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Research Papers

A Novel Method for Quantifying Spatial Correlations Between Patterns of Atherosclerosis and Hemodynamic Factors

[+] Author and Article Information
Véronique Peiffer

Department of Aeronautics and Department of Bioengineering,
Imperial College London,
London SW7 2AZ, UK
e-mail: v.peiffer09@imperial.ac.uk

Anil A. Bharath

e-mail: a.bharath@imperial.ac.uk

Spencer J. Sherwin

Department of Aeronautics,
Imperial College London,
London SW7 2AZ, UK
e-mail: s.sherwin@imperial.ac.uk

Peter D. Weinberg

e-mail: p.weinberg@imperial.ac.uk
Department of Bioengineering,
Imperial College London,
London SW7 2AZ, UK

1Corresponding author.

Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received October 3, 2012; final manuscript received January 6, 2013; accepted manuscript posted January 18, 2013; published online February 7, 2013. Editor: Victor H. Barocas.

J Biomech Eng 135(2), 021023 (Feb 07, 2013) (11 pages) Paper No: BIO-12-1463; doi: 10.1115/1.4023381 History: Received October 03, 2012; Revised January 06, 2013; Accepted January 18, 2013

Studies investigating the relation between the focal nature of atherosclerosis and hemodynamic factors are employing increasingly rigorous approaches to map the disease and calculate hemodynamic metrics. However, no standardized methodology exists to quantitatively compare these distributions. We developed a statistical technique that can be used to determine if hemodynamic and lesion maps are significantly correlated. The technique, which is based on a surrogate data analysis, does not require any assumptions (such as linearity) on the nature of the correlation. Randomized sampling was used to ensure the independence of data points, another basic assumption of commonly-used statistical methods that is often disregarded. The novel technique was used to compare previously-obtained maps of lesion prevalence in aortas of immature and mature cholesterol-fed rabbits to corresponding maps of wall shear stress, averaged across several animals in each age group. A significant spatial correlation was found in the proximal descending thoracic aorta, but not further downstream. Around intercostal branch openings the correlation was borderline significant in immature but not in mature animals. The results confirm the need for further investigation of the relation between the localization of atherosclerosis and blood flow, in conjunction with appropriate statistical techniques such as the method proposed here.

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Figures

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Fig. 1

Illustration of the surrogate data analysis. The images on the left represent (3 × 2) lesion and hemodynamic maps. Corresponding pixels in the two maps are marked with matching numbers of dots. Three sets of surrogate data, generated by shuffling the pixels from the hemodynamic map, are shown (nsur = 3). This results in three surrogate data pairs, as shown on the right. A discriminating statistic is calculated for each of the surrogate pairs, resulting in a histogram in which the percentile of the discriminating statistic calculated for the original pair can be determined (bottom, the histogram shown here is only illustrative). The dotted line indicates the statistic calculated for the original data; its position on the histogram determines the P-value.

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Fig. 7

Comparative analysis of lesion prevalence and WSS maps in segments of the descending thoracic aorta. Greyscale and color maps show lesion prevalence [10] and WSS [11], respectively, averaged for immature (a) and mature (b) animals. Maps are shown en face, and aortic flow is from top to bottom. The graphs summarize the quantitative analysis, and were obtained by combining the normalized histograms for the various samples. Each bin represents the mean+SD of the corresponding bins in the normalized histograms. The null hypothesis states that the lesion and WSS maps are uncorrelated. The red line marks the percentile of the discriminating statistic calculated for the original dataset; the corresponding numerical value is also given.

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Fig. 6

Comparative analysis of lesion prevalence and WSS maps around intercostal ostia. Greyscale and color maps show lesion prevalence [10] and WSS [11], respectively, averaged for immature (left) and mature (right) animals. Maps are shown en face, and aortic flow is from top to bottom. X marks the ostial center in the lesion prevalence maps. The graphs summarize the quantitative analysis, and were obtained by combining the normalized histograms for the various samples. Each bin represents the mean+SD of the corresponding bins in the normalized histograms. The null hypothesis states that the lesion and WSS maps are uncorrelated. The red line marks the percentile of the discriminating statistic calculated for the original dataset; the corresponding numerical value is also given.

