J Pharmacokinet Pharmacodyn DOI 10.1007/s10928-015-9440-2
ORIGINAL PAPER
Pharmacodynamic model for chemoradiotherapy-induced thrombocytopenia in mice Wojciech Krzyzanski1 • Juan Jose Perez-Ruixo2 • John Harrold2
Received: 3 June 2015 / Accepted: 26 August 2015 Ó Springer Science+Business Media New York 2015
Abstract A mechanistic model describing the effects of chemotherapy and radiation on platelet counts and endogenous thrombopoietin (eTPO) in mice was developed. Thrombocytopenia was induced in mice by injection of carboplatin followed by the whole body irradiation on days 0, 28, and 56, with platelet and eTPO samples collected over 84 days. The pharmacodynamic model consisted of a series of aging compartments representing proliferating megakaryocyte precursors, megakaryocytes, and platelets with possible eTPO clearance through internalization. The cytotoxic effects of treatment were described by the kinetics of the effect (K-PD) model, and stimulation of platelet production by eTPO was considered to be driven by receptor occupancy. The proposed PD model adequately described the platelet counts and eTPO concentrations in mice by accounting for nadirs and peaks of platelet count, and rebounds in eTPO time course profiles. The estimates of model parameters were in good agreement with their physiological values reported in literature for mice with platelet lifespan of 4.3 days and 185 cMpl receptors per platelet. The predicted duration of the treatment effect was 0.82 h (approximately 5 carboplatin half-lives in mice). The data was not informative about the
Electronic supplementary material The online version of this article (doi:10.1007/s10928-015-9440-2) contains supplementary material, which is available to authorized users. & John Harrold
[email protected] 1
Department of Pharmaceutical Sciences, University at Buffalo, Buffalo, NY, USA
2
Clinical Pharmacology, Modeling and Simulation, Amgen Inc., One Amgen Center Dr, Thousand Oaks, CA 91360, USA
eTPO stimulatory effect as the nominal precursor production rate was sufficient to account for platelet response to treatment. The model quantified the inverse relationship between eTPO levels and platelet counts and offered an explanation of the tolerance effect observed in the eTPO data. The simulated rebound in free receptors levels correlated with rebounds in eTPO levels. The model suggests that the duration of the toxic effects is determined by the turnover of the proliferating cells in the bone marrow. This indicates that the lifespan of the target cells (megakaryocyte precursors, megakaryocytes and platelets) is a key determinant in the duration of both drug exposure and toxicity due to treatment. The model can be extended to account for pharmacokinetics of exogenous drugs and be applied to analysis of human data. Keywords Chemoradiation Receptor-mediated disposition Thrombocytopenia Thrombopoietin
Introduction Thrombopoietin (TPO) is a hematopoietic growth factor (molecular weight, 95 kD) that is the primary regulator of platelet production. TPO is a ligand of the cMpl receptor that is found on megakaryocyte precursor cells, megakaryocytes and platelets, as well as on stem cells and early bone marrow progenitor cells of all lineages. Upon binding to its receptor, TPO initiates STAT 5, PI3K, and MAPK signaling pathways resulting in increased cell cycling and survival [1]. Activated cMpl receptors expressed on megakaryocyte progenitor cells enhance their survival, proliferation, and differentiation yielding an increase in megakaryocyte polyploidy and numbers, and subsequently in elevated platelet counts. TPO is constitutively produced by liver, kidney and skeletal
123
J Pharmacokinet Pharmacodyn
muscle, while the major mechanism for clearance is binding to the cMpl receptors [2]. Chemotherapy and/or radiotherapy often result in myelosuppression that manifest clinically as cytopenias, which include thrombocytopenia, a below-normal platelet count that can lead to life threatening hemorrhages. Thrombocytopenia is a dose-limiting toxicity for many cytotoxic drugs, including platinum compounds, and radiotherapy [3]. Modeling of myelosuppresive effects of chemotherapy has been widely used to quantify the extent and duration of cytotoxic effects on cells of major hematopoietic lineages: granulocytes, erythrocytes [4], and platelets [5]. TestartPaillet and colleagues provided a review of existing approaches to model hematological toxicities induced by chemotherapy [6]. A general motif found in these mechanistic models is a catenary series of compartments representing stages of cell development. Cells originating in a proliferation compartment in the bone marrow pass through transit compartments representing the different stages of maturation with the final mature cells entering circulation. [7]. The toxic effects of the chemotherapy on proliferating cells are generally driven by drug exposure variables, mainly drug plasma concentrations. Toxicity is represented by either irreversible removal (killing) of the proliferating cells [8] or inhibition of the proliferation rate [7]. Another feature of the myelosuppression models is presence of the negative feedback from the circulating cells affecting the production rate of the proliferating cells [7]. Developing megakaryocytes in the bone marrow and circulating platelets exposed to cytotoxic effects of chemotherapeutic agents and irradiation have been one of myeloid systems successfully described by the transit compartment models [9–11]. Apart from characterizing the toxic effect of treatment, the mechanistic models can be used to quantify and predict the outcomes of the supportive therapies aiming at ameliorating cytopenias [12– 14]. Such an approach can be applied to the design and evaluation of clinical trials to guide selection of optimal sampling times and contribute to knowledge of optimal treatment regimens [15]. The objective of this report was to develop a mechanistic model capable of (1) describing the toxic effects of chemotherapy and radiation treatment (CRT) on platelet count and (2) quantifying the role of endogenous thrombopoietin (eTPO) on platelet production. The model was applied to simultaneously fit eTPO and platelet data from mice with carboplatin and radiation induced thrombocytopenia. To better understand the relationship between the chemoradiation injury and recovery to normal homeostasis the estimated parameter values were used to simulate the cellular responses to treatment. Specifically the impact on cMpl receptor occupancy in the non-observable compartments in the bone marrow was analyzed.
