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. 2015 Jun 2;277(2):489–496. doi: 10.1148/radiol.2015142156

Differentiation of Low- and High-Grade Pediatric Brain Tumors with High b-Value Diffusion-weighted MR Imaging and a Fractional Order Calculus Model

Yi Sui 1, He Wang 1, Guanzhong Liu 1, Frederick W Damen 1, Christian Wanamaker 1, Yuhua Li 1, Xiaohong Joe Zhou 1,✉
PMCID: PMC4627432  PMID: 26035586

The fractional order calculus model provides a set of novel diffusion parameters that improve differentiation of low- and high-grade pediatric brain tumors with a high predictive power.

Abstract

Purpose

To demonstrate that a new set of parameters (D, β, and μ) from a fractional order calculus (FROC) diffusion model can be used to improve the accuracy of MR imaging for differentiating among low- and high-grade pediatric brain tumors.

Materials and Methods

The institutional review board of the performing hospital approved this study, and written informed consent was obtained from the legal guardians of pediatric patients. Multi-b-value diffusion-weighted magnetic resonance (MR) imaging was performed in 67 pediatric patients with brain tumors. Diffusion coefficient D, fractional order parameter β (which correlates with tissue heterogeneity), and a microstructural quantity μ were calculated by fitting the multi-b-value diffusion-weighted images to an FROC model. D, β, and μ values were measured in solid tumor regions, as well as in normal-appearing gray matter as a control. These values were compared between the low- and high-grade tumor groups by using the Mann-Whitney U test. The performance of FROC parameters for differentiating among patient groups was evaluated with receiver operating characteristic (ROC) analysis.

Results

None of the FROC parameters exhibited significant differences in normal-appearing gray matter (P ≥ .24), but all showed a significant difference (P < .002) between low- (D, 1.53 μm2/msec ± 0.47; β, 0.87 ± 0.06; μ, 8.67 μm ± 0.95) and high-grade (D, 0.86 μm2/msec ± 0.23; β, 0.73 ± 0.06; μ, 7.8 μm ± 0.70) brain tumor groups. The combination of D and β produced the largest area under the ROC curve (0.962) in the ROC analysis compared with individual parameters (β, 0.943; D,0.910; and μ, 0.763), indicating an improved performance for tumor differentiation.

Conclusion

The FROC parameters can be used to differentiate between low- and high-grade pediatric brain tumor groups. The combination of FROC parameters or individual parameters may serve as in vivo, noninvasive, and quantitative imaging markers for classifying pediatric brain tumors.

© RSNA, 2015

Introduction

Pediatric brain tumor is one of the most devastating diseases among children, with an annual incidence of approximately three per 100 000 (1). In vivo assessment of pediatric brain tumors is commonly performed with magnetic resonance (MR) imaging, which involves acquiring T2-weighted, fluid attenuation inversion recovery (FLAIR), perfusion (eg, cerebral blood volume), and diffusion-weighted images together with unenhanced and contrast material–enhanced images to evaluate the location, extent, morphologic characteristics, angiogenesis, and cellularity of the tumor (2). Differentiating among low- and high-grade pediatric brain tumors with structural images that are based on T1- or T2-weighted contrast can be challenging because their signal intensity characteristics overlap (1,3). The use of semiquantitative perfusion imaging can considerably improve differentiation of low- and high-grade tumors with good sensitivity (eg, > 90%) (4). However, its specificity remains inadequate and is subject to the choice of cerebral blood volume threshold for different tumor types (5–8). With its ability to probe tissue cellularity, the use of diffusion imaging with apparent diffusion coefficient (ADC) was also attempted to distinguish low- from high-grade brain tumors (9–13). However, its clinical acceptance is limited because of substantial overlap of ADC values among different tumor grades for both adult and pediatric patients (1, 14–23). In addition, in some common locations (eg, the brain stem) of pediatric brain tumors, tumor grading on the basis of MR spectroscopy findings has been difficult because of excessive magnetic susceptibility perturbations and low spatial resolution (24–26). Because of the technical challenges and inadequate sensitivity or specificity of the use of MR imaging for distinguishing low- from high-grade pediatric brain tumors, invasive tissue biopsy remains the reference standard for tumor grading.

