International Journal of
Radiation Oncology biology
physics
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Physics Contribution
Incorporating Cancer Stem Cells in Radiation Therapy Treatment Response Modeling and the Implication in Glioblastoma Multiforme Treatment Resistance Victoria Y. Yu, BS, Dan Nguyen, BS, Frank Pajonk, MD, PhD, Patrick Kupelian, MD, Tania Kaprealian, MD, Michael Selch, MD, Daniel A. Low, PhD, and Ke Sheng, PhD Department of Radiation Oncology, David Geffen School of Medicine, University of California, Los Angeles, Los Angeles, California Received Jun 4, 2014, and in revised form Nov 22, 2014. Accepted for publication Dec 1, 2014.
Summary The role of tumor-intrinsic heterogeneity including cancer stem cells (CSCs) and differentiated cancer cells (DCCs) in radiation therapy treatment response was investigated using 2 pairs of linear-quadratic radiation cell survival models. Using linear-quadratic model-fit parameters, the dynamics of CSC and DCC killing from radiation therapy was simulated using an ordinary differential equation. Simulation indicated the effectiveness of hypofractionation for certain tumors and refractory GBM radioresistance, due to the
Purpose: To perform a preliminary exploration with a simplistic mathematical cancer stem cell (CSC) interaction model to determine whether the tumor-intrinsic heterogeneity and dynamic equilibrium between CSCs and differentiated cancer cells (DCCs) can better explain radiation therapy treatment response with a dual-compartment linear-quadratic (DLQ) model. Methods and Materials: The radiosensitivity parameters of CSCs and DCCs for cancer cell lines including glioblastoma multiforme (GBM), nonesmall cell lung cancer, melanoma, osteosarcoma, and prostate, cervical, and breast cancer were determined by performing robust least-square fitting using the DLQ model on published clonogenic survival data. Fitting performance was compared with the single-compartment LQ (SLQ) and universal survival curve models. The fitting results were then used in an ordinary differential equation describing the kinetics of DCCs and CSCs in response to 2- to 14.3-Gy fractionated treatments. The total dose to achieve tumor control and the fraction size that achieved the least normal biological equivalent dose were calculated. Results: Smaller cell survival fitting errors were observed using DLQ, with the exception of melanoma, which had a low a/b Z 0.16 in SLQ. Ordinary differential equation simulation indicated lower normal tissue biological equivalent dose to achieve the same tumor control with a hypofractionated approach for 4 cell lines for the DLQ model, in contrast to SLQ, which favored 2 Gy per fraction for all cells except melanoma. The DLQ model indicated greater tumor radioresistance than SLQ, but the
Reprint requests to: Ke Sheng, PhD, Department of Radiation Oncology, University of California, Los Angeles, 200 Medical Plaza, B265, Los Angeles, CA 90024. Tel: (310) 853-1533; E-mail: ksheng@ mednet.ucla.edu Int J Radiation Oncol Biol Phys, Vol. 91, No. 4, pp. 866e875, 2015 0360-3016/$ - see front matter Ó 2015 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ijrobp.2014.12.004
Conflict of interest: none. This study was supported in part by NSF grant DGE-1144087. Supplementary material for this article can be found www.redjournal.org.
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radioresistance was overcome by hypofractionation, other than the GBM cells, which responded poorly to all fractionations. Conclusion: The distinct radiosensitivity and dynamics between CSCs and DCCs in radiation therapy response could perhaps be one possible explanation for the heterogeneous intertumor response to hypofractionation and in some cases superior outcome from stereotactic ablative radiation therapy. The DLQ model also predicted the remarkable GBM radioresistance, a result that is highly consistent with clinical observations. The radioresistance putatively stemmed from accelerated DCC regrowth that rapidly restored compartmental equilibrium between CSCs and DCCs. Ó 2015 Elsevier Inc.
Introduction Increasing evidence has suggested that solid tumors are hierarchically organized and contain a small population of cancer stem cells (CSCs) (1, 2). The subpopulations of CSCs are observed to have characteristics of self-renewal, differentiation to non-stem progenies, and unlimited proliferative capacity (3). In in vitro and animal experiments, CSCs are also observed to be more radioresistant than their non-stem counterpartddifferentiated cancer cells (DCCs) (4-11). Therefore, many believe that CSCs are the driving force of cancer progression and that successful therapy must eradicate CSCs. Mathematical models have suggested that perhaps the dynamic equilibrium between the DCC and CSC compartments within a tumor is essential to the treatment outcome (12-15). A “tumor growth paradox”dCSCs driven out of dormancy owing to spontaneous DCC cell death from therapy interventions resulting in accelerated tumor progressiondhas been previously demonstrated with mathematical models and biological experiments (16-18). Classic radiobiological models do not take into account such distinct radiobiology among different groups of tumor cells but instead assume uniform radiosensitivity within a tumor. The limitations of these models came to light as stereotactic ablative radiation therapy (SABR) became successful in the clinic, contrary to predictions of the linearquadratic (LQ) model using conventionally established radiosensitivity parameters (19-21). Modified LQ models that include dose-dependent repair and cell-killing terms, and synthesis of the LQ model with the multitarget model and a dose transition point that moderates the cell survival behavior toward higher fractional doses (19, 21-24) have been developed to address this discrepancy. These models showed superior data fitting of single fractional in vitro cell survival for a wide range of doses compared with the unmodified LQ model. However, these models remain controversial owing to their lack of biological foundations (25, 26), difficulty in determining the additional fit-parameter values for individual patients, and inability to provide insight to the paradoxical treatment outcomes of cancers with known high a/b ratios, such as nonesmall-cell lung cancer (NSCLC) and glioblastoma multiforme (GBM), which respond to hypofractionation differently. Therefore, these modified radiobiological models
have been rarely used in practice, despite the great interest of comparing treatment outcome from different regimens. Because the role of the CSC in cancer progression has become more prevalent, we propose that the incorporation of its properties to radiobiological modeling might improve its performance in predicting radiation therapy treatment response. Therefore, in this study, we performed preliminary exploration with a simplistic mathematical CSC interaction model to determine whether the tumor-intrinsic heterogeneity and dynamic equilibrium between CSCs and DCCs can better explain radiation therapy treatment response with a dual-compartment LQ (DLQ) model.
Methods and Materials Two major components come into play when modeling the distinct radiosensitivity and dynamic interaction between CSCs and DCCs; first, the determination of the radiosensitivity parameters of both compartments, and second, an ordinary differential equation (ODE) that models the CSC self-renewal, differentiation to DCC, and DCC growth and apoptosis. These 2 components were then combined to model CSC and DCC interaction alongside with radiation therapy cell killing of each compartment.
Determination of radiobiological parameters A simple way to include intratumor radiosensitivity heterogeneity is a DLQ model consisting of CSCs and DCCs. For a single fraction of treatment, the model was constructed as 2 2 SFðDÞZFea1 Db1 D þ 1 F ea2 Db2 D ð1Þ with F (0 F 0.2) as the fraction of CSCs out of all cells, and a and b describing the radiobiological properties of each population. The upper bound of F was set to 0.2 owing to the indication from publications that CSCs are a minor subpopulation of a solid tumor (27,28). Least-square fitting of the model was then performed on 8 previously published clonogenic cell survival datasets digitized from multiple human cancer cell lines, including U373MG (GBM), CP3, DU145 (prostate carcinoma) (29), HeLa (cervical cancer) (30), MDA-MB-231 (breast cancer) (31), H460 (NSCLC) (19), TX-4 (osteosarcoma) (20), and melanoma (32) using
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MATLAB R2013a (MathWorks, Natick, MA). All experiments were conducted using x-rays or gamma rays with relative biological effectiveness of 1. To compare the fitting performance and model quality of the DLQ model with the classic LQ model and a modified LQ (universal survival curve [USC]) model (19), we calculated the sum of squares error (SSE), Akaike information criterion (AIC), and Bayesian information criterion (BIC) in log10 scale for all datasets corresponding to all 3 models. The USC model with 3 fit parameters, D0, Dq, and aUSC, is shown in Equation 2. 8 ðaUSC D þ bUSC D2 ; D DT > < lnSFZ 1 Dq > : D þ ; D > DT ð2Þ D0 D0
bUSC Z
ð1 aUSC D0 Þ2 2Dq ; DT Z 4D0 Dq 1 aUSC D0
For the dataset of the breast cancer cell line MDAMB-231, the CSC fraction F was measured to be 0.0204 in the same publication (31) and fixed in curve-fitting.
