 Research
 Open Access
 Published:
A proliferation saturation index to predict radiation response and personalize radiotherapy fractionation
Radiation Oncology volume 10, Article number: 159 (2015)
Abstract
Background
Although altered protocols that challenge conventional radiation fractionation have been tested in prospective clinical trials, we still have limited understanding of how to select the most appropriate fractionation schedule for individual patients. Currently, the prescription of definitive radiotherapy is based on the primary site and stage, without regard to patientspecific tumor or host factors that may influence outcome. We hypothesize that the proportion of radiosensitive proliferating cells is dependent on the saturation of the tumor carrying capacity. This may serve as a prognostic factor for personalized radiotherapy (RT) fractionation.
Methods
We introduce a proliferation saturation index (PSI), which is defined as the ratio of tumor volume to the hostinfluenced tumor carrying capacity. Carrying capacity is as a conceptual measure of the maximum volume that can be supported by the current tumor environment including oxygen and nutrient availability, immune surveillance and acidity. PSI is estimated from two temporally separated routine preradiotherapy computed tomography scans and a deterministic logistic tumor growth model. We introduce the patientspecific pretreatment PSI into a model of tumor growth and radiotherapy response, and fit the model to retrospective data of four nonsmall cell lung cancer patients treated exclusively with standard fractionation. We then simulate both a clinical trial hyperfractionation protocol and daily fractionations, with equal biologically effective dose, to compare tumor volume reduction as a function of pretreatment PSI.
Results
With tumor doubling time and radiosensitivity assumed constant across patients, a patientspecific pretreatment PSI is sufficient to fit individual patient response data (R^{2} = 0.98). PSI varies greatly between patients (coefficient of variation >128 %) and correlates inversely with radiotherapy response. For this study, our simulations suggest that only patients with intermediate PSI (0.45–0.9) are likely to truly benefit from hyperfractionation. For up to 20 % uncertainties in tumor growth rate, radiosensitivity, and noise in radiological data, the absolute estimation error of pretreatment PSI is <10 % for more than 75 % of patients.
Conclusions
Routine radiological images can be used to calculate individual PSI, which may serve as a prognostic factor for radiation response. This provides a new paradigm and rationale to select personalized RT dosefractionation.
Background
Recent advances in radiation oncology have largely focused on the physical characteristics of radiation including beam quality and delivery. Typically, tumor and normal tissue anatomy/geometry are only used as the main parameters with which to enhance therapeutic ratios [1–5]. Adaptive radiotherapy and imageguided radiotherapy have been suggested to primarily reshape the target volume based on changes in tumor volume or position [1–4], rather than as a methodology to adapt to changes in the intrinsic tumorhost biology. Inroads have been made in combining intensitymodulated radiation therapy with functional imaging, such as by [^{18}F]fluoromisonidazole positron emission tomography (FMISOPET) or dynamic contrastenhanced magnetic resonance imaging, for radiation planning and response assessment to direct higher doses of radiation to areas with increased radioresistance [5–8]. In conventional clinical practice, however, most patients treated with definitive radiotherapy receive a similar dose and fractionation scheme based upon primary site and American Joint Committee on Cancer (AJCC) TNM stage (Tumor size, lymph Node involvement, Metastasis presence).
Different fractionation protocols have been tested in prospective clinical trials [9, 10]. Alternative radiation fractionation protocols may improve outcome for some patients but worsen outcome for others. It is important that we begin to understand which tumors respond better to altered fractionation, and how to select the most appropriate fractionation schedule for an individual patient. Innovative models that are based upon cell biology and interactions of the tumor with its unique environment could forecast individual radiation response and justify recommendation of either standard of care or alternative radiation fractionation on a per patient basis.
Tumors grow within a host tissue that both facilitates progression by supplying nutrients and growth factors [11, 12], and inhibits it through physical constraints [13] and immune surveillance [14, 15]. Since many of these factors vary widely across patients, we introduce the concept of tumor carrying capacity as the maximum tumor volume that is achievable in the patientspecific tumor environment, and saturation of tumor proliferation as the tumor approaches its carrying capacity. We propose a noninvasive radiomics measurement of patientspecific carrying capacity and proliferation saturation, which may ultimately help designing more personalized approaches to radiotherapy.
