A Physiological Model of Granulopoiesis to Predict Clinical Drug Induced Neutropenia from in vitro Bone Marrow Studies: With Application to a Cell Cycle Inhibitor

Abstract

Neutropenia is one of the most common dose-limiting toxicity associated with anticancer drug therapy. The ability to predict the probability and severity of neutropenia based on in vitro studies of drugs in early drug development will aid in advancing safe and efficacious compounds to human testing. Toward this end, a physiological model of granulopoiesis and its regulation is presented that includes the bone marrow progenitor cell cycle, allowing for a mechanistic representation of the action of relevant anticancer drugs based on in vitro studies. Model development used data from previously reported tracer kinetic studies of granulocyte disposition in healthy humans to characterize the dynamics of neutrophil margination in the presence of endogenous granulocyte-colony stimulating factor (G-CSF). In addition, data from healthy volunteers following pegfilgrastim and filgrastim were used to quantify the regulatory effects of support G-CSF therapies on granulopoiesis. The model was evaluated for the cell cycle inhibitor palbociclib, using an in vitro system of human bone marrow mononuclear cells to quantify the action of palbociclib on proliferating progenitor cells, including its inhibitory effect on G1 to S phase transition. The in vitro results were incorporated into the physiological model of granulopoiesis and used to predict the time course of absolute neutrophil count (ANC) and the incidence of neutropenia observed in three previously reported clinical trials of palbociclib. The model was able to predict grade 3 and 4 neutropenia due to palbociclib treatment with 86% accuracy based solely on in vitro data.

Introduction

Mathematical modeling of blood cell dynamics has had a rich history, which has paralleled the general application of dynamic systems modeling in biomedicine. Early modeling focused on understanding hematopoiesis in health and disease (e.g., cyclic hematopoietic diseases, chronic myelogenous leukemia, and anemia), while more recent modeling has also been applied to quantify the impact of therapies that alter hematopoiesis (see [1] for a review). A significant application has involved the use of models to describe the myelosuppression observed during chemotherapy. More current interests have focused on translational modeling with the goal of predicting clinical neutropenia from preclinical studies, which is the aim of this work.

The seminal model of drug induced neutropenia introduced by Friberg et al. [2], has been applied to characterize the neutrophil response observed with a number of anticancer agents, both individually and in combination. Extensions to the model presented in [2], as well as other models using different modeling approaches, have been developed (see [3] for a review). These efforts have included models that incorporate the role of endogenous granulocyte-colony stimulating factor (G-CSF) and G-CSF support therapies on neutrophil dynamics. For example, Roskos et al. [4] developed a model describing the granulopoietic effects of pegfilgrastim that included its action on mitotic proliferation and differentiation in bone marrow, mobilization of band cells and segmented neutrophils, as well as circulation of blood neutrophils. Krzyzanski et al. [5] reported a model for plasma neutrophils in response to filgrastim, which included a receptor mediated G-CSF clearance. The model of Friberg et al. was also extended by Quartino et al. [6] to include the autoregulatory action of G-CSF during therapy with cytotoxic agents. Building on their previous work, Schirm et al. [7] presented a model for granulopoiesis and evaluated it using results from 33 clinical trials involving different chemotherapeutic protocols, with and without filgrastim or pegfilgrastim. Craig et al. [8] have reported a comprehensive, mechanistic model of neutrophil dynamics to predict the response of the system to concurrent chemotherapy and G-CSF administration. Other reports have focused on modeling temozolomide myelosuppression in children [9], cell cycle kinetics [10], and other hematopoietic cell lineages [11].

Distinct from these previous modeling efforts, the physiological model proposed in this work incorporates the drug’s mechanism of action on the cell cycle of proliferating bone marrow progenitor cells, as determined from in vitro bone marrow studies. The goal is to provide a model-based prediction of drug induced neutropenia in clinical trials, based solely on in vitro studies of a compound. The overall model is composed of subsystems representing proliferation, differentiation, maturation, and mobilization of neutrophils in the bone marrow, as well as circulating and marginated blood neutrophils, and the kinetics and action of endogenous G-CSF and support therapies. The ability of the model to predict the neutropenia observed in clinical trial patients was evaluated using the cell cycle inhibitor palbociclib.

Methods

Clinical data and in vitro studies

The physiological model of granulopoiesis and its regulation by G-CSF was developed using previously published studies in healthy volunteers following administration of either pegfilgrastim or filgrastim, as well as other studies using labeled granulocytes in the blood. In addition, the mechanism of action of palbociclib was incorporated into the model based on results from an in vitro bone marrow assay system. The granulopoiesis model, including the action of palbociclib, was then used to predict the neutropenia observed in previously conducted clinical trials of palbociclib. Each of these studies is summarized below.

Pegfilgrastim and filgrastim in healthy volunteers

From a previously reported study by Roskos et al. [4] in 32 healthy subjects following single subcutaneous (SC) administration of pegfilgrastim, mean values of pegfilgrastim serum concentrations, absolute neutrophil count (ANC), band cell, and segmented neutrophil counts at the doses of 30 μg/kg, 60 μg/kg, 100 μg/kg, and 300 μg/kg were digitized from published figures. The filgrastim data were taken from the work of Krzyzanski et al. [5] and included the mean filgrastim serum concentration and ANC response following a 5 μg/kg filgrastim intravenous (IV) infusion and a 1 μg/kg subcutaneous (SC) injection from 24 healthy subjects.

Blood neutrophil dynamics

Data from a previously reported tracer kinetic study of granulocyte disposition in healthy volunteers [12] was used to model the neutrophil margination process. In the study design, granulocytes obtained from each of 45 subjects were labeled with radioactive diisopropylfluorophosphate (DFP³²) and reinfused to each subject over 10–15 min. Blood samples were obtained at 0, 3, 6, 10, 24, and 30 h following the end of the infusion to determine the percent of labeled-cells in the circulating blood pool. Among the 45 subjects, data from four were provided in the paper. The time course of labeled granulocytes (as % of labeled cells injected) from these four subjects reported in the paper were digitized and used as described below.

Palbociclib in vitro bone marrow assay

Primary human bone marrow mononuclear cells (Lonza) were cultured in stemline II hematopoietic stem cell expansion medium (Sigma Aldrich), supplemented with 5% FBS and under induction by the following cytokines (R&D Systems): 25 ng/mL stem cell factor (SCF), 10 ng/mL G-CSF, 10 ng/mL granulocyte macrophage-colony stimulating factor, 3 U/mL erythropoietin (EPO), 15ng/mL thrombopoietin (TPO), 10ng/mL IL3, 10ng/mL IL6, and 25ng/mL Flt3 ligand. Cell cultures were maintained in the 37°C, 5% CO₂, and 98% humidity incubator (see [13] for further details). Cells were preincubated for one day and were then exposed to either DMSO or palbociclib for five additional days. Cell counts were determined manually using a hemocytometer from 10 μL aliquots and converted to the total cell number by adjusting for the volume of the cell culture medium in each well. While the total cell count included myeloid, erythroid, and megakaryocytic lineages, neutrophil precursors comprised the majority of the total cell population given the stimulatory conditions of the cell culture. The effects on reduction of total cell counts, therefore, were used as a measure of anti-proliferation (see [13]).

This system was used to conduct both time course and endpoint experiments. In the former, total cell counts were determined daily during the preincubation and incubations periods (days −1–5) at palbociclib concentrations of 10 nM, 100 nM, 1000 nM, or 10000 nM (3 replicates), as well with DMSO. For the endpoint experiments, cells were exposed to either DMSO or palbociclib at one of the following 10 concentrations of: 2.54 nM, 7.62 nM, 22.9 nM, 68.6 nM, 205 nM, 617 nM, 1850 nM, 5560 nM, 16700 nM, 50000 nM. In these experiments, total cell count was measured on day 5 using iQue and expressed as a percent of the DMSO exposure cell count at day 5 (6 replicates at each concentration). As described below, the results from these assays were used to model the in vitro proliferation and differentiation of neutrophils and to quantify the effect of palbociclib, which was then incorporated into the overall physiological model of granulopoiesis.

Palbociclib clinical trials

The model’s ability to predict neutropenia during palbociclib treatment, was evaluated using results from three previously conducted clinical trials [14] totaling 170 patients. The trials involved patients with non-Hodgkin lymphoma, or metastatic solid tumors, or mantle cell lymphoma, or advanced breast cancer [14]. As per the protocol, palbociclib was administered daily on a 3 week on-1 week off schedule with starting dose of 125mg, or a 2 week on-1 week off schedule with starting dose of 200mg [14]. Of the 170 patients in the trials, 141 were dosed following the protocol, while in 29 patients there were recorded dosing modifications. The individual patient data included the palbociclib dose regimen, ANC versus time measurements, as well as covariate information including age, gender, race, height, baseline body weight, and albumin. Based on the ANC versus time measurements, neutropenia was categorized from grade 1 to grade 4 in each patient based on Common Terminology Criteria for Adverse Events [15].

