 Research
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Forecasting tumor and vasculature response dynamics to radiation therapy via image based mathematical modeling
Radiation Oncology volume 15, Article number: 4 (2020)
Abstract
Background
Intraand intertumoral heterogeneity in growth dynamics and vascularity influence tumor response to radiation therapy. Quantitative imaging techniques capture these dynamics noninvasively, and these data can initialize and constrain predictive models of response on an individual basis.
Methods
We have developed a family of 10 biologicallybased mathematical models describing the spatiotemporal dynamics of tumor volume fraction, blood volume fraction, and response to radiation therapy. To evaluate this family of models, rats (n = 13) with C6 gliomas were imaged with magnetic resonance imaging (MRI) three times before, and four times following a single fraction of 20 Gy or 40 Gy whole brain irradiation. The first five 3D time series data of tumor volume fraction, estimated from diffusionweighted (DW) MRI, and blood volume fraction, estimated from dynamic contrastenhanced (DCE) MRI, were used to calibrate tumorspecific model parameters. The most parsimonious and well calibrated of the 10 models, selected using the Akaike information criterion, was then utilized to predict future growth and response at the final two imaging time points. Model predictions were compared at the global level (percent error in tumor volume, and Dice coefficient) as well as at the local or voxel level (concordance correlation coefficient).
Result
The selected model resulted in < 12% error in tumor volume predictions, strong spatial agreement between predicted and observed tumor volumes (Dice coefficient > 0.74), and high level of agreement at the voxel level between the predicted and observed tumor volume fraction and blood volume fraction (concordance correlation coefficient > 0.77 and > 0.65, respectively).
Conclusions
This study demonstrates that serial quantitative MRI data collected before and following radiation therapy can be used to accurately predict tumor and vasculature response with a biologicallybased mathematical model that is calibrated on an individual basis. To the best of our knowledge, this is the first effort to characterize the tumor and vasculature response to radiation therapy temporally and spatially using imagingdriven mathematical models.
Background
Radiation therapy has been a nearly ubiquitous treatment option for cancers for over a century [1]. During that time, there has been steady improvements in treatment delivery driven by increased understanding of radiation biology and engineering advancements in delivery technology. A prime example is for the treatment of gliomas, where complete resection is often practically impossible [2, 3]—necessitating adjuvant radiation (and/or chemo) therapy to treat residual disease. In this setting, developments in radiation therapy technology has facilitated the delivery of more precise irradiation techniques to optimize the dose delivered to the tumor while simultaneously minimizing exposure to healthy tissue. Even with these advances, prognosis for malignant gliomas is overwhelming poor with response to standard therapies typically resulting in disease progression in 7 to 10 months [2]. Emerging imaging [4] and modeling approaches [5,6,7,8], however, may facilitate the improvement of radiation therapy through the assessment of intratumoral heterogeneity, the identification of tumor radiosensitivity, and through in silico trials to optimize therapeutic regimens (e.g., dosing and scheduling) for an individual subject.
Anatomical imaging approaches such as contrastenhanced magnetic resonance imaging (MRI), play a crucial role in identification of treatment volumes and the assessment of response [9]. However, anatomical imaging does not assess physiological or functional properties of the tissue that might be relevant to tumor and radiation biology. Quantitative imaging techniques such as diffusion weighted (DW) MRI and dynamic contrast enhanced (DCE) MRI can provide functional information characterizing changes in tumor cellularity and tissue perfusion, respectively [10]. Both DWMRI and DCEMRI have shown promising potential by providing early imaging biomarkers of response in glioblastoma [11, 12] and have been widely used in other cancers [13,14,15]. The quantitative and noninvasive nature of both DWMRI and DCEMRI makes these techniques wellsuited for mathematical modeling of tumor growth as they can provide the 3D distribution of the tumor cells and vasculature before, during, and after treatment. These temporally defined data sets allow for model initialization and calibration, both of which are required for making tumor specific predictions.
We (and others [6, 16,17,18,19,20,21,22];) have developed a series of increasingly comprehensive biophysical mathematical models of tumor and vasculature growth [5, 23,24,25] to provide individualized forecasts of response to radiation therapy using noninvasive quantitative imaging data. Our overall hypothesis, is that biologicallyrelevant models, informed and calibrated from tumorspecific imaging data, may lead to the early prediction of response, significant improvements in tumor control through the pretreatment optimization of therapy regimens, and the adaptation of treatment regimens by accounting for the spatiotemporal variations in tumor response [26]. Thus, we seek to incorporate the effects of radiation therapy spatially on cellularity and vascularity.
