A GENERAL FRAMEWORK FOR NON-PARAMETRIC SUBJECT-SPECIFIC AND POPULATION DECONVOLUTION METHODS FOR IN VIVO-IN VITRO CORRELATION

Davide Verotta1,2

IDepartment of Biopharmaceutical Sciences and Pharmaceutical Chemistry 2Department of Epidemiology and Biostatistics University of California San Francisco San Francisco, California 94143

1. INTRODUCTION

4

Suppose that a drug is given in some formulation to a system resulting in an (un­known) input A(t), and n observations are collected at different times following the input. The i-th observation takes the form:

Ij

y; = !A ('t)K(t-'t)d't+E; (1)

where E j indicates a random measurement error. The general input identification problem is to estimate A(t ) given K(t ) (see [5]). In

an in vivo in vitro correlation situation there is (partial) knowledge about the input A(t ) which is obtained from in vitro release experiments. However such knowledge is not com­plete because of the changing conditions between in-vitro and in-vivo. Given an in-vitro release profile and observations from an individual receiving a similar formulation of drug, the main problem facing the data analyst is to determine how different is the in-vitro release with respect to the in vivo one. An additional aspect of the problem is that the fonts of variability present in the in-vivo and in-vitro release are different. The in-vitro release is generally observed under strictly controlled experimental situations, while the in-vivo re­lease is observed indirectly in an experiment involving different humans.

In this paper we describe a general modeling framework to determine the degree of difference between the in-vivo and in-vitro release which also describes and allows to esti­mate interindividual differences. The paper is divided in three sections: (I) we describe the

In Vitro-in Vivo Correlations, edited by Young et al. Plenum Press, New York, 1997 43

44 D. Verotta

general framework, (ii) we report results from a set of simulations demonstrating the ap­proach, and (iii) close the paper with a final discussion.

2. GENERAL FRAMEWORK

2.1. Mathematical Model

We propose an approach which partions the input, A(t ), into the observation site (typically plasma) as follows:

AVivo (t )=Achange (t )Avitro (t) (2a)

t A (t )=JAViVO (t)Kabsorption (t-1:)d 1: (2b)

In this formulation the function AVivo(t ) represents the actual profile of drug released (for example in the gastro intestinal tract). This in tum is the product of AVivo<t ), which in­dicates the known (estimated) in-vitro release .input function, and Achange(t ) which indi­cates an unknown component which modifies AVitro(t ) to account for the changed in-vivo conditions.

Finally the convolution of AVivo(t ) with Kabsorption(t ) represents the absorption process following the in-vivo release. The (in-vivo) observations are given (for a system with lin­ear pharmacokinetics) by:

t

Y (t )=JA (1:)Kdisposition (t-1:)d 1: (2c)

where KdisPosition (t ) indicates the disposition function of the drug. Figure (1) depicts the model just described. A few examples of particular instances of the previous general model will help clarify the situation.

Example I: Identical in-vivo in-vitro release; instantaneous absorption In this case

AVivo (t )=Avitro (t) (3a)

and

A (t )=Avivo (t )=Avitro (t) (3b)

which implies AChange(t )=1 and Kabsorption(t ) approaching an impulse (delta Dirac, 8(t» function. This example represents the luckiest situation: a complete equality between in­vitro and in-vivo release in the observation site.

Example 2: Identical in-vivo in-vitro release. First order absorption In this case AVivo(t )=Avitro(t ), and Kabsorption(t )=e·kat where ka is the rate of absorption of released drug into the plasma site. The example represents a somewhat more complicated situation:

Non-Parametrk: Subject-Specific and Population Deconvolution Methods 4S

II In-vitro I / . , _Ltln~-VIVO I ,. A / ~~ I: (0- .~,:)

II

I ' ___ ~ ____ .J

L Kabsorption

K

'~ A !

/ !