Differentiating low- from high-grade brain tumors without the use of invasive biopsy is important to optimize patient management strategies and determine the time point when benign tumors begin to transform into malignant lesions so that timely interventions can be applied without unnecessarily compromising patients’ quality of life. Unlike low-grade brain tumors, high-grade tumors (eg, medulloblastoma and anaplastic ependymoma) can have a higher degree of tissue heterogeneity, a characteristic that may not be adequately captured in a simple ADC value obtained from a monoexponential diffusion model (12,27).

Recognizing the limitation of ADC, several research groups developed a number of more sophisticated diffusion models to extract structural tissue information beyond what ADC can provide (28–40). One such model is known as the fractional order calculus (FROC) model, which yields a new set of parameters to describe the anomalous diffusion process in complex biologic tissues: diffusion coefficient (D), which is measured in square micrometers per millisecond; fractional order derivative in space (β); and a spatial parameter (μ), which is measured in micrometers (30,35). These parameters provide additional avenues for characterizing not only the diffusion process itself (D), but also the tissue structures (β and μ) through which water molecules diffuse. Therefore, the goal of this study is to demonstrate that the new parameters (D, β, and μ) from the FROC model can be used to improve the diagnostic accuracy of MR imaging for differentiating among low- and high-grade pediatric brain tumors.

Materials and Methods

One of the authors (H.W.) is an employee of GE Healthcare. However, this study was not financially supported by GE Healthcare, and the authors who are not GE Healthcare employees or consultants had full control of the data and information that may present a conflict of interest for the author who was a GE Healthcare employee.

Patients

The institutional review board of the performing hospital approved this study, and written informed consent was obtained from the legal guardians of pediatric patients. Seventy one patients with brain tumors were recruited between May 2010 and May 2012. The following inclusion criteria were used: (a) pediatric patients (≤18 years old) with evidence of a brain tumor and (b) no history of surgery, chemotherapy, or radiation therapy. The exclusion criteria comprised a lack of histopathologic results (n = 2) and corrupted diffusion images resulting from excessive motion (n = 2). The final patient group consisted of a total of 67 children (aged 4 months to 13 years old), with 19 girls. According to the 2007 World Health Organization (WHO) classification, 28 patients had low-grade (WHO grade I or II) tumors (11 pilocytic astrocytomas, 11 astrocytic astrocytomas, three ependymomas, and three dysembryoplastic neuroepithelial tumors), and 39 patients had high-grade (WHO grade III or IV) tumors (18 medulloblastomas, four teratoid/rhabdoid tumors, two glioblastoma multiformes, four germinomas, two pineoblastomas, three anaplastic ependymomas, two ependymoblastomas, two primitive neuroectodermal tumors, one anaplastic astrocytoma, and one malignant teratoma) (41).

Image Acquisition

All MR imaging examinations were performed on a 3-T imager (GE Healthcare, Milwaukee, Wis) with a commercial eight-channel phased-array head coil (In Vivo, Gainesville, FL). The imaging protocol included unenhanced T1-weighted images, T2-weighted fast spin-echo images, and diffusion-weighted images with multiple b values for FROC model analysis, followed by contrast-enhanced T1-weighted images. Diffusion-weighted images were obtained with a single-shot spin-echo echoplanar sequence with 12 b values (0, 10, 20, 50, 100, 200, 400, 800, 1200, 2000, 3000, and 4000 sec/mm2). With each b value, a Stejskal-Tanner diffusion gradient was successively applied along the x-, y-, and z-axes to obtain a trace-weighted image to minimize the influence of diffusion anisotropy. The following key data acquisition parameters were used: repetition time msec/echo time msec, 4700/100; separation between two diffusion gradient lobes, Δ = 38.6 msec; duration of each diffusion gradient, δ = 32.2 msec; field of view, 22 cm; section thickness, 5 mm; matrix size, 128 × 128; and imaging time, 3 minutes.

Image Analysis

According to the FROC model, the voxel intensity in a diffusion-weighted image is given by the equation

graphic file with name radiol.2015142156.equ1.jpg

where S0 is the signal intensity without diffusion weighting, Gd is the diffusion gradient amplitude, δ is the diffusion gradient pulse width, and Δ is the gradient lobe separation (30,35). D, β, and μ were previously defined.