Tumors are composites of proliferating and growth arrested cells. Their respective proportions at individual times during radiation contribute to the populationlevel response, in addition to radiation beam and protocol parameters and host tissue properties. In multicompartment mathematical models that distinguish between cycling and growtharrested cells, proliferation and oxygenation statusdependent radiation response could be simulated on the cellular level [16, 17]. Tumor growth in vivo can be approximated by logistic dynamics [18]. Initial exponential growth at low cell densities when most cells have access to ample resources decelerates when cells at the core of the tumor become growtharrested, mainly due to limited space and exhausted intratumoral nutrient supply as resources are consumed by cells closer to the tumor surface [19, 20]. This established the notion of a tumor carrying capacity (K) as the maximum tumor volume (V) that can be supported by a given environment. A tumor carrying capacity may change depending on the oxygen and nutrient supply through tissue vascularization [21], removal of metabolic waste products [22], and evasion of immune surveillance [14]. Greater oxygen supply and removal of metabolic waste increases tumor carrying capacity; in contrast, infiltration of tumor specific cytotoxic T lymphocytes exemplifies a reduction of carrying capacity. Hence, the tumor volumetocarrying capacity ratio (V/K) describes the saturation of tumor cell proliferation at the population level as the tumor approaches its carrying capacity, and as such is deemed the Proliferation Saturation Index (PSI). The PSI at any time reflects the history of the reciprocal changes of a tumor and its environment – and thus can be expected to be patientspecific. Tumor volumes close to their carrying capacity, i.e., with a high PSI, are here assumed to have only a small proportion of proliferating cells that are most sensitive to radiationinduced damage. We therefore hypothesize that a patientspecific PSI may serve as a novel prognostic factor for radiotherapy response.
Methods
Tumor radiation response model and Proliferation Saturation Index (PSI)
Logistic tumor growth is modeled as a deterministic ordinary differential equation between radiation doses
where PSI is the tumor volumetocarrying capacity ratio (V/K), and \( \uplambda =\frac{ \ln 2}{{\mathrm{T}}_{\mathrm{eff}}} \) is the intrinsic tumor growth rate, with the effective tumor doubling time T_{eff} being a composite of the potential doubling time, T_{POT}, reduced by the cell loss fraction, φ [23]. T_{eff} is assumed to be intrinsic and time independent. Radiation response after each application of a single dose d is modeled as an instantaneous volume change \( {\mathrm{V}}_{\mathrm{postIR}}=\mathrm{V}{\upgamma}_d\;\mathrm{V}\left(1\frac{\mathrm{V}}{\mathrm{K}}\right) \) at discrete times during irradiation in the considered treatment protocols (standard of care: once a day [q.d.] at 9 am, no treatment on the weekend; hyperfractionation twice a day [b.i.d] at 9 am and 3 pm; no weekend), where \( {\upgamma}_d=1{e}^{\left(\upalpha {\mathrm{d}}^{\circ }{+}^{\circ}\upbeta {\mathrm{d}}^2\right)} \) represents radiationinduced death following the linearquadratic model [24, 25]. In this form, radiationinduced cell death is only considered for proliferating cells, while quiescent and hypoxic cells are assumed radioresistant. Tumor growth is modeled following (Eqn. 1) between radiation fractions. While other biological effects are without doubt at play before, during and after RT, longitudinal measurements of these effects are currently impossible and thus intentionally not considered explicitly. However, changes in tumor growth rate such as accelerated repopulation as well as other proliferation stimulating radiation effects (reoxygenation, redistribution in the cell cycle) are inherent to the logistic growth and RT model: as the tumor volume shrinks, V/K and thus PSI decreases, which in turn reduces the extrinsically enforced pretreatment reduction in proliferation. It follows from (Eqn. 1) that larger PSI implies a low proliferating cell fraction and thus treatment refractory tumors, whereas tumors with lower PSI are more proliferative and thus more radiosensitive (Fig. 1a), which is in line with the established positive correlation between proliferation rate and radiosensitivity [18]. Therefore, two patients that present with similar tumor volume could have a different tumor environmental conditions and thus different PSI, which results in different responses to the same RT protocol (Fig. 1b).
