Abstract
The goal of this work was to develop a mathematical model to predict Kaplan–Meier survival curves for chemotherapy combined with radiation in NonSmall Cell Lung Cancer patients for use in clinical trial design. The Gompertz model was used to describe tumor growth, radiation effect was simulated by the linearquadratic model with an α/βratio of 10, and chemotherapy effect was based on the logcell kill model. To account for repopulation during treatment, we considered two independent methods: 1) kickoffrepopulation using exponential growth with a decreased volume doubling time, or 2) Gompertzrepopulation using the gradually accelerating growth rate with tumor shrinkage. The input parameters were independently estimated by fitting to the SEER database for untreated tumors, RTOG8808 for radiation only, and RTOG9410 for sequential chemoradiation. Applying the model, the benefit from concurrent chemoradiation comparing to sequential for stage III patients was predicted to be a 6.6% and 6.2% improvement in overall survival for 3 and 5years respectively, comparing well to the 5.3% and 4.5% observed in RTOG9410. In summary, a mathematical model was developed to model tumor growth over extended periods of time, and can be used for the optimization of combined chemoradiation scheduling and sequencing.
Introduction
NonSmall Cell Lung Cancer (NSCLC) is a heterogeneous disease, which not only relates to its anatomical presentation, but also to the diverse biology^{1}. As 40% of patients with NSCLC have locally advanced unresectable disease, combined chemotherapy and radiotherapy (CRT) is considered to be the first choice therapy for most of them^{2,3}.
Though some gains in Overall Survival (OS) have been achieved over the past decades, long term survival in unresectable stage III NSCLC patients remains low and many efforts have been made to improve treatment. Mathematical modeling plays an important role for developing hypotheses to be tested in future clinical trials and for optimizing their design. Especially in the area of accelerated fractionation explored by the CHART and CHARTWEL trials, radiobiological models played a large role in trial design and estimating the therapeutic benefit^{4}.
The trend to stratify patients further into smaller subgroups and intreatment adaptation approaches increase the importance of patientspecific modeling in lung cancer. The characterization of the effect of different treatment regimen is usually limited to the effect of radiation dose^{5,6}, and sometimes include tumor volume^{7}. However, it is well known that the combination and sequencing of chemotherapy with radiotherapy is playing an important role in the treatment of NSCLC^{8}. There are few efforts to mathematically model these combination therapies, which would facilitate patient stratification and optimization of sequencing in combined chemoradiotherapy^{9}.
In this paper, we develop a mathematical model to predict the KaplanMeier survival curve (KMsc), which is basically a cumulative survival frequency distribution curve, for a NSCLC patient population using a “TopBottom” methodology, i.e. using observable patient data as starting point. The aims are to

I.
Develop a NSCLC tumor growth model and patient death model using data of untreated patients, and compare it to clinical observations of tumor size and growth rates.

II.
Derive population distributions of radiation effect parameters using data of radiationonly trials and results from (I).

III.
Derive population distributions of chemotherapy effect parameters using data of sequential chemotherapyradiation trials and results from (I) and (II).
The performance and viability of the model is evaluated and discussed with the published datasets.
Methods
Tumor growth modeling
Multiple methods of modeling the kinetics of tumor growth have been proposed^{10,11}. One of the most popular models is exponential growth, which assumes the tumor can grow exponentially without any capacity constraints until the death of the patient. Exponential growth kinetics can describe tumor growth well when the tumor is relatively small, however, this is not the case for larger tumors considering the insufficient nutrition or vascularity of the tumor^{11}. In order to account for the decreased growth rate with increased tumor volume, other models, e.g. the logistic and Gompertz model, were proposed. In this paper, we employed Gompertz growth, where the evolution of tumor cell number N (i.e. volume times cell density) with growth rate ρ is described by the following differential equation:
The growth rate \(\rho \) and carrying capacity \(K\) are the specific parameters determining the growth curve of a tumor. With this methodology, we are able to simulate one Gompertz growth curve for tumors with different initial volumes (e.g. different stages), enabling us to describe the clinical stages I through IV in one coherent framework.