Historically, the linear quadratic [27] (LQ) model has played a fundamental role in the design and delivery of radiation therapy plans. More recently, there has been an increased interest in the development of tumor (or patient) specific approaches to characterize and/or predict response to radiation therapy [8, 28,29,30,31]. In the preclinical setting, we previously developed [5] a model of response to radiation therapy consisting of immediate (i.e., instantaneous reduction in tumor cells) and longterm (reduced proliferation) effects of radiation therapy. In that study, we demonstrated that combining both immediate and longterm effects outperformed models consisting of single effects. Additionally, we have previously demonstrated that both DWMRI and DCEMRI data could be used to initialize and calibrate a biophysical model of tumor growth and angiogenesis in the absence of treatment that resulted in accurate volume and voxelwise predictions of tumor and blood volume fractions [25]. At the clinical level, one approach by Rockne et al. [6] leveraged anatomical MRI and hypoxiasensitive positron emission tomography images collected in glioblastoma multiforme patients to determine patientspecific radiosensitivity parameters. Rockne et al. demonstrated that incorporation of hypoxiamediated resistance resulted in more accurate predictions of the bulk tumor volume. Prokopiou et al. [8] proposed the use of the proliferation saturation index or simply the ratio of the pretreatment tumor volume to a patientspecific carrying capacity to predict and personalize radiotherapy fractionation. In an in silico clinical trial, Prokopiou et al. demonstrated that patients with proliferation saturation indices ranging from 0.45 to 0.9 were more likely to benefit from the proposed hyperfractionation protocol as opposed to the standard fractionation protocol. These promising approaches represent real progress towards the development of modelbased adaptive radiation therapy. However, we posit that tumor cellularity and volumetric measures alone may be insufficient to capture the spatially and biologically heterogenous response to radiation therapy.
In this contribution, we extend our image driven tumor induced angiogenesis model to incorporate the effects of radiation therapy on cell death and the net proliferation rate. We first develop a family of 10 biologicallyrelevant models to investigate different approaches to characterize the response of the tumor and vasculature to radiation therapy. We then calibrate all 10 models using quantitative DWMRI and DCEMRI data collected in a murine model of glioma growth on an animalspecific basis. The most parsimonious and well calibrated model is then chosen by a statistical model selection scheme. Finally, we assess the model’s ability to accurately predict response in volume and voxelwise measures of tumor and blood volume fraction as observed in posttreatment MRI measures.
Methods
Biophysical model of tumor and vasculature growth
Our mathematical model is built upon the wellstudied reactiondiffusion model framework that has been extensively applied to preclinical [23, 32, 33] and clinical models [29] of glioma growth. In a single species model (i.e., considering only tumor cells of one type) the reaction diffusion model describes the spatial and temporal change in tumor cell number due to proliferation (i.e., the reaction term) and due to the outward movement (i.e., the diffusion term) of tumor cells. In our previous efforts, we extended the standard reactiondiffusion model to incorporate the effect of local tissue stress on tumor cell diffusion [24] as well as characterize the spatialtemporal evolution of both tumor cells and vasculature [25]. We present the salient features of this model system here, while Table 1 summarizes the model parameters and variables and their sources. (A more detailed description can be found elsewhere [24, 25, 34].) The spatial and temporal evolution of the tumor cell fraction is given by Eq. (1):
where \( {\phi}_T\left(\overline{x},t\right) \) is the tumor cell fraction at threedimensional position \( \overline{x} \) and time t, \( {D}_T\left(\overline{x},t\right) \) is the tumor cell diffusion coefficient, \( {\theta}_{T,V}\left(\overline{x},t\right) \) is the summation of tumor and the blood volume fraction carrying capacities, \( {\phi}_V\left(\overline{x},t\right) \) is the blood volume fraction, k_{p,T} is the tumor cell proliferation rate, and \( {\theta}_T\left(\overline{x},t\right) \) is the tumor cell carrying capacity (i.e., the maximum packing fraction that a voxel can functionally support). Tumor cell diffusion is affected by both local tissue mechanical properties, via \( {D}_T\left(\overline{x},t\right) \), and by the space occupied by tumor associated vasculature. The effect of local tissue properties is incorporated by exponentially dampening \( {D}_T\left(\overline{x},t\right) \) as the local mechanical stress increases according to:
where D_{T,0} represents the uninhibited tumor cell diffusion coefficient, λ_{1} is the stresstumor cell diffusion coupling constant, and \( {\sigma}_{vm}\left(\overline{x},t\right) \) is the von Mises stress which reflects the total stress experienced for a given section of tissue. The \( {\sigma}_{vm}\left(\overline{x},t\right) \) is determined by solving for tissue displacement, \( \overrightarrow{u} \), using the linear elastic, isotropic equilibrium equation (Eq. (3)):
where G is the shear modulus,ν is the Poisson’s Ratio, and λ_{2} is the second coupling constant. Literature values are used to assign tissue specific [34] G and v. \( {\theta}_T\left(\overline{x},t\right) \), and \( {\theta}_{T,V}\left(\overline{x},t\right) \), are temporally and spatially varied in response to changes in \( {\phi}_V\left(\overline{x},t\right) \) using the following linear relationship:
where θ_{min} to θ_{max} represents the range of expected carrying capacity values, and ϕ_{V, thresh} represents a critical value for \( {\phi}_V\left(\overline{x},t\right) \) that would begin to change the number of cells a voxel can support. θ_{min} is assigned as the lowest volume fraction within the tumor.