123
Methods Experimental details Healthy female B6D2F1 mice were obtained from The Jackson Laboratory (Bar Harbor, Maine). Both the control and CRT cohorts consisted of animals of 9–11 weeks of age and were given purified water and sterilized food ad libitum. Protocols associated with this study were approved by the Institutional Animal Care and Use Committee within Amgen (Thousand Oaks, CA). Mice assigned to CRT cohort received carboplatin 62 mg/kg intravenously, and 4 h later 5 Gray (Gy) total body radiation from a 137Cs source (Gamma Cell 40 Irradiator, Best Theratronics, Ottawa, Ontario, CA). The doses of carboplatin and radiation were decreased to 90 % of the cycle 1 dose (56.25 mg/kg carboplatin and 4.5 Gy) for cycle 2 and cycle 3, which were administered 28 days apart. Mice assigned to the control cohort were handled and fed as the CRT cohort but were not exposed to any injury. As such the control animals received neither radiation nor chemotherapy. Blood samples were collected to assay eTPO and obtain platelet counts on days 0, 1, 2, 3, 4, 5, 6, 8, 10, 13, 16, 19, 22, 25 and 28 of each cycle (n = 5 per time point). Two samples were taken per mouse, once early in the cycle (via retro-orbital sinus) and a second terminal sample (via cardiac puncture) late in the cycle. For complete blood count analysis, blood was transferred to tubes containing EDTA and analyzed using an Advia 2120 blood analyzer (Siemens, Tarrytown NY). Serum was obtained by allowing blood samples to clot at room temperature for 20 min, followed by centrifugation at 10,000 rpm for 10 min at 40 °C. Serum was stored at -200 °C until analyzed for eTPO levels using the Quantikine Mouse TPO Immunoassay kit (R&D Systems, Minneapolis, MN). Thrombopoiesis model following chemoradiation treatment The thrombopoiesis model developed to describe both the platelet homeostasis and the response to chemoradiation is shown in Fig. 1. The model assumes eTPO is produced at a zero-order rate constant, kTPO, and eliminated by two processes: linear clearance represented by a first-order rate constant, kel, and saturable cMpl receptor binding (kon) and dissociation (koff) followed by internalization (kint) and degradation. The cMpl receptor is expressed on megakaryocyte proliferating precursors encompassing burst-forming unit-megakaryocytes and colony-forming unit-megakaryocytes (Pi); megakaryoblasts, promegakaryocytes, and megakaryocytes (Mi); and platelets (PLTi). Each of these populations consists of n = 10 aging compartments and cells mature from one compartment to the next one at the respective
J Pharmacokinet Pharmacodyn
kTPO
k el
+
+
+
+
R P1
R Pn
R M1
R Mn
k on
k on
k o ff
k on
k o ff
R C Pn
R C P1
2
d iv
2
d iv
...
P1 n /T P
T x ( t)
R C M1
n /T P
2
Pn
k on
k o ff
k in t
k in t
k in
eTPO
+ R PLT1 k on
k o ff
R C Mn
Mn
... n /T M
n /T M
k on
R C PLT1
k o ff
R C PTLn k in t
η M1
R PLTn
k o ff
k in t
k in t
d iv
n /T P
+
PLT1 n /T M
k in t
... n /T P L T n /T P L T
PLTn n /T
PLT
T x ( t)
Fig. 1 Model structure of CRT in mice: Endogenous erythropoietin (eTPO) can bind to receptors on precursor cells (Pi), megakaryocytes (Mi) and platelets (PLTi) to form the complexes RCPi, RCMi, and RCPLTi, respectively. The toxic effects (TX) of CRT shown are killing
the precursor cells. The continuous arrows indicate transfer processes and the dashed arrows stand for control processes. The box represents the stimulatory effect of eTPO on production of P1 cells
first-order rates, n 9 2div/TP, n/TM, and n/TPLT. Where TP, TM, and TPLT are the mean population lifespans for P, M, and PLT populations, respectively, and div = 2 denotes the number of divisions in each proliferating compartment Pi. Additionally, the transition rate n/TM from the last megakaryocyte aging compartment, Mn, to the youngest platelet compartment, PLT1, represented the shedding of platelets to the circulation and, consequently, it was multiplied by a factor g = 4000 accounting for an average number of platelets released by one megakaryocyte [1]. The cMpl receptor binding and dissociation is assumed to be at equilibrium with the bound receptors on each cell population:
The equilibrium dissociation constant, KD = koff/kon was fixed at the literature value of 38 pM reported for TPO in rats [16]. The free (unbound) eTPO concentration is calculated from the total TPO (TPOtot) and the total receptor (Rtot) according to the Eq. (3) [17]: 1 eTPO ¼ Rtot eTPOtot þ KD 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð3Þ þ ðRtot eTPOtot þ KD Þ2 þ 4KD eTPOtot
RCPi ¼ RtotPi RO; RCMi ¼ RtotMi RO; RCPi ¼ RtotPLTi RO ð1Þ where RtotPi, RtotMi, and RtotPLTi are the total receptor concentrations in Pi, Mi, and PLTi compartments, respectively, and RO is the fractional receptor occupancy defined as follows: RO ¼
eTPO KD þ eTPO
ð2Þ
where Rtot is the sum of the total receptors expressed on precursors (RtotP), megakaryocytes (RtotM), and platelets (RtotPLT) as follows: Rtot ¼ RtotP þ RtotM þ RtotPLT
ð4Þ
