Journal of In Silico & In Vitro Pharmacology

Keywords

Limited sampling strategies; Area under the curve; Multiple linear regression; Bayesian; Robustness; Transferability

Introduction

Therapeutic drug monitoring (TDM) is a routine clinical practice for drugs exhibiting significant inter- and intra- individual variability and narrow therapeutic ranges [1]. To optimize treatment efficacy and minimize toxicity, appropriate blood samples are collected for the estimation of therapeutic surrogates. The area under the plasma drug concentration-time curve (AUC) is a typical example [1-4]. Many methods have been proposed to estimate AUC. However, to be reliable, a rich number of samples on individuals is usually needed for the use of these methods [5]. In order to alleviate the associated burden of frequent blood withdrawal, limited sampling strategies (LSS), generally using 3 or less blood samples while keeping reasonable estimation accuracy were proposed and applied in clinical practice [6-9]. Two usual LSS approaches, the multiple linear regression (R-LSS) and the population pharmacokinetic (Pop-PK) model based empirical Bayesian (B-LSS), are equally used for the estimation of AUC [10-12]. In the development of these LSS approaches, a reference dataset of dense sampling, with 6 to 10 blood samples of each individual, is usually provided [13]. Using the trapezoidal method, individual AUC is estimated and set as the reference AUC (AUC_ref). For R-LSS, different subsets of blood samples are tested using multilinear regression method with AUC_ref, and those with the closest predicted values to AUC_ref are identified. Noted as , this prediction is expressed as

where C_i are concentrations sampled at the chosen times t_i, and a_i are the associated regression coefficients, i = 0, . . . k. For its simplicity, R-LSS is a much appreciated methodology. However, its use is highly restrictive since samples are assumed to strictly follow the protocol nominal sampling times, a hypothesis thus excluding any possible time deviations [10-12]. For B-LSS, a well-established Pop-PK model is additionally required for the estimation of AUC. This model, considered as the prior acquired knowledge of drug characteristics, helps to improve the estimation, otherwise solely based on the observed drug concentrations. With the B-LSS method, the individual PK parameters are estimated using the empirical Bayesian approach, then used to predict the individual’s drug concentrations and, consequently, the AUC [14]. One advantage of B-LSS over R-LSS is its flexibility in terms of sampling time deviations which are implicitly included in the framework of the Pop-PK modeling and the estimation of individual PK parameters. Nevertheless, the use of B-LSS can be hampered by the need for trained professionals and specialized software. This situation is however progressively changing since many PK software packages with user-friendly interfaces are now made available [8,11,12,15,16]. The predictive performance of LSS is generally evaluated through the calculation of their bias and precision [17]. Though these criteria inform a certain validity of the LSS, their predictive power and transferability should not be taken for granted when different patient subpopulations and contexts are involved [18]. This fact explains the variety of LSS reported for the same drug by different research groups [9,19]. When a specific LSS is intended to be used in a clinical setting or a population different from the one for which it has been originally established, the degree of its transferability or reproducibility should be of concern [9,20]. Traditionally, this was assessed clinically through a revalidation process using additional subpopulation data [21]. However, running a separate clinical trial for each subpopulation can be unrealistic and even at a large risk of validation failure. To reduce this risk, a modeling-based approach using simulated data is a reasonable choice. We will refer to this transferability to indicate the reliability of a chosen LSS in different settings [9,16,22-24], which includes the robustness and the success rates as will be discussed in the current paper. By taking advantage of both R-LSS and B-LSS, we propose here a hybrid approach to measure the robustness of LSS with respect to the so-called between subject variability (BSV) and the residual variability (RV). Based on this, we will also propose several ways to assess the risk of an identified LSS. The current work concerns the methodology to evaluate the transferability of LSS, which is simulation based. The drug model is chosen from the literature and only used for an illustrative purpose. Therefore, the method we propose here can be easily applied to any drug scenarios with similar needs.

Materials and Methods

Since our methodology is based on simulated data, a published Pop-PK model of cyclosporine (CsA) in clinical renal transplant patients [25] is chosen to illustrate our proposed evaluation procedure of LSS. First, using this model, a full sampling data set will be generated to serve the estimation of AUC_ref. Then, different subsets of blood sampling with all possible combinations containing three time points are used to find the most performing LSS in terms of the closeness of their prediction of AUC to AUC_ref. Once this selection is done, we will propose a simulation process for the assessment of their robustness. An analysis of the LSS robustness in the context of various variability will be investigated as well. Moreover, the problem of how to assess the risk of an LSS in different settings, referred to here as the LSS success rates, will be discussed. These involved steps will be explained in more details below.

