PHARMACODYNAMIC PARAMETERS: INFLUENCE ON DOSE-RESPONSE RELATIONSHIP AND ESTIMATION

Overall drug response is controlled by pharmacodynamic (PD) phase and pharmacokinetic (PK) phase. Over the last twenty years, much greater emphasis has been placed in PK phase because its outcome is much easier to measure and model compared to that of PD. In fact, PD and its parameters play an important role in controlling drug response. This document consists of three studies. The first study demonstrates through computer simulations using STELLA (High Performance System) the manner in which the main PD parameters influence the dose response relationship. A one compartment PK model linked to a sigmoid En.ax model through an effect compartment was used. The results show that as the sigmoidicity constant increases the duration of effect gets shorter. This parameter also impacts the magnitude of the response where its effect depends on the drug concentration and its ratio to the concentration at 50% of the maximum effect (EC50) . Also, it was found that as the ECso increases, the response from a given concentration gets smaller and the duration of effect gets shorter. When an effect compartment is necessary to model drug action, the effect compartment characteristics become more prominent as keo decreases. Thus the delay in response gets larger, the magnitude of response from a given dose gets smaller and the duration of action gets longer as keo decreases. The second study was designed to investigate the effect of different sources of variability, dose, PK and PD parameters, on drug response through computer simulations using STELLA. The different sources of variability were studied separately and in combination using a one-compartment PK model linked to sigmoid Emax and linear PD models. The results show that in presence of similar amount of variability, the response is much more sensitive to variability in PD parameters than variability in PK parameters. It is concluded that variability in PD parameters are clinically important and must be taken into account in order to use the drug effectively and safely. The third study was designed to investigate the optimum sampling design for a PD modeling study through computer simulation using an inhibitory Sigmoid Emax model in NONMEM (Non-linear Mixed Effect Modeling). The bias and precision of parameter estimates were used to judge the performance of various studied designs. The effects of population size and level of inter-individual variabi lity were further studied using the most optimum design. The experimental design for the determination of the equilibrium rate constant associated with an effect compartment was also studied. The most optimum design for determination of PD parameters in the absence of an effect compartment was found to be the one with the following sampling windows: 0.1-0.5, 0.5-1 and 1-2 EC50 units. However, in the presence of high inter-individual variability (60%) estimates of variability parameters, using the most optimum design, were biased and imprecise. More precise estimates of the parameters were obtained with a larger population. The most optimum design for the equilibrium rate constant was found to be the one in which two samples were taken per individual, but it gave poor estimates of the variability parameter.

. Variability (%CV) in PK parameters and its influence on plasma concentration    The results show the manner in which the various PD parameters affect the magnitude and duration of drug response. As n increases the duration of effect gets shorter. Thus the effect dissipates faster at higher values of n. This parameter also impacts the magnitude of the response but the effect of n depends on the drug concentration and its ratio to the EC 50 . If this ratio is greater than one, the drug response gets larger as n increases, but if this ratio is less than one, the drug response gets smaller as n increases. When this ratio equals one, i.e. when the concentration equals to ECso, the response is 50% E.nax and independent of the value of n. As predicted, as the EC50, which reflects the potency of drug, increases, the potency or sensitivity to the drug decreases and the response from a given concentration gets smal ler. Also the effect of the drug decays more rapidly when drug concentrations are low relative to the ECso. Thus as the EC 5 o increases, the effect decays more rapidly and the duration of effect associated with a given effect gets shorter.
When an effect compartment is necessary to model drug action, the effect compartment characteristics become more prominent as k.o decreases. Thus the delay in response gets larger, the magnitude of response from a given dose gets smaller and the duration of action gets longer as k.o decreases. The study demonstrates that the design of rational dosing regimens for clinical therapeutics cannot be performed with knowledge of PK alone. The true optimization of dosage regimens must also take into consideration the PD parameters of the drug. Thus, pharmacological response is usually related to plasma concentration (Cp). This approach appears satisfactory when the drug response is direct, receptor site rapidly equilibrates with plasma and the receptor interaction and response occurs rapidly.
However, in some situations, there is a delay between rise and fall in Cp and rise and fall in response possibly due to a distribution delay. This may necessitate the link between the pharmacokinetic (PK) model and the PD model, using for example an effect compartment.
As early as 1878, Langley suggested that the law of mass action probably governed the drug action and Clark extensively developed this theory in the 1920s < 2 >. According to classical receptor theory, it is assumed that the drug action is proportional to the fraction 3 of receptors occupied. And that maximum effect results when all receptors are occupied C 3 .•> . Using this assumption, the relationship between the drug effect and its concentration is hyperbolic in shape. This hyperbolic function is used to describe the concentrationeffect relationship for many drugs and is now known as the Emax model. The model was expanded to incorporate the possibility that more than one drug molecule may bind to each receptor. This expanded model is known as the sigmoid Emax model and will be discussed in detail later. Clark also used the advantage of the logarithmic transformation of the sigmoid Emax model equation to determine the PD parameters by linear regression method, which is later, modified to the logarithmic model C4>.
The PK of most drugs are described as linear. Thus drug distribution and elimination are generally first order processes, and under the influence of elimination, Cp falls monoexponentially and the half-life is constant. As dictated by receptor theory, the PD of most drugs however are most often non-linear and as discussed above the concentration-effect relationship may be hyperbolic or sigmoid. Thus, as Cp decays, the effect will not necessarily fall in parallel. Thus, the time for the effect to fall by 50% wiII not necessarily be equal to the PK half-life. In consequence, clinically useful dosing guidelines cannot be based on PK models alone but must incorporate a PD model in order to consider nonlinearity in concentration-effect relationship. A delay caused by the time for tbe drug to distribute from the plasma to its site of action may further limit the value of using PK to develop dosing guidelines < 5 >.

Logarithmic model:
The logarithmic model relates the drug response (E) to the logarithmic function of drug concentration at the site of action (C). E = S (log C) + A

I.I
Where: S is a slope parameter, and A is the intercept. This model has the advantage that it linearizes the concentration effect relationship predicted by the more complex Err.ax model Consequently, the Err.ax or sigmoid Emax model, which can describe the whole concentration-effect relationship, should be used whenever possible.

Emu model:
The Emax model is based on receptor theory where the effect is assumed to be proportional to the concentration or the fraction of receptors occupied < 4 >. The Emax model equation is as follows: Where: E max is the maximum effect (efficacy) and EC 50 is the potency, which reflects the sensitivity of organ or tissue to the drug ci q The EC 5 o is the concentration at 50% of the maximum effect.
This model predicts a hyperbolic concentration-effect relationship (Figure I.I). At low concentrations, well below the EC 50 , the receptors are less saturated and the pharmacological action approximates a first order process and the concentration-effect relationship is linear. At higher drug concentrations, as receptor saturation is approached this relationship starts to be non-linear and the law of diminishing returns is observed.
This model has many advantages such as: it incorporates a maximwn effect and incorporates no effect at zero concentration. Additionally, it can be modified to incorporate the situation where more than one drug molecule binds to each receptor (the sigmoid Emax model). inhibition of exercise heart rate in 9 subjects. They found that at concentration well below the EC5o, there was a linear relationship between effect and concentrations.