The mathematical expressions for RtotP, RtotM and RtotPLT are defined later in Eqs. (13, 14). The total eTPO plasma concentration is described by a differential equation [17]: deTPOtot ¼ kTPO kel eTPO kint Rtot RO dt
ð5Þ
123
J Pharmacokinet Pharmacodyn
The dynamics of the P cell populations are controlled by the eTPO stimulation of the production of the youngest precursors, S(RO), and the toxic effect of CRT, Tx(t):
kTPO ¼ kel eTPO0 þ kint Rtot0 RO0
dP1 n n ¼ SðROÞ P1 TxðtÞP1 TP TP dt
ð6Þ
The free eTPO serum concentration at baseline (eTPO0) and stead-state receptor occupancy (RO0) are defined by evaluating Eq. (2) at steady-state:
ð7Þ
RO0 ¼
dPi n2div n ¼ Pi1 Pi TxðtÞPi ; TP dt TP
i ¼ 2; . . .; n
Since there is no eTPO stimulatory or toxic effect of chemoradiation treatment on megakaryocytes, then P cell in nth compartment become M cells as follows: dM1 n n ¼ P1 M1 TP TM dt dMj n ¼ Mj1 Mj ; TM dt
ð8Þ j ¼ 2; . . .; n
ð9Þ
and platelets: dPLT1 gn n ¼ Mn PLT1 TM TPLT dt dPLTj n ¼ PLTj1 PLTj ; TPLT dt
ð10Þ j ¼ 2; . . .; n
ð11Þ
The cell count for P, M and PLT is the sum of cell counts in the aging compartments: P ¼ P1 þ þ P n ; M ¼ M 1 þ þ M n PLT ¼ PLT1 þ þ PLTn
and ð12a; b; cÞ
Only the concentrations of the total receptors are necessary to be known to define eTPO (see Eq. 3). The total receptors on the P cell population are assumed to be proportional to total number of cells: RtotPi ¼ nP P
ð13Þ
The total receptors on platelets are expressed by the power function of PLT: PLT c RtotPLT ¼ nPLT PLT0 ð14Þ PLT0 The total receptors on megakaryocytes are assumed to be proportional to the cell count for younger cells and the power of the cell count for the oldest megakaryocytes: Mn c RtotMn ¼ nM M1 þ þ Mn1 þ Mn0 ð15Þ Mn0 where Mn0 is the baseline value for Mn and PLT0 denotes the baseline value for all circulating platelets. The baseline values for Pi, Mi, and PLTi are determined from steadystate analysis of Eqs. (6)–(11): TM PLT0 ; gTPLT i ¼ 1; . . .; n
Mi0 ¼
123
Pi0 ¼
TP Mi0 ; TM
PLTi0 ¼
PLT0 ; n ð16a; b; cÞ
Similarly, the steady-state solution for Eq. (5) implies
eTPO0 KD þ eTPO0
ð17Þ
ð18Þ
The initial conditions for Eqs. (5)–(11) are the baseline values. The stimulatory function S(RO) is assumed to be on-and-off as reported previously for another cMpl receptor agonist [18]. 1; if RO ROthr SðROÞ ¼ ð19Þ Smax ; if RO [ ROthr where ROthr is the threshold receptor occupancy triggering the maximal effect, Smax. In the absence of the carboplatin plasma concentrations and radiation measurements, the kinetics of treatment effect (K-PD) approach was used to describe the cytotoxic effect [19]. An on-and-off effect was used as described in Eq. (20): Kmax ; if tj t tj þ TTx TxðtÞ ¼ ð20Þ 0; otherwise where tj denotes the start times of chemoradiation and TTx can be interpreted as the duration of the toxic effect. For the cell killing model (20) only Kmax and TTx parameters were estimated. Data analysis The naı¨ve pooled eTPO and PLT data from individual animals were simultaneously fitted to Eqs. (3) and (12c). The eTPO serum concentration was converted to molar concentrations assuming the molecular weight of 95 kD [2]. Observed data in the control group was fitted to determine PLT0 and eTPO0, and these estimates were fixed when fitting the data in the treatment group. The eTPO data were log transformed and a proportional error model was applied to describe eTPO residual variability in the logdomain: logðeTPOðti ÞÞ ¼ logðYeTPOi Þð1 þ eeTPO Þ
ð21Þ
The residual variability for platelet observations was described by the proportional error model as follows: PLTðti Þ ¼ YPLTi ð1 þ ePLT Þ
ð22Þ
where YeTPOi and YPLTi are observed eTPO and PLT values at time ti. The random variables eeTPO and ePLT were assumed to be independent and normally distributed with the mean 0 and variances r2eTPO and r2PLT, respectively.
J Pharmacokinet Pharmacodyn
The model performance was evaluated by the observed versus predicted diagnostic plots as well as superposition of observed versus time and predicted versus time plots. Parameter estimation was performed by the first order conditional estimation method in NONMEM 7.2 (Icon Development Solutions, Ellicot City, MD). Model-based simulations The estimates of the model parameters were used to simulate time courses of eTPO, P, M, and PLT following one and two cycles of CRT using Eqs. (3) and (12). Another set of simulations was performed to obtain the dynamics of the total cMpl receptor pool and the total cMpl receptors expressed on P, M, and PLT cell populations. The latter were calculated from Eq. (13) whereas the total receptors were calculated from Eq. (4). Free receptors were calculated as the difference between total and bound: RP ¼ RtotP RCP ; RM ¼ RtotM RCM ; RPLT ¼ RtotPLT RCPLT
ð23Þ
with the free receptor pool expressed on all cells R ¼ RP þ RM þ RPLT
ð24Þ
Simulations were executed in NONMEM 7.2 applying the control stream used for parameter estimation (See Online Resources 2 and 3).
