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Magn Reson Med. Author manuscript; available in PMC 2012 April 1.

Published in final edited form as:

Published online 2010 November 30. doi: 10.1002/mrm.22688

PMCID: PMC3065939

NIHMSID: NIHMS240489

Benjamin M. Ellingson, PhD,^{1} Peter S. LaViolette, MS,^{2,}^{4} Scott D. Rand, MD, PhD,^{2,}^{3} Mark G. Malkin, MD,^{2,}^{5,}^{6} Jennifer M. Connelly, MD,^{2,}^{5,}^{6} Wade M. Mueller, MD,^{2,}^{6} Robert W. Prost, PhD,^{3,}^{4} and Kathleen M. Schmainda, PhD^{2,}^{3,}^{4}

Send Correspondence to: Kathleen M. Schmainda, Ph.D. Professor, Radiology, Department of Biophysics, 8701 Watertown Plank Road, Milwaukee, WI 53226, Phone: 414-456-4051, Fax: 414-456-6301

The publisher's final edited version of this article is available at Magn Reson Med

See other articles in PMC that cite the published article.

The purpose of the current study was to develop a voxel-wise analytical solution to a glioma growth model using serial diffusion MRI. These cell invasion, motility, and proliferation level estimates (CIMPLE maps) provide quantitative estimates of microscopic tumor growth dynamics. After an analytical solution was found, noise simulations were performed to predict the effects that perturbations in apparent diffusion coefficient (ADC) values and the time between ADC map acquisitions would have on the accuracy of CIMPLE maps. CIMPLE maps were then created for 53 patients with gliomas with WHO grades of II–IV. MR spectroscopy estimates of the Cho/NAA ratio were compared to cell proliferation estimates in CIMPLE maps using Pearson’s correlation analysis. Median differences in cell proliferation and diffusion rates between WHO grades were compared. A strong correlation (R^{2} = 0.9714) and good spatial correspondence were observed between MR spectroscopy measurements of the Cho/NAA ratio and CIMPLE map cell proliferation rate estimates. Estimates of cell proliferation and diffusion rates appear to be significantly different between low (WHO II) and high-grade (WHO III–IV) gliomas. Cell diffusion rate (motility) estimates are highly dependent on the time interval between ADC map acquisitions, whereas cell proliferation rate estimates are additionally influenced by the level of noise present in ADC maps.

Glioblastoma multiforme (GBM) is an infiltrative malignant brain tumor characterized by a very poor prognosis. Despite advancements in surgical procedures, radiation therapy, and chemotherapy, mean survival time is only approximately 14.6 months for these patients (1). Microscopic invasion of tumor cells, which cannot be detected with currently available imaging techniques, is thought to be the a primary reason for such a dismal prognosis (2). Therefore, non-invasive imaging biomarkers, which are sensitive to the presence and activity of invading tumor cells could prove to be a significant step forward in the treatment and prognosis of these patients. In this regard we have developed imaging biomarkers sensitive to tumor cell migration and proliferation. The approach is to use diffusion-weighted MRI (DWI) data in the solution of a glioma growth model.

The rate of water movement, or diffusion, within the brain and other tissues is believed to be a direct reflection of the local microstructure within the tissue. Anatomic magnetic resonance imaging (MRI) can be modified in a way that makes the images sensitive to water diffusion within tissues. This approach, termed diffusion-weighted MRI (DWI), has been shown to be sensitive to changing microenvironments such as the increased cell density that typifies growing tumors (3–8). Simply stated, the higher the cell density, the more boundaries or restrictions the water experiences, and the lower the apparent diffusion coefficient (ADC), a value that is estimated from the DWIs. In this regard, several reports in the literature have demonstrated an inverse correlation between ADC and cell density in brain tumors (3–8). For gliomas, studies have shown that ADC values increase during successful cytotoxic therapy and decrease during tumor growth (9), suggesting ADC values can be used as a biomarker for brain tumor cellularity. Additionally, the National Cancer Institute has recognized the value of DWI estimates of ADC as a cellularity biomarker in cancer research (10). Despite the potential for DWI to provide ADC maps reflecting cell density, to date, DWI has not been used to directly estimate proliferation rates or cell migration rates.

