Introduction

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Empirical models such as the (sigmoid) E_max model have been used to relate the drug exposure, usually AUC, of anticancer drugs to the nadir of leukopenia or neutropenia for clinical (Egorin et al., 1986; Hantel et al., 1990; Erlichman et al., 1991) and preclinical data (Simonsen et al., 2000). Others have used the time above a threshold concentration to relate exposure to the dose-limiting toxicity (Huizing et al., 1993). A more general empirical model, which can predict dependence of AUC, dependence of a time above a threshold concentration, and relationships between these two extremes, has been proposed (Karlsson et al., 1998). Also, the time course of leukopenia has been described and quantified using an empirical model with spline functions (Karlsson et al., 1995).

Empirical models are relatively easily derived and often useful to describe the data within a study. However, to be able to better characterize the mechanisms underlying the effects and get better predictions of other doses and schedules than those investigated, more mechanistic models are preferable. An effort in this direction was taken when an indirect model with a lag time was used to describe leukopenia after paclitaxel administration (Minami et al., 1998). We have recently developed a semiphysiological model of neutropenia after different schedules of 2'-deoxy-2'-methylidenecytidine (DMDC), where we used transit compartments to mimic the maturation chain in the bone marrow (Friberg et al., 2000). However, neither of these presented models is able to characterize rebound phenomena.

5-Fluorouracil (5-FU) is a widely used anticancer agent and despite its long clinical use, the optimal dosage regimen of 5-FU is still debated. Myelotoxicity is dose limiting to patients after an i.v. injection, whereas for an i.v. infusion gastrointestinal toxicity is dose limiting (Pratt et al., 1994). Fractionated injections of 5-FU over several days are more toxic to rats than a single dose (Looney et al., 1978), whereas the same relationship between AUC and the leukopenic effect in rats has been shown for single and fractionated injections within 1 day (Simonsen et al., 2000). However, the fact that the pharmacokinetics of 5-FU is nonlinear (Collins et al., 1980) with circadian variations (Petit et al., 1988) is often neglected in discussions of 5-FU's schedule dependence. Therefore, after different rates of administration, deviations from AUC dependence, rather than deviations from dose dependence, should be used when considering schedule-dependent effects.

As far as we know, no model has been described that can characterize the whole time course of anticancer drug effects on the leukocytes, i.e., no model that can also account for the rebound effects. Limitations in developing such a model from clinical data are that the next course of treatment is generally started before the leukocytes return to baseline and schedules challenging the adverse event systems are not ethical to administrate. This type of model is therefore preferably developed from animal data. In the present study we followed the decrease and the rebound in leukocytes after a single injection and after two fractionated injection schedules of 5-FU in rats and extended the model with transit compartments to also account for the rebound phenomena.

	Materials and Methods

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Animals. Twenty-four male Sprague-Dawley rats [Crl:CD(SD)BR-rats; Charles River, Uppsala, Sweden] weighing 288 ± 21 g (mean ± S.D.) at the first day of treatment, were used. During the whole study, the animals had free access to food and water and they were kept in a room illuminated from 7:00 AM to 7:00 PM. The study was approved by the Animal Ethics Committee in Uppsala.

Treatment. The rats were randomized into three treatment groups and one control group with six rats in each group. One group (single group) received a single 5-FU injection (Flurablastin; Pharmacia & Upjohn, Stockholm, Sweden) of 127 mg/kg on average on day 0. In a pilot study the nominal dose of 125 mg/kg was shown to produce a significant, but acceptable, hematological toxicity. The double group received two injections of 63 mg/kg each, on day 0 and 2, and the triple group was given three injections of 49 mg/kg, on day 0, 2, and 4. The double and triple groups were expected to receive 67 and 60% of the AUC in the single group, respectively. Their doses were calculated from pharmacokinetic parameters derived in a previous study (Simonsen et al., 2000). The lower exposures in the repeated regimens were due to gastrointestinal toxicity (under Results). A control group received single, double, or triple injections of normal saline (Pharmacia & Upjohn) according to a similar schedule. All injections were i.p. at 2:00 PM.