PLT baseline) that starts on day 38 and extends for 3 days. The next peak of 1021 9 109 cells/L (84.0 % of PLT baseline) occurs on day 56, which coincides with the beginning of the third treatment cycle. The nadir of 30 9 109 cells/L (2.5 % of PLT baseline) is achieved on day 64 and lasts for 5 days followed by a gradual increase to reach the count of 408 9 109 cells/L (33.6 % of PLT baseline) on day 84, the last observation time. The eTPO concentrations follow the PLT time course in the opposite direction implicating that the binding eTPO to PLT plays a significant role in its disposition. The mean pretreatment eTPO0 concentration (eTPO baseline) persists at 3086 pg/mL for 5 days after the first CRT administration. Subsequently, eTPO rapidly increases to reach a peak of 6202 pg/mL (200 % of eTPO baseline) on day 10, followed by a decline which reaches the baseline between days 16 and 19 and continues to decrease to achieve a nadir of 2440 pg/mL (79.1 % of eTPO baseline) on day 19. The nadir lasts for 12 days and the eTPO levels rebound to reach another peak of 6360 pg/mL (206 % of eTPO baseline) on day 38. The sharp peak is followed by a rapid decline towards a nadir of 1080 pg/mL (35.0 % of eTPO baseline) on day 53. The increase following the nadir ended at the peak of 5448 pg/mL (177 % of eTPO baseline) reached on day 68. After the last peak eTPO serum concentrations decrease to a level of 2300 pg/mL (74.5 % of eTPO baseline) on day 78 at which they remain until the last measurement on day 84.
Results
Model fittings
Observed data
The fitted curves overlaid with observed data are shown in Fig. 2. The model well describes the lag time in the PLT data, the level of the nadir and peak PLT counts, and the duration of the first two nadirs. The second peak is slightly under predicted and the predicted duration of the third nadir is shorter than the observed one. The lag time in the eTPO data was also adequately described by the model. The above mentioned assessment of model performance is evaluated by the observed versus predicted diagnostic plots for eTPO and PLT are shown in Supplementary Fig. 1 (Online Resource1).
Carboplatin and irradiation were administered on days 0, 28, and 56 resulting in a decrease in platelet counts and an increase in endogenous TPO levels. However, the time courses of PLT and eTPO shown in Fig. 2 exhibit more complex patterns revealing information about control of platelet production by eTPO and impact of PLT on eTPO clearance. Initially PLT have a mean pretreatment baseline of 1,215 9 109 cells/L (PLT baseline). Following the first cycle of CRT, there is a 4 day lag in observable PLT response before it rapidly decreases to a nadir of 18 9 109 cells/L (1.5 % of PLT baseline) on day 10. The presence of the lag time indicates that the toxic effects take place on early platelet precursor cells in the bone marrow, and validates the assumption that megakaryocytes are insensitive to the treatment during their maturation. PLTs subsequently increase to reach a peak of 918 9 109 cells/L (75.5 % of PLT baseline) on day 30, just 2 days after the next CRT cycle. The PLT time course during the second cycle resembles that observed during the first cycle, with a slightly prolonged nadir of 35–40 9 109 cells/L (3.3 % of
Model development The model presented in the Methods is a result of several refinements that were necessary in order to improve model predictive performance and ensure parameter identifiability. The expression of cMpl receptors was originally same for all ages of megakaryocyte populations. However, in the final model the oldest megakaryocytes express significantly more cMpl receptors than the younger cells (g 9 nM vs. nM). This increase in the receptor density with cell age is
123
J Pharmacokinet Pharmacodyn
Fig. 2 Time courses following first treatment (TX) of mean eTPO plasma concentrations and platelet counts in mice for three cycles of chemotherapy and irradiation (left) and fittings of naı¨ve pooled individual animal data by the PD model (right). Symbols represent
observed data (open for the placebo group, closed for the treatment group) and lines are fitted model predictions. The bars are the standard errors
consistent with previously published studies [20]. Further, given the lack of actual P and M counts and in order to enable estimation of the cMpl expression on all cells, the parameters nP, nM, and nPLT were set to be equal and represented by a single parameter n. The relationships between the total platelet receptors and PLT as well as the total megakaryocyte receptors were originally linear. The introduction of nonlinear power relationships, as described by Eqs. (14) and (15), significantly improved the fittings when the power coefficient c was allowed to be estimated. One more modification was necessary to better account for a stronger cytotoxic effect observed in the PLT data for the second and third cycle. The value of Kmax was allowed to change for times greater than 28 days, resulting in two estimates Kmax1 for cycle 1 and Kmax2 for cycles 2 and 3.
development. The simplistic one compartment model for eTPO production allowed for estimation of the elimination rate constant kel = 0.00808 h-1, corresponding to a half-life of 85.8 h. This value exceeds by many folds the half-life for recombinant human TPO for humans reported in the range of 18–32 h [21]. Given the known eTPO baseline, one can use Eq. (17) to determine the eTPO production rate constant kTPO = 0.25 pM/h. The parameter n accounts for the number of cMpl receptors per platelet, which was estimated to be 185 receptors/cell. This value is consistent with the value reported for rats (the 233 receptors per platelet) [22] and it remains close to a n estimate of 696 receptors/cell reported for rats [18]. We postulated a nonlinear dependence between RtotPLT and PLT modeled by a power function. Since the estimate of c is less than 1, the number of receptors per platelet calculated as RtotPLT/PLT increases if PLT falls below the baseline (see Eq. 14). The estimated mean lifespan for the precursors, megakaryocytes, and platelets were TP = 20.5 days, TM = 2.6 days, and TPLT = 4.3 days, respectively. The durations of the mouse thrombopoietic cell
