Model prediction of treatment planning for dose-fractionated radioimmunotherapy

Authors


Abstract

BACKGROUND

Clinical trials of radioimmunotherapy (RIT) often use dose fractionation to reduce marrow toxicity. The dosing scheme can be optimized if marrow and tumor cell kinetics following radiation exposure are known.

METHODS

A mathematic model of tumor clonogenic cell kinetics was combined with a previously reported marrow cell kinetics model that included marrow stromal cells, progenitor cells, megakaryocytes, and platelets. Reported values for murine tumor and marrow cellular turnover rates and radiosensitivity were used in the model calculation.

RESULTS

Given a tolerated level of thrombocytopenia, there is a fractionation scheme in which total radioactive dose administration can be maximized. Isoeffect doses that had different numbers of fractions and total radioactivity, but induced identical platelet nadirs of 20%, were determined. Assuming identical tumor uptake for all dose fractions, six tumor types were examined: early-responding tumors, late-responding tumors, and tumors that lacked a late-responding effect, with either constant or accelerated doubling time. For most tumor types, better tumor control (tumor growth delay and nadir of survival fraction) was predicted for a dosing scheme in which total radioactive dose was maximized. For late-responding tumors with accelerated doubling time, tumor growth delay increased, but the nadir of survival fraction became shallower as the number of fractions increased.

CONCLUSIONS

A mathematic model has been developed that allows prediction of the nadir and duration of thrombocytopenia as well as tumor clonogenic cell response to various RIT doses and fractionation schemes. Given a maximum tolerated level of thrombocytopenia, the model can be used to determine a dosing scheme for optimal tumor response. Cancer 2002;94:1264–9. © 2002 American Cancer Society.

DOI 10.1002/cncr.10295

The purpose of dose fractionation is to maximize tumor control for a given level of normal tissue toxicity.1 Marrow is often the dose-limiting organ in radioimmunotherapy (RIT) in the absence of marrow reconstitution.2–5 Therefore, marrow cell survival and proliferation in response to various RIT dosing schemes need to be addressed. Because of the complicated hierarchical structure of the hematopoietic system6 and time-varying dose rates to marrow, the description of survival and proliferation in response to various RIT dosing schemes is difficult. Consequently, analysis of RIT dosing optimization for tumor control versus marrow toxicity has been limited.

A purely experimental approach would be very time-consuming and costly because of the large number of dose-and-fractionation combinations that would have to be tested. To circumvent this problem, we used a previously reported mathematic model to predict marrow toxicity7 and developed a simple model for tumor clonogenic cell kinetics that considers cell kill by radiation, cell proliferation, and cell loss with constant as well as accelerated tumor doubling time during treatment. This modeling tool allows prediction of bone marrow toxicity and tumor control at any time after dose administration for any given fractionation scheme and set of marrow and tumor radiopharmacokinetics. Based on this modeling, an optimal dosing scheme can be determined for tumor control for a given level of marrow toxicity. The model was developed for mouse marrow and tumor because this model should be relatively easy to validate in future experiments.

Materials and Methods

Tumor Clonogenic Cell Kinetics Model

In the current analysis, tumor control or therapeutic efficacy was described by the number of tumor clonogenic cells (Fig. 1). The number of clonogenic cells was modeled as a result of radiation kill, self-renewal, and cell loss by differentiation or apoptosis. Radiation kill was described using the linear quadratic formula. In the absence of proliferation, the fraction of surviving cells, equation image, after a single dose D is8

equation image(1)

where N is the number of surviving cells, N0 is the initial cell number, and α and β are respective radiosensitivities. For RIT, radiation dose rate to tissue, [r (t)] at time t after injection changes as a result of physical decay and biologic clearance. For clonogenic cells that lack a late-responding effect (β = 0), cell death due to radiation can be described as9

equation image(2)

For clonogenic cells for which β ≠ 0, cell death due to radiation can be described as10

equation image(3)

where μ is a tissue repair time constant.

Figure 1.

Tumor clonogenic cell number at any moment is modeled as a result of radiation kill, self-renewal, and cell loss by differentiation or apoptosis. During radioimmunotherapy, the rate of tumor regrowth can be at constant or accelerated doubling time.

