Forecasting chemoradiation response mid-treatment for high-grade gliomas through patient-specific biology-based modeling

Abstract

Purpose:

The entire course of radiotherapy (RT) for high-grade glioma (HGG) is currently derived from pre-RT MRI. While it is possible to adapt RT during the course of treatment, it is often guided only by anatomical changes to the tumor. This study seeks to determine if a biology-based mathematical model, parameterized by patient-specific, multi-parametric magnetic resonance imaging (mpMRI) data, can accurately forecast HGG response during RT.

Methods and Materials:

21 patients with HGG planned for 6 weeks of concurrent RT and chemotherapy were imaged weekly with mpMRI during RT and at 1-month, 2-month, 3-months post-RT. Each patient’s MRI data from baseline to mid-treatment was used to personalize a family of biology-based mathematical models, from which the most parsimonious was selected and used to predict response at the volume and voxel levels at the remaining mpMRI visits. The model family consists of varied descriptions of how tumor cells proliferate, diffuse, and respond to RT and chemotherapy.

Results:

At the volume level, Pearson correlation coefficients (PCC) greater than 0.86 (p < 0.0001) were observed between the predicted and observed total tumor cellularity and volume up to the 2-month post-RT. A high-level of spatial overlap was measured between the predicted and observed tumor extent with Dice values greater than 0.87 and 0.74 during and following RT, respectively. At the voxel level, PCCs were greater than 0.90 and 0.71 (p < 0.0001) during and following RT, respectively.

Conclusions:

By leveraging patient-specific mpMRI data before and during adaptive RT, this biology-based computational framework yields accurate spatiotemporal forecasts of tumor response at the volume and voxel levels during and following RT.

1. Introduction

High-grade gliomas (HGG) are the most aggressive and rapidly progressive brain malignancies that include anaplastic gliomas and glioblastoma based on molecular and genotypic expressions of gliomas[1]. The standard treatment for patients with newly diagnosed HGG consists of maximal safe resection (gross total or a subtotal resection) followed by concurrent adjuvant chemotherapy (CT) and radiotherapy (RT)[2], [3, p. 2005]. Despite the effectiveness of systemic CT and RT on delaying disease recurrence, most patients have a dismal prognosis with universally rapid disease progression and a median survival of less than 15 months[4], [5].The treatment resiliency of HGG owes to uncertainty of disease boundaries, local invasiveness and diffuse microscopic infiltration of disease leading to failure of localized treatments (i.e., surgical resection and RT)[6]. Additionally, the high levels of genetic and epigenetic heterogeneity expressed by the microscopic subregions of the HGGs leads to variability in treatment response [6]. Even though the tumor might appear to be shrinking in size, surviving resistant subregions are the main drivers of disease resistance and recurrence[7], [8].

Currently, the standard RT approach defines the radiation target volumes for the entire course of treatment using pretreatment anatomical imaging scans from computed tomography with MRI integration for more precise delineation. The gross target volume (GTV) consists of the resection cavity and/or the enhancing HGG region (as seen on contrast-enhanced T₁-weighted MRI) and receives the highest dose of radiation. Accounting for the microscopic disease infiltration, an additional anatomical volume called the clinical target volume (CTV) is defined extending 1.5 to 2 cm beyond the enhancing volume and receives a lower dose of radiation. The exact methodology of volumetric delineation is subject to variability in institutional best-practices with no identification of a biologically relevant target [9]. While this pretreatment planning approach based on anatomy may be effective in treating RT-responsive tumor subregions, it does not identify or inform the status of persistently active RT-resistant subregions. Recognizing this limitation, ongoing efforts are designed to adapt treatment to account for the anatomical tumor changes seen on imaging.

The first assessment of disease status after RT is done using contrast enhanced MRI about 4 weeks after completion of the treatment course as recommended by the National Comprehensive Cancer Network. These images are used as a new baseline to which further follow up imaging is compared; however, there are no clinical trial data that have identified the proper imaging follow up frequency. The Radiological Assessment in Neuro-Oncology working group has provided a clear definition for disease status assessment, yet timely interventions have been limited by the possibility of pseudo-progression, delaying definite decision to beyond 6 months post treatment[7]. These current treatment planning and follow-up approaches ignore the inherent heterogenous biology in HGG which drives resistance and progression that can only be captured retrospectively using advanced MR imaging[8], [10].

Biology-based mathematical modeling have been shown to make accurate predictions of tumor growth and treatment response in both pre-clinical and clinical studies[11], [12], [13], [14], [15], [16]. When applied to patients with HGG, a personalized two-species model for tumor growth, calibrated on patient-specific pre-surgery and 1-month post-RT MRI data, was able to predict tumor volumes at 3 and 5 months with a median error of less than 2.5%[12]. Using such a model to predict, rather than assess, response to RT would allow for anticipatory adaptive RT[17], [18] (based on anticipated disease progression rather than confirmed progression). Towards this goal, the purpose of this study is to evaluate the predictive accuracy of a personalized, mechanism-based model to spatially predict the response of HGG to chemoradiotherapy prior to the conclusion of RT. The novelty of this research lies in the application and evaluation of well-studied mathematical models of tumor growth to patients with HGG imaged before, during, and after RT. To our knowledge, this is the first example of applying image-based modeling to enable spatial and temporal forecasts of treatment response during RT.

2. Materials and Methods

2.1. Patient cohort

The patient cohort in this study consists of 21 patients enrolled in a single-arm prospective clinical trial approved by the institutional review board at MD Anderson Cancer Center. All the methods described below are in accordance with appropriate guidelines and regulations and informed consent was collected from eligible patients that were enrolled into this trial. All patients had histologically confirmed glioblastoma, negative for IDH 1 and IDH 2 mutation (IDH wild-type), and had undergone maximal safe surgical resection followed by concurrent adaptive radiotherapy and Temozolomide (TMZ) as per the Stupp protocol [3]. The RT prescription was 60 Gy to the gross tumor volume (GTV) and then 50 Gy to the GTV with a 2 cm margin for the clinical target volume (CTV) [19]. For each patient, radiotherapy was adapted on week 3 by redefining the GTV to include new enhancing areas. Supplemental Table 1 summarizes the clinicopathologic characteristics of the cohort, and Figure 1a provides an overview of the clinical timeline.