, i Kdisposilion

\/ '--1 Observations

~--~-Figure 1. The general model for in vivo-in vitro correlation. The input into the observation site, A(t), depends on the in vitro release Avuro' the change of release between in vivo and in vitro Achang.(t), the absorption process Kab,orp, non' and the dispostion function of the drug Kd"pn",wn'

there is a complete correspondence between in vitro and in vivo release, but a slow ab­sorption process delays the input into the plasma resulting in an apparent discrepancy be­tween in vitro release and the observed release. The model is:

AVivO (t )=A~itro (t) (4a)

t A (t )=!AViVO ('t)e-ka (t-"C)d 't (4b)

Example 3: Different in-vivo-in-vitro release. First order absorption An example of this situation is obtained if we substitute AChange(t ) with e- kt obtaining:

AVivo (t )=e-kt AVitro (t) (5a)

46 D. Verotta

t A (t)=!Avitro('t)e-kte-ka(t-'t)d't (5b)

where now e-kt represents a time dependent attenuation of the in-vitro release profile, a con­tinuous approximation to, e.g., the elimination of the formulation due to gastric emptying.

The three examples show the flexibility of the general model described above. In general it is easy to see how the approach can describe a variety of situations, and how in particular non-parametric functions (as opposed to the parametric exponentials used in the examples above) can further improve the flexibility of the approach (see the simulations below). We now show how to incorporate inter-individual variability in the in-vivo in-vi­tro correlation model just described.

2.2. Inter-Individual Variability

Inter-individual variability can be present in both the processes altering the in­vivolin-vitro correlation. Individuals can differ in the process changing the in-vitro release and in the absorption process. Accordingly, the model for the in-vivo release rate from the j-th individual (A vivo ' j) takes the form:

A vivo J (t )=Achange J (t )Avitro (t) (6a)

t

A j (t )= !AViVO J (1: )Kabsorption J (t -1:)d 1: (6b)

An example of incorporating inter-individual variability in the mathematical frame­work is provided by the third example above, now:

(7a)

t Aj (t )=!Avitro ('t)e -kj'te-kaJ(t-'t)d 't (7b)

where kj and ka•j represent the j-th individual attenuation and absorption constants.

2.3. Semi-Parametric Representation

A semi-parametric representation of the general model just described is given by:

Achange (t )=Spline 1 (t) (8a)

Kabsorption =e -ka t +Spline 2(t ) (8b)

Non-Parametric Subject-Specific and Population Deconvolution Methods 47

where Spline1 and Spline2 indicate spline functions. Spline 1 is always constrained to be be­tween zero and one and to be monotonic non-increasing. Spline2 is always constrained to be greater than _e-kat. [Splines are the sum of ni polynomials of a certain degree (cubic degree, for a cubc spline), which satisfy continuity conditions of all the derivative up to the degree minus at the 2nd, ... , ni-th point of an increasing sequence of points called breakpoints. For example a linear spline which goes throug~ a set of points results in the familiar broken line connecting the points.] The semi-parametric representation allows to investigate the in-vivolin-vitro cor­relation without having to assume a priori a particular model.

3. SIMULATED DATA

To test the different methods we simulated three data sets. In all of them there are 20 subjects with 20 serum samples taken in each person at log-equispaced times between two minutes and 20 hours after dose administration. Each subject has a biexponential disposi­tion function:

(9)

with c\' 82j , 83j, and 84j log-normally distributed with mean 21 = 0.04, 22 = 15.0,23 = 0.02 and 24 = 0.4, respectively, and covariance matrix:

[0.04 ] 0.02 0.04 0.0001 0.0001 0.04 0.0001 0.00001 0.02 0.04

In all the simulated data sets the in vitro release rate function is given by the (uni­modal, increasing and than decreasing) biexponential:

A. (t) = 87 [e-est - e-e6t] vitro 86-85 (10)

with 2s = 2.0, 26 = 2.6 and 27 = 104, respectively.