Modeling proliferation, differentiation, and maturation in the bone marrow

Figure 1 is a diagram of the complete physiological model of granulopoiesis. In the bone marrow, the pool of long-lived hematopoietic stem cells (HSC) is maintained at an adequate size throughout the lifespan of the organism by offsetting differentiation with self-renewal [16]. HSCs rarely enter the cell cycle and primarily reside in the quiescent G0 state for prolonged periods of time [17]. Moreover, the total pool of HSC is multipotent, producing all types of blood cells in addition to neutrophils. For these reasons, multipotent HSC were modeled as producing a constant rate (PC0, cells/L/d) of committed granulocyte progenitor cells (PC).

Fig. 1:

The schematic diagram of the physiological neutropenia model. The dashed boxes represent the three major subsystems of the model: bone marrow (Eqs. (1)–(4), (9)), blood (Eqs. (5), (10)), and G-CSF (Eqs. (6)–(8)). The dotted lines indicate G-CSF’s action on proliferation, maturation, mobilization, and margination.

Progenitor cell cycle

In contrast to the relatively quiescent HSC, a greater proportion of the committed progenitor cells in granulopoiesis are in active cell cycles. Since many anticancer drugs (e.g., cytotoxics, cell cycle inhibitors) induce neutropenia via their actions on proliferating PC, a mechanistic model of drug-induced neutropenia should incorporate the drugs’ actions on the relevant distinct cell cycle phases of PC.

The proliferating PC were modeled as three cell populations (cells/L) representing the G1, S, and G2/M phases of the cell cycle (Fig. 1). Transitions from G1 to S, S to G2, and M back to G1 were described as first-order processes (rate constants, K₁, K₂, K₃ − 1/d). Since the majority of proliferating cells are thought to withdraw from the cell cycle (fail to pass restriction point) during the G1 phase and then proceed to differentiate [18], the differentiation from PC to metamyelocytes was modeled as originating from the G1 pool (rate constants K₄ − 1/d). Any phase specific apoptosis was assumed to be negligible. With these assumptions, the simplified progenitor cell cycle model shown within Fig. 1, was used to represent the relative pool sizes and net transition rates among the three cell cycle phases.

It is generally accepted that recombinant human granulocyte-colony stimulating factor (rhG-CSF) induces proliferation of promyelocytes and myelocytes (two cell types/stages of PC) by 2–3 fold in healthy subjects [19, 20], and that it expedites the transit time of PCs [21], in part by inducing the G-CSF receptor-coupled STAT3 intracellular pathway [19, 22]. There is no direct evidence, however, that G-CSF alters the relative pool size of the cell cycle phases. Consequently, G-CSF was not incorporated in the cell cycle transitions in the PC component of the model. Instead, G-CSF’s actions were lumped as potentiating the differentiation from stem cells/common precursors (modulating PC0) to PC [23] and inducing the differentiation from the G1 phase of PC to metamyelocytes [21].

Based on the foregoing analysis, the following differential equations were used to represent the concentration of cells (cells/L) in each of the cell cycle phases (G1, S, G2/M):

$$\begin{gathered} \frac{dG1}{dt}=PC\mathit{0}\cdot \left(1+\frac{E{\mathit{max}}_{GCSF}^{PC0}\cdot GCSF}{EC{50}_{GCSF}^{PC}+GCSF}\right)+k_3\cdot G2M-k_1\cdot G1-k_4\cdot G1\cdot \left(1+\frac{{\mathit{Emax}}_{GCSF}^{PC-M}\cdot GCSF}{EC{50}_{GCSF}^{PC}+GCSF}\right) \\ \frac{dS}{dt}=k_1\cdot G1-k_2\cdot S \\ \frac{dG2M}{dt}=k_2\cdot S-k_3\cdot G2M \end{gathered}$$ (1)

where GCSF represents the concentration of G-CSF in the plasma, either endogenous or endogenous plus exogenous, as described below. Both actions of G-CSF were modeled as sigmoidal functions with the same $EC{50}_{GCSF}^{PC}$ but different maximal effects $(Ema{x}_{GCSF}^{PC0}\text{and}{\mathit{Emax}}_{GCSF}^{PC-M}).$

The nominal values used for the cell cycle model parameters K₁, K₂, K₃ were determined based on measurements of cell cycle phase times in fetal PC [24]. Using the reported values, and assuming the duration of S phase was twice the length of the combined G2/M phase [25], yields the following cell division times: G1 − 1.19 h, S − 3 h, G2/M − 1.41 h. The transition rate constants in Eq. (1) were taken as the inverse of their corresponding cell division time (K₁ = 20.17 1/d, K₂ = 8.0 1/d, and K₃ = 17.02 1/d). The overall differentiation time from early PC (myeloblast, promyelocyte, and myelocyte) to metamyelocytes has been reported to be 143h in humans [26–28]. This transit time, however, does not account for the proportion of quiescent progenitor cells, approximately 2/3 of total PCs, that do not actively undergo cell proliferation or differentiation [29]. Also, in our 3-state cell cycle model, only 1/3 of these actively proliferating PCs are in G1 phase of the cell cycle and available to differentiate. Thus, the transit time from the actively dividing G1 pool of PCs to metamyelocytes is 143/9=15.9 h, which yields K₄ = 1.51 1/d.

The progenitor cell production rate (PC0) and the parameters related to the action of G-CSF $\left(EC{50}_{GCSF}^{PC},Ema{x}_{GCSF}^{PC-M}\right)$ were estimated using the data from the pegfilgrastim and filgrastim studies as described below. The maximum G-CSF-stimulated production rate of progenitor cells $(Ema{x}_{GCSF}^{PC0})$ was fixed to be twice the value of the PC to metamyelocyte differentiation rate $(Ema{x}_{GCSF}^{PC-M})$, based on studies reporting that the G-CSF-induced net progenitor cell expansion from the proliferation and differentiation is approximately two folds [20].

Maturation

The model used to represent the maturation of neutrophils (Fig. 1) follows the mechanistic model (and parametrization) proposed by Roskos et al. [4]. The differentiation and maturation of metamyelocytes were described by three compartments in series as follows:

$$\begin{gathered} \frac{dM1}{dt}=k_4\cdot G1\cdot \left(1+\frac{Ema{x}_{GCSF}^{PC-M}\cdot GCSF}{EC{50}_{GCSF}^{PC}+GCSF}\right)-\frac{3}{\tau meta\cdot \left(1-\frac{Ima{x}_{GCSF}^{BM}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}\right)}\cdot M1 \\ \frac{dM2}{dt}=\frac{3}{\tau meta\cdot \left(1-\frac{Ima{x}_{GCSF}^{BM}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}\right)}\cdot M1-\frac{3}{\tau meta\cdot \left(1-\frac{Ima{x}_{GCSF}^{BM}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}\right)}\cdot M2 \\ \frac{dM3}{dt}=\frac{3}{\tau meta\cdot \left(1-\frac{Ima{x}_{GCSF}^{BM}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}\right)}\cdot M2-\frac{3}{\mathit{\tau meta}\cdot \left(1-\frac{Ima{x}_{GCSF}^{BM}\cdot GCSF}{EC{50}_{GCSF}^{BM}+CGCSF}\right)}\cdot M3 \end{gathered}$$ (2)

In the above equations, M1, M2, and M3 represent the concentration of metamyelocytes (cells/L) in each compartment, and τmeta is the overall metamyelocyte mean maturation time. The actions of G-CSF on the maturation time of metamyelocytes is represented by a sigmoidal function with parameters, $EC{50}_{GCSF}^{BM}\text{and}Ima{x}_{GCSF}^{BM}.$

Similarly, the differential equations that describe the maturation and mobilization of band cells in the bone marrow are as follows:

$$\begin{gathered} \frac{dB1}{dt}=\frac{3}{\tau meta\cdot \left(1-\frac{Ima{x}_{GCSF}^{BM}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}\right)}\cdot M3-\left(\frac{3}{\tau band\cdot \left(1-\frac{Ima{x}_{GCSF}^{BM}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}\right)}+\frac{Ema{x}_{GCSF}^{band}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}\right)\cdot B1 \\ \frac{dB2}{dt}=\frac{3}{\tau band\cdot \left(1-\frac{Ima{x}_{GCSF}^{BM}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}\right)}\cdot B1-\left(\frac{3}{\tau band\cdot \left(1-\frac{Ima{x}_{GCSF}^{BM}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}\right)}+\frac{Ema{x}_{GCSF}^{band}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}\right)\cdot B2 \\ \frac{dB3}{dt}=\frac{3}{\tau band\cdot \left(1-\frac{Ima{x}_{GCSF}^{BM}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}\right)}\cdot B2-\left(\frac{3}{\tau band\cdot \left(1-\frac{Ima{x}_{GCSF}^{BM}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}\right)}+\frac{Ema{x}_{GCSF}^{band}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}\right)\cdot B3 \end{gathered}$$ (3)

where B1, B2, and B3 represent the concentration of band cells (cells/L) in each compartment, and τband is the overall band cell mean maturation time. The action of G-CSF on maturation of bone marrow band cells is assumed to be the same as modeled above for metamyelocytes. A sigmoidal function is also used to model G-CSF’s action on the mobilization of band cells with the same EC50 as used for G-CSF’s maturation effect $(EC{50}_{GCSF}^{BM})$ and distinct Emax $(Ema{x}_{GCSF}^{band}).$