Figure 1: Overview of the clinical timeline, image processing, and modeling approaches.

(A) Following surgery, patients are consented to participate in the adaptive RT trial. mpMRI is collected at several time points including post-surgery and pre-RT (MRI₁), weekly during RT (MRI_2–7), and at standard-of-care follow-up time points (MRI_i for $i>7$). mpMRI with contrast is collected at MRI₁, MRI₄, and at MRI_i ($i>7$). Patients receive RT with or without CT consisting of 60 Gy delivered over 30 fractions. After MRI₄ the RT plan is adapted based on disease progression as observed on anatomical images. (B) The mpMRI (T₁-weighted images with and without contrast, T₂-FLAIR, and DWI) and RT plans are analyzed to obtain tumor and tissue segmentations. Normalized tumor cell density is obtained within the tumor segmentations via Eq. [1] using the measured ADC from the DWI data. The longitudinal mpMRI series are then registered to MRI₁. (C) A model family is constructed by considering three main modules: number of species modeled (two variations), assignment of $k$ (two variations), and treatment description (11 variations). The treatment description module consists of four sub-modules (i.e., TxM1 to TxM4) which describe how the efficacy of RT, CT, RT and CT, or RT/CT (combined as a single term) are described spatially.

2.2. Multiparametric MRI

As shown in Figure 1a, patients were imaged with multiparametric MRI (mpMRI) at baseline within a week of initiating RT (MRI₁), weekly during radiotherapy (MRI_2–7), and at post-radiotherapy visits (MRI_8–10). An intravenous gadolinium-based contrast agent was administered only at MRI₁, MRI₄, and MRI_8–10. All other imaging visits were collected without contrast administration. The mpMRI protocol (Figure 1b) consisted of T₁-weighted with and/or without contrast administration (T₁+C), T₂-weighted fluid-attenuated inversion recovery (T₂-FLAIR), and diffusion weighted imaging (DWI). The T₁+C image was typically collected at the highest resolution with an in-plane resolution of approximately 0.5 mm ×0.5 mm and a slice thickness from 1 to 2 mm. DWI scans had the lowest resolution of approximately 2 mm × 2 mm and a slice thickness of 2.5 mm. DWI scans were collected with b values of 0 s/mm ² and 1000 s/mm². All images were collected on a 1.5 T magnet.

2.3. Image Processing

T₁+C and T₂-FLAIR images were imported into Raystation 10 B DTK or 11 B DTK (RaySearch Laboratories, Stockholm, Sweden) for tumor segmentation and annotation. The enhancing disease region was manually segmented and accounted for abnormal enhancement on the T₁+C. The non-enhancing infiltrative disease was segmented using a semi-automated intensity-based tool (Raystation-integrated region growing tool) followed by manual editing. All segmentations received a second review by a senior radiation oncologist. Parametric apparent diffusion coefficient (ADC) maps were generated using an in-house script from the DWI, respectively, using standard methods[20]. All images were then aligned to the T₁+C image collected at MRI₁ via a rigid image registration using the imwarp function in MATLAB R2022a (Mathworks, Nattick, MA) to preserve the observed mass effect during modeling. Within the enhancing disease region of the registered images, the normalized tumor density was estimated from the ADC maps as described in [12], [21] via:

$$\widehat{N}\left(\mathit{x},t\right)=\frac{{ADC}_w-ADC\left(\mathit{x},t\right)}{{ADC}_w-{ADC}_{min}},$$ (1)

where $\widehat{N}\left(\mathit{x},t\right)$ is the tumor volume fraction at 3D position $\mathit{x}$ and time $t,{ADC}_w$ is the ADC of water (set to 3×10⁻³ mm²/s at 37° C)[22], and ${ADC}_{min}$ is the minimum ADC observed within the tumor across all patients and time points. In the non-enhancing disease regions, the tumor volume fraction was set to a fixed value of 0.16[12], [23]. Additionally, on imaging visits without contrast agent administration, the most recent enhancing disease segmentation was utilized, and tumor volume fraction was estimated using the ADC-based approach. While there are numerous studies that have demonstrated relationships between tumor cellularity and ADC[24], we recognize this is a simplification of the underlying tissue structure and in some scenarios, changes in ADC may not reflect true changes in cellularity but changes in other tissue properties (e.g., cell size, cell permeability, tortuosity, inflammatory response, edema). Figure 1b depicts the key components of our image processing framework.

2.4. Tumor growth model family

Rather than advocating for a single model a priori, a family of 44 models of tumor growth and response were developed. The model family was built upon the well-studied reaction-diffusion (RD) model of tumor growth [12], [16], [25], and was constructed in a nested fashion with three overall modules as shown in Figure 1c. The first module consists of two base models of tumor growth, the second module consists of two approaches to assign the tumor proliferation rate, and the third module consists of eleven approaches for coupling imaging data to treatment efficacy. In the first module, the two base models of tumor growth consist of a single-species (Eq. (2)) and a two-species (Eqs. (3) and (4)) RD model describing the spatial and temporal evolution of the normalized tumor density over time due to tumor cell movement (diffusion) and proliferation (the reaction term):

$$\frac{\partial {\widehat{N}}_T\left(\mathit{x},t\right)}{\partial t}=\stackrel{\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}}{\overbrace{\nabla ·\left(D_T\nabla {\widehat{N}}_T\left(\mathit{x},t\right)\right)}}+\stackrel{\mathrm{P}\mathrm{r}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}{\overbrace{k_{p,T}{\widehat{N}}_T\left(\mathit{x},t\right)\left(1-{\widehat{N}}_T\left(\mathit{x},t\right)/{\theta }_T\right)}},$$ (2)