Sigmoid Emax model:
The sigmoid Emax model is a modification of the Emax model , which accounts for the probability that more than one drug molecule binds to each receptor, using the term n, sigmoidicity. The sigmoid Emax model is derived from the Hill equation < 14 >_ It has been proposed as a useful model to describe the in-vivo relationship between dose/concentration and continuous pharmacological effect for many drugs < 15 >_ The 8 sigmoid Emax is the most widely used today because it solves all the limitations of previous described models. This model predicts a sigmoidal concentration-effect relationship (Figure 1.2). The sigmoid Emax model equation is as follows: The sigmoid Emax model has three PD parameters that control the drug response. These parameters are the efficacy CEmax). the potency (EC5o) and the sigmoidicity constant (n).
As with Emax model, the efficacy and potency are the same as Emax model parameters. The efficacy represents the maximum effe~t that occurs when all the receptors are occupied.
Sometimes the efficacy is assigned a value of I 00%. The EC 5 o is the concentration at 50% of the maximum effect and n is the number of drug molecules bound to each receptor and it determines the sigmoid shape of the concentration-effect relationship. For example if n = l, concentration-effect relationship will be hyperbolic (the Emax model), but when n is greater than l the curve becomes sigmoid with a steeper slope in its central region as shown in Figure 1.2. If n is less than I, the slope is steeper at low concentration and shallower at high concentration < 1 >.
Although this model is based on receptor theory it cannot be assumed, even if the concentration-effect data fits the model, that the underlying pharmacological process is truly described by the model. It must be kept in mind that data drive the model. For example, sigmoidicity has been found to be a non-integer in some cases even though the receptor theory would predict it to be always an integer <  analgesia in children where they found that the PD of acetaminophen could be described using a fractional sigmoid Emax model that is with n is less than I (0.54).
In some cases (for example, heart rate and blood pressure), a baseline response is incorporated in the model. The modeled response may be added or subtracted from the baseline effect depending upon whether the response is stimulatory or inhibitory. If the drug has an inhibitory effect on a physiologic response, such as lowering of the exercise heart rate with a beta-blocker, the response is subtracted from the baseline effect. The baseline effect (Eo) is the response at zero drug concentration and it is measured in absence of drug or during placebo administration. It is added to or subtracted from the model equation as follows:    influx of potassium ions, resulting in the temporary reduction of plasma potassium levels (40)

1.3.l. Introduction:
The magnitude and duration of response produced by a given dose of a drug is a function of the sequential PK and PD phases. Consequently the parameters used to model each of the phases are critical in controlling overall drug response and the design of suitable dosage regimens to produce optimum outcomes. Over the last 25 years PK principles have been universally applied to target specific concentrations or concentration ranges.
However, owing to the paucity of information on the PD characteristics of drugs, this phase has been simplified and condensed down to the concept of the therapeutic range.
The limitations of this approach combined with the increase in number of sophisticated PD models published in literature are now permitting alternative, more thorough approaches to the consideration of PD phase in dose optimization < 3 >_ As these models and their associated parameters become more integrated in clinical practice, it becomes critical that practitioners fully appreciate the manner in which PD parameters impact response and thus dosage optimization in much the same way that the importance of the PK parameters, clearance, volume of distribution and half-life, are known.
This study was designed to demonstrate through computer simulations the impact of the major PD parameters, EC 50 , n and k.:o on the magnitude and duration of drug response.
The magnitude of response was evaluated by comparing the maximum observed effect (MOE) from a series of doses. The duration of effect was assessed by measuring the time for the effect to fall by 50% (effect half-life) at different concentrations. Simulations were performed using a one-compartment PK model linked to a sigmoid Emax model with and without an effect compartment. The goal of the study was to provide practitioners and pharmaceutical scientists with information about the PD parameters comparable to that of major PK parameters of clearance, volume of distribution and half-life.

Methods:
An integrated PK-PD model was constructed in STELLA (High Performance Systems, Inc, Hanover, NH). Response data were generated using the sigmoid Emax model. Simulations using different values of n (0.5, 1, 2, 3 & 5) and EC 50 (0.5 , 1, 2, 3 & 5) were performed in order to evaluate the influence of these parameters on the magnitude and duration of response.
The response data were generated by numerical integration every 0.01 hr.

I.3.2.3. Effect Compartment Model:
In addition to a direct link, the PD model was also linked to the PK model through an effect compartment. Illustration of the hypothetical effect compartment is shown in The effect of different values of n on the response is shown in Table I. I, Figures 1.6 and I. 7. As n increases, the slope of the concentration-response curve gets steeper (Figure 1.6). The parameter n is often referred to as the steepness parameter. As n increases the concentration-response curve becomes steeper. The impact of thi s on the MOE from a specific dose depends on the ratio of the concentration to the EC 50 . If this ratio is less than one i.e. at low drug concentration, the MOE from a given dose gets smaller as n increases. In contrast, at ratio greater than one, the MOE gets larger with increase in n.
Note in Figure 1.6, when n is small, very large concentration would be necessary to achieve maximum response. When the ratio is equal to one, i.e. when concentration = EC 50 , the effect is 50% Emax irrespective of the value of n. The effect of n on the duration of effect is shown in Figure 1.7, where again it can be seen that as n increases, the slope of the effect-time curve gets steeper. Thus, as n increases the response dissipates faster and the duration of response decreases. Thus, n affects both the magnitude and duration of response from a given dose.  50 shift the effect-concentration curve along the x-axis (Figure 1.8). As ECso increases, the curve is shifted to the right.
Thus as EC 50 increases, the response from a given concentration decreases. The effecttime relationship after a standard single dose is also considered (Figure 1.9), it can be seen that as EC 50 increases, the MOE from a given dose gets smaller and the rate of fa ll of effect is more rapid with higher values of EC 50 . As mentioned earlier, at low concentration relative to ECso, the fall in effect with time approximates first order process. At high concentration relati ve to ECso however, when the receptors display a 20 greater degree of saturation the rate of fall of effect with time is less than for a linear first order process.

Single IV dose with an effect compartment:
The influences of k.o, n and EC50 on drug response were studied when the integrated PK/PD model incorporated an effect compartment. The effects of n and EC 50 were studied at two values of the keo (0.2 and I et112·1).

Effect of k.o on the drug response:
The equilibrium rate constant (k.o) quantifies tbe delay between plasma concentrations and pharmacological response caused by the time required for drug distribution to its site of action. Compared to models where the distribution process proceeds essentially instantaneously, tbe MOE from a given dose is less but the duration of action is longer The time for the peak effect gets longer as keo decreases. The effects of n and EC 50 on drug response in case of multiple dosing were studied in terms of MOE. The drug response at initiation of therapy is less than that at steady state, because of the accumulation of drug that occurs during the build up to steady state.

Effect of n on the response:
The effect of n on drug response using multiple IV doses is shown in The effect of different keo on the drug response is shown in Table I. I 0. The EC50 and n were kept at I and k1c was kept at 0.0 I et1n" 1 • As expected, there was no initial response after dose administration. The delay gets longer as keo decreases. Compared to single dose, the magnitude of maximum response is slightly affected by the decrease in kco. The fall in drug response is much longer than that in absence of the effect compartment as shown in figure I.I I, where the slope of the timeresponse relationship in presence of the effect compartment is much shallower than that in absence of the effect compartment.

