A tumour control probability model for radiotherapy of prostate cancer using magnetic resonance imaging-based apparent diffusion coefficient maps

Radiotherapy and Oncology xxx (2016) xxx–xxx

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Original article

A tumour control probability model for radiotherapy of prostate cancer using magnetic resonance imaging-based apparent diffusion coefficient maps Oscar Casares-Magaz a,⇑, Uulke A. van der Heide b, Jarle Rørvik c,d, Peter Steenbergen b, Ludvig Paul Muren a a Department of Medical Physics, Aarhus University Hospital/Aarhus University, Denmark; b Department of Radiation Oncology, The Netherlands Cancer Institute, Amsterdam, The Netherlands; c Department of Clinical Medicine, University of Bergen; and d Department of Radiology, Haukeland University Hospital, Bergen, Norway

a r t i c l e

i n f o

Article history: Received 20 April 2015 Received in revised form 13 January 2016 Accepted 12 February 2016 Available online xxxx Keywords: Prostate cancer Tumour control probability (TCP) Apparent diffusion coefficient (ADC) Magnetic resonance imaging (MRI) Cell density

a b s t r a c t Background and purpose: Standard tumour control probability (TCP) models assume uniform tumour cell density across the tumour. The aim of this study was to develop an individualised TCP model by including index-tumour regions extracted form multi-parametric magnetic resonance imaging (MRI) and apparent diffusion coefficient (ADC) maps-based cell density distributions. Materials and methods: ADC maps in a series of 20 prostate cancer patients were applied to estimate the initial number of cells within each voxel, using three different approaches for the relation between ADC values and cell density: a linear, a binary and a sigmoid relation. All TCP models were based on linearquadratic cell survival curves assuming a/b = 1.93 Gy (consistent with a recent meta-analysis) and a set to obtain a 70% of TCP when 77 Gy was delivered to the entire prostate in 35 fractions (a = 0.18 Gy 1). Results: Overall, TCP curves based on ADC maps showed larger differences between individuals than those assuming uniform cell densities. The range of the dose required to reach 50% TCP across the patient cohort was 20.1 Gy, 18.7 Gy and 13.2 Gy using an MRI-based voxel density (linear, binary and sigmoid approach, respectively), compared to 4.1 Gy using a constant density. Conclusions: Inclusion of tumour-index information together with ADC maps-based cell density increases inter-patient tumour response differentiation for use in prostate cancer RT, resulting in TCP curves with a larger range in D50% across the cohort compared with those based on uniform cell densities. Ó 2016 Elsevier Ireland Ltd. All rights reserved. Radiotherapy and Oncology xxx (2016) xxx–xxx

Current treatment options for localised prostate cancer include radical prostatectomy, brachytherapy and external beam radiotherapy (RT), with 60% of all men diagnosed with prostate cancer being referred to RT at some stage [1]. Different RT dose and fractionation regimes have been explored, including hyper-, standardand hypo-fractionated treatment. Long-term outcomes for these schedules in terms of 5- or 10-year biochemical disease-free survival of between 70% and 80% have been reported [2–4]. These treatment protocols have prescribed a homogeneous radiation dose to the entire prostate, regardless of tumour grade or volume. More recently, RT dose escalation for prostate cancer has been pursued, enabled by the development of new RT planning and delivery techniques, such as intensity-modulated RT (IMRT) or arc-based therapies, in combination with image-guidance for increased treatment precision. These techniques have allowed for

⇑ Corresponding author at: Aarhus University Hospital/Aarhus University, Department of Medical Physics, Nørrebrogade 44, Building 5, 8200 Aarhus, Denmark. E-mail address: [email protected] (O. Casares-Magaz).

the delivery of a higher radiation dose to the entire organ [2–5], or only inside part of the prostate (up to 95 Gy) [6], while maintaining or even decreasing the doses delivered to surrounding organs at risk (OARs). Several trials have documented long-term disease-free survival rates around 90% following dose-escalated RT [2–4]. Under the hypothesis that local disease control is also related to the risk of distant metastases and mortality, further dose escalation of the entire prostate is challenging while keeping the risk of treatment-induced morbidities in the OARs at acceptable levels. Trials such as FLAME [6] are therefore investigating the benefit of a boost to the macroscopic tumour within the prostate. In the experimental arm of this randomised trial the dose is escalated to 95 Gy (delivered in 35 fractions, 5 fractions per week) to the macroscopically visible tumour while the remaining part of the prostate receives 77 Gy concomitantly. Tumour control probability (TCP) modelling is an important tool to evaluate the effects of RT strategies addressing dose fractionation, escalation, boosting or painting. Conventional TCP modelling studies have usually focused on exploration of alternative

http://dx.doi.org/10.1016/j.radonc.2016.02.030 0167-8140/Ó 2016 Elsevier Ireland Ltd. All rights reserved.