Parameter estimates Table 1 summarizes the parameters that were fixed and the final parameter estimates that were found during model
123
J Pharmacokinet Pharmacodyn Table 1 Estimates and standard errors (SE) for parameters of the PK/PD model for chemotherapy and irradiation induced thrompocytopenia in mice Parameter
Estimate
SE
Description
kel, h-1
0.00808
0.000911 a
eTPO turnover rate
KD, pM
38.0
FIXED
kint, h-1
0
FIXEDa
eTPO0, pM
32.1b
FIXEDa
Baseline eTPO concentration at steady-state
TP, h
492
43.9
Mean lifespan of megakaryocyte precursor cells
TM, h
62.0
32.4
Mean lifespan of megakaryocytes
TPLT, h PLT0, 109 cells/L
103 1220b
57.8 FIXEDa
Mean lifespan of platelets Baseline platelet concentration at steady-state
n, 10-9 pmol/cell
0.308
0.067
Ratio of receptor to cell concentration
c
0.324
0.0392
Sensitivity of receptors to deviations in cell concentrations from baseline
TTx, h
0.818
0.0171
Duration of CRT effect
Kmax1, h-1
3.37
0.400
Maximum killing effect for first cycle of CRT
Kmax2, h-1
4.95
0.105
Maximum killing effect for cycles 2 and 3 of CRT
ROthr
0.804
FIXEDa
Smax
2.91
FIXEDa
Receptor occupancy threshold required to for eTPO to stimulate platelet production Maximum stimulatory effect of eTPO
r2eTPO r2PLT
0.0925
0.00199
Variance of the residual error for eTPO
0.439
0.0399
Variance of the residual error for PLT
a
Parameter was fixed for the final estimation procedure
b
Parameter estimates obtained from fitting the control data
populations reported in the literature are TP = 15–19 days [23] TM = 0.2–2.3 days [24], and TPLT = 4 days, respectively [25]. Initial estimates of kint approached zero and kint was fixed at this value for further analysis. The estimation of Smax and ROthr resulted in a singular covariance matrix possible due to insufficient information in the data allowing resolution of these parameters. Subsequently, Smax was fixed at a literature value 2.91 obtained for an cMpl receptor agonist romiplostim in rats [18] and ROthr was estimated and fixed at 0.804 to allow calculation of standard errors for the remaining parameters. The final set of parameters resulted in estimates with relatively good precision with the highest CV % of 58 and 51 for TPLT and TM, respectively. Simulations Although the observable data consist of eTPO and PLT, the model permits predictions of time courses of unobservable compartments following CRT. Figure 3 shows the time courses for the proliferating precursors and megakaryocytes with the corresponding eTPO and PLT time courses. Neither precursors nor megakaryocytes show a lag time. Immediately following treatment, precursor and megakaryocyte counts rapidly decline to reach nadirs of 7.5 and 6.6 % of the pretreatment baseline at times less than 24 h and 6 days,
Equilibrium dissociation rate constant for eTPO and cMpl receptor cMpl internalization rate constant
respectively. The duration of the nadir is approximately 5 days for precursors and 8 days for megakaryocytes. Both responses return to 95 % of the baseline after 23 and 33 days, respectively. The analogous time for PLT is 37 days. The eTPO response to the treatment results in a rebound with a nadir of 70.6 % of the baseline at 31 days. The eTPO response subsequently slowly returns to the baseline reaching its 95 % on day 76. Another simulation was performed to determine eTPO, P, M, and PLT responses following two cycles of CRT (Fig. 3). The time courses for the second cycle starts below the baseline values owing to incomplete return of the responses to the baseline at the time of treatment (28 days). Consequently, the nadirs for P, M, and PLT, and eTPO are lower than their values for the first cycle: 2.0, 1.8, 2.2, and 60.5 %, respectively. The durations of the nadirs are comparable to these after the first treatment cycle. This finding reflects the importance of the timing of the CRT with regards to the duration of the cytotoxic effect. The expression of the cMpl receptor is critical for the disposition of eTPO as well as for the mediation of the thrombopoietic signaling. The presented model described the dynamic response of the total (free and bound) cMpl receptor pools for each of the cell population of interest. Simulations shown in Fig. 4 represent the total cMpl receptor time courses following a single cycle of the CRT. According to our model the majority of the cMpl receptors
123
J Pharmacokinet Pharmacodyn Fig. 3 Simulated time courses of non-observable cell populations in the bone marrow along with eTPO and PLT for one (continuous lines) and two (broken lines) cycles of CRT. The first treatment (TX) occurs at time zero, and the parameter values used for simulations are shown in Table 1
are expressed on platelets. At baseline the platelet receptor pool consists of approximately 94 % of the total, whereas the megakaryocyte and precursor pools are 6 and 0.01 % (effectively zero) of the total, respectively. Therefore, the time course of the total cMpl receptor is dominated by the changes platelet receptors over time. The cMpl receptors exhibit a similar time profile to platelets with a lag time of 3 days, a nadir of 44 % of the baseline at 11 days, and a time to reach 95 % of the baseline of 32 days. The noticeable difference between PLT and cMpl receptor time courses is the short duration of the nadir phase for the latter resulting in a distinct minimum. The time to reach the nadir for the total cMpl receptors at 11 days is delayed 2 days after the peak of eTPO. The time courses of free cMpl
123
receptors mimic those for the total cMpl receptors with one distinct difference. Instead of returning to the baseline, free receptors on P, M, and PLT continued to increase, overshooting the baseline to reach a peak of 115 % of the baseline at 36 days, before returning to baseline.
