Research Papers

Comparison of Statistical Methods for Assessing Spatial Correlations Between Maps of Different Arterial Properties

[+] Author and Article Information
Ethan M. Rowland

Department of Bioengineering,
Imperial College London,
London SW7 2AZ, UK;
Department of Aeronautics,
Imperial College London,
London SW7 2AZ, UK
e-mail: ethan.rowland09@imperial.ac.uk

Yumnah Mohamied

Department of Bioengineering,
Imperial College London,
London SW7 2AZ, UK;
Department of Aeronautics,
Imperial College London,
London SW7 2AZ, UK
e-mail: yumnah.mohamied08@imperial.ac.uk

K. Yean Chooi

Department of Bioengineering,
Imperial College London,
London SW7 2AZ, UK;
Department of Aeronautics,
Imperial College London,
London SW7 2AZ, UK
e-mail: yean.chooi06@imperial.ac.uk

Emma L. Bailey

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

Peter D. Weinberg

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

1Corresponding author.

Manuscript received March 16, 2015; final manuscript received July 15, 2015; published online August 6, 2015. Assoc. Editor: Ender A. Finol.

J Biomech Eng 137(10), 101003 (Aug 06, 2015) (15 pages) Paper No: BIO-15-1112; doi: 10.1115/1.4031119 History: Received March 16, 2015

Assessing the anatomical correlation of atherosclerosis with biomechanical localizing factors is hindered by spatial autocorrelation (SA), wherein neighboring arterial regions tend to have similar properties rather than being independent, and by the use of aggregated data, which artificially inflates correlation coefficients. Resampling data at lower resolution or reducing degrees-of-freedom in significance tests negated effects of SA but only in artificial situations where it occurred at a single length scale. Using Fourier or wavelet transforms to generate autocorrelation-preserving surrogate datasets, and thus to compute the null distribution, avoided this problem. Bootstrap methods additionally circumvented the errors caused by aggregating data. The bootstrap technique showed that wall shear stress (WSS) was significantly correlated with atherosclerotic lesion frequency and endothelial nuclear elongation, but not with the permeability of the arterial wall to albumin, in immature rabbits.

Copyright © 2015 by ASME
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Grahic Jump Location
Fig. 1

Average maps for each variable. White areas represent the branch ostium and were excluded from the analysis. WSS was normalized as described in Ref. [12]. In the bottom row, data are coded by rank (the pixel with the lowest intensity is given a rank of 1, the next lowest a rank of 2, etc.). Maps of WSS and permeability are shown at their 1.8 × 1.8 mm extents with two ROIs marked with black lines (1.2 × 1.2 and 1.2 × 1.68 mm).

Grahic Jump Location
Fig. 2

A selection of individual maps for each variable. For lesion prevalence, individual maps represent left or right side averages for each rabbit as the raw data were unavailable.

Grahic Jump Location
Fig. 3

Computing Ldec using the z score. (a) 1/4 of the ring neighborhood which is compared with the center pixel (black), for radii of 1, 3, 5, and 7 pixels. (b) Checkerboard image where each square is 10 × 10 pixels. (c) 10% Gaussian noise added to (b). (d) z scores computed for the images in (b) and (c) for radii up to half the image width. (e) Image of scattered disks with radius 10 pixels. (f) 10% Gaussian noise added to (e). (g) z scores computed for the images in (e) and (f) up to half the image width.

Grahic Jump Location
Fig. 4

Individual-level versus aggregate-level correlation. A model was defined for a 300 × 300 pixel map as a 2D Gaussian with a standard deviation of 60 pixels, whose average spatial location is the center of the map with a standard deviation of 40 pixels in x and y. Two samples (of size 100) were drawn from this model and paired. Three example pairs are shown. Also shown are the average maps of each sample. The mean of 100 pairwise correlation rmean = 0.044. In contrast, the correlation between average maps is 0.975.

Grahic Jump Location
Fig. 5

Scatter plots of the relationships between WSS and each variable. In the top row, Pearson’s r values are given. In the bottom row, data were ranked before plotting; Spearman’s ρ values are given.

Grahic Jump Location
Fig. 6

Z score for 1.2 × 1.68 mm maps at 6 μm resolution of (a) WSS and (b) permeability. Local maxima and curvature maxima (circle markers, found by fitting a curve and computing the second derivative) are labeled. In (b), the z score computed for the permeability map after smoothing with a Gaussian filter (15 × 15 pixels, standard deviation 3) is also shown (dashed line).

Grahic Jump Location
Fig. 7

Variation of p value and ρ with Ldec for the correlation of the WSS and permeability maps. Three methods were used to compute these p values: the median of the subsample p values, p values derived from CIs on the sampling distribution of ρ (circle markers), and Dutilleul’s modified t-test.

Grahic Jump Location
Fig. 8

ρ and p value distributions for the correlation of WSS and permeability obtained using Ldec based sampling. In (a) and (c), Ldec = 34 pixels, m = 43. In (b) and (d), Ldec = 92, m = 6. (e) An example of the scatter plot when m = 6. In (f), the data were ranked before plotting and Spearman’s ρ is given.

Grahic Jump Location
Fig. 9

Example WSS surrogate maps generated by (a) IAAFT and (b) DT-CWT, at resolutions matching those of the lesion (120 μm), nuclear L/W (100 μm), and permeability (6 μm) maps. Three examples are given at the finest resolution.

Grahic Jump Location
Fig. 10

Assessing the randomization constraints for IAAFT and DT-CWT. (a) Average map of the IAAFT surrogates when only the phases are preserved (all amplitudes were set to 1). Extreme values localize to ringing artifacts and the ostium edge. (b) Locations of the maximum value in 1000 surrogates. (c) Locations of the minimum value in 1000 surrogates. (d) Average map of 10,000 IAAFT surrogates. (e) Average map of 10,000 DT-CWT surrogates.

Grahic Jump Location
Fig. 11

Null distributions for the restricted randomization tests using IAAFT and DT-CWT (top two rows) for each correlation (vertical line marks the observed correlation). The third row shows the distributions after randomly flipping the DT-CWT surrogates by multiples of 180 deg. The bootstrap sampling distributions are shown in the bottom row with 95% CI marked (vertical lines).

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Fig. 12

Example bootstrap average maps of WSS and permeability

Grahic Jump Location
Fig. 13

CI coverage computed by double bootstrap for each correlation. Values are given for the confidence level that gives 95% coverage (circle markers).

Grahic Jump Location
Fig. 14

Permutation distributions for the difference between permeability maps from immature and mature rabbits using (a) Spearman’s ρ and (b) the sum of the squared pixel differences. The observed association between the average maps is marked by the left hand vertical line. The right hand vertical line marks the critical region of the null distribution (5%, one-tailed).

Grahic Jump Location
Fig. 15

Hierarchical bootstrap distributions for the correlation of permeability with itself using four resampling schemes. (a) Independent, (b) cluster, (c) nested, and (d) multilevel.




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