Effect of n on the response in presence of an effect compartment:
The effect of non the response in presence of the effect compartment is shown in Tables I.I I and 1.12 for keo of0.2 and I et1n· 1 , respectively. As with single dose, the delay gets longer as n increases and the effect of n on drug response depends on drug concentration and its relation to EC 50 .

Effect of EC50 on the response in presence of an effect compartment:
The effect of different values of ECso on drug response in presence of an effect compartment is shown in Tables 1.13and1.14 for kco of0.2 and I et112• 1 , respectively. As with single dose, the MOE from given doses gets larger as EC 50 decreases with all doses studied. The influence of the effect compartment appeared to be more prominent on the impact of EC5o on drug response than that of n.

DISCUSSION:
In recognition of the importance of PD in determining the response achieved by a given dose of a drug and by extension, the determination of optimum dosage regimen to achieve a desired response, PD studies are receiving increased emphasis in various branches of pharmaceutical science. The importance of these studies in optimizing drug dosage during drug development was recognized and stressed by Reigner et al < 41 >.
Recently PD have been used together with PK studies to optimize clinical trial design.
More recently, the use of integrated PK/PD models has been proposed as a superior method of individualizing doses of drugs in clinical use to achieve the desired therapeutic 25 effect. Traditionally the process of therapeutic drug monitoring aims to have the Cp within the therapeutic range, which is an empirically chosen range in which the average person would experience optimum response. However, therapeutic drug monitoring uses the passive concept of monitoring and fails to explicitly take drug effects into account < 3 >.
As PD models aod their associated parameters become more integrated in clinical practice, it becomes increasingly important that practitioners fully appreciate aod understand the relevance of the various PD parameters to drug response in the same way that PK parameters are well related to Cp. For example, it is generally accepted that

26
The study of the effect of n on the drug response shows that the effect depends on the ratio between drug concentration and EC5o. If this ratio is more than 1, the drug response from a given dose gets larger as n increases as shown with single and multiple doses that give MOE of90 % at n = I in Tables I. I and l.8 respectively. At concentrations less than EC 50 , the response from a given dose gets lower as n increases as shown with single dose and multiple doses that give MOE of 25 % at n = I in Table I. I and Table l.8, respectively. When this ratio equals I, i.e. at 50% effect, the MOE is not influenced by n, because the effect is independent of n when the concentration at the site of action (C) is equal to EC 50 . The independence of the effect on n at this concentration can be proven below: At 50% effect, drug concentration= EC5o.
The value of n varies from drug to drug and from individual to individual. It was found that n for tocainide varied from 2.3 to 20 ci si. The study of the effect of theophylline on eosinopenia and hypokalemia showed that n varied from 3.9 to 8.5 for the eosinopenic effect and 4.2 to 9.4 for the hypokalemic effect < 19 l. Also, the study of the PD of remi fentanil < 20 l showed that n varied from 1.2 to 3.9. In addition, the study of preoperative PD of acetaminophen analgesia in children showed that n varied from 0.3 l to 0.77 < 21 l. Individuals with low n values will experience a greater effect and longer duration of action, at low concentration than individuals with higher n values.
Conversely, at higher concentration, when the receptors are approaching saturation, individuals with low n values will experience lesser response but still experience longer duration of action than those with higher n values.
The study of the effect ofEC 50 on the drug response shows that EC 50 or potency shifts the concentration-response relationship up and down the x-axis as shown in Figure I.8. As ECso increases, the drug gets less potent and a smaller response is achieved from a given dose as shown in Table 1.2 and Table 1.9 with single and multiple doses, respectively.
Likewise as the potency decreases, the EC 5 o increases and the duration of action of the drug gets shorter. This is very important clinically, because when EC 50 increases or decreases it will result in a lower or higher drug response respectively.
Jonkers et al C 42 > studied the changes over time in the concentration-effect relationship of the beta 2-adrenoceptor-agonist, terbutaline. A sigmoid Emax model was used to relate drug concentrations to the response. After one week on oral terbutaline the concentrationeffect relationship was shifted to the right with a higher EC 50 of terbutaline, which resulted in a higher drug concentration to produce a given response. Minto et al C 20 > studied the influence of age on the PD of remifentanil where they found that age was a significant covariate of EC 5 o. The ECso decreased by approximately 50% for the age range studied (20-85 years) as the individual gets older, which resulted in less drug concentration was required to produce the drug response in elderly people who were more sensiti ve to the effect of the drug.

28
The equilibrium rate constant (k.o) is associated with effect compartment, which often added to account for a delay between the rise and fall in Cp and the rise and fall in response. As the value of k.o gets smaller, the time to reach the equilibrium between the drug concentration in plasma and its concentration at the site of action gets longer and the influence of the effect compartment becomes more prominent as the time to reach the equilibrium is very long. This study demonstrated that as k.o decreases the MOE from a given dose gets smaller and the duration of response gets longer. This is because the smaller the value of k.o, the slower the drug distribution to the effect site and the lower the concentration at the site of action from a given dose which results in a lower MOE.
The duration of action gets longer which may be due to the slower redistribution of the The manner in which the values of n and EC 50 affect drug response in presence of an effect compartment are similar to that as in absence of the effect compartment with single and multiple doses.

Conclusion:
The PD parameters namely sigmoidicity constant, potency and equilibrium rate constant, influence the magnitude and duration of drug response. It has to be born in mind that the overall drug action consists of two phases, PK phase and PD phase. Thus, PD parameters should be taken into consideration in evaluating the drug response. Prospective implementation of large-scale population PD evaluation is feasible in early drug development and this approach generates clinically relevant findings < 46 > . The expanded use of PK/PD-modeling is found to be highly beneficial for drug development as well as applied pharmacotherapy and will most likely improve the current state of applied      The multiple doses used gave MOE of 90% at steady state at n and EC50 of 1 in absence of the effect compartment . Theophylline is widely used in treatment of bronchial asthma. It has a narrow therapeutic range and displays wide variability in its pharrnacokinetic (PK) and pharrnacodynamic (PD) parameters. As a result, the PK and PD of theophylline have been extensively studied and published in the literature.
In this study, simulated data were used to investigate the effect of variability arising from the dose and the intra-individual variability in PK parameters (rate of absorption and clearance) and several PD parameters on theophylline plasma concentration and response. A one compartment PK model with zero-input linked to linear and sigmoid Emax models was used to simulate data. For each set of model parameters, the response was measured every 12 hours over a I 0-day period. Hundred replications were performed giving a total of 2000 responses. The influence of PK/PD variability on theophylline response was studied at low, moderate and high levels of intra-individual variability. The effect of variability in dose, PK and PD parameters was studied separately and in combination and their impacts on theophylline response were assessed by estimating the % coefficient of variation in the response using Excel 2000.

