1521-009X/46/11/1796–1804$35.00 https://doi.org/10.1124/dmd.118.082636 DRUG METABOLISM AND DISPOSITION Drug Metab Dispos 46:1796–1804, November 2018 Copyright ª 2018 by The American Society for Pharmacology and Experimental Therapeutics Commentary
Finding Tmax and Cmax in Multicompartmental Models Yi Rang Han, Ping I. Lee, and K. Sandy Pang Department of Pharmaceutical Sciences, Leslie Dan Faculty of Pharmacy, University of Toronto, Toronto, Canada
Received May 30, 2018; accepted August 16, 2018
ABSTRACT Drug absorption data are critical in bioequivalence comparisons, derivative(s) of the concentration-versus-time profiles and the
and factors such as the maximum drug concentration (Cmax), time Newton-Raphson iteration method for the determination of Tmax Downloaded from to achieve Cmax (or Tmax), as well as the area under the curve and Cmax. We show that the method holds for multicompartmen- (AUC) are important metrics. It is generally accepted that the AUC tal oral dosing under single or steady-state conditions in the is a meaningful estimate of the extent of absorption, and Tmax or absence of known microconstants, even for flip-flop (ka < b)
Cmax may be used for assessing the rate of absorption. But models. Simulations showed that the Cmax and Tmax estimates estimation of the rate of absorption with Tmax or Cmax is not obtained with the Newton-Raphson method were more accurate always feasible, as explicit solutions relating T and C to the than those based on the noncompartmental, observation-based max max dmd.aspetjournals.org absorption (ka) and elimination rate (k) constants exist only for method recommended by the US Food and Drug Administration. the one and not multicompartmental oral model. Therefore, the The %Bias attributable to sampling frequency and assay error determination of Tmax or Cmax for multicompartmental models were less than those determined by the noncompartmental is uncertain. Here, we propose an alternate, numerical ap- method, showing that the Newton-Raphson method is viable for proach that uses the point-slope method for the first and second the estimation of Tmax and Cmax. at ASPET Journals on September 24, 2021
Introduction As previously noted (Endrenyi and Al-Shaikh, 1995; Basson et al., Drug absorption data are often used in bioequivalence compari- 1996), the Cmax is affected by both the extent and the rate of sons. The US Food and Drug Administration (FDA) recommends two absorption. In addition, the value of Cmax,obs is further influenced by measures: the area under the curve (AUC) and the maximum drug the frequency of the sampling scheme and the magnitude of the assay errors. As the sampling frequency or assay error is increased, C concentration (Cmax) (CDER/FDA, 2015), with AUC as the primary max, is increased as well (Endrenyi and Al-Shaikh, 1995). Thus, the and Cmax as the secondary measure. The AUC is known to be obs independent of ka, the absorption rate constant under first-order comparison between Cmax,obs values that are generated by two conditions that reflects the extent of drug absorption, and is normally divergent sampling schemes or assaying methods may be inappro- calculated by the trapezoidal rule, which is straightforward for both priate. An alternate measure of the absorption rate is the time to reach Cmax, compartmental and noncompartmental models. The FDA defines obs or Tmax,obs, which has been suggested to be an unconfounded metric for Cmax as the maximum drug concentration observed in the sampled the rate of absorption (Basson et al., 1996). However, the Tmax,obs is a blood or plasma data, and is henceforth denoted here as Cmax,obs.This categorical variable, one that can take on a limited and usually fixed number Cmax,obs partly serves to provide insight into the rate of absorption. of possible values and its discriminating power depends strongly again on However, an understanding of the absorption rate is more complex. the sampling frequency. Generally speaking, values for Cmax and Tmax may be obtained by This work was supported by NSERC (to P.I.L. and Y.R.H.) and the Centre for fitting data to compartmental models and then computing those values Collaborative Research (CCDR), University of Toronto (to K.S.P.) on the basis of the fitted parameters. In the case of the one-compartment https://doi.org/10.1124/dmd.118.082636. model, the model-guided determination of Tmax,1comp and Cmax,1comp is