$$\frac{\partial {\widehat{N}}_E\left(\mathit{x},t\right)}{\partial t}=\nabla ·\left(D_E\left(\mathit{x},t\right)\nabla {\widehat{N}}_E\left(\mathit{x},t\right)\right)+k_{p,E}{\widehat{N}}_E\left(\mathit{x},t\right)\left(1-\left({\widehat{N}}_E\left(\mathit{x},t\right)+{\beta }_{NE}{\widehat{N}}_N\left(\mathit{x},t\right)\right)/{\theta }_E\right),$$ (3)

$$\frac{\partial {\widehat{N}}_N\left(\mathit{x},t\right)}{\partial t}=\nabla ·\left(D_N\left(\mathit{x},t\right)\nabla {\widehat{N}}_N\left(\mathit{x},t\right)\right)+k_{p,N}{\widehat{N}}_N\left(\mathit{x},t\right)\left(1-\left({\widehat{N}}_N\left(\mathit{x},t\right)+{\beta }_{EN}{\widehat{N}}_E\left(\mathit{x},t\right)\right)/{\theta }_N\right),$$ (4)

where for a given species $i,{\widehat{N}}_i$ is the normalized tumor density, $D_i$ is the tumor cell diffusion coefficient, $k_{p,i}$ is the tumor cell proliferation rate, ${\theta }_i$ is the carrying capacity, and ${\beta }_{i,j}$ is the competition parameter between species $i$ and $j$. For the single-species model, Eq. (2) represents the total tumor burden combining the enhancing and non-enhancing disease components, whereas for the two-species model, Eqs. (3) and (4) capture the enhancing and non-enhancing portions ${\widehat{N}}_E$ and ${\widehat{N}}_N$, respectively. Both base models also consider the effects of local tissue mechanical properties which influences tumor cell diffusion spatially and temporally as employed in[26], [27]. $D_i$ evolves spatially and temporally according to:

$$D_i\left(\mathit{x},t\right)=D_{i,0}\left(\mathit{x}\right)\mathrm{e}\mathrm{x}\mathrm{p}\left({-\lambda }_1{\sigma }_{vm}\left(\mathit{x},t\right)\right),$$ (5)

where $D_{i,0}$ is the diffusion coefficient in the absence of mechanical restrictions and is assigned unique values for white and gray matter, ${\lambda }_1$ is the stress-tumor cell diffusion coupling constant, and ${\sigma }_{vm}$ is the von Mises stress (a measure of the total local stress). The von Mises stress was calculated each iteration by first solving the linear elastic mechanical equilibrium equation for tissue displacement $\overrightarrow{u}$:

$$\nabla ·G\nabla \overrightarrow{u}+\nabla \frac{G}{1-2\upsilon }\left(\nabla ·\overrightarrow{u}\right)-{\lambda }_2\nabla {\widehat{N}}_T\left(\mathit{x},t\right)=0,$$ (6)

where $\nu$ and $G$ are literature values of Poisson’s ratio and shear modulus, respectively, and ${\lambda }_2$ is the second coupling constant (set to 1). The complete details of the numerical implementation and model assumptions are described in [26]. We do, though, we outline the steps for computing ${\sigma }_{vm}$ from Eq. (6). First, we use the estimate of tissue displacement $\overrightarrow{u}$ to calculate the principle and shear strains using finite difference approximations. Next, we use Hooke’s law to relate strains to stresses, and then calculate ${\sigma }_{vm}$ from these stresses. Supplemental material equations S1 to S3 provide the equations used to calculate strain, stress, and ${\sigma }_{vm}$. Response to RT and CT were modeled as discrete events using:

$${\widehat{N}}_{i,post}\left(\mathit{x},t\right)={\widehat{N}}_{i,pre}\left(\mathit{x},t\right){SF}_{RT}\left(\mathit{x},t\right){SF}_{CT}\left(\mathit{x},t\right),$$ (7)

where the normalized tumor density following the treatment event, ${\widehat{N}}_{i,post}$, was assigned to product of the pre-treatment normalized tumor density, ${\widehat{N}}_{i,pre}$, and the surviving fraction of cells due to a single dose of radiotherapy, ${SF}_{RT}$, and a single dose of chemotherapy ${SF}_{CT}$. In addition to the immediate effects of RT, we also assume RT reduces the fraction of actively proliferating cells long-term within a voxel and thereby reduces the proliferation rate of the tumor[28], [29]:

$$k_{p,i}=k_{p,0}{{SF}_{RT,prolif}\left(\mathit{x},t\right)}^n,$$ (8)

where $k_{p,0}$ is the patient-specific proliferation rate, ${SF}_{RT,prolif}$ is the reduction of the proliferation rate per single fraction of RT, and $n$ is the number of fractions delivered. Eqs. (2)–(8) constitute the foundational tumor growth and response components of the model family.

In the second module, the proliferation rates, $k_{p,i}$, are defined either uniformly across the volume or voxel-wise as a field. When defined as a field, $k_{p,i}$ may account for regional phenotypic and physiological differences that result in increased or reduced tumor cell proliferation. Conversely, when assigned globally, it is assumed that all tumor cells proliferate at the same rate.

