Water diffusion heterogeneity index in the human brain is insensitive to the orientation of applied magnetic field gradients

The biology underlying molecular imaging in oncology: from genome to anatome and back again

Cancers are complex, evolving, multiscale ecosystems that are characterized by profound spatial and temporal heterogeneity. The interactions in cancer are non-linear in that small changes in one variable can have large changes on another. These multiple interacting phenotypes and spatial scales can best be understood with appropriate mathematical and computational models. Imaging is central to this investigation because it can non-destructively and longitudinally characterize spatial variations in the tumour phenotype and environment so that the system dynamics over time can be captured quantitatively.

Comparison of apparent diffusion coefficients and distributed diffusion coefficients in high-grade gliomas

PURPOSE

To compare apparent diffusion coefficients (ADCs) with distributed diffusion coefficients (DDCs) in high-grade gliomas.

MATERIALS AND METHODS

Twenty patients with high-grade gliomas prospectively underwent diffusion-weighted MRI. Traditional ADC maps were created using b-values of 0 and 1000 s/mm(2). In addition, DDC maps were created by applying the stretched-exponential model using b-values of 0, 1000, 2000, and 4000 s/mm(2). Whole-tumor ADCs and DDCs (in 10(-3) mm(2)/s) were measured and analyzed with a paired t-test, Pearson's correlation coefficient, and the Bland-Altman method.

RESULTS

Tumor ADCs (1.14 +/- 0.26) were significantly lower (P = 0.0001) than DDCs (1.64 +/- 0.71). Tumor ADCs and DDCs were strongly correlated (R = 0.9716; P < 0.0001), but mean bias +/- limits of agreement between tumor ADCs and DDCs was -0.50 +/- 0.90. There was a clear trend toward greater discordance between ADC and DDC at high ADC values.

CONCLUSION

Under the assumption that the stretched-exponential model provides a more accurate estimate of the average diffusion rate than the mono-exponential model, our results suggest that for a little diffusion attenuation the mono-exponential fit works rather well for quantifying diffusion in high-grade gliomas, whereas it works less well for a greater degree of diffusion attenuation.

Intravoxel water diffusion heterogeneity imaging of human high-grade gliomas

This study aimed to determine the potential value of intravoxel water diffusion heterogeneity imaging for brain tumor characterization and evaluation of high-grade gliomas, by comparing an established heterogeneity index (alpha value) measured in human high-grade gliomas to those of normal appearing white and grey matter landmarks. Twenty patients with high-grade gliomas prospectively underwent diffusion-weighted magnetic resonance imaging using multiple b-values. The stretched-exponential model was used to generate alpha and distributed diffusion coefficient (DDC) maps. The alpha values and DDCs of the tumor and contralateral anatomic landmarks were measured in each patient. Differences between alpha values of tumors and landmark tissues were assessed using paired t-tests. Correlation between tumor alpha and tumor DDC was assessed using Pearson's correlation coefficient. Mean alpha of tumors was significantly lower than that of contralateral frontal white matter (p = 0.0249), basal ganglia (p < 0.0001), cortical grey matter (p < 0.0001), and centrum semiovale (p = 0.0497). Correlation between tumor alpha and tumor DDC was strongly negative (Pearson correlation coefficient, -0.8493; p < 0.0001). The heterogeneity index alpha of human high-grade gliomas is significantly different from those of normal brain structures, which potentially offers a new method for evaluating brain tumors. The observed negative correlation between tumor alpha and tumor DDC requires further investigation.

Langevin equation approach to diffusion magnetic resonance imaging

The normal phase diffusion problem in magnetic resonance imaging (MRI) is treated by means of the Langevin equation for the phase variable using only the properties of the characteristic function of Gaussian random variables. The calculation may be simply extended to anomalous diffusion using a fractional generalization of the Langevin equation proposed by Lutz [E. Lutz, Phys. Rev. E 64, 051106 (2001)] pertaining to the fractional Brownian motion of a free particle coupled to a fractal heat bath. The results compare favorably with diffusion-weighted experiments acquired in human neuronal tissue using a 3 T MRI scanner.

