Response classification based on a minimal model of glioblastoma growth is prognostic for clinical outcomes and distinguishes progression from pseudoprogression

Abstract

Glioblastoma multiforme (GBM) is the most aggressive type of primary brain tumor. GBM growth dynamics vary widely across patients, making it difficult to accurately gauge their response to treatment. We developed a model-based metric of therapy response called Days Gained that accounts for this heterogeneity. Here we demonstrate in 63 newly diagnosed GBM patients that Days Gained scores from a simple GBM growth model computed at the time of the first post-radiation therapy MRI scan are prognostic for time to tumor recurrence and overall patient survival. After radiation treatment, Days Gained also distinguished patients with pseudoprogression from those with true progression. Since Days Gained scores can be easily computed with routinely available clinical imaging devices, it offers immediate potential to be used in ongoing prospective studies.

Introduction

Glioblastoma multiforme (GBM) is a highly aggressive and invasive type of primary brain tumor. The current standard of care for GBM includes multimodality therapy with surgery, radiation and chemotherapy, yet, despite these interventions, newly-diagnosed GBM patients have a dismal median survival time of 14.6 months (1). Throughout the period following standard therapy, decisions to change treatment are made based on the patient’s clinical symptoms and radiographic studies; however, the heterogeneity in the spatial distribution and growth dynamics of GBMs, combined with inter-patient variability in treatment response, makes these decisions difficult. In particular, there is a lack of practical tools that incorporate these patient-specific differences when assessing the impact of treatment response on prognosis. Additionally, patients frequently exhibit changes on imaging that mimic progression (pseudoprogression or radiation changes) in the two to six months following chemoradiation but which later stabilize or improve (2–4). Distinguishing these patients from those with true progression is difficult, but crucial for providing the most appropriate therapy.

Motivated by these challenges, we developed a new response metric called “Days Gained” that uses patient-specific computational models of GBM dynamics to estimate the amount of time a given therapy delayed tumor growth (Figure 1). Days Gained differs from existing, commonly-used response metrics such as the Macdonald (5), Response Evaluation Criteria in Solid Tumors (RECIST) (6), and Response Assessment in Neuro-Oncology (RANO) criteria (7) in that it incorporates, among other patient-specific data, a measure of the pre-treatment tumor growth rate. This rate allows one to compute the trajectory of a simulated, untreated virtual control (UVC) and predict the tumor size at post-treatment time points as if it had remained untreated. Whereas current response criteria compare pre-treatment to post-treatment tumor burdens on MRI to assess a patient’s response to therapy, our method compares the post-treatment tumor burden to the expected tumor burden predicted by our model at that same time point (Figure 1). This provides a more complete, accurate and personalized assessment of a therapy’s effectiveness. For example, a patient with a fast-growing tumor that responds robustly may show a similar pre-treatment-to-post-treatment change in tumor burden on MRI as a patient with a slow-growing tumor that responds poorly. Accounting for the tumor’s rate of growth provides a means of avoiding such false negatives.

Figure 1.

Computing the Days Gained metric of response with models of varying complexity. We use patient-specific simulations of untreated tumor growth to estimate tumor burden on T1Gd MRI imaging at post-treatment time points. 4D-anatomical, 4D-spherical and Linear models simulate GBM growth with decreasing levels of computational complexity. Days Gained scores can be computed from each of the three model types using the simulated T1Gd spherically equivalent radius curve. Top series shows model solutions from a 4D-anatomical UVC for a 53 year-old patient with a left temporal lobe GBM. Bottom plot illustrates concept of Days Gained score as previously published (8).