ABBREVIATIONS: a, hybrid constant for distribution for two-compartment model; b, hybrid constant for terminal decay for two-compartment th model; li, hybrid constant for an i compartment model; t, dosing interval in a multiple dosing regimen; Cmax, maximum drug concentration in blood/plasma; Cmax,obs, maximum drug concentration observed among the blood/plasma sampling points; Cmax,2comp, maximum blood/plasma drug concentration in (central) compartment from two-compartment model fit; Cmax,true, true maximum drug concentration reached in the blood/plasma (central) compartment; ka, first-order absorption rate constant; k10, first order elimination rate constant from compartment 1; k12, first order transfer rate constant from compartment 1 to 2; k21, first-order transfer rate constant from compartment 2 to 1; these constants are used in both two- and three-compartment models; RSD, relative standard deviation; Tmax,obs, time observed to reach the Cmax,obs in blood/plasma (central) compartment; Tmax,true, true time needed to reach the maximum drug concentration in the blood/plasma (central) compartment; Tmax,2comp, time observed to reach the Cmax,2comp in the blood/plasma (central) compartment on the basis of the two- compartment model; u, time elapsed since last dose in multiple dosing; V, volume of central compartment for one compartment model; V1, volume of the central compartment for multicompartment models.
1796 Tmax and Cmax for Multicompartmental Models 1797 clear. The concentration C1comp may be expressed in terms of the two-compartment model can be accurately determined, the analysis of Tmax absorption rate constant, ka, and elimination rate constant, k, in a and Cmax in drugs that exhibit multiple compartmental characteristics, biexponential expression, as follows: however, cannot be solved explicitly. Often, the Loo-Riegelman method (Loo and Riegelman, 1968) is used first to obtain ka,thenCmax and Tmax, kaFsysDosepo C ¼ e-kt - e-kat ð1Þ but the microconstants k ,k ,andk and the assumption that elimination 1comp Vðk - kÞ 12 21 10 a occurs from the central compartment are required. In this article, we describe the use of a numerical approach for model- where V is the central volume of the distribution. The condition at guided determination of Tmax and Cmax in multicompartment models, and Tmax,1comp yields the Cmax,1comp. compare the estimates obtained to those from direct observations (Tmax,obs k F Dose a sys po -kTmax;1comp -kaTmax;1comp and Cmax,obs), a method encouraged by regulatory agencies such as the Cmax;1comp ¼ e - e ð1AÞ Vðka - kÞ FDA. This method utilizes the second derivative of the concentration- versus-time profile and the point-slope and Newton-Raphson methods. The Explicit solution for Tmax,1comp is obtained by solving dC1comp/dt = bias resulting from assay error and sampling frequency, as well as its 0(eq.2). performance for different ka values, are discussed. = lnðka kÞ Tmax;1comp ¼ ð2Þ Theory and Methods ka - k
Finding Tmax and Cmax in Multicompartment Models Downloaded from The rate constant, ka, may be estimated via curve stripping or with the The Cmax and Tmax for oral dosing for a two-compartment model may Wagner-Nelson method (Wagner and Nelson, 1963) to obtain the fraction ð Þ ð Þ be found in a manner similar to that described for the one-compartment. AA ‘ - AA T -kat remaining to be absorbed (FRA), FRA ¼ ð Þ ¼ e . Together AA ‘ kaFsysDosepo The Cmax,2comp occurs when the first derivative (eq. 6) is zero. with k and ð Þ ,Cmax,1comp and Tmax,1comp may be calculated V ka - k (eqs. 1 and 2). The equation holds in the case of flip-flop when k , k. dC a 2comp -kat -at -bt ¼ Nð-k Þe þ Lð-aÞe þ Mð-bÞe ¼ 0 at T ; ð6Þ The one-compartment analytical equation for finding T and C a max 2comp max max dt dmd.aspetjournals.org can be extended to a scenario with absorption lag