Finally, the maturation and mobilization of segmented neutrophils in the bone marrow are described as follows:

$$\begin{gathered} \frac{dS1}{dt}=\frac{3}{\tau band\cdot \left(1-\frac{Ima{x}_{\mathit{\text{GCSF}}}^{BM}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}\right)}\cdot B3-\left(\frac{3}{\tau seg\cdot \left(1-\frac{Ima{x}_{GCSF}^{BM}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}\right)}+\frac{Ema{x}_{GCSF}^{seg}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}\right)\cdot S1 \\ \frac{dS2}{dt}=\frac{3}{\tau seg\cdot \left(1-\frac{Ima{x}_{GCSF}^{BM}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}\right)}\cdot S1-\left(\frac{3}{\tau seg\cdot \left(1-\frac{Ima{x}_{GCSF}^{BM}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}\right)}+\frac{Ema{x}_{GCSF}^{seg}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}\right)\cdot S2 \\ \frac{dS3}{dt}=\frac{3}{\tau seg\cdot \left(1-\frac{Ima{x}_{GCSF}^{BM}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}\right)}\cdot S2-\left(\frac{3}{\tau seg\cdot \left(1-\frac{Ima{x}_{GCSF}^{BM}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}\right)}+\frac{Ema{x}_{GCSF}^{seg}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}+k_{ig}\right)\cdot S3 \end{gathered}$$ (4)

where S1, S2, and S3 represent the concentration of segmented neutrophils (cells/L) in each compartment, and τseg is the overall mean maturation time of segmented neutrophils. The action of G-CSF on maturation of bone marrow segmented neutrophils is the same as modeled above for metamyelocytes and band cells. The mobilization of segmented neutrophils by G-CSF is modeled similarly to that of band cells $(EC{50}_{GCSF}^{BM},Ema{x}_{GCSF}^{seg}).$ In the equation for the last of the three segmented neutrophil compartments, K_ig is the rate constant representing the loss of segmented neutrophils within the bone marrow before mobilization to the blood.

The values for the mean maturation time constants were based on radio-labeled tracer analysis studies in humans, as reported in [26] (τmeta = 40 h, τband = 66 h, τseg = 95 h). The value for K_ig (6.23, 1/d) and $Ima{x}_{GCSF}^{BM}$ (0.434) are that used by Roskos et al. [4]. Table 1 lists values for the parameters of the bone marrow sub model that were obtained from literature report, along with their interindividual variability (standard deviation as CV%). The values for the parameters related to the action of G-CSF $(EC{50}_{GCSF}^{PC},Ema{x}_{GCSF}^{PC0},EC{50}_{GCSF}^{PC-M},EC{50}_{GCSF}^{BM},Ema{x}_{GCSF}^{seq},Ema{x}_{GCSF}^{seq})$ were estimated using the data from the pegfilgrastim and filgrastim studies as described below.

Table 1:

Bone marrow subsystem model parameter values obtained from literature.

Parameter (unit)	Definition	Value	Source	SD (CV%)	SD sourcea
k1 (1/d)	Rate constant for transition from G1 to S	20.2	[24]	10	[24]
k2 (1/d)	Rate constant for transition from S to G2	8.00	[24]	10	[24]
k3 (1/d)	Rate constant for transition from M to G1	17.0	[24]	10	[24]
k4 (1/d)	Differentiation rate from progenitor cells to metamyelocytes	1.51	[26]	45	[52]
τmeta (h)	Mean maturation time of metamyelocytes	40.0	[26]	78	[52]
τband (h)	Mean maturation time of band cells	66.0	[26]	69	[52]
τseg (h)	Mean maturation time of segmented neutrophils	95.0	[26]	62	[52]
kig (1/d)	Rate constant for precursor cell loss in marrow due to ineffective granulopoiesis	6.23	[4]	-	-

^aSee supplemental materials for details

Modeling circulating and marginated neutrophil dynamics

Neutrophils in the blood exist in both circulating and marginated pools, where the former includes both band and segmented neutrophils. During margination, segmented circulating neutrophils migrate toward vessel walls, which allows their adhesion to the vascular endothelium via mechanical contact and interaction with endothelial cells, leading to their extravasation [30, 31]. Margination is an essential step in neutrophil pathophysiology and there are several reasons, including the following, for including a model of the margination process in a physiological model of drug induced neutropenia. The pool of marginated neutrophil is similar in size to that of circulating neutrophils, the two pools exhibits different dynamics [12, 32–34], and G-CSF directly alters the margination process. In addition, common anticancer treatments, such as corticosteroids, are best understood as acting to induce the demargination process [35]. Moreover, many common infections that develop during anticancer therapy in cancer patients, directly alter the margination process. Incorporating the margination process dynamics in the model, in addition to providing a more mechanistic representation of the action of G-CSF, allows for subsequent model expansion to include other therapeutic interventions.

As indicated in Fig. 1, neutrophil margination in the blood is represented by first-order rate processes governing the exchange of neutrophils between the segmented (Sb, cells/L) and marginated (Mb, cells/L) pools (rate constants K_SM and K_MS). The processes of rolling, firm adhesion, and trans-endothelial migration were lumped and represented by a first-order exit rate of marginated neutrophils from the peripheral blood (rate constants K_ME). The conversion in the blood of band cells (Bb, cells/L) to segmented neutrophils was also modeled as first-order processes (rate constant K_BS).

Although the precise mechanism of G-CSF’s action on margination process is unclear, several studies have demonstrated transient decreases in circulating neutrophil number starting almost instantaneously following G-CSF injection [36–38]. Accordingly, G-CSF was modeled as directly enhancing the margination rate constant K_SM.

The model equations describing blood neutrophil dynamics and their link to the associated processes in the bone marrow are as follows:

$$\begin{gathered} \frac{dBb}{dt}=\sum _{i=1}^3\left[\left(\frac{Ema{x}_{GCSF}^{band}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}\right)\cdot Bi\right]-k_{BS}\cdot Bb \\ \frac{dSb}{dt}=\sum _{i=1}^3\left[\left(\frac{Ema{x}_{GCSF}^{seg}\cdot GCSF}{EC{50}_{GCSF}^{BM}+GCSF}\right)\cdot Si\right]+k_{BS}\cdot Bb+\frac{3}{{\tau }_{seg}}\cdot S3-k_{SM}\cdot \left(1+\frac{Ema{x}_{GCSF}^{marg}\cdot GCS{F}^{Hmarg}}{{\left(EC{50}_{GCSF}^{marg}\right)}^{Hmarg}+GCS{F}^{Hmarg}}\right)\cdot Sb+k_{MS}\cdot Mb \\ \frac{dMb}{dt}=k_{SM}\cdot \left(1+\frac{Ema{x}_{GCSF}^{marg}\cdot GCS{F}^{Hmarg}}{{\left(EC{50}_{GCSF}^{marg}\right)}^{Hmarg}+GCS{F}^{Hmarg}}\right)\cdot Sb-k_{MS}\cdot Mb-k_{ME}\cdot Mb \end{gathered}$$ (5)

All the rate constants governing the margination and extravasation process (K_SM, K_MS, K_ME, 1/d) were determined, as described below, using the data from neutrophil dynamic tracer studies referenced above. Also, the parameters related to the action of G-CSF on the margination of blood segmented neutrophils $(EC{50}_{GCSF}^{marg},Ema{x}_{GCSF}^{marg},Hmarg)$ and the parameter related to band cells (K_BS) were estimated using the data from the pegfilgrastim and filgrastim studies (see below).

Modeling regulation by G-CSF

Endogenous G-CSF

The principal regulator of neutrophil development is the endogenous hematopoietic cytokine G-CSF [39]. Healthy subjects express low levels of G-CSF in serum from below 30 pg/mL to 163 pg/mL, but infection can elevate the level to be up to 3200 pg/mL [40]. G-CSF regulates the population of circulating neutrophils by binding to G-CSF receptors (G-CSFR), which further induces homodimerization of the receptors forming a tetrameric structure with two receptors [39]. The dissociation constant, K_D, of G-CSF from the tetramer is 2–7 ng/mL [39]. The bound complex ultimately internalizes to endosomal compartments where it undergoes either recycling or degradation [5, 41], which represents a significant pathway for clearance of G-CSF (receptor or target mediated clearance) especially in the blood given the abundance of neutrophils (both circulating and marginated).

