**3.1. Compartmental pharmacokinetics**

The process of fitting PK data to a given mathematical description, or model, is known as compartmental modeling. This modeling is carried out with specialized software applications and [15, 16] **Figure 2** shows the simplest one-compartment PK model where drug is introduced by an intravenous bolus injection into a representative volume compartment and the differential and integrated equations that can be fitted to actual data to determine the values of the constants as defined. Multiple-compartment PK models, such as a 2-compartment model, or a 3-compartmental model, commonly describe a concentration-time course adequately, but more complex models may contain more compartments. The mathematical models are based on the processes which move drug into or out of the compartments; these may be a constant rate of infusion or elimination (a zero-order kinetic process) or concentration-driven diffusion processes (first-order kinetics) or by saturable active transport or metabolic processes (Michaelis–Menten kinetics), or combinations thereof [17–19]. The intention of a compartmental model can be as straightforward as to find the simplest model which describes the PK and predicts drug exposures under new conditions, like a higher dose, or when administered in multiple doses over time, or when administered under a different route

**Figure 2.** One-compartment pharmacokinetic model. graphic presentation of a one-compartment PK model with an intravenous bolus injection of dose (D) into a single central compartment of volume V, with a first-order elimination rate of kel. Where Cp is the concentration in the compartment which decreases over time (t). The model is described by a single differential (1) dCp/dt = D/V × (−kel). The integrated Eq. (2) is Cp = (D/V) e−kelT. Secondary parameter CL (clearance) is calculated by Eq. (3) CL = V\*kel.

ethics, patient safety, and/or patient comfort limit the number of PK samples that can be col-

**Figure 3.** Population pharmacokinetics. Population model predictions (solid line) with 95% CI (shaded area) with

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Compartmental and non-compartmental (to be discussed Section 3.2) PK parameter estimates tend to vary within (intra-patient variability) and between individuals (inter-patient variability), with some drugs having more variability than others. Commonly these estimates vary by at least ±15–20% in normal volunteers which can make interpretations challenging, especially when only a few subjects have been evaluated. PK estimates in patient populations typically have even more variability. Population pharmacokinetics, once data is obtained in enough patients and subjects, can help identify and characterize the various sources of variability.

Non-compartmental pharmacokinetics include a number of calculations performed with a series of PK samples usually with plasma- or serum-concentration-time data. These parameters provide a model-free description of how the drug is dispersed and eliminated from the body. These types of analysis can be done very quickly with limited numbers of subjects, where in compartmental or population modeling can take quite some time to build a model. Since no assumptions of which type of compartmental model fits the data best are required, a non-compartmental approach is applied in most Phase I PK studies, and is quite useful in

lected, such as in neonates, pediatrics, and patients with advanced diseased states.

**3.2. Non-compartmental pharmacokinetics**

observed data (circles).

of administration. However, sometimes the purpose may be more complex, to elucidate additional processes such as metabolism mechanisms or drug effects, and these models may contain many compartments.

Compartmental modeling in the very early stages of drug development might be used for supplemental information or to set the initial assumptions for further additional modeling later in the drug development program; known as population pharmacokinetics [20]. Population pharmacokinetics (termed in the industry as 'Pop PK') is the systemic analysis of compiled data from specific studies or from the entire drug development program. These analyses are used to better understand the concentration-time course of the drug and to explain potential sources of PK variability. These models take either sparse PK data (limited numbers of samples) from large numbers of subjects and patients, or both, and/or rich sampling data (full serial PK sampling profiles) from early PK studies, and often incorporate development of individual patient covariates (e.g., BMI, race, genotypes, concomitant medications, disease status, etc.) to predict exposure and effects in individual patients. **Figure 3** shows that this type of analysis not only allows individual patient predictions, but also provides average PK parameters for the population. A population PK approach is also useful in clinical research situations where

**Figure 3.** Population pharmacokinetics. Population model predictions (solid line) with 95% CI (shaded area) with observed data (circles).

ethics, patient safety, and/or patient comfort limit the number of PK samples that can be collected, such as in neonates, pediatrics, and patients with advanced diseased states.

Compartmental and non-compartmental (to be discussed Section 3.2) PK parameter estimates tend to vary within (intra-patient variability) and between individuals (inter-patient variability), with some drugs having more variability than others. Commonly these estimates vary by at least ±15–20% in normal volunteers which can make interpretations challenging, especially when only a few subjects have been evaluated. PK estimates in patient populations typically have even more variability. Population pharmacokinetics, once data is obtained in enough patients and subjects, can help identify and characterize the various sources of variability.

### **3.2. Non-compartmental pharmacokinetics**

of administration. However, sometimes the purpose may be more complex, to elucidate additional processes such as metabolism mechanisms or drug effects, and these models may con-

**Figure 2.** One-compartment pharmacokinetic model. graphic presentation of a one-compartment PK model with an intravenous bolus injection of dose (D) into a single central compartment of volume V, with a first-order elimination rate of kel. Where Cp is the concentration in the compartment which decreases over time (t). The model is described by a single differential (1) dCp/dt = D/V × (−kel). The integrated Eq. (2) is Cp = (D/V) e−kelT. Secondary parameter CL (clearance)

Compartmental modeling in the very early stages of drug development might be used for supplemental information or to set the initial assumptions for further additional modeling later in the drug development program; known as population pharmacokinetics [20]. Population pharmacokinetics (termed in the industry as 'Pop PK') is the systemic analysis of compiled data from specific studies or from the entire drug development program. These analyses are used to better understand the concentration-time course of the drug and to explain potential sources of PK variability. These models take either sparse PK data (limited numbers of samples) from large numbers of subjects and patients, or both, and/or rich sampling data (full serial PK sampling profiles) from early PK studies, and often incorporate development of individual patient covariates (e.g., BMI, race, genotypes, concomitant medications, disease status, etc.) to predict exposure and effects in individual patients. **Figure 3** shows that this type of analysis not only allows individual patient predictions, but also provides average PK parameters for the population. A population PK approach is also useful in clinical research situations where

tain many compartments.

is calculated by Eq. (3) CL = V\*kel.