time. This may be The above problem may be solved numerically using the Newton- derived by substituting the observed time (t + tlag) and then solving the 2 Raphson method (Galantai, 2000), since the second derivative, (d C2comp) analogous problem of the form, dC1comp,lag/dt = 0 in eq. 2. The one- dt2 compartment concentration profile is: may be easily computed: h i 2 d C a b k F Dose ð Þ ð Þ 2comp 2 -kat 2 - t 2 - t a sys po -k t-tlag -ka t-tlag ¼ Nk e þ La e þ Mb e ð7Þ C1comp;lag ¼ e - e ð3Þ 2 a Vðka - kÞ dt at ASPET Journals on September 24, 2021 and the associated T is: We will illustrate the solution with examples, first with the two- max compartment model, then with multicompartmental models, for drugs ð = Þ with a fast versus a slow absorption rate constant. ln ka k Tmax;1comp;lag ¼ þ tlag ð4Þ ka -k Example 1: Oral Dosing for the Two-Compartment Model Drugs For the two-compartmental model, a triexponential expression now de- The following example demonstrates use of this method to obtain scribes the drug concentration with oral dosing (Gibaldi and Perrier, 1982): Cmax,2comp and Tmax,2comp. The following parameters were selected for the scenario of fast versus slow absorption: k ¼ 2(fast)or0.1 -kat -at -bt a C ¼ Ne þ Le þ Me 21 21 21 (slow) h , with common values for b = 0.2 h , a = 0.8 h ,k21 = 0.5 k F Dose ðk - k Þ k F Dose ðk - aÞ –1 ¼ a sys po 21 a ; ¼ a sys po 21 ; h , and [FsysDosepo/V1] = 100. Microconstants k10,k12, and k21 will where N ða Þðb Þ L ð aÞðb aÞ V1 - ka - ka V1 ka - - remain unchanged when b, a, and k21 are kept constant, owing to the kaFsysDosepoðk21 - bÞ relations that exist between a, b,k10,k12,andk21 (a*b =k10*k21, and a M ¼ ð5Þ b V1ðka - bÞða - bÞ + =k12 +k10 +k21). In fact, the hybrid constants correspond to k10,k12, 21 and k21 values of 0.32, 0.18, and 0.5 h , respectively. After substitution An accurate estimation of the parameters is more complex for such a of the selected values into eq. 5, values of the drug concentration, C , 2 2comp triexponential expression and for the two-compartment, oral case (eq. in the central compartment, the derivatives dC2comp and d C2comp ,andthe dt dt2 5), several scenarios are possible. First is the scenario when ka is much Tmax,2comp estimates were computed and summarized in Table 1. The log- faster than the hybrid decay constant b,i.e,ka ..b,andthethree linear concentration-time profile and the two-compartmental model are exponential components are easily and accurately separated upon shown in Fig. 1A showing a prominent “nose,” and components of curve stripping (Gibaldi and Perrier, 1982). With k12,first-order absorption (ka), distribution (a), and elimination (b)forfast transfer rate from compartment 2 to 1 (k21), k10, rate constant for absorption, or an apparent one-compartment profile for slow removal in compartment 1 and V1, the blood/plasma volume of absorption (Fig. 2A). For fast absorption, the ka may be obtained distribution of drug in the central compartment known from in- by the method of residuals. However, for slow absorption (Fig. 2A), travenous dosing, ka can be obtained by the Loo-Riegelman method ka may only be estimated with the Loo and Riegelman method (Loo (Loo and Riegelman, 1968) with the usual assumption that elimination and Riegelman, 1968) when the microconstants k12,k21,andk10 are occurs in the central compartment. However, for cases when b$ka,the known. In contrast, for both fast and slow absorption, the Newton- distinguishing feature of a prominent “hat” or “nose” that occurs with rapid Raphson method (Galantai, 2000) may be used to obtain Tmax,2comp absorption and slow elimination now disappears, and the oral profile without knowledge of the microconstants. shrinks to one resembling that for the one-compartment model (Chan and Fast ka. A time-expanded view of Fig. 1B showed that the dC2comp/dt Gibaldi, 1985). Under this circumstance, the ka soobtainedbythecurve profile intersected at the zero line (obtained when eq. 6 = 0), stripping procedure is no longer accurate. Even if the parameters for the then continued negatively before rebounding upwards, rising 1798 Han et al.