This receptor mediated clearance of G-CSF results in an autoregulation of neutrophils in the plasma. Low levels of absolute neutrophil count (ANC) (or neutropenia), with the associated reduction in G-CSFR number, lead to a reduction in G-CSF clearance and an increase in G-CSF concentration, which in turn enhances neutrophil production and mobilization [39]. Conversely, high levels of ANC (neutrophilia) result in greater receptor mediated clearance and a reduction in G-CSF concentration, resulting in a reduction in neutrophil production [39]. Accordingly, it is essential to incorporate in our model the mechanisms associated with the regulation of neutrophil dynamics by G-CSF.

In the model, G-CSF was assumed to rapidly bind to G-CSFR with dissociation constant, K_D, to form drug-receptor complex (RC_GCSF). The production of endogenous G-CSF was modeled as a zero-order rate process (K_GCSF, ng/ml/day), yielding a basal value of G-CSF $(C_{GCSF}^0).$ G-CSF is eliminated both renally (rate constant $k_{el}^{GCSF},$ 1/d) and via internalization/degradation of the RC complex (rate constant $k_{int}^{GCSF},$ 1/d). The internalized RC complex was assumed to be non-recyclable. The following equations describe total concentration of G-CSF $(C_{GCSF}^{tot})$ given a rapid binding assumption:

$$\frac{d{C}_{GCSF}^{tot}}{dt}=K_{GCSF}-k_{el}^{GCSF}\cdot C_{GCSF}-k_{\mathrm{int}}^{GCSF}\cdot R{C}_{GCSF}$$ (6a)

where the free G-CSF concentration (C_GCSF) is given by

$$C_{GCSF}=0.5\cdot \left(\left(C_{GCSF}^{tot}-R_{tot}-K_D\right)+\sqrt{{\left(C_{GCSF}^{tot}-R_{tot}-K_D\right)}^2+4\cdot K_D\cdot C_{GCSF}^{tot}}\right)$$ (6b)

and $R{C}_{GCSF}=C_{GCSF}^{tot}-C_{GCSF}$

To insure steady-state, K_GCSF is derived as

$$K_{GCSF}=k_{el}^{GCSF}\cdot C_{GCSF}^0+k_{\text{int}}^{GCSF}\cdot R{C}_{GCSF}^0$$ (6c)

The total concentration of receptors (R_tot = R+RC_GCSF) is determined by the number of neutrophils in the blood both circulating (B_b and S_b) and marginated (M_b), which is given as

$$R_{tot}=Den\cdot \left(B_b+S_b+M_b\right)$$ (6d)

where Den represents the density of G-CSF receptors (mass/cell).

The parameters $K_D,C_{GCSF}^0,$ and Den were fixed at literature reported values [39, 40, 42], while parameters $k_{el}^{GCSF}\text{and}{k}_{int}^{GCSF}$ were estimated as described below using the data from the aforementioned studies with the two major forms of human recombinant G-CSF, filgrastim and pegfilgrastim.

Filgrastim

Filgrastim is a recombinant methionyl type of human G-CSF for treating chronic neutropenia, chemotherapy-induced neutropenia, acute myeloid leukemia with chemotherapy, and bone marrow transplantation with chemotherapy [43]. Since filgrastim is functionally and structurally the same as endogenous G-CSF, it was assumed that regardless of its source, G-CSF follows the same disposition pathways, including clearance, receptor binding and internationalization [5, 19, 39, 44].

When filgrastim is administered exogenously, either by SC injection or IV infusion, the following equation applies to the total concentration of G-CSF $(C_{GCSF}^{tot}):$

$$\frac{d{C}_{GCSF}^{tot}}{dt}=\frac{InputFil(t)}{V_{Fil}}+K_{GCSF}-k_{el}^{GCSF}\cdot C_{GCSF}-k_{\mathrm{int}}^{GCSF}\cdot R{C}_{GCSF}$$ (7)

where V_Fil represents filgrastim’s distribution volume (L) and C_GCSF, K_GCSF, R_tot, RC_GCSF are as defined in Eqs. (6b)–(6d) along with the other parameters. The delivery rate via IV or SC administration, InputFil(t) (ng/d), was modeled as reported in [5]. For the case of SC administration, the linear first-order absorption model parameters, $k_a^{Fil}$ and f_Fil, were estimated along with V_Fil using the using the mean plasma concentration data following SC and IV administration obtained from [5] as described below.

Pegfilgrastim

Another human recombinant form of G-CSF used clinically is pegfilgrastim, which has a significantly longer half-life (15 to 80 h) in comparison to filgrastim. Pegfilgrastim and filgrastim have been shown to have the same mechanism of action based on in vitro receptor binding and effect studies [45]. Following the modeling approach describing the target mediated disposition of two drugs competing for the same receptor reported by Yan et al. [46], a model for the kinetics and actions of both endogenous G-CSF and pegfilgrastim was constructed. It has been demonstrated that the addition of PEG to filgrastim, does not alter the filgrastim binding domain nor the corresponding biological activity [45, 47]. Thus, the dissociation constant was assumed to be the same for filgrastim and pegfilgrastim. In the model, both the endogenous free G-CSF, C_GCSF, and free pegfilgrastim, C_PEG, were assumed to competitively bind to G-CSF receptors forming their respective complexes, RC_GCSF and RC_PEG, with the same dissociation constant, K_D.

Following SC administration of pegfilgrastim, the following equation describes the total concentration of pegfilgrastim in the blood $(C_{PEG}^{tot}):$

$$\frac{d{C}_{PEG}^{tot}}{dt}=\frac{InputPEG(t)}{V_{PEG}}-k_{el}^{PEG}\cdot C_{PEG}-k_{\mathrm{int}}^{PEG}\cdot R{C}_{PEG}$$ (8a)

where $k_{el}^{PEG}(1/\text{d})$ represents the elimination rate constant of free pegfilgrastim $(C_{PEG}^{tot},\text{ng}/\text{ml})$ and $k_{int}^{PEG}(1/\text{d})$ is the internalization/degradation rate constant of the pegfilgrastim receptor complex $(R{C}_{PEG},\text{ng}/\text{ml},R{C}_{PEG}=C_{PEG}^{tot}-C_{PEG}).$ The concentration of free pegfilgrastim (C_PEG, ng/ml) is given below. The input function following SC administration, InputPEG(t) (ng/d), was modeled as reported in [4], and accounted for the delayed and dose-dependent deliver of SC pegfilgrastim (see [4] for model equations and parameter values). Under the competitive binding model, the equations describing the kinetics of free pegfilgrastim and endogenous G-CSF are as follows:

$$\begin{gathered} C_{PEG}=\frac{2\cdot C_{PEG}^{tot}\cdot K_D}{\left(\left(R_{tot}-C_{PEG}^{tot}-C_{GCSF}^{tot}+K_D\right)+\sqrt{{\left(R_{tot}+C_{PEG}^{tot}+C_{GCSF}^{tot}+K_D\right)}^2-4\cdot \left(C_{PEG}^{tot}+C_{GCSF}^{tot}\right)\cdot R_{tot}}\right)} \\ {C}_{GCSF}=\frac{2\cdot C_{GCSF}^{tot}\cdot K_D}{\left(\left(R_{tot}-C_{PEG}^{tot}-C_{GCSF}^{tot}+K_D\right)+\sqrt{{\left(R_{tot}+C_{PEG}^{tot}+C_{GCSF}^{tot}+K_D\right)}^2-4\cdot \left(C_{PEG}^{tot}+C_{GCSF}^{tot}\right)\cdot R_{tot}}\right)} \end{gathered}$$ (8b)

where all terms have been defined previously.

Table 2 lists the values for the parameters associated with endogenous/exogenous G-CSF kinetics that were obtained from literature sources, as well as values for their interindividual variability (standard deviation as CV%). The values for $k_{el}^{GCSF},k_{int}^{GCSF},k_{el}^{PEG}\text{and}{k}_{int}^{PEG},$ as well as those associated with the model for SC filgrastim absorption, $k_a^{Fil},$ f_Fil, and V_Fil, were estimated using the data from the pegfilgrastim and filgrastim studies as described below. The parameters for the pegfilgrastim SC absorption model are those reported by Roskos at al. [4].

Table 2:

G-CSF subsystem model parameter values obtained from literature.