ODE model: The interplay of CSCs and DCCs and radiation therapy The interaction between CSCs and DCCs was modeled according to an ODE model developed by Hillen et al (13) and used by Bachman et al (12). The simulated tumor was assumed to be spatially homogenous, with equal cell density, growth, and apoptosis throughout the tumor region. The ODE with a CSC and a DCC compartment is shown in Equation 3. :
calculated on the basis of the parameter F (fraction of CSC out of total tumor volume) obtained from data fitting of the DLQ model, as shown in Equation 1. Following Bachman et al (33), growth and apoptosis rates (mV, mU, aV) were set to ln(2)/Tpot day1, where Tpot represents the tumor potential doubling time of the simulated cancer at question. The Tpot values were set to be 23, 23, 4, 7.1, 8.2, 11, 1.3, and 3.9 days, following published data for cell lines CP3, DU145 (34), H460 (35), HeLa (36), MDA-MB-231 (37), melanoma (38), TX-4 (39), and U373MG (40), respectively. P was determined to maintain a dynamic equilibrium between CSCs and DCCs. In this study using the ODE formula, this requires the probability of CSC self-renewal to be slightly greater than 0.5. A probability of 0.505 was typically used in previous studies (12, 13) and adopted in this study. In addition, within a reasonable range of 0.5005 to 0.55, we studied the sensitivity of the conversion probability P in the conventional 2 Gy 30 and hypofractionated 10 Gy 5 fractionation schemes. Radiation therapy was modeled by applying the classical LQ equation using the corresponding radiobiological parameters to each compartment at times of treatment. To simplify, we assumed a spatially homogeneous yet biologically heterogeneous tumor receiving a spatially homogeneous dose. To compare the difference between our model and the SLQ model, treatment response was simulated with the same procedure as described above, but only using the set of radiobiological parameters obtained from classical LQ fit. All ODE simulations were performed in MATLAB. Dose fraction sizes of 2, 3, 4, 5.1, 6.5, 7.7, 9.7, and 14.3 Gy were used, and the number of dose fractions
SelfRenewal
U ðtÞZð2P 1ÞmU kðWðtÞÞUðtÞ :
V ðtÞZ 2ð1 PÞmU kðWðtÞÞUðtÞ þ mV kðWðtÞÞVðtÞ aV VðtÞ Differentiation from CSC
DCC Growth
Apoptosis
ð3Þ
WðtÞZUðtÞ þ VðtÞ KðWÞZmaxf1 W 4 ; 0g
where U(t), V(t), and W(t) are the volume fractions of the CSCs, DCCs, and total tumor with respect to a specified volume of interest, P represents the probability that a CSC gives rise to 2 CSCs, and 1-P is the probability that a CSC gives rise to 2 DCCs. The growth rates of CSC and DCC are mU and mV, respectively, and aV is the apoptosis rate of the DCCs. The apoptosis rate of CSCs was set to 0, assuming that CSCs had unlimited replicative potential. k(W ) is a volume constraint that keeps the total tumor volume fraction within the range of 0 and 1. At time t Z 0, the total tumor cell number was set to be 1.3 107 within a volume of interest of 4.2 109 cells. The initial starting fractions of the CSC and DCC compartments were then
required to achieve a tumor control probability (TCP) of 0.9 was determined. Treatment was administered once per day, every weekday. The corresponding biological effective doses (BEDs) to the surrounding normal tissue were also evaluated using Equation 4, with n, d, and a/b representing the number of fractions, dose fraction size, and a/b ratio of the surrounding normal tissue, respectively. The a/b ratio was set to 3 for all BED calculations. The TCP for the DLQ and SLQ models, indicated as TCPDLQ and TCPSLQ, were calculated according to a Poisson distribution as shown in Equation 5, with NDCCþCSC and NCSC representing the average number of remaining total tumor cells and remaining CSCs after treatment, respectively.
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d BEDZnd 1 þ a =b
ð4Þ
TCPDLQ ZexpðNCSC Þ TCPSLQ Zexpð NCSCþDCC Þ
ð5Þ
Because of the representative treatment outcome from GBM and NSCLC, we compared GBM and NSCLC using parameters obtained from cell lines U373MG and H460, respectively. We selected NSCLC for comparison because hypofractionation in NSCLC has been remarkably successful (41), whereas the same approach has been ineffective in treating GBM (42). Using the DLQ model, we applied currently utilized or previously applied GBM treatment schemes, including 2 Gy 30, 1.8 Gy 33, 1 Gy 78 b.i.d. (43), 2 Gy 45 (44), 1.3 Gy 60 b.i.d., 1.5 Gy 40 b.i.d. (44, 45), and 5 Gy 10 b.i.d. (46), to study the tumor response to treatment.
Results DLQ fit results The radiobiological parameters obtained from DLQ, SLQ, and USC fitting results to all 8 clonogenic survival datasets are shown in Table 1. The AIC, BIC, and SSE values of all 3 models are shown in Table 2. The a values of CSCs were smaller than those of their DCC counterparts for all except Tx-4, whose CSC compartment had a smaller b. As shown in Figure 1, the original LQ model resulted in overprediction of cell death in the high-dose range. Assessing by SSE, both DLQ and USC models more accurately described the cell survival behavior for the entire dose range than LQ, with the exception of the melanoma cells. Factoring in the larger number of fitting parameters in the DLQ and USC models, the average AIC and BIC values of the DLQ and USC models were lower than that of the LQ, as shown in Table 2. The melanoma cell was an outlier, showing an extremely low a/b ratio (0.16) in the LQ fit. For Table 1
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MDA-MB-231, F was fixed according to literature. The fitobtained b value for CSCs was close to zero, agreeing with the publication (31).
ODE simulations and therapy schedule optimization The number of dose fractions and BED required to achieve tumor control (TCP Z 0.9) for each dose fraction size and cell line are shown in Tables 3 and 4. As evident from Table 3, DLQ indicated a greater dose for tumor control than the SLQ model. However, the greater radioresistance can still be overcome with hypofractionation, except for GBM cell U373MG. The GBM cells were remarkably resistant to hypofractionated treatment, requiring a prohibitively high dose to control. In Table 4, the lowest possible BED values that achieved tumor control for each cell line are indicated with asterisks. Here the dose fraction size that attained the lowest normal tissue BED was considered the optimal treatment fractionation schedule for each cell line. The SLQ model indicated conventional 2-Gy fractionation schemes to result in the least normal tissue effect while attaining tumor control for all cell lines except melanoma, for which a 7.7-Gy fraction was optimal owing to its low a/b ratio. In comparison, DLQ indicated hypofractionation approaches (>7.7 Gy) to be preferable for U373MG, CP3, HeLa, and H460. Figure 2a demonstrates the difference between the SLQ and DLQ treatment outcome of the GBM and NSCLC cells. With a regularly fractionated treatment of 60 Gy, SLQ indicated similar outcomes between the 2 different cells, but in DLQ the number of remaining GBM cells was 2 orders of magnitude greater than the remaining NSCLC cells. Figure 2b compares the conventional 2 Gy 30 and 5 Gy 10 SABR treatment outcome. For NSCLC, SABR resulted in significantly more effective tumor control than conventional fractionation schedules. In stark contrast, for
DLQ, SLQ, and USC model clonogenic survival fit parameters DLQ fit parameters
USC model fit parameters
Single LQ fit
Cell line
F
a1
b1
a2
b2
a
b
a/b
Dq
D0
aUSC
CP3 DU145 HeLa H460 MDA-MB-231 Melanoma TX-4 U373MG
0.047 0.010 0.052 0.010 0.020 0.166 0.200 0.016
0.021 0.099 0.010 0.010 0.125 0.038 0.244 0.010
0.036 2.22E-05 0.071 0.042 2.43E-06 0.059 0.022 1.77E-07
0.098 0.191 0.197 0.010 0.271 0.013 0.128 0.125
0.057 0.017 0.203 0.079 0.032 0.061 0.105 0.028
0.15 0.22 0.54 0.16 0.36 0.01 0.50 0.17
0.04 0.01 0.06 0.05 0.01 0.06 0.01 0.02
3.45 17.53 8.89 2.95 32.74 0.16 38.90 9.49
3.78 2.28 1.50 4.21 1.50 4.39 1.50 2.05
1.01 2.19 0.84 0.76 1.75 0.89 1.34 2.30
0.07 0.19 0.52 0.01 0.31 0.01 0.18 0.12
Abbreviations: DLQ Z dual-compartment linear-quadratic; SLQ Z single-compartment linear-quadratic; USC Z universal survival curve. F is the fraction of cancer stem cells within the tumor. (a1, b1) and (a2, b2) are the radiobiological parameters of the cancer stem cell and differentiated cancer cell compartment, respectively. Single LQ fit indicates the parameters obtained from a classic LQ model fit. The last 3 columns show the fit parameters of the USC model, Dq, D0, and aUSC.
