A mathematical model of tumor dynamics during stereotactic body radiation therapy for non-small cell lung cancer

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1 A mathematical model of tumor dynamics during stereotactic body radiation therapy for non-small cell lung cancer Russell Injerd, Emma Turian Abstract Image guided radiotherapy allows tumor volume dynamics to be measured at intervals during treatment. This has improved our ability to model certain types of tumors, like non-small cell lung cancer, during treatment. Previous one and two population ODE models of tumor volume dynamics in response to stereotactic ablative radiotherapy (SABR) were created using exponential and logistic growth. Previous studies indicated that a two population exponential model provided the best balance between t and mathematical complexity and may serve a functional role in clinical practice. Our study reevaluates previous ndings and tests the suitability of Gompertz growth models by assessing the goodness of t versus its mathematical complexity. The exponential, Velhurst, and Gompertz models are calibrated using data from 11 patients extracted from a previous study and evaluated using statistical methods, such as the Akaike information criterion and the Wilcoxon rank sum test. Model comparison indicates that Gompertz growth model does not signicantly improve model t, however it may oer advantages as an inferential tool for clinical use. Keywords Image-guided radiotherapy, dynamic modeling, model selection, non-small cell lung cancer, stereotactic ablative radiotherapy. 1 Introduction Tumor modeling has undergone many changes that have resulted in improvements in the description and understanding of tumor dynamics. Complex models can include morphology, pressure and nutrient gradients, and genetic probability [3]. These methods have been most luminary when applied to in vitro spheroids, but are increasingly used in multi-phase and in vitro models with success. The complexity and variability of tumor growth models in vivo require specic patient information which makes regular clinical use problematic. There remains a use for simple mathematical models that make use This study was a requirement for Math 371, Spring Department of Mathematics, NEIU, Chicago, IL, 60625, math@neiu.edu. 1

2 R. Injerd 2 of regularly collected clinical data. Image guided radiotherapy, like stereotactic ablative radiotherapy (SABR), provides precise delivery of radiation using imaging during treatment. This allows the use of higher fractional doses of radiation and reduces the time patients are exposed to radiation [4]. As a result of these new therapies, tumor volume is recorded at several points during treatment. This provides researchers the data to apply mathematical models to describe tumor volume dynamics. Because tumor volume has been found to be a signicant predictor in cancer prognosis [5], simple models, like ordinary dierential equations, that incorporate tumor volume measurements collected during treatment could be a practical clinical tool. A previous study examined 11 patients undergoing SABR treatment and evaluated exponential and logistic models using the Akaike Information Criterion [5]. The study demonstrated that simple ODE models can adequately describe tumor volume of non-small cell lung cancer (NSCLC) patients receiving SABR. This study reassesses these earlier models and introduces Gompertz growth to model the tumor volume dynamics. 2 Data Patient data was extracted from a previous clinical study of SABR treatment for NSCLC [5]. Patients received targeted doses of radiation, higher than the conventional 2 Gy per fraction, aided by advanced image guidance techniques [4]. Images of the tumor are captured prior to each fractional dose. Gross tumor volume (GTV) is estimated by an experienced medical professional by a combination of automation and manual adjustment techniques. A second physicist conducts a second independent estimation of the GTV. The average variability between each clinician's GTV measurements was 3.7% [5]. Patients with issues during GTV contouring due to tissue density classication were excluded from the study. Patients receiving less than 5 fractional doses were also excluded. The study includes 11 patients with NSCLC receiving 5 or 8 fractional doses of SABR treatment with tumors that allow accurate contouring. Patient characteristics are summarized in table 1. A detailed table of GTV can be found in appendix A.

3 R. Injerd 3 Table 1: Patient characteristics Patient no. Age Gender No. of fractions Dose per fraction (Gy) Initial GTV (cc) 1 81 Male Male Female Male Female Female Female Female Female Female Male Mathematical Modeling The main objective of this study is to assess the Gompertz growth function as an alternative model to describe NSCLC tumor volume in patients receiving SABR. Previous studies [5] have assessed exponential and Velhurst models as pragmatic methods to describe tumor dynamics in NSCLC SABR patients. These non-spatialized models make use of available data and omit highly detailed microscopic processes to balance model accuracy and model complexity. Gompertz growth is also an accepted and widely used model for describing tumor growth [3]. Gompertz growth does not introduce any new parameters. Using it to describe NSCLC tumors receiving SABR may improve model accuracy without introducing additional complexity. Previous models were reassessed alongside Gompertz models using both one and two population methods. 3.1 One Population Models Figure 1: One population tumor growth models. growth Tumor Cells Fractional Dose Surviving Fraction GTV One population models assume tumors are composed of homogeneous cells that can be described by a single variable. Though it has been shown through