66
The drug response was found to be more sensitive to variability in PD parameters than to variability in PK parameters or dose. The drug response was more sensitive to the changes in the dose and PK parameters when the sigmoid Emax model was used compared to when the linear model was used. In conclusion, variability in PD parameters is clinically important and must be taken into account in order to use the drug effectively and safely.

INTRODUCTION:
Pharmacokinetics (PK) is the study of the relationship between the dose of a drug and the manner in which its plasma concentrations change over time. Models for PK are used to provide a mathematical representation of this relationship and relate the independent variables of time and dose to the dependent variable, plasma concentration. In a one compartment model the value of plasma concentration is controlled by three PK parameters, clearance (CL), volume of distribution and bioavailability factor.
Pharmacodynarnics (PD) is the study of the biological effects resulting from the interaction between drugs and biological systems. Models for PD are used to provide a simplified description of the drug action and relate the independent variable of drug concentration at site of action to the dependent variable of drug response.
A thorough understanding of PK and PD is the scientific foundation of clinical therapeutics Cl>. Variability in the PK and PD parameters may be small, moderate or large depending on the drug and the pathological state of the patient. Variability in PK and PD parameters will lead to a variation in the drug response. Drugs used for chronic diseases with a proven PK-PD relationship, a small therapeutic range, large PK/PD variability and severe adverse effects are likely to be good candidates to study the impact of different sources of variability on drug response. An example · of this category is theophylline, which has been widely used in the treatment of bronchial asthma.
Theophylline has a wide variability in its CL, which controls the steady state plasma concentration and is critical for determining the maintenance dose. Theophylline's CL is affected by many factors such as diet, disease state and smoking. The resulting variation in CL may lead to substantial variation in plasma concentration and drug response.
Theophylline is mainly eliminated by hepatic metabolism, mediated by cytochrome P450 liver enzymes < 2 > (CYP1A2 and 3A4). Diet influences the metabolism oftheophylline for example; high-protein, low carbohydrate diets generally metabolize theophylline more rapidly, presumably because the diet induces hepatic enzymes (3). Charcoal broiling induces CYP1A2, so it increases theophylline's CL < 4 >. Cigarette smoking increases theophylline's CL by 1.5 to 2 times that of non-smokers < 5 • 6 > . It was found that the effect of smoking appear to last several months after the cigarettes have been discontinued (7).
Also some diseases affect theophylline's CL. Congestive heart failure reduces theophylline's CL to about 40% of normal 18 >. Hepatic cirrhosis can significantly reduce theophylline's CL. Also severe pulmonary disease significantly reduces theophylline's CL 1 6 > . It has been found that CL displays 20% < 9 >, 25% {lo>, 30% (IO) coefficient of variation (% CV) in patients with respiratory diseases. There is not much variability in theophylline's volume of distribution and it is usually kept at a constant value of 0.5 L/Kg (6) The rate of drug absorption differs among various slow-release formulations < 11 • 12 > and occasionally between lots ( I I } of the same brand. Differences in rates of absorption may be clinically important < 13 >. Also, dose to dose variation in plasma concentration have been observed for some theophylline products, probably as a result of intra-individual 69 variability in gastrointestinal function < 1 •>. Food may decrease the rate of absorption of theophylline from many sustained release products < 15 • 16 >. This is probably a result of delayed gastric emptying rate where the drug is held in the stomach fo r longer time before entering the alkaline medium of small intestine where dissolution is more rapid Food also may cause dumping of large amounts oftheophylline from some sustained release products < 18 • 19 >. This may be due to dissolution of the sustained release film coat rapidly at the pH of small intestine after a meal (pH 7.4i 18 >. Theophylline is an example of drug that has a narrow therapeutic range. Traditionally a range of I 0-20 mg/L has been used. However, more recently concentrations at the lower end of the range have been advocated since there is significant and serious adverse effects are more common at higher theophylline concentrations and recent studies indicate that 10 mg/L is as effective as 20 mg/L < 20 >. The effect of theophylline on bronchodilation can be measured in terms of forced vital capacity (FVC) and peak expiratory flow rate (PEFR). However, the FVC is usually used because it assesses ventilatory response as it closely reflects the patency of small airways and it represents the spirometric index, which is the most reproducible measure. A linear model has been used to describe the effect of theophylline on FVC < 21 >. A sigmoid Emax model has been used to describe theophylline's effect on PEFR < 22 >.
The linear model is derived from the Emax model , since when concentrations are small relative to the potency (EC 50 ), the Emax model will collapse into a linear model in which the effect [E (L)] is proportional to steady state plasma concentration (Cpss)-The linear model, will predict no effect when concentrations are zero but its major limitation is that it cannot predict a maximum response. This model can be modified to evaluate data with baseline (Eo) as follows: Where m is the slope parameter, which will approach the value of the ratio of efficacy to potency (EmaxlECso). The baseline effect is tbe response at zero drug concentration and it is measured in absence of the drug or during placebo administration. The concentrationeffect relationship using this model is linear. The advantage of linear model is that it can be used for some relatively toxic drugs when the Emax cannot be approached < 21 The sigmoid Emax model is a modification of the Emax model tbat accommodates the probability that more than one drug molecule, binds to each receptor by using the term sigmoidicity. According to sigmoid Emax model , the effect is related to drug concentration (C) in the following manner:

EC~0 +CY
Where Emax is the efficacy, EC5o is the potency, which reflects the sensitivity of organ or tissue to the drug, and y is the sigmoidicity constant, which is the number of drug molecules bound to each receptor. Using this model, the concentration-effect relationship is sigmoidal in shape. Although this model is based on receptor theory it cannot be assumed that it is the basis of drug action. Sigmoidicity has been found to be a non-integer in some cases even though the receptor theory would predict it to be always an integer < 9 • 22 • 24 -26 > . Also, this model can incorporate a baseline response (Eo). Examples of the use of the sigmoid Emax model in the literature include the study of PD of theophylline side effects (eosinopenia and hypokalemia) < 9 >, the study of PD of theophylline bronchodilation in asthmatic patients < 22 >, the study of the PD of tocainide on suppression of ventricular ectopic depolarizations in patients < 24 >, the study of the influence of age and gender on the PD of remifentanil < 25 >, the study of the preoperative PD of acetaminophen analgesia in children < 26 > and the study of the PD of intravenous diltiazem in patients with atrial fibrillation or atrial flutter < 2 7l.
It has been found that individuals can vary with respect to the baseline, maximum response, potency and the slope of the concentration-effect relationship. These are the main determinants of PD variability < 23 >_ In the past, it was assumed that PK variability is primarily responsible for quantitative differences in drug response < 29 >_ This is probably because PD studies in human were rare until the last couple of decades > is larger than that of PK < 9 • 10 >.
In this work, theophylline was used as a model drug to study the impact of different sources of variability, namely dose in term of content uniformity, PK parameters and PD parameters, on its response. The relative impact of different sources of variability on theophylline response using published models for the linear < 21 > and the sigmoid En.ax < 22 > models was compared.

ll.3 METHODS:
An integrated PK-PD model of theophylline was constructed in STELLA (High Performance Systems, Inc, Hanover, NH) (3Sl .