Here we assume that the tumor carrying capacity is constant during the 5–7 week course of treatment. This is, of course, a gross oversimplification of the underlying biology, but without longitudinal measurements of carrying capacity biomarkers, calibration of dynamic carrying capacity changes during RT is impossible and would introduce additional uncertainty. The model was simulated in MATLAB using the analytical expression for the solution of (Eqn. 1) (MATLAB R2013b, The MathWorks Inc., Natick, MA).
Prospective estimation of patientspecific pretreatment PSI
The analytical solution of pretreatment tumor growth (Eqn. 1) is given by \( \mathrm{V}=\frac{\mathrm{K}\times \mathrm{V}(0)\times {\mathrm{e}}^{\lambda \mathrm{t}}}{\mathrm{K}+\mathrm{V}(0)\times \left({\mathrm{e}}^{\lambda \mathrm{t}}1\right)} \), with V(0) being the initial tumor volume at time t=0. From two different radiological scans, routinely taken at diagnosis and at radiotherapy treatment planning simulation (Δt=45 days in private hospitals [26]), we obtain two distinct tumor volumes (V(0) =V_{diagnosis} and V=V_{simulation}) along the logistic growth trajectory. Then, the analytical solution of the logistic model can be solved explicitly for K, giving the following analytic expression for pretreatment PSI value.
Data fitting
Thirteen longitudinal tumor volume measurements across four NSCLC patients treated with fractionated RT (2 Gy x 30 fractions) were taken from the literature [27]. Tumor volumes are available at the beginning of treatment (V_{simulation}={7.6, 27.4, 97.7, 189.3} cm^{3}) and at least two subsequent RT fractionations for each patient. We assume that the variations in intrinsic effective tumor growth rate \( \lambda \) and radiosensitivity γ_{2Gy} are negligible between patients compared to the variation in individual carrying capacity K, given the spread of tumor volumes across two or three orders of magnitude. We therefore estimate constant λ and γ_{2Gy} values for all patients, and individual pretreatment proliferation saturation indices (PSI_{s}).
We utilize a genetic algorithm that mimics the processes of evolution and natural selection [28] to derive a combination of parameters that best fits patient data. We generate an initial ‘population’ (N=500 ‘individuals’) of parameter sets {\( \lambda \), γ_{2Gy}, PSI_{1}, PSI_{2} , PSI_{3} , PSI_{4}}, with each element of each individual parameter set drawn at random from a uniform distribution [0,1]. In each algorithm iteration (‘generation’) we first evaluate the fitness of each individual in the population by calculating the sum of residuals between data and corresponding simulation results (‘cost’ C)
where t _{ k,i } represents the time point when tumor volume V _{ k } was measured for kth patient, and M is the simulated tumor volume at that time point. We then select the 50 % fittest individuals (i.e., parameter combinations) that have the smallest calculated cost (Eqn. 3), and discard unfit individuals. To maintain a constant population size N in subsequent generations, additional individuals are generated using crossover (25 %) and mutation (25 %) of survived selected individuals. For crossover, two survived selected individuals are chosen at random to generate a new individual ‘offspring’, by randomly selecting ‘parental’ parameter combinations (pairwise mixing). For mutation, a new individual is generated from a survived individual with either λ, γ_{2Gy} or PSI_{ i } being modified randomly by changing its current value randomly up to ±10 %.
The 500 individuals after 500 iterations of the selection, crossover, and mutation procedure of the genetic algorithm (i.e., best sets of parameter combinations with the smallest sum of residuals between data and simulation results) is then refined with a trustregionreflective algorithm (deterministic, gradient based optimization procedure) implemented in the MATLAB lsqnonlin function (MATLAB R2013b with Optimization Toolbox, The MathWorks Inc., Natick, MA), which uses a quadratic approximation for the minimized function in a neighborhood (trust region) around the current point. In order to avoid finding a parameter set yielding only local minimum in the fitness landscape we reiterated the whole fitting procedure 20 times and compared the resulting parameter sets.