4 R. Injerd 4 CT imaging that tumor volume is heterogeneous in terms of density, it is practical and useful to assume homogeneity for model simplicity. The rst model is simple exponential growth given by the following equation: dn kn (1) dt where t is time, and k > 0 is the proliferation rate. The initial tumor volume will be taken from the rst CT measurement. This model assumes that all vital nutrients are available in abundance and places no constraints on growth. This assumption is reasonable for small tumors like stage 1 NSCLC with tumor diameter less than 5cm [5]. The linear-quadratic equation will be used to model the eects of radiotherapy. After each fractional dose a proportion of the cells will die and the remainder of the cells survive treatment. S expp αd βd 2 q (2) where S represents the surviving fraction, d is the dose per fraction, and α and β are the linear and quadratic coecients for cell killing. To limit the possible combinations of α and β the ratio is xed to 10Gy 1, thereby requiring estimation of α only [5]. After a dose of radiotherapy at time t, an innitesimal time later p tq the tumor volume will be: N pt tq N ptqs (3) which will continue to proliferate until the next treatment. Equations (1)-(3) complete the rst model which will be referred to as "POP1EXP". When nutrients are thought to be limited, which is more realistic for tumor cell proliferation, growth can be shown to slow as cells reach a carrying capacity. This can be modeled using the Velhurst growth model: and the Gompertz growth model: dn dt kn 1 N θ dn dt kn ln N θ where k > 0 is the proliferation rate and θ > 0 is the carrying capacity of the cell population. These models have fast initial growth that slow as the carrying capacity is approached. Equations (4), (2), and (3) complete model 2 which will be referred to as POP1LOG. Equations (5), (2), and (3) complete model 3 which will be referred to as POP1GOM. The owchart shown in Fig. 1 illustrates the change in GTV using one population models. (4) (5)

5 R. Injerd Two population models. Figure 2: Two population tumor growth model. Living Cells Fractional Dose Surviving Fraction growth GTV Dead Cells clearance Dead Cells not cleared from tumor Modeling tumors using a single population shows instantaneous shrinking of the tumor volume after a fractional dose of radiotherapy as shown in Eq.(3). Actual reduction in tumor volume after radiotherapy requires dead cells to be cleared from the tumor site, which may take appreciable time [5]. To model this process, it becomes useful to model two populations of cells, living N l and dead N d. Using this methodology exponential growth becomes: the Velhurst growth: or the Gompertz growth: dn l dt dn l dt dn l dt kn l (6) kn l 1 N l θ kn l ln Nl θ After receiving a fractional dose, a proportion of the cells survive irradiation shown by Eq.(2) and the rest become dead cells: (7) (8) N l pt tq N l ptqs (9) N d pt tq N d ptq N l ptqr1 Ss (10) Dead cells will be cleared from the tumor exponentially: dn d dt cn d (11)

6 R. Injerd 6 where c > 0 is the clearance rate. The total tumor volume is the sum of the living and dead cells: N N l N d (12) Using two populations is a useful method to incorporate a delay into dead cell clearance. However, CT images cannot distinguish living and dead cells. The only volume collected at each measurement is the gross tumor volume. Using two populations is merely a mathematical technique to incorporate delay into volume reduction caused by radiotherapy. Equations (6), (2), and (9)-(12) complete model 4 which will be referred to as POP2EXP. Equations (7), (2), and (9)-(12) complete model 5 which will be referred to as POP2LOG. Equations (8), (2), and (9)-(12) complete model 6 which will be referred to as POP2GOM. The owchart shown in Fig. 2 illustrates the change in GTV using two population models. 3.3 Model Calibration & Selection The models are calibrated to the GTV of each patient by estimating the parameters which minimize the sum of squared errors (SSE): SSE Ţ t1 N ptq ˆN ptq 2 (13) where N ptq is the measured GTV right before the t th fraction, T is the total number of fractions for an individual patient, and ˆN ptq is the corresponding modeled GTV. Constraints are placed on each of the parameters to omit impossible values, while providing enough exibility to allow the tting algorithm to nd a suitable t [5]. Sensitivity analysis will be conducted using Monte Carlo simulations. The coecient of variation CV will be calculated to determine which parameters the modeled GTV are most sensitive to. Table 2: Model parameters Parameter Description Units Constraints k proliferation rate d 1 [ ] α cell killing coecient Gy 1 [ ] θ carrying capacity cc [N p0q - 100] c clearance rate for dead cells d 1 [ ] To assess goodness of t the % root mean squared error (%RMSE) is calculated: RMSE a SSE{T (14) %RMSE RMSE{pNmax Nminq 100% (15)