The spatialtemporal evolution of \( {\phi}_V\left(\overline{x},t\right) \), is described with a similar reactiondiffusion model consisting of a diffusion, angiogenesis, and death terms as shown in Eq. (5):
where \( {D}_V\left(\overline{x},t\right) \) is the vascular diffusion coefficient, k_{p,V} is the vascular growth rate, θ_{V} is the blood volume fraction carrying capacity, d is a normalized parameter describing the distance to the periphery of the tumor, and k_{d,V} is the vascular death rate. \( {D}_V\left(\overline{x},t\right) \) is also coupled to local tissue stress as in a similar fashion as \( {D}_T\left(\overline{x},t\right) \). d ranges from 1 for voxels at the periphery of the tumor to 0 for voxels furthest from the periphery.
Modeling response to radiation therapy
We assume the response to radiation therapy is observed over twotime frames: immediate and long term. First, for the immediate response following radiation therapy, some cells die instantaneously due to irreparable damage or early mitotic catastrophe [35, 36]. Second, for long term response following radiation therapy, some cells will have a reduced net proliferation rate due to DNA repair processes or delayed mitotic catastrophe [35, 36]. The immediate effects of radiation therapy are modeled as a direct reduction of \( {\phi}_T\left(\overline{x},t\right) \) at the time of radiation therapy:
where \( {\phi}_{T, postRT}\left(\overline{x},t\right) \) is the postradiation therapy value of the tumor cell fraction, \( {\phi}_{T, preRT}\left(\overline{x},t\right) \) is the preradiation therapy value of the tumor cell fraction, α_{I} is an treatment efficacy parameter for the immediate effects, and C_{I} is one of five coupling approaches (described below). The postradiation effects of \( {\phi}_V\left(\overline{x},t\right) \) are handeled in an identical fashion as Eq. (6). The long term effects R_{LT}(t) of radiation therapy are incorporated in an amended Eq. (1):
Eq. (5) is amended in a similar fashion as Eq. (7). The term R_{LT}(t) is a piecewise function represented by Eq. (8):
where α_{LT} is a treatment efficacy parameter, C_{LT} is one of five coupling approaches (described below), and t_{rt} is the time that radiation therapy is administered.
To systematically investigate different approaches for characterizing the efficacy of radiation therapy spatially or as a function of treatment dose, we implemented five different coupling approaches as shown in Table 2. The logistic growth approach, C_{1}, assumes the efficacy of radiation therapy decreases as \( {\phi}_T\left(\overline{x},t\right) \) approaches \( {\theta}_T\left(\overline{x},t\right) \) due to a slower proliferation (and thus less susceptible) cells. The LQ approach, C_{2}, assumes no spatial variance in treatment efficacy, but relates efficacy to the radiosensitivity parameters (α and β) and the treatment dose (Dose). The vascular coupled approach, C_{3}, relates the treatment efficacy spatially to \( {\phi}_V\left(\overline{x},t\right) \). Here, we assume areas that are wellvascularized (potentially normoxic) and have increased radiosensivity relative to poorlyvascularized (potentially hypoxic) regions. The oxygen enhanced approach, C_{4}, combines elements from C_{2} and C_{3} to spatially vary treatmenty efficacy as a function of \( {\phi}_V\left(\overline{x},t\right) \) while also utilizing a common radiobiology adaptation of the LQ model. While \( {\phi}_V\left(\overline{x},t\right) \) is not a direct measure of tissue oxygenation, we assume that tissue oxygenation is proportional to the blood volume fraction; thus, when \( {\phi}_V\left(\overline{x},t\right) \) is greater than the average pretreatment \( {\phi}_V\left(\overline{x},t\right) \), \( \overline{\phi_{V, pre treatment}} \), the oxygen enhancement ratio (OER) will be greater than 1. Finally, the fifth coupling approach, C_{5}, assumes no spatial variability, and the effect of radiation therapy is evenly applied.
The spatialtemporal evolution of \( {\phi}_T\left(\overline{x},t\right) \) and \( {\phi}_V\left(\overline{x},t\right) \) was deterimined using a finite difference approximation implemented in MATLAB R2018a (Mathworks, Natick, MA) using a fully explicit in time differentiation (time step = 0.01 days) and three dimension in space (Δx = 250 μm, Δy = 250 μm, Δz = 1000 μm) central difference spatial differentiation. No flux (Neumann) boundary conditions were used for \( {\phi}_T\left(\overline{x},t\right) \) and \( {\phi}_V\left(\overline{x},t\right) \) at the skull boundary. The boundary condition for \( \overrightarrow{u} \) was assumed to be zero displacement in the normal direction, while it was assumed that the tissue in the tangential directions was free to move (i.e., slip condition). Complete numerical details on the mechanicallycoupled model, Eq. (3), can be found elsewhere [34].