Alternative radiotherapy protocols
From the estimation of radiationinduced cell death, γ_{2Gy} we can approximate the radiosensitivity parameters α and β to derive γ_{d} for any dose d using the linearquadratic model \( {\upgamma}_d=1{e}^{\left(\upalpha \mathrm{d}+\upbeta {\mathrm{d}}^2\right)} \).
Virtual patient cohort
We create a cohort of n=1,000 in silico virtual patients P _{ i } for which we randomly assign tumor growth rate λ _{ i } ∈ [λ * (1 − x %), λ * (1 + x %)] and radiationinduced cell death \( {\upgamma}_2{{}_{\mathrm{Gy}}}_{{}_i}\in \left[{\upgamma_2}_{\mathrm{Gy}}\;*\;\left(1\mathrm{x}\%\right),{\upgamma_2}_{\mathrm{Gy}}*\left(1+\mathrm{x}\%\right)\right] \) from uniform distributions, where x represents the level of uncertainty (width of the uniform distribution support). For each virtual patient we randomly assign tumor volume and PSI at beginning of treatment, and simulate tumor volume reduction after standard of care 2 Gy x 30 fractionation using Eqn. 1 . We calculate the coefficient of determination, R^{2}, to investigate how different degrees of uncertainty (value of x) impact the predictive power of PSI.
Results
Logistic tumor growth and radiation response model fits retrospective data
The logistic tumor growth and radiation response model (Eqn. 1) fits retrospective longitudinal patientspecific data from the literature [27] with highest accuracy (R^{2}=0.98) for λ=0.045 day^{−1} and γ_{2Gy}=0.084 (Fig. 2). The estimate for λ suggests T_{eff}=15.4 days, which indicates a cell loss factor of φ=50 % for T_{POT}=7.7 days for lung adenocarcinoma [29]. Similar to oropharyngeal cancer cells, fast proliferating NSCLC cells are believed to have a relatively high α/β=20 [30–33]. With the α/β=20 approximated from literature, γ_{2Gy}=0.084 suggests a radiosensitivity parameter α=0.0487 and thus S(2Gy)<91.6 %, which is similar to that reported for A549 if plated before irradiation (83.3 %; [34]) with additional consideration for senescence and transient cell cycle arrest.
Patientspecific carrying capacities are estimated at K={72, 892, 131, 1329} cm^{3}, with pretreatment PSIs ranging from 0.03 (patient 2) to 0.75 (patient 3) with an average of 0.26±0.33 (Fig. 2). Interestingly, although the initial tumor volume of patient 4 is 25 times larger than that of patient 1, the reduction of tumor volume is comparable in both patients (53.4 % and 52.1 %) as their PSIs are similar (0.11 and 0.14). Patient 3, with an intermediate initial tumor volume but large PSI (0.75), has a significantly smaller reduction of tumor volume (20.4 %). Table 1 summarizes patientspecific tumor volumes V_{i}, derived carrying capacities K_{i} and proliferation saturation indices PSI_{i}, and other model parameter values. The minima obtained in each of the 20 independent iterations of the data fitting procedure were indistinguishable (standard deviation/mean = 6x10^{−6}) with negligible differences between the estimated growth rates and radiosensitivities (λ=0.045±3x10^{−5} and γ_{2Gy} = 0.084±6x10^{−5}). The differences in the estimated PSI_{i} values between independent data fitting iterations did not exceed 10 %.
Pretreatment PSI is a prognostic factor for radiation response
For less than 8.7 % uncertainty in intrinsic tumor parameters, pretreatment PSI serves as a prognostic factor with a high coefficient of determination (R^{2}>0.8; Fig. 3a). R^{2} falls below 0.6 for 14.1 % uncertainty in \( \lambda \) and γ_{2Gy}, but remains a better prognostic factor for radiation response than tumor growth rate for uncertainty in intrinsic tumor parameters up to around 25 %. Pretreatment PSI correlates inversely with tumor volume reduction during RT (Fig. 3b).