7 R. Injerd 7 The Shapiro-Wilks normality test will be conducted on the residuals. Graphical analysis will also be conducted using quantile-quantile plots (QQ plots). Care must be taken when using models as inferential tools if residuals are not normal; it implies that the predictive value of the model changes across the range of data. Pearson's correlation analysis will also be conducted to determine whether the parameters are independent or dependent on each other. Dependent parameters may not be fully identiable from available data and inferential use may be problematic. To balance model t and model complexity Akaike Information Criterion (AIC) is calculated using the mean sum of dierences (MSD): MSD 1 T Ţ t1 2 ˆN ptq N ptq (16) N ptq AIC T ln pmsdq 2p (17) where p is the number of free parameters. The AIC rewards goodness of t, while penalizing the number of free parameters. The Wilcoxon paired rank sum test will be used to determine whether the %RMSE and the AIC values calculated for the models dier signicantly. 4 Results Generally, ts are similar in the one population models. Median %RM SE for POP1EXP, POP1LOG, and POP1GOM are %, %, and % respectively. The Wilcoxon paired rank sum test shows signicant dierence in the medians of POP1GOM and POP1EXP with a p-value below.05. Two population models appear to improve in t moving from POP2EXP, to POP2LOG, and to POP2GOM with median % RMSE of %, %, and % respectively. The Wilcoxon paired rank sum test did not produce p-values low enough to reject the null hypothesis of equal medians. There was signicant improvement in t moving from one population models to two population models. The t data is summarized in Fig. (3) and Wilcoxon p-values can be found in appendix B.

8 R. Injerd 8 Figure 3: Box plots of each model comparing t for all 11 patients When complexity is accounted for by penalizing free parameters, one and two population models appear not to benet from the inclusion of a carrying capacity. There is improvement in AIC scores of two population models over one population models. Median AIC values for POP1EXP, POP1LOG, and POP1GOM are , , and respectively. Median values for POP2EXP, POP2LOG, and POP2GOM are , , respectively. Few of the models had p-values low enough to reject the null hypothesis of equal medians. The AIC data is summarized in Fig. (4) and Wilcoxon p-values can be found in appendix C.

9 R. Injerd 9 Figure 4: Box plots of each model comparing t and complexity for all 11 patients Sensitivity analysis conducted using 10 7 iterations for each model indicate α produces the most variation in 4 of the 6 models. The results of the sensitivity analysis are shown in appendix D. The Pearson correlation analysis indicate that most of the parameters are dependent with p-values above.05. The results of the Pearson correlation analysis are shown in appendix F. The Shapiro-Wilks test and QQ plots indicate that the residuals of several patients are not normally distributed. The results of the Shapiro-Wilks test are found in appendix E and QQ plots are included in appendix H. 5 Concluding remarks As reported in previous studies, two population models t the patient data better than one population models [5]. Two population models also produce α values closer to reported values. The POP2EXP had the lowest AIC value indicating that the added θ parameter included in POP2LOG and POP2GOM increases information loss. Though POP2GOM did not outperform POP2EXP when accounting for complexity, it appeared to provide a better t when judging by the %RM SE alone. The Shapiro-Wilks test and the QQ plots also suggest that residuals are more consistently close to being normally distributed for POP2GOM. Depending on the patient, using a two population Gompertz model may improve the utility of inferential tests. Because the study only in-

10 R. Injerd 10 cluded 11 patients and the Wilcoxon rank sum tests failed to show signicance in median dierences, more research and data are necessary to determine which two population model is best suited for clinical application. 6 Acknowledgments This study was produced during the Cancer Modeling Seminar, Math 371, as part of its required work load.

11 R. Injerd 11 A Patient Tumor Volume Day GT V 1 GT V 2 GT V 3 GT V 4 GT V 5 GT V 6 GT V 7 GT V 8 GT V 9 GT V 10 GT V B RMSE% p-values Paired Wilcoxon Rank Sum POP1EXP POP2EXP POP1LOG POP2LOG POP1GOM POP2GOM POP1EXP POP2EXP POP1LOG POP2LOG POP1GOM POP2GOM C AIC p-values Paired Wilcoxon Rank Sum POP1EXP POP2EXP POP1LOG POP2LOG POP1GOM POP2GOM POP1EXP POP2EXP POP1LOG POP2LOG POP1GOM POP2GOM