Multi-b-value diffusion images were fitted to the FROC diffusion model pixel by pixel by using a Levenberg-Marquardt nonlinear fitting algorithm (42). In the fitting, D (which reflects intrinsic diffusivity) was estimated by a monoexponential model with data acquired at lower b values (≤1200 sec/mm2). After D was determined, β and μ were subsequently obtained from the pixel-wise nonlinear fitting by using all b values.

Regions of interest (ROIs) were placed on both the normal-appearing gray matter (eg, the contralateral thalamus) as an internal control and the solid part of tumors by an experienced neuroradiologist (G.L., with 6 years of experience in clinical MR imaging) to avoid possible confounding effects introduced by necrosis, cyst, hemorrhage, edema, or calcification. The tumor ROIs were selected on the T2-weighted echoplanar images (ie, diffusion-weighted images in which b = 0) and then propagated to the corresponding D, β, and μ maps for statistical analysis. All image processing and analysis were performed with customized software developed with Matlab (MathWorks, Natick, Mass).

Statistical Analysis

For each patient, mean values and standard deviations of D, β, and μ were calculated from the normal-appearing gray matter and tumor ROIs. On the basis of these values, the low- and high-grade brain tumor groups were compared by using the Mann-Whitney U test, with significance set at P < .05.

Multivariate logistic regression was used to combine the FROC parameters (D, β, and μ) for differentiating low- and high-grade tumors. This method assumes that the probability of being a high-grade tumor (P0) follows the logistic model P0 = exp (a0 + a1D + a2β + a3μ)/[1 + exp (a0 + a1D + a2β + a3μ)], where a0 is a constant, and a1, a2, and a3 are the regression coefficients of D (in μm2/msec), β (dimensionless; 0 < β≤1), and μ (in μm), respectively. The regression coefficients were estimated by using a maximum likelihood method (43). Receiver operating characteristic (ROC) analysis was performed to determine the area under the ROC curve (AUC) and assess the performance of each individual FROC parameter, as well as the combination of selected FROC parameters (represented by P0), for tumor differentiation. All statistical analyses were carried out by using SPSS software (SPSS, Chicago, Ill).

Results

Comparison of Representative Patients in Each Group

Figure 1 shows a set of images obtained in a representative patient with a low-grade brain tumor (WHO grade II astrocytoma) in the left cerebellum. Axial unenhanced T1-weighted FLAIR (Fig 1, A), contrast-enhanced T1-weighted FLAIR (Fig 1, B), and T2-weighted echoplanar (Fig 1, C) MR images show an enhancing lesion with accompanying cysts. D, β, and μ maps (Fig 1, D–F) show higher values in the solid tumor than in the surrounding healthy tissues. Figure 2 shows a set of images obtained in a representative patient with a high-grade tumor (WHO grade IV medulloblastoma) in the right cerebellum. On unenhanced and contrast-enhanced T1-weighted FLAIR (Fig 2, A and B) and T2-weighted echoplanar (Fig 2, C) images, the lesion had similar contrast and signal characteristics as those in the lesion in Figure 1. However, in this high-grade tumor, the D, β, and μ values (Fig 2, D–F) were considerably lower than they were in the low-grade tumor (Fig 1, D–F).

Figure 1:

Figure 1:

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Low-grade brain tumor (WHO grade II astrocytoma) in the left cerebellum of a 4-year-old boy. Top row: A, axial unenhanced T1-weighted FLAIR; B, contrast-enhanced T1-weighted FLAIR; and, C, T2-weighted echoplanar MR images show an enhancing lesion surrounded by cysts. Bottom row: MR images with quantitative FROC maps show, D, D values; E, β values; and, F, μ values. The solid tumor is enclosed within the ROI (red area in C), which was initially drawn on the T2-weighted echoplanar image with the aid of T1-weighted images and propagated into the FROC maps. The D, β, and μ values within the ROI are higher than those for a high-grade brain tumor (cf Fig 2).