3.1. First Simulated Data Set

Identical in-vivo in-vitro release; instantaneous absorption. In this data set

AVivo (t) = 1 ,Kabsorption J (t )=8(t) and Aj (t) = Avitro (t) (lla)

3.2. Second Simulated Data Set. Identical in-Vivo in-Vitro Release; First Order Absorption

In this simulation AVitro(t )= is the same of the first simulation, Kabsorptionit )= e-kajt

where ka is log normally distributed with mean .5 and approximately 25% variance. The resulting individual input function is:

48 D. Verotta

t A j (t )=!AViVO ('t)e-kaJ(t-i:)d 't (lIb)

3.3. Third Simulated Data Set

Different in-vivo in-vitro release; first order absorption In this simulation Achang • . (t )(t )=e-kjt where k. is log-normally distributed with mean

oJ J 0.05 and variance 0.0625, and Kabsorption,j is the same of simulation 2.

AVivo J (t )=e-kjt Avitro (t) (12a)

t Aj (t )=!Avitro ('t)e-kji:e-kaJ(t-r:)d 't (12b)

In all the simulations observations are obtained by calculating the convolution of Ap ) with IS(t) and adding a random proportional error of 10%. The data corresponding to the three simulations are shown in figure (2).

In the analysis of the simulated data, the subject disposition function is assumed to be known exactly, i.e. the correct subject disposition functions are used during the analy­sis. To simulate a situation where the data analyst does not know the correct model under­lying the data, six kinds of in-vivo/in-vitro correlation models are fitted to each of the different data sets. The most general model fit to the data (M 1) is as follows:

A j (t )=l1lj IAvitro ('t)Spline I ('t) [e-kaJ 112j (t-i:)+113j Spline 2(t -'t)] d't (13)

where 'll'j' 'll2j' and 'll3j are fixed to 1, for method 1, and are log normally distributed for method 2. The other five models (M2 to M6) are characterized as follows:

M2: Splineit )=0 (in-vivo change; first order absorption) M3: Spline,(t )= 1 (no in-vivo change; complex absorption) M4: Splineit )=0, e-ka,j T]2j(t-t)=1 (in-vivo change; fast absorption) M5: Spline,(t )=1, Splineit )=0 (no in-vivo change; first order absorption) M6: Spline,(t )=1, Splineit )=0, e-ka,j T]2j(t-t)=1 (no in-vivo change; fast absorption)

3.4. Estimation Methods

We use two different estimation methods to obtain the desired estimates, Method 1 is a subject-specific two-stages method which takes advantage of the richness of the data available in a in- vitro/in-vivo correlation study. Method 2 is a mixed effect population method which in the present context assumes the parameters OJ to be log-normally distrib­uted and uses maximum-likelihood to obtain the estimates of the parameters in the input model, and the variance covariance matrix of the random effects. Individual estimates are in this case empirical Bayes ones. We used the computer program [2], to obtain the de­sired estimates in both cases. To select between the different models we used the methods reported in [3] for method 2, and the Akaike criterion [1] for method 1.

Non-Parametric Subject-Specific and Population Deconvolution Methods

c: .2 o c: .2 5 a. o!:

:5 L()

o

o

r t f , I I , I I

i

, I , , ,

I , , ,

, ' , ' I ' , ' , '. , ,

\ .

!/ I ! I , ,. 'I

" " I·

': " :J ,:

( , , • r

2

· · · · · , · · · \

• · · · , . , . \ , . .

\ . . , . , .

'.

, . . . ,

4

, , ,

.................. ............. ::.:."::' "':: ---

6

49

8 10

Figure 2. Simulated A(t). Solid line, first simulation (identical in vivo-in vitro release; instantaneous absorption); widely dashed line, third simulation (different in vivo-in vitro release; first order absorption).