Pharmacokinetic Model:
Initially, a one-compartment PK model, with first order gastrointestinal absorption (ka = 0.09 hr · 1 ) < 36 > was used . Subsequently, a zero order absorption in which the dose was assumed to be absorbed at a constant rate over 12 hours was used. For the study of rate of drug absorption, a zero order input with an infusion rate was used in which the infusion time was 7.7 hours.
The model was based on an average 70 Kg individual. The volume of distribution was 35 L based on 0.5 L/Kg < 6 >, CL was 2.8 L/hr, based on 0.04 L/kg/hr < 9 • 10 >. The drug was assumed to undergo first order elimination; thus its elimination rate constant (k , ) = 0.08 hr · 1 and its half-life was 8.7 hours. Simulation was based on a 400 mg oral dose of theophylline administered every 12 hours. This dosage regimen gave steady state plasma concentration of 11.9 mg/L.

Pharmacodynamic Model:
A. The linear model: The Thus, all drug levels and responses were constant. The different sources of variability were studied separately and in combination.
The proportional error (constant coefficient of variation) model was used to describe the variability in the dose, PK and PD parameters as follows:

II .4
Where Pi is the dose or PK or PD parameter at time j, Pm is the mean dose or population mean value of the PK or PD parameter and Epj is a normally distributed random variable with an average value of 0. The standard deviation of Epj represents the CV for variability in the dose, PK or PD parameter.
For the dose, the% CV was set at 3, 6 or 10%. For intra-individual PK variability,% CV in CL was set at 5, I 0, 15, 20 ( 9 >, 25  The response was measured every 12 hours over a IO-day period. Hundred replications were performed giving a total of 2000 responses. The plasma concentration and drug response just before the administration of a dose were tabulated and the coefficient of variation of the 2000 values was determined.

RESULTS:
A One compartment PK model with first order absorption, in absence of the variability model was used to study theophylline's steady state plasma concentrations at different doses (250, 300, 350, 400, 450, 500 mg) every 12 hours. A dose of 400 mg of sustained release theophylline every 12 hours was selected for the study since in the model used, it gave a steady state plasma concentration of 11.9 mg/L, which is near the lower end of the therapeutic range (10-20 mg/L) and is.a common therapeutic target. Using the zero input model, a dosage regimen of 400 mg of theophylline every 12 hours gave the same plasma concentration and drug response as that of the first order absorption model.
The variability in dose, PK and PD parameters was translated to theophylline response in terms of E (L) for FVC and E (S) for PEFR with the linear and sigmoid Emax models, respectively.

Variability in the Dose:
Variability in the dose resulted in essentially equivalent amounts of variability in plasma concentration and E (S), but E (L) was less affected. For example, 6% variability in the dose gave 5.78% and 5.48% CV in plasma concentration and E (S), respectively, while % CV in E (L) was 1.29. The effect of variability in dose on plasma concentration and drug response is shown in Table II.I.

Variability in PK parameters:
Variability in CL resulted in essentially equivalent amounts of variability in plasma concentration and E (S), but E (L) was less affected.

II.4.4. Combined PK-PD Variability:
After the effect of variability of each parameter was studied separately, combinations of PK and PD variability were studied. The effect of combined PK/PD variability on plasma concentration and drug response is shown in Table ll Variability in PK parameters resulted in almost same % CV in plasma concentration, which was not affected at all by any changes in PD parameters. High level of intraindividual variability (45% CV) in the rate of drug absorption could not be studied with large variability in CL/PD because the simulation was unsuccessfully terminated possibly due to division by zero or a value that has become too large to represent < 35 >. Using the sigmoid Emax model , drug response was much more sensitive to the changes in the PK 80 and PD parameters than that of the linear model. The drug response from the two models was much more affected by the changes in PD parameters than that of PK parameters.

Combination of all sources of variability (Dose, PK & PD):
Finally, a combination of all sources of variability (dose, PK and PD) was studied. The effect of combined variability in dose PK and PD parameters on plasma concentration and drug response is shown in Table II.6.
The addition of variability in the dose to the combined PK/PD variability had very small and negligible effects on the variability in plasma concentration and drug response, respectively .

II.5. DISCUSSION:
Computer simulations have been successfully applied in support of clinical drug development for predicting clinical outcomes of planned trials < 37 >_ The use of PK/PD models has been found to be useful in analyzing and integrating data from clinical trials Whiting et al, < 21 > found that variability in both theophylline PK and PD must be taken into account if the drug is to be used to its best advantage. When looking at the PK parameters, the volume of distribution was kept constant because it had no effect on steady state plasma concentration in this model and only a small degree of variability in theophylline's volume of distribution is reported in the literature < 6 > .  Combined variabi lity in CL and rate of drug absorption resulted in slight increase in %CV in plasma concentration achieved from variability in CL alone. The drug response was much more sensitive to the changes in PD parameters than that of PK parameters.
For example 30% CV in all PD parameters resulted in 23.8 and 53.7% CV in drug response from linear and sigmoid Emax models, respectively while same amount of variability in PK parameters resulted in 7.74 and 27.2% CV in drug response from linear and sigmoid Emax models, respectively. Addition of PK variability to PD variability did not cause much difference in% CV in drug response resulted from PD variability alone.
By looking at the combined PK/PD variability, PD variability was found to be the main contributing factor to the changes in the drug response.
The addition of tablet-to-tablet content variability to the combined PK/PD variability resulted in negligible variations in plasma concentration and drug response obtained from PK/PD variability alone. For example, 30% CV in PK/PD resulted in 30

CONCLUSION:
Simulation study can reveal the effect of different sources of variability on the plasma concentration and drug response. This is very important to be considered in designing a PK/PD study.
From this study, the followi ng conclusions can be drawn: I-The impact of the variability in dose and PK parameters on the drug response was less when the linear model rather than the sigmoid Emax model was used.
2-Drug response was more sensitive to the PD variability rather than the PK variability.
3-In the linear model, drug response was most sensitive to variabi lity in baseline FVC.
4ln the sigmoid Emax model , drug response was most sensitive to variability in the maximum attainable effect.
5-Variability in the tablet content resulted in a negligible variability in the response when added to the combined PK/PD variability. 88 6-Drug response estimated from the sigmoid Emax model was much more sensitive to different sources of variability, dose, PK and PD parameters, than that of the linear model.
In summary, variability in PD of a drug is clinically important and must be taken into account in order to use the drug effectively and safely.   The response was measured every 12 hours over a 10-day period. Hundred replications were performed giving a total of 2000 responses.
97 The response was measured every 12 hours over a 10-day period. Hw1dred replications were performed giving a total of2000 responses. The response was measured every 12 hours over a 10-day period. Hundred replications were performed giving a total of2000 responses. The response was measured every 12 hours over a I 0-day period. Hundred replications were performed giving a total of2000 responses.

