Assessment of Multi-Exponential Diffusion Features as MRI Cancer Therapy Response Metrics

Abstract

The aim of this study was to empirically test the effect of chemotherapy-induced tissue changes in a glioma model as measured by several diffusion indices calculated from non-monoexponential formalisms over a wide range of b-values. We also compared these results to the conventional two-point apparent diffusion coefficient (ADC) calculation using nominal b-values.

Diffusion weighted imaging was performed over an extended range of b-values (120–4000s/mm²) on intracerebral rat 9L gliomas prior to and following a single dose of 1,3-bis(2-chloroethyl)-1-nitrosourea. Diffusion indices from three formalisms of DW signal decay ((a) two-point analytical calculation using either low or high b-values, (b) a stretched exponential formalism and (c) a biexponential fit) were tested for responsiveness to therapy-induced differences between control and treated groups.

Diffusion indices sensitive to “fast diffusion” produced the largest response to treatment, which resulted in significant differences between groups. These trends were not observed for “slow diffusion” indices. Although the highest rate of response was observed from the biexponential formalism, this was not found to be significantly different from the conventional monoexponential ADC method. In conclusion, parameters from the more complicated non-monoexponential formalisms did not provide additional sensitivity to treatment response in this glioma model beyond that observed from the two-point conventional monoexponential ADC method.

Introduction

Diffusion-weighted magnetic resonance imaging (DWI) shows promise as an imaging biomarker for treatment response in glioma patients (1–9) as well as in a variety of other clinical tumor types (10–16). Routine in almost all preclinical and clinical scanners, diffusion maps can be generated from a minimum of two images acquired at low (b-value ~100s/mm²) and high (b-value ~1000s/mm²) diffusion weightings. Assuming monoexponential signal attenuation with b-value, the apparent diffusion coefficient (ADC) can be calculated analytically. The application of diffusion MRI for the detection of early tumor treatment response was first reported using a rodent glioma model using diffusion weightings at nominal b-values (≤1000s/mm²) (17). This initial report has been verified and expanded by ensuing publications using different tumor models and therapeutic agents (1,18,19), supporting the use of diffusion MRI as a sensitive imaging biomarker capable of detecting early cellular changes in treated tumors which precede macroscopic volumetric response.

The efficacy of this technique lies in its sensitivity to the molecular motion of water which is affected by cellular, sub-cellular and macromolecular elements that impede otherwise free diffusion of water. Thus therapeutic changes within the tumor at the cellular level can be monitored by serial diffusion measurements (20–23). Through thermal random motion, water molecules sample the surrounding micro-architecture within tissues at length scales (few microns) much smaller than typical MRI voxel resolution (~millimeter). The theoretical basis for diffusion analysis is that cell membranes and other structures hinder the diffusion of molecules (20,24). The magnitude of diffusion-driven displacement is altered by tortuosity and hindering effects, and can therefore be used to infer their presence and density. Studies have revealed that in biological systems water proton signal attenuation due to diffusion weighting does not follow monoexponential decay, and the deviation from monoexponential behavior is best observed at relatively high b-values (≥3000s/mm²). A more accurate description of signal attenuation with b-value over this wide b-value range requires more complex biophysical models (25–28).

An early interpretation of multi-exponential diffusion patterns was that water moves within two or more compartments representing pools of “fast” (extracellular) and “slow” (intracellular) diffusion components in the signal. At low b-values the “fast” diffusion pool dominates signal attenuation whereas at high b-values the “slow” diffusion pool dominates leading to a biexponential form for signal decay. Biexponential signal attenuation in DWI has been studied extensively in a variety of biological systems, and the physical mechanisms that govern non-monoexponential decay continue to be an area of debate. An alternative formalism for the non-monoexponential decay incorporates the underlying complexity in the diffusion medium as a continuous distribution of diffusion coefficients arising from a multiplicity of pools. Termed the “stretched-exponential” formalism, Bennett et al. (29) provided an analytical representation of the signal attenuation as a function of the probability density with a particular diffusion coefficient. Although this formalism can be used to infer the intravoxel diffusion heterogeneity within a biological system, it does not lend itself to straightforward association between biophysical compartments and signal decay. Although the “stretched-exponential” formalism has not been evaluated for its sensitivity to treatment response in tumors, this method has shown promise for characterizing tumors in brain cancer patients (30,31).