The creation of existing response metrics has been largely motivated by the need to standardize the clinical discourse around response to therapy, and their strength lies more in their definitions of response and progression than in their predictive capabilities. Our purpose in creating the Days Gained metric was to develop a quantitative measure of response that is prognostic for patient outcomes, one that can aid clinical decision-making by translating measurable treatment response to survival benefit. Our development of this metric was guided by the intuition that, given the heterogeneity of GBM dynamics, maximizing a response metric’s prognostic power will require incorporating as much patient-specific data regarding tumor kinetics as possible. In our initial examination of the prognostic capabilities of our metric, we used spatially-resolved, highly-individualized models of GBM growth to compute Days Gained scores (8). These models, referred to here as “4D-anatomical” models, account for the full, complex geometry of a patient’s tumor and simulate tumor growth within the unique anatomical confines of the patient’s brain. These models simulate tumor dynamics in four dimensions: length, breadth, height and time. From this initial research we found that patients with higher Days Gained scores had improved overall survival (OS) and progression-free survival (PFS) outcomes compared with lower-scoring patients. However, these 4D-anatomical simulations are computationally intensive, requiring several hours or days to complete. To maximize the ease of clinical translation of our approach, we performed the current study to determine how much detail we could remove from the model and still retain the ability to discriminate OS and PFS.

Additionally, we investigated whether Days Gained might provide a means of addressing the challenges posed by pseudoprogression. Pseudoprogression occurs frequently (4), obscures true patient response, and delays clinical decision-making at a time point when a patient’s course of therapy is often re-evaluated. We therefore examined whether Days Gained scores computed from the first MRI studies performed after radiation could discriminate pseudoprogression from true progression.

The ultimate goal of our research is to validate and translate our response metric in a prospective trial to provide a clearer and more quantitative assessment of patients’ responses to therapy. However, the introduction of complex computational methods into the clinical setting can be disruptive and burdensome to workflow, so our approach was to identify a minimal model of GBM growth that could be used to calculate Days Gained and still retain discriminatory power. We applied two simplified models that reduce the computational cost of simulation: a 4D model that treats the tumor as an isotropic sphere (referred to here as the “4D-spherical” model), and a second, purely linear model of growth (the “Linear” model). The 4D-spherical model is simpler and computationally faster than the 4D-anatomical model, but still requires software that can solve partial differential equations, such as MATLAB® (The MathWorks, Natick, MA). Generally, Days Gained scores from the 4D-spherical model can be computed for one patient on the order of seconds to minutes on the 3.5 GHz desktop computer used for this study. Scores from the simpler Linear model can be computed almost instantaneously using one algebraic equation (see Materials and Methods). In theory, they can be generated using any software that processes algebraic relationships, or even from paper-based nomograms or look-up tables.

For this study we computed Days Gained scores from the first MRI scans taken after standard-of-care radiotherapy for 63 newly-diagnosed GBM patients. We computed scores using both the 4D-spherical and Linear models and compared them against scores generated using the more computationally intensive 4D-anatomical model that we collected from a 31-patient subset of our cohort. We used linear regression and Bland-Altman (BA) analyses (9) to compare the scores and performed Kaplan-Meier analyses (10) to determine their ability to discriminate PFS and OS outcomes. We used Days Gained scores calculated from the minimal model in attempting to distinguish pseudoprogression from true progression among the 36 patients who had radiographic evidence of progression within the first 180 days after completion of radiotherapy.

Materials and Methods

Patients

We collected written informed consent from a total of 63 newly diagnosed GBM patients at the University of Washington (UW) Medical Center and the University of California Los Angeles (UCLA) Medical Center on a protocol approved by the UW and UCLA institutional review boards. Patients began treatment between September, 2001 and September, 2011 (mean: October, 2006). Inclusion in the study required the existence of two pre-treatment MRIs, one pre-treatment T2 weighted sequence (T2), at least one gadolinium contrast-enhanced, T1-weighted (T1Gd) MRI taken following radiation therapy, a positive difference between the later and earlier pre-treatment T1Gd tumor volumes, and a pathological diagnosis of primary GBM. The two pre-treatment T1Gd and T2 imaging sequences are routine in most GBM patients because day-of-presentation (diagnostic) MRI scans and day-of-operation (neuro-navigational) guidance scans are typically taken several days apart, thus providing a temporal interval to measure the tumor’s growth rate. Among our cohort, this pre-treatment interval ranged from 2 to 266 days with a median of 11.