Reliable estimation of patientspecific pretreatment PSI must be achievable despite interpatient variation in tumor growth rate \( \lambda \), as well as limitation in radiological image resolution and noise in tumor volume measurements [35]. For each virtual patient P _{ i } we reverse calculate exact tumor volume at diagnosis using Eqn. 1 (t45 days; [26]), and introduce noise of ±5 % in the tumor volume measures at both diagnosis and treatment planning. The absolute error between exact pretreatment PSI and the proposed estimate using Eqn. 2 from noisy input data for different levels of uncertainty in tumor growth rate \( \lambda \) is shown in Fig. 3c. For uncertainties less than 20 %, the absolute estimation error of pretreatment PSI is <10 % for more than 75 % of patients.
Pretreatment PSIdependent response to hyperfractionation
We consider tumors with fixed estimated radiosensitivity α=0.045 Gy1, fixed α/β=20 Gy, and varying pretreatment PSI. We simulate response to standard of care (2 Gy/fx x 30; q.d. 9 am; no weekend), and compare final tumor sizes to simulated responses to hyperfractionated treatment with 1.2 Gy/fx x 58; b.i.d. (9 am and 3 pm; no weekend), as prescribed in the experimental arm of an RTOG phase III trial in regionally advanced unresectable NSCLC [36]. Model simulations predict an average of 27.6 % improved tumor volume reduction after hyperfractionation for all PSI, due to the larger biologically effective dose (BED) (73.8 Gy_{20} vs. 66 Gy_{20} for standard of care) (Fig. 4a). To demonstrate which patients benefit from hyperfractionation, we compare the clinically applied hyperfractionation protocol to daily fractionation with equal BED (2.21 Gy x 30; q.d. 9 am; no weekend). An improvement in tumor volume reduction of >5 % is predicted for patients with intermediate PSI (0.45–0.9; Fig. 4b). This demonstrates that a patientspecific PSI may inform which patients are most likely to benefit from alternative radiation fractionation prior to clinical intervention.
Conclusions
Despite interpatient variability and the differences in tumor biology across disease sites, radiotherapy is conventionally fractionated at 180–200 cGy daily for 5–7 weeks. Current RT fractionation selection is based on average responses from large historical data sets and clinical trial cohorts without consideration of patientspecific characteristics. The potential uniqueness of each patient at diagnosis due to variation in tumor intrinsic properties such as radiosensitivity and host responses such as angiogenesis or immunoediting [14, 21], leads to the establishment of highly patientspecific circumstances, which can greatly affect clinical response. Tumors growing in tissues are faced with harsh biological and chemical conditions as well as physical forces, which all influence the maximum tumor volume that can be achieved in the current condition – the tumor carrying capacity. As a tumor population approaches its carrying capacity, overall proliferation rate saturates dependent on patient specific tumor history. As such, nominal tumor size alone is insufficient to predict growth dynamics.
It is conceivable that different tumor growth rates in vitro and in vivo are not primarily cell intrinsic; rather the impact of the in vivo environment may be the dominant mechanism that modulates cell behavior, which is absent when expanded in optimal in vitro conditions. The key feature of the patientspecific volumetocarrying capacity ratio and the herein proposed proliferation saturation index, PSI, is a uniform cell growth rate as an intrinsic property that is modulated by host tissue conditions. This leads to different tumor population growth rates before, during and after radiation, including accelerated repopulation during radiotherapy. From two temporally separated radiological scans, routinely taken at diagnosis and treatment simulation, the change in individual in vivo tumor volume can be estimated and compared to the expected in vitro tumor growth, which allows for the estimation of the tumor environmentenforced patientspecific proliferation saturation index, PSI, using the analytic solution to the logistic growth model (Eqn. 1). If diagnostic images are unavailable, two subsequent images taken at later time points during therapy may be used to forecast the response to the remainder of the treatment schedule, and to adapt the protocol if necessary.
