Drug Targeting Organ-Specific Strategies

340 13 Pharmacokinetic/Pharmacodynamic Modelling in Drug Targeting

Usually, the differential equations are written in a different form, by relating the rate of transport from a compartment to the quantity of the drug in that compartment and a rate constant (dimension: time-1), as depicted in Figure 13.2. and formulated as follows.

where A1 is the quantity of the drug in compartment 1, k12 and k21 are distribution rate constants and k10 is the elimination rate constant.

Comparing Eq. 13.1 and 13.2, it follows that a rate constant kxy is equal to CLxy / Vx. From the assumption that there is no net transport between two compartments if the con-

centrations in both compartments are equal, it follows that

k21 · V2 = k12 · V1 (= CL21 = CL12).

Eq. 13.1 and 13.2 are mathematically equivalent, and thus may be used arbitrarily without affecting the modelling results. However, Eq. 13.1 (and Figure 13.1) is preferred since it reflects better the mechanistic basis, as drug transport is governed by drug concentration, both for passive diffusion according to Fick’s Law, and for carrier-mediated transport. In the case of the latter, the terms referring to the rate of transport from compartment x to compartment y,

should be replaced by their Michaelis–Menten equivalent

where Vmaxxy is the maximum transport rate between compartments x and y, and Kmxy is the Michaelis–Menten constant of the transport between x and y.

An example of the use of Michaelis–Menten kinetics in a compartmental model is given in the model of Stella and Himmelstein [5], depicted in Figure 13.3.

13.2.4.2 Physiologically-based Pharmacokinetic (PB-PK) Models

These are relatively complex models describing drug transport between blood and a series of physiological and/or anatomical entities, for example, organs, tissues, or cells [15–20]. PB-PK models are characterized by a relative large number of parameters. In many cases, several of these can be estimated from physiology or anatomy (for example, blood flow and volumes), others may be obtained from in vitro experiments (for example, partition coefficients between water and tissue), or by experiments in isolated tissues (for example, binding and metabolism in isolated liver cells or slices; see Chapter 12). In principle, PB-PK models are well adapted to take into account the extracellular and/or intracellular events in the disposition of the targeting device.

The number of compartments in a physiologically-based pharmacokinetic model may vary between two (in drug targeting: a target compartment and a non-target compartment) and 10 or more, depending on the desired degree of differentiation. The more compartments, the greater the ability of the model to define the true behaviour of the drug. However, the in-

creased number of parameters increases the problem of assigning reliable values to these parameters, both in simulation (Section 13.2.7) and in data analysis (Section 13.2.8). As a general rule, the number of compartments should be chosen carefully, according to the parsimony principle: start the modelling with the simplest model that can discriminate the processes of interest. If the chosen model does not provide satisfactory results (in terms of credible predictions or satisfactory goodness-of-fit), the model can be explored further by adding compartments or connections in a step-by-step procedure.

On the other hand, PB-PK models are frequently used in toxicokinetics for a different purpose, that is, the model should be able to explain the drug distribution over a large number of tissues as measured from in vivo animal studies, with the eventual goal of data extrapolation to man. In this case, the starting point is a model including each organ and tissue from which measurements are available. If necessary, the number of compartments can be reduced by combining compartments with similar properties. A detailed description of the process of explicit (or formal) combining has been given by Nestorov et al. [19] and Weiss [20].

The principles of PB-PK modelling will be explained using the model of Hunt [6], depicted in Figure 13.4, a PB-PK model suited for evaluation of drug targeting strategies (Sections 13.3.2 and 13.4). For this model, the following set of differential equations describing the drug transport (mass per unit of time) can be written according to mass balance:

where V is the volume of the compartment, C is the drug concentration, Q is the blood (or plasma, whichever is the reference fluid) flow, K is the tissue/blood partition coefficient, CL is the (elimination) clearance, and R is the rate of drug input; subscripts C, R, T and E refer to the central compartment, response (or target) compartment, toxicity compartment, and elimination compartment, respectively (Note: Hunt et al. [6] did not include the partition coefficient K as such, in their equations. Rather, their tissue concentrations refer to a blood or plasma concentration which is in equilibrium with the tissue concentration, equal to the ratio C/K; consequently, their tissue volumes refer to apparent volumes, equal to the product K · V).