We computed Days Gained scores for these patients using the 4D-spherical and Linear models and compared them against the 4D-anatomical model scores, our “gold standard” (11, 12). Our studies with the 4D-anatomical model, which requires a significant investment in time and manpower to parameterize and execute, included 31 of the 63 patients in the current study. Therefore, when comparing the 4D-anatomical Days Gained scores to the other models’ scores, we used this 31-patient subset. This subset is similar to the 33-patient set used in the preliminary study that identified the prognostic power of the 4D-anatomical model (8) but it excludes patients with a hedged pathological assessment of GBM. Table 1 presents the clinical characteristics of the 63 patients used to compute 4D-spherical and Linear Days Gained scores and the characteristics of the subset used to compute the 4D-anatomical scores. We observed no significant statistical differences between the 63- and 31-patient groups among the categories listed in the table.

Table 1.

Clinical characteristics of patients in the primary study cohort, the patients included in the 4D-anatomical simulation subset as well as those patients who had progression and pseudoprogression on imaging.

Clinical characteristic	4D-spherical and Linear simulations (n=63)	4D-anatomical simulations (n=31)	Patients with progression on imaging (n=24)	Patients with pseudoprogression on imaging (n=12)
Age (years) (median, range)	55, 19–89	58, 40–89	57, 37–76	54, 19–74
Gender	44% Female	39% Female	42% Female	33% Female
KPS at diagnosis (median, range)	90, 50–100	90, 60–100	90, 60–100	90, 50–100
RTOG RPA classification	III (21%)	III (13%)	III (17%)	III (8%)
	IV (52%)	IV (48%)	IV (50%)	IV (67%)
	V (27%)	V (39%)	V (33%)	V (17%)
Resection treatment
Biopsy only	24%	26%	37.5%	8%
Sub-total resection	27%	26%	20.8%	25%
Gross-total resection	49%	48%	41.7%	67%
Radiation therapy dose (cGy) (median, range in cGy)	6000, 2900–6900	6080, 5000–6900	6000, 2900–6900	6000, 5940–6120
Chemoradiation with temozolomide	42 (67%)	20 (65%)	14 (58%)	8 (67%)
Chemoradiation with BCNU	5 (8%)	4 (13%)	2 (8%)	2 (17%)
Radiation without temozolomide or BCNU	16 (25%)	7 (22%)	8 (34%)	2 (16%)
Days between end of radiotherapy and next MRI scan (mean, range)	27, 0–117	29, 1–72	21, 1–56	30, 11–117

ADT assessed all 63 patients for imaging evidence of progression in the first 180 days post-radiation. Pseudoprogression was defined in those patients with early changes in enhancement consistent with progression that showed subsequent radiographic stabilization or improvement (3). The reviewer examined the entire clinical record of each patient to identify true progression versus pseudoprogression; this included radiology reports and all MRI images taken from the time of diagnosis through time of last clinical follow-up. Table 1 shows the clinical characteristics of the progression and pseudoprogression patient groups. We observed no significant statistical differences in their clinical characteristics.

Mathematical models

The three models used in this study simulate the growing tumor at different levels of detail. The 4D-anatomical model incorporates the patients’ neuroanatomy and the complex geometry of their tumor to simulate temporal changes in tumor cell densities throughout the brain. The 4D-spherical model is a simplified version of the 4D-anatomical model and treats the tumor as an expanding sphere, unconstrained by anatomy. This model simulates the temporal dynamics of tumor cell densities within three spatial dimensions, but does not account for the tumor’s complex geometry nor the anatomical confines of the brain. The third model is a purely linear, empirical model of radial expansion of the T1Gd abnormality. Rather than simulate cell densities within a continuum, it models the growth of the spherically-equivalent (SE) radius of the tumor, which is the radius of the measured tumor volume, assuming a spherical distribution as measured from the tumor burden visualized on T1Gd MRI. (For multifocal tumors this radius is computed from the aggregated volume of the disparate foci.)

$$SE\mathit{radius}=\sqrt[3]{\frac{3\ast \mathit{volume}}{4\ast \pi }}$$ (1)

We provide details on each model type below.