In the third and final module, the surviving fraction terms in Eq. (7) and Eq. (8) are described in 11 ways to incorporate different assumptions on how the efficacy of RT and CT is modeled within the tumor as detailed in Figure 1c. These 11 variations are divided between four sub-modules which assume that treatment efficacy is coupled to the enhancement ratio (TxM1), treatment efficacy is coupled to cell density (TxM2), treatment efficacy is uniform throughout the tumor (TxM3), or treatment is not modeled (TxM4). The assumptions of TxM1 and TxM2 are then applied to either the RT term alone, the CT term alone, both the RT and CT terms, or as a combined RT/CT term, while TxM3 is applied to either both the RT and CT terms or to the combined RT/CT term. For the combined RT/CT term, ${SF}_{CT}$ is fixed to 1 while ${SF}_{RT}$ and ${SF}_{RT,prolif}$ remain as free variables. As treatment is not modeled for TxM4, the effects of RT and CT are absorbed by other model parameters. We note, that while there are other approaches to vary the efficacy of treatment spatially, the selected approaches are based on coupling approaches observed in the literature[11], [12], [15], [28], [30].

To determine ${SF}_i$ for TxM1, TxM2, and TxM3, we employ the linear quadratic model of response to RT:

$${SF}_i(\mathit{x},t)=e^{-{\alpha }_i·Dose(\mathit{x},t)\left(1+\frac{Dose(\mathit{x},t)}{\alpha /\beta }\right)},$$ (9)

where ${\alpha }_i$ is a treatment sensitivity term, $Dose(\mathit{x},t)$ is the dose of RT given in a single fraction, and $\alpha /\beta$ is the ratio of the linear and quadratic sensitivity terms set to a fixed value of 10 Gy[31]. For CT, we assume the quadratic component $\beta$ is zero, thereby reducing the model to a linear form; this reflects a damage mechanism dominated by single-event effects rather than dose-squared effects. Additionally, the chemotherapy dose is normalized to 1 arbitrary unit, so ${SF}_{CT}$ simplifies to: $e^{-{\alpha }_{CT}},$ where ${\alpha }_{CT}$ represents the chemotherapy-specific cytotoxicity. For TxM1, Eq. (9), is modified to account for tissue vascularization which we assume is proportional to tissue oxygenation:

$${SF}_i(\mathit{x},t)=e^{-\frac{\alpha ·Dose(\mathit{x},t)}{OER(\mathit{x},t)}\left(1+\frac{Dose(\mathit{x},t)}{OER(\mathit{x},t)·\alpha /\beta }\right)},$$ (10)

$$OER(\mathit{x},t)=\max \left(ER(\mathit{x},t)\right)-ER(\mathit{x},t)+1,$$ (11)

where $OER(\mathit{x},t)$ is the oxygen enhancement ratio [32] and $ER(\mathit{x},t)$ is the ratio of the post-contrast T₁-weighted image to the pre-contrast T₁-weighted image. While some studies [33], [34], [35] have reported a correlation between relative signal intensity or contrast enhancement and tissue hypoxia, we acknowledge that the $ER$ remains an indirect proxy for oxygenation. An $OER$ of 1 represents a voxel that maximizes $ER$ and we assume is well perfused and normoxic (most responsive to treatment), whereas an $OER$ greater 1 indicates voxels that may be poorly perfused and hypoxic (less responsive to treatment).

For the second sub-module, TxM2, the efficacy of RT and/or CT varies spatially as a function of the normalized tumor density using Eq. (12):

$${SF}_i(\mathit{x},t)={SF}_{i,min}+\left(1-{SF}_{i,min}\right)\left(\frac{\sum _1^{ns}{\widehat{N}}_j(\mathit{x},t)}{\sum _1^{ns}{\theta }_j}\right),$$ (12)

where ${SF}_{i,min}$ is minimum surviving fraction calculated using Eq. (9), ${\widehat{N}}_j$ and ${\theta }_j$ are the normalized tumor density and carrying capacity for species $j$, respectively, and $ns$ is the number of species considered (1 or 2). The underlying assumption is that as the region reaches its carrying capacity it begins to proliferate less frequently and therefore is less susceptible to death via mitotic catastrophe. For the third sub-module, TxM3 uses Eq. (9) as written and assumes that the efficacy of RT and/or CT is spatially uniform. These three modules yield a total of 44 members within the model family. Eqs. (2)–(12) were numerically solved using a 3D finite difference approximation implemented in MATLAB R2022a on a computational domain discretized identically to the acquisition matrix of the images. Zero flux boundary conditions were assumed for normalized tumor density at the skull boundary, while $\overrightarrow{u}$ boundary conditions allowed tangential displacement (slip condition) but not normal displacement. The median and interquartile range for one forward solve of the model from MRI₁ to MRI₇ was 39.5 seconds and 30.8 seconds, respectively, when evaluated on a personal computer with an Apple M1 Max chip (10-core CPU at 3.2 GHz, 32-core GPU, and 64 GB unified memory).

2.5. Model parameter calibration and selection

The parameters (listed in Supplemental Table 2) within each member of the model family were estimated for each patient via the Levenberg-Marquardt algorithm using the data from MRI₁ to MRI₄. We note, that because the model is calibrated to each individual patient, biological and clinical features such as sex, IDH mutation status, and MGMT promoter methylation are not explicitly included as inputs, but their effects are implicitly captured through the patient-specific calibration process. The remaining visits MRI₅ to MRI₁₀, were used to evaluate model predictions. The Levenberg-Marquardt algorithm[21], [26] iteratively updates model parameters to minimize the sum squared errors between the observed and model-calibrated tumor growth at each available imaging visit used for calibration. A more detailed description of the algorithm and its technical details may be found in [26]. Following each calibration, the estimated parameters and their parameter co-variance matrix (calculated using the Jacobian from the Levenberg-Marquardt algorithm) were used to define a multi-variate normal distribution truncated within the parameter bounds. This distribution was sampled 100 times for each model and patient, then used in a forward simulation of the modeling framework to construct confidence intervals for the model-calibrated tumor growth and response. To reduce the computational complexity for models with a spatially-varying proliferation rate, we calibrated only a subset of points within the tumor and interpolated elsewhere (i.e., for a 3 × 3 grid only the corner and center nodes are calibrated, and the remaining four points were interpolated from the nearest value[12], [36]). Additionally, for the two-species model (i.e., Eqs. (3)–(4)), $D_N$ was assigned as $D_E\times f_{NE}$ where $f_{NE}$ is a scaling factor relating the mobility of the enhancing component to the non-enhancing component of the disease.