13.2.4.3 Compartmental Models Versus Physiologically-based Models

Although compartmental models and physiologically-based models may at first, seem quite different, and are usually treated as two different classes of models, both approaches are actually similar [17].When appropriately defined, probably any PB-PK model can be written as a compartmental model and vice versa. This can be seen by comparing the models in Figures 13.1 and 13.3, and their mathematical descriptions in Eq. 13.1 and 13.5.

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The major difference between both approaches is not in the mathematical or pictorial description, but in the interpretation of the parameters. In compartmental modelling, the starting point is the parameters that do not necessarily have a particular anatomical or physiological meaning. This meaning, however, may become clear after a careful analysis of the data, including measurements in different organs and tissues. On the other hand, PB-PK starts with a model with physiologically meaningful parameters. It should be stated that, when applying a PB-PK model to real data, the identification of the parameters may become a major problem in the interpretation (see Section 13.2.8.4).

13.2.4.4 Principles of Modelling

In both types of models, the quantity or concentration of the drug in various sites of the body is described by mathematical equations quantifying drug administration, drug transport and drug elimination. These mathematical representations are usually in the form of differential equations, which can be solved numerically. In some simple cases an explicit analytical solution of the differential equations can be obtained, thus facilitating the calculations. The numerical procedure of solving the differential equations is more generally applicable, but is complicated by the necessity to find a compromise between accuracy and speed of execution. However, using modern, user-friendly software and fast-performing hardware, this is much less of an issue today (see Section 13.7).

13.2.5 Pharmacodynamic Models

Pharmacodynamic (PD) models are used to describe the relationship between drug concentration and drug effect.An overview of various PD models can be found in the literature [21]. The essential elements will be treated in the following sections.

13.2.5.1 Sigmoid Emax Model

For simplicity, a linear relationship between concentration and effect is often assumed, reducing the problem of PK/PD to the pharmacokinetics. However, the concentration–effect relationship of any drug tends towards a plateau, and a sigmoidal model (sigmoid Emax model or Hill equation) is more appropriate [21–24]:

where E is the drug effect (arbitrary unit; same unit as E0 and Emax), E0 is the drug effect in the absence of drug (typically zero, or baseline effect), Emax is the maximum achievable drug effect, Ce is the drug concentration at the effector site, γ is a dimensionless value, indicating the gradient of the concentration–effect relationship, and EC50 is the drug concentration at which the drug effect is 50% of the maximum effect Emax.

In some cases more complex relationships between drug concentration and effect may occur, for example, when indirect drug effects, threshold concentration, all-or-none effect, time effects, or development of tolerance have to be taken into account.

13.2.5.2 Growth/Kill Models

For antibiotic and anti-tumour drugs, more complex models should be applied, taking into account the growth of microorganisms and tumour cells in the absence and presence of the drug.

The following PD model has been proposed for antibiotic drugs [25,26]:

where N is the number of microorganisms, λ is the microbial growth rate in the absence of drug, Nmax is the maximum number of microorganisms that can be reached, Emax is the maximum achievable killing rate, Ce is the drug concentration at the effector site, γ is a dimensionless value, indicating the gradient of the concentration–effect relationship, and EC50 is the drug concentration at which the killing rate is 50% of its maximum value Emax. If N0 is the initial number of microorganisms at time zero, N reflects the number of microorganisms at time t.

For anti-tumour drugs, Ozawa et al. [27] proposed the following models. For cell cycle phase non-specific drugs (type I drug), the cytotoxic activity depends on the drug exposure, as reflected in the area under the intracellular concentration–time profile (AUC), and can be modelled using the following formula [2,28]:

where N is the number of tumour cells, ks is the cell proliferation rate constant of the tumour cells in the absence of drug, kr is the rate constant of physiological cell degradation, k is the drug-induced cell killing rate constant, fuT is the unbound fraction (unbound concentration divided by total concentration) within the cells, CT is the drug concentration within the cells, and Cmin is the minimum concentration required for the cell killing effect [28].

The cytotoxic activity of cell cycle phase specific drugs (type II drugs) is time-dependent, and is different for cells in the sensitive phase (NS) and in the resistant phase (NR), as described as follows [2,27,28]:

where k1 and k2 are the drug-induced cell killing rate constants for sensitive and resistant cells, respectively, and kSR and kRS are the cell-cycle traverse rate constants from S-phase to R-phase, and from R-phase to S-phase, respectively.