In the Gompertz model, the volume doubling time is variable and not well defined as in the exponential growth model, where it is constant. In order to compare to data from the literature, we therefore used the same expression for the VDT as the exponential growth model, which is,
where t1 is the time of the first examination and t2 is the time of the second examination, \({v}_{t1}\) and \({v}_{t2}\) is the volume at t1 and t2 respectively. We assumed t2 occurs 1 year after t1 in this paper to compare to the literature values, which is similar to the observation periods used in most lung screening papers^{12,13}.
Radiation effect modeling
The most commonly used tool for quantitative predictions of radiation effects is the linear quadratic (LQ) model^{14,15}. The LQ model describes that the number of surviving cells after being irradiated by a certain dose of radiation takes the form of an exponential function with a linear and a quadratic term. In the typical dose range (under 10 Gy per fraction) and fractionation of clinical interest in stage III NSCLC, the LQ model shows good performance in terms of describing the radiation effect as a function of prescription dose. The formula can be expressed in differential equation as,
In our approach we assumed \(\alpha /\beta \) = 10, which is consistent with assessments from clinical trials for NSCLC patients^{16}. The cell kill from radiation was determined by the patientspecific radiosensitivity parameter \({\rm{\alpha }}\) and \({\rm{\alpha }}/{\rm{\beta }}\). Radiosensitivity refers to the relative susceptibility of cells to the harmful effect of ionizing radiation.
Considering that cells with faster growth rate have shown to be more radiationsensitive^{17,18,19}, we also introduced a possible correlation between radiosensitivity \(\alpha \) and the growth rate parameter \(\rho \).
Chemotherapy effect modeling
The logcell kill model was used to simulate chemotherapy effects. The logcell kill model subtracts a certain fraction of cancer cells based on the drug concentration regardless of the tumor size at the time of administration^{9,20,21}. It can be expressed as,
where, \({\beta }_{c}\) represents the chemotherapy effect per dose, and \(C(t)\) is the drug concentration at a certain time point. In our model, the concentration was assumed as an exponential decay process,
KaplanMeier survival curve estimation
Death and tumor control conditions
In the literature the time of death is estimated to be about 41 doubling times, which corresponds to a tumor 13 cm in diameter^{22}. In the present model, we therefore used the 13 cm diameter as the death condition. Besides the death from disease we implemented a 1.48% survival reduction according to life expectancy tables from the SEER data to account for unexpected natural death events.
To determine whether the tumor was controlled (i.e. no single clonogenic cell survives), we used the Poisson probability
which determines the probability of tumor control given N remaining cells. A random value was then generated between 0 and 1 to determine whether the specific tumor in the patient is controlled.
Initial volume
For the volume distribution in each stage, a lognormal distribution was assumed, as often seen in clinical patient series^{23}. The range for each stage was defined based on the AJCC standard criteria^{24}. The maximum of the volume distribution for stage I is 5 cm in diameter. For other stages, the volume can be any size, while it should be lower than the death condition (i.e. 13 cm). Considering the detection ability of the diagnostic technology, we use a diameter of 0.3 cm as the minimal detectable volume for all stages^{25}. The tumor cell density was assumed to be 5.8 × 10^{8} per cm^{3}, as determined experimentally for lung cancer^{26}. The choice of the density parameter should not be crucial in our model, as it simply scales everything in an equal manner and a different choice would not have an impact on the ability of the model to fit clinical data.
Monte Carlo Patient Population
To obtain a KaplanMeier survival curve, we used a Monte Carlo method to sample an initial patient population, where each patient has specific tumor and treatment properties, and survival time after treatment should be obtained for each patient. During the sampling process, normal distribution was used for growth and treatment effect parameters. The delay time, which is the time between diagnosis and the start of the treatment, was uniformly sampled from 2–3 weeks. In the fitting procedure, 10000 patients were sampled in each iteration in order to reduce the uncertainty. In the prediction studies, the same number as patients on the trial were sampled each time, which was repeated 100 times to estimate the uncertainty stemming from the limited populationvolume of the specific trial.
