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Finding Tmax and Cmax in Multicompartmental Models Yi Rang Han, Ping I

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Finding Tmax and Cmax in Multicompartmental Models Yi Rang Han, Ping I 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  à (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  à 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.
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