Research investigating the sensitivity of high b-value DWI for treatment assessment has shown promising results (5,32). Mardor and colleagues have demonstrated in patients with malignant brain lesions that the ratio of the diffusion coefficient from the “fast” pool and the “slow” pool signal fraction is highly sensitive to radiation induced changes in the tumor. This parameter not only demonstrated a significant change from baseline as early as one week post-treatment initiation but was capable of predicting clinical outcome in all of the studied patients (5). In contrast, conventional monoexponential ADC (in their study low and high b-values were 120 and 1200s/mm²) was found to be predictive of outcome in only half of the patient population studied. Sensitivity of high b-value DWI to treatment was also observed in a colon cancer mouse model (32). These authors used the area under the normalized non-monoexponential diffusion curve to quantify the diffusion characteristics of the tissue. This diffusion index was found to provide early prognostic information on animal responsiveness to treatment.

In this study, three non-monoexponential diffusion formalisms applied over an extended range of b-values (120–4000s/mm²) were tested against the conventional two-point ADC measurement to determine their sensitivity to therapy-induced changes of tissue using a rodent brain tumor model. Results showed similar time response curves for all diffusion indices following treatment. Although the highest fractional change following treatment was observed using the biexponential formalism, these results were not significantly different from those observed using the conventional two-point ADC calculation.

Methods

Animal Tumor Models

9L gliosarcoma cells were obtained from the Brain Tumor Research Center at the University of California in San Francisco. The cells were grown as monolayers in 10cm² sterile plastic flasks in DMEM with 10% fetal bovine serum, 100IU/mL penicillin and 100mg/mL streptomycin, and 2mmol/L L-glutamine in an incubator held at 37°C and 95%/5% air/CO₂ atmosphere. Before implantation, cells were harvested by trypsinization, counted, and re-suspended in serum-free medium for injection.

Tumor implantation was performed on Male Fischer 344 rats (Harlan Sprague-Dawley, Indianapolis, IN), weighing approximately 125–150g, as previously described (33). Briefly, animals were anesthetized with a ketamine/xylazine mixture (87/13mg/kg) administered intraperitoneal. A small incision was then made over the right hemisphere of the cranium. A 1 mm diameter burr hole was drilled through the skull using a high-speed surgical drill, and a 5μL suspension containing 1×10⁵ 9L cells was injected through the burr hole to a depth of 3mm. After injection of the cells, the burr hole was filled with bone wax to prevent extra-cranial extension of the tumor, and the surgical area was cleaned using 70% ethanol. Vetbond (3M, St. Paul, MN) was used to close the incision until healed.

Chemotherapy

Once the tumors reached 40–80mm³ as quantified using T₂-weighted MRI, pre-treatment diffusion-weighted images (details below) were acquired for all animals. At the time of treatment, 1,3-bis(2-chloroethyl)-1-nitrosourea (BCNU) (LKT Laboratories, St. Paul, MN) was freshly prepared and formulated to a final concentration of 5mg/mL BCNU in 10% ethanol. Subsequent to their pre-treatment DWI scan, animals either received a single bolus intraperitoneal injection of BCNU (9.98mg/kg; n=13) or 10% ethanol as the control vehicle (n=10). Typically tumors increased in volume by 400% over the duration of the study (two weeks post-treatment initiation) and euthanasia was accomplished by CO₂ overdose.