Although most tumors grow exponentially with constant doubling time (some tumors decelerate as they become larger), there is strong evidence for accelerated repopulation during radiotherapy.11 In this study, constant or accelerated tumor doubling times were considered. Accelerated tumor doubling was modeled by increasing the rate of self-renewal and decreasing the rate of cell loss in response to clonogenic cell cytopenia.12

Platelet Kinetic Model

In the current analysis, platelet counts were used to represent normal tissue toxicity because thrombocytopenia is often the dose-limiting factor for RIT. Here, a previously reported compartment model (Fig. 2) consisting of progenitor cells, differentiated cells (megakaryocytes), and functional peripheral blood cells (platelets) was used to describe the platelet kinetic system.7 Briefly, for low-dose-rate irradiation, characteristic of RIT, the radiosensitive progenitor cells suffer the most radiation damage. Radiation damage to platelets and megakaryocytes is relatively insignificant.13 The daily platelet counts during or after RIT are the result of the balance between normal destruction and production from megakaryocytes and progenitor cells. As progenitor cells lack a late-responding effect (β = 0),14 cells killed by radiation can be described by Equation 2.

Figure 2.

Compartmental model describing thrombopoiesis and the effects of irradiation from radioimmunotherapy with time-varying dose rate. The solid arrows represent cell flow from one compartment to another. The dashed arrows symbolize factors that regulate the rate of cell flow as a result of the feedback mechanisms.

Progenitor cells are supported and grow in a stromal microenvironment. Because stromal cells are not of hematologic origin, these cells were modeled separately, as previously described.7 Briefly, a three-compartment model based on Jones and Morris15, 16 was used to describe the processes of sublethal injury, repair of sublethal injury, “one-hit” killing, “two-hit” killing, and proliferation (Fig. 2). The effect of the stromal microenvironment on hematopoiesis was modeled as a regulation factor for progenitor cell self-renewal.

Mouse Model Parameters

For tumors with constant doubling time, a doubling time of 4 days was selected.12 For tumors with accelerated doubling time, the shortest doubling time was assumed to be 1 day, while the rate of self-renewal increased twofold and the rate of cell loss decreased twofold. Six tumor types were examined: early-responding, late-responding, and non-late-responding tumors with either constant or accelerated doubling time. The early-responding tumors had an α of 0.5 Gy−1 and an α/β of 10 Gy. The late-responding tumors had an α of 0.2 Gy−1 and an α/β of 3 Gy. Non-late-responding tumors had an α of 0.5 Gy−1 and a β of 0 Gy. For the purposes of illustration, the tumor repair half-time of 1 hour was used in all calculations.1 The selection of the initial population in each marrow cellular pool, marrow cellular turnover rates, and radiosensitivities have been described previously in detail.7 Briefly, a life span of 5 days was used for platelets. An initial doubling time of 1 day was used for self-renewal of progenitor cells. The doubling time could be shortened to 0.5 days, as progenitor cells were severely depleted. An α value of 0.65 Gy−1 and a β value of 0 were used for progenitor cell radiosensitivity. The published rate constants for stromal cells were based on maximum likelihood estimates from experimental data.16

Pharmacokinetics

Clearance of the radiolabeled antibody in marrow was assumed to be biexponential, with an effective half-life of 2 hours for the fast phase and 2 days for the slow phase. The intercept was assumed to be 70% for the fast phase and 30% for the slow phase. Tumor pharmacokinetics were assumed to be biexponential, with an effective clearance time of 2.05 days and an uptake half-life of 0.167 days (with biologic uptake peak at 1 day). The radiation dose to tumor was assumed to be 5 times the dose to marrow.

Effect of Dose Fractionation on Thrombocytopenia

Assuming identical marrow clearance for each fraction, 4 dosing schemes were compared: (1) 2 fractions 6 weeks apart, (2) 3 fractions at 3-week intervals, (3) 4 fractions at 2-week intervals, and (4) 7 fractions at 1-week intervals. Thrombocytopenia was compared for the same amount of total radioactivity dose but various dosing schemes. Isoeffect doses were defined as doses that had different total radioactivity and dosing schemes but induced identical platelet nadirs of 20% of initial baseline counts. The fractionation scheme that was determined allowed for maximum total injected radioactivity.

Optimal Dose Fractionation for Tumor Control with Identical Platelet Nadir

Tumor growth delay and nadir of surviving fraction were used as indicators of tumor control. Tumor growth delay was defined as the length of time during which the tumor volume was smaller than the initial volume. Nadir was defined as the lowest fraction of the surviving cells encountered during the active course of treatment. Assuming identical tumor uptake and clearance for each dose fraction, effects on tumor control were computed for different dose fractionation schemes that induced identical platelet nadirs of 20%.

Results

Effect of Dose Fractionation on Thrombocytopenia

For a given amount of total injected radioactivity, fractionated administration led to less marrow toxicity than single-dose injection (Fig. 3). While a single dose induced severe thrombocytopenia (nadir of 1.7%), the platelet nadir was 20% if the dose was given in 2 fractions 6 weeks apart. Splitting the dose into 4 fractions at 2-week intervals raised the nadir to 32%. However, increasing the number of fractions to 7 fractions given at 1-week intervals resulted in a nadir of 29% (Fig. 3).