Please cite this article in press as: Casares-Magaz O et al. A tumour control probability model for radiotherapy of prostate cancer using magnetic resonance imaging-based apparent diffusion coefficient maps. Radiother Oncol (2016), http://dx.doi.org/10.1016/j.radonc.2016.02.030

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ADC based TCP model for prostate cancer radiotherapy

fractionation schemes and prediction of isoeffect relations, using the Linear Quadratic (LQ) model [7–12]. These TCP studies have been based on assumptions of homogeneous (radio)biological characteristics within the tumour, usually in combination with a treatment that delivered uniform dose distributions to the target volumes [13]. In recent years, some TCP modelling studies have explored the benefit of non-uniform irradiation of the tumour, boosting areas with lower radio-sensitivity (e.g. hypoxic volumes) [14–16]. More recently, functional imaging has been incorporated into radiobiological models in order to add specific tumour features that may improve modelling accuracy [14,17,18]. In particular, some TCP modelling studies have investigated assigning different oxygen enhancement ratios (OERs) for different cell subpopulations [19–21]. So far, no study has explored the importance of introducing tumour-index volumes and image-based cell density information into TCP models. The aim of the present study was therefore to develop a method to include tumour-index regions obtained from multi-parametric magnetic resonance imaging (MRI) and cell density distributions estimated from apparent diffusion coefficient (ADC) maps in TCP models. Different approaches for the relation between the ADC values and cell density inside the prostate were explored. Finally, the model was applied on realistic dose distributions for both conventional, homogenous dose distributions as well as distributions with a focal boost such as applied in the FLAME trial.

the index lesion, and therefore the number of cells, based on the ADC value and on the above assumptions (see Supplementary material A for details). The second approach was a binary model where we considered a homogeneous volumetric cell density for all voxels inside the index lesion. To compute this volumetric cell density we assigned for all voxels the mean ADC value inside the index volume found for the whole patient group; then the cell density inside the index volume was calculated using the aforementioned linear relation between ADC and cell density. The third approach was based on previous TCP studies [11,19,26,27], where volumetric tumour cell densities in the range of 105 to 107 cell/cm3 were used. We assumed a sigmoid relation between ADC and this cell density range (Fig. 1b), where conventional values for volumetric cell density defined the variation range. TCP calculations were carried out for the linear, the binary and the sigmoid models for the voxels inside the index lesion, while the remaining prostate volume was considered to have a constant cell density of 105 cell/cm3. A number of other approaches for using the ADC maps to obtain cell densities were also explored (see Supplementary material A for details). These three ADC-based approaches were compared to the conventional approach where a constant tumour cell density of 107 cells/cm3 across the whole prostate was assumed for all patients.

Materials and methods

TCP models based on cell density distributions

Patient and image materials All patients included in this study underwent MRI acquisition using an integrated endorectal and pelvic phased-array coil using a 1.5 T whole body MRI unit Siemens Avanto (Siemens Medical Systems, Erlangen, Germany). Diffusion weighted imaging (DWI) covering the entire prostate and seminal vesicles was performed using axial scans with a section-thickness of 3 mm, an intersection gap of 0.8 mm and an acquisition time of 5:33 min. The DWI sequence was acquired using an echo planar sequence (repetition time 3000 ms, echo time 72 ms, iPAT 2, a matrix of 128  128 pixels, field of view 128  128 mm and three b-values: 50, 400 and 800 s/mm2), from which ADC maps were calculated by the scanner software using all b-values. The ADC maps had an image size of 256  256 pixels, with a 250  250 mm field of view for all slices where the prostate was present. Further details of image acquisition, post-processing and patient characteristics were described by Reisæter et al. [22]. Using the post-processing tool Oxiris the index lesion and the whole prostate were delineated for each patient by one radiologist with more than 10 years of experience in evaluating multi-parametric MRI for prostate cancer. Voxel-based cell density estimation from ADC map values Estimation of cell densities based on ADC values was done using three approaches for voxels inside the index lesion; the first was based on a published linear relation between ADC and cell density [23], where a significant negative correlation between ADC values and cell density in prostate cancer was found (correlation coefficient R = 0.697, p < 0.01). The mean (±standard deviation, SD) values obtained in that study for the ADC and cell density at the prostate carcinoma areas were 1.43
10 3 ± 0.19
10 3 mm2/s and 19.8% ± 5.3%, respectively; similar relations have also been found by others [23,24]. Since the cell density was given as the fraction of cell dish surface occupied by tumour cells [23], we assumed a circular shape for the cells in the plate, with an average diameter of 13 lm, based on the results of Park et al. [25] (Fig. 1a). Finally we computed a volumetric cell density for each voxel inside