100
.. 45% CV in PK/PD was studied using 45% CV in CL only (with no variability in rate of drug absorption) together with 45% CV in all PD parameters. The response was measured every 12 hours over a 10-day period. Hundred replications were performed giving a total of 2000 responses. 45% CV in PK/PD was studied using 45% CV in CL only (with no variability in rate of drug absorption) together with 45% CV in all PD parameters. The response was measured every 12 hours over a IO-day period. Hundred replications were performed giving a total of2000 responses.  Simulation studies are useful for providing convincing objective evidence of tbe merits of a proposed study design and analysis. In this study, data were simulated and used to investigate the optimum sampling design for a pharmacodynamic (PD) modeling study.
The various designs were evaluated by consideration of tbe bias and precision of tbe PD parameters and tbeir associated variability parameters.
Response data were simulated from concentration input data for an inhibitory sigmoid Emax model using NONMEM (Non-linear Mixed Effect Modeling) from a population of l 00 individuals. Subsequently, tbese data were used to estimate tbe PD and variability parameters using tbe first order conditional method (FOCE) in NONMEM. The estimation step was based on tbe population of I 00 individuals each providing three concentration-effect data sets from specific concentration-sampling windows. Four sets of concentration sampling windows were initially investigated. The accuracy of parameter estimates, obtained after l 00 replications was assessed using mean and standard deviation of percent prediction error as measures of bias and precision, respectively. The effects of population size and level of inter-individual variability were further studied using tbe most optimum design. The optimum design for the determination of the equilibrium rate constant associated with an effect compartment was also studied. Response data were also simulated from time input data for an inhibitory sigmoid En.ax model. The equilibrium rate constant and its variability parameter were then estimated using the first order method (FO) in NONMEM. Two designs were investigated.
The most optimum design for determination of PD parameters in the absence of an effect compartment was found to be the one with low concentration input in which samples were taken from the following concentration windows: 0.1-0.5, 0.5-1 and 1-2 EC 50 units.
However, in the presence of high inter-individual variability (60%) estimates of variability parameters, using the most optimum design, were biased and imprecise. More precise estimates of the parameters were obtained with a larger population. All designs failed to give accurate estimates of the variability in the sigmoidicity parameter. The most optimum design for the equilibrium rate constant was found to be the one in which two samples were taken from the following sampling windows: 0.25-l.5 and l.5-3 equilibrium half-life units (using 50 individuals) but it gave poor estimates of the variability parameter. In conclusion, accurate estimates of all PD parameters were obtained when samples were taken from 0.1-0.5, 0.5-1 and 1-2 EC 50 units. Increasing the level of inter-individual variability to 60% in the most optimum design gave precise estimates of all PD parameters but variability parameters were poorly estimated. Accurate estimates of the equilibrium rate constant were obtained but not of its variability parameter.

IIl.2. INTRODUCTION
In drug development, the application of population phannacodynamic (PD) modeling can help increase understanding of the quantitative relationships among drug-input patterns, patient characteristics, and drug response. This approach is useful when wishing to identify factors that affect drug behavior, or explain variability in a target population. The population approach can be used to estimate population parameters in many phases of clinical drug development, where information is gathered on how drug will be used in subsequent stages of drug development < 1 > . The population approach is designed to take advantage of observational, randomly obtained data. It can be used to analyze sparsely sampled data ( 24 >. lt, also, encompasses the identification and measurement of variability during drug development and evaluation.
The design of a PD study is critical in determining the accuracy of parameter estimates, especially when data are sparse < 5 -6 >. When designing a population study, practical design limitations such as sampling times, number of samples per individual, and number of individuals should be considered. Also, it is important to consider factors such as the clinician-time, the time spent by the patient in the clinic, especially if the study is conducted on an outpatient basis, and the sampling assay cost. Consequently, a study design that involves taking as few samples as possible from each indi vidual is preferable.
Simulation is a useful tool to provide convincing objective evidence of the merits of a proposed study design and analysis (7). Simulation enables the pharmacometrician to better predict the results of a population study and to choose the study design that will best meet the study objectives < 8 • 13 >_ Several PD models have been used to describe the drug pharmacological response. These models describe the relation between the drug response (dependent variable) and its concentration (independent variable). The sigmoid Emax model has been proposed as a useful model to describe the in vivo relationship between dose/concentration and continuous pharmacological effect for many drugs < 14 >_ This model has three PD parameters, namely efficacy (Emax), potency (EC50) and sigmoidicity constant (y) that control the drug response. Using this model, the concentration-effect relationship is sigmoidal in shape. The sigmoid Emax model has the advantage over other PD models in that it incorporates the sigmoidicity constant, which is the number of drug molecules bound to each receptor. Although this model is based on receptor theory it cannot be assumed, even if the concentration-effect data fits the model, that the drug response is truly described by the model. It must be kept in mind that data drive the model. For example, sigmoidicity has been found to be a non-integer in some cases even though the receptor theory would predict it to be always an integer < 15 • 18 >. In some cases a baseline response (E 0 ) can be incorporated in the model. If the drug has an inhibitory effect on a physiologic response, such as reduction of the heart rate or reduction of the number of eosinophils, the model equation is subtracted from the baseline effect.
The inhibitory sigmoid Emax model has been widely used to describe the PD of many drugs such as suppression of ventricular ectopic depolarizations by tocainide< 15 >, Ill theophylline's induced eosinopenia and hypokalemia< 16 >, percent reduction in heart rate by diltiazem from the baseline< 19 > and reduction of the pain score by the analgesic effect of acetaminophen< 13 >_ ln early PD studies, investigators often made the assumption that drug concentrations measured in plasma were in equilibrium with those at the effect site. This assumption may be valid, if the drug effect is direct, receptor site rapidly equilibrates with plasma and the receptor interaction and response occurs rapidly . However, sometimes a delay between the pharmacological effect and the plasma concentration occurs < 20 > . Sheiner et al 1979 (20)  hr (18).
In population studies, the variability in parameters and the search for factors controlling variability is also an important focus. A lot of information in the literature is available on the study designs for PK < 1 • 7 • 9 · 12 • 23 • 31 > . However, there is very little information about the study designs for PD. Only too often major emphasis is placed on the PK, rather than on PD, however plasma concentration (the PK output) is no more than a surrogate for the pharmacological and/or clinical effects, which require information about the PD of the drug < 32 >_ The incorporation of PD in drug development leads to a more informative drug development program especially in identification of drug dosage regimens for optimal therapeutic outcome through strategies for individualization of dosage < 33 >_ Recognition of PD importance on drug response recently increases. Failure to appreciate the magnitude of variability in PD of a drug can compromise fixed dose clinical trial outcomes making the drug appear less effective or more toxic < 34 >_ The objectives of this study were: ( 1) To determine the optimum sampling design of a PD study. (2) To study the effect of total sample size on the accuracy of parameter estimates.
(3) To study the effect of high level of inter-individual variability on the accuracy of parameter estimates. (4) To determine the optimum sampling design of the equilibrium rate constant using the effect compartment method.
In this study, response data were simulated from concentration input data to determine the accuracy of PD and variability parameters with different sampling designs. The effect compartment model was used to link a one compartment PK model with the inhibitory sigmoid Emax, model to study the optimum sampling design for the equilibrium rate constant where the response data were simulated from time input data.

Simulation of response data from concentration input data:
An integrated PD model oftheophylline was constructed in NONMEM (version 5) (Nonlinear Mixed Effect Modeling) < 35 >.

Pharmacodynamic Model:
An inhibitory sigmoid Emax model, as shown in Equation III.I , was used to simulate response data from concentration input data.