Parameter (unit)	Definition	Value	Source	SD (CV%)	SD sourcea
kD (ng/ml)	Dissociation constant	4.50	[39]	26	[59]
$C_{GCSF}^0$ (ng/ml)	Endogenous G-CSF concentration	0.03	[40]	67	[40]
Den (fg/cell)	G-CSF receptor density	0.108	[42]	25	[5]

^aSee supplemental materials for details

Palbociclib in vitro bone marrow assay model

To incorporate the action of palbociclib, the physiological model of granulopoiesis presented above (Eqs. (1)–(4)) was modified to represent the in vitro bone marrow assay system. Since the degradation of G-CSF is negligible in vitro, the concentration of G-CSF was taken to be constant (10 ng/ml, see above) in the in vitro model. In addition, the cell pools of bone marrow metamyelocytes, band cells, and segmented neutrophils in the in vitro model were each assumed to be subject to a first-order degradation/natural loss rate (K_deg) term, in contrast to the physiological model which included a loss term for only the last neutrophil pool (see the S3 equation in Eq. (4)). Similarly, the mobilization term ((3/τseg)·S₃) from S₃ to blood was removed in Eq. (4) given the in vitro bone marrow system.

Palbociclib is a selective, reversible cyclin-dependent kinase (CDK) 4/6 inhibitor that blocks the cell division by inhibiting the cell cycle G1 phase to S phase transition [14]. In bone marrow, palbociclib’s inhibitory action on CDK 4/6 results in arrested or incompletely differentiated G1 cells (inactive) that cannot proceed to DNA synthesis or differentiation to metamyelocytes [13]. Palbociclib’s action can be viewed as producing a pharmacologic quiescence, where upon withdrawal of the drug, the quiescent bone marrow cells return to the cell cycle [13]. Accordingly, the pharmacology of palbociclib was modeled with a pool representing cells in an inactive resting state, G1r, which is reversible. At the level of a single cell, palbociclib prevents the G1 to S transition, while instead shunting the cell to this resting state. At the level of the population, the total rate of cells exiting G1 would then be unaffected in the presence of the drug, rather the proportion of cells proceeding to G1 and G1r would be altered. Palbociclib’s action was modeled as a reduction of the cell cycle transition rate constant K₁, with a corresponding increase in the transition rate constant from the functional G1 pool to the inactive resting G1r pool. The reverse flux from the G1r to the G1 pool was assumed independent of drug and modeled as following first-order kinetics with the rate constant of K_r. The overall action of palbociclib is depicted in Fig. 2.

Fig. 2:

Mechanism of action of palbociclib incorporated in the cell cycle model. Minus sign represents palbociclib inhibition of k₁, plus sign represents palbociclib induction of the inactivation process.

The resulting progenitor cell model equations incorporating the action of palbociclib are as follows:

$$\begin{gathered} \frac{dG1}{dt}=PC\mathit{0}\cdot \left(1+\frac{Ema{x}_{GCSF}^{PC0}\cdot GCSF}{EC{\mathit{50}}_{GCSF}^{PC}+GCSF}\right)+k_3\cdot G2M-k_1\cdot G1-k_4\cdot G1\cdot \left(1+\frac{Ema{x}_{GCSF}^{PC-M}\cdot GCSF}{EC5O_{GCSF}^{PC}+GCSF}\right)+k_r\cdot G1r \\ \frac{dG1r}{dt}=k_1\cdot \frac{C_{palbo}}{IC{\mathit{50}}_{palbo}+C_{palbo}}\cdot G1-k_r\cdot G1r \\ \frac{dS}{dt}=k_1\cdot \left(1-\frac{C_{palbo}}{IC{\mathit{50}}_{palbo}+C_{palbo}}\right)\cdot G1-k_2\cdot S \\ \frac{dG2M}{dt}=k_2\cdot S-k_3\cdot G2M \end{gathered}$$ (9)

where C_palbo and IC50_palbo represent palbociclib’s concentration and IC50 (ng/ml), K_r (1/d) is the rate constant from the resting to the active G1 state. All other symbols are as defined for Eq. (1). The in vitro bone marrow assay model is defined by Eq. (9) together with Eqs. (2)–(4) (modified as noted above). In this model, the parameters PC0, K_deg, IC50_palbo, and K_r were estimated from the in vitro data as described below, while values for all other parameters are those listed in Table 1.

Parameter estimation

Blood neutrophil dynamics

The dynamics of the DFP³² labeled blood neutrophils in the tracer kinetic study reported in [12] were modeled as follows (see Fig. 1. and Eq. (5)):

$$\begin{gathered} \frac{d{S}_b}{dt}=-k_{SM}\cdot S_b+k_{MS}\cdot M_b \\ \frac{d{M}_b}{dt}=k_{SM}\cdot S_b-k_{MS}\cdot M_b-k_{ME}\cdot M_b \end{gathered}$$ (10)

with all symbols as defined above and initial values of labeled cells set to zero. Because the trace injection of neutrophils does not alter the basal steady state, the band cells and the resting G-CSF concentration were assumed to be negligible, and the relative size of the segmented and marginated neutrophil pools was assumed to be undisturbed from its baseline value measured in [32] (M_b/S_b = 1.13). Therefore, K_SM can be written as 1.13(K_MS+K_ME). The parameters K_MS and K_ME (and the secondary parameter K_SM) were estimated using the circulating labeled neutrophil-time data from each of the four subjects whose measurements were presented in [12]. The individual subject parameter estimation application (ID) in ADAPT (version 5) [48] was used to obtain the maximum likelihood estimates for each subject assuming an additive plus proportional error variance model. The means and standard deviations of the estimates were used as the population parameter values in the simulations described below.

Pegfilgrastim and filgrastim in healthy volunteers

All the data from the pegfilgrastim (mean values of pegfilgrastim serum concentrations, ANC, band cell, and segmented neutrophil counts following four SC doses) and filgrastim (mean filgrastim serum concentration and ANC response following IV and SC doses) healthy volunteer studies extracted from the literature were pooled and used to obtained maximum likelihood estimates of the remaining parameters of the complete granulopoiesis model (NPD application in ADAPT, with separate additive and proportional error variance terms for each measured variable). The model was defined by Eqs. (1)–(6), plus Eq. (7) for filgrastim administration (GCSF = C_GCSF) or Eq. (8) for pegfilgrastim administration (GCSF = C_GCSF+C_PEG). The estimated model parameters included the progenitor cell production rate (PC0), blood neutrophil dynamics (K_BS), the G-CSF effect parameters $(EC{50}_{GCSF}^{PC},Ema{x}_{GCSF}^{PC0},Ema{x}_{GCSF}^{PC-M},EC{50}_{GCSF}^{BM},Ema{x}_{GCSF}^{band},Ema{x}_{GCSF}^{seg},EC{50}_{GCSF}^{marg},{\mathit{Emax}}_{GCSF}^{marg},H_{marg}),$ as well as those defining the kinetics of endogenous G-CSF and filgrastim $(k_{el}^{GCSF},k_{int}^{GCSF},k_a^{Fil},f_{Fil},V_{Fil})\text{and pegfilgrastim}(k_{el}^{PEG},k_{int}^{PEG}).$ Since the baseline value of each state in the model depends on the unknown parameter values, a pre-dose period was added to allow the steady state baseline values to be established.

The resulting physiological model of granulopoiesis was also evaluated by comparing the predicted baseline values of model states to available measured values in health subjects reported in the literature. The predictions were obtained via a population simulation (n=1000) of the model (Eqs. (1) – (6)), using the parameter values and intersubject standard deviations given in Tables 1 and 2, along with the values of the other model parameters estimated from both the neutrophil dynamic tracer study and the filgrastim and pegfilgrastim study (see results). Model parameters were assumed to be independent lognormally distributed and 1000 simulated subjects were generated using the simulation application (SIM) in ADAPT (version 5) [48], with the simulations performed to achieve steady state in all model states.

Palbociclib in vitro bone marrow assay

The data from both the time course and endpoint experiments were pooled and used to estimate the unknown parameters of the in vitro bone marrow assay model (PC0, K_deg, IC50_palbo, K_r) given by Eq. (9) and Eqs. (2)–(4) (maximum likelihood estimates using the NPD application in ADAPT assuming an additive plus proportional error variance model). All other parameters were fixed as noted above in the presentation of the in vitro bone marrow assay model. The initial conditions for each of the cell pools (progenitor cell pools (G1, S, G2/M), metamyelocytes, band cell, and segmented neutrophils) were set equal to one twelfth of the cell number at day 2 (328055/12=27338 cells/mL). During the first three days of the assay (day −1 to day 2), no measurable net cell growth was detected (see results) and the different cell pools in the model were set at their initial values during this time.