Discussion Platelet production is controlled by eTPO and platelets (and to a lesser extent megakaryocytes) control the disposition of eTPO. Therefore a PD model describing such data needs to account for both mechanisms. Since the plasma levels of eTPO are regulated by binding to the cMpl
250
500
400
300 Total Platelets Megakaryocytes Precursors
200
100
Free cMpl Receptor Conc., pM
Total cMpl Receptor Conc., pM
J Pharmacokinet Pharmacodyn
1000
Free cMpl Receptor Conc., pM
Total cMpl Receptor Conc., pM
0
100
10
1
0.1
0.01
200
150 Total Platelets Megakaryocytes Precursors
100
50
0
1000
100
10
1
0.1
0.01
0.001
0.001 0
14
28
42
56
Time Since TX, days
0
14
28
42
56
Time Since TX, days
Fig. 4 Simulated time courses of total (left) and free (right) cMpl receptors expressed on indicated cells (broken lines) and overall sum (continuous lines) following a single cycle of CRT. The lower panels
are the curves from the upper panels in semi-logarithmic scales. The parameter values used for simulations are shown in Table 1
receptors expressed on platelets, a special case of a targetmediated drug disposition model, called a pharmacodynamic-mediated drug disposition model (PDMDD), has been proposed to describe the observed data [26]. We assume that RtotPLT is a power function of the platelet count. Receptor binding and disassociation contributes to nonlinear distribution of eTPO. Based on the equation for the eTPOtot, our model accounts for linear clearance and elimination by internalization and degradation. The estimate of the internalization rate constant kint became negligible and it was ultimately set to 0, leaving the linear clearance as the major pathway of eTPO elimination. Overall, the model adequately described the nadirs in the platelet counts and the tolerance in eTPO levels which is attributed to the rebound values (below the baseline) in eTPO time course prior to a new cycle of therapy (see Fig. 3). This eTPO rebound is caused by a rebound (above the baseline) in the amount of free receptors R (see Fig. 4). However, a distinct feature of PDMDD models is that both PK and PD data must be fitted simultaneously. This implies that model imperfections in describing one data set result in misfits of another to compensate for the error. This is a source of discrepancies between observed and predicted data where the failure of our model to fit the peak of
platelet counts preceding the third cycle of CRT results in over-predictions of the nadir in eTPO plasma concentration. Another misfit occurs at the end of the third cycle where platelet data is over-predicted. However, this discrepancy is caused by the inability of the model to capture the increase in the duration of nadirs between cycles of treatment. The duration of the nadir is controlled by the lifespan of proliferating precursor cells [27]. Each cycle of chemotherapy might leave only longer lived proliferating cells. This can be attributed to the proliferation-dependent cytotoxicity [28, 29]. In the absence of the PK data for carboplatin and no defined model of radiological toxicity in mice, a kinetics of treatment effect (K-PD) approach was used as a minimalistic approach accounting for exposure [19, 30]. In this methodology the cell killing effects of CRT can be represented by a hypothetical state with the on-and-off effects that attained the maximal value Kmax over time TTx following the chemoradiation. Consequently, TTx can be interpreted as a measure of exposure. Our estimates of TTx are 0.818 h which is 5.8-fold longer than the half-life of carboplatin in mouse plasma that was reported as 0.14 h [31].This type of delayed cell killing effect is not uncommon and was previously reported numerous times for both
123
J Pharmacokinet Pharmacodyn
chemotherapy [15, 32] and radiation exposure [26]. Another feature of the cytotoxic model was cycle-dependent value of the killing rate constant Kmax that was higher for the second and third cycle compared to the first one. This phenomenon might be explained by an increase sensitivity of the bone marrow cells to CRT or incomplete recovery following the first treatment. The toxic effects of CRT on proliferating cells result from complex biological processes. In order to better understand the CRT dose– response relationship additional cohorts of varying carboplatin dose and/or radiation dose would need to be explored. This information would allow any potential synergy of the chemo-radiation to be evaluated. However, these data were not available in the present study and the K-PD approach provides a parsimonious understanding of the effects of CRT on sensitive precursor cells The volume of distribution for eTPO could not be estimated in the absence of administration of exogenous TPO. The data were moderately informative about the stimulatory effect of eTPO on the production of the progenitor cells. This mechanism was modeled similarly to the romiplostim effect [18], where the stimulation was driven by the receptor occupancy by an on-and-off function. The threshold occupancy for turning the stimulation on was fixed at 80.4 % with the maximal stimulation (Smax) fixed at 2.91, both values obtained for rats [18]. Given that at the baseline RO0 = 45.1 % (see Eq. 18), the stimulation was 1.0, the maximal stimulation results in 1.91-fold increase in the progenitor production. Such ROthr level corresponds to eTPO values of 155.9 pM which is higher than the eTPO peaks. Consequently, according to our model the eTPO increase due to CRT is insufficient to cause an increase in progenitor production. The PD effect on platelets and megakaryocytes was driven by the baseline values of eTPO and the changes in eTPO concentrations have no impact on the PLT response. This has been confirmed by simulations of PLT time course with Smax = 1 (data not shown). Additional PK data with cMpl receptor agonist are necessary to obtain estimates of Smax and RO50. The receptor expression per cell (platelet and megakaryocyte) decreases as the cell count goes below the baseline level. This behavior would indicate that in thrombocytopenic patients the cMpl expression is lower than in normal subjects. However, this is inconsistent with a reported observation that the expression of cMpl receptors is higher on platelets and megakaryocytes in thrombocytopenic patients with myelodysplastic syndrome than in normal subjects [33, 34]. Such a discrepancy might be explained by a different origin or pathophysiology of the thrombocytopenia and/or its chronicity. More relevant data on cMpl receptor expression levels in patients receiving CRT is unavailable. Most of the current models of chemotherapy induced myelosupression contain a negative feedback process from
123
the circulating cells affecting the production of the proliferating cells [7]. It is necessary to describe a rebound in the circulating cell count observed in the end of the treatment cycle. The rebound can be attributed to PDMDD where suppression of cells expressing receptors for an endogenous hematopoietic growth factor leads to a decrease in its clearance and accumulation that causes more than a baseline stimulation of the precursor cells in the end of the treatment cycle. However, this mechanism can be identified only if the endogenous hormone data are available together with the circulating cell counts. PK/PD models of chemotherapy induced neutropenia have been developed where the negative feedback is replaced by a receptor mediated disposition models of endogenous G-CSF [35, 36]. We applied the same concept to account for the negative feedback due to downregulation of megakaryocytes and platelets despite the absence of the rebound in the platelet count in our multi-cycle data. Additionally, the present model can be expanded to include the effect of exogenous cMpl ligands on platelet count. The major challenge for the current model will be calculation of the fraction RO based, not on Eq. (2), but the Gaddum equation accounting for the competitive binding between eTPO and the exogenous drug [37]. Such an interaction between an endogenous hormone and an exogenous drug have been modeled for erythropoiesis stimulating agents and proved to be essential in an adequate characterization of the drug clearance [38]. In summary, the proposed PD model adequately describes the eTPO levels and platelet count in mice with CIT. The model components reflect the pharmacodynamicmediated disposition of eTPO, K-PD description of exposure to CRT, and an on-and-off stimulatory effect of eTPO interacting with both platelets and megakaryocytes. The model adequately described the nadir of platelet count which is the major marker of toxicity of thrombocytopenia. The model quantifies the inverse relationship between eTPO levels and platelet counts, and offers the explanation of the tolerance effect observed in eTPO data. The simulated rebound in free cMpl receptors levels correlates with below the baseline levels of eTPO despite sub-baseline values of platelets. The model can be extended to account for PK of an exogenous drug and be applied to analysis of human data. Acknowledgments The authors would like to thank Graham Molineux, Ping Wei and Trish McElroy for conducting the experiments to generate the data used in this modeling exercise and Murad Melhem for critical review of the manuscript. Financial support This study was sponsored by Amgen Inc., which was involved in the study design, data collection, analysis, interpretation, writing the manuscript, and the decision to submit the manuscript for publication.