TABLE 1
Results (error free; assay error E/ = 0) arising from the Newton-Raphson method for estimating of Tmax from two- or three-compartmental models
21 21 21 Two-Compartment, Oral (a = 0.8 h , b = 0.2 h ;k21 = 0.5 h ;FsysDosepo/V1 = 100)
21 21 Fast Absorption (ka =2h ) Slow Absorption (ka = 0.1 h ) -2t -0:8t -0:2t -0:1t -0:8t -0:2t C2comp -139e þ83:3e þ55:6e 57:1e 2 7:14e -50:0e -2t -0:8t -0:2t -0:1t -0:8t -0:2t dC2comp/dt 278e -66:7e -11:1e -5:71e þ5:71e þ10:0e 2 2 -2t -0:8t -0:2t -0:1t -0:8t -0:2t d C2comp/dt -556e þ53:3e þ2:22e 0:571e -4:57e -2:00e
Tmax,2comp,true (h) 0.972 5.77 Cmax,2comp,true (mg/ml) 64.2 16.3
21 21 21 21 21 Three-Compartment, Oral (l1 = 1.0 h ; l2 = 0.8 h ; l3 = 0.2 h ;k21 = 0.5 h ;k31 = 1.5 h )
21 21 Fast Absorption (ka =2h ) Slow Absorption (ka = 0.1 h ) -2t -1:0t -0:8t -0:2t -0:1t -1:0t -0:8t -0:2t C3comp -69:4e -312e þ 292e þ 90:3e 88:9e þ 17:4e -25:0e -81:2e -2t -1:0t -0:8t -0:2t -0:1t -1:0t -0:8t -0:2t dC3comp/dt 139e þ312e -233e -18:1e -8:89e -17:4e þ20:0e þ16:2e 2 2 -2t -1:0t -0:8t -0:2t -0:1t -1:0t -0:8t -0:2t d C3comp/dt -278e -312e þ187e þ3:61e 0:889e þ17:4e -16:0e -3:25e Downloaded from
Tmax,3comp,true (h) 1.28 6.25 Cmax,3comp,true (mg/ml) 82.4 24.2
21 Multicompartment, Oral (t = 6; parameters same as above; ka =2h )
Two-Compartment (Fast Absorption) Three-Compartment (Fast Absorption) -2t -0:8t -0:2t -2t -1:0t -0:8t -0:2t Cmulti -139e þ84:0e þ79:6e -69:4e -313e þ294e þ129e dmd.aspetjournals.org -2t -0:8t -0:2t -2t -1:0t -0:8t -0:2t dCmulti/du 278e -67:2e -15:9e 139e þ313e -235e -25:8e 2 2 -2t -0:8t -0:2t -2t -1:0t -0:8t -0:2t d Cmulti/du -556e þ53:8e þ3:18e -278e -313e þ188e þ5:17e
Tmax,multi,true (h) 0.899 1.13 Cmax,multi,true (mg/ml) 84.3 114
Dosepo, oral dose administered; Fsys, systemic oral bioavailability, product of fraction absorbed (Fabs) and availabilities of intestine, liver and lung from the first-pass effect; k31, first-order transfer rate from compartment 3 to 1 (three-compartment model). at ASPET Journals on September 24, 2021 asymptotically to parallel and approach 0. The first derivative Fig. 1C), one would obtain the true Tmax,2comp,true at convergence. (Fig.1B),whenset=0ineq.6,providedtheTmax,2comp estimate, which However, if the initial Testimate1 was selected from the red zone (Fig. 1C), equaled 0.97 h graphically from Fig. 1B (see value in Table 1). Such a fast the numerical method would not converge to Tmax,2comp,true. Hence, it is absorption can occur in the case of immediate-release oral dosage forms of important to select an appropriate value for T . For good initial 2 estimate1 morphine and cefaclor (Barbhaiya et al., 1990; Collins et al., 1998). Even estimates between 0 and 1.9 h (where d C2comp is 0 in Fig. 1C, with the dt2 faster Tmax is possible for bolus subcutaneous routes of morphine and other Tmax,2comp,true at 0.97 h), the numerical method would converge and drugs (Home et al., 1999; Stuart-Harris et al., 2000), for which the method yield the true Tmax,2comp value at the minimum. Estimates larger than presented here is also applicable. 