Hematology Determination. Blood (250 µl) for determination of number of leukocytes, platelets, and hemoglobin was drawn from a hind paw vein. Samples were collected at baseline (1 day before the first injection), on the day after the first injection (day 1) in half of the rats, on day 2 in the other half of the rats, and then in all rats every other day from day 3 to 23 or 25. Then the leukocytes were considered to have returned to baseline. The blood was collected in EDTA-prepared Microtainer tubes (Becton Dickinson, Franklin Lakes, NJ) between 9:00 and 11:00 PM and analyzed in a Coulter counter (Coulter Electronics Ltd., Luton, England) within 4 h. The results are expressed as mean ± S.D.

Pharmacokinetic Sampling. Pharmacokinetics was studied on the first day of drug administration. Two or three blood samples (250-500 µl) from each rat were drawn in a sampling interval of 15 to 120 min after the injection. The control rats were treated similarly. The samples were collected in EDTA-prepared Microtainer tubes and directly chilled and centrifuged for 5 min at 10,000 rpm. The plasma was frozen on dry ice and kept at 80°C until analysis.

Chemical Assay. 5-FU used for preparation of standards and quality controls was purchased from Sigma Chemical Co. (St. Louis, MO). Drug concentrations were determined in 150 µl of plasma by high-performance liquid chromatography with ultraviolet detection after an extraction step with ethyl acetate (Merck, Darmstadt, Germany). The method has previously been described (Koks et al., 1990) but we modified the mobile phase (0.05 M phosphate buffer, pH 4.6) slightly by adding 0.75 ml of tetrahydrofuran (Merck) per 1000 ml of phosphate buffer to increase the selectivity in rat plasma. 5-Bromouracil (Sigma Chemical Co.) was used as internal standard. At three investigated concentrations (51, 770, and 39,000 ng/ml), the coefficient of variation within and between days was 2 to 3% and the accuracy was 93 to 101%. At the limit of quantification, 25 ng/ml, the coefficient of variation was 4% and the accuracy was 94%. Plasma samples with concentrations expected to be above the upper limit of the assay, 50,000 ng/ml, were diluted with blank rat plasma before workup.

Pharmacokinetic Model Building. Apart from the plasma concentration data derived from the present study, additional data from a previous 5-FU study (Simonsen et al., 2000), performed by us under similar experimental conditions with sparse sampling from a hind paw vein was included in the population pharmacokinetic model building. Models with linear, nonlinear, and mixed linear and nonlinear elimination were tried. For the best model, individual parameters, total AUCs, and individual concentration-time profiles were determined as empirical Bayes estimates.

Semiphysiological Model Building. A model for the studied system was constructed, where compartments in series mimicked the leukocyte maturation stages (Fig. 1). The only loss of cells within the delay chain is into the next compartment when no chemotherapy is present. However, in the presence of chemotherapy, the cells in the replicating compartments, i.e., the sensitive cells, can be destroyed. A delay chain with transit compartments delays and spreads out the effects of cell production and chemotherapy (Kahn et al., 1991). For models with the same total transit time, an increased number of compartments reduces the variability in the cellular transit time. Consequently, the entire time course of leukopenic effects can be characterized by means of compartments in series.

View larger version (10K): [in this window] [in a new window]   Fig. 1.   Model with transit compartments to mimic leukocyte maturation. A number of compartments with replicating cells are connected in series with a number of compartments of nonreplicating cells and a compartment of circulating leukocytes. kin is the rate into the first compartment and the rate constants from the delay chain compartments (k) can all be the same or differ for different compartments. The half-life of the circulating leukocytes determines the elimination rate constant (ln2/t1/2,circ). From the compartments with replicating cells, i.e., sensitive cells, an elimination rate constant describes the killing rate due to chemotherapy.

Data Analysis. The first order method within NONMEM, version VI (beta) (Beal and Sheiner, 1992), was used in the pharmacokinetic and pharmacodynamic model building. To discriminate between hierarchical models the difference in objective function values (2log likelihood) was used because this difference is approximately chi square distributed. The criterion for inclusion of a parameter was a decrease in the objective function value of 10.83 (P < .001), whereas a decrease of 3.84 was applied as a criterion for adding an extra compartment in the delay chain, which requires no extra parameters. Graphic analysis of the predictions and residuals was performed within the program Xpose, version 2.0 (Jonsson and Karlsson, 1999). Interindividual variability on the parameters was considered and additive and/or proportional error models were tried. It was also tested to use log-transformed data in the modeling to satisfy the assumption of symmetrical errors.