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(Note the typing errors in Nakai et al. [28]; in their equations 16 and 19, the second term kSR should be replaced by kRS; similarly, in equations 17 and 20, the second term kRS should be replaced by kSR).

13.2.5.3 Empirical PK/PD Relationships

Many empirical relationships between PK and PD have been described in the literature. Several of these empirical relationships have been reviewed by Kobayashi et al. [29].

13.2.6 Pharmacokinetic/Pharmacodynamic (PK/PD) Models

PK/PD models are obtained by combining a PK model (Section 13.2.4) and a PD model (Section 13.2.5), allowing the quantification of the relationship between drug administration and drug action.The principles of PK/PD modelling will be dealt with briefly. For a more detailed treatise, some excellent reviews can be found in the literature [21].

Usually, the target compartment of the PK model is the site where the active drug is released. If there is a negligible diffusion barrier between the site of drug release and the site of drug action (effector site), the drug action (Eq. 13.9) is governed by the concentration in the target compartment. In other cases the site of action may be more remote from the site of drug release, and the concentration at both sites may be different due to a diffusion barrier. In such cases, an extra compartment (effect compartment, effector site) can be added to the model as a link between the ‘driving’ compartment (here, the site of drug release) and the drug effect [23,24]:

where Ce is the drug concentration in the effect compartment, C is the concentration in the driving compartment, and ke0 is the transfer rate constant between the effect compartment and the driving compartment.

13.2.7 Simulations

If an appropriate model is selected and the model parameters are known, the time course of the drug concentration in each compartment (PK models) and the drug effect (PK/PD models) can be calculated for any dosing regimen. In addition, the relevant measures of the effectiveness of drug targeting can be calculated (see Section 13.4).

Usually, the most appropriate model and values of model parameters are not known, in which case, the relevant information is obtained from simulations with various models and parameter values, based on reasonable estimates and on previously obtained experimental data. Despite their limitations, such simulations can be helpful in drug design and development, including the prediction and evaluation of drug targeting strategies. Using PK or PK/PD models, the effect of drug targeting can be quantified, taking into account not only

the process modified in order to target the drug, but also the kinetics of the carrier–drug conjugate and the active drug after liberation. Such a modelling process might precede the experimentation process, in order to gain insight into the potential benefit of a drug targeting concept in comparison to traditional administration of the drug.

13.2.8 Data Analysis by Modelling

Data analysis by modelling may be applied for various reasons, for example:

•To condense the data, thus obtaining a model with relatively few model parameters instead of one with a large number of measurements.

•To explore mechanisms involved in the process under investigation (for example, carriermediated transport).

•To make predictions (for example, to predict the dose needed to maintain the concentration of the drug at the target site within a therapeutic window).

13.2.8.1 Model Building

The process of data analysis by modelling implies building a model from measurements, which involves two steps: (a) building the model structure, and (b) assessment of model parameters.

First, the simplest model which includes the minimum number of compartments and model parameters must be defined. For this model, the parameters are estimated from a set of measurements obtained by non-linear regression or curve-fitting (Section 13.2.8.3). The purpose of this process is to find a set of model parameters which best fits the measurements (Section 13.2.8.2). If the goodness-of-fit is acceptable (Section 13.2.8.5), the model can be evaluated by comparison with other models (Section 13.2.8.6).

13.2.8.2 Defining the Objective Function

The first step in ‘curve fitting’ is to define the ‘best fit’. Usually, the criterion for ‘best fit’ is a weighted least-squares criterion, based on statistical grounds [30–34]. Assuming that the errors in the measured concentrations are normally distributed, the best fitting set of parameters is obtaining by minimization of the following objective function, OBJ:

where

n = total number of concentration measurements

Cmeas,i = measured concentration at time point i (i = 1,2,...,n) Ccalc,i = calculated concentration at time point i

σi = standard deviation of the measurement at time point i

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In Eq. 13.15, the squared standard deviations (variances) act as ‘weights’ of the squared residuals.The standard deviations of the measurements are usually not known, and therefore an arbitrary choice is necessary. It should be stressed that this choice may have a large influence of the final ‘best’ set of parameters. The scheme for appropriate weighting and, if appropriate, transformation of data (for example logarithmic transformation to fulfil the requirement of homoscedastic variance) should be based on reasonable assumptions with respect to the error distribution in the data, for example as obtained during validation of the plasma concentration assay. The choice should be checked afterwards, according to the procedures for the evaluation of goodness-of-fit (Section 13.2.8.5).