We introduced a patientspecific tumor carrying capacity in the logistic tumor growth model, and showed that simulations of tumor volume changes during RT using individual pretreatment PSI (equivalent to tumor volumetocarrying capacity ratio) can reproduce historical radiation response data with high confidence (R^{2}=0.98). Henceforth, in silico trials may be performed to predict which patients benefit from altered treatment protocols dependent on individual PSI and tumor volumes. The ability to forecast the response of individual tumors to different fractionations may pave the way for clinical trials to recommend either standard of care or alternative radiation fractionation on a per patient basis.
The presented model underestimates radiationinduced cell kill as nonproliferative cells are assumed to be completely radioresistant. Before consequential conclusions on alternative fractionations can be drawn, better approximations of radiation effects on the nonproliferative compartment have to be derived. We have refrained from such considerations in the present study in order to keep the number of unknown parameters small and thus preserve validity of the discussed concept. For simplicity we have limited our analysis to the effects of tumor environment on tumor properties and neglected variability in tumorintrinsic properties. It is conceivable that future studies may integrate patientspecific molecular identifiers of intrinsic radiosensitivity [37], such as RSI [38, 39].
While simulation results using the concept of PSI fit retrospective data with high confidence, additional caution is warranted. The tumor carrying capacity is a dynamic entity that is unlikely to remain constant during the course of radiotherapy. Numerous biological processes such as vascular density, neovascularization, or immune surveillance contribute to variation in the carrying capacity, and the effects of radiation protocols on each of these processes individually and in combination are yet to be fully understood. Whilst highresolution longitudinal measurements of contributors to carrying capacity, including those presently unidentified, are elusive, frequent tumor volume measurements during fractionated radiotherapy will provide patientspecific data to further fit the tumor growth model. It may also help us develop models that can form the basis for adaptive radiation therapy, which are greatly needed. For example, insights into the evolution of PSI during a course of therapy have the potential to help us understand the predictability of response. With a better understanding of the evolution of such dynamic factors as tumor response and PSI during the initially prescribed radiation protocol, different protocols could be simulated and the treatment protocol dynamically adapted to hopefully provide better outcomes. This will be the subject of future investigations.
Abbreviations
 PSI:

Proliferation saturation index
 V:

Tumor volume
 K:

Tumor carrying capacity
 RT:

Radiotherapy
 T_{POT} :

Potential doubling time
 T_{eff} :

Effective doubling time
References
 1.
Yan D, Lockman D, Brabbins D, Tyburski L, Martinez A. An offline strategy for constructing a patientspecific planning target volume in adaptive treatment process for prostate cancer. Int J Radiat Oncol Biol Phys. 2000;48:289–302.
 2.
Woodford C, Yartsev S, Dar AR, Bauman G, Van Dyk J. Adaptive radiotherapy planning on decreasing gross tumor volumes as seen on megavoltage computed tomography images. Int J Radiat Oncol Biol Phys. 2007;69:1316–22.
 3.
Feng M, Kong FM, Gross M, Fernando S, Hayman JA, Ten Haken RK. Using fluorodeoxyglucose positron emission tomography to assess tumor volume during radiotherapy for nonsmallcell lung cancer and its potential impact on adaptive dose escalation and normal tissue sparing. Int J Radiat Oncol Biol Phys. 2009;73:1228–34.
 4.
Veiga C, McClelland J, Moinuddin S, Lourenço A, Ricketts K, Annkah J, et al. Toward adaptive radiotherapy for head and neck patients: Feasibility study on using CTtoCBCT deformable registration for “dose of the day” calculations. Med Phys. 2014;41:031703.
 5.
Sharma N, Neumann D, Macklis R. The impact of functional imaging on radiation medicine. Radiat Oncol. 2008;3:25.
 6.