Ill. I 114
Where E is the drug response, Eo is the baseline, Emax is the drug efficacy, which is the maximum drug effect, C is the drug concentration and EC 50 : is the drug concentration at 50% of the Emax (potency). Gamma (y): is the sigmoidicity constant.

Statistical Model:
An exponential model was used to describe the inter-individual variability in all PD parameters as follows: S; =Sm • (EXP(TJe;)) III.2 Where, S; is the estimate for a PD parameter in the ith individual, Sm represents the population mean value of this parameter and T]e; is a normally distributed random variable with an average value of 0 and variance of o}.
Intra-individual variability (Residual error) was also described by exponential error model as follows: Where, Eij is the observed effect for the ith individual at time j, Emij is the modelpredicted effect for the ith individual at time j and Eij is the residual error that represents the difference between the observed response and the model predicted response. Eij is a normally distributed random variable with an average value of 0 and variance of cr 2 .
For the residual error, the formula for the exponential model was written in NONMEM control file as follows: As a result, the simulated response values were in the log form . Consequently, the data had to be protected from zero to avoid error resulting from log zero, by using IF-ELSE statement.
When the exponential model is used to describe inter-individual and intra-individual variability, ro and cr may be regarded as approximate coefficients of variation. The coefficient of variation of the inter-individual variability was chosen to be 30% for all PD parameters. The coefficient of variation of the residual error was chosen to be at the moderate variability level of25% < 21 >.

IIl.3.1.3. Data:
For each design studied, a data set was created based on 100 "individuals" each of whom contributed three samples. Thus, one data set consisted of 300 observations of response data.

IH.3.1.4. Sampling Schedules:
Initially, four basic designs were investigated using windows of increasing concentrations. To mimic a real life situation, in which it is unrealistic to take samples at exactly the same concentration for each individual, sampling windows were used < 9 > . The sampling windows for each of the four basic designs are shown in Table III. I. In Design 1 and 2, samples were taken at low to moderate concentration levels. While in Design 3 and 4, samples were taken at moderate to high concentration levels and high concentration levels, respectively. In order to generalize the results to any drug, not specifically theophylline, the designs were created based on EC 50 units.

Data Simulation:
For each design of the basic designs, 3 random concentration points from within the appropriate sampling window were generated for each individual in Excel. The corresponding response data were then simulated using NONMEM. For each scenario, 100 data sets were replicated. The PRED type model < 35 > was used to simulate the effect data directl y from concentration input data. 117

Effect of Changing Total Sample Size:
As outlined above, initially each data set consisted of I 00 individuals and I 00 replications. Once the most optimum design was identified, the effect of total sample size was further studied using population of 50, 200 and 1000 "individuals". The total number of observation data was 150, 600 and 3000 with 50, 200 and 1000 individuals, respectively.

Effect oflncreasing Inter-individual Variability:
Initially inter-individual variability in PD parameters was set at 30% for the basic four designs. Additionally, the performance of the most optimum design was further assessed at a level of inter-individual variability of 60%.

Parameter Estimation:
For each simulated data set, estimation of PD parameters (ECso, Emax, Eo and y) and variability parameters (variability parameter in EC 50 (roEcso), variability parameter in Emox (roEmax), variabil ity parameter in Eo (roEo) and variability parameter in y (roy) was carried out in NONMEM using the first order conditional estimation method (FOCE). Although this method is time consuming compared to the first order (FO) method, it is the only way to get accurate estimates for the variability parameters using exponential error model (JS)

Bias and Precision of Parameter Estimates
The accuracy of the estimates from each data set were evaluated using the percent prediction error (%PE) as described by the following equation: %PE= 9 •m -e,ru, *100 etruc 111.5 Where 8,;m is the estimated population value of the parameter from one simulated data set and 81ruc is the true population value for the parameter. The %PE was calculated for the 100 simulated data sets in each scenario. The mean and standard deviation of%PE were used to measure bias and precision of parameter estimates respectively. A mean of %PE for a parameter estimate $ 15% was accepted as being unbiased < 11 >. A standard deviation of%PE for a parameter estimate$ 35% was accepted as being precise < 11 >.

III.3.2. Simulation of response data from time input data:
An integrated PK/PD model of theophylline was constructed using NONMEM. A onecompartment PK model was linked to the inhibitory sigmoid Emax PD model through an effect compartment.

Pharmacokinetic Model:
A one-compartment PK model with intravenous bolus input was used. Response data were simulated following a single dose of 300 mg based upon the population PK/PD parameters of theophylline induced eosinopenia. The values used for PK parameters were 2.75 L/hr for clearance (CL) and 28.4 L for volume of distribution (VD) < 16 >.

III.3.2.2. Effect Compartment Model:
The inhibitory sigmoid Emax model, as previously described in Section IIl.3.1.1 was linked to the PK model through an effect compartment with equilibrium rate constant, k.o, of 2.04 hr" 1 < 16 > to account for the lag between plasma concentration and pharmacological response.

III.3.2.4. Statistical Model:
A proportional model was used to describe the inter-individual variability in all PK and PD parameters as follows: 0; = Sm * (I +!)a;) III.6 Where, 9; is the estimate for a PD parameter in the ith individual, 9m represents the population mean value of this parameter and lje; is a normally distributed random variable with an average value ofO and variance ofco 2 .
Intra-individual variability (Residual error) was described by exponential error model as The coefficients of variation of inter-individual variability and residual error were chosen to be at 30% and 25%, respectively.

III.3.2.5. Data:
Two designs were studied as shown in Table III

Ill.3.2.6. Sampling Schedules:
The sampling times in the two designs were chosen from the range of three equilibrium half lives (3 keo At the beginning of this study, the FOCE method was used for the estimation step but it took a very long time (more than 3 hours) for each run due to the structural complication of this method with the complex model, ADV AN 3, used. Also, an enormous amount of overflow error was encountered which delayed the runs remarkably. Thus the estimation of the equilibrium rate constant (k.o) and its variability parameter was carried out in NONMEM using the FO method with each simulated data set.

122
The accuracy of the estimates from each data set was evaluated using %PE as described by Equation Ill.5, Section Ill.3.1.9.

III.4.1. PD Parameters:
Bias and precision of parameter estimates were used to judge the performance of the designs studied where a mean and standard deviation of %PE for a parameter estimate ~ 15% and 35% was accepted as being unbiased and precise, respectively < 11 >.
Design I gave unbiased estimates of all PD parameters; however y was only just achieved unbiased status (Tables III.3  Regarding the precision of the parameter estimates, Designs I and 2 gave imprecise estimates of ffiEmax · Designs 3 and 4 gave imprecise estimates of OJEcso-All but Design 4 gave precise estimates of ffiEO · All designs failed to give accurate (unbiased and precise) estimates of the variability parameter in y (roy). Estimate of the residual error (cr) was unbiased and precise with all the basic designs studied.
From the initially studied four designs, Design 1 was found to be the most optimal design. It was the only design to give accurate (unbiased and precise) estimates of all PD parameters. However, it did give inaccurate estimates of OlEmax and roy. Consequently, it was further studied for the effect of total sample size and the effect of increasing interindividual variability on the accuracy of the parameter estimates.