Usually, the standard deviation of the measurement is dependent on the magnitude of the concentration. The most commonly applied assumption is that the standard deviation is proportional to the concentration, which is either the measured concentration (also referred to as ‘data-based weighting’), or the calculated concentration (model-based weighting).

Many software packages provide only a limited selection of weighting procedures. The most commonly applied weighting procedure is based on the assumption that the standard deviation of the concentration is proportional to the measured concentration. In that case, the objective function may be simplified to:

Alternatively, the following objective function may be used, assuming that the errors in the measured concentrations are log-normally distributed:

Since both Eq. 13.16 and 13.17 assume a constant coefficient of variation of the measurement error, these equations provide similar (but not identical) results.

13.2.8.3 Searching the Best-fitting Set of Parameters

The best-fitting set of parameters can be found by minimization of the objective function (Section 13.2.8.2). This can be performed only by iterative procedures. For this purpose several minimization algorithms can be applied, for example, Simplex, Gauss–Newton, and the Marquardt methods. It is not the aim of this chapter to deal with non-linear curve-fitting extensively. For further reference, excellent papers and books are available [18].

The fitting procedure may be performed by any suitable minimization algorithm. In theory, the final parameter set depends only on the objective criterion (Section 13.2.8.2) and is not dependent on the minimization algorithm, nor on the initial set of parameter values (except for rounding-off errors). However, in practice, the minimization algorithm may fail to reach the minimum of the objective function, and may end in a local minimum. In this respect, minimization algorithms may vary widely. An algorithm which is insensitive to the choice of the initial estimates, is said to be robust, which is a highly desirable property.To lower the risk of convergence to a local minimum of the objective function, the convergence criterion (or stop criterion, for example the relative improvement of the objective function)

chosen should be sufficiently small, and the fitting procedure may be repeated with a different set of initial estimates, or a different minimizing algorithm.

13.2.8.4 Identification of Model Parameters

The procedure to obtain the best fitting set of model parameters (Section 13.2.8.3) can be performed only if each model parameter is uniquely identifiable from the measurements [35–38]. This implies that the same set of model parameters is obtained, irrespective of the initial set (Section 13.2.8.3). In some cases one or more model parameters cannot be identified uniquely, because the measurement data do not contain enough ‘information’ on that particular parameter, for example:

•In the model depicted in Figure 13.1, the parameters CL10 and CL20 cannot be obtained uniquely if only measurement data from compartment 1 are available. There is an infinite number of parameter sets yielding exactly the same concentration profile in compartment 1. Only if certain constraints are imposed (for example, CL20 = 0 or CL20 = CL10), can the model parameters be identified uniquely.

•In the case of Michaelis–Menten kinetics (Eq. 13.4), Vmaxxy and Kmxy cannot be assessed uniquely if the concentration Cx is far below the value of Kmxy; in that case, Eq. 13.4 reduces to Eq. 13.3, and only one parameter CLxy, or the ratio Vmaxxy/Kmxy can be calculated uniquely.

•In the model shown in Figure 13.1, if CL12 is very large compared to CL10, and if no data shortly after administration of a bolus dose are available, the model would behave as a single compartment model, and CL12 will not identifiable; also, only the sum of the volumes rather than both volumes separately can be assessed.

The problem of identification grows rapidly with increasing complexity of the model. In some cases this problem can be solved by an appropriate experimental design. As a general rule, the problem is reduced if the concentrations in compartments other than the central plasma compartment can be measured. Also, simultaneous measurement of drug and drug–carrier conjugate or pro-drug is a condition for identification of the models described in Section 13.3.

Jacquez and Perry [37] developed the program IDENT to investigate the identification of model parameters. In most cases, problems of identification can be detected by inspection of the standard errors of the model parameters (the standard error of a model parameter is a measure of the credibility of the parameter value, which is provided by the most fitting programs). A high standard error (for example, more than 50% of the parameter value) indicates that the parameter value cannot be assessed from the data, most likely due to an identification problem. In that case, the parameter value itself is meaningless, and thus the parameter set should be discarded (see Section 13.2.8.5).

