Clin Drug Investig (2014) 34:43–52 DOI 10.1007/s40261-013-0148-z
ORIGINAL RESEARCH ARTICLE
Comparison of Algorithms for Oral Busulphan Area Under the Concentration–Time Curve Limited Sampling Estimate Fredrik Sjo¨o¨ • Ibrahim El-Serafi • Jon Enestig • Jonas Mattsson • Johan Liwing • Moustapha Hassan
Published online: 1 November 2013 Ó Springer International Publishing Switzerland 2013
Abstract Background and Objectives Therapeutic drug monitoring (TDM) of the first dose of busulphan during conditioning prior to allogeneic stem cell transplantation provides the possibility of improving the clinical outcome via dose adjustment of subsequent doses. The plasma area under the concentration–time curve (AUC) for busulphan is generally accepted as the parameter that gives the best exposure estimate; however, the sampling frequency needed for reliable AUC calculation remains controversial. The aim of the present investigation was to develop and evaluate a limited sampling model for oral busulphan. Methods We have compared models using three to four samples with standard WinNonlinÒ adaptive compartment modeling based on eight samples as reference. The evaluated study population included both adult and pediatric Electronic supplementary material The online version of this article (doi:10.1007/s40261-013-0148-z) contains supplementary material, which is available to authorized users. F. Sjo¨o¨ I. El-Serafi J. Liwing M. Hassan (&) Department of Laboratory Medicine, Experimental Cancer Medicine, Karolinska Institutet Huddinge, Novum, 141 86 Stockholm, Sweden e-mail: [email protected] F. Sjo¨o¨ (&) J. Enestig Hematology Section, Capio S:t Go¨rans Hospital, S:t Go¨ransplan 1, 112 81 Stockholm, Sweden e-mail: [email protected] J. Mattsson Center for Allogeneic Stem Cell Transplantation (CAST), Karolinska University Hospital-Huddinge, Stockholm, Sweden M. Hassan Clinical Research Center, Novum, Karolinska University Hospital-Huddinge, Stockholm, Sweden
patients, but the linear model was devised using analysis of only pediatric patient plasma concentrations. The present model was developed using data from 23 patients with a mean age of 38 years (range 13–59 years) and was evaluated in 20 pediatric patients with a mean age of 6 years (range 0.1–13 years) as well as 23 adult patients (mean age 43 years; range 18–67 years). Results In 23 patients, the mean AUC from a curve fitting model (Purves method) and a single compartment model had an intraclass correlation coefficient (ICC) of 0.947. From a log–log plot of AUC values it was evident that using this estimate of the AUC would affect dose adjustment decisions for very few of the patients. Applying the linear model using three samples resulted in an ICC of 0.932, mostly due to worse performance in the adult population. Conclusions The present results support the use of limited sampling in clinical TDM for oral busulphan provided adequate algorithms and sampling times are used. Moreover, they also demonstrate the caution that is needed when transferring a pharmacokinetic model from a pediatric population to an adult population.
1 Background Busulphan is an alkylating agent that has been used in low doses for the treatment of chronic myeloid leukemia since the 1950s. At present, busulphan is primarily used in high doses as a myeloablative agent in conditioning prior to stem cell transplantation. Busulphan, like many other cytostatic agents, is characterized by a narrow therapeutic index, wide inter- and intrapatient variability of pharmacokinetic parameters, and high toxicity. Nevertheless, doses for individual patients have traditionally been
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established according to body weight alone. A high-dose busulphan regimen, commonly used in preparative regimens for patients undergoing allogeneic bone marrow transplantation, is often given orally for 4 days in divided doses with a total dose of 16 mg/kg. Pharmacokinetic studies of busulphan have shown that bioavailability [area under the concentration–time curve (AUC) oral/intravenous] varies two- to fivefold in pediatric patients and twofold in adults [1]. Other studies have shown that the AUC varies three- to sevenfold in patients receiving a conditioning regimen [2]. Several investigations have focused on studying alternative administration routes and therapeutic drug monitoring (TDM). Intravenous formulations of busulphan have recently become available and studies have confirmed reduced toxicity problems, apparently due to elimination of the unpredictable absorption pharmacokinetics of oral busulphan [3–5]. However, it is likewise clear that substantial interpatient variability in pharmacokinetic parameters remains due to differences in, for example, metabolism [6–9]. Recently, Malar et al. [10] have shown an increase in sinusoidal obstructive syndrome among pediatric patients under the age of 2 years who have been treated with intravenous busulphan, reminding us that intravenous administration will not eliminate the occurrence of dose-related toxicity. TDM has become a commonly used clinical strategy for maintaining busulphan exposure within the target AUC. Numerous studies have correlated the busulphan AUC with regimen-related toxicity, engraftment, and relapse in patients receiving the busulphan/cyclophosphamide preparative regimen. Busulphan is administered as 1 mg/kg four times daily over 4 days or 2 mg/kg twice daily over 4 days. A typical AUC of busulphan when administered four times daily is 3,600–5,400 ngh/mL, while the AUC ranges between 9,000 and 12,000 ngh/mL when busulphan is given twice daily [9, 11, 12]. Several methods for calculating AUC values have been reported. In general, it can be said that the results of different methods converge when the sampling intervals are reduced. Given sufficient sampling frequency and duration, a simple method such as the trapezoidal rule to the maximum (peak) concentration (Cmax) and log-trapezoidal approximation during elimination will be highly accurate. For practical reasons, however, more than eight to ten samples are seldom used for TDM. Software such as WinNonlinÒ and NONMEMÒ for fitting the results to a model are necessary to achieve the best possible AUC estimate [13]. The use of TDM in cancer therapy and/or in stem cell transplantation is complicated by collection of multiple blood samples in a short time span from anemic patients and young children, and because the analysis and evalua-
tion have to be carried out rapidly and often during offhours in order to allow early dose adjustment. Most importantly, the physician has only one chance to achieve a successful treatment. To achieve a successful TDM, a practical limited sampling protocol in combination with a reliable algorithm that compensates for the missing data compared with a regular rich set sampling is needed. During recent decades, several attempts have been made to develop limited sampling models (LSMs) [14–16], but recent developments in computer hardware and software offer new possibilities. In the present study, we investigate simultaneously fitting plasma concentration data to both a non-compartment model and a one-compartment model. The present model was also compared using the same data in a verified linear formula used at our center for limited sampling data AUC calculations.
2 Patients and Methods 2.1 Model Building In the present investigation, we have employed the following models: multiple regression models, compartment models, and plain curve fitting models. The present model was developed using 23 patients and validated in 20 pediatric and 23 adult patients (Table 1). 2.1.1 Multiple Regression Models The strategy of analyzing the correlation between AUC and each concentration sample from a full sample set using a multiple regression procedure has been described in our previous study [15], where the concentration samples found to have the strongest correlation with AUC were retained while the other concentrations were declined. Stepwise regression was used to construct a linear equation (Eq. 1): AUC ¼ k0 þ k1 C1 þ k2 C2 þ þ kn Cn
ð1Þ
where kn is the calculated constant associated with the nth sample and Cn is the plasma concentration of busulphan in the nth sample. The number of concentration samples is limited to n and the sampling times after dose administration must be identical in all subjects. The formula is verified using a new sample and the correlation is often satisfactory, provided the same population is studied at the same center using the same procedures [17, 18]. The main advantage is the simplicity of the produced equation, and it is very easy to implement the model in clinical practice [19].
Limited Sampling Model for Oral Busulphan Table 1 Patient characteristics
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Group
Mean age (range) [years]
Initial patient group (23 patients)
38 (13–59)
Diagnosis (n)
Sex (male/female) [n]
AML (12)
10/13
CML (9) Ewing sarcoma (1) Precursor B-ALL (1) Pediatric evaluation group (20 patients)
6 (0.1–13)
AML/MDS (10)
8/12
Neuroblastoma (6) MPD (1)
ALL acute lymphoblastic leukemia, AML acute myeloid leukemia, CML chronic myelogenous leukemia, JMML juvenile myelomonocytic leukemia, MDS myelodysplastic syndrome, MPD myeloproliferative disorder
Hurler syndrome (1) JMML (1) Fanconi anemia/MDS (1) Adult evaluation group (23 patients)
43 (18–67)
Drug dynamics within the body in the present model were approximated by kinetically defined compartments. Mathematical formulas describe plasma concentration over time based on absorption, elimination, and distribution rates in compartment models. One or two compartments are most commonly used. Due to the distribution properties of busulphan it is reasonable to implement a one-compartment model for this drug [20]. The formula for calculating AUC then takes the form: AUC ¼
F Dose kabs kel t e ekabs t dt Vd ðkabs kel Þ
10/13
CML (2)
2.1.2 Compartment Models
t¼1 Z
AML (18) MDS (3)
ð2Þ
t¼0
where F is the bioavailability, Dose the drug dose administered, kabs the absorption rate constant, kel the elimination rate constant, t the time, and Vd is the volume of drug distribution. By fitting the absorption rate kabs, elimination rate kel, and the absorbed dose (Fdose), drug distribution ratio FDose Vd , it is possible in theory to calculate the exact AUC. In practice, problems arise from erratic absorption patterns and noisy data that affect the result. This means that, in particular, the absorption rate is not always constant and this may affect the validity of the formula and can result in a major deviation of the AUC estimate. The estimate is further very sensitive to even single errors in concentration measurements. 2.1.3 Non-Compartment Curve Fitting Models In the present model, the third strategy was to fit a mathematical formula to the plasma concentrations.