Rockne RC, Trister AD, Jacobs J, HawkinsDaarud AJ, Neal ML, Hendrickson K, et al. A patientspecific computational model of hypoxiamodulated radiation resistance in glioblastoma using 18FFMISOPET. J R Soc Interface. 2015;12.
 7.
Bhatnagar P, Subesinghe M, Patel C, Prestwich R, Scarsbrook AF. Functional imaging for radiation treatment planning, response assessment, and adaptive therapy in head and neck cancer. Radiographics. 2013;33:1909–29.
 8.
Clausen MM, Hansen AE, Lundemann M, Hollensen C, Pommer T, Munck Af Rosenschöld P, et al. Dose painting based on tumor uptake of CuATSM and FDG: a comparative study. Radiat Oncol. 2014;9:228.
 9.
Ahmed KA, Correa CR, Dilling TJ, Rao NG, Shridhar R, Trotti AM, et al. Altered fractionation schedules in radiation treatment: a review. Semin Oncol. 2014;41:730–50.
 10.
Prasanna A, Ahmed MM, Mohiuddin M, Coleman CN. Exploiting sensitization windows of opportunity in hyper and hypofractionated radiation therapy. J Thorac Dis. 2014;6:287–302.
 11.
Folkman J, Merler E, Abernathy C, Williams G. Isolation of a tumor fractor responsible for angiogenesis. J Exp Med. 1971; 133:275288.
 12.
Kallinowski F, Okunieff P. Blod flow, oxygen and nutrient suppy, and metabolic microenvironment of human tumors: a review. Cancer Res. 1989; 49:64496465.
 13.
Roose T, Netti PA, Munn LL, Boucher Y, Jain RK. Solid stress generated by spheroid growth estimated using a linear poroelasticity model. Microvasc Res. 2003;66:204–12.
 14.
Dunn GP, Bruce AT, Ikeda H, Old LJ, Schreiber RD. Cancer immunoediting: from immunosurveillance to tumor escape. Nat Immunol. 2002;3:991–8.
 15.
Wilkie KP, Hahnfeldt P. Mathematical models of immuneinduced cancer dormancy and the emergence of immune evasion. Interface Focus. 2013;3:20130010–0.
 16.
Enderling H, Park D, Hlatky LR, Hahnfeldt P. The Importance of Spatial Distribution of Stemness and Proliferation State in Determining Tumor Radioresponse. Math Model Nat Phenom. 2009;4:117–33.
 17.
Gao X, McDonald JT, Hlatky LR, Enderling H. Acute and fractionated irradiation differentially modulate glioma stem cell division kinetics. Cancer Res. 2013;73:1481–90.
 18.
Rockne R, Rockhill JK, Mrugala M, Spence AM, Kalet I, Hendrickson K, et al. Predicting the efficacy of radiotherapy in individual glioblastoma patients in vivo: a mathematical modeling approach. Phys Med Biol. 2010;55:3271–85.
 19.
Folkman J, Hochberg M. Selfregulation of growth in three dimensions. J Exp Med. 1973;138:745–53.
 20.
Almog N, Henke V, Flores L, Hlatky LR, Kung AL, Wright RD, et al. Prolonged dormancy of human liposarcoma is associated with impaired tumor angiogenesis. FASEB J. 2006;20:947–9.
 21.
Hahnfeldt P, Panigrahy D, Folkman J, Hlatky LR. Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy. Cancer Res. 1999;59:4770–5.
 22.
Freyer JP, Sutherland RM. Regulation of growth saturation and development of necrosis in EMT6/Ro multicellular spheroids by the glucose and oxygen supply. Cancer Res. 1986;46:3504–12.
 23.
Dale RG, Jones B: Radiobiological Modelling in Radiation Oncology. British Inst of Radiology; 2007.
 24.
Chougule A, Supe SJ. Linear quadratic modelestimation of alpha/beta ratio for mucosal reaction. Strahlenther Onkol. 1993;169:427–30.
 25.
Fowler JF. The linearquadratic formula and progress in fractionated radiotherapy. Br J Radiol. 2014;62:679–94.
 26.