MRI Scans

During MRI examinations, animals were anesthetized with 1–2% isoflurane/air, and body temperature was maintained by blowing warm air through the bore of the magnet using an Air-Therm (World Precision Instruments, Sarasota, FL). MR scans were performed immediately prior to treatment and every three days thereafter using a 9.4T, 16cm horizontal bore Varian (Palo Alto, CA) Direct Drive system with a quadrature rat head coil (Doty Scientific, Inc., Columbia, SC). Diffusion-weighted (DW) images were acquired using a spin-echo sequence, with a navigator echo and gradient waveforms sensitive to isotropic diffusion (34), with the following parameters: repetition time (TR)/echo time (TE) = 4000/41ms, field of view (FOV) = 30mm, matrix size = 64×64, slice thickness = 2mm, number of slices = 8, sweep width = 50kHz, gradient pulse width = 10.5ms, gradient pulse separation = 25ms and b-values (x-gradient, y-gradient, z-gradient amplitudes) of 120 (5.3, 4.8, 4.2G/cm), 1200 (16.6, 15.0, 13.1G/cm), 1600 (19.1, 17.3, 15.6G/cm), 2000 (21.3, 19.2, 17.4G/cm), 3000 (25.9, 23.4, 21.3G/cm) and 4000s/mm² (29.7, 26.9, 24.6G/cm) with averages of 1, 1, 1, 1, 4 and 16, respectively. DWI scans were constrained to a total scan time of 2 hours based upon an institutionally approved animal protocol. In addition to the time constraint, only voxels in the tumor with a signal to noise (SNR) >6 at high diffusion weighting (i.e. b=4000s/mm²) were evaluated. The first step in maintaining this constraint was to determine a voxel size that provides adequate SNR and resolution within a 2 hour MR experiment. As observed in Chenevert et al., (3), 9L rodent brain tumors treated with 13.3mg/kg BCNU can exhibit an ADC as high as 1.5×10⁻³mm²/s. A sucrose phantom of 15% sucrose/water, with a measured ADC of 1.4×10⁻³mm²/s over the b=120–1200s/mm² range, was used to determine sequence parameters (i.e. FOV, slice thickness and averages) that provide a SNR>6 at b=4000s/mm² (35).

Post-Processing of Diffusion-Weighted Images

All MRI data were transferred to a PC, interpolated to a matrix size of 256×256 and analyzed using in-house software developed in MATLAB (The MathWorks, Inc., Natick, MA). Diffusion signal decay, found to follow a non-monoexponential trend in healthy brain tissue and tumor from our animal model, was analyzed using three diffusion approaches.

1. Two-Point Analytical Formalism

The simplest of all three techniques investigated captures the non-monoexponential trends observed in the DW images from a two-point subsampling of the signal decay curve using the following equation:

$${\text{ADC}}_{1-2}=ln\left(\frac{{\text{S}}_1}{{\text{S}}_2}\right)/({\text{b}}_2-{\text{b}}_1)$$ [1]

where S₁ and S₂ are the signal intensities at b-values b₁ and b₂, respectively, and ADC_1–2 is the diffusion coefficient obtained using b₁ and b₂. The conventional monoexponential ADC was calculated using b-values of 120 and 1200s/mm² (ADC_120–1200) which captures the rapid diffusion decay in the nominal b regime while avoiding perfusion effects observed at very low b-value (<100s/mm²). Slow diffusion decay, observed in the high b regime, was captured by determining the ADC using b-values of 2000 and 4000s/mm² (ADC_2000–4000). The ratio of ADC_2000–4000/ADC_120–1200, defined as R_TP, was used as an empiric index of non-monoexponential behavior derived from the piece-wise two-point formalism. An R_TP close to one implies monoexponential behavior whereas a decreasing R_TP implies greater disparity in signal decay between low-b and high-b regimes, thus greater multi-exponential behavior.

2. Stretched Exponential Formalism

This formalism defines the divergence of a diffusible particle from monoexponential trends as anomalous diffusion (29,36). Referred to as the stretched exponential, this formalism portrays molecular diffusion in a locally non-homogeneous environment, which is represented by the equation:

$$S(b)=S_{0}exp[-{(b·\mathit{DDC})}^{\alpha }]$$ [2]

S is the signal intensity at a given b-value, S₀ is the signal intensity with no diffusion-weighting, DDC is the distributed diffusion coefficient and α is the anomalous exponent bound between 0 to 1 (29,36). By inspection of equation [2], it should be clear an α = 1 is equivalent to mono-exponential diffusion signal decay. Conversely an α approaching 0 indicates a high degree of multi-exponential signal decay, thus α will be used as the non-monoexponential index derived from the stretched exponential formalism. This convention maintains consistency with Bennett et al.’s [31,32] definition of α as a diffusion heterogeneity index, although we remind the reader that a numerically high α value (~1) represents a low intravoxel diffusion heterogeneity approaching mono-exponential decay, whereas a numerically low α value represents a high degree of diffusion heterogeneity exhibited as multi-exponential decay. It is also worth emphasis that the term “heterogeneity” in this context refers to intravoxel heterogeneity of exponential decays, as opposed to intervoxel heterogeneity of diffusion coefficients as often is the case, particularly in tumor. Parameter maps of DDC and α were calculated by linearizing the stretched exponential equation and then fitting it to the DW images in a pixel-wise manner over all b-values using a linear least-squares technique.