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Table 2
AIC, BIC, and SSE comparison AIC
BIC
SSE
Cell line
DLQ
LQ
USC
DLQ
LQ
USC
DLQ
LQ
USC
CP3 DU145 HeLa H460 MDA-MB-231 Melanoma TX-4 U373MG Average
50.30 104.84 34.51 13.19 17.38 44.22 40.17 92.19 49.60
40.05 98.37 20.56 4.03 12.54 50.76 18.33 75.91 39.06
53.25 106.66 18.25 17.80 13.97 44.59 31.80 93.94 47.53
42.10 96.63 30.67 12.71 18.42 38.55 35.92 85.64 45.08
35.95 94.27 18.64 4.26 13.17 47.92 16.20 72.64 36.82
47.78 101.19 15.69 17.48 14.80 40.81 28.97 89.57 44.54
0.198 0.030 0.030 0.020 0.004 0.058 0.027 0.011 0.047
0.347 0.046 0.123 0.366 0.016 0.056 0.174 0.031 0.145
0.205 0.033 0.126 0.019 0.009 0.070 0.062 0.013 0.067
Abbreviations: AIC Z Akaike information criterion; BIC Z Bayesian information criterion; SSE Z sum of square error. Other abbreviations as in Table 1. AIC, BIC, and SSE values in log10 scale for all 3 models on all clonogenic survival datasets.
GBM, SABR fractionating did not noticeably improve tumor control. To understand the unique GBM radioresistance to hypofractionated treatment, we plotted the CSC and DCC compartment growth over time to explore their relationship and interactions (Fig. 2c). The interaction between the 2
A
compartments was simulated and compared for fractionation schedules of 2 Gy 30, 5 Gy 10, and 10 Gy 3. From Figure 2c, the dynamic equilibrium was disrupted when the DCC population became substantially smaller than the CSC population as a result of aggressive treatment. Consequently, rapid DCC regrowth occurred to restore the
B
100
100
Survival fraction
Survival fraction
10-1
-2
10
10-3
10-4
CP3 Dual compartment fit USC fit LQ fit
10-5 0
1
2
3
4
5
6
7
10-1
10-2
HeLa Dual compartment fit USC fit LQ fit
10-3
8
9
0
10 11 12 13 14
2
1
3
Dose (Gy)
D
100
Survival fraction
Survival fraction
C
10-1
0
1
2
3
4
5
5
7
6
8
100
10-1
U373MG Dual compartment fit USC fit LQ fit
MDA-MB-231 Dual compartment fit USC fit LQ fit
10-2
4
Dose (Gy)
6
Dose (Gy)
7
8
9
10
11
0
1
2
3
4
5
6
7
8
9
10
11
Dose (Gy)
Fig. 1. Dual-compartment linear-quadratic (DLQ), universal survival curve (USC), and linear-quadratic (LQ) model fit comparison. (a) Prostate carcinoma cell line CP3. (b) Cervical cancer cell line HeLa. (c) Breast cancer cell line MDA-MB231. (d) Glioblastoma multiforme cell line U373MG. The DLQ, USC, and LQ fit results are represented by the solid, dashed, and dotted lines, respectively.
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Number of fractions needed for each dose fractionation scheme for DLQ and SLQ 2 Gy
3 Gy
4 Gy
5.1 Gy
6.5 Gy
7.7 Gy
9.7 Gy
14.3 Gy
Cell line
DLQ
SLQ
DLQ
SLQ
DLQ
SLQ
DLQ
SLQ
DLQ
SLQ
DLQ
SLQ
DLQ
SLQ
DLQ
SLQ
CP3 DU145 HeLa H460 MDA-MB-231 Melanoma TX-4 U373MG
85 71 52 75 60 54 30 >150
39 39 15 35 25 73 20 45
41 48 24 35 40 26 19 >100
22 25 9 20 16 33 13 28
24 36 14 20 30 16 13 >75
15 18 6 13 12 19 9 19
16 28 9 13 24 10 10 >59
10 13 5 9 9 12 7 14
10 22 6 8 19 7 7 >46
7 10 4 6 7 7 6 10
7 19 4 6 16 5 6 >39
5 8 3 5 6 5 5 8
5 15 3 4 13 3 4 >31
4 6 2 3 5 4 4 6
3 10 2 2 9 2 3 >21
2 4 1 2 3 2 2 3
Abbreviations as in Table 1.
equilibrium. For conventionally fractionated treatments, the equilibrium was disrupted to a lesser degree, resulting in slower regrowth that offset the disadvantage of tumor cell killing compared with the hypofractionated approaches. The sensitivity of the DLQ model to the conversion probability P (PZ.5005-.55) on cell lines U373MG (GBM) and H460 (NSCLC) at the 2 Gy 30 (solid) and 10 Gy 5 (dashed) fractionation schemes is shown in Figure 2e and f. The results show that increasing P leads to more rapid GBM growth but minimally impacts the NSCLC cells (and other non-GBM cells that are not shown in the figure). The sensitivity study proved that a small variation in P does not affect the superior treatment outcome from hypofractionation on NSCLC and definitive treatment failure in GBM. The results of a sample of currently used or previously tested fractionation schedules are shown in Figure 2d. The 5 Gy 10 and 2 Gy 45 fractionation schemes resulted in the worst and best outcome at 100 days, respectively. However, the tumor still recurred for all treatment schedules, consistent with clinical trial results.
radiobiological parameters when explaining patient treatment outcome. For example, fitting of clinical prostate treatment response to conventionally fractionated and hypofractionated treatment using LQ results in either unreasonably low a or fast repopulation time (47). Modifications, including the USC, linear-quadratic linear (LQL), and generalized LQ models, have been made to moderate the LQ function toward the ablative dose range to resolve the apparent discrepancy between model prediction and measured cell survival data. However, these modifications are highly controversial because the modifications are incompatible with the underlying mechanisms of the original LQ model. These modified LQ models were further challenged by their failure to show superior modeling of the clinical data (26). Fowler (48) showed that these synthetic modifications can be avoided by assuming a much higher a/b value in the LQ model when fitting experimental cell survival data. However, increasing a/b values would increase the difficulty of explaining the success of SABR. Instead of assuming dose-dependent radiobiological parameters of a uniform cell population, we showed that when the intact LQ model is applied to tumor cell subpopulations, previously published single-fraction cell survival data can be accurately represented throughout the entire dose range. The DLQ model naturally leads to a more radioresistant subpopulation, which we referred to as
Discussion The classic LQ model has been challenged to explain response to SABR doses. In addition to its deviation from in vitro cell survival data, it often uses unreasonable
Table 4
BED to surrounding normal tissue required to achieve tumor control for DLQ and SLQ simulations 2 Gy
Cell line
DLQ
CP3 283 DU145 237* HeLa 173 H460 250 MDA-MB-231 200* Melanoma 180 TX-4 100* U373MG >500
3 Gy
4 Gy
5.1 Gy
6.5 Gy
SLQ DLQ SLQ DLQ SLQ DLQ SLQ DLQ SLQ 130* 246 130* 288 50* 144 117* 210 83* 240 243 156 67* 114 150* >600
132 224 150 336 54 131 120 187 96 280 198 149 78 121 168 >700
140 220 168 386 56 124 121 179 112 330 177 138 84 138 177 >812
138 206 179 453 69 124 124 165 124 391 165 144 96 144 193 >947
Abbreviation: BED Z biological effective dose. Other abbreviations as in Table 1. * Lowest possible BED value achieved for each cell line.