All TCP modelling was based on the Linear-Quadratic (LQ) cell survival curves, combined with a Poisson based TCP model (see Supplementary material B for the basic equations). TCP calculations were performed assuming a/b = 1.93 Gy, based on the recent meta-analysis of Vogelius and Bentzen [28]. The value for a was chosen such that the mean TCP value for the linear model was 70% (for the whole patient cohort) when the dose delivered to the entire prostate of each patient was 77 Gy in 2.2 Gy/fraction (as in the standard arm of the FLAME trial). These survival ratios were based on a range of studies reporting control levels of biochemical free-disease survival rates after 5 years in the range of 69–72%, with dose schedules of 68–79.3 Gy [3,4,29]. To account for variations in the radiobiological parameters across the different voxels, we introduced a positive term into the survival fraction of the LQ model. This variation term involved higher powers of the product d
n, where d is the dose per fraction (2.2 Gy) and n is the number of fractions. This term is governed by the standard deviation of the radiobiological parameters (ra = rb = 15%) and by the correlation factor between them (r = 0.01) [9,30] (see Supplementary material C for details). We also explored inclusion of a term for repopulation in the LQ model in order to show the generalisability of our approach, but these were not included in the final model (again see Supplementary material D for details). Finally, we also explored the introduction of variations in the hypoxic volumes across the tumours. Model application using FLAME target dose distributions For all 20 patients we simulated dose distributions for the prostate corresponding to both the standard and the experimental arm of the FLAME trial [6]. In the experimental arm, index lesion voxels were assigned a dose of 2.7 Gy, while all other voxels outside the index lesion were assigned a dose of 2.2 Gy; in the standard arm all voxels were assigned the latter dose. For voxels near the border of the index lesion were assigned dose values according to gradients, where prostate to normal tissue gradients were steeper (38.0 cGy/mm) compared to index to prostate gradients (17.4 cGy/mm), in order to simulate the existence of constraints

Please cite this article in press as: Casares-Magaz O et al. A tumour control probability model for radiotherapy of prostate cancer using magnetic resonance imaging-based apparent diffusion coefficient maps. Radiother Oncol (2016), http://dx.doi.org/10.1016/j.radonc.2016.02.030

3

ADC - Cell Density Linear Relation

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Linear relation ADC-CD% Data set from Gibbs

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O. Casares-Magaz et al. / Radiotherapy and Oncology xxx (2016) xxx–xxx

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Fig. 1. (Left panel) Scatter data set extracted from Gibbs et al. [23] (scatter black), and linear relation used between ADC values and superficial cell density (red line). (Right panel) Sigmoid relation used between ADC maps and volumetric cell density. The reader should be aware there is a two order of magnitude difference in the volumetric cell density scale (Y-axis) between panels.

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1.4 0 Fig. 2. (Left panel) ADC map for one of the MR slice through the centre of the prostate of one the patients. (Right panel) Simulated dose per fraction (Gy) distribution according with FLAME protocol, for voxels inside the prostate of one of the patients analysed. Red dark area inside the prostate corresponds with the index lesion. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Table 1 Volume and ADC values for the prostate and index lesion, and initial number of cells for the different approaches of cell density for the 20 patients. For the different approaches of the initial number of cells, in brackets can be seen the percentage of cells allocated in the index lesion. (*Assuming 105 cell/cm3 for prostate volume outside index lesion).