A spatio-temporal mathematical model of glioma growth was developed based on the uncontrolled proliferation potential of gliomas, along with their ability to invade (11,12). Since gliomas do not typically metastasize outside of the brain, a partial differential equation can be used to describe brain tumor growth dynamics. Specifically, the rate of change in glioma cell density at a particular position in the brain can be described as being equal to the net motility of glioma cells plus the net proliferation. This model has led to a number of simulation studies meant to describe the *macroscopic* growth and invasion of gliomas under a variety of treatment conditions (13–15). These simulations typically ascribe a singular value (or mathematical tensor) for “cell diffusion rate” and a singular value for “proliferation rate” to describe growth and invasion of the tumor as a whole (Fig. 1). These values are then applied to the glioma growth equation to estimate cell density throughout the brain and predict radiographic recurrence.

Based on the strong spatial correlation between tumor cell density and ADC (16), we hypothesized that the glioma growth equation could be solved analytically, on a voxel-wise basis, for use in characterizing *microscopic* tumor growth and infiltration. Such cell invasion, motility, and proliferation level estimates (CIMPLE maps) that spatially quantify tumor growth dynamics would be invaluable for the characterization of human brain tumors, determining the level of aggressive malignant behavior, and quantifying the effects of treatment. In the current study, we describe a voxel-wise analytical solution to the glioma growth model using estimates of ADC, characterize the contribution of acquisition parameters and noise on the resulting measurements, and then demonstrate their potential when applied to a group of patients with gliomas.

Starting with a well-established glioma growth model first developed by Swanson *et al.* (12,17), we see the rate of change in glioma cell density is equal to the net invasion of glioma cells plus the net proliferation

Eq. 1

where *c* is cell density, *D* is the diffusion coefficient of migrating cells, *ρ* is the cell proliferation rate, and *t* is time. Based on evidence of a strong negative correlation between tumor cell density and ADC of water measured using DWI (3–8,18), ADC can be substituted into Eq. 1 to yield

Eq. 2

where *ADC* is the apparent diffusion coefficient of water as a three-dimensional scalar field (i.e. ADC image intensity acquired using DWI), *D* is the diffusion coefficient of migrating cells as a three-dimensional scalar field, and *ρ* is the cell proliferation rate as a three-dimensional scalar field. Expanding the definition of the *invasion* term in Eq. 2 results in the following equation (see Appendix A):

Eq. 3

The three-dimensional scalar fields *D* and *ρ* can now be theoretically solved for directly by applying this equation to two different time series of ADC maps; however, since a *rate of change in ADC* is necessary in Eq. 3, a minimum of three independent 3D ADC image intensities are necessary for a direct analytic solution. This implies that *ρ* and *D* describe the tumor dynamics over the particular time interval represented by the three ADC images. Solving for the proliferation rate of cells, *ρ*, for time point *n* − 1 in Eq. 3 yields

Eq. 4

where

Eq. 5

describes the rate of change in ADC for the previous time point (day *n − 1*) with respect to the time point at *n − 2* (oldest scan date). Substituting the value of *ρ* into Eq. 3 defined for the current time point, *n*, then solving for *D* results in:

Eq. 6

where

Eq. 7

The resulting equation (Eq. 6) is a nonhomogeneous, linear first order partial differential equation with respect to *D*. The *Method of Characteristics* (19) can now be used to solve the partial differential equation (see Appendix A) with respect to *D* along a set of characteristic curves. An estimate of *D* can be found as:

Eq. 8

Thus, using three ADC maps collected on days *t ^{n}, t^{n−1}*, and

Equation 8 suggests both the timing between ADC map acquisition (*dt*) and the noise in the ADC map itself may contribute to erroneous measurements of the cell diffusion coefficient, *D*, and proliferation rate, *ρ*. In other words, if the time between acquisitions of ADC maps is relatively small, but the noise content in the ADC maps is high, results may be falsely interpreted as high cell migration or proliferation rates. To explore the dependence of CIMPLE map parameter estimates on noise and the time between acquisition of ADC maps, we added Gaussian distributed random noise to the standard ADC map calculated from the diffusion tensor imaging atlas generated from 81 normal subjects as part of the International Consortium of Brain Mapping collaboration (ICBM-DTI-81 atlas, http://www.loni.ucla.edu/Atlases). These DWIs were originally collected on a Siemens 1.5T MR scanner and acquired using a single-shot echo planar imaging (EPI) sequence with sensitivity encoding (SENSE), a parallel imaging factor of 2.0, and two averages per subject. During acquisition an imaging matrix of 96×96 was used with a field-of-view (FOV) of 240×240 mm and a slice thickness of 2.5 mm acquired parallel to the anterior commissure-posterior commissure line (AC-PC), resulting in a total of 60 sections covering the entire cerebral hemispheres and brainstem without gaps between slices. Diffusion weighting was applied along 30 independent directions with a *b*-value of 1000 s/mm^{2}. The population-averaged DWIs and *b* = 0 s/mm^{2} image dataset, interpolated to an isotropic resolution of 1 mm, were used to generate a standard ADC atlas for simulation in the current study. Further details regarding this image dataset have been published previously (20,21).