Figure 3.

For a given level of total injected radioactivity, fractionation reduces the marrow toxicity compared with single-dose injection. In comparison of multiple dose fractions, the platelet nadir induced by 7 fractions given in 1-week intervals is worse than that induced by 4 fractions at 2-week intervals (nadir of 29% vs. 32%).

For isoeffect doses that induced platelet nadirs of 20%, the scheme involving 4 fractions at 2-week intervals allowed for the maximum total dose (Table 1). If the number of fractions increased from 4 to 7, the total radioactive dose had to be reduced in order to maintain the nadir. At the same time, the duration of the platelet depletion was prolonged.

Table 1. Isoeffect Doses of Radioimmunotherapy That Would Induce Identical Platelet Nadirs of 20% from Initial Baseline Countsa
No. of fractionsTotal radioactivity (relative)Time interval (weeks)Days platelet counts < 40%Days platelet counts < 30%
  • a

    Maximum total radioactivity could be given in 4 fractions at 2-week intervals.

2100627.018.7
3125339.023.9
4128242.330.3
7121141.231.8

Optimal Dose Fractionation for Tumor Control with Identical Platelet Nadir

Tumor responses were compared for the 4 isoeffect doses that induced platelet nadirs of 20%: (1) 2 fractions, 100-unit dose; (2) 3 fractions, 125-unit dose; (3) 4 fractions, 128-unit dose; and (4) 7 fractions, 121-unit dose. For tumors with constant doubling times, better tumor control (growth delay and nadir) was achieved for 4 dose fractions for the early-responding tumors and 3 dose fractions for the late-responding tumors (Table 2).

Table 2. Tumor (with Constant Doubling Time) Responses to Isoeffect Doses That Induced Platelet Nadirs of 20%a
Fraction no.Total radioactivity (relative)Interval (week)Early-responding tumors α/β = 10 Gy, α = 0.5 Gy−1Late-responding tumors α/β = 3 Gy, α = 0.2 Gy−1
Delay (days)NadirDelay (days)Nadir
  • a

    Tumor doubling times remained unchanged during treatment. Better tumor control was obtained by doses given in 4 fractions at 2-week intervals for early-responding tumors and in 3 fractions at 3 week intervals for late-responding tumors.

210061515.44 × 10−8601.46 × 10−2
312531871.08 × 10−10834.12 × 10−3
412821896.51 × 10−11824.27 × 10−3
712111764.80 × 10−10741.42 × 10−2

For early-responding tumors with accelerated doubling times, better tumor control was achieved by 3 dose fractions (Table 3). For late-responding tumors with accelerated doubling times, tumor growth delay increased and the nadir became shallower as the number of fractions increased.

Table 3. Tumor (with Accelerated Doubling Time) Responses to Isoeffect Doses That Induced Platelet Nadirs of 20%a
Fraction no.Total radioactivity (relative)Interval (weeks)Early-responding tumors α/β = 10 Gy, α = 0.5 Gy−1Late-responding tumors α/β = 3 Gy, α = 0.2 Gy−1
Delay (days)NadirDelay (days)Nadir
  • a

    Tumor doubling times accelerated during treatment. While deeper survival nadir was obtained by doses given in 2 fractions, longer tumor growth delay was obtained by doses given in 3 fractions for early-responding tumors and in 7 fractions for late-responding tumors.

21006443.70 × 10−4147.54 × 10−2
31253632.56 × 10−3301.53 × 10−1
41282616.05 × 10−3373.01 × 10−1
71211581.44 × 10−2496.13 × 10−1

For tumors that lacked a late-responding effect (β = 0) and had a constant doubling time, tumor control was only dependent on the total radioactivity injected (Table 4). For tumors that lacked a late-responding effect and had an accelerated doubling time, a deeper nadir was obtained by a dose given in 2 fractions, and longer delay was obtained by a dose given in 4 fractions.

Table 4. Tumor (with No Late-Responding) Response to Isoeffect Doses That Induced Platelet Nadir of 20%a
Fraction no.Total radioactivity (relative)Interval (weeks)Constant doubling time tumorsAccelerated doubling time tumors
Delay (days)NadirDelay (days)Nadir
  • a

    Tumor lacked a late-responding effect (α = 0.5 Gy−1, β = 0). For tumors with constant doubling time, better tumor control was obtained by dose with maximum total radioactivity. For tumors with accelerated doubling time, deeper nadir was obtained by doses given in 2 fractions and longer delay was obtained by doses given in 4 fractions.