4D-anatomical model

The general methods for generating our 4D UVC simulations are well-documented (13–21) and are followed here. Two patient-specific parameters control the model’s dynamics: the tumor’s net diffusive capacity (D) and the tumor’s net proliferation rate (ρ). We compute D and ρ using measures of tumor volume on T1Gd and T2 MRIs at two pre-treatment time points. To solve these terms, we first compute tumor growth rate by dividing the difference of the two pre-treatment T1Gd volumes by the amount of time between their measurements. We then compute the invasiveness index of the tumor, D/ρ, using a non-linear relationship that maps a concomitant pair of T1Gd and T2 SE radii to D/ρ values (14–16, 18, 20, 21). Using this relationship and Fisher’s equation for traveling wave velocity ( $2\sqrt{D\rho }$) we identify D and ρ separately. We then simulate the growth of the tumor to the post-treatment time point, accounting for its unique diffusive and proliferative dynamics and the complex architecture of the brain with the following reaction-diffusion partial differential equation:

$$\frac{\partial c}{\partial t}=\nabla ·(D\nabla c)+\rho c(1-\frac{c}{k})$$ (2)

The equation describes the density of glioma cancer cells c in terms of D, ρ and the tumor cell carrying capacity of the tissue k (10⁸ cells/cm³) (16).

In our 4D-anatomical simulations we simulate tumor growth within a spatial discretization that is based on the patient’s day-of-operation, neuro-navigational, high resolution T1Gd MRI. We use the Statistical Parametric Mapping (22) toolbox in MATLAB to automatically segment the gray matter, white matter and cerebro-spinal fluid (CSF) regions of the brain. Each voxel in our discretization is then assigned a value of D based on this segmentation. To constrain the tumor growth to neuroanatomical boundaries, we set D to zero in the CSF. To simulate the increased diffusive capacity of gliomas in white matter, we set D in areas of gray matter to a value 1000 times less than in white matter. From our simulations we are able to visualize the tumor’s volumetric growth within the patient’s brain as shown in the serial, 3D images of Figure 1.

4D-spherical model

Applying Equation (2 but assuming spherical symmetry generates a full 4D simulation, but with growth only in the radial direction, creating a spherical tumor. We performed simulations for this simplified geometry using the built-in pdepe.m function in MATLAB. We parameterize these simulations with D and ρ values that are computed in the same way as the 4D-anatomical simulations but execute them within a homogenous spatial domain that, unlike the 4D-anatomical models, is not informed by patient-specific anatomy.

Linear model

The Linear model assumes that the T1Gd SE radius of the tumor increases linearly from the date of diagnosis onward. This trajectory is represented by the equation

$$y=mx+y_{\mathit{PRE}}$$ (3)

where y is the simulated SE T1Gd radius, x is the amount of time since the final T1Gd MRI taken prior to cytotoxic treatment, m is the pre-treatment rate of T1Gd radial growth, and y_PRE is the SE radius computed from the tumor volume observed on the final pre-treatment T1Gd MRI.

To compute m we find the difference between the tumor volumes on two pre-treatment T1Gd, and divide it by the amount of time between the scans. For this study we used the first and final pre-treatment scans to compute m, regardless of how many additional pre-treatment scans were performed. As an alternative, m can also be computed using pre-treatment T2 scans. We repeated all our analyses using T2-derived velocities where available and the implications of our results remained consistent with the T1Gd-derived velocities (data not shown).

Days Gained scores are only applicable for positive values of m, as our method assumes positive tumor growth during the period immediately prior to treatment. Scores will be meaningless within the context of clinical treatment if pre-treatment T1Gd imaging shows a negative tumor growth rate.

Computing Days Gained scores from Untreated Virtual Controls

To generate Days Gained scores from the simulation results of the 4D-spherical and 4D-anatomical models, we first compute the T1Gd SE radius time curve from the volumetric solutions. This curve simulates the radius of the SE volume as would be observed on T1Gd MRI. We compute it by calculating the total volume of voxels with tumor cell densities at or above 80% of the total cell carrying capacity of the tissue (14, 16), and then determine the SE radius using Equation (1. To compute a Days Gained score we find the time point on the T1Gd SE curve where it best matches the actual post-treatment tumor size. The Days Gained score is the amount of time between this time point and the final (post-treatment) time point on the curve (Figure 1). In some 4D-anatomical simulation studies, therapy reduced the tumor to a size smaller or larger than what was computed on the curve. In these cases we used a linear interpolation between the T1Gd SE radii values at the last pre-treatment time point and the post-treatment time point to compute the Days Gained score. This is consistent with the radial expansions predicted by the model and already observed in a spectrum of gliomas (17, 23).