Simulations consisted of adding Gaussian distributed noise to the standard ADC atlas, with a standard deviation of 0.1, 0.2, 0.5, or 1 μm^{2}/ms (approximately 15%, 25%, 75%, and 150% of average brain tissue ADC), and examining results generated assuming time intervals (*dt*) between ADC maps of 1 day, 1 week (7 days), 2 weeks (14 days), 1 month (30 days), or 3 months (90 days). In our analysis, we chose to focus primarily on the effects of time intervals between scans using a standard deviation of ADC = 0.2 μm^{2}/ms, since this was approximately the standard deviation of ADC within normal-appearing white and gray matter in our patient population (22). The particular timing of ADC acquisition (*dt*) chosen for simulations reflects typical follow-up MRI intervals for glioma patients at our Institution.

A total of 53 patients with gliomas who were previously enrolled in a study of MR perfusion imaging at our institution were enrolled in the current retrospective study. Three of these subjects, two patients with a WHO grade IV tumor (Patients A&B) and one with a WHO grade II tumor (Patient C), had concurrent disease progression and confirmed radiation necrosis, respectively, during the time span represented by CIMPLE maps. This was based on radiographic evidence, neurological deterioration, and biopsy in the case of confirmed radiation necrosis. Patient B and C showed evidence of disease progression and radiation necrosis following adjuvant temozolomide, whereas Patient A was evaluated prior to any treatment. These patients were excluded from quantitative analysis (total number of patients with quantitative analysis = 50), but the results are visualized in Fig. 4 for illustrative purposes. Of the 53 patients enrolled, a total of 15 had a WHO grade IV (glioblastoma multiforme), 17 had a WHO grade III (Anaplastic Astrocytoma, AA; Anaplastic Oligodendroglioma, AO; or mixed histological type), and a total of 21 patients had a WHO grade II (Astrocytoma, A; Oligodendroglioma, O; or mixed histological type). All patients gave informed consent according to the guidelines approved by the Institutional Review Board at our Institution. Table 1 summarizes the patient population used in the current study.

Representative post-contrast T1-weighted images, FLAIR images, cell diffusion coefficient estimates (*D*), and cell proliferation rate estimates (*ρ*) for a patient with an untreated glioblastoma multiforme (top row, WHO IV), recurrent glioblastoma **...**

Routine clinical MRI scans were collected for each patient, including a spoiled gradient recalled (SPGR) anatomical scan, pre-contrast T1-weighted (T1) scan, post-contrast T1-weighted (T1+C) scan, and a fluid-attenuated inversion recovery scan (FLAIR) collected on a 1.5-T MR scanner (Signa Excite, CVi, or LX; GE Medical Systems, Milwaukee, WI). 3D SPGR images were acquired with echo-time (TE)/repetition time (TR) = 3.16 msec/8.39 msec, number of averages (NEX) = 2, slice thickness = 1.3 mm collected contiguously, flip angle = 10 degrees, field-of-view (FOV) = 240 mm, and a matrix size of 256 × 192 (zero-padded and interpolated to 256 × 256) resulting in a total of 123 to 128 images. Diffusion weighted images (DWIs) were collected pre-contrast with TE/TR = 102.2 msec/8,000 msec, NEX = 1, slice thickness of 5 mm with 1.5 mm interslice gap, flip angle = 90 degrees, FOV = 240 mm, and a matrix size of 128 × 128 (reconstructed images were zero-padded and interpolated to 256 × 256) using either an EPI or PROPELLER readout. DWIs were acquired with *b* = 0 and 1,000 s/mm^{2}, using all gradients applied equally (isotropic). Effective diffusion times averaged 22 ms. After collecting the images, the apparent diffusion coefficient (ADC) images were calculated from the *b* = 1,000 s/mm^{2} and *b* = 0 images using AFNI.