The most parsimonious model that balanced model complexity and agreement with the observations was selected using the Akaike Information Criterion (AIC)[37]:

$$AIC=2k+n\ln \left(\frac{RSS}{n}\right)+2k\left(\frac{k+1}{n-k-1}\right),$$ (13)

where $k$ is the number of calibrated parameters for a given model, $n$ is the number of data points used for calibration, and $RSS$ is the residual sum squares error between the observed and model-calibrated tumor growth. To select the most parsimonious model to be used for all patients, we first compute each patient’s $AIC$ weights for each model:

$$w_i=\frac{exp\left(-\left({AIC}_i-{AIC}_{min}\right)/2\right)}{\sum _{j=1}^{44}exp\left(-\left({AIC}_j-{AIC}_{min}\right)/2\right)},$$ (14)

where $w_i$ is the AIC based weight for the $i^{th}$ model, ${AIC}_i$ is the AIC for the $i^{th}$ model, and ${AIC}_{min}$ is the minimum $AIC$. For a given model $i,AIC$ weights can be interpreted as the probability that model $i$ is the best model given the data and model family[37]. The weights for each model were then averaged, and the model with the highest average weight was selected for the cohort and used for all error analysis described in Section 2.6.

2.6. Error and statistical analysis

Model estimates of the spatial tumor distribution were compared directly to the observed tumor distribution as estimated via DWI at both the volume and voxel levels. At the volume level we calculated the correlation and agreement between the model estimated and measured total tumor cell count (TTC) and total tumor volume (TTV) using the respective Pearson correlation coefficient (PCC) and concordance correlation coefficient (CCC). TTC was calculated by multiplying the normalized tumor density at each voxel by an estimate of the maximum number of tumor cells that can fit within a given voxel assuming a spherical packing fraction within a cube of 0.7405[38], and average tumor cell radius 6.8 μm[39], and voxel volume determined for each patient in the image segmentation step. Additionally, we assessed the level of overlap between the observed tumor volume and the model-predicted tumor volume using the Dice similarity coefficient. At the voxel-level we calculated the correlation and agreement between voxel-wise estimates of the normalized tumor density using the PCC and CCC, respectively. In addition, we evaluated the performance of our model to estimate statistically significant changes in tumor cellularity via its accuracy, specificity, sensitivity. All error metrics were calculated for the total tumor volume (enhancing and non-enhancing volumes) and the enhancing volume. To enable comparisons between the model and the observation, we first construct a 3D tumor mask using a threshold of 0.16 to the simulated tumor cell density to identify voxels containing detectable tumor [12], [23]. This threshold corresponds to the value used to initialize the non-enhancing tumor region and reflects the minimum detectable tumor burden. To construct the 3D tumor mask, we identified the outermost contour defined by this threshold and filled all voxels within this boundary, thereby including regions such as necrosis and resection cavities that lie within the tumor extent. This approach aligns with our ground truth segmentations reviewed and approved by the treating radiation oncologist, which also encompass these regions. For both PCC and CCC we also report the p-value and number of data points used to perform the test. For the PCC, the p-value is calculated using the built in MATLAB function corrcoef which uses a t-test to test if PCC follows a Student’s t-distribution hypothesis of zero correlation. For the CCC, the p-value is estimated using an asymptotic z-test[40], where a Fisher’s z-transformation of the CCC is assumed to be approximately normally distributed.

To calculate model accuracy, specificity, and sensitivity of predicted statistically significant changes in cellularity we applied an approach similar to functional diffusion mapping[41] to our model estimated normalized tumor density maps. First, the model estimated normalized tumor density maps are converted back to $ADC$ (via Eq. (1)) and the relative changes in ADC from MRI₁ to MRI_i for $i>2$ were calculated. Then, the relative changes in ADC were divided into significant increases (corresponding to an decrease in cellularity) and decreases (corresponding to an increase in cellularity) based on a change in ADC of 0.4mm²/ms as recommended in [41]. We then defined a positive outcome as a significant relative decrease in ADC (i.e., increase in cellularity) and negative outcome as an increase in ADC (i.e., decrease in cellularity) from MRI₁ to MRI_i for $i>2$.

For each parameter in the selected model, the median and interquartile range (IQR) were calculated for the entire cohort and stratified by sex. Significant differences between parameter distributions were evaluated using the Wilcoxon rank sum test with continuity correction, and linear relations between model parameters explored using PCC. Statistical significance was assigned at p <= 0.05.

3. Results

3.1. Model selection and cohort parameters

Figure 2a shows a heat map depicting the AIC weights, $w_i$, for each model and patient as well as the average value at the bottom. (Description of each model and their features are reported in Supplemental Table 4). Model 43, which is a two-species model with a voxel-wise assigned $k$, and spatially-uniform treatment efficacy (TxM3) applied to a combined RT/CT parameter, had the highest weight of 0.164 and was used for all subsequent analysis. Figure 2b shows the sum of model weights for the three main model modules (i.e., number of species, assignment of $k$, and treatment description), indicating that 78.5% of the model weight came from two-species models for the first module, 69.2% of the model weight came from assignment of voxel-wise $k\mathrm{’}\mathrm{s}$ for the second module, and 43.3% of the weight came from models using TxM1 for the third module. Supplemental Tables 4–7 report the median parameter values as well as the correlation between parameter values for the entire cohort as well as by sex. In Supplemental Table 4 only $k_{p,N}$ showed a statistically significant increase for females compared to males (p = 0.008), while $D_{E,g}$ was nearing statistically significance (p = 0.103). The values of $k_{p,E},D_{E,w},D_{E,g},D/k$ and $a_{RT}$ reported within Supplemental Table 2 are within the ranges reported in literature[15], [42]. Supplemental Tables 5–7 show moderate correlation between the diffusion and proliferation parameters to ${\alpha }_{RT}:k_{p,E}$ vs. $a_{RT}$ (0.710/0.590/0.740; all/female/male), $D_{E,w}$ vs. ${\alpha }_{RT}$ (0.05/−0.620/0.360), $D_{E,g}$ vs. ${\alpha }_{RT}$ (−0.440/−0.770/0.100), and $D/k$ vs. ${\alpha }_{RT}$ (−0.480/−0.750/−0.410).