In vivo experimental methods
All experimental procedures were approved by our Institutional Animal Care and Use Committee. Thirteen female Wistar rats (weights from 240 to 286 g) were anesthetized and inoculated intracranially with 10^{5} C6 glioma cells (obtained from SigmaAldrich, St. Louis, MO, USA). Rats were imaged with MRI on days 10, 12, 14, 16.5, 18.5, 20.5, and 22.5. A permanent jugular catheter was placed in each rat up to 48 h prior to their first MRI study. Rats received a single fraction 20 Gy (n = 5) or 40 Gy (n = 8) of whole brain radiation therapy on day 14.5 using a Therapax DXT 300 xray machine (300 kVp/10 mA, Pantak Inc., East Haven, CT, USA) at a dose rate of 2.3 Gy/min. During the irradiation procedure, the animals were anesthetized with a mixture of 2% isoflurane in 98% oxygen, and lead shielding was used to minimize radiation exposure outside of the brain.
MRI data was acquired using a 9.4 T horizontalbore magnet (Agilent, Santa Clara, CA) with a 38 mm diameter Litz quadrature coil (Doty Scientific, Columbia, SC, USA) over a 32 × 32 × 16 mm^{3} field of view sampled with a 128 × 128 × 16 matrix. Figure 1 summarizes the experimental and computational framework used in this study. Following the initial imaging visit, all subsequent imaging visits were registered using a mutual information based rigid registration algorithm to provide imaging spatial offsets and rotations to minimize retrospective registration [23] (Fig. 1a). The same registration algorithm was used to perform retrospective registration as needed to maximally align the imaging data across time. DWMRI and DCEMRI experiments were performed at each visit. For the DWMRI experiment, a pulsed gradient, fast spin echo sequence with three orthogonal diffusion encoding directions with bvalues of 150, 500, 1100 s/mm^{2}, TR = 2000 ms, TE = 30 ms, number of excitations = 10, Δ = 25 ms, and δ = 2 ms. A standard monoexponential model was fit to the DWMRI data to estimate the apparent diffusion coefficient (ADC) voxelwise within the tumor. ADC measures (Fig. 1a) at each voxel were then used to estimate \( {\phi}_T\left(\overline{x},t\right) \) assuming the linear relationship shown in Eq. (9):
where ADC_{w} is the ADC of free water at 37 °C [37], \( ADC\left(\overline{x},t\right) \) is the ADC value at position \( \overline{x} \) and time t, and ADC_{min} is the minimum ADC observed within the tumor regionsofinterest (ROIs) across all animals. We have previously used Eq. (9) to provide noninvasive estimates of tumor cellularity [5, 24, 38,39,40]; however, we acknowledge that the assumption that all changes in ADC are related to changes in cellularity is a simplification of the complexity of the cellular (e.g., cell size and permeability) and tissue properties (e.g., tortuosity) in the presence and absence of treatment that also contributes to changes in ADC (see discussion in [5, 25]). For the DCEMRI experiment, we first collected a T_{1} map using an inversionrecovery snapshot with TR = 5000 ms, TE = 3 ms, 8 inversion times logarithmically spaced between 200 and 4000 ms, and two averaged excitations. DCEMRI data was then acquired using a spoiled gradient echo sequence with TR = 45 ms, TE = 1.4 ms, two averaged excitations, and a flip angle of 20°. A 200 μL bolus (0.05 mmol kg^{− 1}) of GadoDTPA™ (BioPhysics Assay Lab, Worcester, MA) was injected after 25 image sets were acquired. Relative blood volume fraction, \( {\phi}_V\left(\overline{x},t\right) \), was calculated by computing the ratio of the area under the curve for the concentration of the contrast agent in tissue time course to the arterial input function [41] over the first 60 s (Fig. 1a). Tumor ROIs were identified using a relative enhancement map (postcontrast image divided by precontrast image) derived from DCEMRI data.