All images for each patient were registered with a rigid method to their own baseline SPGR anatomical images (*Day 0*) using a mutual information algorithm and a 12-degree of freedom transformation using FSL (FMRIB, Oxford, UK; http://www.fmrib.ox.ac.uk/fsl/). Fine registration (1–2 degrees & 1–2 voxels) was then performed using a Fourier transform-based, 6-degree of freedom, rigid body registration algorithm (23) followed by visual inspection to ensure adequate alignment.

The creation of CIMPLE maps was incorporated into an AFNI pipeline using a combination of bash and AFNI calculation commands. Nearest neighbor interpolation was implemented in AFNI and used to estimate the spatial gradients of ADC. Resulting cell diffusion coefficient maps, *D*, and proliferation rate maps, *ρ*, were smoothed using a 3×3 median filter. Although the analytical solution allows for all possible values of *D* (negative *D* represents *divergence* of cells from the image voxel and positive *D* represents *convergence* of cells into the image voxel), we chose to report simply the absolute value of *D*, which corresponds to the total cell diffusion magnitude, in order to simplify visualization and interpretation. To restate, the absolute value of *D* represents the overall magnitude of tumor invasion and motility, whether or not it is into or out of a specific region (i.e. if tumor cells diffuse out of one voxel, they must be diffusing into another voxel. This represents tumor invasion in both voxel locations). Simulations were performed to determine the 95% confidence intervals for *D* and *ρ* using the precise number of days between each ADC map, which may be slightly different for each patient depending on imaging follow-up times. Voxels exhibiting a cell diffusion coefficient, *D*, and proliferation rate, *ρ*, beyond the 95% confidence interval (*p* < 0.05) were retained for visualization and analysis. Voxels having values less than the 95% confidence interval were excluded and presumed to be due to noise.

A total of three ADC maps from each patient were used in the analysis, except for the two patient examples outlined in the discussion section. It is important to note that neither tumor progression, as determined radiographically, nor a change in the treatment paradigm occurred during the time interval represented by the three ADC maps, unless otherwise noted (in 5 patients). Regions of FLAIR signal abnormality were chosen for analysis because many of the patients included in the study did not exhibit contrast-enhancement on post-contrast T1-weighted images. Further, T2 and FLAIR signal abnormalities are routinely used to visualize the presumed malignant tumor boundary in non-enhancing gliomas (24–26). “Hot spot” analysis was performed within FLAIR abnormal regions on 50 of the 53 patients to isolate the location of both the highest estimate of cell diffusion coefficient and proliferation rate for each patient. Kruskal-Wallis test and Dunn’s test for multiple comparisons were performed for cell diffusion coefficient and proliferation rate in order to compare values across different tumor grades.

Single-voxel MR spectroscopy (MRS) and/or chemical shift imaging (CSI) was performed in 9 of the 53 patients enrolled in the current study using a 1.5T MR scanner (Signa Excite, CVi, or LX; GE Medical Systems, Milwaukee, WI) and either a point resolved spectroscopy (PRESS) technique with echo times of 35 ms (for MRS) or a chemical shift selective imaging (CHESS) technique (for CSI). MRS/CSI data was collected during the same time frame reflected by CIMPLE maps.

In order to validate the quantitative estimate of proliferation rates obtained from CIMPLE maps, linear regression was performed between the mean proliferation rate (*ρ*) and the choline-to-N-Acetylaspartate (Cho/NAA) ratio obtained within similar image voxels on MRS/CSI. Specifically, Cho/NAA was calculated within the voxel chosen for MRS/CSI. An equivalent region was selected on *ρ* maps, corresponding directly to the precise voxel location chosen for MRS/CSI. The average value of *ρ* within this region was computed for each patient. Linear regression was then performed between Cho/NAA ratio and the average *ρ* value. Recent reports have also suggested there may be a relationship between cell motility and the choline-to-creatine ratio (Cho/Cr) as seen in gliomatosis cerebri (27,28), since higher creatine content suggests more available ATP and thus higher cell motility. To test whether the average cell diffusion coefficients (*D*) correlate with the ratio of Cho/Cr in a particular image voxel obtained with MRS/CSI, linear regression was again performed.

