci. 61:467-470 (1972). 4. V. F. Smolen. Determination of time course of in-vivo pharmacological effects from in-vitro drug-release testing. J. Pharm. Sci. 60:878-882 (1971). 5. V. F. Smolen and W. A. Weigland. Optimally predictive in-vitro drug dissolution testing for in-vivo bioavailability. J. Pharm. Sci. (in press). 6. V. F. Smolen, E. J. Williams, and P. B. Kuehn. Bioavailability and pharmacokinetic analysis of chlorpromazine in humans and animals using pharmacological data. Can. J. Pharm. Sci. 10:95-106 (1975). 7. V. F. Smolen, H. R. Murdock, and E. J. Williams. Bioavailability analysis of chlorpromazinc from pupilometric data. J. Pharmacol. Exp. Ther. 195:404-415 (1975). 8. V. F. Smolen and R. D. Schoenwald. Drug absorption analysis from pharmacological data. I. The method and its confirmation exemplified for a mydriatic drug, tropicamide. J. Pharm. Sei. 60:96-103 (1971). 9. V. F. Smolen. Quantitative determination of drug bioavailability and biokinetic behavior from pharmacological data for ophthalmic and oral administration of a mydriatic drug. J. Pharm. Sci. 60:354-365 (1971). 10. V. F. Smolen. Optimal control of drug input and response dynamics: A role for biomedical engineering in pharmaceutical science. Am. J. Pharm. Ed. 37:107-125 (1973). 11. R. D: Schoenwald and V. F. Smolen. Drug absorption analysis from pharmacological data. II. Transcorneal biophasic availability of tropicamide. J. Pharm. Sci. 60:1039-1045 (1971). 12. V. F. Smolen, V. D. Turrie, and W. A. Weigand. Drug input optimization: Time optimal control of simultaneously occurring multiple pharmacological effects. J. Pharrn. Sci. 61:I941-1952 (1972). 13. V. F. Smolen, P. B. Kuehn, and E. J. Williams. Idealized approach to the optimal design, development, and evaluation of drug delivery systems. I. Drug bioavailability inputpharmacological response output relationships. Drug Dev. Comm. 1:143-172 (1975). 14. V. F. Smolen. The determination of drug bioavailability characteristics from pharmacological data. Can. J. Pharm. Sci. 1 : 1 4 (1972).

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15. A. K. Jhawar, Ph.D. dissertation, Department of Chemical Engineering, Purdue University, August 1974. 16. P. B. Kuehn. Ph.D. dissertation, Industrial and Physical Pharmacy Department, Purdue University, August 1974. 17. A. K. Jhawar, P. B. Kuehn, V. F. Smolen, and W. A. Weigand. Mathematical methods for evaluation of drug delivery systems. I. A frequency response method for pharmacokinetic model identification. J. Pharmacokin. Biopharm. (in review). 18. A. K. Jhawar, P. B. Kuehn, V. F. Smolen, and W. A. Weigand. Mathematical methods for evaluation of drug delivery systems. II. A generalized numerical deconvolution procedure for computing the amount of drug absorbed. J. Pharmacokin. Biopharm. (in review). 19. S. J. Pizer, A. B. Ashare, A. B. Callahan, and G. L. Brownel. Fourier transform analysis of tracer data. In F. Heinmets (ed.), Concepts and Models of Biomathematics: Simulation Techniques and Methods, Dekker, New York, 1969, pp. 105-129. 20. J. G. Wagner and E. Nelson. Kinetic analysis of blood levels and urinary excretion in the absorptive phase after single doses of drug. J. Pharm. Sci. 53:1392-1403 (1961). 21. J. C. K. Loo and S. Riegelman. New method for calculating the intrinsic absorption rate of drugs. J. Pharm. Sci. 57:918-928 (1969). 22. M. Hanano. Studies of absorption and excretion of drugs. VII. A new estimation method for the release of drugs from dosage forms and the availability in-vivo. Chem. Pharm. Bull. 15:994-1001 (1967). 23. L. Z. Benet. The use and application of deconvolution methods in pharmacokinetics. Abstracts of papers presented at the 13th National Meeting of the APhA Academy of Pharmaceutical Sciences, Chicago, November 5-9, 1972, pp. 169-171. 24. C. D. McGillem and G. R. Cooper. Continuous and Discrete Signal and System Analysis, prelim, ed., Chap. 4. Purdue University Press, Lafayette, Ind., 1973, Chap. 4. 25. M. Silverman and A. S. V. Burgen. Application of analog computer to measurement of intestinal absorption rates with tracers. J. AppL Physiol. 16:911-913 (1961). 26. J. O. Osburn. Inverse simulation. Inst. Control Syst. 40:131 133 (1967). 27. J. S. Mattson, H. B. Mark, and H. C. MacDonald. Computer Fundamentals for Chemists, Dekker, New York, 1973, pp. 255-275. 28. V. F. Smolen. A study of drug bioavailability as related to physiological response. Quarterly Report of Research Progress No. 4 Food and Drug Administration Contract No. 73-23, May 30, 1974. 29. V. F. Smolen, Theoretical and computational basis for drug bioavailability determinations using pharmacological data. II. Drug input ~ response relationships. J. Pharmacokin. Biopharm. 4:355-375 (1976).