Generalized Breast Density Metrics

Abstract

Mammograms represent data that can inform future risk of breast cancer. Data from two case-control study populations were analyzed. Population 1 included women (N = 180 age matched case-control pairs) with mammograms acquired with one indirect x-ray conversion mammography unit. Population 2 included women (N = 319 age matched case-control pairs) with mammograms acquired from 6 direct x-ray conversion units. The Fourier domain was decomposed into n concentric rings (radial spatial frequency bands). The power in each ring was summarized giving a set of measures. We investigated images in raw, processed and calibrated representations and made comparison with the percentage of breast density (BD) determined with the operator assisted Cumulus method. Breast cancer associations were evaluated with conditional logistic regression, adjusted for body mass index and ethnicity. Odds ratios (ORs), per standard deviation increase derived from the respective breast density distributions and 95% confidence intervals (CIs) were estimated.

A measure from a lower radial frequency ring, corresponding 0.083–0.166 cycles/mm and BD had significant associations with risk in both populations. In Population 1, the Fourier measure produced significant associations in each representation: OR = 1.76 (1.33, 2.32) for raw; OR = 1.43 (1.09, 1.87) for processed; and OR = 1.68 (1.26, 2.25) for calibrated. BD also provided significant associations in Population 1: OR = 1.72 (1.27, 2.33). In Population 2, the Fourier measure produced significant associations for each representation as well: OR = 1.47 (1.19, 1.80) for raw; OR = 1.38 (1.15, 1.67) for processed; and OR = 1.42 (1.15, 1.75) for calibrated. BD provided significant associations in Population 2: OR = 1.43 (1.17, 1.76). Other coincident spectral regions were also predictive of case-control status.

In sum, generalized breast density measures were significantly associated with breast cancer in both FFDM technologies.

1. Introduction

Breast density is a strong breast cancer risk factor, typically assessed from mammograms [1–6]. There are various methods of measuring breast density [3, 6–14]. Many of these measures capture the degree of bright tissue (mammographic breast density) in a mammogram. The breast cancer risk associated with the percentage of mammographic breast density (BD) estimated by operator assisted methods is robust across studies [4, 15]. Various automated texture measures have also shown to correlate with breast cancer [9]; examples of these metrics include image domain filtering, co-occurrence measures, and higher order (above one) moments derived from the pixel distribution. We refer to measures of this kind as generalized breast density metrics because they do not capture dense tissue directly. Generalized automated measures may be more sensitive to data representations because their consistency across imaging platforms can be dependent upon the spatial frequency reproduction capabilities of the respective imaging systems.

Although breast density is an accepted risk factor [1–6], it has not been integrated into the clinical management for breast care [16] and there is no recognized standard for breast density determination clinically [17]. Moreover, the relationships between breast density and the underlying biological processes are also not well understood [18]. Texture measures can capture breast structure over varying spatial scales. Metrics capable of capturing relevant spatial scales could be useful for informing studies focused on understanding the related biological processes with breast structure, in addition to risk prediction purposes.

We have shown that variation measured in mammograms is associated with breast cancer whether based on digitized film mammograms [19] or mammograms acquired with a specific indirect x-ray detection full field digital mammography unit [20] (FFDM). We have also shown that measures taken over annular regions in the Fourier domain are associated with breast cancer using digitized film mammograms [21]. This methodology decomposes the image domain variation into annular regions (rings) in the Fourier domain with well-defined radial spatial frequency bandwidths resulting in a set of measures. The measure from each ring can also be considered as a texture feature when viewed in the image domain. The set cardinality is adjustable depending on the endpoint and detector element or pixel spacing. In this report, we examined the Fourier spectral properties of mammograms acquired with two FFDM technologies using raw, processed (for presentation), and calibrated mammograms to find spectral regions associated with breast cancer and document similarities. We also made comparisons with the percentage of breast density measure (BD) determined with the Cumulus method, considered as the reference standard. Two case-control studies were used to evaluate these metrics.

2. Materials and Methods

2.1. Population and Imaging

Population 1:

Study 1 included 180 individually matched case-control pairs of adult women that attended the breast clinics at the Moffitt Cancer Center (MCC) between 2006 and 2011. Study images were acquired with a General Electric (GE) Senographe 2000D FFDM unit (General Electric Medical Systems, Milwaukee, WI, USA). This unit uses indirect x-ray detection and has Δ = 100μm pitch. Raw images are in monochrome 1 format with 14 bit per pixel dynamic range and clinical display (for presentation) images are in monochrome 2 format with 12 bit dynamic range. We used mammograms in cranial caudal (CC) orientation as study images. Raw images were used for calibration purposes.

Population 2:

Study 2 included 319 individually matched case-control pairs of adult women that attended the breast clinics at MCC between 2011 and 2017. Two-dimensional (2D) study mammograms were acquired from one of six Hologic (Hologic, Inc., Bedford, MA) mammography units: three conventional 2D Selenia FFDM units (H units) and three Dimensions digital breast tomosynthesis (DBT) units (D units). The 2D FFDM component from the combo-HD mode was used to supplement the dataset derived from the 2D FFDM units because MCC phased out conventional 2D units in the recent past. These units use direct x-ray detection and have Δ = 70μm pitch. Raw images are in monochrome 1 format with 14 bit per pixel dynamic range and processed images are in monochrome 2 format with 12 bit dynamic range.

Both populations were developed with the same IRB-approved protocol. Cases (unilateral disease) were either: (i) women attending the breast clinics at MCC diagnosed with breast cancer or (ii) attendees of surrounding area clinics sent to MCC for breast cancer treatment or diagnostic purposes and found to have breast cancer. Cases had pathology verified unilateral (first time) breast cancer. Controls were attendees of MCC with no history of breast cancer. Controls were individually matched to cases on age (± 2 years), hormone replacement therapy (HRT) usage and current duration, screening history, and mammography unit. The HRT match was based on status of current users or non-users. Non-users included women that have not taken HRT for at least two years. If a case was a current HRT user, the control was matched on this duration (± 2 years). Controls were matched by screening history using a three category classification. Group 1 included women with prior screening history by any means; the duration between the last screening and the study image date must be no more than 30 months. Group 2 included women with a screening history that does not fit within in Group 1 or Group 3. Group 3 included women with no screening history. We used mammograms in cranial caudal (CC) orientation as study images. The unaffected breast was used as the study image for cases (image acquired before treatment) and the matching lateral breast for controls. Population characteristics are provided in Table 1. Women that had breast implants were excluded from this study.