Our 4D UVC simulations provide a number of outputs for quantifying patient response to treatment. Each UVC can be used to estimate the tumor volume as it would be observed on T1Gd and T2 imaging at any time between the pre- and post-treatment time points along with the total number of tumor cells distributed throughout the brain. Thus, we have several metrics available for computing Days Gained scores. We have found that the T1Gd SE radius discriminates OS most significantly (24) and also discriminates PFS (8); therefore, our focus here is on the predictive value of Days Gained scores based on this output.

To compute Days Gained from the Linear model we use Equation (3) to solve for x when y is the measured post-treatment T1Gd SE radius (y_POST), then subtract this value from the number of days that have passed since the final pre-treatment scan (x_POST):

$$\mathit{Days}\mathit{Gained}=x_{\mathit{POST}}-\frac{(y_{\mathit{POST}}-y_{\mathit{PRE}})}{m}$$ (4)

For all three model implementations tested there were some patients that had a post-treatment tumor burden that was larger than the model-predicted burden. Days Gained scores are negative in these cases, indicating that the burden increased beyond what would be expected in the absence of treatment.

Image processing

Computing a Days Gained score requires a volumetric segmentation of the tumor on two pre-treatment and one post-treatment T1Gd MRI scan. Spherically equivalent radii, which are used to parameterize each UVC and to compute Days Gained scores, can be computed from these volumes using Equation (1. We measured tumor volumes on T1Gd and T2 patient MRIs using semi-automated segmentation software developed in MATLAB. This software facilitates tumor volume measurement by employing a background subtraction algorithm that helps automate the tumor segmentation process by allowing the user to identify the tumor region with a polygon of interest. An automated edge detection algorithm then defines the imaging abnormality within the enclosed area. Although our segmentation pipeline relies on custom software not currently available to clinicians, there are several readily-available, commonly used clinical visualization tools available for performing this same kind of segmentation, including ImageJ (25), OsiriX (26), and Mipab (27).

All images were segmented by at least two independent reviewers who received training in segmenting tumor volumes from various MRI modalities. All measurements were reviewed by a third, independent reviewer who ensured all scans were accurately measured. All measurers and reviewers received training in neuroanatomy and MRI abnormalities and are familiar with common MRI signals such as resection, stroke, cerebrospinal fluid, and normal anatomical variants and features.

Statistical analyses

To quantify the agreement between Days Gained scores computed from the three different models, we performed linear regression and Bland-Altman analyses (9). We computed the coefficient of regression statistic in R version 2.14.1 and computed Bland-Altman biases and limits of agreement using MATLAB.

For each set of Days Gained scores that we computed from our models, we performed iterative Kaplan-Meier analyses (10) to find an optimal Days Gained threshold that would maximally discriminate patients based on PFS and OS. We computed PFS as the interval between the patient’s start of cytotoxic therapy and the confirmation of progression using standard criteria as gathered from radiology reports. We computed OS as the interval between the patient’s date of diagnosis and their date of death. We censored observations at the time of last follow-up if the outcome in question was not observed. To ensure that our analyses included a substantial number of patients with and without the observed outcome, we limited it to cases where both groups discriminated by the Days Gained threshold contained at least 20% of the total patient number. We used the “survival” package for R software version 2.14.1 (R Foundation for Statistical Computing, Vienna, Austria) to perform the Kaplan-Meier analyses (log-rank test).

Once we identified the simplest model that significantly discriminated outcomes according to these Kaplan-Meier analyses, we validated its overall prognostic power using bootstrap validation (28) in R (version 2.15.1). We created training sets by randomly resampling the original dataset with replacement, then computed the optimal Days Gained threshold that discriminated the outcome in question. We then applied this threshold to a Kaplan-Meier analysis of the original dataset and recorded the chi-squared statistic. After 10,000 iterations of this procedure, we averaged the resulting chi-squared statistics, then computed the corresponding p-value to validate our metric for discriminating PFS and OS outcomes.