	Results

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General Toxicity. The rats in the triple group lost the most weight, on average 16% of the weight at baseline. Two rats in this group had pronounced diarrhea. In the single and double group the average maximum weight losses were 5 and 12%, respectively. However, one rat in the double group died unexpectedly 10 days after the first dose. The cause of death is unknown. No signs of abdominal toxicity were found in this rat or any other rat by the day of sacrifice.

Hematology. At baseline the respective observed leukocyte counts in the single, double, triple, and placebo groups were 15.1 ± 3.1, 14.7 ± 2.1, 17.6 ± 4.2, and 15.3 ± 3.0 × 10⁹/l. There was no significant difference in the baseline values between the groups (P > .05, one-factor ANOVA; Statistica, Statsoft, Tulsa, OK). All 5-FU-treated rats showed a transient decrease in leukocytes (Fig. 2) and in all but one rat in the triple group, leukocyte levels over the baseline value were found, 13 to 15 days after the first dose. The observed absolute leukocyte counts at nadir were 5.5 ± 1.1, 5.3 ± 1.2, and 5.0 ± 2.2 × 10⁹/l in the single-, double-, and triple-dose groups, respectively, and correspond to 37 ± 8, 36 ± 7, and 28 ± 9% of baseline. The respective maximum leukocyte counts were 159 ± 19, 156 ± 23, and 212 ± 85% of baseline.

View larger version (34K): [in this window] [in a new window]   Fig. 2.   Observed number of leukocytes (), a Loess smoother (S-PLUS, version 2000; Mathsoft Inc., Seattle, WA) through the observations (---), and the model population predicted number of leukocytes over time () in control rats and after three injection schedules of 5-FU. Arrows indicate days of 5-FU dosing.

Pharmacokinetics. A one-compartment model with capacity-limited elimination using the Michaelis-Menten relationship described 5-FU pharmacokinetics the best (Fig. 3). For rats dosed in the afternoon, as in the present study, the population values of the maximal rate of elimination, V_max, and the concentration at half of V_max, K_m, were estimated to 108 mg/l/h (RSE = 5.5%) and 22 mg/l (RSE = 7.7%), respectively. For rats dosed in the morning, which was the case for some rats in the other study used in the pharmacokinetic analysis, V_max was found to be 12% higher. The interindividual variability of V_max and K_m showed a positive correlation and was estimated to 21% (CV). The volume of distribution was 0.24 liter (RSE = 2.4%) and inclusion of interindividual variability on this parameter did not improve the model significantly. An additive and a proportional component were included in the residual error model and they were estimated to 380 mg/l and 14%. Derived mean ± S.D. AUCs were 137 ± 16, 76 ± 7, and 72 ± 3 mg · h/l in the single, double, and triple group, respectively.

View larger version (13K): [in this window] [in a new window]   Fig. 3.   Observed concentrations () and predicted concentration-time curve () after the first 5-FU injection in each group.

Semiphysiological Model. The final semiphysiological model could well describe the leukocyte data (Fig. 2). Two 5-FU-sensitive, two 5-FU-insensitive transit compartments, and a circulating leukocyte compartment were included in the model (Fig. 4; Table 1). Allowing different delay rate constants (k_delay) from the sensitive and from the insensitive compartments did not improve the model. However, second order delay rate constants gave a marked improvement of the fit over first order rate constants. When estimated, the order (amplification factor) of k_delay was 1.98. Inclusion of a tissue compartment in equilibrium with the circulating leukocyte compartment did not improve the model significantly.

View larger version (15K): [in this window] [in a new window]   Fig. 4.   Semiphysiological model of circulating leukocytes after 5-FU injections. A slope times the concentration of 5-FU (slope · conc5-FU) described the 5-FU-dependent elimination rate of cells from the sensitive compartments. kin was determined to the number of leukocytes in the circulating compartment at baseline times the rate constant of disappearance of the circulating cells (ln2/t1/2,circ). The rate constants from the delay chain compartments (kdelay) were of second order. A negative feedback from the circulating leukocytes to affect kin was included.