Figure 2: Model selection and Akaike information criterion weights.

For each patient, the Akaike information criterion weights were calculated for each model. In the top panel, these weights are visualized for each patient (vertical axis) and model (horizontal axis). The bottom row represents the average AIC weight across all 21 patients. Note that the Akaike weights vary across patients, however visually more of the weight is on the right most side of the heatmap. The heatmap is summarized in the bottom panel which reports the total weight contributed by different model components (number of species, assignment of proliferation rates, description of treatment terms). Notably, most of the weight (78.5%) comes from the right-hand side of panel (a) indicating a two-species model is an important component of the model family.

3.2. Evaluation of model calibration and predictions

Figure 3 shows results for the cohort and for patient 1 over the model calibration time points MRI₂ to MRI₄. Figure 3a and 3b present the correlation and agreement between the predicted and observed TTC and TTV, respectively, for the entire cohort. A very strong level of correlation and agreement (PCC > 0.977, CCC > 0.963, p-value < 0.0001 for both, n = 21) was seen between the predicted and observed data for both TTC and TTV. We note Supplemental Figure 1, shows a version of Figure 3a but separated by extent of resection (gross total resection vs. partial or biopsy) and patient sex, where we do not observe statistically significant differences in the strength of the model calibrated TTC between groups. Figure 3c, visualizes the tumor distribution over time for both the observation (top row) and model (bottom row) showing that the model captures the low cell density interior and the high cell density periphery of this tumor. At the voxel level, Figure 3d shows a strong level of agreement (PCC > 0.906, CCC > 0.842, p-value < 0.0001 for both, n > 4376) in voxel-wise cell count is observed from MRI₂ to MRI₄.

Figure 3: Representative results from the calibration scenario.

(A) The observed and model calibrated total tumor cell count (${TTC}_{obs}$ and ${TTC}_{model}$, respectively) are shown for each patient at the three visits used for model calibration (MRI₂ to MRI₄, color coding by visit). (B) In a similar fashion, the observed and model calibrated total tumor volume (${TTV}_{obs}$ and ${TTV}_{model}$, respectively) are shown for each patient at the three visits used for model calibration (MRI₂ to MRI₄). For panels (a) and (b) a high level of correlation (PCCs > 0.977, p-value < 0.0001, n = 21) and agreement (CCCs >0.963, p-value < 0.0001, n = 21) are observed between the model and observed TTV and TTC. (C) The central tumor bearing slice is shown for patient 1 from MRI₁ to MRI₄ for the observed and model calibrated normalized cell density demonstrating a high level of agreement and correlation. (D) The level of agreement and correlation between the model and observed voxel-wise cell count is further visualized and quantified through 2D histograms resulting in PCCs > 0.906 and CCCs > 0.842 (p-value < 0.0001 for both, n > 4376). The 2D histograms in panel D are normalized to report the probability rather than the counts.

Figure 4 reports the results for the cohort and for patient 2 over the model prediction time points MRI₅ to MRI₁₀. Figure 4a and 4b report the correlation and agreement between the predicted and observed TTC and TTV, respectively, for the entire cohort. A very strong level of agreement (PCC > 0.973, CCC > 0.945, p < 0.0001, n = 21) were observed for both TTC and TTV until the last week of RT (MRI₇). However, for the follow-up visits, MRI₈ to MRI₁₀ a lower correlation was generally observed between predicted and observed data with PCCs ranging from 0.729 to 0.954 (MRI_8–9: p < 0.0001, n ≥11 MRI₁₀: p =0.0031, n ≥11). Similarly, the agreement between the observed and predicted TTC and TTV was lower for the follow-up visits, with CCCs ranging from 0.263 to 0.760 (MRI_8–9: p < 0.0001, n ≥11, MRI₁₀: p = 0.054, n = 14). Figure 4c visualizes the tumor distribution over time for both the observation and model showing that the model generally overestimates the tumor cell distribution. For Figure 4d, we generally observed a non-significant decrease in agreement and correlation between voxel-wise cell count overtime with PCCs ranging from 0.859 to 0.943 (p < 0.0001, n > 21661) and CCCs ranging from 0.802 to 0.896 (p < 0.0001, n > 21661).

Figure 4: Representative results from the prediction scenario.

(A) The observed and model predicted total tumor cell count (${TTC}_{obs}$ and ${TTC}_{model}$, respectively) are shown for each patient at the prediction time points (MRI₅ to MRI₁₀, color coded by visit). (B) In a similar fashion, the observed and model predicted total tumor volume (${TTV}_{obs}$ and ${TTV}_{model}$, respectively) are shown for each patient at the prediction time points (MRI₅ to MRI₁₀). For panels (a) and (b) a high level of correlation (PCCs > 0.860, p < 0.0001, n ≥11) is observed up to MRI₉ between the model and observed TTV and TTC. A high level of agreement (CCCs >0.945, p < 0.0001, n =21) is observed up to MRI₇ between the model and observed TTV and TTC. (C) The central tumor bearing slice is shown for one patient from MRI₅ to MRI₁₀ for the observed and model predicted normalized cell density demonstrating a high level of agreement and correlation. (D) The level of agreement and correlation between the model and observed voxel-wise cell count is further visualized and quantified through 2D histograms with PCCs > 0.859 (p < 0.0001, n > 21661) and CCCs > 0.802 (p < 0.0001, n >21661). The 2D histograms in panel D are normalized to report the probability rather than the counts.

