Forecasting tumor and vasculature response dynamics to radiation therapy via image based mathematical modeling

Our mathematical model is built upon the well-studied reaction-diffusion model framework that has been extensively applied to pre-clinical [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 reaction-diffusion model to incorporate the effect of local tissue stress on tumor cell diffusion [24] as well as characterize the spatial-temporal 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 three-dimensional 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, λ₁ is the stress-tumor 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 λ₂ 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 spatial-temporal evolution of \( {\phi}_V\left(\overline{x},t\right) \), is described with a similar reaction-diffusion 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.

We assume the response to radiation therapy is observed over two-time 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 post-radiation therapy value of the tumor cell fraction, \( {\phi}_{T, preRT}\left(\overline{x},t\right) \) is the pre-radiation 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 post-radiation 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, C₁, 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, C₂, 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, C₃, relates the treatment efficacy spatially to \( {\phi}_V\left(\overline{x},t\right) \). Here, we assume areas that are well-vascularized (potentially normoxic) and have increased radiosensivity relative to poorly-vascularized (potentially hypoxic) regions. The oxygen enhanced approach, C₄, combines elements from C₂ and C₃ 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 pre-treatment \( {\phi}_V\left(\overline{x},t\right) \), \( \overline{\phi_{V, pre- treatment}} \), the oxygen enhancement ratio (OER) will be greater than 1. Finally, the fifth coupling approach, C₅, assumes no spatial variability, and the effect of radiation therapy is evenly applied.

The spatial-temporal 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 mechanically-coupled model, Eq. (3), can be found elsewhere [34].

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 10⁵ C6 glioma cells (obtained from Sigma-Aldrich, 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 x-ray 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 horizontal-bore magnet (Agilent, Santa Clara, CA) with a 38 mm diameter Litz quadrature coil (Doty Scientific, Columbia, SC, USA) over a 32 × 32 × 16 mm³ 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. DW-MRI and DCE-MRI experiments were performed at each visit. For the DW-MRI experiment, a pulsed gradient, fast spin echo sequence with three orthogonal diffusion encoding directions with b-values of 150, 500, 1100 s/mm², TR = 2000 ms, TE = 30 ms, number of excitations = 10, Δ = 25 ms, and δ = 2 ms. A standard mono-exponential model was fit to the DW-MRI data to estimate the apparent diffusion coefficient (ADC) voxel-wise 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 regions-of-interest (ROIs) across all animals. We have previously used Eq. (9) to provide non-invasive estimates of tumor cellularity [5, 24, 38‐40]; 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 DCE-MRI experiment, we first collected a T₁ map using an inversion-recovery snapshot with TR = 5000 ms, TE = 3 ms, 8 inversion times logarithmically spaced between 200 and 4000 ms, and two averaged excitations. DCE-MRI 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 Gado-DTPA™ (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 (post-contrast image divided by pre-contrast image) derived from DCE-MRI data.

In addition to the five coupling scenarios described above (i.e., C₁ to C₅), 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 sub-region 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 C₂ and C₄, 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 simulated-annealing Levenberg-Marquardt algorithm [25]. Briefly, an initial guess of model parameters and the initial conditions at t₁, \( {\phi}_T\left(\overline{x},{t}_1\right) \) 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 Levenberg-Marquardt algorithm typically only accepts changes in model parameters that result in a decrease in the objective function. Here, we use a simulated-annealing 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.

We utilized the Akaike Information criterion (AIC [43]) to select the model (Fig. 1c) that optimally balances model complexity and model-data 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.

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 (t₆ and t₇). 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 two-sample t-test for distrubutions with equal means and variances was used to evaluate the differences between the two separate model predictions. A p-value of < 0.05 was considered significant.

Table 3 shows the results of the model selection process. The C₃ radiation therapy coupled (or vasculature coupled) approach with global parameters had the lowest average rank (2.62) and was tied with the C₄ (or OER coupled) approach as the most frequently selected model (5 out of 13). Additionally, the C₂ radiation therapy coupled (or LQ coupled) approach was selected for 3 out of 13 animals. The C₁ radiation therapy coupled (or logistic growth coupled) approach with local parameters had the highest average rank (9.69). In the following analysis, model C₃ with global parameters (most selected) was compared to model C₁ 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).

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 post-treatment time points. High \( {\phi}_V\left(\overline{x},t\right) \) is observed near the brain-skull 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 t₆ and t₇. 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 t₆. 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 t₇, 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 t₆ and t₇, 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 t₇, 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 t₆ and t₇, 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 t₆ and t₇, wheras the RT2 model has greater than 32% error. Statistically significant (p < 0.05) differences are observed between the RT1 and RT2 model at t₆ and t₇. 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 t₆ and t₇. Similarily, the RT2 has a modest level of agreemennt with CCCs greater greater than 0.63 for both t₆ and t₇. 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.