1.9 h, however, may lead to the asymptotic value of 0 (Fig. 1C). When The iterative scheme of the Newton-Raphson method to find where all the Testimate values are plotted on the C-versus-time plot (shown as the function is zero is illustrated in detail in Fig. 1B. An initial time colored points in inset of Fig. 1A), it may be seen that the estimates estimate (Testimate1) is required for the numerical method. For demon- readily approach Tmax,2comp,true. With known Tmax,2comp,true,thecorre- stration purposes, an arbitrary time of 0.25 h was selected for Testimate1. sponding Cmax,2comp,true may be calculated (eq. 5). For practical applications, Tmax,obs may be used as the initial estimate for Slow ka. By contrast, the log-linear concentration profile and the two- Testimate1.AtTestimate1, the tangent line of the function dC2comp/dt was compartmental model with slow ka, shown in Fig. 2A, revealed an constructed using the “point-slope method,” where the slope was apparent, one-compartment characteristic. The absorption constant (k ) 2 a essentially d C2comp evaluated at T From the line equation of the could be obtained only with the Loo and Riegelman (1968) method with dt2 estimate1. tangent (y = mx + b), the x-intercept at y = 0 (i.e., the time value a1) was known microconstants k12,k21,andk10 or simultaneously curve fitting computed, yielding the corresponding Testimate2 on the dC2comp/dt curve with intravenous data involving these microconstants. Again, the (see Fig. 1B, green). The point-slope procedure was repeated as shown iterative Newton-Raphson method could be readily applied here. By in the blue point in Fig. 1B to provide a2 and the third estimate, Testimate3. selecting Testimate1, finding a1, then Testimate2, the point-slope procedure The continued iterations lead to Testimate7 value for which the difference was applied to provide a2 and Testimate3, then eventually Tmax,2comp,true between dC2comp/dt and 0 is less than a chosen tolerance level («). When (Fig. 2C, showing comparable green and red regions as in Fig. 1C), 21 this criterion is met, the numerical method has converged to the true time similar to the iteration scheme of Fig. 1 when ka is faster (2 h ). The needed to reach the maximum drug concentration in the blood/plasma only difference here was that the procedure provided a “shallow well” 2 (central) compartment (T ). when d C2comp was set = 0 in eq. 7 (Fig. 2C). The first derivative (Fig. 2C), max,true dt2 21 Convergence may be achieved numerically in the critical point that may when set = 0, revealed the true Tmax,2comp estimate of 5.8 h . The points not be of interest. We illustrate this with the scenarios below in Fig. 1C, on C2comp-versus-time plot (Fig. 2A) showed the progression in which shows the behavior of the dC2Comp/dt-versus-time curve in a more approaching the true Cmax,2comp and Tmax,2comp when ka is slow 21 expanded time scale. When the initial Testimate1 (within the shaded green (0.1 h ). Again, when the initial Testimate1 (within the shaded green region) was appropriately selected before the minimum for the dC2comp/dt region, Fig. 2C) was appropriately selected before the minimum for the curve (at 1.9 h, or the boundary between shaded green and red regions in dC2comp/dt curve (at 12.5 h, or boundary between shaded green and red Tmax and Cmax for Multicompartmental Models 1799 Downloaded from dmd.aspetjournals.org at ASPET Journals on September 24, 2021
