# The Mathematics of Quasi-Diffusion Magnetic Resonance Imaging

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## Abstract

**:**

^{2}s

^{−1}), and a fractional exponent, $\alpha $. Here we investigate the mathematical properties of the QDI signal and its interpretation within the quasi-diffusion model. Firstly, the QDI equation is derived and its power law behaviour described. Secondly, we derive a probability distribution of underlying Fickian diffusion coefficients via the inverse Laplace transform. We then describe the functional form of the quasi-diffusion propagator, and apply this to dMRI of the human brain to perform mean apparent propagator imaging. QDI is currently unique in tissue microstructural imaging as it provides a simple form for the inverse Laplace transform and diffusion propagator directly from its representation of the dMRI signal. This study shows the potential of QDI as a promising new model-based dMRI technique with significant scope for further development.

## 1. Introduction

## 2. Theory

#### 2.1. Quasi-Diffusion Imaging

^{2}s

^{−1}. The extension to the CTRW diffusion model is provided by the fractional partial diffusion equation,

^{β}s

^{−α}. It is assumed that at time, $t=0$, all the material is located at the origin and is given by the Dirac Delta function $P\left(x,0\right)=\delta \left(x\right)$. In the special case of quasi-diffusion, the Gaussian scaling relationship, $\frac{2\alpha}{\beta}=1,$ is substituted into (4) to give,

^{2α}s

^{−α}. In the quasi-diffusion case, a Gaussian diffusion coefficient in mm

^{2}s

^{−1}can be recovered from the effective normal diffusion coefficient,

^{−1}), γ is the gyromagnetic ratio of hydrogen (for quantification of water diffusion), and $g$ is the diffusion encoding gradient strength (in mTm

^{−1}). The effective diffusion time of the pulse sequence is denoted as $\overline{\mathsf{\Delta}}=\mathsf{\Delta}-\frac{\delta}{3}$ (in s) for a given diffusion gradient pulse duration, δ, and separation, Δ. In clinical applications, dMRI are typically acquired by keeping $\overline{\mathsf{\Delta}}$ constant while altering $q$ by changing the diffusion encoding gradient strength, $g$.

^{−1}) and r-space (in mm) is only exact in this limit. In practice, this assumption is routinely violated on clinical MRI systems due to technical and safety limitations of in vivo MRI, meaning that gradient pulses have finite duration usually in the range $20\mathrm{ms}\delta \mathsf{\Delta}70$ ms.

^{−2}), and $S\left(0\right)$ is the signal intensity at $b=0$ s mm

^{−2}. Equation (10) is a stretched Mittag–Leffler function that describes Gaussian diffusion when $\alpha =1$ and an effective normal diffusion for $0<\alpha <1$. The diffusion coefficient, ${D}_{1,2},$ and fractional exponent, $\alpha ,$ are independent parameters that together parameterise a family of decay curves according to the rate of diffusion signal decay (${D}_{1,2}$) and the shape of the power law tail, $\alpha $.

^{2}s

^{−1}and $0.5<\alpha <1$ in typical healthy brain tissue [1]. Figure 1 illustrates the family of signal decay curves described by (10). Figure 1a shows the quasi-diffusion signal attenuation parameterised by $b$ for an arbitrary diffusion coefficient, ${D}_{1,2}=1.5\times {10}^{-3}$ mm

^{2}s

^{−1}for $0.1\le \alpha \le 0.99$ with Figure 1b showing the quasi-diffusion signal attenuation parameterised by $q$.