J Pharmacokinet Pharmacodyn Compliance with ethical standards Potential conflicts of interest The following authors are employees of and own stock in Amgen Inc.: Juan Jose Perez Ruixo and John Harrold. Wojciech Krzyzanski is a consultant for Amgen and received consultation fees for this work.
References 1. Kaushansky K, Roth GJ (2004) Megakaryocytes and platelets. Wintrobe’s Clinical Hematology 2. Kuter DJ (2009) Thrombopoietin and thrombopoietin mimetics in the treatment of thrombocytopenia. Annu Rev Med 60:193–206 3. Salmon SE, Sartorelli AC (2001) Cancer chemotherapy. McGraw-Hill, New York 4. Woo S, Krzyzanski W, Jusko WJ (2008) Pharmacodynamic model for chemotherapy-induced anemia in rats. Cancer Chemother Pharmacol 62:123–133. doi:10.1007/s00280-007-0582-9 5. Bernstein SH, Jusko WJ, Krzyzanski W, Nichol J, Wetzler M (2002) Pharmacodynamic modeling of thrombopoietin, platelet, and megakaryocyte dynamics in patients with acute myeloid leukemia undergoing dose intensive chemotherapy. J Clin Pharmacol 42:501–511 6. Testart-Paillet D, Girard P, You B, Freyer G, Pobel C, Tranchand B (2007) Contribution of modelling chemotherapy-induced hematological toxicity for clinical practice. Crit Rev Oncol Hematol 63:1–11. doi:10.1016/j.critrevonc.2007.01.005 7. Friberg LE, Henningsson A, Maas H, Nguyen L, Karlsson MO (2002) Model of chemotherapy-induced myelosuppression with parameter consistency across drugs. J Clin Oncol 20:4713–4721. doi:10.1200/JCO.2002.02.140 8. Friberg LE, Freijs A, Sandstro¨m M, Karlsson MO (2000) Semiphysiological model for the time course of leukocytes after varying schedules of 5-fluorouracil in rats. J Pharmacol Exp Ther 295:734–740 9. Bender BC, Schaedeli-Stark F, Koch R, Joshi A, Chu Y-W, Rugo H, Krop IE, Girish S, Friberg LE, Gupta M (2012) A population pharmacokinetic/pharmacodynamic model of thrombocytopenia characterizing the effect of trastuzumab emtansine (T-DM1) on platelet counts in patients with HER2-positive metastatic breast cancer. Cancer Chemother Pharmacol 70:591–601. doi:10.1007/ s00280-012-1934-7 10. du Rieu QC, Fouliard S, Jacquet-Bescond A, Robert R, Kloos I, Depil S, Chatelut E, Chenel M (2013) Application of hematological toxicity modeling in clinical development of abexinostat (S-78454, PCI-24781), a new histone deacetylase inhibitor. Pharm Res 30:2640–2653. doi:10.1007/s11095-013-1089-1 11. Kobuchi S, Ito Y, Hayakawa T, Nishimura A, Shibata N, Takada K, Sakaeda T (2015) Semi-physiological pharmacokinetic-pharmacodynamic (PK-PD) modeling and simulation of 5-fluorouracil for thrombocytopenia in rats. Xenobiotica 45:19–28. doi:10.3109/00498254.2014.943335 12. Perez-Ruixo JJ, Green B, Doshi S, Wang Y-M, Mould DR (2012) Romiplostim dose response in patients with immune thrombocytopenia. J Clin Pharmacol 52:1540–1551. doi:10.1177/ 0091270011420843 13. Perez-Ruixo JJ, Doshi S, Wang Y-MC, Mould DR (2013) Romiplostim dose-response in patients with myelodysplastic syndromes. Br J Clin Pharmacol 75:1445–1454. doi:10.1111/bcp. 12041 14. Hayes S, Mudd PN, Ouellet D, Johnson BM, Williams D, Gibiansky E (2013) Population PK/PD modeling of eltrombopag in subjects with advanced solid tumors with chemotherapy-induced
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25. 26.
27.
28. 29.
30.