The final equation for calculation of the cell diffusion coefficient (Eq. 8) suggests that even a small change in ADC over a short period of time (*dt*) may be falsely interpreted as cells diffusing in or out of the image voxel space. Simulation results after applying Gaussian distributed noise with a standard deviation of 0.2 μm^{2}/ms to the ADC atlas confirmed this hypothesis. As the time interval between ADC maps increased, the noise distribution in both the proliferation rate (Fig. 2A) and cell diffusion coefficient (Fig. 2C) approached zero. Conversely, as the time interval between ADC maps decreased the noise was more widely distributed over all proliferation rates and cell diffusion coefficients.

Probability density functions for proliferation and diffusion rate. A) Probability density functions (PDF) of proliferation rate extracted from simulated data for various time intervals (*dt*) using Gaussian distributed noise with standard deviation of **...**

Qualitatively, the probability distribution represented by the cell proliferation rates (*ρ*) appeared to follow a Lorentzian distribution of the form:

Eq. 10

where *γ* is a scale parameter describing the half-width at half-maximum (HWHM) and *A* is an amplitude scaling factor. Quantitatively, non-linear, least-squares estimates using a Lorentzian distribution model implemented in MATLAB sufficiently described distributions represented by the cell proliferation rates at the various time points (see Fig. 2B). Alternatively, the probability distribution represented by the absolute value of cell diffusion coefficients (*D*) appeared to follow a Rayleigh distribution of the form:

Eq. 11

where *σ* is the mode of the distribution. This distribution is likely the result of quantifying the absolute value of *D*, as opposed to all values of *D*. Quantitative, non-linear, least-squares estimates using a Rayleigh distribution implemented in MATLAB sufficiently described the distributions represented by the cell diffusion coefficients at the various time points (see Fig. 2D).

The noise variability in the ADC maps affected the cell proliferation rate estimates (Fig. 2E), but estimates of cell diffusion coefficient (*D*) were largely unchanged with increasing noise (Fig. 2F). As expected, the estimate of cell proliferation rates converged toward zero as the variability of ADC approached zero, suggesting that ADC maps with minimal noise will result in the best estimates of cell proliferation rate. Table 2 summarizes the probability density function parameter estimates for the simulation studies.

MRS and/or CSI analysis was performed in 9 out of the 53 patients enrolled in the current study. Spatial regions with relatively high Cho/NAA on CSI images also appeared to have high estimates of cell proliferation rate (Fig. 3). Quantitatively, a strong linear correlation was found between Cho/NAA and estimates of cell proliferation rate (*Pearson’s Coefficient of Determination, R ^{2} = 0.9714, P < 0.0001*); however, no significant correlation was found between Cho/Cr and estimates of cell diffusion coefficient (

In general, CIMPLE maps appeared to demonstrate increasing cell diffusion and proliferation rates with increasing malignancy (Fig. 4 and Fig. 5). When CIMPLE maps were applied to patients with progressive disease (top two rows in Fig. 4), cell diffusion and proliferation rate estimates were particularly high. Spatial regions containing the highest cell proliferation rates appeared to be isolated on the rim of the contrast enhancing lesions, which is consistent with the literature (29). Alternatively, cell diffusion rates appeared the highest in regions central to FLAIR signal abnormality, suggesting a high rate of cell migration during recurrence. For a stable patient of the same tumor grade (third row in Fig. 4), cell diffusion rates were still elevated but more heterogeneous throughout the regions of FLAIR abnormality. Proliferation rate estimates were again highest in regions of contrast enhancement. In stable patients without contrast-enhancement (fourth row, Fig. 4), CIMPLE maps show low cell diffusion and proliferation rates. In a patient with a low grade glioma (fifth row in Fig. 4) showing slight contrast enhancement within central gray matter regions, CIMPLE maps suggest high cell diffusion and proliferation rates in the areas of slight contrast enhancement. As a qualitative validation, CIMPLE maps in a patient with histologically confirmed radiation necrosis show low estimates of cell diffusion and only negative proliferation rates (sixth row in Fig. 4). Additionally, the bottom row in Fig. 4 illustrates scatter plots showing the voxel-wise relationship between proliferation rate and cell diffusion coefficient in two patients with WHO grade IV, one with a relatively high proliferation rate and one with a relatively high cell diffusion rate.