Parameter estimation
We expressed the combined growth, chemotherapy and radiotherapy treatment for one patients as,
With this formula, we can derive the KMsc for a sampled patient population, that is for each patient we have the specific initial volume, the growth parameter, and the treatment response parameters. To derive the parameters based on the clinical observed KMsc, we formulized the cost function as,
where, \({{SF}}_{{model}}({Mo}{n}_{i})\) represent the survival fraction of patients who can survive longer than \({Mo}{n}_{i}\). The parameter of the model was derived by fitting to the observations from clinical trials, first only the growth term, then the radiation and chemotherapy terms separately. Consequently, the parameter estimation was a three stage process, as shown in Fig. 1, that is from top to bottom, the previously determined parameters were fixed and used as input for the next stage.
To derive the parameters that determine the tumor volume distribution and the growth pattern, the KMsc were fit to that of an untreated population collected in Detterbeck et al. in 2008 for stage I–IV disease^{22}. The dataset includes the survival of 23954 patients with NSCLC who did not receive surgical resection, chemotherapy or radiotherapy.
With the resulting parameters being fixed we further derived the radiation effect parameter using a radiationonly clinical trial (RTOG 8808), with a prescribed dose of 60 Gy in 2 Gy fractions^{27}.
For the parameters of the chemoeffect model, we used the 60 Gy arm of the RTOG 9410 trial which was designed to compare concurrent and sequential chemoradiation therapy^{8}. The enrolled patient numbers is over 200 per arm, prescription includes cisplatin at 100 mg/m^{3} on days 1 and 29 and vinblastine at 5 mg/m^{3} per week for 5 weeks. 60 Gy TRT started on day 50 and day 1 for the sequential and concurrent arm respectively. For the two drugs, we used the same \({\beta }_{c}\) for simplicity, with a unit of (mg/m^{3})^{−1}. The halflife of the drugs was assumed to be 24 h, as measured in pharmacokinetic studies^{28,29}.
Table 1 lists the complete list of the parameters that are used in the model. The subscripts of \({\rm{\mu }}\) and \({\rm{\sigma }}\) in the table represent the mean and sigma of the distribution for each variable. The optimization was constrained to avoid clearly implausible combinations of parameter values (as listed in Table 1). For example, the μ and σ for each parameter (including volume, \(\rho \), \(\alpha \) and \({\beta }_{c}\)) can not be negative, and the minimum detectable tumor diameter is ~3 mm. The model was implemented in python 3.5, for the fit process the lmfit package was applied using Powell’s method.
Results and Discussion
Tumor growth
Based on the survival curve of untreated population, the volume distribution and growth pattern for stage I–IV patients was derived. Considering that this procedure contains 13dimensional (i.e. listed under the growth model section in Table 1) parameter space, which cannot be exhaustively explored in a heuristic fashion, we performed a twostage optimization procedure to find a sensible and clinically relevant solution (i.e. within the constraints listed in Table 1). In the first stage, we used a predefined initial volume distribution taken from the literature (i.e. Stage I: 2.5 ± 2.5 cm, Stage II: 3.5 ± 3 cm, Stage III:6.6 ± 3 cm) to obtain the optimal parameters (i.e. carrying capacity K and growth parameter \(\rho \)) for the Gompertz growth model. Figure 2 illustrates the K and \(\rho \) surface with respect to the residuals for \({\rho }_{\mu }\) = 7 \(\times \) 10^{−5}, revealing a valley with a minimum at K = 30 and \({\rho }_{\sigma }\) = 7.23 \(\times \) 10^{−3}. In the second step we kept these optimal parameters (i.e. K and \(\rho \)) for the Gompertz growth model constant and reoptimized (i.e. finding the ones resulting in the minimal residual of the cost function) the initial volume distribution that was kept constant before.
Figure 3 shows the predicted KaplanMeier survival curves (KMsc) for the untreated populations for different stages together with the observed data. With the fitted initial volume distribution and the growth parameters (K and \(\rho \)), we obtained the distribution of volume doubling time (VDT) (based on equation 2) for patient populations of each stage during the first year after diagnosis. The one year interval was chosen because it is commonly used in most lung screening papers^{12,13}. Table 2 lists the tumor volume distributions as predicted by the model fit for the median tumor diameter for stage I/II/III/IV, i.e. 1.2/3.5/6.9/9.7 cm, respectively. As an example, Figure 4 shows the volume and volume doubling time distributions for stage I patients. The values for stages I–III are close to observations from studies based on CT or radiographic imaging^{23,30}. The estimated capacity is 30 cm in diameter, which has to be interpreted as a parameter to describe the slowing growth with increased tumor size, and not as a size to be reached or a realistic volume limit. Similarly, for the stage IV patients, the tumor volume here should not be interpreted as the solid volume of the primary tumor, but potentially multiple tumor nodules and metastases, and is more a description of “tumor load” in the entire patient. The median value of VDTs for the tumor were predicted to be 59/89/198/311 days for stage I/II/III/IV, which compares well to literature values^{12} ^{,} ^{31} even though deviations can be expected due to the dependence on the time interval^{32}. For example, the mean VDT is between 93–452 days according to several publications^{12,31,33,34,35,36}.