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Fig. 5

Scatter plot of lesion prevalence versus WSS around intercostal ostia of immature animals, and regression line showing how inappropriate a linear analysis is for this data

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Fig. 4

Flow diagram of the surrogate sample data analysis for quantitative comparison of lesion and hemodynamic maps

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Fig. 3

The average distance at which pixels become de-correlated was determined from summary statistics of the correlation between pixels and their direct neighbors (left), their diagonal neighbors (middle left), pixels at a distance of 2 pixels lengths (middle right), pixels at a distance of 2√2 pixels lengths (right), and so on, up to an inter-pixel distance of 4.

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Fig. 2

Illustration of the surrogate analysis on samples of [lesion, hemodynamic] data. Samples of the original dataset (indicated by the grid squares outlined in bold lines) are selected [left, nsmp = 2], and a surrogate data analysis (nsur = 2) is applied on each of these samples. The median of the results of all surrogate data analyses (not shown) is taken as a measure for the spatial correlation between the lesion and hemodynamic maps.

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Fig. 8

Model for the synthesis of test images that are used in the validation experiments of Sec. A.1. The test images at the rightmost ends of both chains (top and bottom) mimic two different imaging processes of the same spatial distribution, set by the spatial points and smooth trace on the left. By comparing the pixel similarity, or differences, between both images, and against those arising from nonsimilar shapes, the statistical inference process can be confirmed to give meaningful results. See text for further details.

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Fig. 9

Illustration of a single spatial geometry, represented by the spline curve in (a), which has been converted into two simulated images, shown in (b) and (c). The image (b) is produced by one type of physical process/imaging model which is linear, while (c) is produced from the same geometry, but is created from a strongly nonlinear imaging model. It is important to stress that both (b) and (c) arise from the same spatial shape. Such models are used in the method validation over an extensive ensemble of 100 model pairs. Note that (c), coincidentally, exhibits some similarity to the patterns that might be seen in a WSS map. This demonstrates that the generative model for spatial distributions used in the additional validation experiments of section A.1 is suitable for the purposes of this paper. Images (b) and (c) are used directly in the statistical tests reported for the validation experiments of Sec. A.1.

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Fig. 13

(a) A patch taken from one image in the Berkeley image database and (b) the same patch following contrast inversion and a nonlinear histogram equalisation process. A smaller patch (c) extracted from (a) is compared to a patch (d) extracted from the same neighborhood (within 5 pixels distance). The surrogate data analysis test on these two shows that the patches are strongly related (e). Testing against the null hypothesis (surrogate data) shows that we should reject the null hypothesis: the patches are strongly correlated. Note that for this test, the alternative hypothesis (red) is well to the right of the main hypothesis, and sharply peaked; the fact that it is well away from the main bell-curve of the null hypothesis provides a strong measure of confidence.

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Fig. 10

(a) Distribution of the mean absolute difference (MAD) statistic (c) for two different images, (a) and (b). The image pair is clearly strongly correlated, because the “driving” geometry is identical for both cases. The images are very similar, but contain different noise realizations and different degrees of spatial blurring. Because of the strong relationship, the distributions of distances for the surrogate and unscrambled data (c) are very different. The null hypothesis—that the images are uncorrelated—may be rejected. (b) The trio of figures on the right concern quite unrelated image pairs, (d) and (e). Their distinctly different nature is clearly reflected in the surrogate and real distributions, (f), which are nearly indistinguishable. One cannot reject the null hypothesis.

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Fig. 11

(a) The similarity of two images (a) and (b) with different contrast mechanisms, but similar “source” shapes are captured through a confident rejection of the null hypothesis [(c); red trace relative to blue trace]. (b) On the other hand, the less clearly related examples of (d) and (e) show strong overlap of the null hypothesis (H0) with the alternative hypothesis (H1) yielding close, but not quite identical, distributions (f). The null hypothesis cannot be rejected. The distributions do not quite overlap because there is a small region of spatial overlap of the light regions of (d) and (e) around region R in both patches.

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Fig. 12

Images from the Berkeley Natural Image Database, showing patches of the same size selected at random. These patches were then distorted. Patches from nearby (< 6 pixels) from the same region (R0, R1) were considered as representing the same spatial distribution; those from other regions of the same image (R2) or other images (R3) were considered as being different. Note that pairs of patches were always compared by mapping them through distortions that include small rotations, blurring and color shifts.

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