Figure 5 shows the results of the functional diffusion mapping analysis indicating areas of significant increases and decreases in ADC for patient 9. Figure 5a shows the transformation from the normalized cell density map for the observed and predicted (left column) to a scatter plot of the individual voxel values of ADC (middle column) to a mapping of increasing (red), decreasing (blue), and no significant change (green) areas back to their spatial location. Visually, these results demonstrate a high level of overlap between the observed and predicted (significant) changes at MRI₇ in ADC relative to MRI₁. Figure 5b, shows 2D histograms of the change in ADC between MRI₁ and MRI₇ to MRI₉. For all three visits, a high level of accuracy (greater than 0.768) and sensitivity were observed (greater than 0.852). Specificity was generally lower and ranged from 0.626 to 0.849.

Figure 5: Observed versus predicted functional diffusion maps.

(A) The left column shows the normalized cell density for the observed and model at MRI₇ for a representative patient. In the middle column, the corresponding cell density maps are converted back to ADC the fDM analysis. In the right column, the fDM maps are overlayed on an anatomical image to spatially indicate voxels with significantly increased (red), significantly decreased (blue), and no significant change (green) in ADCs from baseline to MRI₇. Visually, we see a high level of agreement in the predicted areas of increasing and decreasing cell count relative to baseline. (B) 2D histograms depicting the model predicted and observed change in ADC (DADC) at MRI₇ to MRI₉ are shown for the same patient. The accuracy (ACC) and sensitivity (SE) were greater than 0.768, while the specificity (SP) was greater than 0.626. The 2D histograms in panel B are normalized to report the probability rather than the counts.

Summary statistics from the calibration and prediction scenarios are shown for the entire patient cohort for the calibration visits (MRI₂ to MRI₄) and the prediction visits (MRI₅ to MRI₁₀) in Figure 6. Figure 6a reports the Dice similarity coefficient for the enhancing and total tumor volumes, where a high level of overlap was observed for MRI₂ to MRI₇ (median Dice > 0.871). For the post-RT follow-up visits (MRI₈ to MRI₁₀), lower overlap was observed (median Dice > 0.716). We note that some of the outliers in these plots represent patients who had multi-focal disease where new lesions appeared after the calibration phase. Figure 6b and Figure 6c report the correlation (PCC) and agreement (CCC) at the voxel level where a strong level of correlation (median PCC > 0.870) and a high level of agreement (median CCC > 0.776) was observed for visits during RT (MRI₂ to MRI₇). However, for post-RT visits, model predictions had reduced correlation (median PCC >0.636) and agreement (median CCC > 0.570). For Figure 6b and 6c, the number of data points used to calculate the PCC and CCC ranged from 2652 to 55292 and all p values were less than 0.0001. For Figure 6a to Figure 6c, error generally increased over time resulting in lower Dice, PCC, and CCC value. Figure 6d, summarizes the accuracy, sensitivity, and specificity obtained from the analysis of the functional diffusion maps to identify spatial locations with significant increase and decreases in cellularity. For all visits, the median accuracy was greater than 0.932, the median specificity was greater than 0.916, and the median sensitivity was greater than 0.844. For some visits, sensitivity was lower than accuracy and specificity which is due to our model erroneously predicting an increase in cellularity where it was not observed in the measurement.

Figure 6: Summary statistics from the calibration and prediction scenarios.

(A) The Dice similarity coefficient quantifies the level of spatial overlap between the enhancing tumor (blue) and total tumor (orange) volumes during the calibration time points (MRI₂ to MRI₄) and prediction time points (MRI₅ to MRI₁₀). In each plot, the vertical dashed line separates the time points used for calibration and for prediction. Generally, the Dice value decreased over time; however, the median value was greater than 0.716 for all visits. Panels (B) and (C) show the voxel-level error analysis describing the level of correlation (PCC) and agreement (CCC). The median PCC and CCC were greater than 0.636 and 0.570, respectively for all visits ($n$ ranged from 2652 to 55292, p < 0.0001). (D) The results of the functional diffusion mapping analysis are reported as the accuracy, sensitivity, and specificity across the entire tumor volume. The median value for the accuracy, sensitivity, and specificity were greater than 0.844 for all visits.

4. Discussion

The long-term vision of this work is to develop a digital twin for high-grade glioma enabling the personalization and adaptation of RT plans for individual patients. One of the key components of a digital twin is the set of mathematical models and physical constraints that accurately capture the characteristics of a patient and its tumor in the digital space[43]. Towards this end, we developed and implemented a family of mathematical models of tumor growth and treatment response and then calibrated them for each individual patient using their mpMRI data obtained prior to- and during RT. Model selection identified the most parsimonious model that balanced model complexity and goodness-of-fit, resulting in the selection of a two-species model of tumor composition (enhancing and non-enhancing disease), voxel-wise proliferation rate, and a combined RT/CT treatment term with uniform efficacy across the tumor. We then evaluated the model’s predictive accuracy at the remaining weeks of RT and at standard-of-care follow-up visits. Using only mpMRI data up to the 3^rd week of RT, we achieved high accuracy in tumor-wise predictions (TTC and TTV had PCC > 0.729, p ≤ 0.0031, n ≥11), and voxel-wise predictions (CCC > 0.697, 0.0001, n >2652) up to 1-month post-RT. It is important to note that these predictions were made using data prior to the completion of therapy; thus, this computational approach could enable and guide timely adaptations to RT. As higher accuracy was observed for near-term predictions, embedding this biology-based model within a digital twin framework[17], [18] could enable higher accuracy of predictions moving forward through continuous updates of tumor state throughout treatment.

