Fractal dimension based carotid plaque characterization from three-dimensional ultrasound images

Abstract

Irregularity of the plaque surface associated with previous plaque rupture plays an important role in the risk estimation of stroke caused by carotid atherosclerotic lesions. Thus, the aim of this study is to develop and validate novel vulnerability biomarkers from three-dimensional ultrasound (3DUS) images by analyzing the surface morphological characteristics of carotid plaque using fractal geometry features. In the experiments, a total of 38 3DUS plaque images were obtained from two groups of patients treated with 80 mg of atorvastatin or placebo daily for 3 months respectively. Two types of 3D fractal dimensions (FDs) were used to describe the smoothness of plaque surface morphology and the roughness from intensity of 3DUS images. Student’s t test showed that the two fractal features were effective for detecting the statin-related changes in carotid atherosclerosis with p < 0.00023 and p < 0.0113 respectively. It was concluded that the 3D FD measurements were effective for analyzing carotid plaque characteristics and especially effective for evaluating the impact of atorvastatin treatment.

Appendix: Texture features

The gray-level co-occurrence matrix included all slices from the 3DUS images. Ten texture measurements were exacted from the co-occurrence matrix in the horizontal, vertical, and diagonal orientations with θ = 0°, 45°, 90°, 135°, including contrast, correlation, dissimilarity, energy, entropy, homogeneity, maximum probability, means, variance, and inverse different moment. For each orientation, the texture measurements were defined as follows:

• 1. Mean and variance:

$$ \mu =\frac{1}{N}\sum \limits_i\sum \limits_j\sum \limits_kP\left(i,j,k\right) $$

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$$ {\sigma}^2=\frac{1}{N}\sum \limits_i\sum \limits_j\sum \limits_k\left(P\left(i,j,k\right)-\mu \right) $$

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• 2. Contrast:

$$ Contrast=\sum \limits_i\sum \limits_j\sum \limits_k{\left(\left(i-j\right)\left(i-k\right)\left(j-k\right)\right)}^2P\left(i,j,k\right) $$

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• 3. Correlation:

$$ Correlation=\frac{\left[\sum \limits_i\sum \limits_j\sum \limits_k\left((ijk)P\right(i,j,k\left)-{\mu}_x{\mu}_y{\mu}_z\right)\right]}{\sigma_x{\sigma}_y{\sigma}_z} $$

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• 4. Dissimilarity

$$ Dissimilarity=\sum \limits_i\sum \limits_j\sum \limits_k\left[\left|i-j\right|\cdot \left|i-k\right|\cdot \left|j-k\right|\cdot P\left(i,j,k\right)\right] $$

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• 5. Energy:

$$ Energy=\sum \limits_i\sum \limits_i\sum \limits_k{\left[P\left(i,j,k\right)\right]}^2 $$

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• 6. Entropy:

$$ Entropy=-\sum \limits_i\sum \limits_j\sum \limits_kP\left(i,j,k\right)\ln P\left(i,j,k\right) $$

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• 7. Homogeneity

$$ Homogeneity=\sum \limits_i\sum \limits_j\sum \limits_k\frac{P\left(i,j,k\right)}{1+{\left(i-j\right)}^2+{\left(i-k\right)}^2+{\left(j-k\right)}^2} $$

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• 8. Maximum probability:

$$ maxProbability=\underset{i,j,k}{\max }P\left(i,j,k\right) $$

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• 9. Inverse different moment:

$$ IDM=\sum \limits_{i,j,k;i\ne j,i\ne k,j\ne k}\frac{P\left(i,j,k\right)}{{\left(i-j\right)}^2+{\left(j-k\right)}^2+{\left(i-k\right)}^2} $$

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