7.7 Gy DLQ
144 192* 206 522 82 110* 124 165 144 439 144 137 124 165 206 >1071
SLQ
9.7 Gy DLQ
137 205 220 616 82 123 137 164* 165 534 137* 123* 137 164 220 >1273
SLQ
14.3 Gy DLQ
164 247 246 825 82 165 123 165 205 742 164 165 164 247 246 >1732
SLQ 165 330 82 165 247 165 165 247
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Survival cell number
106
104
102
GBM 2 Gy x 30 GBM 5 Gy x 10 NSCLC 2 Gy x 30 NSCLC 5 Gy x 10
108 106 104 102
0
C
GBM and NSCLC (conventional vs. hypofractionation) 1010
GBM dual compartment GBM single compartment NSCLC dual compartment NSCLC single compartment
108 Survival cell number
B
GBM and NSCLC comparison, 2Gy x 30
10
20
30
40 50 60 Time (Days)
70
80
D
GBM compartment interaction
108
100
90 100
0
10
20
30
40 50 60 Time (Days)
10
2 Gy x 30 DCC 2 Gy x 30 CSC 5 Gy x 10 DCC 5 Gy x 10 CSC 10 Gy x 3 DCC 10 Gy x 3 CSC
104 103 0
E
1.8 Gy x 33 2 Gy x 30 1 Gy x 78 B.I.D 2 Gy x 45 1.3 Gy x 60 B.I.D 1.5 Gy x 40 B.I.D 5 Gy x 10
10
20
30
40 50 60 Time (Days)
70
80
107
106
105
90 100
0
F
H460:P sensitivity analysis 2Gy x 30 (solid) 10Gy x 5 (dashed)
108
Surviving CSC+DCC
Survival cell number
Survival cell number
5
90 100
GBM used fractionation schedules
108
106
80
109
7
10
70
8
10
20
30
40 50 60 Time (Days)
70
80
90 100
U373MG:P sensitivity analysis 2Gy x 30 (solid) 10Gy x 5 (dashed)
0
P =.5005 P =.505 P =.515 P =.535 P =.55
106
07
06
104
P =.5005 P =.505 P =.515 P =.535 P =.55
05 2
10
0
20
40 60 Time (days)
80
100
04
0
20
40 60 Time (days)
80
100
Fig. 2. (a) Glioblastoma multiforme (GBM) (black) and nonesmall-cell lung cancer (NSCLC) (red) 2 Gy 30 singlecompartment linear-quadratic (dashed) and dual-compartment linear-quadratic (solid) comparison. (b) GBM and NSCLC comparison between conventional 2 Gy 30 and hypofractionation 5 Gy 10. (c) GBM cancer stem cell (CSC) and differentiated cancer cell (DCC) compartment interaction with fractionation schedules of 2 Gy 30 (black), 5 Gy 10 (red), and 10 Gy 3 (blue). CSC, dashed; DCC, solid. (d) Current and previous treatment schedules for GBM. (e, f) Sensitivity analysis of the stem cell conversion factor P for GBM and NSCLC cell lines. A color version of this figure is available at www.redjournal.org. CSCs, whose greater radioresistance results in their increasing weight with high-dose fractions, straightening the survival curve and providing a duality in the cell biological behavior. Therefore, our model explains the cell
survival curve without violating the underlying biological mechanisms of the LQ model. A mathematical model is essential to describe the dynamic equilibrium of CSCs and DCCs. To achieve this
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goal, we adopted an integro-differential model previously used to test the efficacy of combination radiation and differentiation therapy (12, 14). The original model set the same a for both CSCs and DCCs, a b value of zero for CSCs, and an equal number of CSCs and DCCs (F Z 0.5) at the beginning of treatment. These assumptions oversimplify the heterogeneous radiobiology because the CSC fraction is known to vary between tumors, and is usually a minor population within the tumor (27, 28). Our model is also different from a previous study investigating the glioma stem cell division kinetics modulated by acute and fractionated radiation treatment, which assumed the same a/b for both the stem and non-stem cells (18). We improved these models by adopting individual radiobiological parameters obtained from the DLQ fit to cell survival data. We showed that the improved DLQ model led to possible insights in tumor response to various treatment fractions. Among the simulated dose fraction sizes, the dual-compartment model generally indicated greater radioresistance to treatment as an effort to restore cell subpopulation equilibrium. We found that SLQ indicated conventional fractionation schemes to result in the least normal tissue BED while attaining tumor control, whereas DLQ preferred hypofractionation approaches of 7.7 or 9.9 Gy per fraction for 4 tumor sites. The only exception is the melanoma cells, with an extremely low a/b ratio, which unsurprisingly prefer hypofractionation. This study may provide an alternative angle to understand the recent success of SABR, in addition to significantly improved dose conformity. Although clinical treatment fractionations were used as reference doses in this study, because of the uncertainties in estimating the radiobiological parameters and then the significant simplification involved in using these parameters to simulate actual tumor response to treatment, DLQ in its current form may be better understood as a “surprise implication” model instead of a treatment planning model. Bearing the limitation in mind, this study suggests that radioresistance can be generally overcome by using hypofractionated treatment regimens, with the exception of a GBM cell line that is resistant to both conventionally fractionated and hypofractionated treatment. The significance of this finding not only lies in the consistence with the particularly high GBM local recurrent recurrence rates, despite aggressive treatment (44, 49); it may also be used to explain the discrepancy of GBM cell in vitro and in vivo radiosensitivity (50, 51). Our model suggests that that the poor treatment outcome in GBM is perhaps driven by a CSC population that is much more radioresistant than its DCC counterpart, with a1/a2 Z 0.08 and b1/ b2 Z 6.26 106 (b1 on the order of 107). This significant difference between radiosensitivities resulted in the rapid regrowth of the DCC compartment, fueled by its depletion without simultaneously eliminating the more radioresistant CSCs, despite their small fraction at the beginning of treatment (F Z 0.016). Cancer stem cell b values close to zero were also observed for DU145 and
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MDA-MB-231. However, the ratios of a for both cell lines were approximately 6-fold larger than that of U373MG. The less dramatic difference between compartmental a values enabled hypofractionation approaches to overcome the regrowth of DCCs. Therefore, this model provides a possible explanation for previously failed hypofractionated and SABR GBM treatments. There are several limitations in this study. The DLQ model increases the number of fitting parameters, which may increase complexity in calculating tumor BED doses. However, different from existing modified LQ models, they can be measured on the basis of individual biopsy. For example, Lagadec et al (31) showed that not only the CSC fraction can be quantified using flow cytometry; the radiosensitivity of subpopulations can be separately measured. Therefore, our model parameters can be experimentally determined for an individual tumor or patient. The DLQ model can still be overly simplistic because a tumor may or may not present the CSC phenotype, or there may be more than 2 types of cells. Environmental factors, such as tumor vasculature, oxygen content in its microenvironment, and endothelial cell damage, can greatly affect tumor response to radiation therapy but are not modeled by the proposed method (26). Furthermore, tumors are modeled under a spatial homogeneous assumption. The mitotic rates of CSCs and DCCs and the apoptosis rate of DCCs were also set to be equal, which may not realistically simulate tumor growth and cell death. Additionally, the statistical error of in vitro cell survival measurement data was not reported, preventing us from including it into data fitting. Furthermore, a shift in CSC differentiation probability and CSC cell cycle may be considered after fractionated radiation (18). Finally, dose-dependent reprogramming of DCCs to CSCs by radiation has recently been shown (52). Incorporation of reprogramming may provide more insight in treatment outcome modeling (53). However, as the first step to incorporate intratumor heterogeneity in radiobiological response modeling, a simple model is better to shed light on the subject at hand. As a natural extension of the study, we will use the new model to optimize treatment schedules, particularly for the tumors that conventional and hypofractionated treatment failed to control.
Conclusion A dual-compartment model for cell survival was studied on the basis of coexisting CSCs and DCCs. Without modifying underlying LQ cell survival behavior, the model was shown to be capable of describing the clonogenic cell survival behavior for a wide dose range. By using ODEs that simulate the dynamics of CSC and DCC differentiation and apoptosis, we found tumor responses to conventional and hypofractionated treatments that were consistent with clinical observations. Most remarkably, we demonstrated that the dynamic equilibrium between DCC and CSC
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compartments within a GBM tumor might contribute to the poor clinical outcome after radiation therapy despite its apparently low in vitro radioresistance.
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International Journal of Radiation Oncology Biology Physics 21. Astrahan M. Some implications of linear-quadratic-linear radiation dose-response with regard to hypofractionation. Med Phys 2008;35: 4161-4172. 22. Guerrero M, Li XA. Extending the linear-quadratic model for large fraction doses pertinent to stereotactic radiotherapy. Phys Med Biol 2004;49:4825-4835. 23. Kavanagh BD, Newman F. Toward a unified survival curve: In regard to Park et al. (Int J Radiat Oncol Biol Phys 2008;70:847-852) and Krueger, et al. (Int J Radiat Oncol Biol Phys 2007;69:1262-1271). Int J Radiat Oncol Biol Phys 2008;71:958-959. 24. McKenna F, Ahmad S. Toward a unified survival curve: In regard to Kavanagh and Newman (Int J Radiat Oncol Biol Phys 2008;71:958959) and Park et al. (IntJ Radiat Oncol Biol Phys 2008;70:847-852). Int J Radiat Oncol Biol Phys 2009;73:640. author reply 640e641. 25. Fowler JF. Linear quadratics is alive and well: In regard to Park et al. (Int J Radiat Oncol Biol Phys 2008;70:847-852). Int J Radiat Oncol Biol Phys 2008;72:957. author reply 958. 26. Brown JM, Carlson DJ, Brenner DJ. The tumor radiobiology of SRS and SBRT: Are more than the 5 Rs involved? Int J Radiat Oncol Biol Phys 2014;88:254-262. 27. Visvader JE, Lindeman GJ. Cancer stem cells in solid tumours: Accumulating evidence and unresolved questions. Nat Rev Cancer 2008;8:755-768. 28. Enderling H, Hlatky L, Hahnfeldt P. Cancer stem cells: A minor cancer subpopulation that redefines global cancer features. Front Oncol 2013;3:76. 29. Garcia LM, Wilkins DE, Raaphorst GP. Alpha/beta ratio: A dose range dependence study. Int J Radiat Oncol Biol Phys 2007;67:587593. 30. Puck TT, Marcus PI. Action of x-rays on mammalian cells. J Exp Med 1956;103:653-666. 31. Lagadec C, Dekmezian C, Bauche L, et al. Oxygen levels do not determine radiation survival of breast cancer stem cells. PLoS One 2012;7:e34545. 32. Barranco SC, Romsdahl MM, Humphrey RM. The radiation response of human malignant melanoma cells grown in vitro. Cancer Res 1971;31:830-833. 33. MacArthur BD, Lemischka IR. Statistical mechanics of pluripotency. Cell 2013;154:484-489. 34. Haustermans KM, Hofland I, Van Poppel H, et al. Cell kinetic measurements in prostate cancer. Int J Radiat Oncol Biol Phys 1997; 37:1067-1070. 35. Shimomatsuya T, Tanigawa N, Muraoka R. Proliferative activity of human tumors: Assessment using bromodeoxyuridine and flow cytometry. Jpn J Cancer Res 1991;82:357-362. 36. Trott KR. Experimental results and clinical implications of the four R’s in fractionated radiotherapy. Radiat Environ Biophys 1982;20: 159-170. 37. Rew DA, Campbell ID, Taylor I, et al. Proliferation indices of invasive breast carcinomas after in vivo 5-bromo-2’-deoxyuridine labelling: A flow cytometric study of 75 tumours. Br J Surg 1992;79: 335-359. 38. Sasaki T, Sato Y, Sakka M. Cell population kinetics of human solid tumors: A statistical analysis in various histological types. Gann 1980;71:520-529. 39. Crnalic S, Panagopoulos I, Boquist L, et al. Establishment and characterisation of a human clear cell sarcoma model in nude mice. Int J Cancer 2002;101:505-511. 40. Struikmans H, Rutgers DH, Jansen GH, et al. S-phase fraction, 5-bromo-2’-deoxy-uridine labelling index, duration of S-phase, potential doubling time, and DNA index in benign and malignant brain tumors. Radiat Oncol Investig 1997;5:170-179. 41. Timmerman R, Paulus R, Galvin J, et al. Stereotactic body radiation therapy for inoperable early stage lung cancer. JAMA 2010;303: 1070-1076. 42. Combs SE, Widmer V, Thilmann C, et al. Stereotactic radiosurgery (SRS): Treatment option for recurrent glioblastoma multiforme (GBM). Cancer 2005;104:2168-2173.