3. Biexponential Model

Calculation of the biexponential diffusion components was performed by a pixel-wise fit to all DW images of the following equation:

$$\text{S}(\text{b})={\text{S}}_0\left({\text{V}}_1{\text{e}}^{(-\text{b}{\text{D}}_1)}+{\text{V}}_2{\text{e}}^{(-\text{b}{\text{D}}_2)}\right).$$ [3]

S and S₀ are signal intensities at a given b-value and no diffusion-weighting, respectively, D₁ and D₂ are the fast and slow diffusion coefficients, respectively, and V₁ and V₂ are the fast and slow signal fraction contributions, respectively. The fractional signal components are related by the expression V₂=1−V₁. The fit was performed using a non-linear least-squares technique.

Image Analysis

Volumes of interest (VOI) over the tumors were manually contoured on the low b-value DWI which exhibits T₂-weighted contrast and serves for quantification of tumor volume. Low SNR voxels were excluded prior to calculation of mean parameter values within the VOI from each diffusion formalism. To accomplish this, voxels having SNR≤6 on the b=4000s/mm² DWI were identified by software in a binary 3D mask. The mask was then applied to all DW images guaranteeing that only those voxels with a SNR>6 were evaluated. Regions of necrosis or blood pools, typically observed as hypointense on T₂-weighted images, were manually omitted from the VOIs. Parameter change with respect to treatment was assessed using the percent change of the mean of each parameter (100×[Post-therapy–Pre-therapy]/Pre-therapy).

Histology

An additional six animals were used for obtaining histology of the tumors for control (n=3) and treated animals (n=3) 6 days post-treatment. 9L tumors from these animals were placed in buffered formalin overnight, dehydrated in 70% ethanol, and subsequently embedded in paraffin. Tissue sections were prepared for histological processing by routine techniques. Briefly, paraffin sections (5μm thick) were cut on a microtome and heated for 20min at 65°C. Slides were de-paraffinized in xylene with three changes for 5min each, then re-hydratedthrough an alcohol gradient for 2min each (100% alcohol, 95% alcohol, 70% alcohol). Sections were first stained using a Gill’s 2X hematoxylin solution and then subsequently stained with eosin.

Statistics

A paired Student’s t-test was used to assess significance between the percent change in each parameter post-treatment initiation from pre-treatment values and between the percent change in similar parameters for each formalism at individual time points in the treated group. Group comparisons were assessed for each parameter at individual time points using an independent sample Student’s t-test. Treatment efficacy on overall survival was assessed by log-rank test and displayed using a Kaplan-Meier survival curves. All statistical computations were performed with a statistical software package (SPSS Software Products, Chicago, IL). Statistical significance was assessed at p<0.05.

Results

Representative DW images, acquired at b-values of 120, 1200, 3000 and 4000s/mm², are demonstrated in Figure 1. Using low b-value images, delineation of tumor extent allowed for tumor volumes to be measured over time. Although the tumor volume in treated animals did appear to have a slower rate of growth than controls, this did not result in statistical differences in tumor volume between groups at individual time points (Figure 2A). Nevertheless, as presented in Figure 2B the median survival of treated animals (9 days with a 95% confidence interval of 8.2–9.8 days) was found to be significantly longer than control animals (5 days with a 95% confidence interval of 3.8–6.2 days; p=0.001). Increased longevity in treated animals was consequent to tumor cell death, which was verified by histology. Histological sections of representative control and treated animals at day 6 post-treatment initiation are presented in Figure 3. Following BCNU treatment, fewer nuclei were observed in the treated tumor than control suggesting massive cell kill in the tumor volume of treated animals. An increase in pleomorphism and giant cells were also evident in the treated tumors. Tumor growth rate kinetics and histology were consistent with previous findings using the 9L gliosarcoma rat brain tumor model (5).

Figure 1.

Representative diffusion-weighted images of a rat brain harboring a 9L gliosarcoma acquired at b-values of (A) 120, (B) 1200, (C) 2000 and (D) 4000 s/mm². Images were independently scaled for better visualization at higher b-values.

Figure 2.

(A) Plot over time of the mean tumor volume. Data presented as mean±SEM. Significant difference in mean tumor volume between groups was assessed using an unpaired Student’s t-test. P-values are provided at individual time points. (B) Kaplan-Meier survival plot for overall survival is presented for control and treated animals. Controls are shown as solid line with diamond markers and treated are shown as dashed lines with square markers. Significant differences in overall survival were observed between groups as assessed using a log-rank test (p=0.001).