Model parameter calibration
In addition to the five coupling scenarios described above (i.e., C_{1} to C_{5}), we considered two additional parameterization approaches. First, we assumed k_{p,V} was assigned as a global parameter (uniform throughout the domain). In a previous study, this model was selected as the model that best balances model fit and model complexity [25]. Second, we considered the case where k_{p,V} was assigned locally within the tumor. With these two additional parameterization approaches, we generated a total of 10 models to calibrate per animal. For the locally varying approach, parameter values were calibrated within the tumor at a subset of the points and then interpolated elsewhere. For example, for a given 3 × 3 voxel subregion within the tumor ROI, the parameter values were calibrated at the corner and center positions, while the remaining four points were interpolated from the nearest calibrated values. This parameter calibration approach both significantly reduced the number of individual parameters that needed to be calibrated while also smoothing or regularizing the parameter field spatially. For models C_{2} and C_{4}, which incorporate the LQ model of response to radiation therapy, α_{I} and α_{LT} are both assigned to 1 and the LQ parameter α is calibrated. The parameter β is calculated assuming an α/β ratio of 14 (assigned from literature values [42]). While the calibration approach (Fig. 1b) is briefly discussed here, the interested reader is referred to previous studies [25, 34] for a more complete description on the algorithm used. To personalize model predictions for individual animals, model parameters were calibrated on an animal specific basis using data from time points 2 to 5 via a hybrid simulatedannealing LevenbergMarquardt algorithm [25]. Briefly, an initial guess of model parameters and the initial conditions at t_{1}, \( {\phi}_T\left(\overline{x},{t}_1\right) \) and \( {\phi}_V\left(\overline{x},{t}_1\right) \), are used to return estimates of \( {\phi}_T\left(\overline{x},{t}_2\to {t}_5\right) \) and \( {\phi}_V\left(\overline{x},{t}_2\to {t}_5\right) \) via a finite difference simulation of the model system. Then, a Jacobian is numerically determined and used to determine the appropriate change in model parameters to decrease the objective function (simply, the sum of the squared errors between each of the measured and simulated \( {\phi}_T\left(\overline{x},t\right) \) and \( {\phi}_V\left(\overline{x},t\right) \) values). A standard LevenbergMarquardt algorithm typically only accepts changes in model parameters that result in a decrease in the objective function. Here, we use a simulatedannealing approach to accept changes in model parameters that result in a decrease in the objective function value and occasionally accept increases in the objective function based off of the simulated annealing criterion. By incorporating a stochastic element, this hybrid algorithm allows this approach to escape potential local minimum and find the global minimum. The algorithm ceases when either the error in the objective function stagnates (less than 0.5% change in successive iterations) or when 1000 iterations are reached.
Model selection
We utilized the Akaike Information criterion (AIC [43]) to select the model (Fig. 1c) that optimally balances model complexity and modeldata agreement. The AIC is defined as:
where k is equal to the number of parameters calibrated for a given model, n is the number of data points used to calibrate the model, and RSS is the residual sum squares between the measured and model \( {\phi}_T\left(\overline{x},t\right) \) and \( {\phi}_V\left(\overline{x},t\right) \). For each animal, we calculated the AIC for each model and ranked the models from 1 to 10. We then calculated the average rank for each model and selected the model with the lowest average rank.
Model prediction and error analysis
The model prediction step is summarized in Fig. 1d. We considered two models in the prediction stage: (1) the lowest ranked model and (2) the highest ranked model. The calibrated model parameters were used to simulate each model forward in time to predict tumor growth at the remaining time points not used in the model calibration (t_{6} and t_{7}). The model predicted \( {\phi}_T\left(\overline{x},t\right) \) and \( {\phi}_V\left(\overline{x},t\right) \) were compared directly to the measured \( {\phi}_{T, meas}\left(\overline{x},t\right) \) and \( {\phi}_{V, meas}\left(\overline{x},t\right) \) at the global and local levels. At the global level, we calculated the percent error in predicted tumor volume and the Dice coefficient. The Dice coefficient describes the degree of overlap of the predicted and measured tumor volumes with a Dice value of 1 indicating perfect overlap. At the local level we calculated the concordance correlation coefficient (CCC) which describes the level of agreement between the predicted and measured values at each voxel location.
The Lilliefors test (‘lillietest’ function in MATLAB) was used to determine if the set of error observations from each time point, model, and error metric comes from a distribution in the normal family. No signficant p values (at a 5% significance level) were observed; thus we are unable to reject the null hypothesis that the observations come from a distribution in the normal family. All results are presented as the mean and 95% confidence interval when appropriate. A twosample ttest for distrubutions with equal means and variances was used to evaluate the differences between the two separate model predictions. A pvalue of < 0.05 was considered significant.
Results
Model selection
Table 3 shows the results of the model selection process. The C_{3} radiation therapy coupled (or vasculature coupled) approach with global parameters had the lowest average rank (2.62) and was tied with the C_{4} (or OER coupled) approach as the most frequently selected model (5 out of 13). Additionally, the C_{2} radiation therapy coupled (or LQ coupled) approach was selected for 3 out of 13 animals. The C_{1} radiation therapy coupled (or logistic growth coupled) approach with local parameters had the highest average rank (9.69). In the following analysis, model C_{3} with global parameters (most selected) was compared to model C_{1} with local parameters (least selected). For the following analysis, we will abbreviate these two models as RT1 (most selected model) and RT2 (least selected model).