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Radiation Oncology biology
physics
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Physics Contribution
Incorporating Cancer Stem Cells in Radiation Therapy Treatment Response Modeling and the Implication in Glioblastoma Multiforme Treatment Resistance Victoria Y. Yu, BS, Dan Nguyen, BS, Frank Pajonk, MD, PhD, Patrick Kupelian, MD, Tania Kaprealian, MD, Michael Selch, MD, Daniel A. Low, PhD, and Ke Sheng, PhD Department of Radiation Oncology, David Geffen School of Medicine, University of California, Los Angeles, Los Angeles, California Received Jun 4, 2014, and in revised form Nov 22, 2014. Accepted for publication Dec 1, 2014.
Summary The role of tumor-intrinsic heterogeneity including cancer stem cells (CSCs) and differentiated cancer cells (DCCs) in radiation therapy treatment response was investigated using 2 pairs of linear-quadratic radiation cell survival models. Using linear-quadratic model-fit parameters, the dynamics of CSC and DCC killing from radiation therapy was simulated using an ordinary differential equation. Simulation indicated the effectiveness of hypofractionation for certain tumors and refractory GBM radioresistance, due to the
Purpose: To perform a preliminary exploration with a simplistic mathematical cancer stem cell (CSC) interaction model to determine whether the tumor-intrinsic heterogeneity and dynamic equilibrium between CSCs and differentiated cancer cells (DCCs) can better explain radiation therapy treatment response with a dual-compartment linear-quadratic (DLQ) model. Methods and Materials: The radiosensitivity parameters of CSCs and DCCs for cancer cell lines including glioblastoma multiforme (GBM), nonesmall cell lung cancer, melanoma, osteosarcoma, and prostate, cervical, and breast cancer were determined by performing robust least-square fitting using the DLQ model on published clonogenic survival data. Fitting performance was compared with the single-compartment LQ (SLQ) and universal survival curve models. The fitting results were then used in an ordinary differential equation describing the kinetics of DCCs and CSCs in response to 2- to 14.3-Gy fractionated treatments. The total dose to achieve tumor control and the fraction size that achieved the least normal biological equivalent dose were calculated. Results: Smaller cell survival fitting errors were observed using DLQ, with the exception of melanoma, which had a low a/b Z 0.16 in SLQ. Ordinary differential equation simulation indicated lower normal tissue biological equivalent dose to achieve the same tumor control with a hypofractionated approach for 4 cell lines for the DLQ model, in contrast to SLQ, which favored 2 Gy per fraction for all cells except melanoma. The DLQ model indicated greater tumor radioresistance than SLQ, but the
Reprint requests to: Ke Sheng, PhD, Department of Radiation Oncology, University of California, Los Angeles, 200 Medical Plaza, B265, Los Angeles, CA 90024. Tel: (310) 853-1533; E-mail: ksheng@ mednet.ucla.edu Int J Radiation Oncol Biol Phys, Vol. 91, No. 4, pp. 866e875, 2015 0360-3016/$ - see front matter Ó 2015 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ijrobp.2014.12.004
Conflict of interest: none. This study was supported in part by NSF grant DGE-1144087. Supplementary material for this article can be found www.redjournal.org.
at
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tendency to maintain cell subpopulation equilibrium.
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radioresistance was overcome by hypofractionation, other than the GBM cells, which responded poorly to all fractionations. Conclusion: The distinct radiosensitivity and dynamics between CSCs and DCCs in radiation therapy response could perhaps be one possible explanation for the heterogeneous intertumor response to hypofractionation and in some cases superior outcome from stereotactic ablative radiation therapy. The DLQ model also predicted the remarkable GBM radioresistance, a result that is highly consistent with clinical observations. The radioresistance putatively stemmed from accelerated DCC regrowth that rapidly restored compartmental equilibrium between CSCs and DCCs. Ó 2015 Elsevier Inc.
Introduction Increasing evidence has suggested that solid tumors are hierarchically organized and contain a small population of cancer stem cells (CSCs) (1, 2). The subpopulations of CSCs are observed to have characteristics of self-renewal, differentiation to non-stem progenies, and unlimited proliferative capacity (3). In in vitro and animal experiments, CSCs are also observed to be more radioresistant than their non-stem counterpartddifferentiated cancer cells (DCCs) (4-11). Therefore, many believe that CSCs are the driving force of cancer progression and that successful therapy must eradicate CSCs. Mathematical models have suggested that perhaps the dynamic equilibrium between the DCC and CSC compartments within a tumor is essential to the treatment outcome (12-15). A “tumor growth paradox”dCSCs driven out of dormancy owing to spontaneous DCC cell death from therapy interventions resulting in accelerated tumor progressiondhas been previously demonstrated with mathematical models and biological experiments (16-18). Classic radiobiological models do not take into account such distinct radiobiology among different groups of tumor cells but instead assume uniform radiosensitivity within a tumor. The limitations of these models came to light as stereotactic ablative radiation therapy (SABR) became successful in the clinic, contrary to predictions of the linearquadratic (LQ) model using conventionally established radiosensitivity parameters (19-21). Modified LQ models that include dose-dependent repair and cell-killing terms, and synthesis of the LQ model with the multitarget model and a dose transition point that moderates the cell survival behavior toward higher fractional doses (19, 21-24) have been developed to address this discrepancy. These models showed superior data fitting of single fractional in vitro cell survival for a wide range of doses compared with the unmodified LQ model. However, these models remain controversial owing to their lack of biological foundations (25, 26), difficulty in determining the additional fit-parameter values for individual patients, and inability to provide insight to the paradoxical treatment outcomes of cancers with known high a/b ratios, such as nonesmall-cell lung cancer (NSCLC) and glioblastoma multiforme (GBM), which respond to hypofractionation differently. Therefore, these modified radiobiological models
have been rarely used in practice, despite the great interest of comparing treatment outcome from different regimens. Because the role of the CSC in cancer progression has become more prevalent, we propose that the incorporation of its properties to radiobiological modeling might improve its performance in predicting radiation therapy treatment response. Therefore, in this study, we performed preliminary exploration with a simplistic mathematical CSC interaction model to determine whether the tumor-intrinsic heterogeneity and dynamic equilibrium between CSCs and DCCs can better explain radiation therapy treatment response with a dual-compartment LQ (DLQ) model.
Methods and Materials Two major components come into play when modeling the distinct radiosensitivity and dynamic interaction between CSCs and DCCs; first, the determination of the radiosensitivity parameters of both compartments, and second, an ordinary differential equation (ODE) that models the CSC self-renewal, differentiation to DCC, and DCC growth and apoptosis. These 2 components were then combined to model CSC and DCC interaction alongside with radiation therapy cell killing of each compartment.