Table 1a.

Population 1 Characteristics: This table provides Population 1 characteristics by either distribution mean for a given measure or percentages of the population. Where applicable, the standard deviation of the respective distribution is provided parenthetically. This population has images acquired with a Senographe 2000D unit. Breast density quantities were derived from log-transformed data for modeling.

Population 1
Measure	p-values	Case N	Case Mean standard deviation or percentage	Control N	Control Mean standard deviation or percentage	Total N	Total Mean standard deviation or percentage
Age	0.25	180	58.6 (10.5)	180	58.5 (10.4)	360	58.6 (10.4)
Race
Caucasian	0.74	159	88.3%	162	90.0%	321	89.2%
African-American	<0.0001	7	3.9%	13	7.2%	20	5.6%
Asian	0.16	7	3.9%	3	1.7%	10	2.8%
More than One	N/A	2	1.1%	0	0.0%	2	0.6%
Other	N/A	4	2.2%	0	0.0%	4	1.1%
Unknown	<0.0001	1	0.6%	2	1.1%	3	0.8%
Ethnicity
Non-Hispanic	0.70	165	91.7%	162	90.0%	327	90.8%
Hispanic	0.84	14	7.8%	16	8.9%	30	8.3%
Unknown	<0.0001	1	0.6%	2	1.1%	3	0.8%
BMI	0.011	179	26.6 (4.6)	180	25.3 (4.3)	359	25.9 (4.5)
Screening Group
Group 1	N/A	162	90.0%	162	90.0%	324	90.0%
Group 2	N/A	13	7.2%	13	7.2%	26	7.2%
Group 3	N/A	5	2.8%	5	2.8%	10	2.8%
HRT Usage
Current	N/A	36	20.0%	36	20.0%	72	20.0%
Not Currently	N/A	144	80.0%	144	80.0%	288	80.0%
MS	0.08
Pre-Menopausal		38	21.1%	48	26.7%	86	23.9%
Menopausal		142	78.9%	132	73.3%	274	76.1%
BD	0.017	180	2.9 (0.7)	180	2.7 (0.8)	360	2.8 (0.8)
P1 (raw)	0.0010	180	4.3 (0.8)	180	4.0 (0.8)	360	4.1 (0.8)
P1 (processed)	0.075	180	5.5 (0.8)	180	5.3 (0.8)	360	5.4 (0.8)
P1 (calibrated)	0.022	180	−1.5 (1.1)	180	−1.7 (1.2)	360	−1.6 (1.2)
p1 (raw)	0.0017	180	−0.9 (0.3)	180	−1.0 (0.3)	360	−0.9 (0.3)
p1 (processed)	0.0010	180	−0.9 (0.3)	180	−1.0 (0.3)	360	−0.9 (0.3)
p1 (calibrated)	0.0005	180	−0.9 (0.3)	180	−1.0 (0.3)	360	−0.9 (0.3)

2.2. Breast Density Measures

The work involves comparing the same Fourier domain spectral measurements from raw, processed (for presentation used for clinical display purposes), and calibrated mammograms (raw mammograms) from two populations with mammograms acquired with different FFDM technologies. Each technology and units have their own calibration data described previously [22–28]. Briefly, calibration adjusts for the acquisition technique differences, resulting in a normalized intensity scale over this range [0,100]. This Fourier domain approach is a set of measures that summarize the power of a given mammogram in a radial spatial frequency system, referred to as P (Power), stemming from our earlier work [29]. The power spectrum of a given mammogram is decomposed radially into n concentric rings, where n is an adjustable integer parameter. The power within each ring is summed producing n spectral measurements, referred to as the ring analysis. To apply the Fourier transform to the breast area, the analysis was constrained to relatively large rectangular regions of interest (ROI) inscribed within the breast area for each image. These techniques were developed earlier to compare the spectra of mammograms in different pixel representations [29] outside of the epidemiologic context and was subsequently evaluated for breast cancer risk using digitized film [21].

To describe the ring analysis specifics, we define f_c as the highest resolvable spatial frequency component in the Cartesian spatial frequency coordinate system, which differs for the two technologies used in this work. Coordinates in this Cartesian spatial frequency domain are denoted by (f_x, f_y). We calculate f_c using the Nyquist criteria [30, 31] expressed as ${\mathrm{f}}_{\text{c}}=\frac{1}{2\mathrm{\Delta }}$, which applies in both the f_x and f_y directions. This gives f_c = 5 cycles/mm for the GE unit and f_c ≈ 7.14 cycles per mm for the Hologic units. Figure 1 shows the basic ring architecture in the two-dimensional Fourier plane using a coarse illustration (relatively large ring widths) with the relevant quantities labeled. In this example, n = 11 (integer ring indexing from r = 0, to n-1) giving ring widths in the radial direction defined as ε = f_c/n = f_c/11. The support region for any given ring measure is more easily described by considering a radial frequency variable f_r = r × ε, that ranges in radial spatial frequencies over this interval, [(r × ε, (r + 1) × ε] measured in cycles/mm. Summarizing the power in each ring gives 11 different ring measurements with each ring corresponding to a specific radial frequency band noting the center is a disk with diameter = 2 × ε. The portion of the power spectrum exterior to the rings can be considered separately, summarized, and used as another measure. These exterior regions are referred to as the corners. The ring-width lower limit is dependent upon the breast area size. The number of divisions, n, must be less than half of the ROI shorter dimension (i.e., the x-dimension) to prevent degenerate ring widths, which scales with the breast area. In our work, we ensure that the ring-width is not less than the width of four pixels in the Fourier domain measured along the f_x coordinate axis. In general, the ring-width is an adjustable experiment parameter. Two measures were derived from the ring analysis. One measure is the summarized power in a given ring referenced as P_r. The other related measure is normalized at the patient level given by p_r = P_r/(total power from the rings excluding the center + the power in the corners). Several automated processing steps are required to apply this methodology to mammograms: first, the breast is segmented from the background creating a binary mask; determine largest rectangular region that fits within the breast area with a margin trimmed off the interior and exterior sides [29], discussed in more detail below; apply a separable two-dimensional Hanning window [31]; remove the ROI mean; take 2D Fourier transform (FT) and form power spectrum (the square of the FT magnitude); and apply the ring analysis. We note, applying the Hanning window reduces spectral leakage with the tradeoff of spreading (blurring) the frequency resolution [31]. The breast area segmentation is applied to the corresponding clinical display (for presentation) mammograms rather than the raw images. These images have only two non-zero regions: the breast area (large area) and view maker (relatively small area). A simple threshold was used to form a binary image comprised of two separate regions; the smaller region is set to zero leaving the binary breast area mask. The sum over all rings plus the corners is equivalent to the image variance (in our case the largest rectangle). The ring analysis essentially decomposes the image variation. This relationship is an approximation in this situation because of the Hanning window application.