Figure 2:

Figure 2:

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High-grade brain tumor (WHO grade IV medulloblastoma) in the right cerebellum of a 6-year -old boy. Top row: A, axial unenhanced T1-weighted FLAIR; B, contrast-enhanced T1-weighted FLAIR; and, C, T2-weighted echoplanar MR images show an enhancing lesion. Bottom row: MR images with quantitative FROC maps show, D, D values; E, β values; and, F, μ values. The solid tumor is enclosed with the ROI (red area in C), which was initially drawn on the T2-weighted echoplanar image with the aid of T1-weighted images and propagated into the FROC maps. The signal characteristics of anatomic images (A–C) are similar to those of the low-grade tumor (cf Fig 1). However, the FROC maps (D–F) show substantial differences between low- and high-grade tumors.

Group Comparison on the basis of FROC Parameters

After calculating the mean FROC parameter values for each tumor ROI, the group mean and standard deviation for each patient group were obtained (Table 1). Because D in the FROC model was obtained from a monoexponential fitting with lower b values (≤1200 sec/mm2), this parameter was also used to approximate ADC, which is typically obtained with b = 0 and b = 1000 sec/mm2. This approximation was justified because D agreed with ADC within 4%. The comparison of the two tumor groups is shown in Figure 3. Typically, for each of the three FROC parameters, low-grade tumors had higher values than did high-grade tumors. For all three FROC parameters, there was a significant difference between low- and high-grade tumors (P ≤ .007). In comparison, for the internal control (normal-appearing gray matter), there was no significant difference in the FROC parameters between the two tumor groups (P ≥ .14) (Table 2).

Table 1.

Values of FROC Diffusion Parameters for Low- and High-Grade Pediatric Brain Tumors

graphic file with name radiol.2015142156.tbl1.jpg

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Note.—Unless otherwise indicated, data are the mean plus or minus the standard deviation. P values were determined with the Mann-Whitney U test.

Figure 3:

Figure 3:

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Scatter diagrams and boxplots show the mean values of FROC parameters (D, β, and μ) for low- and high-grade pediatric brain tumors. Boxes show the 25th and 75th percentiles, as well as the median (red line). Vertical end bars denote the range of data except the outliers (ie, values more than 1.5 times the box length, larger than the 75th percentile, or smaller than the 25th percentile; +). * = significant differences (P < .05) between low- and high-grade tumors.

Table 2.

Values of FROC Diffusion Parameters for Normal-Appearing Gray Matter in Patients with Low- and High-Grade Pediatric Brain Tumors

graphic file with name radiol.2015142156.tbl2.jpg

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Note.—Unless otherwise indicated, data are the mean plus or minus the standard deviation. P values were determined with the Mann-Whitney U test.

ROC Analysis

Multivariate logistic regression analysis showed that D and β were significant variables (P = .028 and P = .004, respectively), but μ was not (P = .252) (44). Thus, μ was excluded from the logistic model. The constant and coefficients of D and β for the logistic model were 25.1, −3.8, and −25.9, respectively. For each tumor, the probability of being high-grade (P0) was calculated from its D and β values with the formula P0 = exp (25.1–3.8D-25.9β)/[1 + exp (25.1–3.8D−25.9β)]. Thereafter, P0 was used to represent the combination of D and β parameters in the ROC analysis.

Figure 4 shows the ROC curves for each FROC parameter used to differentiate low- and high-grade pediatric brain tumors. β (0.943) had a higher AUC than both D (0.910) and μ (0.763). Also shown in Figure 4 is the ROC curve for the combined β and D values. This combination produced the highest AUC (0.962), indicating an improved performance for tumor differentiation. Table 3 summarizes the area, 95% confidence interval, and asymptotic significance (P value) for each ROC curve.

Figure 4:

Figure 4:

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Graph shows the ROC curves for D, β, μ, and the combination of D and β for differentiating low- and high-grade pediatric brain tumors. The AUCs for the corresponding ROC curves are 0.910 (D), 0.943 (β), 0.763 (μ), and 0.962 (D+β).

Table 3.

Area, 95% Confidence Interval, and Asymptotic Significance of ROC Curves by using FROC Parameters for Differentiating Low- and High-grade Pediatric Brain Tumors

graphic file with name radiol.2015142156.tbl3.jpg

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Note.—P0 = a combination of D and β.