Determination of radiobiological parameters A simple way to include intratumor radiosensitivity heterogeneity is a DLQ model consisting of CSCs and DCCs. For a single fraction of treatment, the model was constructed as 2 2 SFðDÞZFea1 Db1 D þ 1 F ea2 Db2 D ð1Þ with F (0 F 0.2) as the fraction of CSCs out of all cells, and a and b describing the radiobiological properties of each population. The upper bound of F was set to 0.2 owing to the indication from publications that CSCs are a minor subpopulation of a solid tumor (27,28). Least-square fitting of the model was then performed on 8 previously published clonogenic cell survival datasets digitized from multiple human cancer cell lines, including U373MG (GBM), CP3, DU145 (prostate carcinoma) (29), HeLa (cervical cancer) (30), MDA-MB-231 (breast cancer) (31), H460 (NSCLC) (19), TX-4 (osteosarcoma) (20), and melanoma (32) using
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MATLAB R2013a (MathWorks, Natick, MA). All experiments were conducted using x-rays or gamma rays with relative biological effectiveness of 1. To compare the fitting performance and model quality of the DLQ model with the classic LQ model and a modified LQ (universal survival curve [USC]) model (19), we calculated the sum of squares error (SSE), Akaike information criterion (AIC), and Bayesian information criterion (BIC) in log10 scale for all datasets corresponding to all 3 models. The USC model with 3 fit parameters, D0, Dq, and aUSC, is shown in Equation 2. 8 ðaUSC D þ bUSC D2 ; D DT > < lnSFZ 1 Dq > : D þ ; D > DT ð2Þ D0 D0
bUSC Z
ð1 aUSC D0 Þ2 2Dq ; DT Z 4D0 Dq 1 aUSC D0
For the dataset of the breast cancer cell line MDAMB-231, the CSC fraction F was measured to be 0.0204 in the same publication (31) and fixed in curve-fitting.
ODE model: The interplay of CSCs and DCCs and radiation therapy The interaction between CSCs and DCCs was modeled according to an ODE model developed by Hillen et al (13) and used by Bachman et al (12). The simulated tumor was assumed to be spatially homogenous, with equal cell density, growth, and apoptosis throughout the tumor region. The ODE with a CSC and a DCC compartment is shown in Equation 3. :
calculated on the basis of the parameter F (fraction of CSC out of total tumor volume) obtained from data fitting of the DLQ model, as shown in Equation 1. Following Bachman et al (33), growth and apoptosis rates (mV, mU, aV) were set to ln(2)/Tpot day1, where Tpot represents the tumor potential doubling time of the simulated cancer at question. The Tpot values were set to be 23, 23, 4, 7.1, 8.2, 11, 1.3, and 3.9 days, following published data for cell lines CP3, DU145 (34), H460 (35), HeLa (36), MDA-MB-231 (37), melanoma (38), TX-4 (39), and U373MG (40), respectively. P was determined to maintain a dynamic equilibrium between CSCs and DCCs. In this study using the ODE formula, this requires the probability of CSC self-renewal to be slightly greater than 0.5. A probability of 0.505 was typically used in previous studies (12, 13) and adopted in this study. In addition, within a reasonable range of 0.5005 to 0.55, we studied the sensitivity of the conversion probability P in the conventional 2 Gy 30 and hypofractionated 10 Gy 5 fractionation schemes. Radiation therapy was modeled by applying the classical LQ equation using the corresponding radiobiological parameters to each compartment at times of treatment. To simplify, we assumed a spatially homogeneous yet biologically heterogeneous tumor receiving a spatially homogeneous dose. To compare the difference between our model and the SLQ model, treatment response was simulated with the same procedure as described above, but only using the set of radiobiological parameters obtained from classical LQ fit. All ODE simulations were performed in MATLAB. Dose fraction sizes of 2, 3, 4, 5.1, 6.5, 7.7, 9.7, and 14.3 Gy were used, and the number of dose fractions
SelfRenewal
U ðtÞZð2P 1ÞmU kðWðtÞÞUðtÞ :
V ðtÞZ 2ð1 PÞmU kðWðtÞÞUðtÞ þ mV kðWðtÞÞVðtÞ aV VðtÞ Differentiation from CSC
DCC Growth
Apoptosis
ð3Þ
WðtÞZUðtÞ þ VðtÞ KðWÞZmaxf1 W 4 ; 0g
where U(t), V(t), and W(t) are the volume fractions of the CSCs, DCCs, and total tumor with respect to a specified volume of interest, P represents the probability that a CSC gives rise to 2 CSCs, and 1-P is the probability that a CSC gives rise to 2 DCCs. The growth rates of CSC and DCC are mU and mV, respectively, and aV is the apoptosis rate of the DCCs. The apoptosis rate of CSCs was set to 0, assuming that CSCs had unlimited replicative potential. k(W ) is a volume constraint that keeps the total tumor volume fraction within the range of 0 and 1. At time t Z 0, the total tumor cell number was set to be 1.3 107 within a volume of interest of 4.2 109 cells. The initial starting fractions of the CSC and DCC compartments were then
required to achieve a tumor control probability (TCP) of 0.9 was determined. Treatment was administered once per day, every weekday. The corresponding biological effective doses (BEDs) to the surrounding normal tissue were also evaluated using Equation 4, with n, d, and a/b representing the number of fractions, dose fraction size, and a/b ratio of the surrounding normal tissue, respectively. The a/b ratio was set to 3 for all BED calculations. The TCP for the DLQ and SLQ models, indicated as TCPDLQ and TCPSLQ, were calculated according to a Poisson distribution as shown in Equation 5, with NDCCþCSC and NCSC representing the average number of remaining total tumor cells and remaining CSCs after treatment, respectively.
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d BEDZnd 1 þ a =b
ð4Þ
TCPDLQ ZexpðNCSC Þ TCPSLQ Zexpð NCSCþDCC Þ
ð5Þ
Because of the representative treatment outcome from GBM and NSCLC, we compared GBM and NSCLC using parameters obtained from cell lines U373MG and H460, respectively. We selected NSCLC for comparison because hypofractionation in NSCLC has been remarkably successful (41), whereas the same approach has been ineffective in treating GBM (42). Using the DLQ model, we applied currently utilized or previously applied GBM treatment schemes, including 2 Gy 30, 1.8 Gy 33, 1 Gy 78 b.i.d. (43), 2 Gy 45 (44), 1.3 Gy 60 b.i.d., 1.5 Gy 40 b.i.d. (44, 45), and 5 Gy 10 b.i.d. (46), to study the tumor response to treatment.
Results DLQ fit results The radiobiological parameters obtained from DLQ, SLQ, and USC fitting results to all 8 clonogenic survival datasets are shown in Table 1. The AIC, BIC, and SSE values of all 3 models are shown in Table 2. The a values of CSCs were smaller than those of their DCC counterparts for all except Tx-4, whose CSC compartment had a smaller b. As shown in Figure 1, the original LQ model resulted in overprediction of cell death in the high-dose range. Assessing by SSE, both DLQ and USC models more accurately described the cell survival behavior for the entire dose range than LQ, with the exception of the melanoma cells. Factoring in the larger number of fitting parameters in the DLQ and USC models, the average AIC and BIC values of the DLQ and USC models were lower than that of the LQ, as shown in Table 2. The melanoma cell was an outlier, showing an extremely low a/b ratio (0.16) in the LQ fit. For Table 1
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MDA-MB-231, F was fixed according to literature. The fitobtained b value for CSCs was close to zero, agreeing with the publication (31).