Predicting neutropenia in palbociclib clinical trials

Population PK model of palbociclib

To evaluate the ability of the model to predict the ANC response observed in clinical trials of palbociclib, the physiological model of granulopoiesis incorporating the action of palbociclib (Eq. (9) plus Eqs. (1)–(6)) was linked to a population PK model of palbociclib reported previously [49]. Palbociclib PK was described using a linear two-compartment model with first-order absorption, with body weight (kg) and age (years) found to be significant covariates for clearance, while central compartment distribution volume was explained by body weight (see Table 3). This model was used to simulate the palbociclib plasma concentration time profile for each of the 170 patients in the palbociclib clinical trials cited above, using each patient’s dose regimen and typical values of model parameters as indicated in Table 3 (individual patient parameter values were not available).

Table 3:

Palbociclib PK model parameter values (from [49]).

Parameter (unit)	Definition	Value	SD (CV%)c
aCL (L/h)	Apparent clearance	60.2	37
bV1 (L)	Apparent distribution volume central compartment	2710	31
Q (L/h)	Apparent intercompartmental clearance	10.6	127
V2(L)	Apparent distribution volume peripheral compartment	61300	-
ka (1/h)	Absorption rate constant	0.367	84

^aCL =60.2(Age/61)^−0.45(BW/72.8)^0.484;

^bV₁ =2710(BW/72.8)^0.00906

^cFrom population analysis reported in [49]

Patient-specific simulation of ANC time course

Using the dose regimen for each patient, a Monte Carlo simulation (n=1000) of the linked granulopoiesis and palbociclib PK models was performed using the simulation application (SIM) in ADAPT (version 5) [48]. Model parameters were assumed to be independent lognormally distributed with mean and intersubject standard deviations values as given in Tables 1 and 2, along with the palbociclib PK parameter values given in Table 3. The values for the remaining model parameters were obtained from estimation using both the neutrophil dynamic tracer study and the filgrastim and pegfilgrastim study (see results). A pre-dose period was added to allow the steady state baseline values to be established.

To allow for comparison of the model-predicted ANC-time profiles and neutropenia grade to measured values, the basal ANC measured in each patient must be matched with a baseline ANC from the model simulation. This was accomplished by matching each patient’s measured pre-dose ANC to the closest baseline ANC value obtained from the Monte Carlo simulation. Owing in part to the model structure, multiple simulations will yield very similar baseline ANC values to that of a particular patient, but the predicted ANC time profiles will in general be different. To account for this, we determined a median profile based on all the simulations that had a baseline ANC “similar” to that of the particular patient: the median of the ANC-time profiles from the 100 nearest simulated baseline ANC values closest to patient’s baseline value was then calculated to represent the ANC-time profile for that patient.

Comparing patient-specific model predicted and observed ANC time course and neutropenia grade

The patient-specific, baseline ANC-matched ANC predictions from the model were compared to the measured ANC values via linear regression analysis. The model was further evaluated based on its ability to predict the occurrence of grade 2 (nadir ANC, 1.0×10⁹ cells/L to 1.5×10⁹ cells/L), grade 3 (nadir ANC, 0.5×10⁹ to 1.0×10⁹ cells/L), and grade 4 (nadir ANC, < 0.5×10⁹ cells/L) neutropenia in each patient.

Model predicted population ANC time profiles and neutropenia grade

Populations simulations (n=1000) were also performed for each of the two main palbociclib dosing cohorts: 125 mg, 3 week on 1 week off dosing cohort and 200 mg, 2 week on 1 week off dosing cohort. The results were used to predict the ANC time course in the population with each of these two dose regiments, as well as to predict incidence of neutropenia.

Results

Circulating and marginated neutrophil dynamics

The resulting parameter estimates obtained from fitting the tracer kinetic margination model (Eq. (10)) to the labeled neutrophil measurements obtained from each of the four subjects reported in [12] are shown in Table 4 (mean and standard deviation). The average of the four model fits is shown in Fig. 3 along with the mean and standard deviation of the labeled neutrophil activity (overall observed versus predicted r² = 0.98). The estimated parameters of the margination model k_SM and k_MS indicate that circulating and marginated neutrophils are in rapid equilibrium, and that this is considerably more rapid than the maturation, mobilization, and extravasation processes, which is consistent with literature reports [12, 32, 34].

Table 4:

Margination subsystem model parameter estimates from tracer experiments.

Parameter (unit)	Definition	Estimate (RSE%)	SD as CV%b
kMS (1/d)	Rate constant for demargination from Mb to Sb	123 (22)	52
kME (1/d)	Rate constant for extravasation of marginated cells	4.45 (1.2)	15
akSM (1/d)	Rate constant for margination from Sb to Mb	144 (−)	51

^aSecondary parameter: k_SM=1.13(k_MS +k_ME)

^bSee Methods

Fig. 3:

Tracer kinetic margination model fit (average of four subjects – line) to the labeled neutrophil measurements (mean and standard deviation – symbols and error bars).

Pegfilgrastim and filgrastim in healthy volunteers

The physiological model of granulopoiesis defined by Eq. (1)–(6) was fitted to the pooled filgrastim data (PK model, Eq. (7)) and pegfilgrastim data (PK model, Eq. (8)), with fixed model parameters as given in Tables 1–4. Figure 4 shows the model predictions and data for pegfilgrastim, while the results for filgrastim are shown in Fig. 5. The model describes the dose-dependent plasma PK of pegfilgrastim and filgrastim, as well as their respective effects on the dynamics of ANC. For pegfilgrastim, in addition, the model also reflects the observed dose-dependent increase in both band cells and segmented neutrophils. For filgrastim, the time courses of the ANC response for both SC and IV administration are captured by the model. Overall, the results from the physiological model with one set of parameters, are comparable to the model results for pegfilgrastim reported in [4] (compare to figures 4,5 in [4]) and the model results for filgrastim presented in [5] (compare to figures 2 in [5]).

Fig. 4:

Model predicted (lines) and observed (symbols) pegfilgrastim serum concentration (upper panel left) and ANC (upper panel right) following the SC injection of 30, 60, 100, and 300 μg/kg pegfilgrastim. Lower panels show model predicted (lines) and observed (symbols) band cells (solid lines) and segmented neutrophils (dashed lines) for each of the four doses as indicated. Overall observed versus predicted r2 values: serum pegfilgrastim – 0.99; ANC – 0.95; band cells – 0.86; segmented neutrophils – 0.92.

Fig. 5:

Model predicted (lines) and observed (symbols) filgrastim serum concentration (left panel) and ANC (right panel) following the IV (5 μg/kg) and SC (1 μg/kg) of injections. Overall observed versus predicted r2 values: serum filgrastim – 0.80; ANC – 0.90.

Table 5 shows the estimated parameter values of the physiological model along with their standard errors. Given the wide range of the G-CSF exposure in the filgrastim and pegfilgrastim studies, both in magnitude and time course (due to their vastly different kinetics), the collective use of the resulting system response data allowed reliable estimation of the model parameters (Table 5). The remaining filgrastim PK parameters were estimated as follows: $k_a^{Fil}$ − 15.9 1/d (RSE% − 5.2); f_Fil − 0.245 (RSE% − 3.2); V_Fil − 1.33 L (RSE% − 5.0). The estimates imply that G-CSF can induce a maximum two-folds differentiation in metamyelocytes and a maximum four-folds increase in proliferation of progenitor cells compared to the basal state. Also, the estimate of $Ema{x}_{GCSF}^{marg}$ indicates that G-CSF induces a maximum 0.6-fold increase in the rate constant of margination of segmented blood neutrophils; this is responsible for the 30% rapid decrease from baseline ANC values following G-CSF predicted by the model.

Table 5:

Model parameter estimates obtained from filgrastim and pegfilgrastim studies.

Parameter (unit)	Definition	Estimate (RSE%)	SD (CV%)	SD sourcea
PC0 (cells/L/d)	Basal production rate of progenitor cells	17.5×l010 (2.4)	48	[60]
$EC{50}_{GCSF}^{PC}$ (ng/ml)	Concentration of G-CSF eliciting a half-maximal effect on progenitor cells	36.7 (22)	36	[5]
b$Ema{x}_{GCSF}^{PC0}$(−)	Maximum G-CSF-stimulated production of progenitor cells	3.78 (−)	22	[20]
$Ema{x}_{GCSF}^{PC-M}$ (−)	Maximum G-CSF-stimulated differentiation to metamyelocyte	1.89 (5.5)	11	[20]
$EC{50}_{GCSF}^{BM}$ (ng/ml)	Concentration of G-CSF eliciting a half-maximal effect on all maturation times and on mobilization of bone marrow cells	2.47 (6.0)	36	[5]
$Ema{x}_{GCSF}^{band}$ (1/d)	Maximum G-CSF-stimulated mobilization of bone marrow band cells into blood	0.034 (4.4)	18	[20]
$Ema{x}_{GCSF}^{seg}$ (1/d)	Maximum G-CSF-stimulated mobilization of bone marrow segmented neutrophils into blood	0.307 (7.3)	20	[20]
$EC{50}_{GCSF}^{marg}$ (ng/ml)	Serum concentration of G-CSF eliciting a half-maximal effect on margination of segmented blood neutrophils	0.664 (24)	36	[5]
$Ema{x}_{GCSF}^{marg}$ (−)	Maximum G-CSF on margination of segmented blood neutrophils	0.600 (15)	-	-
Hmarg (−)	Hill constant for G-CSF’s margination effect	1.73 (27)	-	-
kBS (1/d)	Rate constant for maturation of band cells in blood to segmented neutrophils	2.43 (5.4)	-	-
$k_{el}^{GCSF}$ (1/d)	G-CSF elimination rate constant	8.84 (3.4)	47	[5]
$k_{int}^{GCSF}$ (1/d)	Receptor-mediated G-CSF internalization rate constant	0.965 (76)	106	[6]
$k_{el}^{PEG}$ (1/d)	Pegfilgrastim elimination rate constant	0.660 (3.6)	-	-
$k_{int}^{PEG}$ (1/d)	Receptor-mediated pegfilgrastim internalization constant	41.5 (2.3)	-	-
c$Ima{x}_{GCSF}^{BM}$ (−)	Maximum G-CSF-inhibition of mean maturation times	0.434 (−)	16	[20]