^{−2}in 6 non-collinear gradient encoding directions evenly distributed across the sphere. Experimental diffusion times were $\delta =23.5$ ms and $\mathsf{\Delta}=43.7$ ms giving an effective diffusion time of $\overline{\mathsf{\Delta}}=35.9$ ms. Data were acquired on a clinical 3T MR scanner at St George’s, University of London (SGUL) with a voxel size of $1.5$ mm $\times 1.5$mm $\times 5$mm in 35 min 12 s. Full image acquisition parameters are given in Appendix A. Data analysis were performed using the technique described in [1] to estimate ${D}_{1,2}$ and $\alpha $ values in each diffusion encoding direction and their mean values within each image voxel. The top row of Figure 2 shows the exceptional quality of fit of the quasi-diffusion model to observed data in individual grey (Figure 2a) and white matter voxels (Figure 2b) across the full range of $b$-factors. The ${D}_{1,2}$ and $\alpha $ maps exhibit similar mean apparent diffusion coefficients in brain tissue, with the bright signal pertaining to cerebrospinal fluid-filled (CSF) spaces where diffusion is Gaussian with ${D}_{FW}=3\times {10}^{-3}$ mm

^{2}s

^{−1}. The bottom row of Figure 2 shows the quality of data reconstruction from three points, $b=\left\{0,1080,5000\right\}$ s mm

^{−2}within the dMRI data acquisition, which were chosen as their modelled quasi-diffusion signal attenuation closest to that of the full 29 $b$-value dataset across the entire image [55]. Overall, Figure 2 highlights how well the quasi-diffusion model fits acquired data and demonstrates that high quality images can be acquired in a clinically feasible time of 120 s without the need for an extensive set of different b-value images to accurately define the signal decay curve.

#### 2.2. General Properties of the Mittag–Leffler Function

#### 2.3. Asymptotic Properties of the Quasi-Diffusion Characteristic Equation

^{2}s

^{−1}and has an upper limit of ${D}_{FW}$ for free water at human body temperature. In this case, the behaviour of (12) and (15) will be dominated by $b$-value as both $b\to 0$ and $b\to \infty $.

^{2}s

^{−1}and $0.1\le \alpha \le 0.99$.

#### 2.4. The Laplace Transform of the Quasi-Diffusion Characteristic Equation

^{2}s

^{−1}). This provides further support for quasi-diffusion being a natural generalisation and fractionalisation of diffusion dynamics when the process occurs within complex structures that hinder or restrict free diffusion.

^{2}s

^{−1}and $0.1<\alpha <0.99$. At $\alpha $ close to unity (e.g., $\alpha $ = 0.99), the probability density function tends towards the Dirac delta function with a characteristic ADC equivalent to the resultant mean of an unrestricted Gaussian diffusion process. This indicates that diffusion dynamics within the structure being studied are Gaussian with no boundaries that hinder or restrict diffusing particles. As $\alpha $ decreases (from 0.9 to 0.7), the probability density function broadens to include significant contributions at low ADCs representing characteristics of diffusion in a restricting environment and increasing tissue heterogeneity. For $\alpha <0.65$, the probability density function smoothly deforms to a hyperbolic shape that becomes dominated by large contributions from restriction and greater tissue heterogeneity. It should be noted that in practice for dMRI in brain tissue, we are typically in the range $0.5<\alpha \le 1$ [1].

^{2}s

^{−1}. The slight apparent anisotropy in measurements is likely related to partial volume effects with brain tissue that are not fully removed in the CSF segmentation, combined with the effects of noise which have broadened the distribution from being a Dirac-delta function. In grey matter (Figure 3c), which contains layers of neurons, the probability density functions represent an almost isotropic diffusion within a heterogeneous tissue microstructure. In white matter (Figure 3d), which consists of axons that provide the wiring of the brain, there is anisotropic diffusion such that diffusion parallel to tissue microstructure is through a more homogeneous diffusion environment (i.e., along axons surrounded by myelin sheaths) than diffusion perpendicular to axons which is more hindered and/or restricted by tissue microstructural boundaries. The white matter spectra indicate considerable anisotropy in the axial and radial ADCs, with large contributions of restricted diffusion to the radial quasi-diffusion coefficient, ${D}_{1,2}$, representing a low $\alpha $.