Figure 1.

Fourier Ring Architecture Illustration: This shows the ring layout in the two-dimensional Fourier plane. We use a relatively coarse ring-width example with n = 11 for illustrative purposes. Cartesian spatial frequency coordinates are referenced as (f_x, f_y). The highest resolvable special frequency, f_c (red), is applicable to both frequency directions. This also shows a radial frequency variable defined as f_r (green). The radial width of a given ring is given by $\frac{{\mathrm{f}}_{\text{c}}}{\text{n}}=\frac{{\mathrm{f}}_{\text{c}}}{11}$ cycles/mm. Each ring corresponds to a specific radial frequency band. Ring labeling starts at r = 0 (center disk) to r = n-1= 10 (inner border of the outer ring); rings are gray shaded, where the inner ring is white and regions exterior to the last ring are black. The radial frequency bounds for the r^th ring are expressed as r×ε for the inner ring border and (r+1) × ε for the outer ring border; in this, example f_r points to the inner boundary of the fourth ring that is 4×ε from the origin. This mask is used as an overlay for the frequency spectrum of a given mammogram. Summing over the spectrum within each ring area produces n = 11 measures in this example designated as P_r. The portion of the spectrum exterior to the last ring (black), where f_r > n × ε = 11× ε, is referred to as the corners; the sum over this area produces an additional measure.

For replication purposes, we provided details for the algorithm used to locate the inscribed rectangular ROI. The method is a general approach that applies to both digitized film and FFDM images. This algorithm determines the rectangle with approximately the largest area that fits within a given breast area and then reduces this area by preset margins; this applies to rectangles that have sides paralleling the image borders. We define A_L as the area of the largest rectangle that can be inscribed within a given breast area. As discussed above, the number of rings is limited by the x-direction spatial extent of a given breast area. To mitigate this limitation, we prefer ROIs with larger x-dimensions. This is achieved by first finding the area for all rectangles that can be inscribed within a given breast area, which are plotted as a function of their x-dimension. Figure 2 provides a graphical illustration of the steps: (a) a typical mammogram; (b) segmented binary mask; (c) the binary mask with representative color-coded rectangles inscribed selected from approximately 3000 rectangles; (d) plot of all possible areas as a function of their x-dimension. The area curves were smoothed with a box car averaging filter of length 21, and color coded points correspond with the areas of the selected rectangles shown in (c). The area-curve is searched starting at the 0.67× A_L intersection on the right portion of the plot for a derivative sign change. The search is performed in the –x-direction. The reverse search order prefers regions with larger x-dimensions and initiating the search at this intersection rather than finding A_L directly is an empirical correction that accounts for poor breast area segmentation (i.e. most often with digitized film) where the area curve may not be smooth or have a clear maximum. In previous applications with similar FFDM images, the search found a rectangle within 5% of A_L in 100% of the samples (i.e. the curves were very smooth), which was also the case for the mammograms used in this study. Figure 3 shows the detected region (black border) and the region with the trimmed margins (white border). The trimmed region was determined removing these margins: 0.10 × y-dimension from both the top and bottom, 0.10 × x-dimension from the exterior side, and 0.01 × x-dimension from the interior side along the image border. The top, bottom, and exterior reductions remove breast area corresponding to where the breast may not have been uniformly compressed.

Figure 2.

Region of Interest Algorithm Illustration: (a) this shows a typical mammogram; (b) this shows the segmented breast area; (c) this shows 6 color coded rectangles inscribed with the breast region selected from approximately 3000 possible rectangles; and (d) this shows the breast area for all possible rectangles as function of their x-dimension with the 6 selected color-coded rectangles marked with color-coded dots. The horizontal line marks 0.67 × A_L and the vertical line marks the detected rectangle (black point).

Figure 3.

Region of Interest Algorithm Output: This shows the box algorithm output for the rectangle for the mammogram shown in Figure 2 with approximately the largest area outlined (black border) and the final box used in the analysis with the margins trimmed (white border).

The ring analysis can be adjusted, producing many measurements per patient. Based on our earlier findings [21], we initially performed a constrained low frequency search over (0.0, 1.0) cycles/mm using raw mages from Population 1 to determine rings associated with breast cancer. We let the number or rings vary within this spectral range and found that ε = 0.083 produced two inner ring measures that were associated with breast cancer by comparisons across cases and controls with a paired t-test. In the more general spectral analysis, we used ε = 0.083 giving n = 61 for Population 1 and n = 86 for Population 2. This choice keeps ε constant across populations and is within the x-dimension width limitation discussed above. In both populations, we appended the power in the corners as an additional last point (r = 61 and r = 86, respectively). The goal is to find common spectral regions related to breast cancer status. This comparison analysis was applied to both raw and processed images. For each ring set, we performed a paired t-test at each ring across the case and controls (e.g., P_r from the cases was compared with P_r from the controls for all r) giving a set of comparisons internal to each population. Plotting the p-values for a given ring set provided a graphical technique to determine important regions in the Fourier spectra within a given population and note similar regions across populations.