View larger version (28K): [in this window] [in a new window]   Fig. 5.   External validation of the final semiphysiological model with a data set from a previous study (Simonsen et al., 2000). Shown are mean observed () and from the final model mean predicted (×) leukocyte counts expressed as percentage of baseline. The error bars represent the 90% predicted intervals of the leukocyte count means in each of the 100 simulated data sets. For the multiple dosing regimens, the dosing intervals were 6 and 4 h for the double and triple injections, respectively.

	Discussion

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5-FU disposition in rats is similar to that in humans. As in the present study, the best pharmacokinetic model in patients was a one-compartment model with nonlinear elimination in a study performed by our group (Sandström et al., 1996). V_max, K_m, and V were 105 mg/l/h, 27 mg/l, and 0.35 l/kg, respectively, compared with 108 mg/l/h, 22 mg/l, and 0.83 l/kg in rats in the present study. Moreover, repeated short infusions of 5-FU produce a transient decrease and rebound in leukocytes in patients (Moore et al., 1968), similar to what is observed in the present study. The rat is well suited for in vivo hematology investigations (Ulich and del Castillo, 1991), although a smaller rodent, the mouse, is mainly used. The mouse has been shown to be able to predict myelosuppressive effects of many anticancer drugs well (Schurig et al., 1986). However, rats can withstand serial bleedings. This together with sparse sampling and population analysis offers the opportunity to study pharmacokinetics and follow hematology in the same individual and establish pharmacokinetic-pharmacodynamic relationships. Sparse sampling and population analysis can provide accurate estimates of pharmacokinetic parameters and reliable predictions in dose-exposure-response relationships (Collart et al., 1992; Sheiner and Wakefield, 1999) and have been stressed in preclinical studies (Nedelman et al., 1993; Burtin et al., 1996).

The fractionated schedules of 5-FU showed a similar or a more pronounced decrease in leukocytes than the single group despite the fact that the total AUCs in the fractionated groups were only 52 to 56% of the AUCs in the single group. The previously developed E_max model for 5-FU-injections within 1 day (Simonsen et al., 2000) predicts the nadir values in the present study after the observed total AUCs to 33, 41, and 42% of baseline for the single, double, and triple group, respectively, compared with the observed 37, 36, and 28%. The more than expected decrease in leukocytes in the triple group could be explained by the hypothesis that primitive hematopoietic stem cells cycle too slowly to be sensitive to a single dose of 5-FU (Harrison and Lerner, 1991). However, the first dose might stimulate the cycling activity of the primitive cells and therefore, the cells are vulnerable to doses administered 3 to 5 days after the first dose (Harrison and Lerner, 1991).

As the presented semiphysiological model predicts, 5-FU has been shown to drastically reduce the number of myeloid- and lymphoid-proliferating cells (Yeager et al., 1983; Vetvicka et al., 1986). In contrast to the AUC-dependent model previously derived (Simonsen et al., 2000), the more mechanistic model presented here could predict the in the present study observed schedule-dependent decrease in leukocytes. The fractionated regimens also produced more weight loss and gastrointestinal toxicity compared with the rats in the single group. Therefore, a fractionated injection regimen of 5-FU was concluded to be more toxic, both with regard to hematological and gastrointestinal toxicity, than a single injection and the leukopenic effect of 5-FU was not dependent on AUC when extending the fractionation over several days.

In our study all schedules produced a rebound in leukocytes, implying an extensive production capacity of leukocytes upon stress. However, the complexity of hematopoietic cell regulation is not fully understood. It is known that circulating leukocytes produce hematopoietic growth factors such as colony-stimulating factors and interleukins, continuously or upon stimulation (Testa and Dexter, 1999). Some of these factors stimulate or inhibit the proliferating hematopoietic cells, directly or indirectly, and hence affect the leukocyte production. For example, the grade of chemotherapy-induced neutropenia has been shown to correlate with elevated levels of colony-stimulating factors known to increase the neutrophil production (Cebon et al., 1994). When the number of circulating neutrophils starts to increase, the levels of colony-stimulating factors decrease. It has therefore been proposed that circulating neutrophils are involved in the degradation of colony-stimulating factors (Takatani et al., 1996; Ericson et al., 1997). After a 5-FU injection the cycling activity of primitive cells increases (Harrison and Lerner, 1991), i.e., k_in increases. Our feedback mechanism from the circulating leukocytes on k_in seems therefore reasonable.

