## 1 Introduction  
Glioblastoma is the most common and most lethal primary malignant tumor of the brain [1]. A noted attribute of these tumors is their resistance to virtually all forms of therapy [2; 3; 4; 5], which may be related to underlying cellular phenotypic plasticity and a propensity to invade normal brain tissue [6; 7; 8; 9]. Moreover, this biological attribute may be associated with distributed patterns of proliferation and migration under the influence of local force, nutrients, and growth factors [10; 11]. Large scale analyses of glioblastoma have confirmed that considerable genomic and epigenetic heterogeneity between tumors exists [12; 13; 14; 15], in addition to significant cellular heterogeneity within tumors [6; 16; 17; 18], which together may promote the ability of tumor cells to evade treatment. Furthermore, glioma cells have been shown to closely associate with neuronal components within the brain, including the formation of functional electrical and chemical connections with neighboring cells [19; 20] as well as migration along axonal structures [21; 22; 23]. It is likely that cellular and molecular promoters for glioblastoma growth and invasion are complex and affect the biological behavior of the tumor through spatially distributed regulatory pathways in the neighboring microenvironment [24]. Some evidence also suggests that subsets of glial cells within the tumor, particularly stem cell populations which may also serve as tumor initiating cells in malignant gliomas, function much like migrating neural and glial cells in the perinatal brain [7; 9; 25; 26; 27; 28]. These cells may recapitulate a developmental phenotype which subserves robust migratory interactions with white matter tracts intersecting the tumor [21]. We postulated that the directional migration of glioblastoma cells into the brain parenchyma follows trajectories dependent on programmed interactions with axons and non-malignant glial cells. However, despite the importance of discerning structural mechanisms for tumor migration [29], no methods exist to reliably map the distribution and progression of these invading cells after disease diagnosis. Diffusion-weighted magnetic resonance imaging (dwMRI) characteristically depicts tissue organization based on the orientation of restricted molecular displacement [30; 31]. More broadly, dwMRI is sensitive to the displacement of water molecules in the presence of an oriented magnetic field, whether or not those movements are restricted by compartmentation due to membranes or other cellular, subcellular, or extracellular structures. dwMRI methods have been applied for mapping functionally important white matter tracts of the brain based on preoperative MRI scans, which can then be registered to patient anatomy intraoperatively using surgical navigation units during tumor resection [32; 33; 34; 35]. Diffusional metrics of potential diagnostic and prognostic utility have also been identified [36; 37; 38; 39; 40; 41], including preliminary assessment of tractography-derived metrics [39; 40; 42]. Despite these current applications of dwMRI in neuro-oncology, these methods have not yet been rigorously investigated as a means of directly mapping tumor cell invasion. In this study, we applied Q-space magnetic resonance imaging (QSI), a dwMRI method capable of generating high resolution maps of fiber alignment through the analysis of sub-voxel diffusivity [43; 44], in patients with glioblastoma. We hypothesized that patterns of tumor-associated tractography may be able to predict the location of invading tumor cells along tumor-involved white matter structures using this method [45; 46; 47; 48; 49; 50]. We further hypothesized that the presence of long tumor-associated tracts projecting outward from the tumor may be associated with poorer overall survival in glioblastoma. Ultimately, we identified specific patterns of tractography extending from tumors that appear to colocalize with microscopic and transcriptomic features of invasiveness.