A multi-compartmental SE-BOLD interpretation for stimulus-related signal changes in diffusion-weighted functional MRI

A new interpretation is proposed for stimulus-induced signal changes in diffusion-weighted functional MRI. T(2)-weighted spin-echo echo-planar images were acquired at different diffusion-weightings while visual stimulation was presented to human volunteers. The amplitudes of the positive stimulus-correlated response and post-stimulus undershoot (PSU) in the functional time-courses were found to follow different trends as a function of b-value. Data were analysed using a three-compartment signal model, with one compartment being purely vascular and the other two dominated by fast- and slow-diffusing molecules in the brain tissue. The diffusion coefficients of the tissue were assumed to be constant throughout the experiments. It is shown that the stimulus-induced signal changes can be decomposed into independent contributions originating from each of the three compartments. After decomposition, the fast-diffusion phase displays a substantial PSU, while the slow-diffusion phase demonstrates a highly reproducible and stimulus-correlated time-course with minimal undershoot. The decomposed responses are interpreted in terms of the spin-echo blood oxygenation level dependent (SE-BOLD) effect, and it is proposed that the signal produced by fast- and slow-diffusing molecules reflect a sensitivity to susceptibility changes in arteriole/venule- and capillary-sized vessels, respectively. This interpretation suggests that diffusion-weighted SE-BOLD imaging may provide subtle information about the haemodynamic and neuronal responses.

Longitudinal diffusion tensor imaging in a rat brain glioma model

In order to investigate the properties of water motion within and around brain tumors as a function of tumor growth, longitudinal diffusion tensor imaging (DTI) was carried out in a rat brain glioma (C6) model. As tumors grew in size, significant anisotropy of water diffusion was seen both within and around the tumor. The tissue water surrounding the tumor exhibited high planar anisotropy, as opposed to the linear anisotropy normally seen in white matter, indicating that cells were experiencing stress in a direction normal to the tumor border. When tumors were sufficiently large, significant anisotropy was also seen within the tumor because of longer-range organization of cancer cells within the tumor borders. These findings have important implications for diffusion-weighted MRI experiments examining tumor growth and response to therapy.

From diffusion-weighted MRI to anomalous diffusion imaging

We present a novel interpretation of non-monoexponential diffusion-weighted signal decay with b-value in terms of the theory of anomalous diffusion. Anomalous diffusion is the theory of diffusing particles in environments that are not locally homogeneous, such as brain tissue. In such environments the model of restricted diffusion commonly employed in the analysis of diffusion MR data is not valid, leading to a nonlinear time dependence for the mean-squared displacement of spins, and to a prediction of a stretched exponential form for the signal decay. We show that this prediction leads directly to a new parameter, the anomalous exponent, which may be measured from scan data and from this we can estimate a fractal dimension, d(w), which categorizes the complexity of the excursions of diffusing spins. We construct images of the anomalous exponent and fractal dimension from in vivo human brain data. Distributions of exponents and dimensions are constructed in grey and white matter and cerebrospinal fluid. We observe that these distributions peak at biologically plausible values consistent with previous studies: grey matter dw = 2.366 +/- 0.31, white matter dw = 2.587 +/- 0.39, CSF dw = 1.970 (mode). Marked contrast is observed between grey and white matter when compared with lateral ventricle CSF. We then consider the anisotropy of the value of the anomalous exponent and define quantities analogous to the mean diffusivity and fractional anisotropy that are commonly generated from diffusion tensor images.

Anomalous diffusion expressed through fractional order differential operators in the Bloch-Torrey equation

Diffusion weighted MRI is used clinically to detect and characterize neurodegenerative, malignant and ischemic diseases. The correlation between developing pathology and localized diffusion relies on diffusion-weighted pulse sequences to probe biophysical models of molecular diffusion-typically exp[-(bD)]-where D is the apparent diffusion coefficient (mm(2)/s) and b depends on the specific gradient pulse sequence parameters. Several recent studies have investigated the so-called anomalous diffusion stretched exponential model-exp[-(bD)(alpha)], where alpha is a measure of tissue complexity that can be derived from fractal models of tissue structure. In this paper we propose an alternative derivation for the stretched exponential model using fractional order space and time derivatives. First, we consider the case where the spatial Laplacian in the Bloch-Torrey equation is generalized to incorporate a fractional order Brownian model of diffusivity. Second, we consider the case where the time derivative in the Bloch-Torrey equation is replaced by a Riemann-Liouville fractional order time derivative expressed in the Caputo form. Both cases revert to the classical results for integer order operations. Fractional order dynamics derived for the first case were observed to fit the signal attenuation in diffusion-weighted images obtained from Sephadex gels, human articular cartilage and human brain. Future developments of this approach may be useful for classifying anomalous diffusion in tissues with developing pathology.