“Hot spot” analysis comparing proliferation and diffusion rate estimates across tumor grades. A) Scatter-plot showing microscopic proliferation rate vs. diffusion rate for all patients (*n* = 50). Gray boundaries outline estimates from Swanson **...**

The cell proliferation rate versus cell diffusion coefficient chart in Fig. 5A illustrates “hot spot” analysis results for 50 of the 53 patients in the graphical form first presented by Swanson *et al*. (12,17). Interestingly, our *microscopic* estimates of *D* and *ρ* are on the same order of magnitude as parameters estimated from the *macroscopic* glioma growth simulation studies. Also, estimates of cell diffusion and proliferation rates in high grade gliomas (WHO III–IV) from the current study are well distributed within the regions that typify macroscopic simulation estimates for high grade gliomas (see “high grade” label in Fig. 5A). Low grade gliomas, however, appear to have higher cell diffusion and proliferation rates than those estimated in *macroscopic* glioma growth simulations (see “low grade” label in Fig. 5A); however, this may be partially due to nature of the “hot spot” analysis, which isolates only the highest values of each.

Analysis of cell diffusion coefficients in “hot spots” suggests significant differences between tumor grades (*Kruskal-Wallis, P<0.0001; *Fig. 5B). Specifically, WHO III and WHO IV tumors had significantly higher cell diffusion coefficients compared with WHO II tumors (*Dunn’s Multiple Comparison Test*, *P < 0.001*); however, no significant differences were found between WHO III and WHO IV tumors (*Dunn’s Multiple Comparison Test, P > 0.05*). The Spearman Rank Correlation test suggested a significant linear correlation between WHO grade and cell diffusion coefficients (*Spearman, R = 0.7633, P < 0.0001*). Similar to cell diffusion estimates, cell proliferation rates were significantly different between tumor grades (*Kruskal-Wallis Test, P=0.0160; *Fig. 5C). Again, WHO III and WHO IV tumors demonstrated slightly higher cell proliferation rate estimates compared to WHO II (*Dunn’s Multiple Comparison Test, P < 0.05*); however, no differences were found between WHO III and WHO IV tumors (*Dunn’s Multiple Comparison Test, P > 0.05*). Similar to estimates of cell diffusion, Spearman Rank Correlation test suggested a significant linear correlation between WHO grade and cell proliferation rate (*Spearman, R = 0.3815, P = 0.0063*).

Although DWI estimates of ADC are known to reflect the underlying tumor cell density, to date, no efforts have been made to estimate *microscopic* cell diffusion and proliferation rates on a voxel-wise basis using information collected in serial DWIs. Results from the current study suggest the noise variability in the ADC maps and the time between DWI scans influences measurement of proliferation rates, whereas only the time between DWI scans appears to influence measurements of cell diffusion coefficient (motility). Based on these findings, the CIMPLE maps with the highest degree of accuracy can be achieved when using high quality DWI datasets (high signal-to-noise ratio) gathered over a long time period. Intuitively, if tumors are scanned over a very short interval the tumor cells don’t have enough time to proliferate and diffuse at the magnitudes detectable on DWI. Based on simulation results, we recommend the inter-image interval of at least 2 weeks (14 days) in order to accurately estimate low grade glioma cell diffusion and proliferation rates from serial ADC maps (assuming a noise standard deviation of 0.2 μm^{2}/ms). At our institution, the standard deviation of ΔADC within typical DWIs has been shown to be approximately 0.2 μm^{2}/ms (22) and the time between DWI datasets is approximately 1–3 months for high-grade gliomas and 6 months to many years for low-grade gliomas. Based on these clinical guidelines and the results from the current study, the CIMPLE map estimates of proliferation and cell diffusion rates in suspected tumor regions are conveniently well beyond that believed to occur due to noise or acquisition parameters.

The correlation between spatially matched regions on CIMPLE maps to MRS/CSI provides strong validation for this new technique. As expected, cell turnover from rapidly dividing (and/or degrading) cell membranes resulted in a high Cho/NAA ratio in the same spatial locations exhibiting a high cell proliferation on CIMPLE maps. Similarly, regions of high cell proliferation rates appeared to be isolated in regions of contrast enhancement on post-contrast T1-weighted images, consistent with biopsy observations (30,31). Although not statistically significant, CIMPLE map estimates suggest a possible inverse correlation between Cho/Cr and cell diffusion rates, consistent with the empirical observations in infiltrative gliomatosis cerebri tumors (27,28). Interestingly, CIMPLE map estimates of cell diffusion rates appear to be highest at distinctly different spatial locations compared to regions of high proliferation rate estimates, which is consistent with the theory that proliferation and migration of astrocytomas are mutually exclusive behaviors (32). The scatter plots in Fig. 4 also illustrate this behavior.