ODE simulations and therapy schedule optimization The number of dose fractions and BED required to achieve tumor control (TCP Z 0.9) for each dose fraction size and cell line are shown in Tables 3 and 4. As evident from Table 3, DLQ indicated a greater dose for tumor control than the SLQ model. However, the greater radioresistance can still be overcome with hypofractionation, except for GBM cell U373MG. The GBM cells were remarkably resistant to hypofractionated treatment, requiring a prohibitively high dose to control. In Table 4, the lowest possible BED values that achieved tumor control for each cell line are indicated with asterisks. Here the dose fraction size that attained the lowest normal tissue BED was considered the optimal treatment fractionation schedule for each cell line. The SLQ model indicated conventional 2-Gy fractionation schemes to result in the least normal tissue effect while attaining tumor control for all cell lines except melanoma, for which a 7.7-Gy fraction was optimal owing to its low a/b ratio. In comparison, DLQ indicated hypofractionation approaches (>7.7 Gy) to be preferable for U373MG, CP3, HeLa, and H460. Figure 2a demonstrates the difference between the SLQ and DLQ treatment outcome of the GBM and NSCLC cells. With a regularly fractionated treatment of 60 Gy, SLQ indicated similar outcomes between the 2 different cells, but in DLQ the number of remaining GBM cells was 2 orders of magnitude greater than the remaining NSCLC cells. Figure 2b compares the conventional 2 Gy 30 and 5 Gy 10 SABR treatment outcome. For NSCLC, SABR resulted in significantly more effective tumor control than conventional fractionation schedules. In stark contrast, for
DLQ, SLQ, and USC model clonogenic survival fit parameters DLQ fit parameters
USC model fit parameters
Single LQ fit
Cell line
F
a1
b1
a2
b2
a
b
a/b
Dq
D0
aUSC
CP3 DU145 HeLa H460 MDA-MB-231 Melanoma TX-4 U373MG
0.047 0.010 0.052 0.010 0.020 0.166 0.200 0.016
0.021 0.099 0.010 0.010 0.125 0.038 0.244 0.010
0.036 2.22E-05 0.071 0.042 2.43E-06 0.059 0.022 1.77E-07
0.098 0.191 0.197 0.010 0.271 0.013 0.128 0.125
0.057 0.017 0.203 0.079 0.032 0.061 0.105 0.028
0.15 0.22 0.54 0.16 0.36 0.01 0.50 0.17
0.04 0.01 0.06 0.05 0.01 0.06 0.01 0.02
3.45 17.53 8.89 2.95 32.74 0.16 38.90 9.49
3.78 2.28 1.50 4.21 1.50 4.39 1.50 2.05
1.01 2.19 0.84 0.76 1.75 0.89 1.34 2.30
0.07 0.19 0.52 0.01 0.31 0.01 0.18 0.12
Abbreviations: DLQ Z dual-compartment linear-quadratic; SLQ Z single-compartment linear-quadratic; USC Z universal survival curve. F is the fraction of cancer stem cells within the tumor. (a1, b1) and (a2, b2) are the radiobiological parameters of the cancer stem cell and differentiated cancer cell compartment, respectively. Single LQ fit indicates the parameters obtained from a classic LQ model fit. The last 3 columns show the fit parameters of the USC model, Dq, D0, and aUSC.
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Table 2
AIC, BIC, and SSE comparison AIC
BIC
SSE
Cell line
DLQ
LQ
USC
DLQ
LQ
USC
DLQ
LQ
USC
CP3 DU145 HeLa H460 MDA-MB-231 Melanoma TX-4 U373MG Average
50.30 104.84 34.51 13.19 17.38 44.22 40.17 92.19 49.60
40.05 98.37 20.56 4.03 12.54 50.76 18.33 75.91 39.06
53.25 106.66 18.25 17.80 13.97 44.59 31.80 93.94 47.53
42.10 96.63 30.67 12.71 18.42 38.55 35.92 85.64 45.08
35.95 94.27 18.64 4.26 13.17 47.92 16.20 72.64 36.82
47.78 101.19 15.69 17.48 14.80 40.81 28.97 89.57 44.54
0.198 0.030 0.030 0.020 0.004 0.058 0.027 0.011 0.047
0.347 0.046 0.123 0.366 0.016 0.056 0.174 0.031 0.145
0.205 0.033 0.126 0.019 0.009 0.070 0.062 0.013 0.067
Abbreviations: AIC Z Akaike information criterion; BIC Z Bayesian information criterion; SSE Z sum of square error. Other abbreviations as in Table 1. AIC, BIC, and SSE values in log10 scale for all 3 models on all clonogenic survival datasets.
GBM, SABR fractionating did not noticeably improve tumor control. To understand the unique GBM radioresistance to hypofractionated treatment, we plotted the CSC and DCC compartment growth over time to explore their relationship and interactions (Fig. 2c). The interaction between the 2
A
compartments was simulated and compared for fractionation schedules of 2 Gy 30, 5 Gy 10, and 10 Gy 3. From Figure 2c, the dynamic equilibrium was disrupted when the DCC population became substantially smaller than the CSC population as a result of aggressive treatment. Consequently, rapid DCC regrowth occurred to restore the
B
100
100
Survival fraction
Survival fraction
10-1
-2
10
10-3
10-4
CP3 Dual compartment fit USC fit LQ fit
10-5 0
1
2
3
4
5
6
7
10-1
10-2
HeLa Dual compartment fit USC fit LQ fit
10-3
8
9
0
10 11 12 13 14
2
1
3
Dose (Gy)
D
100
Survival fraction
Survival fraction
C
10-1
0
1
2
3
4
5
5
7
6
8
100
10-1
U373MG Dual compartment fit USC fit LQ fit
MDA-MB-231 Dual compartment fit USC fit LQ fit
10-2
4
Dose (Gy)
6
Dose (Gy)
7
8
9
10
11
0
1
2
3
4
5
6
7
8
9
10
11
Dose (Gy)
Fig. 1. Dual-compartment linear-quadratic (DLQ), universal survival curve (USC), and linear-quadratic (LQ) model fit comparison. (a) Prostate carcinoma cell line CP3. (b) Cervical cancer cell line HeLa. (c) Breast cancer cell line MDA-MB231. (d) Glioblastoma multiforme cell line U373MG. The DLQ, USC, and LQ fit results are represented by the solid, dashed, and dotted lines, respectively.
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Number of fractions needed for each dose fractionation scheme for DLQ and SLQ 2 Gy
3 Gy
4 Gy
5.1 Gy
6.5 Gy
7.7 Gy
9.7 Gy
14.3 Gy
Cell line
DLQ
SLQ
DLQ
SLQ
DLQ
SLQ
DLQ
SLQ
DLQ
SLQ
DLQ
SLQ
DLQ
SLQ
DLQ
SLQ
CP3 DU145 HeLa H460 MDA-MB-231 Melanoma TX-4 U373MG
85 71 52 75 60 54 30 >150
39 39 15 35 25 73 20 45
41 48 24 35 40 26 19 >100
22 25 9 20 16 33 13 28
24 36 14 20 30 16 13 >75
15 18 6 13 12 19 9 19
16 28 9 13 24 10 10 >59
10 13 5 9 9 12 7 14
10 22 6 8 19 7 7 >46
7 10 4 6 7 7 6 10
7 19 4 6 16 5 6 >39
5 8 3 5 6 5 5 8
5 15 3 4 13 3 4 >31
4 6 2 3 5 4 4 6
3 10 2 2 9 2 3 >21
2 4 1 2 3 2 2 3
Abbreviations as in Table 1.
equilibrium. For conventionally fractionated treatments, the equilibrium was disrupted to a lesser degree, resulting in slower regrowth that offset the disadvantage of tumor cell killing compared with the hypofractionated approaches. The sensitivity of the DLQ model to the conversion probability P (PZ.5005-.55) on cell lines U373MG (GBM) and H460 (NSCLC) at the 2 Gy 30 (solid) and 10 Gy 5 (dashed) fractionation schemes is shown in Figure 2e and f. The results show that increasing P leads to more rapid GBM growth but minimally impacts the NSCLC cells (and other non-GBM cells that are not shown in the figure). The sensitivity study proved that a small variation in P does not affect the superior treatment outcome from hypofractionation on NSCLC and definitive treatment failure in GBM. The results of a sample of currently used or previously tested fractionation schedules are shown in Figure 2d. The 5 Gy 10 and 2 Gy 45 fractionation schemes resulted in the worst and best outcome at 100 days, respectively. However, the tumor still recurred for all treatment schedules, consistent with clinical trial results.
radiobiological parameters when explaining patient treatment outcome. For example, fitting of clinical prostate treatment response to conventionally fractionated and hypofractionated treatment using LQ results in either unreasonably low a or fast repopulation time (47). Modifications, including the USC, linear-quadratic linear (LQL), and generalized LQ models, have been made to moderate the LQ function toward the ablative dose range to resolve the apparent discrepancy between model prediction and measured cell survival data. However, these modifications are highly controversial because the modifications are incompatible with the underlying mechanisms of the original LQ model. These modified LQ models were further challenged by their failure to show superior modeling of the clinical data (26). Fowler (48) showed that these synthetic modifications can be avoided by assuming a much higher a/b value in the LQ model when fitting experimental cell survival data. However, increasing a/b values would increase the difficulty of explaining the success of SABR. Instead of assuming dose-dependent radiobiological parameters of a uniform cell population, we showed that when the intact LQ model is applied to tumor cell subpopulations, previously published single-fraction cell survival data can be accurately represented throughout the entire dose range. The DLQ model naturally leads to a more radioresistant subpopulation, which we referred to as
Discussion The classic LQ model has been challenged to explain response to SABR doses. In addition to its deviation from in vitro cell survival data, it often uses unreasonable
Table 4
BED to surrounding normal tissue required to achieve tumor control for DLQ and SLQ simulations 2 Gy
Cell line
DLQ
CP3 283 DU145 237* HeLa 173 H460 250 MDA-MB-231 200* Melanoma 180 TX-4 100* U373MG >500
3 Gy
4 Gy
5.1 Gy
6.5 Gy
SLQ DLQ SLQ DLQ SLQ DLQ SLQ DLQ SLQ 130* 246 130* 288 50* 144 117* 210 83* 240 243 156 67* 114 150* >600
132 224 150 336 54 131 120 187 96 280 198 149 78 121 168 >700
140 220 168 386 56 124 121 179 112 330 177 138 84 138 177 >812
138 206 179 453 69 124 124 165 124 391 165 144 96 144 193 >947
Abbreviation: BED Z biological effective dose. Other abbreviations as in Table 1. * Lowest possible BED value achieved for each cell line.