We performed a receiver operator curve (ROC) analysis to find the optimal Days Gained threshold that would distinguish our true progression and pseudoprogression cohorts. After identifying this cutoff, we performed a Welch’s t-test to determine if the two groups had significantly different Days Gained scores and also assessed the positive and negative predictive value of the cutoff. To validate our approach more generally, we performed LOOCV (28), a validation method more suited to dichotomous tests and reduced datasets. From the LOOCV we computed the overall positive and negative predictive value to validate our method’s discriminatory power.

We performed uni- and multi-variate Cox proportional hazard analyses on Days Gained, age, RTOG RPA classification, enrollment in clinical trial, and gender to identify potential confounders in the study and to determine the value of Days Gained independent of other clinical factors.

We performed the ROC, t-test, LOOCV, predictive value and Cox proportional hazards analyses in R version 2.15.0. We used the pROC package for ROC analysis (29). For all statistical tests, we considered p values less than or equal to 0.05 to be significant.

Results

Agreement between Days Gained scores from 4D-anatomical, 4D-spherical and Linear models

Figures 2 shows the agreement between Days Gained scores computed from T1Gd MRI scans at the first post-radiation therapy time point for the three model implementations. Figures 2B, D and F show the Bland-Altman biases and limits of agreement (9) across the model comparisons. We observed very strong correlation and agreement between scores generated by the three models: R² values were greater than 0.9 and absolute Bland-Altman biases were less than 14 days.

Figure 2.

Linear regression (A, C, E) and Bland-Altman (B, D, F) analyses comparing Days Gained scores computed using the 4D-anatomical, 4D-spherical and Linear model implementations. R² values in A, C and E are the coefficients of determination computed from linear regression analysis; the solid line is the linear regression best fit and the dotted line is unity. Bland-Altman bias, upper limit of agreement (ULA) and lower limit of agreement (LLA) are indicated by dashed lines in B, D and F.

Progression-free and overall survival analyses

Figure 3 shows the results of our Kaplan-Meier analysis where we identified the Days Gained thresholds that would most significantly discriminate PFS and OS among the Linear, 4D-spherical and 4D-anatomical model studies. We found that the three sets of Days Gained scores all have a maximal discriminating threshold that separates the patient population into groups with significantly different PFS and OS outcomes. For PFS, this level of significance was slightly higher for the Linear and 4D-spherical model scores (p<0.001 for both) than for the 4D-anatomical scores (p=0.004). For OS, the 4D-anatomical scores showed a slightly higher level of significance (p=0.004) than the Linear (p=0.005) and 4D-spherical (p=0.006) models. Maximally discriminating Days Gained thresholds and associated p-values from the Kaplan-Meier analyses are shown in Figure 3 and Table 2. Table 2 also shows the differences in median PFS and OS between patient groups above and below the optimal Days Gained thresholds that we identified. These optimal thresholds differentiated patients into cohorts with statistically significant median PFS and OS. Across the three model implementations, the PFS of high-scoring patients was 102–154% greater than that of lower-scoring patients. OS of high-scoring patients was 65–85% greater than that of lower scoring patients. Figure 4 illustrates the complete set of p-values from our Kaplan-Meier analyses for PFS and OS across model implementations. The figure illustrates that our OS and PFS results are robust across a range of Days Gained thresholds. Bootstrap validation of the minimal, Linear model confirmed its significant prognostic power for discriminating both PFS (p<0.001) and OS (p=0.01).

Figure 3.

Kaplan-Meier analyses comparing the PFS and OS of patients classified according to Days Gained score. Scores from the minimal “Linear” model discriminated outcomes with a level of significance similar to the other, more detailed, model implementations. Statistics for these analyses are shown in Table 2.

Table 2.

Statistics for Kaplan-Meier analyses of progression-free survival (PFS) and overall survival (OS) outcomes using Days Gained scores from the three model implementations.

Model	n	Optimal Days Gained threshold for PFS	p-value for PFS	Difference in median PFS (months, % improvement)	Optimal Days Gained threshold for OS	p- value for OS	Difference in median OS (months, % improvement)
Linear	63	93	<0.001	6.4, 154%	78	0.005	9.4, 85%
4D-spherical	63	86	<0.001	6.4, 154%	84	0.006	9.4, 85%
4D-anatomical	31	117	0.004	7.1, 102%	117	0.004	8.1, 65%

Figure 4.

p-values from iterative Kaplan-Meier analyses on PFS and OS. White boxes in the figure correspond to Days Gained thresholds that revealed significant differences (p≤0.05) in the outcomes of higher and lower-scoring patient groups. Figure 3 and Table 2 present the optimal thresholds found for each model implementation.