The Cumulus (version 3, University of Toronto) application was implemented by an experienced operator in the batch mode to label processed images only. Cases and controls were randomly mixed and the operator was blinded to all patient information and case-control status to ensure objectivity. This measure provides a performance metric for comparison because it has been investigated and validated on a wide-variety of datasets over extended timeframes [1, 2, 4, 32–34] and it does not require data processing expertise.

Conditional logistic regression modeling was used to estimate breast cancer associations. Associations were developed for the same images in the raw, processed, and calibrated formats. Breast density distributions were log-transformed. Odds ratios (ORs) from both continuous and quartile models were used as the association metrics with 95% confidence intervals (CIs). Continuous ORs are presented as per standard deviation (SD) increase determined from the respective breast density measurement distribution. In the quartile models, cut points were determined from the breast density distributions of the controls. We also considered the area under the receiver operating characteristic curve (Az) for each model with 95% CIs. Models will be presented unadjusted and adjusted for body mass index (BMI) and ethnicity. When comparing proportions, we used the McNemar’s (exact) test for within population comparisons. When comparing continuous measures, we used the t-test. Image processing was performed in the IDL environment (Version 8.6, Exelis Visual Information Solutions, Inc., Jersey City, NJ) and regression analyses in the SAS environment (V9.4, SAS Institute Inc., Cary, NC).

3. Results

Patient characteristics for Population 1 are provided in Table 1a. Race varied across cases and controls to some degree. Caucasians (p = 0.74) and Asians (p = 0.16) were represented similarly, whereas African Americans had a greater representation in the control group (7.2%) compared with the case group (3.9%) [p < 0.001]. Ethnicity was similar across cases and controls: Hispanics (p = 0.84) and non-Hispanics (p = 0.70). Cases had higher mean BMI (p = 0.011) than controls, but menopausal status was similar (p = 0.076). Cases had higher mean levels of breast density than controls for all measures except P₁-processed (p = 0.075): BD (p = 0.017); P₁-raw (p = 0.001); P₁-calibrated (p = 0.022); p₁-raw (p = 0.002); p₁-processed (p = 0.001); and p₁-calibarted (0.001).

Patient characteristics for Population 2 are provided in Table 1b. Both Caucasian (p = 0.91) and African American (p = 0.25) women were represented similarly across cases and controls. Controls had a greater proportion of Hispanic women compared with the cases (p < 0.001), whereas differences in BMI (p=0.072) and menopausal status (p = 0.46) were not statistically significant. Cases had higher mean levels of breast density than controls for all measures except P₁-processed (p = 0.108): BD (p = 0.023); P₁-raw (0.002); P₁-calibrated (p = 0.008); p₁-raw (p = 0.003); p₁-processed (p = 0.003); and p₁-calibarted (p = 0.008).

Table 1b.

Population 2 Characteristics: This table provides Population 2 characteristics by either distribution mean for a given measure or percentages of the population. When applicable the standard deviation of the respective distribution is provided parenthetically. Breast density quantities were derived from log-transformed data for data modeling. This population has two-dimensional (2D) images from both conventional 2D and digital breast tomosynthesis Hologic units.

Population 2
Measure	p-values	Case N	Case Mean standard deviation or percentage	Control N	Control Mean standard deviation or percentage	Total N	Total Mean standard deviation or percentage
Age	0.13	319	58.8 (11.3)	319	58.7 (11.3)	638	58.8 (11.3)
Race
Caucasian	0.91	273	85.6%	271	85.0%	544	85.3%
African-American	0.25	26	8.2%	36	11.3%	62	9.7%
Asian	1.00	8	2.5%	8	2.5%	16	2.5%
More than One	0.63	3	1.0%	1	0.3%	4	0.6%
Other	N/A	3	1.0%	0	0.0%	3	0.5%
Unknown	0.51	6	1.9%	3	0.9%	9	1.4%
Ethnicity
Non-Hispanic	0.0008	287	90.0%	258	80.9%	545	85.4%
Hispanic	0.0003	29	9.1%	59	18.5%	88	13.8%
Unknown	1.00	3	0.9%	2	0.6%	5	0.8%
BMI	0.07	319	28.7 (6.0)	314	27.8 (6.6)	632	28.3 (6.3)
Screening Group
Group 1	N/A	224	70.2%	224	70.22%	448	70.22%
Group 2	N/A	58	18.9%	58	18.2%	116	18.2%
Group 3	N/A	37	11.6%	37	11.6%	74	11.6%
HRT Usage
Current	N/A	18	5.6%	18	5.6%	36	5.6%
Not Currently	N/A	301	94.4%	301	94.4%	602	94.4%
MS	0.46
Pre-Menopausal		79	24.8%	73	22.9%	152	23.8%
Menopausal		240	75.2%	246	77.1%	486	76.9%
BD	0.023	319	3.1 (0.5)	319	3.0 (0.5)	638	3.1 (0.5)
P1 (raw)	0.0023	319	2.6 (1.0)	319	2.4 (1.0)	638	2.5 (1.0)
P1 (processed)	0.11	319	8.5 (0.7)	319	8.3 (0.8)	638	8.4 (0.8)
P1 (calibrated)	0.0085	319	−1.2 (1.1)	319	−1.4 (1.3)	638	−1.3 (1.2)
p1 (raw)	0.0027	319	−1.2 (0.4)	319	−1.3 (0.4)	638	−1.2 (0.4)
p1 (processed)	0.0035	319	−1.2 (0.4)	319	−1.3 (0.4)	638	−1.2 (0.4)
p1 (calibrated)	0.0080	319	−1.1 (0.4)	319	−1.2 (0.4)	638	−1.2 (0.4)

The Fourier ring analysis comparisons were performed with raw and processed images. In the comparison figures (Figure 4, 5, 6 and 7) the upper plots show p-values for each ring. The dashed line marks the significance at 0.05 and 0.02 levels for reference in each plot. The lower plots show the corresponding t-statistics, where the dashed line marks t-statistic = 0 in each lower plot. Points above this line show where a given measure from the case group exhibited stochastic dominance and vice versa.