Model prediction
Figures 2, 3 and 4 report the results of the model prediction phase. Figure 2 shows both the predicted and measured \( {\phi}_T\left(\overline{x},t\right) \) and \( {\phi}_V\left(\overline{x},t\right) \) over the entire tumor volume for a representative animal receiving a single fraction of 20 Gy. The distributions for \( {\phi}_T\left(\overline{x},t\right) \) and \( {\phi}_V\left(\overline{x},t\right) \) are also shown for RT1, while the white contours indicate the RT2 prediction on the \( {\phi}_T\left(\overline{x},t\right) \) images. For this particular animal, no clear necrosis (low cell density region) is observed in the posttreatment time points. High \( {\phi}_V\left(\overline{x},t\right) \) is observed near the brainskull interface, while it decreases towards the interior of the brain. Visually, there appears to be a strong agreement in the overall shape and distribution of \( {\phi}_T\left(\overline{x},t\right) \) and \( {\phi}_V\left(\overline{x},t\right) \) at both t_{6} and t_{7}. The fused image showing the differences between the measured and predicted distributions indicate that there are some areas that the model incorrectly predicts tumor cells where there are none (bright green regions). Likewise, there are areas where the model fails to predict tumor cells where they exist in the data (bright pink regions). These visual observations are confirmed through the percent error in tumor volume, Dice coefficient, and CCC reported below. Both RT1 and RT2 models have less than 6% error in the predicted tumor volume at t_{6}. A high degree of overlap is found between the predicted and observed tumor volume with Dice coefficients of 0.83 and 0.82 for the RT1 and RT2 models, respectively. However, at the voxel level, the RT1 model has a strong level of agreement to the measured tumor cell fraction with a CCC of 0.76, while the RT2 model has a CCC of 0.54. At t_{7}, less than 20% error is observed in tumor volume predictions for the RT1 and RT2 models. A high degree of overlap still exists between the predicted and measured tumor volumes resulting in Dice correlation coefficients of 0.81 and 0.79 for the RT1 and RT2 model, respectively. At the local level, the RT1 and RT2 models have CCCs of 0.70 and 0.44, respectively. For \( {\phi}_V\left(\overline{x},t\right) \), CCCs greater than 0.95 are observed for the RT1 model at both t_{6} and t_{7}, while CCCs less than 0.10 are observed for the RT2 model.
Similar to Fig. 2, Fig. 3 depicts a representative animal receiving a single fraction of 40 Gy. Unlike the animal in Fig. 2, this animal appears to have developed necrosis (or low \( {\phi}_T\left(\overline{x},t\right) \) regions) in the prediction time points. Generally, these low \( {\phi}_T\left(\overline{x},t\right) \) correspond to low \( {\phi}_V\left(\overline{x},t\right) \) regions in the right hand panels. Both the model predictions for \( {\phi}_T\left(\overline{x},t\right) \) and \( {\phi}_V\left(\overline{x},t\right) \) appear to recapitulate this phenonoma resulting in increased \( {\phi}_T\left(\overline{x},t\right) \) and \( {\phi}_V\left(\overline{x},t\right) \) at the periphery and decreased \( {\phi}_T\left(\overline{x},t\right) \) and \( {\phi}_V\left(\overline{x},t\right) \) towards the interior of the tumor. Additionally, compared to the 20 Gy animal, the RT2 model seems to have increased overestimation of the future tumor growth. These qualitative observations are supported by the quantitative error assessment. For example, the RT1 model resulted in less than 2% error in tumor volume predictions compared to the RT2 model which has greater than 113% error in tumor volume predictions. The high degree of overlap of the predicted and observed tumor volumes for the RT1 model is reflected in a Dice correlation coefficient of 0.85, while the RT2 model has a value of 0.50. At the local level, RT1 and RT2 models have CCCs of 0.68 and 0.01, respectively. Similarly, for t_{7}, the RT1 model resulted in less than 12% error in tumor volume predictions compared to the RT2 model which has greater than 193% error in predictions. A high degree of overlap between the predicted and observed tumor volume is observed resulting in Dice correlation coefficients of 0.83 and 0.52 for the RT1 and RT2 models, respectively. At the local level, the RT1 model has provided accurate descriptions of the intratumoral heterogeneity resulting in CCCs of 0.70, while the RT2 model resulted in CCCs of 0.05. For \( {\phi}_V\left(\overline{x},t\right) \), CCCs greater than 0.62 are observed for the RT1 model at both t_{6} and t_{7}, while CCCs less than 0.37 are observed for the RT2 model for the same time points.