Several methods have been devised, some involving elaborate calculations such as splines or piecewise polynomial interpolation. The simple numeric trapezoidal rule or trapezoidal rule with log trapezoidal rule during the elimination phase is, however, most commonly used. It is hard to find convincing evidence for the benefit of utilizing complicated calculations. A comparison of 11 numerical curve fitting models found that an interpolation with piecewise parabolas through the origin for concentration intervals until the second concentration or Cmax, with log trapezoidal rule for the remaining intervals, showed the most promising results [21]. The proposed method was chosen for study because it can produce a negative curvature. Purves [21] argued that an interpolation method with negative curvature would presumably be more adequate during the absorption phase after an oral dose than a method, such as the common trapezoidal rule, with zero curvature. Among the methods compared were the Lagrange and cubic spline methods. Both were deprecated due to large variance in their estimates. The AUC using the proposed interpolation for n samples is calculated in Eq. 3: AUC ¼
Zt2
ða1 t þ b1 tÞ dt þ
i¼2
t0
þ
ip 1 Z X
tiþ1
2
n1 X i¼p
ðai t2 þ bi tÞ dt
ti
ðtiþ1 ti ÞðCiþ1 Ci Þ h i log CCiþ1i
ð3Þ
where p is the peak plasma sample concentration, Ti the time point i, Ci the drug concentration in sample i, and n is the total number of samples. The equation ai t2 þ bi t describes a parabola through origin (PTO) and (ti, Ci); (ti?1, Ci?1).
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2.1.4 Implementation of Models Specific models must be designed in order to compare the respective strategies. It can always be argued that there are flaws in the specific model implementation rather than in the strategy itself. However, we have carefully strived to produce the best possible models, in respect to accuracy and efficiency, from each strategy. 2.1.5 Implementation of a Multiple Linear Regression Model Until recently we have used a linear model (Eq. 4) for estimating the AUC using limited sampling at our facility. The model was validated and developed by us and is accordingly well adapted to local circumstances. It originates from studies of busulphan pharmacokinetics in 20 children who underwent bone marrow transplantation for either leukemia or inherited disorders. Based on three plasma concentrations (1, 3, and 6 h) after administration of the first dose, a linear model with high correlation (r = 0.998) was devised using multiple linear regression (Eq. 4): AUC ¼ 1:69 C1 þ 1:45 C2 þ 7:28 C3
ð4Þ
where C1 is the concentration at 1 h, C2 is the concentration at 3 h, and C3 is the concentration at 6 h. The results have been published in Bone Marrow Transplantation, and we refer to the full article for a more detailed description of the development and specifics of this model [15]. 2.1.6 Implementation of a Compartment Model Several kinds of compartment models have been devised for busulphan pharmacokinetics. Two compartments may be used in a model of drug distribution for a more accurate simulation of the actual pharmacokinetics in the human body. The specifics of the drug determine how much is gained in accuracy from introducing the complexity of a two-compartment model. In general, it can be said that sparse input data reduce the feasibility of using a complex model with several deduced parameters. Further, as stated previously, several studies have demonstrated that a onecompartment model provides a good approximation of busulphan pharmacokinetics in the human body [20]. A lag time for the absorption phase can be added to account for a delay before the drug starts appearing in plasma. When using few samplings (just one or two samples) during the absorption phase, it is obvious that it is not possible to determine objectively whether a concentration
results from a slow absorption and small lag time or vice versa. There is also evidence that lag time has a limited effect on the AUC estimate [22]. As a consequence, we have decided not to use lag time. For fitting the parameters of the model to the measured plasma concentrations, we employ one of the most widely used methods, the Levenberg–Marquardt algorithm. It outperforms simple gradient descent and other conjugate gradient descent methods in a wide variety of problems [23]. For the calculations, we rely on an implementation of the algorithm by CenterSpaceTM Software (Corvallis, OR, USA) in the NMath library for the .NET platform. The model has the same timepoints for sampling as the regression model developed earlier. To make the Levenberg– Marquardt algorithm converge with reasonable regularity, we found that at least four plasma concentrations are needed from each patient. We examined the first ten patients using a recently introduced method for finding the most predictive design points in a model [24]. The analysis was done with R using an implementation of the algorithm provided by the authors of the method. We found that the last concentration at 8 h was the most predictive concentration sample for AUC. Sampling at 8 h was accordingly added for both the compartment model and the non-compartment curve fitting model. From some datasets it is not possible to construct a compartment model due to mathematical reasons and the model fails. In this study, this occurred in three adult patients but in none of the pediatric patients. 2.1.7 Implementation of Non-Compartment Curve Fitting Model In our department we have used estimates of total AUC for t0 ? ? for decisions regarding dose adjustment. The comparison by Purves [21] described earlier inspired us to look more closely at the possibility of using piecewise formulas with a PTO until or one step beyond Cmax and then the piecewise log trapezoidal rule. However, we had to adapt the formula to four concentrations, which we consider to be the least number of concentrations necessary to get a meaningful implementation of this strategy. We also had to add an estimate for the AUC tail area from the last concentration to infinity. Since the algorithm involves repeated conditional calculations, it is practical to utilize a computer program. A graphic representation of the resulting plasma concentration simulation is possible because the model involves integration of an actual curve. The algorithm fails if it is not possible to calculate the tail area. This happens if the Cmax is not reached before the last sample is taken. In our study, this occurred in four adult patients and in one pediatric patient.
Limited Sampling Model for Oral Busulphan
2.1.8 Combining the Non-Compartment Curve Fitting Model with the Compartment Model The two LSMs represent different strategies for interpreting the same data and the results of both deviate from estimates using standard rich sampling, but not in the same way. This led us to the idea that a strategy of combining the methods, using the average of the two LSM estimates, could perform better than either model alone. Also, as described above, calculations using either model will fail to produce a result for some data. The risk that both calculations will fail is far lesser (in our material, that did not occur for any subject). If one LSM strategy calculation fails, it is still possible to use the estimate from the other LSM. The performance of this strategy was tested and compared using the different single LSM modeling strategies.
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was added, and the organic phase was removed and analyzed using GC-ECD. The injection temperature was 250 °C, the column was operated isothermically at 135 °C, and the detector temperature was 300 °C. The calibration curve was linear within the range 10–2,600 ng/mL. 2.4 Computer Program 2.4.1 Principal Features of the Program The calculation algorithms are described in the Electronic Supplementary Material. The reader is referred to the corresponding authors for an in-depth description of the other parts of the program. 2.5 Area Under the Concentration–Time Curve and Simulated Plasma Concentration Curve
2.2 Patients Adult and pediatric patients were recruited from the Center for Allogeneic Stem Cell Transplantation at Karolinska University Hospital–Huddinge, Stockholm, Sweden. All patients had been diagnosed with malignant hematological disease and were treated with busulphan as part of conditioning therapy before allogeneic stem cell transplantation. The study was approved by the local ethical committee at Karolinska Institutet, Stockholm, Sweden. According to local guidelines, oral busulphan was administered in two daily doses of 2 mg/kg for 4 days preceding cyclophosphamide. All adult and adolescent patients as well as parents of pediatric patients consented to participation in this protocol according to the Declaration of Helsinki. 2.3 Assay Methodology Patients undergoing stem cell transplantation received oral busulphan as 2 mg/kg twice daily for 4 days. Blood samples were collected before the dose and for measurement at 0.5, 1, 2, 3, 4, 6, 8, and 10 h after administration. Blood was collected (1.5 mL/sample) in heparinized VacutainerÒ tubes. Samples were centrifuged at 3,000 g and plasma was separated, transferred to new tubes, and analyzed using gas chromatography equipped with electron capture detector (GC-ECD) (Varian 3800; Varian, Inc., Palo Alto, CA, USA). An aliquot of 50 lL of internal standard [1,5bis(methanesulfonoxy)pentane] at a concentration of 10 lg/mL dissolved in acetone was added to 0.5 mL of the plasma. 400 lL of n-heptane and 1 mL of 8 M sodium iodide (NaI) were added. The reaction between busulphan and the internal standard and NaI was carried out at 70 °C for 45 min under magnetic stirring. 200 lL of n-heptane
The four measured plasma concentrations are used for calculating the resulting AUC estimates of the compartmental and modified Purves models and the resulting average is shown one graph at a time for illustration of the simulated plasma concentration curve. Figure 1 shows a screen dump from an example session to illustrate the presentation of data.
