Non-compartmental analysis of pharmacokinetic concentrations

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The terms pharmacokinetics (PK) and pharmacodynamics (PD) are often described as “what the body does to the drug” and “what the drug does to the body,” respectively. PK/PD analyses are important for determining an appropriate dosing schedule to ensure that there is a sufficient quantity of the drug at the appropriate biological targets to observe beneficial activity. Ideally, this activity translates into an observable efficacy benefit in a clinical trial. On the safety side, PK/PD analyses are important to ensure that the drug does not reach concentrations that are likely to have negative effects, which could translate into adverse events or other safety issues in a clinical trial. PK studies are often performed in healthy volunteers to guide dosing for subsequent clinical trials in patients (though particularly toxic drugs are always tested in patients). In these latter trials, a subset of patients may be selected for a PK sub-study to determine whether the PK behaves as expected, but it also serves as a means to acquire data on efficacy and safety endpoints, which can then be described in conjunction with concentration data and associated PK parameters. Sometimes, particularly for topical medications, PK analyses are used to confirm that a drug is not detectable in blood, tissues, or other fluids to allay fears of greater systemic exposure beyond the treatment site. Finally, PK parameters are used to establish bioequivalence between branded products and their generic counterparts.

The JMP Fit Curve platform provides one- and two-compartment models for PK analysis. However, many pharmaceutical companies are interested in non-compartmental analysis (NCA), which, similar to non-parametric statistical analyses, applies a minimal number of assumptions to the underlying PK model. For a given patient, their concentration data are plotted across time, and estimates of PK parameters are derived. For example, area under the curve (AUC), a common estimate of drug exposure, is estimated using the trapezoidal rule. The forthcoming Pharmacokinetics report in JMP Clinical 19 performs an NCA using data available from the CDISC pharmacokinetics concentrations (PC) domain. Further, this report produces PK profiles that enable the user to view the data of each individual patient in the context of summary statistics derived across groups of patients.

However, before we can start the Pharmacokinetics report, we need to describe new sample data available in JMP Clinical 19. Longtime users of JMP Clinical know that the sample study Nicardipine does not have concentration data available in a PC domain. JMP Clinical 19 contains a new simulated study for the drug Theophylline. This simulated clinical trial is based on the Theophylline data that were described in Pinheiro & Bates (1995). Theophylline was orally administered to 12 individuals using weight-based doses in mg/kg. Serum concentrations were measured 11 times over 25 hours, at pre-dose and 0.25, 0.5, 1, 2, 3, 5, 7, 9, 12, and 24 hours post-dose. Theophylline is a bronchodilator used to treat the symptoms of asthma and other lung diseases.

The one-compartment model presented in Example 1 of the SAS PROC NLMIXED documentation was used to simulate the serum concentrations of 100 patients using 4.5 mg/kg and 6.0 mg/kg doses to represent two treatment groups. For a 70 kg patient, these weight-based doses represent doses of 315 mg and 420 mg, respectively. For the purposes of the simulated clinical trial, doses of 300 mg and 400 mg of theophylline were assumed for the treatment arms, which is in line with current treatment recommendations.

The simulated trial assumes patients were enrolled across three clinical sites, evenly balanced for males and females with a race distribution that approximates the population of the United States. Weight was randomly generated to occur between 50 and 100 kg. Two visits were assumed with all pharmacokinetic (pk) sampling within 12 hours occurring at Visit 1, with the 24 hour sample collected the following day at Visit 2. Serum concentrations were collected as mg/L.

The Pharmacokinetics report has minimal options. The first set of options has to do with how AUC is calculated.

1. Linear to regression start, then log-linear: Area is estimated using observed concentrations before the regression start and log-concentrations from the start of regression onwards.
2. Linear for the whole curve: Area is estimated using the observed concentrations for the entire curve.
3. Linear ascending, log-linear descending: Area is estimated using a concentration when the previous concentration is less than or equal to the current concentration or the log-concentration when the previous concentration is greater.
4. Log-linear for the whole curve: Area is estimated using the log-concentrations for the entire curve.

The second major set of options determines how graphics are plotted, using either the planned time point or the observed time point. By default, the report uses Linear to regression start, then log-linear and Planned Time. Figure 1 displays observed concentrations for the Theophylline data by the planned time after dose.

Figure 1. Observed concentrations by time

Figure 2 displays concentration data by treatment for each patient (red dots), with shaded areas describing the overall range of concentration data (gray), the treatment-specific range of concentration data (light blue), the treatment-specific interquartile range of concentration data (dark blue), along with the treatment-specific median concentration curve (black) and the individual concentration curve (red), which summarizes the mean concentration when more than one patient is presented.

Figure 2. Observed concentrations by time by treatment

Figure 2 can be subset to an individual patient to view their particular PK concentration profile (Figure 3). This view makes it possible to assess the individual’s pharmacokinetics in the context of the entire treatment arm, as well as across all patients.

Figure 3. PK profile for patient SIMPK001-100-001Concentration data can be further summarized and presented by demographic subgroups determined by sex and race (Figure 4). Here, the subgroup-specific range of concentration data (light blue) and the subgroup-specific interquartile range of concentration data (dark blue) are presented, along with the subgroup-specific median concentration curve (black).