Figure 3.

Hematoxylin–eosin stained sections of intracerebral 9L tumors for representative (A) control and (B) BCNU treated animals on day 6 post-treatment.

Presented in Table 1 is a summary of the parametric indices generated from the three formalisms for control and treated tumor groups acquired at baseline. Significant differences in indices with similar diffusion properties were observed between all formalisms. In contrast, group comparisons did not result in statistical differences for any given parameter (Table 1: parameter values for control and treated animals are summarized in the top and bottom rows, respectively). To verify the accuracy of our biexponential fit to the data, pre-treatment values were calculated in healthy rat striatum. Biexponential results of D₁ and V₁ (0.88×10⁻³mm²/s and 0.79) were found to be comparable with previous values in brain tissue (28). In contrast, D₂ (0.42×10⁻³mm²/s) was 2.5x that of Niendorf and colleagues’ measurement of 0.165×10⁻³mm²/s (28).

Table 1.

Mean Parameter Values at Baseline

Two-Point	Stretched Exponential	Biexponential
ADC120–1200 [×10−3 mm2/s]	DDC [×10−3 mm2/s]	D1 [×10−3 mm2/s]
1.02 (0.02)	0.97 (0.02)	1.39 (0.05)
0.99 (0.03)	0.95 (0.02)	1.38 (0.05)
ADC2000–4000 [×10−3mm2/s]		D2 [×10−3 mm2/s]
0.71 (0.02)		0.68 (0.03)
0.69 (0.01)		0.62 (0.02)
RTP	α	V2
0.71 (0.02)	0.83 (0.01)	0.39 (0.01)*
0.71 (0.02)	0.87 (0.01)	0.35 (0.01)*
ADC120–1200/RTP	DDC/α	D1/V2
1.46 (0.07)	1.10(0.04)	3.99 (0.18)
1.48 (0.05)	1.09 (0.03)	3.74 (0.19)

Pre-treatment data are presented for controls (n=10; top) and treated (n=13; bottom) animals as means (SEM).

^*: The slow fractional signal intensity is provided due to similar trends to R_TP and α post-treatment initiation.

As shown in Figure 4, maps of parameters more sensitive to “fast diffusion” properties pre-treatment (left column) and 6 days following BCNU treatment (right column) were generated over the tumor volume and superimposed on T₂-weighted images. Top-row images (Figure 4A and D) represent ADC_120–1200, middle–row images (Figure 4B and E) are DDC and bottom-row images (Figure 4C and F) are D₁. In addition, the full time course of ADC_120–1200, DDC and D₁ expressed as percent change from pre-treatment values are illustrated in Figure 4G, H and I respectively. D₁ was found to be significantly larger than ADC_120–1200 and DDC at baseline (Table 1) and at day 6 post-treatment initiation as well as having, in absolute terms, a larger dynamic range (~1.5–3.0×10⁻³mm²/s) within the tumor volume allowing easier visualization of tumor features (Fig. 4C,F). As for the responsiveness of these indices to treatment, the percent change from baseline peaked at day 6 post-treatment initiation, followed by a descent toward baseline at day 9 (Fig. 4G,H,I). Near identical trends were observed for ADC_120–1200 and DDC with significant group and baseline value differences observed on days 3 and 6. Similar results were observed for D₁ except for the negligible group differences at day 3 post treatment initiation which is attributed to the slower rate of ascent from baseline (Fig. 4I). Although change in D₁ was found to be most responsive to treatment with a ~25% increase at day 6 from baseline, it was not found to be significantly larger at this time point or any other time point from what was observed for change in ADC_120–1200 and DDC (p=0.204 and p=0.711, respectively, for day 6).

Figure 4.

Representative maps and line plots of percent change in parameters sensitive to “fast diffusion” generated using (A,D,G) two-point, (B,E,H) stretched-exponential and (C,F,I) biexponential formalisms are provided. Diffusion maps, overlaid on T₂-weighted images of a rat brain, were acquired at days 0 (A,B,C) and 6 (D,E,F) post-treatment initiation. Line plots (G,H,I) consist of mean values and standard errors from control and treated groups over the entire experiment. Data are presented as the mean ± the standard error of the mean. The symbols † and * designate significant differences from baseline and between groups, respectively. Statistical significance was assessed at p<0.05.