Figure 4.

Ring Analyses for Population 1: This shows the ring analysis for P applied to images acquired with the Senographe 2000D FFDM unit for the raw on the left and for presentation (proc) images on the right. The spectrum was divided into 61 rings (bands) and the corners. Top plots give the p-values for each band; dashed lines mark 0.05 and 0.02 significance levels for reference in each plot. The bottom plots show the associated t-statics. The dashed line marks where the t-statistic = 0 in each plot; points above this line indicate the measure from the group exhibited stochastic dominance.

Figure 5.

Normalized Ring Analyses for Population 1. This shows the ring analysis for p (normalized) applied to images acquired with the Senographe 2000D FFDM unit for the raw on the left and processed images on the right. The spectrum was divided into 61 rings (bands) and the corners. Top plots give the p-values for each band; the dashed lines mark the 0.05 and 0.02 significance levels for reference in each plot. The bottom plots show the associated t-statistics. The dashed line marks where the t-statistic = 0; points above this line indicate the measure from the case group exhibited stochastic dominance.

Figure 6.

Ring Analyses for Population 2: This shows the ring analysis for P applied to images acquired with the Hologic units for the raw on the left and for processed images on the right. The spectrum was divided into 86 rings (bands) and the corners. Top plots give the p-values for each band; dashed lines mark the 0.05 and 0.02 significance levels for reference in each plot. The bottom plots show the associated t-statistics. The dashed line marks the t-statistic = 0; points above this line indicate the case group measure exhibited stochastic dominance.

Figure 7.

Normalized Ring Analyses for Population 2. This shows the ring analysis for p (normalized) applied to images acquired with the Hologic for the raw on the left and processed images on the right. The spectrum was divided into 86 rings (bands) and the corners. Top plots give the p-values for each band; dashed lines mark 0.05 and 0.02 significance levels for reference in each plot. The bottom plots show the associated t-statistics. The dashed line marks where the t-statistic = 0; points above this line indicate the measure from the case group exhibited stochastic dominance.

Comparisons:

When comparing across populations, rings 0–60 are comparable. Rings above this correspond to spatial frequencies beyond 5 cycles/mm (in a rectangular spatial frequency coordinate system) and do not exist in 100μm images (Population 1). From experience in developing breast density measures we use p-value ~ 0.02 as the upper threshold. For raw P, the first ring (0.083– 0.166 cycles/mm), and rings 16–61 (f_r = 1.66–7.07 cycles/mm) were predictive of case status in Population 1; for processed P, the first ring was predictive of case status, whereas rings 43–45 and 47–61 were predictive of control status. For raw P in Population 2, rings 1–34 (f_r = 0.083 – 2.91 cycle/mm) were in predictive of case status; in contrast, the processed data showed no preference. In sum, rings 1 and 16–34 (1.33–2.91 cycles/mm) were coincident across populations in the raw data and predictive of case status.

For p (normalized), the first ring was predictive of case status (raw and processed) and rings 5–61 (raw and processed) were predictive of control status in Population 1. For Population 2, the normalized measure from the first ring (raw) was also predictive of case status, whereas rings 25–86 (2.1–10.1 cycles/mm) for raw and 18–66 (1.49–5.56 cycles/mm) for the processed were predictive of control status for Population 2. In sum, the first ring was common across populations and predictive of case status. In contrast, rings 25–60 (2.07–5.0 cycles/mm) from raw data and rings 18–60 (1.49–5.0 cycles/mm) for processed data were coincident across populations and predictive of control status (the 61^st ring in Population 1 captures the corners, whereas the 61^st ring in Population 2 is a one ring-band measure indicating the two are not directly comparable). In sum, it is interesting to note that the normalized raw plots for both populations appear similar across populations.

In the following breast cancer association comparisons, ORs and Azs are provided with 95% confidence intervals parenthetically. To limit the presentation, we compared continuous and quartile adjusted models. For continuous models, ORs are cited as per standard deviation increases determined from the log-transformed distributions. For the quartile models, we compared the fourth quartiles across the different image analysis metrics.

Breast cancer associations for BD are shown in Table 2. These were used for standard reference comparisons. BD provided significant ORs in the continuous models: OR = 1.72 (1.27, 2.33) with Az = 0.64 (0.57, 0.71) for Population 1; and OR = 1.43 (1.17, 1.76) with Az = 0.62 (0.56, 0.67) for Population 2. In the quartile models, BD provided significant fourth quartile associations: OR = 3.25 (1.53, 6.88) with Az = 0.63 (0.56, 0.70) and OR = 2.27 (1.29, 4.00) with Az = 0.65 (0.59, 0.70) for Population 1 and 2, respectively.

Table 2.

Percentage of Breast Density: This table gives the percentage of breast density (BD) associations from the Cumulus method for Population 1 (left) and Population 2 (right). Breast density distributions were log-transformed. Continuous ORs are provided in per standard deviation (SD) increase. Models are provided in both unadjusted and adjusted for BMI and ethnicity.