Figure 4. Subgroup concentration profiles by race

There are numerous plots to assess the computed parameters of the non-compartmental analysis. Figure 5 presents a box plot with an overlaid violin plot to communicate the distribution of observed parameters by key covariates. Figure 6 presents a scatter plot to view pairs of PK parameters with non-parametric curves to assess their bivariate relationship.

Figure 5. Box and violin plots of AUCLast by treatment

Figure 6. Scatter plot of Cmax by AUCLast by sex

Figure 7 presents concentrations with a curve fitted to the mean concentration at each time point (blue) and a curve fitted to the mean predicted concentration based on a linear fit over the elimination phase of the curve (red).

Figure 7. Scatter plot with observed concentrations and average predicted curvesFigure 8 presents computed parameters for each patient in a tabular listing.

Figure 8. PK parameters from a non-compartmental analysis

 Computed parameters include the following:

1. C_Max (CMAX): The maximum observed concentration for a patient.
2. C_Min (CMIN): The minimum observed non-zero concentration for a patient.
3. C_Last (CLST): The last measurable concentration for a patient.
4. Ĉ_Last (PREDCN): The last measurable predicted concentration for a patient, which is observable in Figure 7 when subset to an individual patient (Figure 9).

5. T_Max (TMAX): The time of the maximum observed concentration for a patient.

6. T_Last (TLST): The time of the last measurable concentration for a patient.

7. T_Start (TSTART): Observed time where the “start of regression” occurs, which is observable in Figure 7 when subset to an individual patient (Figure 9).

8. λz_N (LAMZNPT): The number of points included in the regression of the elimination constant, which is observable in Figure 7 when subset to an individual patient (Figure 9).
9. R² (RSQUARE): The maximum R² from the best fitting regression line to determine the elimination constant.
10. λz (LAMZ): The elimination constant, which is the rate at which a drug is eliminated from the body.

11. T_1/2 (THALF): The half-life, which is the time required for the drug concentration to decrease by half.

12. AUC_Last (AUCLST): The area under the curve, computed from time 0 to T_Last, the last observable time point, which represents the total drug exposure in the body over time.

13. AUC_∞ (AUCIFO): The area under the curve, computed from time 0 to ∞.

14. AUCPEO: The percentage of AUC_∞ which is extrapolated i.e., (AUC_∞ - AUC_Last) / AUC_∞ x 100.
15. AUCPCX2: The percent of AUC_∞  which is extrapolated relative to AUC_Last, i.e., (AUC_∞ - AUC_Last) /  AUC_Last x 100.

16. AUC_{(Last, ∞)} (AUCEXOB): Extrapolated AUC, from time T_Last to ∞.

17. AUCEXPR: Predicted extrapolated AUC, from time T_Last to ∞.

18. AUMC_Last (AUMCLST): The area under the first moment curve, from time 0 to T_Last, the last observable time point.

19. AUMC_∞ (AUMCIFO): The area under the first moment curve, computed from time 0 to ∞.

20. AUMCPEO: The percent of AUMC_∞, which is extrapolated, i.e., (AUMC_∞ - AUMC_Last) / AUMC_∞ x 100.
21. AUMCPCX2: The percent of AUMC_∞, which is extrapolated relative to AUMC_Last, i.e., (AUMC_∞ - AUMC_Last) / AUMC_Last x 100.

22. AUMC_{(Last, ∞)} (AUMCEXOB): Extrapolated AUMC, from time T_Last to ∞.

23. AUMCEXPR: Predicted extrapolated AUMC, from time T_Last to ∞.
24. Mean Residence Time (MRT): Represents the average time a drug molecule spends in the body.
25. Parameters that are available if drug exposure is available from an EX domain:
    1. C₀ (C0): The initial or extrapolated concentration at time 0 following an intravenous bolus injection.
    2. Plasma Clearance (CLO): The volume of blood or plasma cleared of a drug per unit of time.
    3. Volume of Distribution (VZO): The apparent volume of body fluid into which a drug is distributed.
    4. Volume Steady State (VSSO): The apparent volume of body fluid into which a drug is distributed at steady state, which is the time at which the rate of drug administration and elimination are equal.

Figure 9. Observed concentrations with predicted line for patient SIMPK001-100-001.

Statistical teams in clinical development generally do not receive an in-depth education into PK/PD, though they are often called upon to perform these analyses in practice. Statistical analysis plans, therefore, often limit computations and analyses to a small number of the more easily understood PK parameters, such as AUC_Last, C_Max, and T_Max. JMP Clinical 19 places NCA for PK concentration data within easy reach, providing informative visualizations for straightforward review by the study team.

Note that current functionality assumes that each individual patient provides their own complete PK curve. In other words, sparse-sampling approaches, where an individual provides concentration data for a subset of time points to compute AUC_Last, C_Max, and T_Max from the curve produced by connecting the mean concentrations at each time point, are not supported.

References

Pinheiro JC & Bates DM. (1995). Approximations to the Log-Likelihood Function in the Nonlinear Mixed-Effects Model. Journal of Computational and Graphical Statistics 4: 12–35.