The exponential growth model was also implemented for comparison but resulted in about 40% lower median tumor volumes in stage III patients, not comparing well with the published data (e.g. i.e. Stage I: 2.5 ± 2.5 cm, Stage II: 3.5 ± 3 cm, Stage III:6.6 ± 3 cm)^{22,31} and not in line with published GTV (gross tumor volume) sizes at this stage^{37,38,39}. Furthermore, the VDT is a constant of 80 days for all stages, which does not agree well with the literature especially for later stages (>100 days)^{12} and is not in agreement with the clinical findings that smaller tumors have a shorter VDT^{40}.
Radiation Therapy
The radiation therapy response was modeled with the LQ model. Based on experimental observations, a correlation between the growth rate \(\rho \) and the radiosensitivity \(\alpha \) was considered. The underlying biological interpretation is that faster growing tumor cells spend a higher proportion of their time in mitosis, which impairs double strand break (DSB) repair and makes them more radiosensitive compared to slower growing ones^{17,18,19}. In the SmallCell variant of lung cancer this has also been shown invivo, as the growth rate measured by Ki67 in patient samples correlated with the extent of volumetric response to CRT^{19}. Figure 5 shows the distributions for \(\alpha \) and \(\rho \) on the left, and the survival fraction (i.e. fraction of cells retaining their reproductive integrity) distribution on the right. The survival fraction in Fig. 5c was obtained based on the LQ model from the derived \(\alpha \) distribution. As shown in Fig. 5b, the best fit is gained by assuming a slight correlation (coefficient = 0.87) between a patient’s growth rate and radiosensitvity. Figure 5b demonstrates the superior fit (i.e. lower residual value of the cost function) of the survival curve with correlation comparing to without correlation. The estimated median values for \(\alpha \) for the radiation effect was 0.16 Gy^{−1}, which is similar to previous in vivo estimates of 0.16–0.18^{41}. These estimated values for \(\alpha \) are below the common range of in vitro estimates (0.2–0.5 Gy^{−1})^{16,42,43}. Figure 5c shows the distribution of SF_{2Gy} (the cell survival fraction after 2 Gy), derived from the distribution of the radiosensitivity parameter \(\alpha \).
From clinical observations we know that the growth rate accelerates with the shrinkage of the tumor volume during radiation therapy, a phenomenon termed repopulation^{44,45}. To account for this effect, most of the current models use the kickoffrepopulation approach, which introduces an exponential growth with a VDT of 3 days after a kickoff period of 28 days, counting from the start of therapy^{16}. However, this phenomenon is in principle intrinsically included in the Gompertz model, as tumors shrinking from radiation cell kill grow faster according to Gompertz kinetics. We compared the two approaches in this study.
Figure 6 shows the results for the two methods. We found that both repopulation methods can describe the survival curve after radiation therapy well (see Fig. 6a). This confirms that the parameterless Gompertzrepopulation can naturally account for repopulation (i.e. gradually decreased VDT as the tumor shrinks, see Fig. 6c) during radiation therapy. The estimated median values for the radiation effect \(\alpha \) were 0.16 for Gompertz repopulation and 0.17 for kickoffrepopulation respectively. These values agree well with clinical findings^{46,47}, and underscore that Gompertz growth could naturally account for repopulation without additional parameters.