#### 2.5. The Quasi-Diffusion Propagator

^{2}s

^{−1}and fractional exponents in the range $0.5<\alpha \le 0.99$, which cover the range of $\alpha $ typically found in the human brain. The propagator was calculated based on Equation (38). The shape of the propagator in space (Figure 4a) is such that the probability density function becomes more kurtotic as $\alpha $ decreases from the Gaussian case ($\alpha =1$) corresponding to a higher probability of both smaller and larger step lengths, in correspondence with greater heterogeneity of the diffusion environment. This is consistent with assumptions made in DKI [24,25]. The shape of the propagator in time (Figure 4b) is also consistent with quasi-diffusion representing an effective normal diffusion process as it is unimodal and Gaussian-like.

#### 2.6. Application of Quasi-Diffusion MRI to Mean Apparent Propagator Imaging

^{−3}, $RTAP$ in mm

^{−2}and $RTPP$ in mm

^{−1}from which pore characteristics may be calculated. For the one-dimension quasi-diffusion case, $RTPP$ is given by (29) as,

^{−1}, far higher than the capability of any MR scanner or NMR system. $RTAP$ and $RTOP$ are then given by (28) as,

^{2}s

^{−1}. Consequently, the short time limit is,

## 3. Examples

#### 3.1. Quasi-Diffusion Mean Apparent Propagator Imaging of the Corpus Callosum

^{−2}in 32 non-collinear diffusion encoding directions equally spaced on the sphere with $\delta =28.7$ ms and $\mathsf{\Delta}=43.9$ ms, giving an effective diffusion time of $\overline{\mathsf{\Delta}}=34.33$ ms. Full data acquisition details are provided in Appendix A.

#### 3.2. Quasi-Diffusion Imaging of Brain Tumour

^{−2}in 6 non-collinear diffusion encoding directions equally spaced on the sphere with $\mathsf{\delta}=23.5$ ms, and $\mathsf{\Delta}=43.7$ ms, giving an effective diffusion time of $\overline{\mathsf{\Delta}}=35.9$ ms. Full patient and data acquisition details are provided in Appendix A. Image analysis was performed using the same techniques as Section 3.1 with the exception that effective mean spherical pore radius was calculated.

_{1,2}values (Figure 6b). This region plus a 2 cm margin would be the target for radiotherapy. The quasi-diffusion imaging maps provide different image contrasts with potential to distinguish these regions from a single image modality. Elevated α is evident across the whole lesion (Figure 6c) indicating that the oedema and tumour core are less structured than normal white matter. In addition, α and RTOP

^{1/3}(Figure 6d) allow distinction between grey and white matter for which contrast is present in the T1wGd image but not the D

_{1,2}map. Within the RTOP

^{1/3}hypointense region that delineates tumour oedema/infiltration, there is an area of elevated signal relating to T1wGd enhancement, which is a result of the presence of a greater density of tumour microstructure. Within the T1wGd enhancing region, the effective pore size is approximately 2.5 µm (Figure 6e).

## 4. Discussion

^{−2}in human white matter in vivo [70], and $b=100,000$ s mm

^{−2}in rat and human ex vivo corpus callosum [67]. Furthermore, the quasi-diffusion model predicts a stretched exponential form of the signal as $b\to {0}^{+}$ consistent with low $b$-values ($b$ < 300 s mm

^{−2}) containing a significant signal proportion attributable to blood perfusion and a potentially super-diffusive dynamic [71].

## 5. Conclusions

## 6. Patents

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Diffusion Magnetic Resonance Imaging Data

#### Appendix A.1. Human Participants

**Healthy subjects:**Two healthy participants were recruited (1 male, age 27 and 1 female, age 28 years). Ethics approval was obtained (East London 3 REC 10/H0701/36) and written informed consent was obtained from each participant prior to MR scanning.

**Brain tumour patient:**A brain tumour patient (age 58 years) with a World Health Organisation (WHO) Grade IV glioblastoma was recruited as part of the “Tissue-type magnetic resonance imaging of brain tumours” study at SGUL for which ethical approval was obtained (South Central Hampshire REC 17/SC/0460). Written informed consent was obtained prior to MR scanning.