BD (Population 1)					BD (Population 2)
Qrt	Case N=180	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)		Qrt	Case N=319	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)
1	30	1.00 (Ref.)	1.00 (Ref.)		1	67	1.00 (Ref.)	1.00 (Ref.)
2	52	1.70 (0.93, 3.13)	2.18 (1.14, 4.18)		2	56	0.84 (0.53, 1.34)	0.84 (0.52, 1.38)
3	44	1.51 (0.81, 2.81)	2.18 (1.08, 4.37)		3	102	1.60 (1.02, 2.53)	1.96 (1.19, 3.23)
4	54	1.89 (0.98, 3.67)	3.25 (1.53, 6.88)		4	94	1.53 (0.97, 2.43)	2.27 (1.29, 4.00)
Az		0.56 (0.48, 0.63)	0.63 (0.56, 0.70)		Az		0.55 (0.50, 0.61)	0.65 (0.59, 0.70)
Log Con	SD	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)		Log Con	SD	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)
	0.7668	1.35 (1.05, 1.74)	1.72 (1.27, 2.33)			0.4911	1.21 (1.02, 1.43)	1.43 (1.17, 1.76)
Az		0.57 (0.49, 0.64)	0.64 (0.57, 0.71)		Az		0.56 (0.51, 0.62)	0.62 (0.56, 0.67)

The ring comparison indicated that the first ring was similar across populations. We examined the P₁ and p₁ measures. Population 1 results for the ring analysis are provided in Table 3. Note that associations for P₁ as a continuous variable were statistically significant for all representations and the magnitude of the ORs were nearly identical. Results based on quartiles revealed that p₁ associations with risk were stronger for calibrated images (OR=5.25) than processed (OR=3.74) or raw images (OR=3.64), although the Az values were nearly identical. In total, the associations provided by these measures were at least equivalent to those produced by BD.

Table 3.

Population 1 Breast Cancer Associations for the P₁ and p₁: This table gives quartile (Qrt) and continuous (Con) odds ratios (ORs) for P₁ (top) and normalized p₁ (bottom) for Population 1. The three data formats from left to right are raw, processed and calibrated. The area under the receiver operator character curve (Az) is also provided for each model. Breast density distributions were log-transformed. Continuous ORs are provided in per standard deviation (SD) increase. Models are provided in both unadjusted and adjusted for BMI and ethnicity.

P1 (raw)					P1 (processed)					P1 (calibrated)
Qrt	Case N=180	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)		Qrt	Case N=180	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)		Qrt	Case N=180	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)
1	24	1.00 (Ref.)	1.00 (Ref.)		1	35	1.00 (Ref.)	1.00 (Ref.)		1	32	1.00 (Ref.)	1.00 (Ref.)
2	51	2.20 (1.15, 4.21)	3.00 (1.46, 6.14)		2	38	1.08 (0.57, 2.04)	1.28 (0.66, 2.47)		2	52	1.58 (0.86, 2.91)	2.12 (1.10, 4.11)
3	42	1.89 (0.93, 3.83)	2.96 (1.33, 6.60)		3	48	1.53 (0.80, 2.93)	2.05 (1.01, 4.15)		3	34	1.17 (0.59, 2.32)	1.86 (0.87, 3.99)
4	63	2.96 (1.48, 5.89)	5.34 (2.37, 12.0)		4	59	1.77 (0.96, 3.26)	2.53 (1.28, 4.98)		4	62	1.93 (1.03, 3.61)	3.29 (1.57, 6.89)
Az		0.60 (0.52, 0.67)	0.66 (0.59, 0.73)		Az		0.57 (0.50, 0.64)	0.64 (0.57, 0.71)		Az		0.58 (0.50, 0.65)	0.66 (0.59, 0.72)
Log Con	SD	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)		Log Con	SD	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)		Log Con	SD	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)
	0.8416	1.48 (1.16, 1.88)	1.76 (1.33, 2.32)			0.8044	1.25 (0.98, 1.59)	1.43 (1.09, 1.87)			1.1965	1.31 (1.04, 1.67)	1.68 (1.26, 2.25)
Az		0.61 (0.53, 0.68)	0.67 (0.60, 0.74)		Az		0.56 (0.49 0.63)	0.67 (0.59, 0.74)		Az		0.57 (0.50, 0.64)	0.68 (0.62, 0.75)
p1 (raw)					p1(processed)					p1(calibrated)
Qrt	Case N=180	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)		Qrt	Case N=180	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)		Qrt	Case N=180	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)
1	32	1.00 (Ref.)	1.00 (Ref.)		1	30	1.00 (Ref.)	1.00 (Ref.)		1	27	1.00 (Ref.)	1.00 (Ref.)
2	44	1.43 (0.75, 2.74)	1.68 (0.85, 3.32)		2	43	1.53 (0.79, 2.97)	1.69 (0.84, 3.40)		2	42	1.88 (0.92, 3.84)	1.97 (0.93, 4.19)
3	38	1.20 (0.61, 2.38)	1.66 (0.80, 3.45)		3	40	1.43 (0.73, 2.79)	1.92 (0.93, 3.94)		3	38	1.68 (0.84, 3.36)	2.18 (1.04, 4.60)
4	66	2.29 (1.20, 4.36)	3.64 (1.75, 7.56)		4	67	2.51 (1.30, 4.84)	3.74 (1.81, 7.72)		4	73	3.49 (1.71, 7.13)	5.25 (2.39, 11.5)
Az		0.58 (0.50, 0.65)	0.67 (0.60, 0.74)		Az		0.57 (0.50, 0.64)	0.65 (0.58, 0.72)		Az		0.61 (0.53, 0.68)	0.66 (0.59, 0.73)
LOG CON	SD	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)		Log Con	SD	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)		Log Con	SD	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)
	0.2925	1.46 (1.14, 1.87)	1.75 (1.32, 2.31)			0.2859	1.49 (1.16, 1.91)	1.75 (1.33, 2.30)			0.2821	1.53 (1.19, 1.96)	1.77 (1.34, 2.33)
Az		0.57 (0.49, 0.64)	0.66 (0.59, 0.73)		Az		0.59 (0.52, 0.66)	0.64 (0.57, 0.71)		Az		0.58 (0.51, 0.66)	0.64 (0.57, 0.71)

Population 2 results are provided in Table 4. The associations and Az for P₁ as a continuous variable were statistically significant for all representations and the magnitude of the ORs were similar. The fourth quartile associations were also statistically significant for all representations with similar magnitude and Az. The associations and Az for p₁ were also similar to those produced by P₁ in this population. The associations provided both measurement variants for all representations were at least equivalent to those produced by BD.

Table 4.