Effect of Total Sample Size:
The effect of sample size was studied by using the most optimum design with 50, 100, 200 and 1000 individuals. The bias of PD and variabi li ty parameters appeared to be the same for all population sizes, with the exception of the estimation of y which became more biased as the population size increased. By looking at the confidence intervals for the bias of y estimates with different sample sizes, it was found that as the sample size increased, the confidence intervals got smaller. The confidence intervals for bias of y   Figure 111.5, respectively .

III.4.4. Effect of Increasing Inter-individual Variability:
When inter-individual variability was increased from 30 to 60%, the most optimum design still resulted in unbiased and precise estimates of EC 5

III.S. DISCUSSION
The magnitude and duration of drug response is controlled by the PK and PD phases. The determination of clinically useful guidelines thus must account for the parameters of each phase and also any link models necessary to account for delays in the distribution of drug from the plasma to the site of action < 36 >. The expanded application of PK/PD models has been found to be highly beneficial for establishing doses used during drug development. pharmacology is to determine the PK/PD parameters of a drug in a patient population.
The software NONMEM is commonly used to model response data to build population PK-PD models that characterize the relationship between a patient's PK-PD parameters and other patient specific covariates such as the patient's pathological, physiological conditions and concomitant drug therapy < 46 > .
When planning a population PD study, several aspects must be given careful attention. shown that the sigmoidicity constant causes a lot of problems in the estimation step and results in many overflow errors < 35 >, such as a division by zero or floating point overflow.
In the present study, in some cases more than 600 runs were necessary to get 100 correct runs without an overflow error. Also, the use of the FOCE method causes an unsuccessful termination of the simulation step for many runs < 10 >. However, this method provides more accurate estimates of the variability parameters < 10 • 35 >.
Initially, three basic designs were investigated. These designs differed in their In general, PK/PD parameters are estimated more accurately than the variability parameters. Al-Sanna et al < 7 > and Ette et al < 23 > found that the population PK fixed-effect parameters were efficiently estimated but the inter-individual variability parameters were inaccurate and imprecise for most of the sampling schedules.
The most optimum design identified in this study, was used to study the effect of total sample size on the accuracy of the parameter estimates. Increasing sample size to 1000 individuals gave more precise estimates of all PD and variability parameters. Increasing total sample size resulted in biased estimates of y. By conducting hypothesis testing on the population means C 43 > for the bias of y estimates with different sample sizes (level of significance of 0.05), it was found that as the total sample size increased the power to reject the null hypothesis that y estimate is unbiased increased. Also, the confidence intervals for bias of y were wide for population sizes of 50 and 100 individuals ( 4.31 and 2.62, respectively) compared with that for population sizes of 200 and 1000 individuals (1.88 and 0.64, respectively). Therefore small sample size had a little power to detect a departure from the null hypothesis that the estimate is unbiased. Thus as the total sample size increased, the confidence intervals got smaller and the power to detect the bias increased.
Inter-individual variability in PD may result from variability in receptor density and affinity, formation and elimination kinetics of endogenous ligands, postreceptor transduction processes, Homeostatic responses, the kinetic characteristics of transporters involved in drug transfer between fluids of distribution and the biophase and variability in the baseline (Eo) among population C 34 l_ It is important to identify and quantify the variability in PD parameters for the clinical safety and effectiveness of drug use C 34 >_ When the inter-individual variability was increased from 30% to 60%, the precision and bias of PD parameters was the same, except that of y where the estimate was biased.
However at this higher level of inter-individual variability, all variability parameters were very poorly estimated. Sun et al 1996 (II) found that there was an increase in bias and imprecision in parameter estimation as inter-subject variability was increased. Increasing inter-indi vidual variability in the most optimum design resulted in biased estimate of the residual error. This may be due to the difficulty in partitioning error between inter and intra individual variability at this high level of inter-individual variability. Ette et al C 6 l found that positively biased estimates of residual variability were obtained irrespective of the sample size used (30 to 1000 subjects) at coefficient of variation of 60% or more.
It is accepted that for a fixed sample size, PK/PD parameters are estimated more accurately than the associated variability parameters <  However, Design B in which 2 samples were taken from each individual of 50 subjects gave more precise and unbiased estimates of k.o than that of Design A in which one sample was taken from each individual of I 00 subjects. Estimates of the variability parameter in k.o were very poor for both designs. Estimates of the residual error (cr) were precise with Design B and imprecise with Design A. This is in accordance with the work of Ette et al < 29 >. They used half (50 subjects) the total number of subjects required for accurate parameter estimation with the one sample per subject design and doubling the total number of observations per subject. They found that with one observation per subject, the design yielded biased and imprecise estimates of inter-individual variability, and residual variability could not be estimated. Obtaining a second sample from each subject gave better estimates of the residual error, because it facilitated the partitioning of inter-subject variability and residual intra-subject variability, by introducing information about the latter. Two samples from 50 individuals appeared to be enough to get accurate estimate of one parameter. Breant et al (JO) found that two samples from each individual of 15-20 patients were enough to perform a reasonable population analysis to get accurate estimates of two parameters (CL and VD). They also found that the values of the PK parameters were very similar to those obtained with 3 to 5 blood levels and with more patients.
Large inter-individual variability in keO has been found for many drugs, for example the percent coefficient of variation in k.o of acetaminophen in children undergoing outpatient tonsillectomy is 131% CISJ and that oftheophylline induced eosinopenia is 191% < 16 >. The variability parameter in keO was very poorly estimated in this study, possibly because the percent coefficient of variation in k.o used was very small (30%) compared to the real one (191 %) < 16 > or may be due to misspecification of the error model used .

III .6. CONCLUSION
Simulating a planned study offers a potentially useful tool for evaluating and understanding the consequences of different study designs. Simulation can reveal the effect of input variables and assumptions on the results of a planned population PD study.
From this work, the following general conclusions on a PD study design can be drawn: 1. Optimal sampling and pre-experiment simulation is a useful tool for designing informative population study < 24 J_ 2. Design l , in which concentration samples were taken from the following sampling windows: O.l-0.5, 0.5-l and l-2 EC50 performed best overall and it was considered to be the most optimum design for a PD study specially its input concentration was low which is suitable for a narrow therapeutic range and potent drugs.
3. Increasing the total sample size improved the accuracy of the parameter estimates.
4. When inter-individual variability was increased to 60%, with the exception of the sigmoidicity constant's estimate which was biased, accurate estimates of all PD parameters were found. The variability parameters were very poorly estimated.
5. All the designs failed to give accurate estimates of the variability parameter in the sigmoidicity constant.
6. Accurate equilibrium rate constant estimates were obtained with the two designs studied, however Design B in which 2 samples were taken per individual with total sample size of 50 individuals performed better.
7. Both Designs A and B gave very poor estimates of the variability parameter in the equilibrium rate constant.
In summary, Design I was considered to be the most optimum design for studying the PD parameters, namely efficacy, potency, baseline response and sigmoidicity constant.
Design B was considered to be the most optimum design for studying the equilibrium rate constant but it gave poor estimates of the variability parameter.