3 Statistical Analysis The intraclass correlation coefficient (ICC) assesses agreement as well as consistency (precision). The ICC is based on analysis of variance calculations and, depending on the data, different models are used. The presented case of the ICC is based on a mixed-model analysis of variance (Eq. 5). The equation for the agreement parameter q is seen in Eq. 6. This parameter can be estimated from an analysis of variance table and the sum of squares obtained from this analysis, as described in Eq. 5: ICC ¼
BMS EMS BMS þ ðk 1ÞEMS þ nk ðMMS EMSÞ
ð5Þ
where BMS is the between-patients mean square, EMS the residual error mean square, MMS the methods mean square, k the number of methods compared (in this case 2 each comparison), and n is the number of patients (23). q¼
r2P r2P þ r2M þ r2I þ r2E
ð6Þ
where q is the correlation, r2P the variance of patients, r2M the variance of methods, r2I the variance of patient–method interaction, and r2E is the variance of the residual error.
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Fig. 1 Screen dump from the AUC2 program showing the graphic user interface for calculations of area under the concentration–time curve. AUC area under the concentration–time curve
4 Results 4.1 Log–Log Plot Since we are comparing ratios between AUC estimates, a logarithmic scale is most appropriate. Figure 2 illustrates the three-sample linear LSMs and the combined use of the two four-sample LSMs on the y-axis and the reference WinNonlinÒ adaptive compartment modeling with full sampling on the x-axis. It can be seen that both LSMs correlate reasonably with the reference method, but a closer look reveals that significant underestimation is much more common for the linear model, especially in the upper right corner with the highest AUC estimates. Over-estimation seems to occur to a comparable extent for both the three and the four sample models. More LSM estimates for adults differ significantly from the reference compared with estimates for the children, but the degree to which the ‘‘deviators’’ differ seems to be similar for both adults and children.
Dose reduction is often advocated if the AUC is above 12,000 ngh/mL. The plot shows that there may be a risk that a patient with an AUC in the range 12,000–20,000 ngh/mL measured with the reference method does not receive the appropriate dose reduction, especially when assessed with the regression model using three samples (LSM).
4.2 The Intraclass Correlation Coefficient The ICC, based on a mixed-model analysis of variance, was calculated for the LSM as compared to reference. The results (Table 2) show the performance of the respective methods both for the total study population and for children/adults, respectively. The results essentially confirm the findings in the log– log plot; however, it can be more clearly seen that the linear model performs much better for children than for adults.
Limited Sampling Model for Oral Busulphan
49
reference. Furthermore, the confidence intervals show that it is unlikely that the mean AUC estimate difference from reference would render any of the limited sampling methods examined inadequate for use in clinical decision making. 4.3 Mean Error and Root of Mean Squared Error
Fig. 2 Comparison plot of the three-sample linear limited sampling models and the combined use of the two four-sample limited sampling models on the y-axis and the reference WinNonlinÒ adaptive compartment modeling with full sampling on the x-axis. Filled squares represent combination estimates of limited sampling models and empty squares represent regression limited sampling models in adult patients. Filled circles represent combination estimates of limited sampling models and empty circles represent regression limited sampling models in pediatric patients. AUC area under the concentration–time curve
Another way to compare AUC is to compare backtransformed confidence intervals for the mean log AUC ratio between the method to be examined and reference. This is the preferred method for bioequivalence assessment between drugs, as demanded by regulatory authorities both in Europe and in the USA [25]. A two-sided 90 % confidence interval calculated in this manner for the respective LSM is found in Table 3. This will primarily provide a measure of accuracy but not precision, since the interval width is mostly related to sample size. From Table 3 it can be concluded that there is no statistical evidence that any of the methods used in this study systematically produces a higher or lower AUC than
The root of the mean squared error (RMSE) is a measure of precision and the mean error (ME) is a measure of bias. As can be seen in Table 4, using the average of the Purves modified model and compartmental model produces the best precision, except in a subgroup analysis of the adult patients. However, a tendency to overestimation of AUC is seen in all groups. The Purves model shows the best result in the adult subgroup, but four of 23 adult patients could not be evaluated with the Purves model for mathematical reasons.
5 Discussion Despite several efforts to reduce the toxicity of busulphan by adding protective drugs or finding new drugs for stem cell transplantation conditioning [26, 27], busulphan TDM remains an important issue [28]. The need for a simplified method of monitoring drug pharmacokinetics in clinical practice is obvious. A rich sampling method with, for example, eight samples is practical during phase I studies on healthy volunteers when a 1-week delay in receiving results is not of consequence; however, there is an urgent need for other, more practical, methods for TDM. Standard protocol first-dose pharmacokinetics for guiding TDM is a cumbersome procedure warranting a new approach in itself; moreover, co-medication, auto-induction, or inhibition of busulphan metabolism and drug interaction indicate the need for repeated evaluations during therapy [9, 29]. However, repeated evaluations in this patient group are not feasible as common practice except if using limited sampling strategies. Stem cell transplantation is a curative
Table 2 Intraclass correlation coefficient for the limited sampling models versus rich sampling area under the concentration–time curve estimate. Calculations were made using R and CRAN package irr Multiple regression model
Modified Purves curve fitting model
Simplified compartment model
Average between PM and CM estimatesa
All (n = 43)
0.93 (0.89–0.96)
0.91 (0.84–0.95)
0.95 (0.88–0.97)
0.95 (0.90–0.97)
Children (n = 20)
0.94 (0.86–0.98)
0.89 (0.74–0.96)
0.96 (0.90–0.99)
0.96 (0.85–0.98)
Adults (n = 23)
0.89 (0.76–0.95)
0.93 (0.83–0.97)
0.89 (0.73–0.95)
0.92 (0.82–0.96)
Values are expressed as mean (range) CM compartmental model, PM Purves modified model a
While in no case did both the Purves model and the single-compartment model fail simultaneously, in some cases one of the methods failed to produce an area under the concentration–time curve estimate. In those cases, the available result from the successful estimate was used instead of an average
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50 Table 3 Ratios for limited sampling models versus rich sampling area under the concentration–time curve estimate Multiple regression model
Modified Purves curve fitting model
Simplified compartment model
Average between PM and CM estimatesa
All (n = 43)
1.01 (0.73–1.40)
0.96 (0.67–1.39)
0.93 (0.70–1.22)
0.94 (0.72–1.23)
Children (n = 20)
0.98 (0.69–1.39)
0.90 (0.59–1.39)
0.92 (0.70–1.18)
0.91 (0.70–1.18)
Adults (n = 23)
1.04 (0.77–1.40)
1.03 (0.80–1.33)
0.93 (0.68–1.27)
0.97 (0.75–1.27)
Values are expressed as mean (range) CM compartmental model, PM Purves modified model a
While in no case did both the Purves model and the single-compartment model fail simultaneously, in some cases one of the methods failed to produce an area under the concentration–time curve estimate. In those cases, the available result from the successful estimate was used instead of an average
Table 4 Root of the mean squared error and mean error for limited sampling models versus rich sampling area under the concentration–time curve estimate Multiple regression model All (n = 43)
Modified Purves curve fitting model
Simplified compartment model
Average between PM and CM estimatesa RMSE 2,722
RMSE 3,016
RMSE 2,899
RMSE 3,806
MSE -54
MSE 320
MSE 1,601
MSE 924
Children (n = 20)
RMSE 2,802
RMSE 3,224
RMSE 2,991
RMSE 2,115
MSE 82
MSE 728
MSE 1,271
MSE 969
Adults (n = 23)
RMSE 3,189
RMSE 2,530
RMSE 4,474
RMSE 3,156
ME -173
ME -87
ME 1,932
ME 884
CM compartmental model, ME mean error, PM Purves modified model, RMSE root of the mean squared error a
While in no case did both the Purves model and the single-compartment model fail simultaneously, in some cases one of the methods failed to produce an area under the concentration–time curve estimate. In those cases, the available result from the successful estimate was used instead of an average
treatment for several malignant and non-malignant disorders. The clinical outcome is, to a great extent, dependent on the conditioning regimens that have been reported to be correlated with transplantation-related mortality and morbidity [30]. Busulphan and cyclophosphamide are widely used as a conditioning regimen in stem cell transplantation. The present study focuses on busulphan pharmacokinetics before allogeneic bone marrow transplantation, where dose adjustment guided by the plasma concentration AUC is relevant. This is especially valuable during busulphan therapy, when dose adjustment is needed to avoid treatment-related side effects or the risk of insufficient treatment. The challenge is to retain precision and reliability with limited sampling, even when the data are noisy. It is also important that the model has some robustness against change in population factors so that it can be independently verified by different institutions and used safely over time in a fast-developing field such as clinical stem cell transplantation. The accuracy and precision of different proposed limited sampling strategies have been investigated theoretically by other authors using simulated data. In a recent study using Monte-Carlo simulation, a tendency towards underestimation with multiple linear regression and overestimation with curve fitting algorithms was found [31]. The same
tendency was also seen in our study using real patient data, but was not statistically significant. A 90 % confidence interval of the mean difference indicates that it is not of a magnitude that would impact the usefulness of AUC estimates for clinical decisions. It is not possible for even the most perfect measure of plasma concentration AUC to prevent all cases of toxicity or therapy failure. Pharmacokinetic drug monitoring must be complemented with identification of at-risk subpopulations such as patients with hepatic or renal impairment [32]. It is likewise impossible to believe that there is an algorithm that could perfectly recreate the information lost when sampling is limited to fewer timepoints. TDM is nevertheless an important tool for improving patient outcome during transplant conditioning with busulphan. If LSM can make the TDM tool substantially less cumbersome and more available, with only minor loss of sharpness, it will in many cases be an attractive option. We compared three LSMs and the industry standard, rich sampling WinNonlinÒ adaptive compartment modeling. Our results indicate that limited sampling methods can produce clinically useful estimates. In particular, a combination of a curve fitting model and a simplified singlecompartment model seems to perform well. The low deviation in the present model would rarely affect clinical
Limited Sampling Model for Oral Busulphan
decision making and would not impair the clinical benefit gained from TDM. LSM opens up new possibilities that warrant further study. With limited sampling, monitoring can be performed repeatedly during the induction phase at the same cost and with the same effort as monitoring one single dose with standard rich sampling. This could provide a second chance to catch outliers in absorption or metabolism, which more than makes up for the lost precision in limited sampling protocols. Randomized trials comparing clinical outcome after several LSMs and rich sampling should be carried out to clarify the effectiveness of the respective approach. Using an ordinary WindowsÒ desktop computer without any specialized software, a compartment and a non-compartment curve fitting method can be employed for the AUC estimate, the calculus requiring less than a minute to execute. The possibility of presenting a graph for the simulated concentration curve further facilitates the interpretation. All models provided estimates that, in our opinion, have reasonable accuracy to provide guidance for dose adjustments. Added sampling points increase accuracy, and the decision as to what model to choose must be based on individual needs and means. We believe LSMs have an important place in the drug monitoring of busulphan and that further studies are warranted in this field. When the alternative is no monitoring, our study clearly shows that LSM is likely to be of significant benefit. The most important calculus algorithms are shown in the Electronic Supplementary Material. The code is in C# using software packs from CenterSpace for some statistical and mathematical computations. The full program is available in compiled form from the corresponding authors.