The Pop-PK model

A one-compartment Pop-PK model with first-order elimination and absorption rate for cy-closporine (CsA) is chosen to illustrate our proposed approach [25]. Typical values of model parameters are used in this paper and summarized in Table 1.

Table 1: The one compartmental Pop-PK model for illustration.

In this model, BSV for PK parameters is described with the exponential model:

(2)

Where θ_ij is the j^th PK parameter for the ith individual, θ_j is the typical value of the parameter, η_ij is a random variable assumed to be normally distributed with zero mean and variance ω². The unexplained variability RV is described using the additive and proportional model

(3)

Where, C_obs and C_pred are the observed and predicted values of blood concentrations, respectively;

ε₁ and ε₂ are normally distributed random variables with zero mean and variances σ₁² and σ₂², respectively. For the sake of simplicity, we take in this paper BSV as the average of η of the PK parameters and RV as the value of ε₁, since the impact of ε₂ on the estimation of AUC is very small.

Simulated dataset of full sampling concentrations and the estimation of AUC_ref

Using the above Pop-PK model, 100 full sampling concentration datasets, with 60 subjects in each, were simulated. Using a previously reported protocol [11], 12 sampling points are chosen:

0, 0.5, 1, 1.5, 2, 3, 4, 8, 12, 15, 22 and 24 hours post dose. Using the trapezoidal method, AUC_ref. was estimated for each of these full sampling dataset.

Identification of LSS

A number of 220 possible LSS, which are the combination results from three concentration-time points, were evaluated for their performance. For each dataset k, we performed a multiple linear regression in terms of the chosen combination of (C₁, C₂, C₃) and AUC_ref to determine the corresponding coefficients (a₁, a₂, a₃), which leads to the prediction of AUC_ref, :

(4)

The performance of the 220 LSS was evaluated using the 95^th percentile of the absolute values of relative prediction errors (95^th PAE%) [11,12]. For the given dataset k and the combination m, the relative error E_i, i = 1 • • • , 60, are calculated as

(5)

Where AUC_ref,i and are AUC_ref and for i^th individual, respectively. Then, 95^th PAE% is estimated using the following formula

95^th PAE% = 95^th percentile of the increasingly ordered set of {| E |}_i. (6)

For each dataset k, the best N LSS among the 220 combinations in terms of 95^th PAE% values are kept as appropriate candidates for AUC estimation. In the current paper, we considered the cases of N = 1 or N = 5.

Accordingly, for 100 simulated datasets, we thus have a list of 100 or 500 LSS that can be considered as the most promising LSS for the estimation of AUC. This list will be used to assess an LSS.

Reliability assessment of LSS

Using the above list, we can investigate the robustness of an LSS, and assess the risk of its use as well. We propose the following metrics for their estimations.

• Robustness: For an identified LSS in the list, its robustness refers to its frequency within the list. This can be explained by the fact that when an LSS appears the most in the identification process, it is the one which is likely to perform always the best, thus can be considered as the most robust.

• Absolute success rate: Given a 95^th PAE% threshold Θ, for example 15% in the current paper, we can measure the rate that an identified LSS has 95^th PAE% smaller than Θ within the 100 simulated datasets. Mathematically, the absolute success rate of an LSS is given by

Absolute success rate = #{95^th PAE%(LSS) < Θ}/100 (7)

Where # is the number of occurrences. This rate can be interpreted as the confidence that one can have in an identified LSS to reach the expected accuracy when applied in a new setting.

• Relative success rate: We can also measure the rate that an identified LSS has the best performance compared to other LSS within the 100 datasets. For an identified best LSS_I, its relative success rate compared to another LSS_J can be defined as

Relative success rate = #{k : 95^th PAE%(LSS_J ) - 95^th PAE% (LSS_I ) > 0}/100 (8)

Where # is the number of occurrences.

Analysis of LSS robustness for various variability

A different variability in PK parameters may change the predictive performance of an LSS. Hence we also estimate here the impact of the BSV and RV on the robustness of LSS. This allows a different angle of view to understand the risk of uncertainty of an LSS.