Inclusion of amplification factors, i.e., second order rate constants from the delay chain compartments, improved our model. The effect of multiplicity, described by an amplification factor on a rate constant into a transit compartment, has previously been used in simulations (Sun and Jusko, 1998). In the present model, the multiplicity of cells was accompanied with a decreased maturation time as the number of maturing cells increased. Inclusion of amplification factors in the model can therefore be interpreted as part of the regulation of the leukocyte production because hematopoietic growth factors lead to extra cell divisions and they can also shorten the transit times within the maturation chain (Testa and Dexter, 1999). When the number of cells within a compartment is below its baseline an amplification factor higher than one will lead to a prioritization of increasing the number of cells within the compartment before transferring the cells to the next compartment. Second order rate constants within the delay chain will consequently lead to a rapid and pronounced change in the number of circulating cells, as we observed. A model including multiplicity only from the sensitive compartments and with different effects on the maturation time for sensitive and insensitive compartments would have been more logical, but not deemed possible to characterize with the present data.

A posterior predictive check was performed to assess the validity of the model. The model characterized the time course and magnitudes of the decrease and rebound in leukocytes well, except for an underestimation of the maximum leukocyte level in the triple group. A probable reason for the pronounced rebound observed in the triple dose group is that when primitive stem cells are affected, the system becomes more stimulated. Attempts to find model parameters to characterize this pronounced and quick rebound were not successful. It is likely that more information about the system is necessary to be able to fully characterize the rebound after this type of schedule because the rebound depends on the functional capacity of the system rather than on the drug. Mechanistic models of rebound phenomena have been described for other physiological systems (Lima et al., 1989; Brynne et al., 1999; Gries et al., 1999), but not for rebound effects with magnitudes of this size.

Compared with experimental findings with labeled leukocytes in untreated rats, our estimated leukocyte half-life in systemic circulation of 16 min is somewhat short. In rats, about 85% of the circulating leukocytes are lymphocytes and their half-life in blood has been determined to be around 30 min (Westermann et al., 1988). The respective half-lives of circulating neutrophils and monocytes have been estimated to 5.7 h (Gerecke et al., 1973) and 12 to 13 h (Syren, 1974) in rats. However, with only daily leukocyte counts, the temporal resolution is limited. For neutrophils the total blood pool is known to consist of circulating cells and cells that are marginating along the walls of vessels. The circulating and marginating pools are about equal in size and are in complete and rapid equilibrium with each other (Vincent, 1977). However, it is likely that the equilibrium is too rapid to be captured in a kinetic analysis and inclusion of such an equilibrium tissue compartment did not improve our model fit.

The complexity of the present model is limited by the fact that it is developed from only one type of observation, i.e., number of circulating leukocytes. However, even though the model contains relatively few parameters it can describe the data well. It incorporates many important structural features of the hematopoietic system; cell maturation, 5-FU cytotoxicity, feedback from circulating cells, cell amplification, maturation time changes, rebound, and degradation of circulating cells. The idea to mimic the transient decrease of leukocytes after chemotherapy with transit compartments was presented with simulations by Kahn et al. (1991). We have previously developed a model of the schedule-dependent effects of DMDC on neutrophils with drug-sensitive and drug-insensitive transit compartments (Friberg et al., 2000). However, the model presented here predicted the schedule-dependent leukopenic effects of 5-FU and included a feedback mechanism and second order rate constants to explain the rapid rebound. The model was also shown to be valid for other doses and injection schedules of 5-FU. Therefore, this study adds to the knowledge of the schedule dependence of 5-FU and after adaptation to patients, this model might be a useful tool for quantifying and predicting leukopenia in the clinical setting.