Fig. 4 summarizes global and local error analysis for the cohort. The RT1 model has less than 12% error in tumor volume predictions (panel a) at both t_{6} and t_{7}, wheras the RT2 model has greater than 32% error. Statistically significant (p < 0.05) differences are observed between the RT1 and RT2 model at t_{6} and t_{7}. A high level of spatial agreement is observed between the predicted and observed tumor volumes for the RT1 model resulting in Dice coefficients (panel b) greater than 0.74 at both timepoints. Lower agreement is observed for the RT2 model, which resulted in Dice coefficients less than 0.63 at both time points. The RT1 model resulted in statistically significant (p < 0.05) larger Dice coefficients compared to the RT2 model. At the local level, the RT1 model has a high level of agreement between the predicted and measured \( {\phi}_T\left(\overline{x},t\right) \) with CCCs (panel c) greater than 0.77 for both t_{6} and t_{7}. Similarily, the RT2 has a modest level of agreemennt with CCCs greater greater than 0.63 for both t_{6} and t_{7}. For the blood volume fraction predictions (panel d), a modest level of agreement is observed between the predicted and observed \( {\phi}_V\left(\overline{x},t\right) \) for the RT1 model resulting in CCCs greater than 0.65 at both time points. The RT2 model performed nearly as well as the RT1 model resulting in CCCs greater than 0.59.
Discussion
Ten biologicallybased models of tumor and vasculature response to radiation therapy were developed and tested for their ability to capture variations in the individual animal responses in radiation response. Each model was calibrated for individual animals using serial DW and DCEMRI estimates of tumor and blood volume fractions. The model that coupled radiation response to tumor vascularity using all global model parameters (C_{3}) was selected as the model that optimally balanced model complexity and model fit. This selected model was then used to provide a forecast of the 3D distribution of both tumor and blood volume fractions for each animal. The selected model resulted in low global level errors (percent error in tumor volume < 12%, Dice coefficients > 0.74) and local level errors (CCCs > 0.65) for tumor and blood volume fractions predictions. This study illustrates the ability of this imagedriven, subjectspecific modeling framework to capture and describe tumorspecific response to whole brain radiation therapy. Notably, the two most selected models (C_{3} and C_{4}) both coupled radiation therapy efficacy to vascular distribution.
The varied response of glioma (and other solid tumors) to radiation therapy is linked to the spatial and biological heterogeneity unique to each tumor. Notably in the case of brain tumors [44], heterogeneity in hypoxia status is associated with tumor progression and overall survival due to increased resistance to radiation therapy. Several emerging imaging [45, 46] methods such as ^{18}Ffluoromisonidazole positron emission tomography (^{18}FMISO PET) and oxygen enhanced MRI can quantify the extent of hypoxia pretreatment potentially providing the means to optimize therapy for individual tumors. Using ^{18}FMISO PET data, Rockne et al. demonstrated improved characterization of individual patient radiosensitivity compared to approaches that ignored the effect of hypoxia on treatment sensitivity. However, the dynamics of hypoxia (and thus radiosensitivity) is likely to vary during treatment due to the phenomenon of reoxygenation during radiation therapy [47]. A recent modeling study by Alfonso et al. [48] also demonstrated in silico the importance of characterizing radiosensitivity dynamics during the course of fractionated radiation therapy. Specifically, the initial distribution of intratumoral radiosensitivity and the fractionation scheme significantly impact the efficacy or (overall radiosensitivity) during the course of fractionated therapy. Therefore, to adapt radiation therapy to counteract the temporal and spatial variations in radiosensitivity, characterization of the upstream processes that lead to hypoxia or changes in radiosensitivity are needed. There have been several attempts at describing these dynamics at the volumelevel [49, 50] and with cell scale models [51]; however, to adapt therapy to potentially address spatial variability in radiosensitivity, a spatially and temporally resolved tumorspecific mathematical description is required. In this effort, we developed and systematically characterized a spatiotemporal model of tumor and vasculature response to radiation therapy. It is the inability of the local vasculature to support local tumor cell growth that leads to spatial variations in hypoxia. To the best of our knowledge, this is the first effort to characterize the tumor and vasculature response to radiation therapy temporally and spatially using imaging driven mathematical models.
There are several areas for further development of our experimentalcomputational approach. First, a limitation we have previously discussed in detail elsewhere [25] is the use of DWMRI to estimate tumor volume fraction. While the techniques has been well studied in both the preclinical and clinical settings and appears to be wellsuited for characterizing tumor properties [10], the ADC does provide only a firstorder approximation of true cell density. Second, we acknowledge that the doses of radiation therapy used in this study exceed what is commonly given clinically in a single instance. Nevertheless, in the preclinical setting, large single fraction doses are commonly used to exhibit a strong distinct response between treatment groups. Clinically, large doses of radiation are delivered typically in 2 Gy fractions to maximize the therapeutic effect and minimize damage to healthy tissue. Additionally, fractionated therapy presents the opportunity to target populations of cells that have become more radiosensitivity following reoxygenation. Furthermore, ongoing efforts are focused on applying this modeling approach to more clinically relevant fractionated therapy in a cohort of animals [52]. Third, the mechanical tissue properties are temporally invariant and literature assigned, and the linear relationship between vascularity and carrying capacity may not reflect the true in vivo or clinical scenario. Finally, we acknowledge that the current implementation uses more time points for model calibration than is commonly acquired in the clinical setting. However, MRIguided external beam devices could foreseeably acquire the necessitated quantitative and anatomical image images before and during treatment for model calibration [53, 54].