Discriminating pseudoprogression

Fifty-eight of our 63 patients had more than one post-treatment imaging study available for review, with 12 demonstrating evidence of pseudoprogression and 24 with true progression in the first 180 days following radiotherapy (the remaining 22 showed no evidence of progression). ROC analysis of Days Gained within the population of patients with progression revealed an optimal score threshold of 90 that detects pseudoprogression with a sensitivity of 1.0, specificity of 0.58, positive predictive value of 0.55, negative predictive value of 1.0 and area under the ROC curve of 0.81. Figure 5A shows the significant difference in Days Gained scores between the true progression and pseudoprogression cohorts (p=0.003, Welch’s t-test). To anticipate results from the same study with a higher patient number, we applied a bi-normal smoothing algorithm (30) to our ROC curve. At a threshold of 90 Days Gained, this smoothed curve gives a sensitivity of 0.83, specificity of 0.63, positive predictive value of 0.53, and negative predictive value of 0.88. Area under the curve is 0.79. Figure 5B presents the original and smoothed versions of the ROC curve. LOOCV of our method for discriminating pseudoprogression returned similar predictive value results with a sensitivity of 0.79, specificity of 0.59, PPV of 0.5 and NPV of 0.93.

Figure 5.

A: Boxplot of Days Gained scores computed from the minimal model in 36 patients considered to have progression in the first 180 days after radiation. True progressors had significantly lower scores than pseudoprogressors. Negative Days Gained values indicate that post-treatment tumor burden was greater than that predicted by the simulated, untreated tumor. B: ROC curve for detecting pseudoprogression using Days Gained scores. Bi-normal smoothed curve (dashed) overlaid.

Cox proportional hazard analyses

Uni-variate Cox proportional hazard analysis revealed significant correlations between survival and Days Gained score (p=0.009), enrollment in clinical trial (p=0.006), age (p=0.027) and RTOG RPA classification. Gender was not significantly correlated with survival (p=0.35). Multi-variate Cox proportional hazard analysis only showed a significant correlation between survival and Days Gained score (p=0.03) and enrollment in clinical trial (p=0.001).

Discussion

Patient-specific modeling is currently an active area of research, but there are few examples of modeling applications that have been translated into the clinic for routine use (31). One of the challenges in making the step from the computational biology lab to the bedside is model complexity. Most patient-specific modeling research focuses on the creation of 4D, finite element or finite volume models. These models, including our 4D-anatomical model, require significant computational overhead to simulate and analyze, and interpretation of simulation results often requires expertise with the model. The few patient-specific modeling applications that have demonstrated clinical impact are based on simple models that do not necessitate the use of a computer (31, 32). Because our ultimate goal is to improve the outcomes of GBM patients by applying patient-specific modeling approaches, we sought to find a minimal model of GBM growth that can be readily computed and analyzed. Our results here indicate that Days Gained scores based on our minimal growth model are strongly predictive of scores from more detailed models (Figure 2), and are able to discriminate PFS and OS outcomes with similar statistical power (Figures 3 and 4, Table 2). We are therefore confident that the purely linear model of tumor growth can serve as a minimal model for calculating patients’ Days Gained scores at post-treatment time points. Additionally, we applied Days Gained scores from this model to address a current, difficult clinical problem: discriminating pseudoprogression in patients with disease progression following radiation (Figure 5).

We anticipate that the linear model’s simplicity will help accelerate the translation of our Days Gained response metric into the clinic. Days Gained scores can be computed from this model using nomograms, mobile device applications or desktop software, whichever is most clinically convenient. Most appealing is that computing these Days Gained scores requires no additional clinical data beyond what is routinely collected for GBM patients. However, the required volumetric segmentation of the tumor on two pre-treatment and one post-treatment scan may present a bottleneck to clinical adoption. Although semi-automated tools exist that reduce segmentation effort, this task can be time-consuming, especially when performed on high-resolution MRI data. In future studies we plan to investigate whether simpler measurements of tumor burden, such as those used in clinical response criteria (sum of products of maximally perpendicular diameters, for example) can be used as surrogates for a complete volumetric assessment of the tumor.