7.7 Gy DLQ
144 192* 206 522 82 110* 124 165 144 439 144 137 124 165 206 >1071
SLQ
9.7 Gy DLQ
137 205 220 616 82 123 137 164* 165 534 137* 123* 137 164 220 >1273
SLQ
14.3 Gy DLQ
164 247 246 825 82 165 123 165 205 742 164 165 164 247 246 >1732
SLQ 165 330 82 165 247 165 165 247
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A
Survival cell number
106
104
102
GBM 2 Gy x 30 GBM 5 Gy x 10 NSCLC 2 Gy x 30 NSCLC 5 Gy x 10
108 106 104 102
0
C
GBM and NSCLC (conventional vs. hypofractionation) 1010
GBM dual compartment GBM single compartment NSCLC dual compartment NSCLC single compartment
108 Survival cell number
B
GBM and NSCLC comparison, 2Gy x 30
10
20
30
40 50 60 Time (Days)
70
80
D
GBM compartment interaction
108
100
90 100
0
10
20
30
40 50 60 Time (Days)
10
2 Gy x 30 DCC 2 Gy x 30 CSC 5 Gy x 10 DCC 5 Gy x 10 CSC 10 Gy x 3 DCC 10 Gy x 3 CSC
104 103 0
E
1.8 Gy x 33 2 Gy x 30 1 Gy x 78 B.I.D 2 Gy x 45 1.3 Gy x 60 B.I.D 1.5 Gy x 40 B.I.D 5 Gy x 10
10
20
30
40 50 60 Time (Days)
70
80
107
106
105
90 100
0
F
H460:P sensitivity analysis 2Gy x 30 (solid) 10Gy x 5 (dashed)
108
Surviving CSC+DCC
Survival cell number
Survival cell number
5
90 100
GBM used fractionation schedules
108
106
80
109
7
10
70
8
10
20
30
40 50 60 Time (Days)
70
80
90 100
U373MG:P sensitivity analysis 2Gy x 30 (solid) 10Gy x 5 (dashed)
0
P =.5005 P =.505 P =.515 P =.535 P =.55
106
07
06
104
P =.5005 P =.505 P =.515 P =.535 P =.55
05 2
10
0
20
40 60 Time (days)
80
100
04
0
20
40 60 Time (days)
80
100
Fig. 2. (a) Glioblastoma multiforme (GBM) (black) and nonesmall-cell lung cancer (NSCLC) (red) 2 Gy 30 singlecompartment linear-quadratic (dashed) and dual-compartment linear-quadratic (solid) comparison. (b) GBM and NSCLC comparison between conventional 2 Gy 30 and hypofractionation 5 Gy 10. (c) GBM cancer stem cell (CSC) and differentiated cancer cell (DCC) compartment interaction with fractionation schedules of 2 Gy 30 (black), 5 Gy 10 (red), and 10 Gy 3 (blue). CSC, dashed; DCC, solid. (d) Current and previous treatment schedules for GBM. (e, f) Sensitivity analysis of the stem cell conversion factor P for GBM and NSCLC cell lines. A color version of this figure is available at www.redjournal.org. CSCs, whose greater radioresistance results in their increasing weight with high-dose fractions, straightening the survival curve and providing a duality in the cell biological behavior. Therefore, our model explains the cell
survival curve without violating the underlying biological mechanisms of the LQ model. A mathematical model is essential to describe the dynamic equilibrium of CSCs and DCCs. To achieve this
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goal, we adopted an integro-differential model previously used to test the efficacy of combination radiation and differentiation therapy (12, 14). The original model set the same a for both CSCs and DCCs, a b value of zero for CSCs, and an equal number of CSCs and DCCs (F Z 0.5) at the beginning of treatment. These assumptions oversimplify the heterogeneous radiobiology because the CSC fraction is known to vary between tumors, and is usually a minor population within the tumor (27, 28). Our model is also different from a previous study investigating the glioma stem cell division kinetics modulated by acute and fractionated radiation treatment, which assumed the same a/b for both the stem and non-stem cells (18). We improved these models by adopting individual radiobiological parameters obtained from the DLQ fit to cell survival data. We showed that the improved DLQ model led to possible insights in tumor response to various treatment fractions. Among the simulated dose fraction sizes, the dual-compartment model generally indicated greater radioresistance to treatment as an effort to restore cell subpopulation equilibrium. We found that SLQ indicated conventional fractionation schemes to result in the least normal tissue BED while attaining tumor control, whereas DLQ preferred hypofractionation approaches of 7.7 or 9.9 Gy per fraction for 4 tumor sites. The only exception is the melanoma cells, with an extremely low a/b ratio, which unsurprisingly prefer hypofractionation. This study may provide an alternative angle to understand the recent success of SABR, in addition to significantly improved dose conformity. Although clinical treatment fractionations were used as reference doses in this study, because of the uncertainties in estimating the radiobiological parameters and then the significant simplification involved in using these parameters to simulate actual tumor response to treatment, DLQ in its current form may be better understood as a “surprise implication” model instead of a treatment planning model. Bearing the limitation in mind, this study suggests that radioresistance can be generally overcome by using hypofractionated treatment regimens, with the exception of a GBM cell line that is resistant to both conventionally fractionated and hypofractionated treatment. The significance of this finding not only lies in the consistence with the particularly high GBM local recurrent recurrence rates, despite aggressive treatment (44, 49); it may also be used to explain the discrepancy of GBM cell in vitro and in vivo radiosensitivity (50, 51). Our model suggests that that the poor treatment outcome in GBM is perhaps driven by a CSC population that is much more radioresistant than its DCC counterpart, with a1/a2 Z 0.08 and b1/ b2 Z 6.26 106 (b1 on the order of 107). This significant difference between radiosensitivities resulted in the rapid regrowth of the DCC compartment, fueled by its depletion without simultaneously eliminating the more radioresistant CSCs, despite their small fraction at the beginning of treatment (F Z 0.016). Cancer stem cell b values close to zero were also observed for DU145 and
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MDA-MB-231. However, the ratios of a for both cell lines were approximately 6-fold larger than that of U373MG. The less dramatic difference between compartmental a values enabled hypofractionation approaches to overcome the regrowth of DCCs. Therefore, this model provides a possible explanation for previously failed hypofractionated and SABR GBM treatments. There are several limitations in this study. The DLQ model increases the number of fitting parameters, which may increase complexity in calculating tumor BED doses. However, different from existing modified LQ models, they can be measured on the basis of individual biopsy. For example, Lagadec et al (31) showed that not only the CSC fraction can be quantified using flow cytometry; the radiosensitivity of subpopulations can be separately measured. Therefore, our model parameters can be experimentally determined for an individual tumor or patient. The DLQ model can still be overly simplistic because a tumor may or may not present the CSC phenotype, or there may be more than 2 types of cells. Environmental factors, such as tumor vasculature, oxygen content in its microenvironment, and endothelial cell damage, can greatly affect tumor response to radiation therapy but are not modeled by the proposed method (26). Furthermore, tumors are modeled under a spatial homogeneous assumption. The mitotic rates of CSCs and DCCs and the apoptosis rate of DCCs were also set to be equal, which may not realistically simulate tumor growth and cell death. Additionally, the statistical error of in vitro cell survival measurement data was not reported, preventing us from including it into data fitting. Furthermore, a shift in CSC differentiation probability and CSC cell cycle may be considered after fractionated radiation (18). Finally, dose-dependent reprogramming of DCCs to CSCs by radiation has recently been shown (52). Incorporation of reprogramming may provide more insight in treatment outcome modeling (53). However, as the first step to incorporate intratumor heterogeneity in radiobiological response modeling, a simple model is better to shed light on the subject at hand. As a natural extension of the study, we will use the new model to optimize treatment schedules, particularly for the tumors that conventional and hypofractionated treatment failed to control.
Conclusion A dual-compartment model for cell survival was studied on the basis of coexisting CSCs and DCCs. Without modifying underlying LQ cell survival behavior, the model was shown to be capable of describing the clonogenic cell survival behavior for a wide dose range. By using ODEs that simulate the dynamics of CSC and DCC differentiation and apoptosis, we found tumor responses to conventional and hypofractionated treatments that were consistent with clinical observations. Most remarkably, we demonstrated that the dynamic equilibrium between DCC and CSC
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compartments within a GBM tumor might contribute to the poor clinical outcome after radiation therapy despite its apparently low in vitro radioresistance.
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