Three levels (small, medium, high) of BSV and RV are tested. For BSV, the small, medium and high variability are 0-20%, 20- 50% and 50-80%, respectively. For RV, three levels of 0-10%, 15- 20% and 30-60% are used for the proportional component, and three levels of 0-0.1, 0.1-0.2 and 0.2-0.6 (ng/mL) for the additive component.

Software

The commercial software package MATLAB (2008b, The Math Works Inc, Natick, Massachusetts, USA) and NONMEM (version VII, Icon Development Solutions, Ellicott City, MD) were used for calculations and simulations.

Results

Robustness of the LSS

In the left panel of Figure 1, we display, in a 2-D contour plot heat map style, the LSS robustness in terms of different BSV and RV values. Figure 1A shows the robustness of the best (most frequent) LSS for N = 1. It can be observed that the robustness is very low even when BSV and RV values are very small, which makes the decision difficult regarding the appropriateness of these best LSS.

Figure 1: Robustness and maximum 95^th P AE%. PAE-percentile absolute error; RV-residual variability; BSV-between subject variability.

However, when N = 5, we have a larger list of LSS (500), from which a best LSS can be identified for each value of BSV and RV. It can be observed that there is a significant increase in robustness, as seen in figure 1B, making this situation more appealing. This increase is not homogeneous in terms of BSV and RV. For example, for medium variability as with BSV = 30% and RV = 20%, where the identified LSS is 172 (t₁ = 2h, t₂ = 4h, t₃ = 12h), its robustness is less than 50%. Nonetheless, when BSV is small (< 20%), we always have a high robustness (> 60%), regardless of RV. The estimation error of those LSS should be clinically acceptable. Figure 1C and D, represents the maximum estimation error for the best LSS, for N = 1 and N = 5, respectively. Contrarily to the robustness, there is no significant difference in the estimation error for the best LSS when small variability is present, as indicated in the lower left square of Figure 1C and D.

Absolute success rate

In Figure 2, the left panel shows the robustness of LSS in a decreasing order, for three levels of BSV and RV, the middle panel displays the 95^th PAE% of the most robust LSS of each level, compared to a threshold Θ = 15%, for 100 simulations or clinical settings. For BSV = 5% and RV = 10%, the most robust LSS noted 186 (t₁ = 3h, t₂ = 4h, t₃ = 8h) has a robustness larger than 80 % and a full absolute success rate of 100%. Hence for a threshold of 15%, the LSS186 can be safely transferred to another setting. However, for BSV = 35% and RV = 20%, the most robust LSS, LSS₁₇₂ (t₁ = 2h, t₂ = 4h, t₃ = 12h), has a robustness of 49% and an absolute success rate of 23%, making it more susceptible to errors when applied in different clinical conditions. Finally, for an extreme case of BSV = 40% and RV = 60%, LSS₁₈₆ is again the most robust, with more than 60% of robustness but has null absolute success rate of 0%. If threshold of Θ = 20% is clinically acceptable, our conclusion for the LSS₁₇₂ can be changed, with a success rate of 95%, which may be acceptable. However, the conclusion for the third case remains unchanged.

Figure 2: Robustness, 95^th P AE% and relative success rate of the best LSS. PAE, percentile absolute error; RV, residual variability; BSV, between subject variability.

Relative success rate

The right panel of Figure 2 shows the relative success rate of each of the most robust LSS of the left panel. In all cases, this success rate is higher than 50%. For BSV = 5% and RV = 10%, the most robust LSS noted 186 (t₁ = 3h, t₂ = 4h, t₃ = 8h) has a relative success rate of 60% and up, which supports its superiority as previously indicated. For the case of BSV = 40% and RV = 60%, LSS₁₈₆ still distinguishes itself among the other LSS, though its absolute success rate makes it useless (the less worst). In the middle case of BSV = 35% and RV = 20%, we can see that LSS₁₇₂, has 53% of relative success rate compared to its nearest competitor LSS₁₇₃ (t₁ = 2 h, t₂ = 4 h, t₃ = 15 h), making them almost equal in this aspect. However, the absolute success rate of LSS₁₇₃ is 20%, which is not far away from the one of LSS₁₇₂ which is 23%. In this case LSS₁₇₃ can be considered as an alternative for LSS₁₇₂.