Currently the exponential growth model is still the standard in radiotherapy effect modeling, evidenced for instance by the inclusion of an exponential growth based time factor in the BED model^{15,16}. Sachs et al. described a similar model, and discussed that simple approaches such as the Gompertz growth and linearquadratic models can still capture the complicated interactions present during radiation therapy^{48}. The model described by Sachs et al. is similar with the present model. In a recent study, Jeong (2017) developed a mechanistic mathematical model for radiation therapy, which quantitatively predicted the differences in response seen in early stage lung cancer across the range of clinical dose and fractionation schemes^{49}. They used a compartmental ‘tumourlet’response model to account for the oxygen and tumor repopulation effect, which is more mechanistic and requires more parameters compared to the present model. Models such as this could be implemented with our chemotherapy and growth models to enable prediction of fractionation effects.
Chemotherapy
With consistent parameters for tumor growth and radiation response, we then proceeded to derive the chemoeffect parameter \({{\rm{\alpha }}}_{\mathrm{chemo}}\) using the sequential chemoradiation (sCRT) arm of RTOG 9410. The sCRT included cisplatin at 100 mg/m^{3} on days 1 and 29 and vinblastine at 5 mg/m^{3} per week for 5 weeks with 60 Gy radiation therapy beginning on day 50, as shown in Figure. 7. To explain the observed improvement in survival of sCRT compared to radiation only, the resulting median chemotherapy effect parameter \({{\rm{\beta }}}_{{\rm{c}}}\) was determined to be 0.028/(mg/ml).
As our methodology has not been used before to derive quantitative cell kill parameters for chemotherapy agents, we verified the predictive power using published data. From published data of large cohorts, we know that with doublet chemotherapy in stage IV lung cancer patients, the median survival is about 6–10 months and 1year OS is about 25–35%^{50,51,52}. To test if our model would predict similar values, we generated cohorts of stage IV patients which received 4 cycles, which is the median number of cycles given^{51,52}, of our chemotherapy simulation and predicted the improvement in survival. Our model predicted a median survival of 10.2 months, which is reasonably close to the observed survival.
The Gompertz model also explains the moderate repopulation (i.e. accelerated tumor growth) observed after the chemotherapy segment of sequential chemoradiation, which was observed in clinical trials^{40,53}. Our model predicted a median VDT of 12 days and a mean VDT of 41 days for a time interval of 100 days between two examinations after induction chemotherapy. Sharouni et al. measured the median and mean VDT of 29.4 and 45.8 days after induction chemotherapy, which they described as significantly lower than the normal VDT^{40}. Also they found that small tumors have the shortest VDTs. This is a confirmation of the results given by our model.
Combined modality treatment
We simulated concurrent chemoradiation (cCRT) by combining the parameters obtained above. We took the same chemo and radiotherapy cell kill used in Fig. 7a for sCRT and shifted the RT section to start together with the first cycle of chemotherapy, mimicking the concurrent trial arm of RTOG9410 (see Fig. 7b). The main goal was to determine if an additional radiosensitization parameter would be necessary to fit the observed survival curves, or if the increased cell kill in a shorter treatment time is sufficient to explain the superior clinical results of cCRT.
Figure 7c demonstrates that the combination of chemotherapy and radiotherapy in a concurrent fashion predicts OS improvements of 6.6% and 6.2% for 3 and 5years for stage III patients, which compares well to the 5.3% and 4.5% observed in the cCRT arm (see Fig. 7d) of RTOG9410^{8}.
For the overall survival prediction of concurrent and sequential CRT, our model performs well and predicts results observed in clinical trials (see Fig. 7). There is no need to add a radiosensitization parameter for the interaction between chemo and radiotherapy, indicating that the reduction in treatment time and the resulting suppression of repopulation is the deciding factor for the superiority of cCRT compared to sCRT. Figure 8 demonstrates tumor cell load of a patient who will benefit from the concurrent compared to the sequential chemoradiation regimen. With sCRT the tumor grows back after chemo and radiotherapy, as the cell kill was not dense enough and allowed the tumor to regrow. Since the tumor cell number was reduced to below 1 (high probability of tumor control) with Concurrent chemoradiation therapy, the tumor is controlled according to our model.