The ability to adapt RT[44] offers the opportunity to not only improve the accuracy of dose delivery but also to improve the efficacy of RT through patient-specific optimization of dose and schedule. Imaging plays a crucial role in improving the accuracy of dose delivery [44], but it also could be used for identifying targets for dose escalation (or dose de-escalation) in response to the observed treatment effect. Integrating medical imaging with predictive, personalized mathematical models [17], [45], [46], [47] could enable predictive adaptive radiotherapy, which targets tissues in anticipation of their response. The computational framework presented in this manuscript is a step towards this goal; however, it needs to be coupled with a framework for treatment personalization. One such approach by Zahid et al. called ‘dynamics-adapted RT dose’ provided both model calibration and dose adaptation using weekly tumor volume measurements from computed tomography[46]. In 39 head and neck cancer patients, a patient’s own growth dynamics estimated from pre-RT and early in the course of RT volume measurements were used in an in silico study to identify the optimal dose to achieve locoregional control for an individual patient. Promisingly, Zahid et al.’s in silico study estimated that 77% of patients could achieve similar locoregional control through dose de-escalation, while the remaining patients would require dose escalation to achieve locoregional control. In the HGG setting, a predictive digital twin framework[17] was proposed and evaluated in an in silico cohort of patients that employs Bayesian parameter inference and optimization under uncertainty to identify the optimal dose under varying maximum dose constraints. Unique to this approach is the end-to-end incorporation of uncertainty in the measurement and model predictions into risk-aware dose optimization. The design of this HGG digital twin enabled the optimization of weekly dose and schedule for the spectrum of patients from those who respond well to the total standard-of-care dose of 60 Gy to those who may have a more aggressive or resilient tumor requiring higher doses for tumor control. This digital twin serves as a proof-of-concept for how to enable high-consequence decision making, like those found in oncology, through uncertainty quantification and risk-based optimization.

We emphasize that our approach differs from population-based or data-driven methods in that our mathematical models are personalized to each patient’s imaging data, yielding patient-specific parameters. A key advantage of this approach is that anatomical features—such as the extent of resection and tumor location—are directly incorporated during model initialization. This is especially critical for the resection cavity which we assign based on the radiotherapy-planning MRI (MRI₁), rather than the immediate postoperative scan, to minimize errors from postoperative tissue relaxation and cavity shifts. Notably, our cohort was nearly evenly split between resection status and patient sex, yet we still achieved accurate calibrations across all groups, demonstrating the robustness of our patient-specific calibration (Supplemental Figure 1). Although we do not explicitly incorporate clinical variables such as IDH mutation status[48], MGMT promoter methylation[49], or patient sex [42], [50], these factors may be implicitly captured through their influence on calibrated parameters like proliferation rate and treatment sensitivity (e.g., see Supplemental Tables 4–7 and Whitmire et al. [42] for trends in proliferation rates and patient outcomes). Applying this framework to a larger cohort could uncover population-level trends in parameter distributions, potentially enabling more accurate parameter initialization and predictive modeling prior to the start of radiotherapy.

There are several opportunities for future development of this modeling framework to improve predictive accuracy and its clinical translatability. First, our current models of response to RT and CT may oversimplify the biology and dynamics of HGG response. Future models could, for example, explicitly incorporate the pharmacokinetics and pharmacodynamics of the CT, model the five R’s of radiobiology[51], or employ a multiscale approach to model response dynamics from the cell [52], [53] to tissue scale. Any of these approaches should also include a model of toxicity to healthy-appearing tissue. Additionally, our current model also does not account for changes in steroid dose (which may result in an over- or underestimate of predicted tumor volume) and dynamic changes in carrying capacity [27], [36], [54] (which may influence overestimation of local cell count) due to RT or CT response). Second, the modeling framework presented in this study employs the most recent contour of the enhancing tumor to establish the tumor region for imaging visits that did not include the injection of a contrast agent. This substitution may introduce uncertainty in the observed tumor response and model parameters resulting in predictions that either overestimate or underestimate tumor size and/or cellularity. Future efforts could consider the collection of arterial spin labeled MRI on imaging visits that do not include a contrast injection to provide an updated estimate of the “enhancing tumor” region [55]. Third, weekly [46] (or even daily imaging [47]) may be cost-prohibitive or logistically challenging in the community setting. Optimal experimental design [56] could be used within a digital twin framework [17], [18], [57] on a patient-specific basis to identify both the quantity and the timing of imaging visits to enable timely predictions of response. Fourth, several imaging factors should be considered when planning future studies, including the choice of image resolution and the incorporation of more direct measures of tumor hypoxia. The voxel-level accuracy of our model is inherently limited by the lowest resolution of the imaging data used in our study—in this case, DWI. The relatively low resolution of DWI introduces partial volume effects, which can lead to over- or underestimation of the true tumor burden within individual voxels. While higher-resolution scans could mitigate these effects, they often come at the cost of lower signal-to-noise ratio and increased acquisition time. Additionally, other MRI measurements could provide valuable information on RT response including (for example) oxygen-enhanced MRI and tissue oxygen level–dependent (TOLD) MRI [58], [59] which offer improved characterization of tumor hypoxia, though these techniques are only now beginning to be evaluated in clinical settings [60]. Finally, full-time-course calibration of RT response using MRI_1–7 to predict post-treatment progression is a promising direction; this is an area of active development for our team as we are investigating a scalable, sequential data-assimilation framework to enable this approach.

5. Conclusions

This study leveraged a longitudinal mpMRI dataset to personalize a two-species biologically-based image-informed model of high-grade glioma response, using data collected from baseline to week 3 of RT. Patient specific forecasts, made prior to the completion of RT, resulted in accurate patient-specific forecasts of total tumor cellularity at the tumor-level up to 3-months post-RT using data collected prior to the conclusion of RT. Likewise, accurate predictions at the voxel-level were achieved up to 1-month post-RT. The success of this modeling framework suggests the plausibility of integrating imaging with mathematical modeling to achieve predictive adaptive radiotherapy.

Data Availability Statement for this Work

Research data are not available at this time.