Analogous parametric maps and line plots to Fig. 4 are illustrated in Figure 5 for quantities sensitive to the “slow diffusion” component of the decay curve, namely ADC_2000–4000 (Fig. 5A,C,E) and D₂ (Fig. 5B,D,F). In general, ADC_2000–4000 and D₂ showed little change in day 6 values from baseline (Fig. 5A–D). Percent change in the mean values over time corroborates observations found in the maps from the representative animal (Fig. 5A–D) with ADC_2000–4000 and D₂ peaking at less than 10% of baseline. Group differences were only observed at day 3 for D₂, partly attributed to a drop in control D₂. Interestingly, both ADC_2000–4000 and D₂ resulted in approximately 7% decrease from baseline on day 9 post-treatment initiation (Fig. 5H,I) which correlated with the descent back to pre-treatment values observed in diffusion coefficients sensitive to “fast diffusion” (Fig. 4).

Figure 5.

Representative maps of the “slow diffusion” coefficients and line plots of percentage change in parameters generated using (A,C,E) two-point and (B,D,F) biexponential formalisms are provided. Diffusion maps, overlaid on T₂-weighted images of a rat brain, were acquired at days 0 (A,B) and 6 (C,D) post-treatment initiation. Line plots (E,F) consist of mean values and standard errors from control and treated groups over the entire experiment. Data are presented as the mean ± the standard error of the mean. The symbols † and * designate significant differences from baseline and between groups, respectively. Statistical significance was assessed at p<0.05.

As discussed previously a comparison of the absolute numerical value of non-monoexponential metrics is not meaningful due to differences in how these parameters are defined. Qualitatively, R_TP was most sensitive to treatment exhibiting the largest percent drop from baseline values (Fig. 6). The remaining parameters showed similar qualitative trends from baseline to day 6 post-treatment initiation. A significant drop from pre-treatment values was observed at day 6 for R_TP (−11%), α (−7%) and V₂ (−6%). R_TP and α continued to have significantly lower values to baseline at day 9, which was not established by V₂ due to scatter in the data. Group differences were only found at day 6 for R_TP and α. The ratio of D₂ and D₁, as obtained from the biexponential formalism, provided analogous results to R_TP (data not shown). Although the mean value of D₂/D₁ decreased by more than 15%, these results were not found to be statistically different from the controls.

Figure 6.

Representative non-monoexponential metric maps and line plots of percentage change in parameters generated using (A,D,G) two-point, (B,E,H) stretched-exponential and (C,F,I) biexponential formalisms are provided. Metric maps, overlaid on T₂-weighted images of a rat brain, were acquired at days 0 (A,B,C) and 6 (D,E,F) post-treatment initiation. Line plots (G,H,I) consist of mean values and standard errors from control and treated groups over the entire experiment. Data are presented as the mean ± the standard error of the mean. The symbols † and * designate significant differences from baseline and between groups, respectively. Statistical significance was assessed at p<0.05.

Discussion and Conclusions

Diffusion-weighted MRI has shown potential as a surrogate biomarker for treatment response in cancer patients (37–40). Acquisition of diffusion maps is typically performed at relatively moderate diffusion-weighting, i.e. b-values that typically span the 0 to 1000s/mm² range. It is speculated that water diffusion measurements at higher b-values may provide increased sensitivity to relevant drug-induced changes in tumor composition by virtue of possible therapeutic alteration of cellular constituents responsible for the “slow diffusion” components of signal decay observed at relatively high b-values. The current study sought to determine the sensitivity of diffusion parameters derived from various mathematical formalisms of non-monoexponential water diffusion to treatment-induced tissue alteration following treatment of the 9L glioma model.