12 R. Injerd 12 D Sensitivity Analysis µ x σ CV POP1EXP α k POP2EXP α k c POP1LOG α k θ POP2LOG α k θ c POP1GOM α k θ POP2GOM α k θ c E Residual Shapiro-Wilks Test Statistic Patient no. POP1EXP POP2EXP POP1LOG POP2LOG POP1GOM POP2GOM Omitted values indicate a p-value<.05 indicating likely non-normality

13 R. Injerd 13 F Parameter Correlation α θ c POP1EXP k POP2EXP α k POP1LOG α k θ POP2LOG α k θ POP1GOM α k θ POP2GOM α k θ Omitted values indicate a p-value<.05 indicating likely independence

14 R. Injerd 14 G Patient Plots Figure 5: Patient 1 Figure 6: Patient 2

15 R. Injerd 15 Figure 7: Patient 3 Figure 8: Patient 4

16 R. Injerd 16 Figure 9: Patient 5 Figure 10: Patient 6

17 R. Injerd 17 Figure 11: Patient 7 Figure 12: Patient 8

18 R. Injerd 18 Figure 13: Patient 9 Figure 14: Patient 10

19 R. Injerd 19 Figure 15: Patient 11

20 R. Injerd 20 H QQ Plots Figure 16: Patient 1 POP1EXP Figure 19: Patient 1 POP2LOG Figure 17: Patient 1 POP2EXP Figure 20: Patient 1 POP1GOM Figure 18: Patient 1 POP1LOG Figure 21: Patient 1 POP2GOM

21 R. Injerd 21 Figure 22: Patient 2 POP1EXP Figure 25: Patient 2 POP2LOG Figure 23: Patient 2 POP2EXP Figure 26: Patient 2 POP1GOM Figure 24: Patient 2 POP1LOG Figure 27: Patient 2 POP2GOM

22 R. Injerd 22 Figure 28: Patient 3 POP1EXP Figure 31: Patient 3 POP2LOG Figure 29: Patient 3 POP2EXP Figure 32: Patient 3 POP1GOM Figure 30: Patient 3 POP1LOG Figure 33: Patient 3 POP2GOM

23 R. Injerd 23 Figure 34: Patient 4 POP1EXP Figure 37: Patient 4 POP2LOG Figure 35: Patient 4 POP2EXP Figure 38: Patient 4 POP1GOM Figure 36: Patient 4 POP1LOG Figure 39: Patient 4 POP2GOM

24 R. Injerd 24 Figure 40: Patient 5 POP1EXP Figure 43: Patient 5 POP2LOG Figure 41: Patient 5 POP2EXP Figure 44: Patient 5 POP1GOM Figure 42: Patient 5 POP1LOG Figure 45: Patient 5 POP2GOM

25 R. Injerd 25 Figure 46: Patient 6 POP1EXP Figure 49: Patient 6 POP2LOG Figure 47: Patient 6 POP2EXP Figure 50: Patient 6 POP1GOM Figure 48: Patient 6 POP1LOG Figure 51: Patient 6 POP2GOM

26 R. Injerd 26 Figure 52: Patient 7 POP1EXP Figure 55: Patient 7 POP2LOG Figure 53: Patient 7 POP2EXP Figure 56: Patient 7 POP1GOM Figure 54: Patient 7 POP1LOG Figure 57: Patient 7 POP2GOM

27 R. Injerd 27 Figure 58: Patient 8 POP1EXP Figure 61: Patient 8 POP2LOG Figure 59: Patient 8 POP2EXP Figure 62: Patient 8 POP1GOM Figure 60: Patient 8 POP1LOG Figure 63: Patient 8 POP2GOM

28 R. Injerd 28 Figure 64: Patient 9 POP1EXP Figure 67: Patient 9 POP2LOG Figure 65: Patient 9 POP2EXP Figure 68: Patient 9 POP1GOM Figure 66: Patient 9 POP1LOG Figure 69: Patient 9 POP2GOM

29 R. Injerd 29 Figure 70: Patient 10 POP1EXP Figure 73: Patient 10 POP2LOG Figure 71: Patient 10 POP2EXP Figure 74: Patient 10 POP1GOM Figure 72: Patient 10 POP1LOG Figure 75: Patient 10 POP2GOM

30 R. Injerd 30 Figure 76: Patient 11 POP1EXP Figure 79: Patient 11 POP2LOG Figure 77: Patient 11 POP2EXP Figure 80: Patient 11 POP1GOM Figure 78: Patient 11 POP1LOG Figure 81: Patient 11 POP2GOM

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