Yorio JT, Xie Y, Yan J, Gerber DE. Lung cancer diagnostic and treatment intervals in the United States: a health care disparity? J Thorac Oncol. 2009;4:1322–30.
 27.
Wang P, Feng Y. A mathematical model of tumor volume changes during radiotherapy. Sci World J. 2013;2013:1–5.
 28.
Park LJ, Park CH, Park C, Lee T. Application of genetic algorithms to parameter estimation of bioprocesses. Med Biol Eng Comput. 1997;35:47–9.
 29.
Shibamoto Y, Hara M: Radiobiology of normal lung tissue and lung tumours. Med Radiol Radiat Oncol 2005:59–65.
 30.
Maciejewski B, Withers HR, Taylor JM, Hliniak A. Dose fractionation and regeneration in radiotherapy for cancer of the oral cavity and oropharynx: tumor dose–response and repopulation. Int J Radiat Oncol Biol Phys. 1989;16:831–43.
 31.
Fowler JF, Chappell R. Nonsmall cell lung tumors repopulate rapidly during radiation therapy. Int J Radiat Oncol Biol Phys. 2000;46:516–7.
 32.
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;72:957–8.
 33.
Chi A, Wen S, Liao Z, Fowler J, Xu J, Nguyen NP, et al. What would be the most appropriate α/β ratio in the setting of stereotactic body radiation therapy for early stage nonsmall cell lung cancer. Biomed Res Int. 2013;2013:391021.
 34.
Buch K, Peters T, Nawroth T, Sänger M, Schmidberger H, Langguth P. Determination of cell survival after irradiation via clonogenic assay versus multiple MTT Assay  A comparative study. Radiat Oncol. 2012;7:1.
 35.
Wee CW, Sung W, Kang HC, Cho KH, Han TJ, Jeong BK, et al. Evaluation of variability in target volume delineation for newly diagnosed glioblastoma: a multiinstitutional study from the Korean Radiation Oncology Group. Radiat Oncol. 2015;10:137.
 36.
Sause W, Kolesar P, Taylor IV S, Johnson D, Livingston R, Komaki R, et al. Final results of phase III trial in regionally advanced unresectable nonsmall cell lung cancer: Radiation Therapy Oncology Group, Eastern Cooperative Oncology Group, and Southwest Oncology Group. Chest. 2000;117:358–64.
 37.
Unger K. Integrative radiation systems biology. Radiat Oncol. 2014;9:21.
 38.
Eschrich SA, Pramana J, Zhang H, Zhao H, Boulware D, Lee JH, et al. A gene expression model of intrinsic tumor radiosensitivity: prediction of response and prognosis after chemoradiation. Int J Radiat Oncol Biol Phys. 2009;75:489–96.
 39.
Eschrich SA, Fulp WJ, Pawitan Y, Foekens JA, Smid M, Martens JWM, et al. Validation of a radiosensitivity molecular signature in breast cancer. Clin Cancer Res. 2012;18:5134–43.
Acknowledgements
This publication is supported by the Personalized Medicine Award 09335041405 from the DeBartolo Family Personalized Medicine Institute Pilot Research Awards in Personalized Medicine (PRAPM). We thank Rachel Walker for critical reading of the manuscript.
Author information
Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
SP carried out the mathematical modeling and numerical simulation. JP carried out the mathematical modeling and numerical simulation and helped to draft the manuscript. EM, JC, JFTR, KL, JKL, RM, LB participated in study design and coordination and helped to draft the manuscript. HE conceived of the study, and participated in its design and coordination and helped to draft the manuscript. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
The Creative Commons Public Domain Dedication waiver (https://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated in a credit line to the data.
About this article
Cite this article
Prokopiou, S., Moros, E.G., Poleszczuk, J. et al. A proliferation saturation index to predict radiation response and personalize radiotherapy fractionation. Radiat Oncol 10, 159 (2015). https://doi.org/10.1186/s130140150465x
Received:
Accepted:
Published:
Keywords
 Proliferation saturation index
 Personalized radiotherapy
 Logistic tumor growth
 Mathematical model
 Hyperfractionation