There are several points of consideration for translating these efforts to the clinical setting. One important difference in tumor biology between the preclinical C6 model and high grade gliomas is the presence of both enhancing (clinical disease) and nonenhancing (subclinical or invasive disease) tumor regions [55]. Our current approach could be applied to the enhancing tumor region as it is most similar to what is observed in the C6 murine model of glioma. However, it is less clear how cell density varies throughout the nonenhancing region, and thus assigning cell density is difficult. One solution is to apply fixed cell density values within this region as is done in [56]. Second, high grade glioma patients typically receive surgery, followed by fractionated radiation therapy and temozolomide [57], as opposed to the single fraction radiotherapy used in this study. As currently formulated, the model does not consider these elements; however, it could be easily incorporated by applying Eq. (6) every time radiation therapy is delivered or by incorporating the cytotoxic effects of the systemic therapy into Eq. (7), as we have done clinical breast cancer setting [39]. Third, there is the computational intensity of this approach. For a single subject, calibration of the mostcomplex model is feasible on a personal computer. However, for multiple model calibrations and multiple subjects (in this paper we had 130 individual calibrations) it is more efficient to utilize multiple compute nodes on a highperformance computing system. For reference, a single calibration took less than 5 h on a personal computer (4 GHz Intel Core i76700K with 4 cores), or less than 3 on a compute node (2.6 GHz Intel Xeon E5–2690 v3 with 12 cores). A single simulation time step takes ~ 0.3 s. In the clinical setting, medical images are typically collected with a larger sampling matrix (512^{2} or 256^{2} vs 128^{2}), thus leading to a larger simulation domain (and therefore greater computational intensity). In preliminary clinical efforts, a single simulation time step on the larger simulation domain now takes ~ 0.8 s. Finally, at the clinical level there is the additional opportunity to incorporate additional biological information (e.g., immunohistochemistry analysis, vascular or cellular density) into the model.
Conclusions
We have developed and applied a biologicallybased, mathematical model of tumor and vasculature response to radiation therapy that can be parameterized for individual tumors using noninvasive quantitative imaging measures. Importantly, we have demonstrated an experimentalcomputational framework to noninvasively and spatially resolve response dynamics for individual animals that can be used to forecast future response. Notably, the most selected model described the 3D dynamics of tumor and blood volume fraction and linked tumor vasculature to spatial radiation response. Future efforts should include expanding this modeling framework to the setting of fractionated preclinical and clinical data.
Availability of data and materials
The datasets generated and/or analyzed during the current study are available from the corresponding author on reasonable request.
Abbreviations
 ADC:

Apparent diffusion coefficient
 AIC:

Akaike information criterion
 CCC:

Concordance correlation coefficient
 DCEMRI:

Dynamic contrastenhanced
 DWMRI:

Diffusionweighted
 LQ:

Linear quadratic
 MRI:

Magnetic resonance imaging
 OER:

Oxygen enhancement ratio
 ROI:

Regions of interest
 RSS:

Residual sum squared error
 RT1:

Most selected radiation therapy model
 RT2:

Least selected radiation therapy model
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Acknowledgements
The authors acknowledge the Texas Advanced Computing Center for providing highperformance computing resources. T.E.Y. is a CPRIT Scholar of Cancer Research.
Funding
This work was supported through funding from the National Cancer Institute R01CA138599, U01CA174706, CPRIT RR160005, and AAPM Research Seed Funding.
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DAH performed imaging experiments, analyzed and interpreted the data, developed and implemented the model, and contributed to writing the manuscript. AMJ participated in model development and analysis, and contributed to writing the manuscript. TEY participated in study design and interpretation of the data, and was a contributor in writing the manuscript. All authors read and approved the final manuscript.
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Correspondence to David A. Hormuth II.
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Hormuth, D.A., Jarrett, A.M. & Yankeelov, T.E. Forecasting tumor and vasculature response dynamics to radiation therapy via image based mathematical modeling. Radiat Oncol 15, 4 (2020) doi:10.1186/s1301401914462
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Keywords
 DWMRI
 DCEMRI
 Glioma
 Diffusion
 Mathematical modeling
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