^aSee supplemental materials

^bSecondary parameter: $Ema{x}_{GCSF}^{PC0}$ = 2 · $Ema{x}_{GCSF}^{PC-M}$

^cValue of $Ima{x}_{GCSF}^{BM}$ was fixed as per the model in [4]

The G-CSF (filgrastim) and pegfilgrastim estimated clearance-related parameters, $k_{el}^{GCSF},k_{int}^{GCSF},k_{el}^{PEG},k_{int}^{PEG}$ are listed in Table 5. The resulting value of $k_{el}^{PEG}$ is much smaller than $k_{int}^{PEG}$, indicating that pegfilgrastim is mainly eliminated via receptor-mediated clearance, whereas $k_{el}^{GCSF}$ is greater than $k_{int}^{GCSF}$ which suggests the major elimination pathway for G-CSF is renal linear elimination. Both results agree with previous PK studies of pegfilgrastim and filgrastim [39, 50].

Since proliferation, maturation, and margination processes are distinct physiological processes, three separate EC50 parameters were used to represent these above three processes. The EC50 values used to characterize G-CSF’s action on proliferation, maturation, and margination processes, were estimated as follows: $EC{50}_{GCSF}^{PC}=36.7\text{ng/ml},EC{50}_{GCSF}^{BM}=2.47\text{ng/ml},EC{50}_{GCSF}^{marg}=0.664\text{ng/ml}.$ For comparison, we note that the single G-CSF EC50 in the model reported in [4] for pegfilgrastim is 9.86 ng/mL, while that for the filgrastim model reported in [5] is 3.15 ng/mL.

Baseline physiological values predicted by the model

The physiological model (Eqs. (1) – (6)) was also evaluated by comparing the predicted baseline values of model states, obtained via a population simulation (n=1000), to available measured basal values in healthy subjects reported in the literature. For the bone marrow and G-CSF subsystem models, the parameter means and standard deviations used in the population simulation are those given in Tables 1 and 2, while the mean values (estimated) and standard deviations (obtained from literatures) for the remaining parameters are shown in Tables 4 and 5.

The simulated values of the baseline pool sizes of circulating neutrophils, band cells, and marginated neutrophils in the blood are in good agreements with literature reported values [26, 32, 51, 52]. The model-predicted median value of circulating neutrophils is 4.1×10⁹ cells/L, which is within the observed range of 2.5–6×10⁹ cells/L reported for normal subjects [26, 32, 51, 52]. The model-predicted median value of band cells is 0.81×10⁸ is consistent with literature reports of values less than 0.7×10⁹ cells/L [26, 51, 52]. The ratio of marginated to circulating neutrophils reported in the literature is around 1.1 (ranges from 0.23 to 4.9) [32]. The model-predicted median value of marginated neutrophils is 4.7 × 10⁹ cells/L, resulting in a model predicted marginated/circulating neutrophils of 1.2.

Palbociclib in vitro bone marrow assay

The in vitro bone marrow model (Eqs. (2)–(4) and (9)) was fitted to the complete set of cell viability measurements (time course and end-of-assay), to estimate the model parameter PC0, K_deg, IC50_palbo, K_r. The remaining model parameters were fixed at their values in Table 1 as described in the methods section. Figure 6 shows the model fits to all the end-of-assay measurements (both experiment types) along with the time course results for control, while the estimated parameters are listed in Table 6. We note that the zero-order production of progenitor cells in the model (PC0) accounts for the asymptotic value of cell viability without palbociclib.

Fig. 6:

In vitro model results. Time course measurements for control (left panel) and all end-of-assay measurements (right panel) following palbociclib treatment. Symbol: measurements, lines: model predictions. Overall r²=0.94.

Table 6:

Model parameter estimates from palbociclib in vitro bone marrow assay.

Parameter (unit)	Definition	Estimate (RSE%)
PC0 (cells/mL/d)	In vitro production rate of progenitor cells	2.92×105 (2.2)
kdeg (1/d)	Net rate of natural loss/degradation of bone marrow cells	0.08 (13)
IC50palbo (nM)	Concentration of palbociclib eliciting a half-maximal inhibition of G1 to S transition	203 (9.2)
kr (1/d)	Rate constant from the resting to active G1 state	0.001 (36)

To translate the in vitro estimate of IC50_palbo to an in vivo value for use in the palbociclib clinical trial simulations, the free fractions of palbociclib in the in vitro assay and in human plasma were used. The free fraction of palbociclib in the in vitro bone marrow assay is approximately 75% due to nonspecific binding to the FBS in the culture media as reported in [53]; whereas the free fraction in human plasma is approximately 15% [54]. This results in a projected total drug in vivo IC50 of 1015 nM ((75%/15%)*203 nM) or 454 ng/ml.

Predicting neutropenia in palbociclib clinical trials

Baseline ANC

Comparing the distribution of the patient baseline ANC measurements with the distribution of model predicted baseline ANC values, yields the following results (both measured and predicted): 30^th percentile, 3.3×10⁹; median, 4.1×10⁹; 70^th percentile, 5.1×10⁹ (cells/L). As expected, given the patient-specific simulation procedure (see methods), the measured and predicted baseline ANC distributions are in agreement.

Comparison of patient-specific model predicted and observed ANC time course and neutropenia grade

The predicted ANC values at all measurement times in all patients (using the patient-specific, baseline ANC-matched ANC time profiles) are compared to their measured values in Fig. 7. The model-predicted ANCs correlate with measured ANCs (r²=0.55) and the spread of the observed versus predicted ANCs is distributed symmetrically around the line of identity (slope of regression line 0.90). Figure 8 illustrates the ability of the model to predict ANC-time profiles for selected patients treated with different dose regimens over a range of baseline ANC values. The figure shows three selected patients in each of four baseline ANC groups determined based on the percentiles of the distribution of the measured pre-dose ANC values: < 30^th percentile; 30^th-50^th percentile; 50^th-70^th percentile; > 70^th percentile.

Fig. 7:

Model predicted versus measures ANC values in all subjects (symbols). Line of identity (solid line). Regression line (dashed).

Fig. 8:

Model predicted versus measured ANC-time profiles in 12 selected patients from the four baseline ANC groups (three per group), representing the baseline ranges from below 3.3×10⁹ (first row), between 3.3–4.2×10⁹ (second row), 4.2–5.9×10⁹ (third row), and above 5.9×10⁹ cells/L (bottom row) of the measured baseline ANCs.

Since a primary application of the model is to predict the incidence of neutropenia, we compared the neutropenia observed in the clinical trials patients to model predictions. The grade of neutropenia during treatment with palbociclib was determined from the ANC time course measured in each patient and as predicted by the model using each patient’s baseline ANC-matched ANC time profile. Table 7a shows the number of observed and predicted patients with grade 2, 3, and 4 neutropenia. Given that the incidence of grade 3 and 4 neutropenia is of greatest clinical relevance, the model was further evaluated based on its ability to classify neutropenia into one of two groups: grades 3 or 4 neutropenia versus less than grade 3 (grade 2, 1, none) neutropenia. The results are shown as a classification table (Table 7b). Overall, the observed and predicted incidences of grade 3 or 4 neutropenia and less than grade 3 neutropenia are in good agreement with a total of 146/170 subjects correctly classified by the model. From the classification table, the overall prediction accuracy of the model is 0.86, with a precision of 0.95.

Table 7a:

Overall observed and model predicted number of patients that developed each grade of neutropenia in the palbociclib clinical trials (n=170).

Neutropenia grade	Observed	Predicted
Grade 2	51	62
Grade 3	72	54
Grade 4	4	4

Table 7b:

Classification table of the observed and model predicted number of patients with grade 3 or 4 neutropenia (Grade 3 or 4) versus less than grade 3 neutropenia (< Grade 3). Overall prediction ^aaccuracy of the model is 0.86, with a ^bprecision of 0.95.