Analysis of CIMPLE maps resulted in similar estimates of cell proliferation rates when compared to other estimates from the literature. Proliferation rates estimated from 29 tumors using the *macroscopic* glioma growth model were found to range from 1–32/yr depending on tumor grade (17), which is similar to our estimates of *microscopic* proliferation rates in 50 tumors of various grades (*WHO II, Median = 11.66/yr; WHO III, Median = 19.20/yr; WHO IV, Median = 25.28/yr; Minimum of all tumors = 2.23/yr; Maximum of all tumors = 80.01/yr*). *Macroscopic* diffusion rates estimated from the same 29 tumors yielded values ranging from 6–324 mm^{2}/yr (17), which is also similar to our estimates of *microscopic* diffusion rates (*WHO II, Median = 29.17 mm ^{2}/yr; WHO III, Median = 91.53 mm^{2}/yr; WHO IV, Median = 101.9 mm^{2}/yr; Minimum of all tumors = 7.088 mm^{2}/yr; Maximum of all tumors = 310.3 mm^{2}/yr*). The similarities between our estimates of proliferation and diffusion rates and the literature, combined with the strong correlation found between MR spectroscopy and proliferation rates, suggests that CIMPLE maps may reflect an accurate representation of the underlying

The primary limitations to this novel technique are the need for three ADC maps, and thus the assumption that the proliferation and diffusion do not change over this time frame, in order to provide an analytical solution to the glioma growth equation and the need for adequate time between scans in order to properly decipher cell diffusion and proliferation rates from background noise. For patients who appear radiographically and neurologically stable during three sequential time points this may be an adequate assumption; however, this assumption may not be valid in patients who show evidence of an accelerating disease process over multiple time points. Qualitatively, CIMPLE maps created in patients demonstrating progressive disease (Patients A and B in Fig. 4) appear to reflect the degree of disease progression, suggesting this assumption may be satisfied during disease progression.

An additional limitation to the current study is the strong reliance on precise image registration of ADC maps from the three days. The application of a median smoothing filter allows some misregistration to be tolerated; however, erroneous results may occur in progressing tumors exhibiting significant mass effect. To overcome these challenges, we chose to use two sequential automated registration steps followed by manual inspection; however, it is conceivable that an additional elastic (non-rigid) registration may also be of benefit in registration of problem datasets. Future studies aimed at quantifying the effects of elastic registration on resulting CIMPLE maps is warranted.

A large assumption in the current study is that DWI estimates of ADC are inversely correlated with tumor cellularity. Since many pathologies and clinical scenarios can alter ADC measurements, steps must be taken to rule out localized infection, subacute stroke, substantial gliosis, and tissue swelling from seizure activity prior to interpretation. Further, proper choice of *b*-values used to accurately estimate ADC must be considered. Per the recommendations of the National Cancer Institute Diffusion MRI Consensus Conference (10), three or more *b*-values (0 s/mm^{2}, >100 s/mm^{2}, and >500 s/mm^{2}) should be used for an adequate estimate of ADC that is also perfusion-insensitive (by using at least two *b*-values > 100 s/mm^{2}). Additionally, the choice of *b*-values greater than 1500 s/mm^{2} results in a multi-exponential signal decay, where a single estimate of ADC may not be appropriate. Unfortunately, the current study was performed retrospectively and so many of the consensus recommendations could not be implemented.

We present in the current study a voxel-wise analytical solution to a glioma growth model, which allows for direct spatial quantification of microscopic tumor proliferation and migration based on serial diffusion MRI scans in the same patient. This technique may be invaluable for assessing *in vivo* tumor dynamics and predicting response to treatment, to the extent tumor cellularity can be characterized by ADC.

Grant Support: NIH/NCI R21-CA109820; NIH/NCI R01-CA082500; MCW Advancing Healthier Wisconsin; MCW Translational Brain Tumor Research Program; MCW Cancer Center Fellowship

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