Population 2 Breast Cancer Associations for P₁ and p₁: This table gives quartile (Qrt) and continuous (Con) odds ratios (ORs) for P₁ (top) and normalized p₁ (bottom) for Population 2. The three data formats from left to right are raw, processed and calibrated. The area under the receiver operator character curve (Az) is also provided for each model. Breast density distributions were log-transformed. Continuous ORs are provided in per standard deviation (SD) increase. Models are provided in both unadjusted and adjusted for BMI and ethnicity.

P1(raw)					P1(processed)					P1(calibrated)
Qrt	Case N=319	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)		Qrt	Case N=319	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)		Qrt	Case N=319	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)
1	62	1.00 (Ref.)	1.00 (Ref.)		1	47	1.00 (Ref.)	1.00 (Ref.)		1	52	1.00 (Ref.)	1.00 (Ref.)
2	77	1.28 (0.81, 2.02)	1.47 (0.91, 2.37)		2	89	1.86 (1.17, 2.95)	2.13 (1.30, 3.49)		2	81	1.59 (0.99, 2.57)	1.97 (1.16, 3.33)
3	77	1.34 (0.83, 2.17)	1.69 (1.01, 2.81)		3	99	2.15 (1.33, 3.50)	2.61 (1.54, 4.45)		3	97	1.93 (1.19, 3.11)	2.56 (1.49, 4.39)
4	103	1.91 (1.15, 3.18)	2.36 (1.36, 4.08)		4	84	1.85 (1.13, 3.01)	2.54 (1.45, 4.46)		4	89	1.75 (1.09, 2.83)	2.62 (1.47, 4.69)
Az		0.56 (0.51, 0.62)	0.63 (0.58, 0.69)		Az		0.54 (0.49, 0.59)	0.66 (0.61, 0.71)		Az		0.57 (0.51, 0.62)	0.63 (0.58, 0.69)
Log Con	SD	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)		Log Con	SD	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)		Log Con	SD	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)
	1.0361	1.33 (1.10, 1.60)	1.47 (1.19, 1.80)			0.7668	1.23 (1.05, 1.44)	1.38 (1.15, 1.67)			1.1856	1.25 (1.05, 1.48)	1.42 (1.15, 1.75)
Az		0.59 (0.54, 0.62)	0.63 (0.58, 0.69)		Az		0.57 (0.52, 0.62)	0.61 (0.56, 0.66)		Az		0.59 (0.54, 0.64)	0.61 (0.56, 0.66)
p1(raw)					p1(processed)					p1(calibrated)
Qrt	Case N=319	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)		Qrt	Case N=319	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)		Qrt	Case N=319	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)
1	51	1.00 (Ref.)	1.00 (Ref.)		1	53	1.00 (Ref.)	1.00 (Ref.)		1	49	1.00 (Ref.)	1.00 (Ref.)
2	80	1.55 (0.99, 2.43)	1.74 (1.08, 2.80)		2	83	1.57 (0.99, 2.48)	1.78 (1.09, 2.91)		2	92	1.78 (1.14, 2.78)	2.02 (1.25, 3.25)
3	103	2.09 (1.30, 3.37)	2.54 (1.52, 4.24)		3	87	1.68 (1.05, 2.67)	2.03 (1.23, 3.35)		3	91	1.87 (1.16, 3.03)	2.32 (1.38, 3.90)
4	85	1.73 (1.07, 2.80)	2.34 (1.35, 4.03)		4	96	1.93 (1.18, 3.15)	2.62 (1.52, 4.52)		4	87	1.80 (1.11, 2.93)	2.46 (1.43, 4.25)
Az		0.59 (0.54, 0.64)	0.62 (0.57, 0.68)		Az		0.56 (0.51, 0.62)	0.61 (0.56, 0.66)		Az		0.57 (0.52, 0.62)	0.64 (0.59, 0.70)
Log Con	SD	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)		Log Con	SD	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)		Log Con	SD	unadjusted OR (95% CI)	BMI and ethnicity adjusted OR (95% CI)
	0.4062	1.29 (1.09, 1.52)	1.46 (1.20, 1.77)			0.3799	1.28 (1.08, 1.52)	1.44 (1.19, 1.75)			0.3996	1.25 (1.06, 1.47)	1.40 (1.16, 1.69)
Az		0.58 (0.53, 0.64)	0.63 (0.57, 0.68)		Az		0.57 (0.51, 0.62)	0.63 (0.57, 0.67)		Az		0.57 (0.52, 0.62)	0.61 (0.56, 0.66)

We briefly discuss other coincident rings. Rings 16–34 were similar in the raw format across populations. We summed the power in these rings producing one measure referred to as P_16–34. As a continuous measure from raw data, P_16–34 produced significant associations in the adjusted models: OR = 1.52 (1.11, 1.97) with Az = 0.62 (0.55, 0.69) in Population 1; and OR = 1.48 (1.09, 2.01) with Az = 0.62 (0.56, 0.67) for Population 2. We performed a similar investigation for the normalized measure by summing rings 25–60 for both populations producing p_25–60, where the control measure (raw data) showed stochastic dominance (inverse relationship). As a continuous variable, this measure produced significant associations in the adjusted model: OR = 0.64 (0.50, 0.83) with Az = 0.64 (0.57, 0.71) for Population 1; and OR = 0.74 (0.61, 0.89) with Az = 0.63 (0.57, 0.68) for Population 2.

As noted, the Fourier ring measures can equivalently be viewed as texture measures in the image domain. Specifically, examples are provided illustrating textures (structure). Figure 8 shows a mammogram from Population 2 with the rectangular box outlined. To illustrate texture captured by both an isolated ring and a range of rings, we used the respective rings(s) as a filter by taking the FT of the rectangular region (without the Hanning window application), multiplying by the ring(s), followed by Fourier inversion. Figure 9 (top-left) illustrates the structure captured by the first ring. If we consider a radial spatial frequency variable, f_r, the inner radial boundary for the first ring is given by f_r = Δ= 0.083 cycles/mm for both P₁ and p₁. Likewise, the first ring outer radial boundary is given by f_r = 2 × Δ = 0.166 cycles/mm. The radial spatial frequency components within this band have radial spatial periodicities (i.e. 1/f_r) between 6–12mm. Figure 9 (top-middle) shows the structure captured by rings 16–34, where cases exhibited stochastic dominance across the populations (P_16–34). This band range equates with radial spatial periodicities between 0.34–0.75mm. Figure 9 (top-right) shows the structure captured by rings 25–60, where the controls exhibited dominance for the normalized measure (i.e. p_25–60). The associated band range equates with radial spatial periodicities between 0.20–0.48mm. The associated bands, illustrated in the bottom row, appear as ellipses because the frequency spacing is more resolved in the f_y direction. Figure 9 also demonstrates the multiresolution aspects of the ring analysis. The lower spatial frequency rings account for the more coarse structure, whereas the higher spatial frequencies tend to account for finer detail.