Prostate volume (cm3) Index volume (cm3) Percentage of index volume (%) Mean ADC index (mm2/ms) Mean ADC outside index (mm2/ms) N0 linear ADC index (108 cells)⁄ N0 binary ADC index (108 cells)⁄ N0 sigmo ADC linear (108 cells)⁄ N0 q = 107 cell/cm3 (108 cells)

Mean

SD

Median

Range

44.5 3.7 8.2 1.0 1.3 31.3 (99.6) 32.6 (99.7) 0.9 (71.6) 4.4 (8.2)

12.9 3.8 7.9 0.2 0.1 4.0 (0.4) 3.3 (0.4) 0.3 (25.6) 1.3 (7.9)

40.4 2.5 6.2 1.0 1.3 23.4 (99.2) 22.1 (99.2) 0.2 (81.6) 4.0 (6.1)

32.7–83.5 0.3–17.0 0.9–33.4 0.8–1.3 1.0–1.5 2.5–172.6 (98.5–99.9) 2.8–149.9 (98.6–99.9) 0.0–1.4 (10.1–99.3) 3.2–8.4 (0.9–33.4)

outside the prostate gland (for the rectum and bladder) (Fig. 2). TCP calculations were done for both arms of the trial using cell density distributions obtained with the linear model. All cell density calculations, TCP calculations, as well as simulations of dose distributions were developed using in–house software created in Matlab v.R2011b (The MathWorks, Inc., Natick, MA, USA).

Results The variation in volumes for the prostate and the index lesion, the ADC distribution as well as the initial number of cells resulting from the different approaches used for the volumetric cell density are shown in Table 1. The ADC values inside the index lesion ranged from 0.2
103 to 2.4
103 mm2/ms, while ADC values between

Please cite this article in press as: Casares-Magaz O et al. A tumour control probability model for radiotherapy of prostate cancer using magnetic resonance imaging-based apparent diffusion coefficient maps. Radiother Oncol (2016), http://dx.doi.org/10.1016/j.radonc.2016.02.030

ADC based TCP model for prostate cancer radiotherapy

0 and 3.1
103 mm2/ms were seen outside the index lesion. The population-average of the mean ADC values inside the index lesion (1.0
103 mm2/ms) was significantly lower than the mean ADC outside the index (1.3
103 mm2/ms) (paired t-test, p < 0.01). With both the linear and binary model, a higher fraction (range 98.5– 99.9%) of the initial number of cells was located inside the index lesion (relative to the remaining prostate volume) compared to either using the sigmoid relation or a constant cell density. Using the linear relation and the binary model also resulted in a higher value of the total initial number of cells (31.3
108 and 32.6
108 cells on average, respectively) than according to the sigmoid (0.9
108 cells on average) or to the homogeneous cell density (4.4
108 cells on average). Individual data together with initial number of cells for other explored approaches are shown in detail in Supplementary Table 1. Overall, TCP curves based on ADC maps showed much larger differences between individuals than those assuming a uniform cell density. Using the linear ADC vs. cell density relation for the index lesion, the range of D50% (the dose required to reach 50% TCP) across the population was 20.1 Gy. Using the binary model assuming two different homogeneous cell densities for the index (8.8
108 cells/cm3) and the remaining prostate (105 cells/cm3), the range of D50% across the population was 18.7 Gy, while using the sigmoid relation for the index lesion, the range of D50% was 13.2 Gy, and 4.1 Gy when using a constant density (of 107 cells/ cm3) for the entire prostate (Fig. 3). In all calculations, the linear component (a) of the LQ model was set to 0.18 Gy 1 to obtain a 70% mean TCP for a target receiving 77 Gy in 35 fractions when using the linear relation between the ADC values and the volumetric cell density (Fig. 3). A mean TCP value over the patient cohort of 70% was set using the dose distributions of the standard arm protocol (77 Gy in 2.2 Gy/fraction, in 35 fractions, 5 times per week) assuming the linear relation between ADC and cell density. For the dose distributions of the experimental arm (77 Gy for normal prostate and 95 Gy for the index lesion in 35 fractions) the mean TCP increased to 97%. In addition the D50% 95% CI for the TCP curves of all the patients was reduced from 18.4 Gy for the standard arm to 8.9 Gy for the experimental arm (Fig. 4). The inclusion of the repopulation term in the LQ model (see Supplementary material D) led to differences between 1.2% and 3.1% in the D50% across all TCP calculations. Inclusion and modifica-