6 Conclusion The present results support the use of limited sampling in clinical TDM for oral busulphan. It also shows that adequate algorithms and sampling times are important elements in order to reach reliable results. Moreover, it also demonstrates the caution that is needed when transferring a pharmacokinetic model from a pediatric population to an adult population. Together, this model is reliable and robust and can be used for dose adjustment of busulphan that may lead to decreased drug-related toxicity and improve the clinical outcome. Acknowledgments The present investigation was supported by grants from the Swedish Cancer Foundation (CF) and the Swedish Childhood Cancer Society (BCF). None of the authors has any conflict of interest.
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References 1. Hassan M, Ljungman P, Bolme P, et al. Busulfan bioavailability. Blood. 1994;84(7):2144–50. 2. Grochow LB, Jones RJ, Brundrett RB, et al. Pharmacokinetics of busulfan: correlation with veno-occlusive disease in patients undergoing bone marrow transplantation. Cancer Chemother Pharmacol. 1989;25(1):55–61. 3. Hassan Z, Ljungman P, Ringde´n O, et al. Pharmacokinetics of liposomal busulphan in man. Bone Marrow Transplant. 2001; 27(5):479–85. 4. Hassan M, Nilsson C, Hassan Z, et al. A phase II trial of liposomal busulphan as an intravenous myeloablative agent prior to stem cell transplantation: 500 mg/m(2) as a optimal total dose for conditioning. Bone Marrow Transplant. 2002;30(12):833–41. 5. Kashyap A, Wingard J, Cagnoni P, et al. Intravenous versus oral busulfan as part of a busulfan/cyclophosphamide preparative regimen for allogeneic hematopoietic stem cell transplantation: decreased incidence of hepatic venoocclusive disease (HVOD), HVOD-related mortality, and overall 100-day mortality. Biol Blood Marrow Transplant. 2002;8(9):493–500. 6. Robertson GR, Liddle C, Clarke SJ. Inflammation and altered drug clearance in cancer: transcriptional repression of a human CYP3A4 transgene in tumor-bearing mice. Clin Pharmacol Ther. 2008;83(6):894–7. 7. Bartelink I, Bredius R, Belitser S, et al. Association between busulfan exposure and outcome in children receiving intravenous busulfan before hematologic stem cell transplantation. Biol Blood Marrow Transplant. 2009;15:231–41. 8. Bartelink I, Bredius R, Bierings M, et al. Targeting to an optimal AUC of intravenous busulfan prevents graft failure in transplantation in children with non-malignant diseases. Biol Blood Marrow Transplant. 2009;15:76–7. 9. McCune J, Holmberg L. Busulfan in hematopoietic stem cell transplant setting. Expert Opin Drug Metab Toxicol. 2009;5: 957–69. 10. Malar R, Sjoo F, Rentsch K, et al. Therapeutic drug monitoring is essential for intravenous busulfan therapy in pediatric hematopoietic stem cell recipients. Pediatr Transplant. 2011;15(6): 580–8. 11. Ljungman P, Hassan M, Be´ka´ssy AN, et al. High busulfan concentrations are associated with increased transplant-related mortality in allogeneic bone marrow transplant patients. Bone Marrow Transplant. 1997;20(11):909–13. 12. Vassal G. Pharmacologically-guided dose adjustment of busulfan in high-dose chemotherapy regimens: rationale and pitfalls (review). Anticancer Res. 1994;14(6A):2363–70. 13. Gabrielsson JL, Weiner DL. Methodology for pharmacokinetic/ pharmacodynamic data analysis. Pharm Sci Technol Today. 1999;2(6):244–52. 14. Chattergoon DS, Saunders EF, Klein J, et al. An improved limited sampling method for individualised busulphan dosing in bone marrow transplantation in children. Bone Marrow Transplant. 1997;20(5):347–54. 15. Hassan M, Fasth A, Gerritsen B, et al. Busulphan kinetics and limited sampling model in children with leukemia and inherited disorders. Bone Marrow Transplant. 1996;18(5):843–50. 16. Sandstro¨m M, Karlsson MO, Ljungman P, et al. Population pharmacokinetic analysis resulting in a tool for dose individualization of busulphan in bone marrow transplantation recipients. Bone Marrow Transplant. 2001;28(7):657–64. 17. Schuler U, Schroer S, Ku¨hnle A, et al. Busulfan pharmacokinetics in bone marrow transplant patients: is drug monitoring warranted? Bone Marrow Transplant. 1994;14(5):759–65.
52 18. Vassal G, Deroussent A, Challine D, et al. Is 600 mg/m2 the appropriate dosage of busulfan in children undergoing bone marrow transplantation? Blood. 1992;79(9):2475–9. 19. Balasubramanian P, Chandy M, Krishnamoorthy R, et al. Evaluation of existing limited sampling models for busulfan kinetics in children with beta thalassaemia major undergoing bone marrow transplantation. Bone Marrow Transplant. 2001;28(9):821–5. 20. Nieto Y, Vaughan WP. Pharmacokinetics of high-dose chemotherapy. Bone Marrow Transplant. 2004;33(3):259–69. 21. Purves RD. Optimum numerical integration methods for estimation of area-under-the-curve (AUC) and area-under-themoment-curve (AUMC). J Pharmacokinet Biopharm. 1992;20(3): 211–26. 22. Nerella NG, Block LH, Noonan PK. The impact of lag time on the estimation of pharmacokinetic parameters. 1. One-compartment open model. Pharm Res. 1993;10(7):1031–6. 23. Gill P, Murray W. Algorithms, for solution, of non-linear leastsquares problem. SIAM J Numer Anal. 1978;1978:977–92. 24. Ferraty F, Hall P, Vieu P. Most-predictive design points for functional data predictors. Biometrika. 2010;97(4):807–24. 25. Karalis V, Macheras P. Current regulatory approaches of bioequivalence testing. Expert Opin Drug Metab Toxicol. 2012;8(8): 929–42.
F. Sjo¨o¨ et al. 26. Sjo¨o¨ F, Aschan J, Barkholt L, et al. N-acetyl-L-cysteine does not affect the pharmacokinetics or myelosuppressive effect of busulfan during conditioning prior to allogeneic stem cell transplantation. Bone Marrow Transplant. 2003;32(4):349–54. 27. Sjo¨o¨ F, Hassan Z, Abedi-Valugerdi M, et al. Myeloablative and immunosuppressive properties of treosulfan in mice. Exp Hematol. 2006;34(1):115–21. 28. Chunduri S, Dobogai L, Peace D, et al. Fludarabine/i.v. BU conditioning regimen: myeloablative, reduced intensity or both? Bone Marrow Transplant. 2008;41:935–40. 29. Lindley C, Shea T, McCune J, et al. Intraindividual variability in busulfan pharmacokinetics in patients undergoing a bone marrow transplant: assessment of a test dose and first dose strategy. Anticancer Drugs. 2004;15(5):453–9. 30. Grochow LB. Busulfan disposition: the role of therapeutic monitoring in bone marrow transplantation induction regimens. Semin Oncol. 1993;20(4 Suppl 4):18–25 (quiz 26). 31. Tsuruta H, Fukumoto M, Bax L, et al. Limited sampling strategies to estimate the area under the concentration–time curve. Biases and a proposed more accurate method. Methods Inf Med. 2012;51:383–94. 32. Canal P, Chatelut E, Guichard S. Practical treatment guide for dose individualisation in cancer chemotherapy. Drugs. 1998; 56(6):1019–38.