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13.2.8.5 Goodness-of-Fit

After fitting the parameters of a model to a set of measurement data, criteria for the good- ness-of-fit are required. There will always be some differences between the measured data and the values calculated from the model. These differences may be due to the following causes:

•Measurement errors in the data, for example, inevitable analytical errors implicit in the chosen analytical method. In general, measurement errors are random errors, and their order of magnitude may be known from the precision of the assay, as assessed during the validation of the assay. If the magnitude of the measurement errors is comparable to the precision of the assay, the goodness-of-fit is acceptable. The possibility of problems in the case of measurements close to the detection limit of the assay should be taken into account. In this case, the relative errors in the analysis may be significantly larger than over the usual range.

•Model mis-specification. If an inappropriate model is chosen (for example, a model with too small a number of compartments, or an incorrect structure), it will not be able to describe the measurements adequately, resulting in systematic deviations between the measurements (for example, plasma or tissue concentrations) and the values calculated from the model. Such systematic deviation can be detected by the visual methods described below.

•Other errors in the procedure, such as failure to distinguish between carrier-bound and unbound drug, as well as errors in dosing, deviations in the time of measurement, incorrect sampling procedure, exchange of samples, mistakes in dilution during sample treatment, and so forth. These types of error are the most problematic, and no general solution can be given.

There are several methods available for the assessment of goodness-of-fit, however, there are no exact and objective criteria for its evaluation.This is due to the following: (1) goodness-of- fit is not a single property, and cannot be expressed in a single value, and (2) numerical measures of goodness-of-fit do no have an absolute meaning. Therefore there is a dependence on somewhat subjective criteria.To ensure maximal objectivity, the criteria for accepting a set of model parameters obtained by the fitting procedure as a valid result should be defined explicitly before the analysis is initiated. In practice, however, this condition is hardly applicable during the development of new drug targeting preparations, taking into account the complexity of the modelling procedure.

The following criteria could be used to ensure an acceptable goodness-of-fit:

•Visual inspection of the observed and calculated data should not reveal any significant lack of fit.

•Residuals (difference between observed and calculated data) or normalized residuals (residuals divided by the corresponding standard deviation) should be scattered randomly around zero, by visual inspection.

•Normalized residuals should be neither diverging nor converging when plotted against time or plotted against (logarithm of) concentration, by visual inspection.

•Residuals should not be serially correlated, as identified by visual inspection or by an appropriate statistical test (for example a Run’s test).

•The standard error of each relevant parameter should be lower than a predefined value (for example 50% of the parameter value). High standard errors may reflect problems in the identification (see Section 13.2.8.4).

•In the case of any of the calculated pharmacokinetic parameters being seen as physiologically unfeasible, the analysis should be interpreted with care, and should not be presented without comments.

•Outlying data points should be dealt with explicitly, and should not be discarded unless felt to be physiologically impossible. The impact of elimination of the outlying points on the parameter estimates should be investigated.

Non-compliance with one or more of these criteria may indicate that an inappropriate model or an inappropriate weighting scheme was chosen.

13.2.8.6 Model Selection

There may be more than one plausible model structure that can be used to describe the data. In that case, any plausible model is analysed in a similar way. If the goodness-of-fit of more than one model is acceptable (see Section 13.2.8.5), a procedure for selecting the ‘best’ model is required.

It is common practice to compare the results of different models, each yielding an acceptable goodness-of-fit, according to the following procedure. First, the models are classified hierarchically in a tree structure.The more complex models are considered as extensions of the simpler models, by adding extra parameters, for example, an extra compartment, a extra degradation step, or a time lag. It can be said that the simpler model is a special case of the more complex model, for example because one or more parameters have a fixed value (in general a zero value). Then, starting with the simplest model (see Section 13.2.8.1), the models are compared in pairs according to their hierarchical relationship. Such a comparison can be based on statistical criteria, for example [39]:

(a) An F-test by which the following value is calculated:

where N is the number of measurements, P is the number of model parameters, and WRSS is the weighted residual sum of squares (see Section 13.2.8.2); the subscript s refers to the simpler model and the subscript c to the more complex model.

If the value F exceeds the tabulated value of Fischer’s F-distribution for (Pc – Ps) and (N – Pc) degrees of freedom, and a confidence level of (usually) 95% (α = 0.05), then the complex model fits significantly better to the data than the simpler model. If not, the ‘parsimony’ principle dictates that the simpler model should be accepted as the ‘best’ model.

(b) Akaike Information Criterion (AIC). For each model the AIC is calculated according to the following equation:

The model with the lowest AIC value is accepted as the ‘best’ model.