Figure 5 shows the scatterplots for β versus D (Fig 5, A) and μ versus D (Fig 5, B), with blue crosses and red triangles denoting low- and high-grade tumors (which were histopathologically proved), respectively. Of the 67 patients, tumors were correctly differentiated by using D (Dthreshold, 1.01 μm2/msec) in 54 patients (80.6%), whereas they were correctly differentiated by using β (βthreshold, 0.81) in 59 patients (88.1%) and μ (μthreshold, 8.36 μm) in 47 patients (70.1%) (Fig 5, A and B). The two tumor groups were best differentiated by using a combination of D and β, which had an accuracy of 92.5% (n = 62), as indicated by the blue line in Figure 5, A.

Figure 5:

Figure 5:

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Scatter plots show, A, β versus D and, B, μ versus D for the low- (blue cross) and high- (red triangle) grade pediatric brain tumor groups. The cutoff for D is 1.01 mm2/sec (green line), and the cutoff for β is 0.81 (purple line). A shows a weaker correlation between D and β (R = 0.685, P < .001), whereas B shows a very strong correlation between D and μ (R = 0.862, P < .01). Blue line = cutoff for D and β.

Discussion

Our results demonstrate the feasibility of using a set of novel FROC diffusion parameters (D, β, and μ) to differentiate low- and high-grade pediatric brain tumors. ROC analysis shows that β had the highest predictive accuracy when each parameter was used alone. More importantly, multivariate logistic regression analysis revealed that the combination of D and β further improved predictive accuracy. These results are important because the new FROC parameters improved the predictive power by 7.5–11.9 percentage points compared with the existing approach, in which diffusion coefficient D is used alone. With these results, our study demonstrates the feasibility of using the FROC diffusion model to grade pediatric brain tumors without relying on invasive biopsy.

Although ADC values are increasingly used as an imaging marker to probe tissue changes during the neoplastic process or monitor tumor response to therapy, they overly simplify the anomalous diffusion process in complex biologic tissues without recognizing intravoxel structural heterogeneity, which is particularly relevant in brain tumors (12, 16,45–47). Given the intravoxel structural heterogeneity, it is plausible that diffusion process within a voxel should also exhibit diffusion heterogeneity. Unlike the monoexponential diffusion model, the FROC model recognizes this diffusion heterogeneity through its parameters. Previous studies have suggested that β value inversely correlates with increased intravoxel heterogeneity (30,35,48). Our study established the role of β in distinguishing low- from high-grade pediatric brain tumors on the basis of varying degrees of structural heterogeneity. Recently, Kwee et al (31) studied high-grade gliomas in adults by using a stretched-exponential diffusion model, another non-Gaussian diffusion model that was empirically developed, rather than by fractionalizing Fick diffusion equation (33). The empirical heterogeneity index α introduced by Bennett et al (33) is mathematically equivalent to β in our study. Despite this similarity, by focusing on pediatric brain tumors (mainly medulloblastoma), our study is different than Kwee’s study on adults. Consequently, our β value (0.72 ± 0.06) for the high-grade pediatric tumor group was different than that reported by Kwee et al (α = 0.58 ± 0.08) in adults with high-grade glioma, which primarily consisted of glioblastoma multiforme.

The μ value has been suggested as a measure of diffusion mean free length, but its biologic interpretation remains intriguing (35). In our current study, μ did not significantly contribute to the differentiation of low- and high-grade pediatric brain tumors, a finding that can be explained by the strong coupling between the diffusion coefficient (D) and mean free length (μ) (35,49). By using all of the patient data, a strong linear correlation was observed between D and μ, with a high Pearson correlation coefficient (R = 0.862, P < .001), which demonstrated substantial coupling between D and μ. Our results showed that μ decreased from low-grade (less restricted diffusion) to high-grade (more restricted diffusion) brain tumors, a finding consistent with the fact that the mean free length of diffusing molecules is shortened in high-grade brain tumors because of its relatively higher cellularity and vascular density (13). It is interesting to note that β was less correlated with D (R = 0.685, P < .001) or μ (R = 0.637, P < .001). The weaker correlation is particularly useful because it provides two parameters: diffusion rate through D and heterogeneity through β, which improve sensitivity, specificity, and predictive accuracy.