ORIGINAL RESEARCH ARTICLE
Comparison of Algorithms for Oral Busulphan Area Under the Concentration–Time Curve Limited Sampling Estimate Fredrik Sjo¨o¨ • Ibrahim El-Serafi • Jon Enestig • Jonas Mattsson • Johan Liwing • Moustapha Hassan
Published online: 1 November 2013 Ó Springer International Publishing Switzerland 2013
Abstract Background and Objectives Therapeutic drug monitoring (TDM) of the first dose of busulphan during conditioning prior to allogeneic stem cell transplantation provides the possibility of improving the clinical outcome via dose adjustment of subsequent doses. The plasma area under the concentration–time curve (AUC) for busulphan is generally accepted as the parameter that gives the best exposure estimate; however, the sampling frequency needed for reliable AUC calculation remains controversial. The aim of the present investigation was to develop and evaluate a limited sampling model for oral busulphan. Methods We have compared models using three to four samples with standard WinNonlinÒ adaptive compartment modeling based on eight samples as reference. The evaluated study population included both adult and pediatric Electronic supplementary material The online version of this article (doi:10.1007/s40261-013-0148-z) contains supplementary material, which is available to authorized users. F. Sjo¨o¨ I. El-Serafi J. Liwing M. Hassan (&) Department of Laboratory Medicine, Experimental Cancer Medicine, Karolinska Institutet Huddinge, Novum, 141 86 Stockholm, Sweden e-mail: [email protected] F. Sjo¨o¨ (&) J. Enestig Hematology Section, Capio S:t Go¨rans Hospital, S:t Go¨ransplan 1, 112 81 Stockholm, Sweden e-mail: [email protected] J. Mattsson Center for Allogeneic Stem Cell Transplantation (CAST), Karolinska University Hospital-Huddinge, Stockholm, Sweden M. Hassan Clinical Research Center, Novum, Karolinska University Hospital-Huddinge, Stockholm, Sweden
patients, but the linear model was devised using analysis of only pediatric patient plasma concentrations. The present model was developed using data from 23 patients with a mean age of 38 years (range 13–59 years) and was evaluated in 20 pediatric patients with a mean age of 6 years (range 0.1–13 years) as well as 23 adult patients (mean age 43 years; range 18–67 years). Results In 23 patients, the mean AUC from a curve fitting model (Purves method) and a single compartment model had an intraclass correlation coefficient (ICC) of 0.947. From a log–log plot of AUC values it was evident that using this estimate of the AUC would affect dose adjustment decisions for very few of the patients. Applying the linear model using three samples resulted in an ICC of 0.932, mostly due to worse performance in the adult population. Conclusions The present results support the use of limited sampling in clinical TDM for oral busulphan provided adequate algorithms and sampling times are used. Moreover, they also demonstrate the caution that is needed when transferring a pharmacokinetic model from a pediatric population to an adult population.
1 Background Busulphan is an alkylating agent that has been used in low doses for the treatment of chronic myeloid leukemia since the 1950s. At present, busulphan is primarily used in high doses as a myeloablative agent in conditioning prior to stem cell transplantation. Busulphan, like many other cytostatic agents, is characterized by a narrow therapeutic index, wide inter- and intrapatient variability of pharmacokinetic parameters, and high toxicity. Nevertheless, doses for individual patients have traditionally been
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established according to body weight alone. A high-dose busulphan regimen, commonly used in preparative regimens for patients undergoing allogeneic bone marrow transplantation, is often given orally for 4 days in divided doses with a total dose of 16 mg/kg. Pharmacokinetic studies of busulphan have shown that bioavailability [area under the concentration–time curve (AUC) oral/intravenous] varies two- to fivefold in pediatric patients and twofold in adults [1]. Other studies have shown that the AUC varies three- to sevenfold in patients receiving a conditioning regimen [2]. Several investigations have focused on studying alternative administration routes and therapeutic drug monitoring (TDM). Intravenous formulations of busulphan have recently become available and studies have confirmed reduced toxicity problems, apparently due to elimination of the unpredictable absorption pharmacokinetics of oral busulphan [3–5]. However, it is likewise clear that substantial interpatient variability in pharmacokinetic parameters remains due to differences in, for example, metabolism [6–9]. Recently, Malar et al. [10] have shown an increase in sinusoidal obstructive syndrome among pediatric patients under the age of 2 years who have been treated with intravenous busulphan, reminding us that intravenous administration will not eliminate the occurrence of dose-related toxicity. TDM has become a commonly used clinical strategy for maintaining busulphan exposure within the target AUC. Numerous studies have correlated the busulphan AUC with regimen-related toxicity, engraftment, and relapse in patients receiving the busulphan/cyclophosphamide preparative regimen. Busulphan is administered as 1 mg/kg four times daily over 4 days or 2 mg/kg twice daily over 4 days. A typical AUC of busulphan when administered four times daily is 3,600–5,400 ngh/mL, while the AUC ranges between 9,000 and 12,000 ngh/mL when busulphan is given twice daily [9, 11, 12]. Several methods for calculating AUC values have been reported. In general, it can be said that the results of different methods converge when the sampling intervals are reduced. Given sufficient sampling frequency and duration, a simple method such as the trapezoidal rule to the maximum (peak) concentration (Cmax) and log-trapezoidal approximation during elimination will be highly accurate. For practical reasons, however, more than eight to ten samples are seldom used for TDM. Software such as WinNonlinÒ and NONMEMÒ for fitting the results to a model are necessary to achieve the best possible AUC estimate [13]. The use of TDM in cancer therapy and/or in stem cell transplantation is complicated by collection of multiple blood samples in a short time span from anemic patients and young children, and because the analysis and evalua-
tion have to be carried out rapidly and often during offhours in order to allow early dose adjustment. Most importantly, the physician has only one chance to achieve a successful treatment. To achieve a successful TDM, a practical limited sampling protocol in combination with a reliable algorithm that compensates for the missing data compared with a regular rich set sampling is needed. During recent decades, several attempts have been made to develop limited sampling models (LSMs) [14–16], but recent developments in computer hardware and software offer new possibilities. In the present study, we investigate simultaneously fitting plasma concentration data to both a non-compartment model and a one-compartment model. The present model was also compared using the same data in a verified linear formula used at our center for limited sampling data AUC calculations.
2 Patients and Methods 2.1 Model Building In the present investigation, we have employed the following models: multiple regression models, compartment models, and plain curve fitting models. The present model was developed using 23 patients and validated in 20 pediatric and 23 adult patients (Table 1). 2.1.1 Multiple Regression Models The strategy of analyzing the correlation between AUC and each concentration sample from a full sample set using a multiple regression procedure has been described in our previous study [15], where the concentration samples found to have the strongest correlation with AUC were retained while the other concentrations were declined. Stepwise regression was used to construct a linear equation (Eq. 1): AUC ¼ k0 þ k1 C1 þ k2 C2 þ þ kn Cn
ð1Þ
where kn is the calculated constant associated with the nth sample and Cn is the plasma concentration of busulphan in the nth sample. The number of concentration samples is limited to n and the sampling times after dose administration must be identical in all subjects. The formula is verified using a new sample and the correlation is often satisfactory, provided the same population is studied at the same center using the same procedures [17, 18]. The main advantage is the simplicity of the produced equation, and it is very easy to implement the model in clinical practice [19].
Limited Sampling Model for Oral Busulphan Table 1 Patient characteristics
45
Group
Mean age (range) [years]
Initial patient group (23 patients)
38 (13–59)
Diagnosis (n)
Sex (male/female) [n]
AML (12)
10/13
CML (9) Ewing sarcoma (1) Precursor B-ALL (1) Pediatric evaluation group (20 patients)
6 (0.1–13)
AML/MDS (10)
8/12
Neuroblastoma (6) MPD (1)
ALL acute lymphoblastic leukemia, AML acute myeloid leukemia, CML chronic myelogenous leukemia, JMML juvenile myelomonocytic leukemia, MDS myelodysplastic syndrome, MPD myeloproliferative disorder
Hurler syndrome (1) JMML (1) Fanconi anemia/MDS (1) Adult evaluation group (23 patients)
43 (18–67)
Drug dynamics within the body in the present model were approximated by kinetically defined compartments. Mathematical formulas describe plasma concentration over time based on absorption, elimination, and distribution rates in compartment models. One or two compartments are most commonly used. Due to the distribution properties of busulphan it is reasonable to implement a one-compartment model for this drug [20]. The formula for calculating AUC then takes the form: AUC ¼
F Dose kabs kel t e ekabs t dt Vd ðkabs kel Þ
10/13
CML (2)
2.1.2 Compartment Models
t¼1 Z
AML (18) MDS (3)
ð2Þ
t¼0
where F is the bioavailability, Dose the drug dose administered, kabs the absorption rate constant, kel the elimination rate constant, t the time, and Vd is the volume of drug distribution. By fitting the absorption rate kabs, elimination rate kel, and the absorbed dose (Fdose), drug distribution ratio FDose Vd , it is possible in theory to calculate the exact AUC. In practice, problems arise from erratic absorption patterns and noisy data that affect the result. This means that, in particular, the absorption rate is not always constant and this may affect the validity of the formula and can result in a major deviation of the AUC estimate. The estimate is further very sensitive to even single errors in concentration measurements. 2.1.3 Non-Compartment Curve Fitting Models In the present model, the third strategy was to fit a mathematical formula to the plasma concentrations.