Previous work by our group using the 9L brain tumor model has shown that ADC calculated using moderate b-values can increase by up to 60% within a week following a single dose (13.3mg/kg) of BCNU (3). As measured in this study, parameters sensitive to “fast diffusion” showed similar trends following a single bolus (9.98mg/kg) of BCNU; all peaking by day 6 post-treatment initiation. The maximum percent change in parameter value from baseline was observed in D₁. This is expected since D₁ is a more specific measurement of “fast diffusion” than DDC and ADC_120–1200 which are not completely devoid of the “slow diffusion” properties in the signal decay curve. A positive therapeutic effect was confirmed by an increased overall survival (Fig. 2) as well as direct evidence from histological tumor sections comparing treated versus untreated tumors (Fig. 3). Another characteristic trend of ADC following treatment, that has been observed here and by others, is the temporally evolving descent to baseline values. This has been found to correlate with tumor cell repopulation, which has been reported in the literature (28). In contrast, those indices specifically sensitive to “slow diffusion” exhibited a negligible change post-treatment initiation until day 9 where a drop had occurred in both parameter values when compared to baseline. The lack of response following treatment and the sudden drop in ADC_2000–4000 and D₂ during cell repopulation, which is reflected in the diffusion coefficients sensitive to “fast diffusion” descent to baseline, is quite perplexing, suggesting independent mechanisms affecting the “fast” and “slow” diffusion properties of the tissue. Additional mechanisms, such as macrophage infiltration and clearance of macromolecules, may also contribute to our observations. Despite subtle variations in the trends of the parameters with either “fast” or “slow” diffusion properties, there was no significant difference in the percent change from baseline between diffusion parameters with like properties.

As opposed to the diffusion coefficients, the non-monoexponential metrics, R_TP, α and V₂, are defined differently and thus cannot have the same interpretation, even though they have similar trends. Consequent to the negligible change in ADC_2000–4000, R_TP is driven almost exclusively by ADC_120–1200 for most of the study. Not until day 9, did we see a divergence from this dependence, which is partly attributed to the mirrored descent observed in ADC_2000–4000 to ADC_120–1200 resulting in a negligible change in R_TP from day 6 to day 9 post-treatment initiation. Analogous but not significant results were observed for D₂/D₁. Large variability in the measurement of D₁ and D₂ (Figure 4 and 5) from the nonlinear fit most likely contributed to the non-statistical differences in D₂/D₁ between groups. In contrast, the slow diffusion signal fraction, V₂, defines the proportion of water signal in the slow compartment independent of water diffusivity. The drop in V₂ suggests shrinkage of the slow compartment volume fraction, conversely an expansion of the fast compartment volume fraction, following treatment initiation. The anomalous exponent α represents the deviation of signal attenuation from monoexponential behavior (α=1). This perturbation is assumed to be attributed to increased heterogeneity within the tissue. The decrease in α seen in Fig. 6H suggests an increase in tumor intravoxel heterogeneity that maximizes at day 6 and continues to day 9. Following treatment of the tumor with BCNU, a loss in tumor cellularity (Figure 3) further deviated the attenuation curve away from monoexponential behavior than what was observed from control and baseline values (Table 1). Due to similar trends in the non-monoexponential metrics and the lack of response to treatment of the “slow diffusion” indices, one may speculate that the “fast diffusion” properties within the tumor dominate what we observe for R_TP, α and V₂ following treatment. Various theories have been proposed to provide a physical account of the deviation in diffusion sensitive signal attenuation from monoexponential behavior in biological tissue (20,21,41). Use of these theories to determine the exact physical properties that govern non-monoexponential water diffusion warrants further investigation but is beyond the scope of this study.

Irrespective of the mechanisms driving non-monoexponential behavior in diffusion-sensitive signal attenuation, high b-value DWI provides additional advantages over conventional monoexponential ADC measurements that may provide a more sensitive biomarker for tumor treatment response and characterization. The conventional approach for measuring ADC, b-values around 1000s/mm², are hindered by the presence of highly diffuse tissue, such as cysts and necrotic areas, which may reside within or around the tumor volume adding increasing difficulty in localizing viable tumor. At higher b-values, these rapidly diffusing regions within tumors are essentially filtered out leaving only densely packed tumor which have lower ADC values. Recent research investigating the sensitivity of high b-value DWI for treatment assessment has shown promising results. Mardor and colleagues have demonstrated in patients with malignant brain lesions that the ratio of D₁ and V₂ (defined as R in Eqn. [1] of Mardor et al. (5)) is highly sensitive to radiation induced changes in the tumor. This parameter not only demonstrated a significant change from baseline as early as one week post-treatment initiation but was capable of predicting clinical outcome in all of the studied patients. In contrast, conventional monoexponential ADC (comparable to ADC_120–1200 in this study) was only capable of predicting response in about half of their patient population. It is not clear whether R, as presented by Mardor and colleagues, is driven by D₁ or V₂, or if D₁ or V₂ alone would provide ample sensitivity to predict tumor response to treatment since this analysis was not provided in their study. We further evaluated the approach proposed by Mardor and colleagues (5) (D₁/V₂) using our data. The percentage change in D₁/V₂ from baseline was approximately 42% in the treated group at day 6 post-treatment which was significantly different from controls (−2% (p=0.002)). Although, D₁/V₂ demonstrated a percentage change 1.6× greater than that generated by D₁, this increase was not statistically different (p=0.15). The probable cause for the lack of significance was the additional scatter in the data as a result of the nonlinear fit. Unlike DWI at moderate b-values, acquisition of diffusion-sensitized signal at b-values of >2000s/mm² is not trivial. This is attributed to the exponential loss of signal due to increased attenuation at high b-values. As signal approaches the noise floor artificial non-monoexponential trends in the signal profile are observed, adversely affecting the slow diffusion measurements. To accommodate these losses images must be acquired with sufficient SNR resulting in longer scan times, which may not result in patient compliance. Additional computational time is also required when fitting the biexponential formalism to the DWI data. In the current study approximately 30min per dataset were required for the voxel-wise nonlinear fit. The stretched-exponential formalism does not suffer from this deficiency since it can be linearized and solved using an algebraic solution of the linear least-squares. Numerically fitting two parameters for the stretched exponential model can also be more stable relative to fitting three parameters required by the biexponential model.