Observed╲Predicted	Grade 3 or 4	< Grade 3
Grade 3 or 4	55	21
< Grade 3	3	91

^aCalculated as (55+91)/170;

^bCalculated as 55/(55+3)

Model predicted population ANC time course and neutropenia grade

To illustrate how the model could be used to predict the ANC response in drug development, a population simulation was performed for each of the two main dosing cohorts in the palbociclib clinical trials. The results are shown in Fig. 9, with the 125mg dose on a 3/1 schedule in the left panels and the 200mg dose on a 2/1 schedule in the right panel. In the figures, the solid lines represent the overall ANC profile for the median baseline ANC in the population (4.2×10⁹ cells/L), while the upper dashed lines indicate the model predictions for a baseline ANC of 5.9×10⁹ cells/L and the lower dashed line for a baseline ANC of 3.3×10⁹ cells/L. As with the individual subject simulations, the predicted time course for each of the three baseline ANC values was calculated as the median of the 100 nearest baseline ANC values (see Methods). To provide context, all the measured values of ANC from the clinical trial are shown (open symbols). We note that these simulations use the protocol dose regimens, not the actual patient-specific dose regimen with its dose adjustments, nor do they use the subject matched baseline ANC value. (these were used for the model prediction evaluation in the previous section). For the 125mg 3/1 schedule, ANC declines and reaches a nadir between day 14 (end of 2^nd week of doses in cycle 1) and day 28 (start of cycle 2), and then rebounds during the subsequent week (a little earlier than the start of cycle 2). For the 200mg 2/1 schedule, ANC reaches its nadir around day 18 (off-treatment week of cycle 1) and begins to rebound thereafter through day 28 (end of first week of treatment in cycle 2). The model predicts that for both schedules, ANC is fully recovered in 4–6 weeks after cessation of drug treatment.

Fig. 9:

Data and model predictions for the two main dose groups: 125mg starting dose on a 3/1 schedule (left panel); 200mg starting dose on a 2/1 schedule (right panel). Model predictions for the median baseline ANC group 4.2×10⁹ cells/L – solid lines; model predictions for the 3.3×10⁹ cells/L baseline ANC group – lower dashed lines; model predictions of 5.9×10⁹ cells/L baseline – upper dashed lines. All measurements – open symbols.

The prediction of the incidence of neutropenia in the population using each of these two dose regimens is shown in Fig. 10, stratified on baseline ANC. The incidence of grade 2, 3 or 4 neutropenia (rows) is shown for each of four baseline ANC ranges (columns), as a percent of simulated patients in each baseline ANC group.

Fig. 10:

For the 125 mg 3/1 schedule (left panel) and 200mg 2/1 schedule (right panel) dosing cohorts, the model-predicted percent of patients with grade 2, 3 or 4 neutropenia in each of the four baseline ANC groups. The gray scale reflects the percent of subjects in each column, (e.g., dark grey – greater than 50%, intermediate grey – 25% to 50%, light grey – less than 25%).

Discussion

A physiological model of granulopoiesis and its regulation has been developed that includes the bone marrow progenitor cell cycle, allowing for a mechanistic representation of the action of relevant anticancer drugs as informed by in vitro bone marrow studies. The complete integrated model also accounts for the maturation and mobilization of bone marrow neutrophils, the dynamics between circulating and marginated blood neutrophil pools, the neutrophil modulated plasma kinetics of endogenous G-CSF, as well as the action of G-CSF support therapies. Model development built upon the work of Roskos et al. [4] and Krzyzanski et al. [5] on the action of pegfilgrastim and filgrastim, respectively. The model was evaluated using the cell cycle inhibitor palbociclib, including bone marrow toxicity (BMT) studies to first incorporate palbociclib’s mechanism of action in the model, and then to predict the ANC response in clinical trial patients.

The overall model results presented in this work, including the basal values, margination dynamics and the results following stimulation with G-CSF, support the applicability of the model to describe major elements of granulopoiesis and its regulation. To characterize the regulation of neutrophils by endogenous G-CSF, as well as the effects of filgrastim and pegfilgrastim support therapies, a receptor mediated disposition model for two drugs competing for the same receptor was implemented [46]. This composite model was able to describe the neutrophils response following filgrastim and pegfilgrastim. Future model extensions could incorporate the pharmacodynamic feedback mechanism responsible for G-CSF disposition, as proposed by [55] in their general theory for pharmacodynamically mediated drug disposition.

In the model, the dynamic relationship between circulating and marginated neutrophils, and the extravasation of blood neutrophils is described by a two-compartment model. Model rate constants were estimated using data from a previous study using radioactive DFP32 labeled granulocytes in healthy volunteers [3]. While the available data included time course measurements in only four subjects with sparse initially sampling, our resulting model reflects margination and extravasation kinetics adequately, which cannot be described by the empirical models [56, 57]. In addition, our model predicts margination process kinetics comparable to the model reported in [58]. We note, however, that our results are based on data from healthy men, and given the pathophysiological and treatment factors that are known to affect the margination process, further investigation into how these factors might alter the margination subsystem model is needed.

To model the action of palbociclib in the in vitro bone marrow studies, the bone marrow sub-system model shown in Fig. 1 was modified to represent the in vitro assay by distributing the first-order degradation (K_deg) across all maturation cell types and eliminating the neutrophil blood mobilization term. This model was then combined with the progenitor cell model incorporating the action of palbociclib and used to estimate the in vitro system parameter PC0, as well as K_deg and the drug specific parameters IC50_palbo and K_r. Using the pooled BMT data, IC50_palbo was estimated to be 203 nM. We emphasize that IC50_palbo is the concentration of palbociclib that elicites a half-maximal inhibition of the G1 to S transition rate constant, in contrast to the BMT cell viability-palbociclib concentration IC50, which is the concentration that inhibits growth of the total cells in the culture by 50%. As described in results, the value of IC50_palbo estimated from the in vitro studies was then scaled to account for differences in the protein binding (free fraction) between in vitro and in vivo studies.

The predicted patient specific ANC time course results (Fig. 7 and 8), as well as the predicted and observed incidences of neutropenia (Table 7a and 7b) demonstrate the model’s predictive ability. The model was able to successfully translate in vitro bone marrow assay results to predict the clinically observed neutrophil response by accounting for the differences in free fraction and the faster rate of proliferation in vitro due to cytokine stimulation. To further assess possible combination effects of palbociclib and G-CSF support therapies, it would be necessary to explore whether the in vitro cytokines affect palbociclib’s IC50 in the assay (e.g., via cytokine titration). A sensitivity analysis of the model using the 125 mg, 3/1 dosing schedule, found that the change in ANC nadir value is most sensitive to the parameters PC0, τseg, k_ME, k₁, and k₄, suggesting the importance of the underlying physiological processes represented with these parameters in determining neutropenia.

To illustrate how the model could be used to predict the ANC response of a prospective dose regimen as part of a clinical trial design effort, a population simulation was performed for each of the two main dosing cohorts in the palbociclib clinical trials. From Fig. 10, we observe that patients following the 200mg 2/1 schedule develop more grades 3 or 4 neutropenia than those on the 125mg 3/1 schedule. For the 125mg 3/1 schedule, most of the patients (83.7%) with baseline ANC below 3.3×10⁹ cells/L develop grades 3 or 4 neutropenia, while for other baseline ANC groups, there is a much lower probability of developing grades 3 or 4 neutropenia. For the 200mg 2/1 schedule, nearly all the patients (96%) with baseline ANC below 3.3 ×10⁹ cells/L develop grades 3 or 4 neutropenia, along with more than one third of the patients (41%) with baseline ANC between 3.3×10⁹ cells/L and 4.2×10⁹ cells/L. For both the 125mg 3/1 and 200mg 2/1 schedules, patients with baseline ANC above 5.9×10⁹ cells/L have low probabilities, 1.4% and 5.8% respectively, of developing grades 3 or 4 neutropenia. As shown in Fig. 9, the neutropenia is reversible with ANC returning to baseline values upon discontinuation of palbociclib treatment.

The physiological model of granulopoiesis presented in this report provides a framework for using information on the mechanism of action of a drug, as obtained from in vitro bone marrow studies, to predict the myelosuppressive effects of the drug in patient populations. This model-based in vitro to clinical translations was demonstrated and evaluated using the cell cycle inhibitor palbociclib. Following a similar approach, the model may be used to predict the incidence of neutropenia for other anticancer agents, individually or in combination. This is of high value in drug discovery where translational efforts are focused on nominating safe and efficacious compounds for testing in patients. Because of the modular structure of the integrated model and its physiological basis, the individual model subsystems can be expanded based on other studies into the underlying system mechanisms of operation.