#### Appendix A.2. Acquisition of Diffusion Magnetic Resonance Imaging Data

^{−1}at a slew rate of 100 mTm

^{−1}ms

^{−1}). Fat suppression was achieved using Spectral Presaturation by Inversion Recovery (SPIR) and Slice Selection Gradient Reversal (SSGR). A SENSE factor of 2 in the anterior-posterior direction and a half-scan factor of 0.891 were applied to minimise echo-train length and overall acquisition time.

**Acquisition of imaging data illustrated in Figure 2:**Quasi-diffusion tensor imaging data were acquired with: Echo Time ($TE)=90$ ms, Repetition Time $\left(TR\right)=6000$ ms, $\delta =23.5$ms, $\mathsf{\Delta}=43.9$ ms, field of view 210 mm $\times $ 210 mm with 22.5 mm thick slices acquired at $2.3$ mm $\times $ 2.3 mm $\times $ 5 mm resolution that were zero-filled (by use of the Fourier transform) to provide $1.5$ mm $\times $ 1.5 mm $\times $ 5.0 mm voxels. dMRI were acquired twice in 29 $b$-values with $b=\left\{0,180,360,\dots \mathrm{step}180\dots ,4860,5000\right\}$ s mm

^{−2}. Images without diffusion-sensitisation $(b=0$ s mm

^{−2}) were acquired 16 times. Diffusion encoding gradients were applied in 6 non-collinear directions. Data were acquired in 35 min 12 s.

**Acquisition of imaging data illustrated in**

**Section 3.1**

**and Figure 5:**Quasi-diffusion tensor imaging data were acquired with: $TE=90$ ms, $TR=9000$ ms, $\delta =28.7$ms, $\mathsf{\Delta}=43.9$ms, field of view 224 mm $\times $ 224 mm with 50.2 mm thick slices acquired at $3$ mm $\times $ 3 mm $\times $ 2 mm resolution that were zero-filled (by use of the Fourier transform) to provide $2$ mm isotropic voxels. dMRI were acquired once in 3 $b$-values with $b=\left\{0,1100,4000\right\}$ s mm

^{−2}. Images without diffusion-sensitisation were acquired 8 times. Diffusion encoding gradients were applied in 32 non-collinear directions. Data were acquired in 10 min 48 s.

**Acquisition of imaging data illustrated in**

**Section 3.2**

**and Figure 6:**Quasi-diffusion tensor imaging data were acquired with: $TE=90$ ms, $TR=6000$ ms, $\mathsf{\delta}=23.5$ms, $\mathsf{\Delta}=43.9$ ms, field of view 210 mm $\times $ 210 mm with 22.5 mm thick slices acquired at $2.3$ mm $\times $ 2.3 mm $\times $ 5 mm resolution that were zero-filled (by use of the Fourier transform) to provide $1.5$ mm $\times $ 1.5 mm $\times $ 5 mm voxels. dMRI were acquired once in 3 $b$-values with $b=\left\{0,1100,5000\right\}$ s mm

^{−2}. Images without diffusion-sensitisation were acquired 8 times. Diffusion encoding gradients were applied in 6 non-collinear directions. Data were acquired in 120 s.

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**Figure 1.**The family of quasi-diffusion imaging signal attenuation curves for an arbitrary diffusion coefficient of ${D}_{1,2}=1.5\times {10}^{-3}$mm

^{2}s

^{−1}and a range of fractional exponents, $0.1\le \alpha \le 0.99$. Normalised signal is shown parameterised by (

**a**) diffusion-sensitisation, $b$, and (

**b**) $q$. Graph (

**c**) shows the power law behaviour of the signal decay parameterised by $b$ on a logarithmic scale.

**Figure 2.**Quasi-diffusion model fits to experimental diffusion magnetic resonance imaging data acquired in the brain of a young, healthy subject (age $28$ years). The top row shows the results of fitting the quasi-diffusion model to an acquisition of $29b$-values over the range $0<b\le 5000$ s mm

^{−2}, with the bottom row showing the results of fitting the quasi-diffusion model to $3$ $b$-values, $b=\left\{0,1080,5000\right\}$ s mm