Impact of different variability on the robustness of LSS

Figure 3 and 4 display the robustness of best LSS in terms of BSV and BSV/RV, for increasing levels of RV. The results of Figure 3 show that, for any BSV, the robustness decreases with the increase in RV. However, for a fixed RV, the robustness decrease first when BSV increases, but then the trend is reversed at some point. We think this could be attributed to the different intrinsic natures of BSV and RV, while the former can be explained by the PK heterogeneity of the population and can be predicted by the Pop-PK model, while the latter refers to the unexplained part of the variability as a white noise of the model. Figure 4 informs further by linking the ratio of BSV and RV to the robustness in presence of variability. We can see that, for all RV tested, the variation of robustness are almost in the same range and the mentioned trend changes around the point where BSV is almost double of RV.

Figure 3: Robustness vs. BSV for different RV. RV, residual variability; BSV, between subject variability.

Figure 4: Robustness vs. BSV/RV for different RV. RV, residual variability; BSV, between subject variability.

Discussion

Limited sampling strategies have been proposed to reduce the invasiveness of therapeutic drug monitoring for drugs having large variability and narrow therapeutic ranges. A large number of studies have been performed for the selection of best LSS using data collected in a particular clinical setting. Many LSS have been identified to encompass a variety of compounds and populations. At the same time, new methodologies have been proposed; mainly using the multilinear regression or Pop-PK based empirical Bayesian approaches. Beyond the LSS selection, additional aspects have recently been addressed, such as the impact of sampling time deviation on the prediction performance of LSS [12]. The LSS validation is an important issue. Many methods have been proposed regarding this respect, such as the Leave One out Cross Validation (LOOCV), Jackknife, bootstrapping, etc. All these methodologies are based on the use of a unique dataset, which is generally split into two parts: one for learning and the other for confirming. However, the limitation of these methods is that they are based on data collected bearing similar conditions. The generalizability and transferability of these developed LSS to another setting remains an open problem [9,16,18]. This is what we addressed in the current paper. In fact, there exist two main concerns: 1) how to facilitate the selection of the best LSS; 2) how to inform the associated risk for their extended use. To address these two points, we proposed here a hybrid approach that takes advantage of both R-LSS and B-LSS and can be used to evaluate the reliability of an LSS. Through a simulation approach and exemplified by a drug model, we here introduced the concept of LSS robustness and success rate. This enabled us to quickly identify a best LSS and quantitatively document its associated risk when the intent is to use it elsewhere. Since the variability is the major factor influencing the prediction capacity of LSS, we also studied the impact of different levels of BSV and RV on this prediction and found that the ratio of BSV and RV is a determinant factor. For illustration, we have chosen a relatively simple Pop-PK model with some modifications of the reported variability in order to easy our calculations. Doing so, we were able to quickly identify the problems and find the solutions. However, the proposed method can be easily adapted to more realistic situations and the developed procedure can be implemented within home developed software for its routine use in LSS development.

Acknowledgements

This work was supported by FRQNT postdoctoral fellowship, held by Leila K. Asl, NSERC Discovery Grant, NSERC-Industrial Chair in partnership with Novartis, Pfizer and inVentiv Health Clinical as well as FRQNT Projet d’équipe, held by F Nekka and J Li.

