Fractional diffusion as a probe of microstructural change in a mouse model of Duchenne Muscular Dystrophy
Matt G Hall1, Paola Porcari2, Andrew Blamire2, and Chris A Clark1

1Institute of Child Health, University College London, London, United Kingdom, 2Newcastle Magnetic Resonance Centre, Newcastle University, Newcastle, United Kingdom

### Synopsis

We apply a fractional diffusion model to preclinical data from a mouse model of Duchenne Muscular Dystrophy, and compare to histological measurements of the underlying tissue. We find that the alpha exponent of the model provides contrast which is indicative of the microstructural changes associated with DMD. We observe contrast between the wild type and mdx mouse model.

### Purpose

Duchenne Muscular Dystrophy (DMD) is a genetic muscle-wasting disorder affecting 1 in 3600 boys worldwide [1] categorised by progressive degeneration of muscle tissue. The current gold-standard for assessing the progress of the pathology in tissues is biopsy, which is both invasive and localised in tissue. More recently, efforts have been made to develop reliable image-based biomarkers using diffusion-weighted MRI (DWI) [2] which have suggested that diffusion-weighted measurements may be sensitive to microstructural changes in muscle due to DMD. Muscle tissue structure is known to be hierarchical: myofibres are bundles of myofibrils which are in turn bundles of myofilaments [3] and thus diffusion exhibits restriction across a wide range of length scales. Fractional diffusion [4] is a model of the diffusion signal which assumes spin displacements and waiting times are well described by power-law distributions – this approach is consistent with hierarchical structure. These models are good candidates for analysing diffusion-weighted measurements in muscle tissue. Here we apply a fractional diffusion model to preclinical data from wild type and mdx mouse models of DMD. The resulting parameter maps are then compared to histological data for the underlying tissue microstructure. We find that parameters of the fractional diffusion model are sensitive to microstructural changes associated with DMD.

### Methods

Imaging

Employing a diffusion-weighted STEAM sequence we acquire images at six diffusion times and four different gradient strengths including an unweighted acquisition per diffusion time (see Table-1 for details). All acquisitions have the same TE/TR (4000/20ms) and gradient pulse duration of 3ms. Samples are positioned such that muscle fibres align with the slice select direction and diffusion encoding gradients are all applied parallel to the phase encode direction. Data are acquired at 7T using a Varian preclinical scanner.

Histology

Follow in vivo measurements, the gastrocnemius muscle was carefully removed from the left hindlimb of each mouse. The muscle was then mounted on a small mound of Tissue-Tek OCT on a cork disc and frozen using cooled isopentane (2-methylbutane). Cross sections of frozen muscle (8um thick) were stained with laminin alpha-2 chain, rat antibody (1:1000 dilution), fluorescently labelled secondary antibody, goat anti-rat alexa 488 (1:1000 dilution) and DAPI. Five areas on each of the two sections at comparable proximal levels were photographed at 20x magnification and the Feret's diameter of each muscle fibre measured using Fiji (Image-J).

Modelling

Diffusion-weighted data were normalised using the unweighted measurement from the appropriate diffusion time and a three parameter Mittag-Leffler function

$$S=S(0)\sum_{k=0}^{\infty}\frac{(-D_{\alpha,\beta}q^\beta \Delta^\alpha)^k}{\Gamma(\alpha k+1)}$$

Where $D_{\alpha,\beta}$ is the fractional diffusivity, $\alpha$ is the temporal exponent and $\beta$ is the temporal exponent, both associated with the underlying diffusion process. Results here concentrate on the $\alpha$ parameter.

Fits are performed using a Levenberg-Marquardt algorithm. Initial parameters were derived from an approximate, two-parameter Mittag-Leffler model, which is fitted with initial parameters from a linear exponential fit. Fits are performed 100 times in each voxel with random perturbations to initial parameters and the combination chosen that has the smallest sum of squared residuals.

### Results

Fig-1 shows maps of the values of the $\alpha$ parameter fitted in each voxel in a wild type (top row) and Mdx mouse (bottom row). The Mdx show darkened features which are not present in the wild type mice, such as those circle in red. Fig-2 shows histograms of muscle fibre diameter from the centre of the Gastrocnemius muscle (top) in Wildtype and Mdx mice, and the $\alpha$ exponent from an ROI covering the darkened region circled in Fig-1 compared to a similar region in wildtype tissue. Histology reveals a shift n the distribution of muscle fibre radius, and imaging reveals a similar change in the values of fitted exponents.

### Discussion & Conclusions

The darkening in the $\alpha$ maps is present in both Mdx mice and not in either wildtype. Histology of similar regions shows a change in tissue microstructure which supports the idea that the fractional diffusion model used is sensitive to the microstructural changes caused by pathology. This work has a small sample size, but these preliminary results show localised darkened regions which may be associated with disease regions.

This work considers only one parameter of the model and as such there may be further insight to be drawn from considering all three, either singly or in combination. This may lead to increased contrast and improved image-based biomarkers.

There is also scope to optimise the acquisition. The current dataset contains 24 images, which is similar to the number required DTI (for example) but there may be scope to reduce this and reduce acquisition time.

### Acknowledgements

This work was supported by the European Union NMD-Bioimage project. Seventh Framework Programme (FP7/2007-2013) under grant agreement 602485 Bioimage-NMD project.

### References

[1] Bushby et al, Lancet Neurology 9 (1) (2010)

[2] Hooijmans et al, NMR in Biomedicine, 28 (11) (2015)

[3] Saladin, Anatomy and Physiology (3rd ed.), Watnik (New York) (2010)

[4] Magin et al, Microporous and Mesoprous Materials, 178: 39–43 (2013)

### Figures

Table-1: Paramters used in diffusion-weighted acquisitions. Data were acquired at several different gradient strengths $|\mathbf{G}|$ and diffusion times $\Delta$ in milliseconds. All gradient pulse durations ($\delta$) were 3ms.

Fig-1: Maps of the fitted $\alpha$ paramter in wildtype (top row) and Mdx mice (bottom row). The Mdx maps exhibit localised darkening which is not present in the wildtype maps. Circles indicate the feature used in the histograms in Fig-2.

Fig-2: Histograms of fibre size distribution (top) in the central Gastronemius from histology and the $\alpha$ exponent fitted to diffusion-weighted measurements in an ROI covering the darkened region shown in Fig-1.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)
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