There are several limitations to our experimental approach that must be discussed. Scan time was limited to no more than two hours. This in turn limited the signal averaging, i.e. SNR, and range and number of b-values used per scanning session. As discussed, large slice thicknesses and small matrix sizes were used to maintain our self imposed constraint of SNR>6. This likely resulted in unavoidable partial volume averaging in the tumor, which would be less with thinner slices. Another area of concern was the lack of sufficiently high b-values, which are most sensitive to “slow diffusion” rates. This could have possibly led to an overestimation of the D₂ in the biexponential fit. Using the mean D₁ and D₂ determined at day 6 post-treatment initiation, we found in treated tumor tissue (1.7×10⁻³mm²/s and 0.6×10⁻³mm²/s), less than 4% (exp(−2000*0.0017) ~ 3.3%) of the fast diffusion signal was still present at a b-value of 2000s/mm², whereas 29.9% of the slow diffusion component signal was still available. Finally, the biexponential diffusion coefficients, D₁ and D₂, acquired here for healthy striatum varied by only a factor of 2, contrary to the factor of 5 to 10 typically observed in the literature. Based on observations of biexponential diffusion in rodent models, D₁ as measured in this study is in accordance with literature results (0.88 here, compared to 0.82 and 0.77 ×10⁻³mm²/s in the literature), whereas D₂ appears to be overestimated by a factor of 2 (0.43 here, compared to 0.17 and 0.18 ×10⁻³mm²/s in the literature) (28,41). The discrepancy in D₂ is most probably due to the lack of sufficiently high b-values used in this study. Finally, the filtering of low SNR voxels from our whole-tumor analysis may have inadvertently removed necrotic regions in the tumor. Signal intensity within regions of high diffusivity or short T2 due to blood products is prone to have low signal at high b-value. As discussed earlier, the contribution of noise in our data was reduced by filtering voxels whose SNR<6 on the highest weighted DWI (4000s/mm²). This maintained voxels having high SNR at high b-values but removed regions of high diffusivity or low SNR (i.e. necrosis, cystic and blood products) from the whole-tumor analysis. To avoid excessive loss of tumor volume while maintaining SNR≥6, image matrix size and slice thickness were set to maintain adequate SNR at a cost of resolution. The volume fraction of tumor analyzed at days 6 and 9 post-treatment initiation in treated animals was 93.5±11.3% and 94.5±9.9% (means±SD), respectively. Based on these values, filtering tumor regions that contribute to low SNR in DWI at high b-value did not result in excessive loss of tumor volume for our analysis in this study.

We have demonstrated the sensitivity of various non-monoexponential diffusion formalisms for monitoring early response to chemotherapeutic treatment for brain tumors in an animal model. The extent of the response varied, with the fast diffusion component of the biexponential formalism exhibiting the largest percent change from baseline than other diffusion coefficient; slightly more than was observed in the conventional monoexponential ADC and DDC measurements. However, for this 9L glioma model treated with a single dose of BCNU the more complicated formalisms provided no additional sensitivity to treatment response over what was observed using conventional monoexponential ADC measured over the standard modest b-value range.