We have previously developed a novel patient-specific Proliferation-Invasion (PI) mathematical model¹ to estimate tumor cell invasion in glioblastoma (GBM). The PI model is parameterized with tumor volumes delineated on clinical MRIs and has been implemented to determine effectiveness of treatment^2,3, assess surgical resection benefit⁴, predict IDH1 mutation status⁵, and optimize radiotherapy⁶. The apparent diffusion coefficient (ADC) calculated from diffusion-weighted MRI is implicated as predictive of tumor cellularity and microstructure in GBM. The purpose of our study was to investigate whether tumor ADC values are associated with levels of tumor cell invasion, as predicted by the PI model.

MRI: Our retrospective study included six patients (mean age: 65.5 years, 50% female) with contrast-enhancing GBMs. Each subject underwent MRI with T1Gd, FLAIR and DWI sequences acquired at 1.5T or 3T. ADC maps were calculated on a pixel-by-pixel basis at the console. The T1Gd images and ADC maps were first registered to the FLAIR images for each patient (Mirada Medical, Denver, CO). The volume of abnormality was determined separately on T1Gd and FLAIR images for each tumor using in-house software developed in MATLAB (Natick, MA). T1Gd regions of interest (ROIs) were drawn to exclude necrosis. A FLAIR penumbra ROI was created by subtracting the T1Gd ROI from the FLAIR ROI. The T1Gd, FLAIR, and FLAIR penumbra ROIs were then used to mask the ADC maps. Summary statistics were calculated for the ADCs within each ROI: mean, median, minimum, maximum, standard deviation, range, lower quartile (25^th quantile), upper quartile (75^th quantile), interquartile range, mode, and skewness. For each patient, a circular ROI (5 pixel radius) was drawn in the contralateral normal appearing white matter (cNAWM) on three consecutive slices depicting the T1Gd abnormality. PI Model: The PI model relates tumor cell density (c) to tumor cell diffusion (D) and tumor cell proliferation (ρ)

$$ \frac{\partial a}{\partial b}=\triangledown \cdot(D(x)\triangledown c)+\rho c(1-\frac{c}{k})$$

where t is time, x is location, and k is cell carrying capacity. The edge of the T1Gd volume is associated with 80% cell density and the edge of the FLAIR ROI is associated with 16% cell density⁷. A feature of this model is that the solution asymptotically approaches a traveling wave when solved in spherical symmetry. The invasiveness index, D/ρ, is reflective of the shape of this wavefront (Figure 1), independent of the speed at which it is travelling⁴. Thus, it can be estimated given two points on the wavefront from a single time point, which we obtain from T1Gd and T2/FLAIR MRIs. For the results presented here, the D/ρ values for each tumor were estimated via a Bayesian technique. By holding the product constant, asymptotic wavefront velocity, the Bayesian technique allowed us to find the most likely value of D/ρ which minimized error between the model prediction and observed radial values while allowing for 0.5 mm spherically-equivalent radial error in both the observation and in the MRI acquisition. Statistics: Dunn’s multiple comparisons test was used to compare the ADC values in the cNAWM to different tumor ROIs. We used linear regression modeling to evaluate the relationship between ADC metrics for each ROI and D/ρ. A p-value of 0.05 was used to determine significance.

For the 6 GBMs evaluated, D/ρ ranged from 0.89 to 3.40 mm². T1Gd volumes ranged from 22.0 to 72.6 cm³ and FLAIR volumes ranged from 74.7 to 200.3 cm³. Figure 2 shows that the mean ADC within the cNAWM ROIs were significantly lower than in the FLAIR and FLAIR penumbra ROIs (adjusted p=0.005 and 0.011, respectively). Figures 3 and 4 show that the lower quartile of ADC values within the FLAIR and FLAIR penumbra ROIs were positively correlated with D/ρ (p=0.041 and p=0.026, respectively). Additionally, the skewness of ADC values within the T1Gd abnormality negatively correlated with D/ρ (p=0.021, Figure 5).Generally, tumors with higher D/ρ are thought to be more invasive than tumors with lower D/ρ. The FLAIR region, particularly outside of the T1Gd, is the more invasive portion of tumor that includes a complex interplay between normal tissue, tumor cells, and edema. Within the FLAIR abnormality, it is possible that the leading edge of tumor is beginning to breakdown normal tissue structure and the degree to which this occurs is best reflected by lower quartile ADC values. Previous work has shown that evaluating the lower portion of ADC histograms can be indicative of response to treatment^8,9. We plan explore this further in a greater number of patients.Understanding the relationship between D/ρ and ADC could be important for targeting brain tumor therapies, quantifying response to treatment, and further refining the PI model.