References

1. Touw DJ, Neef C, Thomson AH, Vinks AA (2005) Cost-Effectiveness of Therapeutic Drug Monitoring Com- mittee of the International Association for Therapeutic Drug M, Clinical T: Cost-effectiveness of therapeutic drug monitoring: A systematic review. Ther Drug Monit 27: 10-17.
2. Kahan BD (2002) Therapeutic drug monitoring of immunosuppressant drugs in clinical practice. Clinical Therapeutics 24: 1223-1223.
3. Wacke R, Rohde B, Engel G, Kundt G, Hehl EM, et al. (2000) Comparison of several approaches of therapeutic drug monitoring of cyclosporinA based on individual pharmacokinetics. European Journal of Clinical Pharmacology 56: 43-48.
4. Wavamunno MD, Chapman JR (2008) Individualization of immunosuppression: Concepts and rationale. Current Opinion in Organ Transplantation 13: 604-608.
5. Yeh KC, Kwan KC (1978) A comparison of numerical integrating algorithms by trapezoidal, La- grange, and spline approximation. J Pharmacokinet Biopharm 6: 79-98.
6. Moore MJ, Bunting P, Yuan S, Thiessen JJ (1993) Development and validation of a limited sampling strategy for 5-fluorouracil given by bolus intravenous administration. Ther Drug Monit 15: 394-399.
7. van Warmerdam LJ, ten Bokkel Huinink WW, Maes RA, Beijnen JH (1994) Limited-sampling models for anticancer agents. J Cancer Res Clin Oncol 120: 427-433.
8. Panetta JC, Iacono LC, Adamson PC, Stewart CF (2003) The importance of pharmacokinetic limited sampling models for childhood cancer drug development. Clin Cancer Res 9: 5068-5077
9. Ting LS, Villeneuve E, Ensom MH (2006) Beyond cyclosporine: A systematic review of limited sampling strategies for other immunosuppressant. Ther Drug Monit 28: 419-430
10. Sprague DA, Ensom MH (2009) Limited-sampling strategies for anti-infective agents: Systematic review. Can J Hosp Pharm 62: 392-401.
11. Sarem S, Li J, Barri`ere O, Litalien C, Th´eorˆet Y, et al. (2014) Bayesian approach for the estimation of cyclosporine area under the curve using limited sampling strategies in pediatric hematopoietic stem cell transplantation. Theor Biol Med Model 11: 39.
12. Sarem S, Nekka F, Ahmed SI, Litalien C (2015) Impact of sampling time deviations on the prediction of the area under the curve using regression limited sampling strategies. Biopharm Drug Dispos.
13. Dasgupta A (2012) Therapeutic drug monitoring: Newer drugs and biomarkers (1st edn). Elsevier, USA.
14. Abd Rahman A, Tett S, Staatz C (2014) How Accurate and Precise Are Limited Sampling Strategies in Estimating Exposure to Mycophenolic Acid in People with Autoimmune Disease? Clinical Pharmacokinetics 53: 227-245.
15. Loh GW, Ting LS, Ensom MH (2007) A systematic review of limited sampling strategies for platinum agents used in cancer chemotherapy. Clin Pharmacokinet 46: 471-494.
16. van der Meer AF, Marcus MAE, Touw DJ, Proost JH, Neef C (2011) Optimal Sampling Strategy Development Methodology Using Maximum A Posteriori Bayesian Estimation. Therapeutic Drug Monitoring 33: 133-146.
17. Sheiner LB, Beal SL (1981) Some Suggestions for Measuring Predictive Performance. Journal of Pharmacokinetics and Biopharmaceutics 9: 503-512.
18. Barraclough KA, Isbel NM, Franklin ME, Lee KJ, Taylor PJ, et al. (2010) Evaluation of limited sampling strategies for mycophenolic acid after mycophenolate mofetil intake in adult kidney transplant recipients. Ther Drug Monit 32: 723-733.
19. David OJ, Johnston A (2001) Limited sampling strategies for estimating cyclosporin area under the concentration-time curve: review of current algorithms. Ther Drug Monit 23: 100-114.
20. Op den Buijsch RA, van de Plas A, Stolk LM, Christiaans MH, van Hooff JP, et al. (2007) Evaluation of limited sampling strategies for tacrolimus. Eur J Clin Pharmacol 63: 1039-1044.
21. Suarez-Kurtz G, Ribeiro FM, Vicente FL, Struchiner CJ (2001) Development and validation of limited- sampling strategies for predicting amoxicillin pharmacokinetic and pharmacodynamic parameters. Antimicrob Agents Chemother 45: 3029-3036.
22. (1994) Drugs Directorate Guidelines, Acceptable Methods. Health Protection Branch - Health and Welfare Canada: 20-22.
23. (1994) Text on Validation of Analytical Procedures. ICH Harmonized Tripartite Guideline prepared within the Third International Conference on Harmonization of Technical Requirements for the Registration of Pharmaceuticals for Human Use (ICH).
24. (1996) Validation of Analytical Procedures: Methodology. ICH harmonized Tripartite Guideline pre- pared within the Third International Conference on Harmonization of Technical Requirements for the Registration of Pharmaceuticals for Human Use (ICH) : 1-8.
25. Wu KH, Cui YM, Guo JF, Zhou Y, Zhai SD, et al. (2005) Population pharmacokinetics of cyclosporine in clinical renal transplant patients. Drug Metab Dispos 33: 1268-1275.