Several methods have been devised, some involving elaborate calculations such as splines or piecewise polynomial interpolation. The simple numeric trapezoidal rule or trapezoidal rule with log trapezoidal rule during the elimination phase is, however, most commonly used. It is hard to find convincing evidence for the benefit of utilizing complicated calculations. A comparison of 11 numerical curve fitting models found that an interpolation with piecewise parabolas through the origin for concentration intervals until the second concentration or Cmax, with log trapezoidal rule for the remaining intervals, showed the most promising results [21]. The proposed method was chosen for study because it can produce a negative curvature. Purves [21] argued that an interpolation method with negative curvature would presumably be more adequate during the absorption phase after an oral dose than a method, such as the common trapezoidal rule, with zero curvature. Among the methods compared were the Lagrange and cubic spline methods. Both were deprecated due to large variance in their estimates. The AUC using the proposed interpolation for n samples is calculated in Eq. 3: AUC ¼
Zt2
ða1 t þ b1 tÞ dt þ
i¼2
t0
þ
ip 1 Z X
tiþ1
2
n1 X i¼p
ðai t2 þ bi tÞ dt
ti
ðtiþ1 ti ÞðCiþ1 Ci Þ h i log CCiþ1i
ð3Þ
where p is the peak plasma sample concentration, Ti the time point i, Ci the drug concentration in sample i, and n is the total number of samples. The equation ai t2 þ bi t describes a parabola through origin (PTO) and (ti, Ci); (ti?1, Ci?1).
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2.1.4 Implementation of Models Specific models must be designed in order to compare the respective strategies. It can always be argued that there are flaws in the specific model implementation rather than in the strategy itself. However, we have carefully strived to produce the best possible models, in respect to accuracy and efficiency, from each strategy. 2.1.5 Implementation of a Multiple Linear Regression Model Until recently we have used a linear model (Eq. 4) for estimating the AUC using limited sampling at our facility. The model was validated and developed by us and is accordingly well adapted to local circumstances. It originates from studies of busulphan pharmacokinetics in 20 children who underwent bone marrow transplantation for either leukemia or inherited disorders. Based on three plasma concentrations (1, 3, and 6 h) after administration of the first dose, a linear model with high correlation (r = 0.998) was devised using multiple linear regression (Eq. 4): AUC ¼ 1:69 C1 þ 1:45 C2 þ 7:28 C3
ð4Þ
where C1 is the concentration at 1 h, C2 is the concentration at 3 h, and C3 is the concentration at 6 h. The results have been published in Bone Marrow Transplantation, and we refer to the full article for a more detailed description of the development and specifics of this model [15]. 2.1.6 Implementation of a Compartment Model Several kinds of compartment models have been devised for busulphan pharmacokinetics. Two compartments may be used in a model of drug distribution for a more accurate simulation of the actual pharmacokinetics in the human body. The specifics of the drug determine how much is gained in accuracy from introducing the complexity of a two-compartment model. In general, it can be said that sparse input data reduce the feasibility of using a complex model with several deduced parameters. Further, as stated previously, several studies have demonstrated that a onecompartment model provides a good approximation of busulphan pharmacokinetics in the human body [20]. A lag time for the absorption phase can be added to account for a delay before the drug starts appearing in plasma. When using few samplings (just one or two samples) during the absorption phase, it is obvious that it is not possible to determine objectively whether a concentration
results from a slow absorption and small lag time or vice versa. There is also evidence that lag time has a limited effect on the AUC estimate [22]. As a consequence, we have decided not to use lag time. For fitting the parameters of the model to the measured plasma concentrations, we employ one of the most widely used methods, the Levenberg–Marquardt algorithm. It outperforms simple gradient descent and other conjugate gradient descent methods in a wide variety of problems [23]. For the calculations, we rely on an implementation of the algorithm by CenterSpaceTM Software (Corvallis, OR, USA) in the NMath library for the .NET platform. The model has the same timepoints for sampling as the regression model developed earlier. To make the Levenberg– Marquardt algorithm converge with reasonable regularity, we found that at least four plasma concentrations are needed from each patient. We examined the first ten patients using a recently introduced method for finding the most predictive design points in a model [24]. The analysis was done with R using an implementation of the algorithm provided by the authors of the method. We found that the last concentration at 8 h was the most predictive concentration sample for AUC. Sampling at 8 h was accordingly added for both the compartment model and the non-compartment curve fitting model. From some datasets it is not possible to construct a compartment model due to mathematical reasons and the model fails. In this study, this occurred in three adult patients but in none of the pediatric patients. 2.1.7 Implementation of Non-Compartment Curve Fitting Model In our department we have used estimates of total AUC for t0 ? ? for decisions regarding dose adjustment. The comparison by Purves [21] described earlier inspired us to look more closely at the possibility of using piecewise formulas with a PTO until or one step beyond Cmax and then the piecewise log trapezoidal rule. However, we had to adapt the formula to four concentrations, which we consider to be the least number of concentrations necessary to get a meaningful implementation of this strategy. We also had to add an estimate for the AUC tail area from the last concentration to infinity. Since the algorithm involves repeated conditional calculations, it is practical to utilize a computer program. A graphic representation of the resulting plasma concentration simulation is possible because the model involves integration of an actual curve. The algorithm fails if it is not possible to calculate the tail area. This happens if the Cmax is not reached before the last sample is taken. In our study, this occurred in four adult patients and in one pediatric patient.
Limited Sampling Model for Oral Busulphan
2.1.8 Combining the Non-Compartment Curve Fitting Model with the Compartment Model The two LSMs represent different strategies for interpreting the same data and the results of both deviate from estimates using standard rich sampling, but not in the same way. This led us to the idea that a strategy of combining the methods, using the average of the two LSM estimates, could perform better than either model alone. Also, as described above, calculations using either model will fail to produce a result for some data. The risk that both calculations will fail is far lesser (in our material, that did not occur for any subject). If one LSM strategy calculation fails, it is still possible to use the estimate from the other LSM. The performance of this strategy was tested and compared using the different single LSM modeling strategies.
47
was added, and the organic phase was removed and analyzed using GC-ECD. The injection temperature was 250 °C, the column was operated isothermically at 135 °C, and the detector temperature was 300 °C. The calibration curve was linear within the range 10–2,600 ng/mL. 2.4 Computer Program 2.4.1 Principal Features of the Program The calculation algorithms are described in the Electronic Supplementary Material. The reader is referred to the corresponding authors for an in-depth description of the other parts of the program. 2.5 Area Under the Concentration–Time Curve and Simulated Plasma Concentration Curve
2.2 Patients Adult and pediatric patients were recruited from the Center for Allogeneic Stem Cell Transplantation at Karolinska University Hospital–Huddinge, Stockholm, Sweden. All patients had been diagnosed with malignant hematological disease and were treated with busulphan as part of conditioning therapy before allogeneic stem cell transplantation. The study was approved by the local ethical committee at Karolinska Institutet, Stockholm, Sweden. According to local guidelines, oral busulphan was administered in two daily doses of 2 mg/kg for 4 days preceding cyclophosphamide. All adult and adolescent patients as well as parents of pediatric patients consented to participation in this protocol according to the Declaration of Helsinki. 2.3 Assay Methodology Patients undergoing stem cell transplantation received oral busulphan as 2 mg/kg twice daily for 4 days. Blood samples were collected before the dose and for measurement at 0.5, 1, 2, 3, 4, 6, 8, and 10 h after administration. Blood was collected (1.5 mL/sample) in heparinized VacutainerÒ tubes. Samples were centrifuged at 3,000 g and plasma was separated, transferred to new tubes, and analyzed using gas chromatography equipped with electron capture detector (GC-ECD) (Varian 3800; Varian, Inc., Palo Alto, CA, USA). An aliquot of 50 lL of internal standard [1,5bis(methanesulfonoxy)pentane] at a concentration of 10 lg/mL dissolved in acetone was added to 0.5 mL of the plasma. 400 lL of n-heptane and 1 mL of 8 M sodium iodide (NaI) were added. The reaction between busulphan and the internal standard and NaI was carried out at 70 °C for 45 min under magnetic stirring. 200 lL of n-heptane
The four measured plasma concentrations are used for calculating the resulting AUC estimates of the compartmental and modified Purves models and the resulting average is shown one graph at a time for illustration of the simulated plasma concentration curve. Figure 1 shows a screen dump from an example session to illustrate the presentation of data.
3 Statistical Analysis The intraclass correlation coefficient (ICC) assesses agreement as well as consistency (precision). The ICC is based on analysis of variance calculations and, depending on the data, different models are used. The presented case of the ICC is based on a mixed-model analysis of variance (Eq. 5). The equation for the agreement parameter q is seen in Eq. 6. This parameter can be estimated from an analysis of variance table and the sum of squares obtained from this analysis, as described in Eq. 5: ICC ¼
BMS EMS BMS þ ðk 1ÞEMS þ nk ðMMS EMSÞ
ð5Þ
where BMS is the between-patients mean square, EMS the residual error mean square, MMS the methods mean square, k the number of methods compared (in this case 2 each comparison), and n is the number of patients (23). q¼
r2P r2P þ r2M þ r2I þ r2E
ð6Þ
where q is the correlation, r2P the variance of patients, r2M the variance of methods, r2I the variance of patient–method interaction, and r2E is the variance of the residual error.