31.
thrombocytopenia. Cancer Chemother Pharmacol 71:1507–1520. doi:10.1007/s00280-013-2150-9 Friberg LE, Karlsson MO (2003) Mechanistic models for myelosuppression. Invest New Drugs. doi:10.1023/A:1023573 429626 Li J, Xia Y, Kuter DJ (1999) Interaction of thrombopoietin with the platelet c-mpl receptor in plasma: binding, internalization, stability and pharmacokinetics. Br J Haematol. 106:345–356. doi:10.1046/j.1365-2141.1999.01571.x Mager DE, Krzyzanski W (2005) Quasi-equilibrium pharmacokinetic model for drugs exhibiting target-mediated drug disposition. Pharm Res 22:1589–1596. doi:10.1007/s11095-005-6650-0 Krzyzanski W, Sutjandra L, Perez-Ruixo JJ, Sloey B, Chow AT, Wang Y-M (2012) Pharmacokinetic and pharmacodynamic modeling of romiplostim in animals. Pharm Res 30:655–669. doi:10.1007/s11095-012-0894-2 Jacqmin P, Snoeck E, van Schaick EA, Gieschke R, Pillai P, Steimer JL, Girard P (2007) Modelling response time profiles in the absence of drug concentrations: definition and performance evaluation of the K-PD model. J Pharmacokinet Pharmacodyn 34:57–85. doi:10.1007/s10928-006-9035-z Debili N, Wendling F, Cosman D, Titeux M, Florindo C, Dusanter-Fourt I, Schooley K, Methia N, Charon M, Nador R (1995) The Mpl receptor is expressed in the megakaryocytic lineage from late progenitors to platelets. Blood 85:391–401 Vadhan-Raj S, Murray LJ, Bueso-Ramos C, Patel S, Reddy SP, Hoots WK, Johnston T, Papadopolous NE, Hittelman WN, Johnston DA, Yang TA, Paton VE, Cohen RL, Hellmann SD, Benjamin RS, Broxmeyer HE (1997) Stimulation of megakaryocyte and platelet production by a single dose of recombinant human thrombopoietin in patients with cancer. Ann Intern Med 126:673–681 Yang C, Li YC, Kuter DJ (1999) The physiological response of thrombopoietin (c-Mpl ligand) to thrombocytopenia in the rat. Br J Haematol. 105:478–485 Long MW, Gragowski LL, Heffner CH, Boxer LA (1985) Phorbol diesters stimulate the development of an early murine progenitor cell. The burst-forming unit-megakaryocyte. J Clin Invest 76:431–438. doi:10.1172/JCI111990 Odell TT, Jackson CW (1971) Length of maturation time. In: Paulus JE (ed) Platelet kinetics, radioisotopic, cytological, mathematical, and clinical aspects. North Holland Publishing Company, Amsterdam Odell TT, McDONALD TP (1961) Life span of mouse blood platelets. Proc Soc Exp Biol Med 106:107–108 Wang Y-MC, Krzyzanski W, Doshi S, Xiao JJ, Perez-Ruixo JJ, Chow AT (2010) Pharmacodynamics-mediated drug disposition (PDMDD) and precursor pool lifespan model for single dose of romiplostim in healthy subjects. AAPS J 12:729–740. doi:10. 1208/s12248-010-9234-9 Krzyzanski W, Jusko WJ (2002) Multiple-pool cell lifespan model of hematologic effects of anticancer agents. J Pharmacokinet Pharmacodyn 29:311–337 Valeriote F, van Putten L (1975) Proliferation-dependent cytotoxicity of anticancer agents: a review. Cancer Res 35:2619–2630 Fernandes DJ, Sur P, Kute TE, Capizzi RL (1988) Proliferationdependent cytotoxicity of methotrexate in murine L5178Y leukemia. Cancer Res 48:5638–5644 Gabrielsson J, Jusko WJ, Alari L (2000) Modeling of dose-response-time data: four examples of estimating the turnover parameters and generating kinetic functions from response profiles. Biopharm Drug Dispos 21:41–52. doi:10.1002/1099081X(200003)21:2\41:AID-BDD217[3.0.CO;2-D van Hennik MB, van der Vijgh WJF, Klein I, Elferink F, Vermorken JB, Winograd B, Pinedo HM (1987) Comparative pharmacokinetics of cisplatin and three analogues in mice and humans. Cancer Res. 47:6297–6301
123
J Pharmacokinet Pharmacodyn 32. Simeoni M, Magni P, Cammia C, De Nicolao G, Croci V, Pesenti E, Germani M, Poggesi I, Rocchetti M (2004) Predictive pharmacokinetic-pharmacodynamic modeling of tumor growth kinetics in xenograft models after administration of anticancer agents. Cancer Res 64:1094–1101 33. Tamura H, Ogata K, Luo S, Nakamura K, Yokose N, Dan K, Tohyama K, Yoshida Y, Hamaguchi H, Sakamaki H, Kuwaki T, Tahara T, Kato T, Nomura T (1998) Plasma thrombopoietin (TPO) levels and expression of TPO receptor on platelets in patients with myelodysplastic syndromes. Br J Haematol. 103:778–784 34. Yoon SY, Li CY, Tefferi A (2000) Megakaryocyte c-Mpl expression in chronic myeloproliferative disorders and the myelodysplastic syndrome: immunoperoxidase staining patterns and clinical correlates. Eur J Haematol 65:170–174 35. Pastor ML, Laffont CM, Gladieff L, Schmitt A, Chatelut E, Concordet D (2013) Model-based approach to describe G-CSF
123
effects in carboplatin-treated cancer patients. Pharm Res 30:2795–2807. doi:10.1007/s11095-013-1099-z 36. Quartino AL, Karlsson MO, Lindman H, Friberg LE (2014) Characterization of endogenous G-CSF and the inverse correlation to chemotherapy-induced neutropenia in patients with breast cancer using population modeling. Phram Res 31:3390–3403. doi:10.1007/s11095-014-1429-9 37. Gaddum JH, Hameed KA, Hathway DE, Stephens FF (1955) Quantitative studies of antagonists for 5-hydroxytryptamine. Q J Exp Physiol Cogn Med Sci 40:49–74. doi:10.1113/expphysiol. 1955.sp001097 38. Yan X, Chen Y, Krzyzanski W (2012) Methods of solving rapid binding target-mediated drug disposition model for two drugs competing for the same receptor. J Pharmacokinet Pharmacodyn 39:543–560. doi:10.1007/s10928-012-9267-z