3.5. Measures of Performance

To compare across methods we used the following measures of performance: RMSE, the square root of the mean of the squared differences between true and the esti­mated individual input rate function. ME, the mean of the differences between true and es­timated individual input rate function. RMSE and ME are evaluated at the sampling points, and divided by the standard deviation of the true input to obtain normalized RMSE (NRMSE) and ME (NME), this simply allows to compare better across simulations. The expected value NME if the estimates of close on average to the true model tends to zero.

50 D. Verotta

Table 1. First simulated data set individuals' input functions

TRUE Ml M2 M3 M4 Ms M6 NRMSE Method I 0.112 0.241 0.228 0.226 0.202 0.192 0.185 Method 2 0.132 0.241 0.227 0.234 0.22 0.191 0.203 NME Method 1 0.0235 0.021 0.0153 0.Q25 0.0136 0.0235 0.0151 Method 2 0.0223 0.0193 0.0147 0.00793 0.00939 0.0173 0.0222

TRUE indicates the model used to generate the data. M 1-M6 the different models described in the text.

4. RESULTS

For all the simulations we will report the results obtained by fitting the "true" model (the same used to generate the data) and models Ml-M6 to the data.

4.1. Simulated Data: First Data Set

Table 1 reports the simulations results corresponding to the first data set. In this simulation model M6 has the same functional shape of the model used to simulate the data (the "true" model), while for all the other models are overspecified in respect to the true model. For example model M2 approximates the true model if Splinej(t )",1, (Spline/t)"'O), and e·ka• j ~2j(t.,)", I. By a relatively small margin model M6 obtains the best results for both methods of estimation. However all models obtain really similar perform­ance, as it should be expected since model M6 is always a special case of models MI-M5.

4.2. Simulated Data: Second Data Set

Table 2 reports the simulations results corresponding to the second data set. In this simulation model M5 has a similar functional shape of the model used to simulate the data, while models MI-M3 are overspecified in respect to the true model, and models M4 and M6 are misspecified. For example model MI approximates the true model Splinej(t)'" e·kt and Splinez<t )",0. Models M4 and M6 on the other end cannot approximate the true model because they do not incorporate an absorption function. Models MI-M3 and M5 obtain similar performance in respect to both NRMSE and NME. Models M4 and M6 ob­tain almost twice the size ofNRMSE.

Table 2. First simulated data set individuals' input functions

TRUE Ml M2 M3 M4 Ms M6 NRMSE Method I 0.192 0.261 0.258 0.266 0.582 0.252 0.605 Method 2 0.212 0.261 0.257 0.274 0.6 0.251 0.623 NME Method I 0.0264 0.023 0.0183 0.029 0.619 0.0295 0.672 Method 2 0.0252 0.0213 0.0177 0.0119 0.614 0.0233 0.679

See legend to table I

Non-Parametric Subject-Specific and Population Deconvolution Methods 51

Table 3. Third simulated data set indiviuals' input functions

TRUE M\ M2 M3 M4 Ms M6 NRMSE Method I 0.179 0.261 0.258 0.576 0.602 0.572 0.555 Method 2 0.199 0.261 0.257 0.584 0.62 0.571 0.573 NME Method 1 0.0267 0.023 0.0183 0.579 0.619 0.599 0.582 Method 2 0.0255 0.0213 0.0177 0.562 0.614 0.593 0.589

See legend to table I

4.3. Simulated Data: Third Data Set

Table 3 reports the simulations results corresponding to the third data set. In this simulation model M I and M2 have a functional shape similar to the model used to simu­late the data (model M2 is closer), models M3-M6 are misspecified in respect to the true model. (they cannot approximate the true model). Model MI approximates the true model if Spline,(t)"'e-kt and Splinez{t)"'O. Model M2 approximates the true model if Spline2(t)"'0. Models MI-M2 obtain similar performance in respect to both NRMSE and NME while models M3-M6 obtain higher values for NRMSE.