Our study has several limitations. First, information on tumor location and patient age was not included as contributing parameters in the multivariate analysis because the limited patient number was inadequate to sample across these parameters. Nonetheless, we evaluated the locations of the 67 pediatric brain tumors with T1-, T2-, and diffusion-weighted images, and, together with patient age, found that 82.1% (n = 55) of the tumors were correctly differentiated. This accuracy is comparable to that of using D alone (80.6%) but lower than that for a combination of D and β (92.5%). Thus, the quantitative approach of using FROC parameters offers an advantage over the conventional method of relying on tumor location and patient age. Second, a direct correlation between MR images and histologic sections was not performed. Although the difference of FROC parameters between the two tumor groups is significant, we were not able to pinpoint the histologic basis (eg, cell size distribution, cytoplasm ratio, or extent of necrosis) for the change in each individual parameter. Correlation between the FROC parameters and histologic characteristics is an area of future investigations. Third, the sample size for each subtype of brain tumors is relatively small, which precluded us from exploring potential applications of FROC parameters for distinguishing among subtypes of brain tumors within the low- or high-grade tumor group. The limited number of patients also compromised our ability to use an independent set of subjects to validate sensitivity and specificity, which resulted in a study prevalence that does not necessarily reflect the prevalence encountered in common clinical practice. Lastly, we intentionally eliminated the effect of diffusion anisotropy by performing the FROC analysis on only the trace-weighted images to limit the acquisition time to 3 minutes, despite the fact that the FROC model is capable of characterizing diffusion anisotropy. We recognize that, even with 3 minutes, the acquisition time needed by the FROC model is longer than what is required for a simple monoexponential ADC analysis. However, the ability to obtain more than one quantitative parameter for improved tumor differentiation well justifies the longer, yet clinically acceptable, acquisition time.

In conclusion, we demonstrated that the FROC model provides a set of novel diffusion parameters that improve differentiation of low- and high-grade pediatric brain tumors with a high predictive power (AUC, 0.962) and accuracy (92.5%). In the future, these parameters will be extended to characterize other types of tumors.

Advances in Knowledge

  • ■ A new set of parameters from a fractional order calculus (FROC) diffusion model provides a measure of intravoxel heterogeneity (β), as well as water diffusivity (D), in tumor tissues.

  • ■ When used together, the FROC parameters can differentiate between low- and high-grade pediatric brain tumors with an area under the receiver operating characteristics (ROC) curve of 0.962 in an ROC analysis.

Implication for Patient Care

  • ■ The combination of multiple as well as individual FROC parameters may become an in vivo, noninvasive, and quantitative imaging marker for classifying pediatric brain tumors.

Acknowledgments

Acknowledgments

The authors are grateful to Drs Kejia Cai, Frederick C. Damen, William Schey, and Hui Xie for helpful discussions and manuscript editing.

Received September 9, 2014; revision requested October 15; revision received December 23; accepted January 29, 2015; final version accepted February 19.

Supported in part by grants from the University of Illinois Center for Clinical and Translational Sciences and Xinhua Hospital, Shanghai, China.

Funding: This research was supported by the National Institutes of Health (grants UL1RR029879, R01 NS057514, and 1S10RR028898).

Disclosures of Conflicts of Interest: Y.S. disclosed no relevant relationships. H.W. disclosed no relevant relationships. G.L. disclosed no relevant relationships. F.W.D. disclosed no relevant relationships. C.W. disclosed no relevant relationships. Y.L. disclosed no relevant relationships. X.J.Z. Aactivities related to the present article: none to disclose. Activities not related to the present article: consultant for Horizon Medical Physics Services; royalties from Elsevier; retirement with General Electric; stock in Apple and Google; travel expenses from the International Society for Magnetic Resonance in Medicine, the Chinese Academy of Sciences, and the National Institutes of Health; owner of Horizon Medical Physics Services. Other relationships: none to disclose.

Abbreviations:

ADC
apparent diffusion coefficient
AUC
area under the ROC curve
FLAIR
fluid attenuation inversion recovery
FROC
fractional order calculus
ROC
receiver operating characteristic
ROI
region of interest
WHO
World Health Organization

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