210061422.48 × 10−741.47.53 × 10−4
312531775.20 × 10−1059.74.11 × 10−3
412821822.23 × 10−1059.91.06 × 10−2
712111728.99 × 10−1057.32.00 × 10−2

Discussion

The main reason for using dose fractionation is to take advantage of the difference between early-responding and late-responding tissues. While the radiation effect on early-responding tissue can be reduced by prolonging treatment time and dose fractionation, the radiation effect on late-responding tissues will not be changed that much if the total dose is not changed.1 This is because cells do not die after irradiation until they undergo division.

In RIT, the advantage of dose fractionation is less obvious.17 This is because the dose-limiting marrow progenitor cell has almost no repair.14 However, the surviving progenitor cells will accelerate the rate of proliferation if there is adequate stromal cell support. Stromal cells are relatively radioresistant and can repair sublethal damage. Characterizing stromal cells as late-responding tissue may provide some rationale for dose fractionation in RIT.

In our model, the rate of stromal cell proliferation increased if the stromal cell number decreased; the rate of progenitor cell self-renewal increased if the progenitor cell number decreased; and the rate of maturation from megakaryocytes to platelets increased if the platelet number decreased. These three feedback mechanisms were modeled for keeping homeostasis in immediate cellular pools. The stromal cell support for progenitor growth was modeled so that the rate of progenitor cell self-renewal would decrease as the normal stromal cell number decreased. Although cellular function and homeostasis through cytokines are very complex, the predictions made by this model were in good agreement with the severity of thrombocytopenia and time of nadir measured in mice receiving RIT.7

Currently, most clinical trials of RIT are limited by marrow toxicity. The goal of these protocols is to establish a fractionated maximum tolerated dose so that maximal radiation can be delivered to tumors.4 For a mean baseline platelet count of 250 K/μL in humans,18 a National Cancer Institute toxicity grade of III (25–50 K/μL) corresponds to nadirs of 10–20%. A nadir of 20% was selected in this study to simulate a Grade III toxicity for tolerable doses. In clinical studies, this 20% nadir may not always be applicable to some patients because of the large variation in initial platelet counts. Another important toxicity index is duration of platelet depletion. While those isoeffect doses induced identical nadirs, larger number of fractions resulted in prolonged duration of the platelet nadir (Table 1). In practice, a combined index may be needed to reflect both the nadir and duration of thrombocytopenia. Nevertheless, isoeffect doses can always be obtained as long as the desired toxicity criterion are determined.

Thrombocytopenia was reduced substantially by splitting a single dose into two, but further splitting did not always reduce thrombocytopenia (Fig. 3). This was also illustrated by comparing isoeffect doses that induced an identical amount (20%) of platelet nadirs (Table 1). The total radioactive dose had to be reduced if the number of fractions increased from 4 to 7. This could be a result of the overlapping of radiation exposure between each fraction when the time interval became comparable to the effective clearance half-life of the radiopharmaceutical in the marrow. The above model prediction was in good agreement with the mouse experiments in that continuous injection of RIT induced more marrow toxicity than bolus injection.19

In the current study, optimal dose fractionation had a time interval of 2–3 weeks for all tumors except the late-responding tumors with an accelerated doubling time (Tables 2–4. These results illustrate the predominant impact of marrow toxicity on dose optimization. Under identical platelet nadirs, tumor control was more dependent on total injected radioactivity and less dependent on the dosing scheme, except for late-responding tumors with accelerated doubling time. Because of larger nonlinear dose effects during an active course of dose fractionation, the dosing scheme had more influence on tumor control for the late-responding tumor with accelerated doubling time (Table 4).

Typical parameters were selected in the current modeling for the purpose of illustration. The numeric results of the model prediction may change as model parameters are changed. For example, the predicted optimal dose fractionation at time intervals of 2–3 weeks can be changed if the clearance of radiopharmaceutical in marrow is much longer or isoeffect doses defined induce a deeper platelet nadir. In such situations, a longer time interval between each fraction can be expected.20 In another example, identical radiopharmacokinetics were assumed for each dose fraction. However, radiopharmacokinetics of the tumor can change between fractions as a result of treatment. Later fractions can have less uptake as a percentage of injected radioactivity dose than the initial fraction.19 Nevertheless, if we can describe the pattern of change between the fractions based on experimental data, the model prediction can still provide insight to guide the designs of clinical trials of optimized dose fractionation.

Conclusions

A mathematic model has been developed that allows prediction of the nadir and duration of thrombocytopenia as well as the tumor clonogenic cell response to various RIT doses and fractionation schemes. The model was developed for mouse marrow and tumor, as this model will be relatively easy to validate in future experiments. Given a maximum tolerated level of thrombocytopenia, the model can be used to determine a dosing scheme for optimal tumor response.

Acknowledgements

The authors gratefully acknowledge Joseph A. O'Donoghue, Ph.D., for his valuable comments and suggestions for improving this manuscript.

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