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Fig. 1 Screen dump from the AUC2 program showing the graphic user interface for calculations of area under the concentration–time curve. AUC area under the concentration–time curve
4 Results 4.1 Log–Log Plot Since we are comparing ratios between AUC estimates, a logarithmic scale is most appropriate. Figure 2 illustrates the three-sample linear LSMs and the combined use of the two four-sample LSMs on the y-axis and the reference WinNonlinÒ adaptive compartment modeling with full sampling on the x-axis. It can be seen that both LSMs correlate reasonably with the reference method, but a closer look reveals that significant underestimation is much more common for the linear model, especially in the upper right corner with the highest AUC estimates. Over-estimation seems to occur to a comparable extent for both the three and the four sample models. More LSM estimates for adults differ significantly from the reference compared with estimates for the children, but the degree to which the ‘‘deviators’’ differ seems to be similar for both adults and children.
Dose reduction is often advocated if the AUC is above 12,000 ngh/mL. The plot shows that there may be a risk that a patient with an AUC in the range 12,000–20,000 ngh/mL measured with the reference method does not receive the appropriate dose reduction, especially when assessed with the regression model using three samples (LSM).
4.2 The Intraclass Correlation Coefficient The ICC, based on a mixed-model analysis of variance, was calculated for the LSM as compared to reference. The results (Table 2) show the performance of the respective methods both for the total study population and for children/adults, respectively. The results essentially confirm the findings in the log– log plot; however, it can be more clearly seen that the linear model performs much better for children than for adults.
Limited Sampling Model for Oral Busulphan
49
reference. Furthermore, the confidence intervals show that it is unlikely that the mean AUC estimate difference from reference would render any of the limited sampling methods examined inadequate for use in clinical decision making. 4.3 Mean Error and Root of Mean Squared Error
Fig. 2 Comparison plot of the three-sample linear limited sampling models and the combined use of the two four-sample limited sampling models on the y-axis and the reference WinNonlinÒ adaptive compartment modeling with full sampling on the x-axis. Filled squares represent combination estimates of limited sampling models and empty squares represent regression limited sampling models in adult patients. Filled circles represent combination estimates of limited sampling models and empty circles represent regression limited sampling models in pediatric patients. AUC area under the concentration–time curve
Another way to compare AUC is to compare backtransformed confidence intervals for the mean log AUC ratio between the method to be examined and reference. This is the preferred method for bioequivalence assessment between drugs, as demanded by regulatory authorities both in Europe and in the USA [25]. A two-sided 90 % confidence interval calculated in this manner for the respective LSM is found in Table 3. This will primarily provide a measure of accuracy but not precision, since the interval width is mostly related to sample size. From Table 3 it can be concluded that there is no statistical evidence that any of the methods used in this study systematically produces a higher or lower AUC than
The root of the mean squared error (RMSE) is a measure of precision and the mean error (ME) is a measure of bias. As can be seen in Table 4, using the average of the Purves modified model and compartmental model produces the best precision, except in a subgroup analysis of the adult patients. However, a tendency to overestimation of AUC is seen in all groups. The Purves model shows the best result in the adult subgroup, but four of 23 adult patients could not be evaluated with the Purves model for mathematical reasons.
5 Discussion Despite several efforts to reduce the toxicity of busulphan by adding protective drugs or finding new drugs for stem cell transplantation conditioning [26, 27], busulphan TDM remains an important issue [28]. The need for a simplified method of monitoring drug pharmacokinetics in clinical practice is obvious. A rich sampling method with, for example, eight samples is practical during phase I studies on healthy volunteers when a 1-week delay in receiving results is not of consequence; however, there is an urgent need for other, more practical, methods for TDM. Standard protocol first-dose pharmacokinetics for guiding TDM is a cumbersome procedure warranting a new approach in itself; moreover, co-medication, auto-induction, or inhibition of busulphan metabolism and drug interaction indicate the need for repeated evaluations during therapy [9, 29]. However, repeated evaluations in this patient group are not feasible as common practice except if using limited sampling strategies. Stem cell transplantation is a curative
Table 2 Intraclass correlation coefficient for the limited sampling models versus rich sampling area under the concentration–time curve estimate. Calculations were made using R and CRAN package irr Multiple regression model
Modified Purves curve fitting model
Simplified compartment model
Average between PM and CM estimatesa
All (n = 43)
0.93 (0.89–0.96)
0.91 (0.84–0.95)
0.95 (0.88–0.97)
0.95 (0.90–0.97)
Children (n = 20)
0.94 (0.86–0.98)
0.89 (0.74–0.96)
0.96 (0.90–0.99)
0.96 (0.85–0.98)
Adults (n = 23)
0.89 (0.76–0.95)
0.93 (0.83–0.97)
0.89 (0.73–0.95)
0.92 (0.82–0.96)
Values are expressed as mean (range) CM compartmental model, PM Purves modified model a
While in no case did both the Purves model and the single-compartment model fail simultaneously, in some cases one of the methods failed to produce an area under the concentration–time curve estimate. In those cases, the available result from the successful estimate was used instead of an average
F. Sjo¨o¨ et al.
50 Table 3 Ratios for limited sampling models versus rich sampling area under the concentration–time curve estimate Multiple regression model
Modified Purves curve fitting model
Simplified compartment model
Average between PM and CM estimatesa
All (n = 43)
1.01 (0.73–1.40)
0.96 (0.67–1.39)
0.93 (0.70–1.22)
0.94 (0.72–1.23)
Children (n = 20)
0.98 (0.69–1.39)
0.90 (0.59–1.39)
0.92 (0.70–1.18)
0.91 (0.70–1.18)
Adults (n = 23)
1.04 (0.77–1.40)
1.03 (0.80–1.33)
0.93 (0.68–1.27)
0.97 (0.75–1.27)
Values are expressed as mean (range) CM compartmental model, PM Purves modified model a
While in no case did both the Purves model and the single-compartment model fail simultaneously, in some cases one of the methods failed to produce an area under the concentration–time curve estimate. In those cases, the available result from the successful estimate was used instead of an average
Table 4 Root of the mean squared error and mean error for limited sampling models versus rich sampling area under the concentration–time curve estimate Multiple regression model All (n = 43)
Modified Purves curve fitting model
Simplified compartment model
Average between PM and CM estimatesa RMSE 2,722
RMSE 3,016
RMSE 2,899
RMSE 3,806
MSE -54
MSE 320
MSE 1,601
MSE 924
Children (n = 20)
RMSE 2,802
RMSE 3,224
RMSE 2,991
RMSE 2,115
MSE 82
MSE 728
MSE 1,271
MSE 969
Adults (n = 23)
RMSE 3,189
RMSE 2,530
RMSE 4,474
RMSE 3,156
ME -173
ME -87
ME 1,932
ME 884
CM compartmental model, ME mean error, PM Purves modified model, RMSE root of the mean squared error a
While in no case did both the Purves model and the single-compartment model fail simultaneously, in some cases one of the methods failed to produce an area under the concentration–time curve estimate. In those cases, the available result from the successful estimate was used instead of an average
treatment for several malignant and non-malignant disorders. The clinical outcome is, to a great extent, dependent on the conditioning regimens that have been reported to be correlated with transplantation-related mortality and morbidity [30]. Busulphan and cyclophosphamide are widely used as a conditioning regimen in stem cell transplantation. The present study focuses on busulphan pharmacokinetics before allogeneic bone marrow transplantation, where dose adjustment guided by the plasma concentration AUC is relevant. This is especially valuable during busulphan therapy, when dose adjustment is needed to avoid treatment-related side effects or the risk of insufficient treatment. The challenge is to retain precision and reliability with limited sampling, even when the data are noisy. It is also important that the model has some robustness against change in population factors so that it can be independently verified by different institutions and used safely over time in a fast-developing field such as clinical stem cell transplantation. The accuracy and precision of different proposed limited sampling strategies have been investigated theoretically by other authors using simulated data. In a recent study using Monte-Carlo simulation, a tendency towards underestimation with multiple linear regression and overestimation with curve fitting algorithms was found [31]. The same
tendency was also seen in our study using real patient data, but was not statistically significant. A 90 % confidence interval of the mean difference indicates that it is not of a magnitude that would impact the usefulness of AUC estimates for clinical decisions. It is not possible for even the most perfect measure of plasma concentration AUC to prevent all cases of toxicity or therapy failure. Pharmacokinetic drug monitoring must be complemented with identification of at-risk subpopulations such as patients with hepatic or renal impairment [32]. It is likewise impossible to believe that there is an algorithm that could perfectly recreate the information lost when sampling is limited to fewer timepoints. TDM is nevertheless an important tool for improving patient outcome during transplant conditioning with busulphan. If LSM can make the TDM tool substantially less cumbersome and more available, with only minor loss of sharpness, it will in many cases be an attractive option. We compared three LSMs and the industry standard, rich sampling WinNonlinÒ adaptive compartment modeling. Our results indicate that limited sampling methods can produce clinically useful estimates. In particular, a combination of a curve fitting model and a simplified singlecompartment model seems to perform well. The low deviation in the present model would rarely affect clinical
Limited Sampling Model for Oral Busulphan
decision making and would not impair the clinical benefit gained from TDM. LSM opens up new possibilities that warrant further study. With limited sampling, monitoring can be performed repeatedly during the induction phase at the same cost and with the same effort as monitoring one single dose with standard rich sampling. This could provide a second chance to catch outliers in absorption or metabolism, which more than makes up for the lost precision in limited sampling protocols. Randomized trials comparing clinical outcome after several LSMs and rich sampling should be carried out to clarify the effectiveness of the respective approach. Using an ordinary WindowsÒ desktop computer without any specialized software, a compartment and a non-compartment curve fitting method can be employed for the AUC estimate, the calculus requiring less than a minute to execute. The possibility of presenting a graph for the simulated concentration curve further facilitates the interpretation. All models provided estimates that, in our opinion, have reasonable accuracy to provide guidance for dose adjustments. Added sampling points increase accuracy, and the decision as to what model to choose must be based on individual needs and means. We believe LSMs have an important place in the drug monitoring of busulphan and that further studies are warranted in this field. When the alternative is no monitoring, our study clearly shows that LSM is likely to be of significant benefit. The most important calculus algorithms are shown in the Electronic Supplementary Material. The code is in C# using software packs from CenterSpace for some statistical and mathematical computations. The full program is available in compiled form from the corresponding authors.