^{−2}. All imaging data were acquired in 6 diffusion encoding directions at an effective diffusion time of $\overline{\mathsf{\Delta}}=35.9$ ms. Normalised signal attenuation is shown for a grey matter voxel indicated by the blue arrow (graphs (

**a**,

**e**)), and a white matter voxel indicated by the red arrow (graphs (

**b**,

**f**)). Axial slices are shown for maps of mean ${D}_{1,2}$ (images (

**c**,

**g**)) and mean α (images (

**d**,

**h**)).

**Figure 3.**Decomposition of the signal into a spectrum of Fickian apparent diffusion coefficients via the inverse Laplace transform. Graph (

**a**) shows the probability density functions for a unit diffusion coefficient, ${D}_{1,2}=1$mm

^{2}s

^{−1}for different fractional exponents, $0.1\le \alpha \le 0.99.$ Spectra are shown for axial, radial and mean ${D}_{1,2}$ and $\alpha $ values in (

**b**) cerebrospinal fluid, (

**c**) grey matter and (

**d**) white matter. ${D}_{1,2}$ and $\alpha $ values used to calculate the spectra are from Barrick et al. [1].

**Figure 4.**The one-dimensional quasi-diffusion propagator for an arbitrary diffusion coefficient, ${D}_{1,2}=1.5\times {10}^{-3}$ mm

^{2}s

^{−1}for $0.5\le \alpha \le 0.99$. The one-dimensional probability density function is shown (

**a**) in space at an effective diffusion time of $\overline{\mathsf{\Delta}}=35.9$ ms, and (

**b**) in time at $x=5$ μm.

**Figure 5.**Quasi-diffusion mean apparent propagator imaging results for a young, healthy subject (age 27 years). The top row shows axial slices through (

**a**) $\sqrt[3]{RTOP}$, (

**b**) $\sqrt{RTAP}$, and (

**c**) $RTPP$ maps. The bottom row shows (

**d**) effective pore radius estimates in a cross-section through the mid-sagittal plane of the corpus callosum, and (

**e**) a probability density function of micron radii for the analysed 10 mm thick section of the corpus callosum (graph (

**e**)). The location of the corpus callosum is identified by red arrows. The red and blue lines in graph (

**e**) indicate the mean and median, respectively.

**Figure 6.**Quasi-diffusion imaging results for axial slices through a high-grade glial tumour (WHO Grade IV glioblastoma). Diffusion magnetic resonance imaging data were acquired at $b=\left\{0,1100,5000\right\}$ s mm

^{−2}in 6 diffusion encoding directions at an effective diffusion time of $\overline{\mathsf{\Delta}}=35.9$ ms. Imaging data were acquired in a clinically feasible time of 120 s. Image (

**a**) shows a T1-weighted image acquired after injection of gadolinium contrast agent, with maps of (

**b**) mean ${D}_{1,2}$, (

**c**) mean $\alpha $, (

**d**) $\sqrt[3]{RTOP}$ and (

**e**) effective mean pore size, $\langle R\rangle ,$ (which in the absence of oedema or necrosis will relate to cell size). Arrows indicate the following regions: white—tumour core, red—necrosis, blue—oedema, yellow—grey matter, green—white matter.

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**MDPI and ACS Style**

Barrick, T.R.; Spilling, C.A.; Hall, M.G.; Howe, F.A. The Mathematics of Quasi-Diffusion Magnetic Resonance Imaging. *Mathematics* **2021**, *9*, 1763.
https://doi.org/10.3390/math9151763

**AMA Style**

Barrick TR, Spilling CA, Hall MG, Howe FA. The Mathematics of Quasi-Diffusion Magnetic Resonance Imaging. *Mathematics*. 2021; 9(15):1763.
https://doi.org/10.3390/math9151763

**Chicago/Turabian Style**

Barrick, Thomas R., Catherine A. Spilling, Matt G. Hall, and Franklyn A. Howe. 2021. "The Mathematics of Quasi-Diffusion Magnetic Resonance Imaging" *Mathematics* 9, no. 15: 1763.
https://doi.org/10.3390/math9151763