Introduction

Fractals are defined by self-similarity at different length scales of evaluation and may be used to evaluate complexity of structures or processes.¹ Fractals have been identified in many areas of human biology such as the airway and vascular trees in the lungs.² Ventilation distribution has also been shown to follow a fractal pattern³ using fluorescence imaging in healthy pigs at ~2cm³ resolution. ¹²⁹Xe MRI now provides a way to visualize and measure pulmonary ventilation with high spatial (~0.15cm³) and temporal resolution in patients with respiratory disease,⁴ however the spatial heterogeneity or complexity of ¹²⁹Xe MRI signal has been challenging to capture in a robust metric. While MRI ventilation heterogeneity and its comparisons with other imaging modalities are dependent on image resolution, fractal dimensions describe scale-dependent variability in a scale-independent manner.³ Our objective here was to investigate whether MRI ventilation heterogeneity follows a fractal pattern in participants with asthma and accordingly measure the fractal dimension.

Methods

Participants and Data Acquisition:
Participants with asthma provided written informed consent to an ethics-board-approved protocol (NCT03733535) and underwent MRI and pulmonary function tests during a single two-hour visit. ¹H and ¹²⁹Xe MRI were performed within five minutes of each other using a whole-body 3.0T Discovery MR750 system (GEHC, USA) with broadband imaging capabilities as previously described.⁵ Subjects were instructed to inhale a gas mixture from a 1.0L Tedlar bag from functional residual capacity and image acquisition was performed under breath-hold conditions. Anatomical ¹H MRI was performed before ¹²⁹Xe, after inhalation of 100% N₂ using the whole-body radiofrequency coil and ¹H fast-spoiled, gradient-recalled echo sequence with partial-echo (8s total acquisition time, repetition-time (TR)/echo-time (TE)/flip-angle=4.7ms/1.2ms/30°, field-of-view (FOV)=40x40cm, matrix=128x80 zero-padded 128x128, 15-17 slices, 15mm slice-thickness, zero gap). ¹²⁹Xe static ventilation images were acquired after inhalation of 400mL ¹²⁹Xe diluted to 1.0L with ⁴He using a quadrature flexible-vest coil and 3D fast-gradient echo sequence with partial-echo (12s total acquisition time, TR/TE/initial flip-angle=5.1ms/1.5ms/1.4°, variable flip-angle, FOV=40x40cm, matrix=128x80 zero-padded 128x128, 15-17 slices, 15mm slice-thickness, zero gap).
Data Analysis:
Static ventilation images were segmented using 3D k-means clustering to classify voxel intensities into five clusters ranging from signal void or ventilation defects (Cluster1) to hyperintense signal (Cluster5).⁶ Ventilation defect percent (VDP) was defined as the ventilation defect volume normalized to the thoracic cavity volume. Fractal analysis was performed for each MRI cluster using the Minkowski-Bougliand box-counting method in MATLAB 2020a (Mathworks, USA)⁷ as shown in Figure 1 (illustrated in 2D). Grids of progressively increasing cube length L were iteratively placed over the 3D image and the number of cubes N containing MRI defect/signal was counted at each iteration. Cube lengths were 32 voxels/100mm, 16 voxels/50mm, 8 voxels/25mm, 4 voxels/12.5mm, 2 voxels/6.25mm, and 1 voxel/3.125mm. Fractal geometry is governed by a power law characterized by the fractal dimension D:
$$N(n)=k\cdot L(n)^{-D}$$
The fractal dimension was calculated as the negative slope from the linear regression of the log-log plot of L vs N. Fractal dimension characterizes the complexity and space-filling capacity of the image; the greater complexity of how MRI defects/signal fill the image space, the greater the fractal dimension. Fractal dimension was only evaluated when MRI defects/signal clusters occupied >4% of the total MRI lung volume to ensure there was sufficient signal for fractal analysis. Fractal dimensions between MRI signal clusters were compared using one-way ANOVA.

Results

We evaluated 25 participants with asthma (19 females/6 males, 57±13-years) with a range of ventilation heterogeneity (VDP=17±13%, range 1-53%). Figure 1 demonstrates visual spatial aggregation of MRI signal clusters, suggestive of fractal MRI signal behaviour. Nineteen participants (19/25, 76%) with VDP (or Cluster1 volume)>4% were used to generate fractal dimension for defect regions, 24 participants (96%) had Cluster2 volume >4%, 24 participants had Cluster3 volume >4%, and all participants had Cluster4 and Cluster5 volume >4%. All MRI clusters showed good fit with power law (all R²=0.99), and fractal dimensions were 1.39±0.24 for defect regions, 1.45±0.20 for Cluster2, 1.80±0.16 for Cluster3, 1.82±0.06 for Cluster4, and 1.56±0.22 or Cluster5 (p<0.0001, Figure 2). While all clusters of MRI signal exhibited fractal behaviour, defect regions had the greatest variance in fractal dimension and Cluster4 had the least variance.

Discussion

In this proof-of-concept demonstration, we measured fractal dimensions of ¹²⁹Xe MRI ventilation heterogeneity for the first time in participants with asthma. Since ventilation heterogeneity is primarily driven by airway tree abnormalities, which itself follows a fractal pattern,² we expected that MRI ventilation would follow a fractal pattern. In fact, MRI ventilation strongly followed power law for fractal analysis (R²=0.99). This suggests spatial clustering of MRI ventilation such that ventilation in a given region is correlated with its neighbouring regions.^3,8 These results open the door for fractal analysis as a new tool to investigate regional MRI ventilation heterogeneity in patients with lung disease, and provide a foundation for future studies to compare with pulmonary fractal structures/processes such as airway and vessel trees,² pulmonary perfusion, and ventilation/perfusion mismatch.³

Conclusions

¹²⁹Xe MRI ventilation distribution follows a fractal distribution in participants with asthma and fractal dimension can be quantified as a novel measure of ventilation heterogeneity. Fractal analysis of imaging measurements can be used to investigate pulmonary structure-function relationships in patients with lung disease.

Acknowledgements

No acknowledgement found.