Mean TCP FLAME Trial 1

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Dose to the prostate (Gy) Fig. 4. Mean (solid line) and 95% CI (shadow area) computed for the TCP curves of the standard (grey) and experimental arm (black) of the FLAME trial. Vertical dashed line corresponds with the prescribed 35 fractions. Upper x-axis indicates the number of fraction, while lower x-axis indicates the dose received by the prostate outside the index lesion in 2.2 Gy/fraction.

tion of the hypoxic volumes (see Supplementary material E) on the other hand caused large differences in the TCP, while varying the specific location of those volumes inside the prostate only led to minor variations (less than 0.1 Gy) in D50%. Discussion To the best of our knowledge this is the first study developing a TCP model including voxel-wise cell density and tumour-index information derived from MRI. In particular, we estimated cell density distributions from ADC maps to determine the initial number of cells in each voxel inside the index lesion volume. Across the cohort included in this study, the ADC values inside the index lesion were lower than outside, in agreement with previous publications [23,24,31]. This is consistent with an increased cell density inside the GTV. The Poisson-based TCP model used LQ cell survival

Mean TCP

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Fig. 3. (Left panel) Individual TCP curves for the 20 patients using the LQ model, computed for the different approaches of the voxel cell density: constant cell density (red curves), sigmoid relation between ADC and cell density inside the index lesion (grey curves), binary model (yellow) and linear relation between the ADC values and cell density inside the index lesion (green curves). (Right panel) Mean values (solid line) and 95% CI (shadow areas) computed for the individual TCP curves in the left panel. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Please cite this article in press as: Casares-Magaz O et al. A tumour control probability model for radiotherapy of prostate cancer using magnetic resonance imaging-based apparent diffusion coefficient maps. Radiother Oncol (2016), http://dx.doi.org/10.1016/j.radonc.2016.02.030

O. Casares-Magaz et al. / Radiotherapy and Oncology xxx (2016) xxx–xxx

relations, where a term to account for the variation in the radiobiological parameters was included. Compared to classical modelling approaches [19,20], the inclusion of cell density and index volume information yielded an increased TCP curve variation across patients. In this study we have explored several approaches for the relation between ADC values and cell density, including the assumption used in most previous modelling studies of a constant and uniform cell density. The linear relation exploits the voxel-wise relation between ADC values and volumetric cell density within the index lesion. The binary model includes mean patient cohort ADC values to generate a homogeneous high cell density inside the index, while the sigmoid relation was used to calculate TCPs based on conventional values for the volumetric cell densities, but modulated by ADC values inside index volumes. All these models consider higher cell densities inside the index, as has been observed by several authors [23,24,26]. This study has shown how the variation in dose–response relations (TCP) between patients is increased when individual tumour information is included in the models. The range of D50% increased up to threefold for the patient cohort when either the linear relation between ADC and cell density or the binary model were used compared to using a constant and uniform cell density; and twofold when using the sigmoid relation. The variation between model approaches, as well as patients, was assessed at the D50% level since this is a well-defined part on the TCP curve. Finally, the linear model adds a slight difference compared to the binary model (7.0% range increment for the D50%). This indicates that the index volume mainly governs individual TCP results, while ADC variability inside the index plays a lesser role. Binary and linear relations between ADC and cell density resulted in cell density values inside the index lesion which were two or three orders of magnitude higher than previous studies [11,19,26]. It should still be pointed out that the linear relation of Gibbs et al. [23] was between ADC and a surface-based and not a volumetric density, and hence a conversion to the latter was needed. In addition, linear and binary models provided realistic tumour cell distributions inside the prostate based on the number of tumour cells allocated inside vs. outside the index [26]. TCP differences owing to variable cell density compared to previous modelling studies were clearly illustrated using the sigmoid model, with the cell density ranging between the upper and lower bound of previous published modelling assumptions [11,19,26,27]. The sigmoid model includes MRI information (tumour-index volumes and ADC-based cell densities) while using classical volumetric cell density ranges instead of those extracted from histopathological studies [23,24]. Then, the sigmoid model exposes inter-patient TCP variability when classical cell density values are based on MRI information. Overall, incorporating ADC maps information together with index volumes were found to lead to a larger variation for the individual TCP curves, potentially carrying information that can be useful for optimisation and evaluation of individualised dose boosting/painting strategies for prostate cancer. In a previous prostate TCP modelling paper of Walsh et al. [26] a = 0.25 Gy 1 and a/b = 2.48 Gy were used; the differences relative to our values are caused by different assumptions in the model, e.g., they assumed a mean prostate volume of 36 cm3 and a constant 10% fraction of dominant intraprostatic lesion. Indeed, there are still uncertainties in the radiobiological parameters for prostate cancer [28,32]. Radiobiological parameters for specific sub-populations are determined by the cell environment, the dynamics of repopulation, the local oxygen pressure as well as several other factors. In Supplementary material E, we explored the effects of modifying the radiobiological parameters for a percentage of the total number of voxels according to different tumour risks. The modifi-