Growth rate based Stratification of Concurrent and sequential CRT: an example application
To demonstrate the application of the developed model in clinical practice, we investigated which population benefits most from concurrent chemoradiation. For most patients who have unresectable disease, combined chemo and radiation therapy is considered as the first treatment of choice. However, it is unknown how to define criteria for selecting a subgroup of patients who would benefit from concurrent CRT compared to treating sequentially, which is predicted to cause less toxicity. We separated the patient population into subgroups according to the growth rate distribution (considering the median of the distribution and the growth parameter \({\rm{\rho }}\)). All parameters for treatment response are given in Table 3. The schedule of combined chemotherapy and radiation treatment was set as identical to RTOG 9410. The model predicted that for patients with a faster tumor growth (above the median), the benefit can be as high as 11.1% for stage III patients, but only 4.7% for those patients with a slower growth rate (below the median) for 3 year overall survival. The benefit can be as high as 14.4% for patients with a growth rate in the top 25^{th} percentile. This could be important in the design of clinical trials evaluating treatment options for older and frail patients that cannot tolerate a concurrent regimen, verifying these predictions in patients might necessitate advanced imaging modalities (e.g. PETMTV (PET derived metabolic tumor volume)) to determine the growth rate (e.g. utilizing the correlation between PETMTV and tumor growth rate^{55}).
Limitations and future directions
There are numerous models that describe tumor growth and treatment response. However, none of the standard models allows simulation and prediction of multimodality regimen for locallyadvanced NSCLC. We developed a mathematical model for the combined treatment of chemotherapy and radiation based on overall survival data from several clinical trials, with the purpose of improving and optimizing treatment strategies for future clinical trials. Notably, parameters in the model were derived from the clinical relevant outcome data. The tumor growth and patient death model can also be used as foundation for the simulation of other therapeutic interventions, such as targeted agents.
As every model has its inherent limitations, we would like to highlight the caveats of our approach.

(1)
The \({\rm{\alpha }}/{\rm{\beta }}\) ratio used was 10 and the cell cycle effect and oxygen enhancement ratio for radiosensitivity was not considered. This might be necessary if one wants to investigate different fractionation regimen^{16}, which was not considered in this study.

(2)
The Gompertz model in use was shown to fit the data and clinical observations better than the exponential model. However, there are other models with the same growth pattern, e.g. the logistic or the Bertalanffy model, which could be expected to have a similar modeling performance^{11}.

(3)
Logcell kill was used to model chemotherapy in the study, which is commonly used and a widely accepted approach. However, there is another main model in use for this purpose, the NortonSimons model, which differs significantly^{56}. We do not think this has an impact on the results of our study, as we investigate only one chemotherapy regimen and only study changes in timing between chemo and radiotherapy. If the aim would be to investigate different chemotherapy regimen and their interaction with radiation, then the model in use could have an impact on the results.

(4)
There is no modeling of normal tissue toxicity, although natural deaths were considered. Toxicity from treatment can result in extra deaths, and RTOG 0617^{57}, which showed lower survival for the higher dose arm, has raised questions about the limits of dose escalation with concurrent CRT in the general NSCLC population. Whatever the exact reasons for the lower survival in this trial may be, toxicity might play a prominent role. Therefore models such as the one presented above should be used with care for regimen that are supposedly pushing the boundaries of normal tissue toxicity. However, a toxicity component could be added to the current model, which would necessitate the analysis of dose distributions for all patients, which were unavailable for these trials.
Conclusion
Combined CRT is a mainstay of treatment for locallyadvanced NSCLC patients. We developed a mathematical model for the combined treatment of chemotherapy and radiation, with the purpose of improving and optimizing treatment strategies for future clinical trials. The established model can predict survival curves for populations of NSCLC patients with or without treatment. The parameterless Gompertz repopulation can naturally account for repopulation during radiation therapy and after induction chemotherapy. The model provides a tool for the optimization of combined chemoradiation scheduling and sequencing.
Change history
17 August 2018
A correction to this article has been published and is linked from the HTML and PDF versions of this paper. The error has not been fixed in the paper.
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C.G. (Changran Geng) and C.G. (Clemens Grassberger) designed and implemented the model, and analysed results. C.G. (Changran Geng) and C.G. (Clemens Grassberger) wrote the main manuscript text based on the discussion with H.P. C.G. (Clemens Grassberger) and H.P. cosupervised the project; All authors reviewed and approved the final manuscript.
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Correspondence to Changran Geng or Clemens Grassberger.
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