Fig. 1. The log-linear blood/plasma concentration-versus-time plot for a two- compartment oral dosing case, with elimination from central compartment, for drugs 21 exhibiting fast absorption with ka of 2 h and .b. Parameter values were: a, b, and Fig. 2. The log-linear blood/plasma concentration-versus-time plot for a two- 21 k21 are 0.8, 0.2, and 0.5 h , respectively. (A) The dashed line is an extrapolated log compartment oral dosing case, with elimination from central compartment, for drugs 21 , b a b linear terminal slope, b; the method of residuals would further provide a and ka.The exhibiting flip-flop kinetics (slow ka = 0.1 h ). Parameter values were: , , 21 inset shows an expanded, zoomed-in version. (B) Plot of the first derivative versus and k21 are 0.8, 0.2, and 0.5 h , respectively, and values for a, b, and k21 were time with the first two iterations or Testimate are highlighted in green and blue; the identical to those for Fig. 1. (A) The colored square symbols are the iteration points inset shows the zoomed-out version of the same plot. A total of five iterations of the Newton-Raphson method; a zoomed-in plot is shown in the inset. (B) Plot of were required and these were replotted on (A) and the inset (see text for details), the first derivative as a function of time; In (B), the green and blue symbols are and (C) an expanded time plot of (B). Initial estimates, when chosen from the points used to obtain the first two iterations of the Newton-Raphson method (see text green and not red region in (C), highlight the convergence of the Newton-Raphson for details). (C) Zoomed-out version of the same plot. Initial estimates made within method with the true or correct value. However, initial estimates chosen from the the green but not red shaded region highlights where Newton-Raphson method will red region will not converge but approach infinity upon reaching the iteration converge with the correct value. limit.
regions in Fig. 2C), then one would obtain the true Tmax,2comp,true at Figures 1 and 2 both illustrate that the Newton-Raphson method convergence. For good initial estimates between 0 and 12.5 h (where works well regardless whether k . b,ork , b. 2 a a d C2comp 2 is 0 in Fig. 2C, with the Tmax,2comp,true at 5.77 h), the numerical dt Example 2: Oral Dosing for Multicompartment Model Drugs method would converge and yield the true Tmax,2comp value at the minimum. Initial estimates larger than 12.5 h, however, may lead to the The general method is applicable not only to examine temporal asymptotic value of 0 (Fig. 2C). Multiple examples of a long Tmax have data after single oral dosing for the two-compartment model cases been observed for the slow, oral absorption profiles of drugs such as but also for multiple dosing at steady-state. The concentration for sertraline (.7 h) (Allard et al., 1999) and zonisamide (5–6 h) (Mimaki, multicompartmental models is shown below, where hybrid con- 1998). stants are denoted as li: 1800 Han et al. Downloaded from dmd.aspetjournals.org at ASPET Journals on September 24, 2021 Fig. 3. The log-linear blood/plasma concentration-versus-time plot for the three-compartment oral dosing, with elimination from central compartment. The right panel shows the derivative dC3comp/dt plot. The method is similar to that described for the two-compartment model. Parameter values for l1, l2, l3,k21, and k31 are 1.0, 0.8, 0.2, 1.5, and –1 21 0.5 h , respectively, with ka of 2 (A) or 0.1 h (B). The square symbols are the evaluation points used for computation of dC3comp/dt by the Newton-Raphson method; these same points also appeared in the right panels of (A) and (B). The squares are the evaluation points, and the method is similar to that described for the two-compartment model cases (see Figs. 1 and 2).