5

cation of radiobiological parameters, even though only for 5% of the volume, was seen to introduce a considerable modification in the TCP curves (Supplementary Fig. 2), while the specific location of these volumes against different cell density values spread all over the prostate implied a variation of 0.1 Gy for the D50% range (Supplementary Fig. 2). In this direction, Titz et al. [33] developed a computational tumour modelling framework where they simulated effects of anti-angiogenic therapy based on multiparametric functional imaging. They underlined the importance of including tumour biology in development of treatment strategies, for drug and radiation treatments. In the aforementioned exercise in the present paper, we have illustrated how sensitive prostate tumours are to variations in radiobiological parameters, due to the low a/b, and its implications in the final TCP. All patients enrolled in this study underwent radical prostatectomy as curative treatment for prostate cancer. These patients were diagnosed with low and intermediate prostate cancer; however, external beam radiotherapy with dose escalation is recommended for intermediate and high prostate cancer. This could be a limitation for higher risk of prostate cancers involved in RT, therefore index volumes and initial number of cells might differ from those calculated in this study. Still the same principles would be applicable although differences in radiobiological parameters compared to those showed in this study may be expected. ADC map accuracy is dependent on the signal-to-noise ratio, which increases with the main magnetic field (B0 = 1.5T). This study was performed using a single snapshot ADC maps for each patient. However, it has been shown that ADC values can vary in time due to changes in cellularity due to external agents such as radiotherapy [34]. Indeed, ADC uncertainties, intrinsic to the acquisition process, may lead to errors in cell density estimation, which in turn, may lead to errors in the final TCP computation. For instance, a 1% ADC deviation may lead up to 4% in the TCP for the fractionation schedules we studied. In addition, index imagebased volumes estimations are affected by inter-observer variability, with direct implication for our method. Also, due to the multifocal nature of prostate cancer, small tumour aggregations are expected outside the index lesion and might not be accounted. The use of other functional imaging MRI sequences may allow for an estimation of the radiosensitivity of different cell subpopulations, and this might be taken advantage of to further improve our TCP model. This could enable us to better identify potentially failing patients as well as to improve treatment design to increase individual TCP. In addition, this method could also give us the opportunity to decrease dose to surroundings organs at risk, resulting in a lower complication probability. Indeed, in the present study we have only considered the estimated TCP when comparing the standard and the experimental arm of the dose escalation trial. In future studies, we will also evaluate the dose delivered to normal tissues, which may be the challenging point when dose escalation is performed, in particular for index tumours close to the dose-limiting normal tissues (rectum [35] and bladder). In conclusion, in this study we have presented a method to include patient-specific cellularity information together with volumes of tumour-indexes into TCP modelling. By acquiring prostate ADC maps, defining index lesions and correlating them with cell density values, this method distinguishes TCP patient individualities, and may predict clinical outcome differences or provide potential failure case information in the early stage of the planning process.

Conflict of interest statement None declared.

Please cite this article in press as: Casares-Magaz O et al. A tumour control probability model for radiotherapy of prostate cancer using magnetic resonance imaging-based apparent diffusion coefficient maps. Radiother Oncol (2016), http://dx.doi.org/10.1016/j.radonc.2016.02.030

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ADC based TCP model for prostate cancer radiotherapy

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Please cite this article in press as: Casares-Magaz O et al. A tumour control probability model for radiotherapy of prostate cancer using magnetic resonance imaging-based apparent diffusion coefficient maps. Radiother Oncol (2016), http://dx.doi.org/10.1016/j.radonc.2016.02.030