6 Conclusion The present results support the use of limited sampling in clinical TDM for oral busulphan. It also shows that adequate algorithms and sampling times are important elements in order to reach reliable results. Moreover, it also demonstrates the caution that is needed when transferring a pharmacokinetic model from a pediatric population to an adult population. Together, this model is reliable and robust and can be used for dose adjustment of busulphan that may lead to decreased drug-related toxicity and improve the clinical outcome. Acknowledgments The present investigation was supported by grants from the Swedish Cancer Foundation (CF) and the Swedish Childhood Cancer Society (BCF). None of the authors has any conflict of interest.
51
References 1. Hassan M, Ljungman P, Bolme P, et al. Busulfan bioavailability. Blood. 1994;84(7):2144–50. 2. Grochow LB, Jones RJ, Brundrett RB, et al. Pharmacokinetics of busulfan: correlation with veno-occlusive disease in patients undergoing bone marrow transplantation. Cancer Chemother Pharmacol. 1989;25(1):55–61. 3. Hassan Z, Ljungman P, Ringde´n O, et al. Pharmacokinetics of liposomal busulphan in man. Bone Marrow Transplant. 2001; 27(5):479–85. 4. Hassan M, Nilsson C, Hassan Z, et al. A phase II trial of liposomal busulphan as an intravenous myeloablative agent prior to stem cell transplantation: 500 mg/m(2) as a optimal total dose for conditioning. Bone Marrow Transplant. 2002;30(12):833–41. 5. Kashyap A, Wingard J, Cagnoni P, et al. Intravenous versus oral busulfan as part of a busulfan/cyclophosphamide preparative regimen for allogeneic hematopoietic stem cell transplantation: decreased incidence of hepatic venoocclusive disease (HVOD), HVOD-related mortality, and overall 100-day mortality. Biol Blood Marrow Transplant. 2002;8(9):493–500. 6. Robertson GR, Liddle C, Clarke SJ. Inflammation and altered drug clearance in cancer: transcriptional repression of a human CYP3A4 transgene in tumor-bearing mice. Clin Pharmacol Ther. 2008;83(6):894–7. 7. Bartelink I, Bredius R, Belitser S, et al. Association between busulfan exposure and outcome in children receiving intravenous busulfan before hematologic stem cell transplantation. Biol Blood Marrow Transplant. 2009;15:231–41. 8. Bartelink I, Bredius R, Bierings M, et al. Targeting to an optimal AUC of intravenous busulfan prevents graft failure in transplantation in children with non-malignant diseases. Biol Blood Marrow Transplant. 2009;15:76–7. 9. McCune J, Holmberg L. Busulfan in hematopoietic stem cell transplant setting. Expert Opin Drug Metab Toxicol. 2009;5: 957–69. 10. Malar R, Sjoo F, Rentsch K, et al. Therapeutic drug monitoring is essential for intravenous busulfan therapy in pediatric hematopoietic stem cell recipients. Pediatr Transplant. 2011;15(6): 580–8. 11. Ljungman P, Hassan M, Be´ka´ssy AN, et al. High busulfan concentrations are associated with increased transplant-related mortality in allogeneic bone marrow transplant patients. Bone Marrow Transplant. 1997;20(11):909–13. 12. Vassal G. Pharmacologically-guided dose adjustment of busulfan in high-dose chemotherapy regimens: rationale and pitfalls (review). Anticancer Res. 1994;14(6A):2363–70. 13. Gabrielsson JL, Weiner DL. Methodology for pharmacokinetic/ pharmacodynamic data analysis. Pharm Sci Technol Today. 1999;2(6):244–52. 14. Chattergoon DS, Saunders EF, Klein J, et al. An improved limited sampling method for individualised busulphan dosing in bone marrow transplantation in children. Bone Marrow Transplant. 1997;20(5):347–54. 15. Hassan M, Fasth A, Gerritsen B, et al. Busulphan kinetics and limited sampling model in children with leukemia and inherited disorders. Bone Marrow Transplant. 1996;18(5):843–50. 16. Sandstro¨m M, Karlsson MO, Ljungman P, et al. Population pharmacokinetic analysis resulting in a tool for dose individualization of busulphan in bone marrow transplantation recipients. Bone Marrow Transplant. 2001;28(7):657–64. 17. Schuler U, Schroer S, Ku¨hnle A, et al. Busulfan pharmacokinetics in bone marrow transplant patients: is drug monitoring warranted? Bone Marrow Transplant. 1994;14(5):759–65.
52 18. Vassal G, Deroussent A, Challine D, et al. Is 600 mg/m2 the appropriate dosage of busulfan in children undergoing bone marrow transplantation? Blood. 1992;79(9):2475–9. 19. Balasubramanian P, Chandy M, Krishnamoorthy R, et al. Evaluation of existing limited sampling models for busulfan kinetics in children with beta thalassaemia major undergoing bone marrow transplantation. Bone Marrow Transplant. 2001;28(9):821–5. 20. Nieto Y, Vaughan WP. Pharmacokinetics of high-dose chemotherapy. Bone Marrow Transplant. 2004;33(3):259–69. 21. Purves RD. Optimum numerical integration methods for estimation of area-under-the-curve (AUC) and area-under-themoment-curve (AUMC). J Pharmacokinet Biopharm. 1992;20(3): 211–26. 22. Nerella NG, Block LH, Noonan PK. The impact of lag time on the estimation of pharmacokinetic parameters. 1. One-compartment open model. Pharm Res. 1993;10(7):1031–6. 23. Gill P, Murray W. Algorithms, for solution, of non-linear leastsquares problem. SIAM J Numer Anal. 1978;1978:977–92. 24. Ferraty F, Hall P, Vieu P. Most-predictive design points for functional data predictors. Biometrika. 2010;97(4):807–24. 25. Karalis V, Macheras P. Current regulatory approaches of bioequivalence testing. Expert Opin Drug Metab Toxicol. 2012;8(8): 929–42.
F. Sjo¨o¨ et al. 26. Sjo¨o¨ F, Aschan J, Barkholt L, et al. N-acetyl-L-cysteine does not affect the pharmacokinetics or myelosuppressive effect of busulfan during conditioning prior to allogeneic stem cell transplantation. Bone Marrow Transplant. 2003;32(4):349–54. 27. Sjo¨o¨ F, Hassan Z, Abedi-Valugerdi M, et al. Myeloablative and immunosuppressive properties of treosulfan in mice. Exp Hematol. 2006;34(1):115–21. 28. Chunduri S, Dobogai L, Peace D, et al. Fludarabine/i.v. BU conditioning regimen: myeloablative, reduced intensity or both? Bone Marrow Transplant. 2008;41:935–40. 29. Lindley C, Shea T, McCune J, et al. Intraindividual variability in busulfan pharmacokinetics in patients undergoing a bone marrow transplant: assessment of a test dose and first dose strategy. Anticancer Drugs. 2004;15(5):453–9. 30. Grochow LB. Busulfan disposition: the role of therapeutic monitoring in bone marrow transplantation induction regimens. Semin Oncol. 1993;20(4 Suppl 4):18–25 (quiz 26). 31. Tsuruta H, Fukumoto M, Bax L, et al. Limited sampling strategies to estimate the area under the concentration–time curve. Biases and a proposed more accurate method. Methods Inf Med. 2012;51:383–94. 32. Canal P, Chatelut E, Guichard S. Practical treatment guide for dose individualisation in cancer chemotherapy. Drugs. 1998; 56(6):1019–38.