Previous studies have shown the linear nature of radial expansion of gliomas on MRI (17, 23), thus it is not surprising that our minimal, linear model performs as well as the 4D-spherical and 4D-anatomical versions. However, while the minimal model may be optimal for computing Days Gained scores in a clinical setting, more detailed models will be required for analyzing tumor cell density profiles, spatial growth dynamics, or the interaction between a patient’s neuroanatomy and the growing tumor. For instance, in the context of customizing radiation therapy for an individual, spatio-temporal tumor dynamics must be resolved to optimize the radiation dose distribution (15, 33).

In this study we computed each patient’s Days Gained score using the first post-radiation T1Gd MRI scan. In the modern era of treatment with surgery, radiation combined with temozolomide (34), we have seen the rate of pseudoprogression as high as 50% of patients (4). As every Days Gained score was measured during this period, we were concerned that the presence of pseudoprogression may lead to spurious results in our predictions. Rather, we found that all patients with pseudoprogression had Days Gained scores that were significantly higher than patients with true progression. These scores discriminated between the two groups, and cross-validation confirmed their high negative predictive value. Furthermore, even if pseudoprogression falsely lowered some Days Gained scores computed from the first post-radiation scan, these scores remained prognostic for PFS and OS. We also recognize that pseudo-response, a spurious decrease in contrast-enhancing tumor burden resulting from anti-angiogenic therapy (35) could have affected Days Gained scores. However, this is unlikely in our patient population given that no patients received anti-angiogenic therapy prior to the post-treatment scan used to compute their score.

In terms of discriminating pseudoprogression from true progression, our statistical analyses show that the power of Days Gained lies in its sensitivity and negative predictive value. Thus, progression observed in a patient scoring below 90 Days Gained is highly likely to be a true increase in tumor burden rather than an artifact of treatment. Days Gained is less powerful at discriminating true progression from pseudoprogression in patients that score above 90. In these cases, using Days Gained in combination with other tests and imaging modalities may help discriminate pseudoprogression.

We acknowledge that some of the shorter pre-treatment intervals used to compute tumor growth rates might have introduced error into Days Gained scores. However, our study goal was to develop a metric of response that could be used with clinical data that is routinely collected in GBM cases. We did not, therefore, seek to remove sources of error in our measurements if they were part of routine patient care, but rather find a response metric that is prognostic for survival outcomes despite the presence of those errors.

We also acknowledge that a small fraction of our patients may not have received the current standard of care because they received treatment prior to the widespread adoption of the Stupp protocol; however, the vast majority of patients represent modern treatment cases. Our Cox proportional hazard analyses showed that Days Gained scores correlated with survival, indicating the independent value of the metric. There are many potential confounders in survival analysis in this population, including clinical trial enrollment and age, however multivariate Cox proportional hazard analysis demonstrates that Days Gained remains significant despite these confounders.

Our results show that Days Gained scores based on a simple linear growth model and computed at an early time point may discriminate patients who may have worse outcomes. With nearly twice the number of patients as in our preliminary work (8), our results corroborate initial findings that suggested Days Gained scores had prognostic power, and our bootstrap validation analyses provide further evidence of these predictive capabilities. We believe that this metric can form the basis of adaptive clinical trial design and, eventually, a method of providing personalized treatment earlier in the disease course. It is our hope that the prognostic power of our response metric will help in assessing treatment effectiveness and ultimately identify those patients who would benefit from changes in their course of treatment.

In future studies we plan to investigate whether Days Gained scores remain prognostic of PFS and OS when calculated from later post-treatment time points, including during the post-salvage therapy period. For such a study we will have to assess the impact that anti-angiogenic agents (bevacizumab, cediranib, etc.) have on Days Gained scores. As addressed by the RANO working group, effective response criteria must account for the use of anti-angiogenic therapies because they can reduce the permeability of tumor vessels and give false indications of response on T1Gd MRI (7). We also plan to include Days Gained in a prospective trial of treatment for GBM.