62 Pharmacokinetics and Adverse Effects of Drugs - Mechanisms and Risks Factors

Non-compartmental pharmacokinetics include a number of calculations performed with a series of PK samples usually with plasma- or serum-concentration-time data. These parameters provide a model-free description of how the drug is dispersed and eliminated from the body. These types of analysis can be done very quickly with limited numbers of subjects, where in compartmental or population modeling can take quite some time to build a model. Since no assumptions of which type of compartmental model fits the data best are required, a non-compartmental approach is applied in most Phase I PK studies, and is quite useful in understanding the drug and indexing its exposure, determining the clinical dose, and designing the final marketed dosage form. The PK parameters obtained from non-compartmental analyses are illustrated in **Figure 4**. Cmax, the peak concentration gives researchers a maximum drug exposure and is also dependent on the absorption rate for extravascular administrations, while the time of Cmax, Tmax, is also indicative of the rate of absorption, but one must understand drug elimination is also occurring at this time. The log-linear slope at the end of the concentration-time curve can be used to estimate the terminal elimination rate constant and the terminal elimination half-life, assuming the curve is well characterized and the PK exhibits first-order elimination. Too short of a sampling interval or limitations of the bioanalytical method may result in missing the terminal elimination phase, so in some cases this slope may be more representative of drug distribution. By calculating an area under the concentration-time curve, called AUC, an index of overall exposure is obtained, and this exposure is independent of the shape of the curve, be it the sharp increase of an intravenous injection with a high Cmax, or lower concentrations observed over a longer amount of time after a slow-release oral formulation. From AUC calculations and the terminal elimination rate constant, estimations volume of distribution and clearance, abbreviated as V and CL, for intravenous doses, or after extravascular doses abbreviated V/F and CL/F, unadjusted for the bioavailable, F, can be made.

**4. Nonclinical pharmacokinetics**

dose (MED) [23].

[21, 24–26].

competition within these transporter systems [29].

Before an investigational drug is ever administered to a human subject, an immense amount of animal and in vitro data are gathered. For example, tests in cell lines and/or animal models are used to determine the potential of the drug's therapeutic action. Other in vitro tests can screen for safety, such as in hERG (Human ether-a-go-go Related Gene) expressed cells, to determine a drug's potential to interact with the potassium channel, IKR, and cause cardiac arrhythmias [21]. Ultimately, single- and repeat-dose toxicity studies, also called 'toxicokinetic or TK' studies, in rodents and at least 1 non-rodent species are needed to support the investigation of the drug in humans [14]. While these studies are mandated by regulatory agencies, they are also useful in the design of the FIH study for a drug's development program. Depending on the type of drug and its apparent risk, several methods of determining the starting dose based on observed toxicity at dose levels can be used. These methods range from simple adjustment and allometric scaling of the non- observed adverse effect level (NOAEL) in the most sensitive animal species studied, with a safety margin [22]. to complex scaling modeling to predict human exposures from animal data. For particularly risky compounds, the starting dose is sometimes carefully based on the minimum biologically active concentration and its associated dose level, also called the minimum effective

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Sometimes detailed PK in animals is available, but generally the PK data from animal studies come from the toxicoketinetic studies. In these studies, the goal is to determine exposure for correlation with toxicity, but qualitative expectations of how the drug will behave in a human are conceived. It would be expected, but not guaranteed, that a quickly absorbed and quickly eliminated drug would also act similarly in humans. Useful predictions of a drug's human PK can be made using computer modeling techniques, called PBPK (physiological based pharmacokinetic modeling) interspecies scaling, which take different species' capacities of absorption, body distribution, and metabolic/excretion into account and simulate PK concentrations based on an analogous human model

Nonclinical studies are also important for providing an idea of the mechanism of the drug's metabolism, whether any cytochrome P 450 enzymes are involved, and identification of metabolites which could be important in humans [27]. Metabolites identified in animals that represent 10% of drug circulating material need to be monitored in toxicology studies and later in clinical studies if still a significant metabolite, is disproportionately produced in humans, or if it is biologically active [28]. In vitro experiments with hepatic enzyme preparations and various chemical probes identify which CYP 450 enzymes are potentially active in the metabolism of a drug. Once these are determined, potential drug-drug interaction pathways are realized; this information is then used to design Phase I drug-drug interaction studies to characterize the clinical significance of these possible interactions in. In vitro experiments also provide the identities of drug-transporters which may move drug into or out of various organs in the body. Drug–drug interactions can also be mediated by inhibition or

**Figure 4.** Non-compartmental model parameters. Cmax = peak concentration, Tmax = time of peak concentration, kel = negative terminal slope from ln concentration versus time regression, T1/2 = 0.693/kel (apparent terminal elimination half-life) AUC0-t = Area un the concentration-time curve from 0 to the last quantifiable concentration estimated by the trapezoidal rule, AUC0-∞ = Area under the curve extrapolated to infinity (AUC0-t + Cp(t)/kel, CL/F = Dose/AUC0-∞ (after a single dose), V/F = apparent volume of distribution after an extravascular dose, calculated by CL/F / kel.