Table 4 reports the fraction of times a particular model (MI-M6) is selected by the model selection criterion, for all the simulations and estimation methods. Overall the cor­rect functional shape is selected in all the simulations, although sometimes an overspeci­fied model is selected. Importantly misspecified models are almost never selected.

5. DISCUSSION

In this paper we describe a general framework for in-vivolin-vitro correlation model­ing. We report a general model to describe possible changes between in-vivo and in-vitro, provide a semi-parametric implementation of the methodology and describe a way to in­corporate random effects in the model, thus providing a way to devise popUlation models for in-vivolin-vitro correlation. The general model we propose allows the partition of the difference between in-vivo in-vitro situations into physiological events: difference due to a change in release of drug, and difference due to the (in-vivo) absorption of drug. The semi-parametric implementation of the model allows the investigation of different modes

Table 4. Percent times a model is selected

M J M2 M3 M4 Ms M6 Method 1 First simulation <0.05 <0.05 <0.05 <.05 <.05 0.98 Second simulation <0.05 <0.05 0.22 0 0.76 0 Third simulation .21 0.76 <0.1 0 0 0 Method 2 First simulation <0.05 <0.05 <0.05 <.05 <.05 0.99 Second simulation <0.05 <0.05 0.32 0 0.64 0 Third simulation .31 0.66 <0.1 0 0 0

52 D. Verotta

of change by making limited assumptions on the functional shape induced by the changes in the in-vivo input function.

In a limited and preliminary set of simulations we test two different estimation methods to obtain the desired estimates: Method 1 is a standard subject-specific method which directly estimates the individuals input functions using least squares. Method 2 is a mixed effect population method which obtains individual estimates as empirical Bayes ones. In respect to the individual estimates the two methods perform similarly and they also obtain similar performance in respect to mean and variance population functions (not shown). These results shows clear evidence that the extra complication of method 2 (which includes random effects and requires a much more elaborate model selection strat­egy) is not necessary in a data rich situation like in-vivolin-vitro correlation studies. More­over, in other data rich situations [4], method 2 can obtain biased estimates of population mean input function, and individual estimates which are too close to the population mean, and therefore biased in respect to the individual true input function.

The same simulations indicate that available model selection criteria allow to cor­rectly select (on average) the correct model for change between in-vivo and in-vitro situ­ations. Overall misspecified models are never selected, but sometimes overspecified models are. The selection of an overspecified model has practical consequences if the model for the a particular individual is important because it will increase the variance of the estimated input function. However if the target is the estimation of a population mean function, and its inter-individual variability, a two-stage estimate (based on individual es­timates) will obtain unbiased mean input function, while the inter-individual variability will be in general more influenced by the intrinsic inter-individual variability than the (comparatively little) extra variability induced by using a overspecified model.

The author is currently investigating further generalizations of the methodology, devis­ing ways to carry over measures of precision ofthe in-vitro release estimates to obtain overall measures of precision for the estimated in-vivo input function, and assessing the performance of the different methods in respect to the estimation of mean population estimates.

REFERENCES

1. Akaike, H., "A New Look at the Statistical Model Identification Problem," IEEE Trans Automat Contr, 19:716-723,1974.

2. Boeckmann, A.J., S.L. Beal, and L.B. Sheiner, NONMEM Users Guides, 1989. 3. Fattinger, K.E. and D. Verotta, "A Non-Parametric Subject-Specific Population Methods for Deconvolu­

tion. l. Description, Internal Validation and Real Data Examples," J Pharmacokin Biopharm, 23: 1996. 4. Fattinger, K.E. and D. Verotta, "A Non-Parametric Subject-Specific Population Methods for Deconvolu­

tion. II. External Validation," J Pharmacokin Biopharm, 23: 1996. 5. Verotta, D., "Concepts, Properties, and Applications of Linear Systems to Describe the Distribution, Iden­

tify Input, And Control Endogenous Substances and Drugs in Biological Systems," Critical Review Bioen­gineers, 1996. In Press.