Figure 8.

Mammogram Illustration: This shows a processed (for presentation) mammogram used for viewing purposes with the rectangular region of interest outlined.

Figure 9.

Structure Captured by Select Spatial Frequency Bands: Regions in the top row correspond to the outlined region in Figure 8. These images show the structure captured by various frequency bands that were common across populations determined without the Hanning window application. The image on the top-left shows structure captured by P₁. The image in the top-middle shows structure captured in rings16–34 and the image on the top-right shows structure captured in rings 25–60. The respective corresponding bands in the Fourier plane are illustrated in the bottom row, where the passbands are illustrated in white. The Fourier plane corresponds to Figure 1.

4. Discussion

Fourier power spectra of mammograms were decomposed with the ring analyses in a radial spatial frequency coordinate system and comparisons were made across FFDM technologies. The data representation influenced the relationships of the spectral components with breast cancer status. This is illustrated in both populations when comparing the raw and processed data representations (see top plots in Figure 4 and 5). Common regions in the Fourier domain predictive of breast cancer across technologies were isolated. The primary regions predictive of case status were related to lower radial spatial frequencies captured by both the P₁ and p₁ metrics and more mid-bands captured by P_16–34. The normalized comparisons showed that p_25–60, a wider band region, was predictive of control status. In comparison with BD, these measures provided at least equivalent associations with breast cancer within populations. For Population 1, this applied to P₁ from raw and calibrated data and to p₁ for all representations. For Population 2 and band = 1, all Fourier measures considered were at least equivalent to BD. We note our findings for continuous BD in this report are in agreement with a recent meta-analysis that examined percentage of breast density measures including BD [4].

There has been considerable work in relating textures to breast cancer risk [9, 35]. There is a duality between applying filters in the image domain and spectral analyses in the Fourier domain. The Fourier domain view is normally not the focus of cancer epidemiologic studies. In contrast, our approach considered both views. Our approach can be considered as a multiresolution analysis measured in a radial direction, consistent across varying conditions. We found both a specific lower radial frequency band (i.e. P₁) corresponding to specific spatial periodicities and a multiple band region (P_16–34) corresponding to a range of spatial periodicities were predictive of case status. The normalized measures are relative (relative scale) to each woman and isolated two important spectral regions; one very narrow band related to case status (p₁) and a wide-band (p_25–80) related to control status. In contrast, often the focus of texture work is multivariate in character [36–39] where the precise description of a given feature is not the primary interest.

The results for the inner ring also agree with our previous work that analyzed digitized film mammograms derived from a screening practice, where associations were also compared with BD [21]. Our findings, past and present, are an indication that the structure captured from this inner band is robust across mammographic imaging platforms and data representations. This lower band similarity is consistent with characteristics common to imaging systems. The modulation transfer function is often unity at very low spatial frequencies for mammography systems and decreases as frequency increases [40]. We also found measures from multiple band regions were consistent across platforms indicating these may also be robust metrics, noting that the MTFs generally vary across FFDM systems and screen-film over the spatial frequency domain [40]. The normalized ring analysis may be the preferred approach when merging data from different sources because it accounts for scaling differences between data sources. Our findings also show that calibration is not required when using these Fourier metrics because the other data representations provided similar magnitudes of breast cancer risk associations. We note, the associations for Population 1 were stronger than Population 2. This may be due population attributes. Population 1 does not include women with the full spectrum of breast sizes due to the limited detector field of view for the specific mammography unit. This variation could also be due to the sample size of Population 1 relative to Population 2.

We made comparisons with BD. In one context, our approach could provide another way of measuring risk from mammograms because the automated algorithm is relatively simple and does not require thresholds. In another context, these Fourier measures emphasize certain spatial scales within the mammographic structure. These metrics may provide additional information associated with risk because our analysis removed the mean intensity, which is more related to average breast density. The biological mechanisms relating breast density (or breast structure) and breast cancer are largely unknown [1, 18, 41]. A more analytical description of breast structure has the potential to better inform future studies in understanding the biology of risk.

There are several limitations with our study. Sampling of cases and controls was not population-based, but rather a mixture of cases ascertained at an NCI-designated comprehensive cancer center inclusive of referrals from the community. There is no evidence that the cases are not representative, but the current findings should be replicated in a population-based study. The analysis was also restricted to relatively large rectangular regions, indicating a portion of the breast area was excluded from the analyses. Mammograms are three-dimensional volumes projected onto two-dimensions producing overlapping structures or anatomical noise, which is a fundamental limitation for clinical purposes. In our analysis, it is not clear if this is a limitation. It may be that the strength of our signal (bands associated with breast cancer) is dependent upon anatomical noise. Using BD as a control reference indicates these artifacts have negligible impact on the findings. The ring analysis results in metrics with excellent radial spatial frequency resolution due to the well-defined and infinity steep boundaries between each ring. The price for this spatial frequency localization is poor spatial localization in the image domain. This tradeoff is not relevant for our current work because the interest is a global breast measurement rather than a localized one. The ring measures consider radial symmetry implying the image structure lacks directionality. This is only a limitation in this study because the ring analysis can be easily modified to consider radial arcs corresponding to directionality in the image domain.

5. Conclusion

There is a critical need to measure breast density accurately because it is instrumental for various clinical applications. These applications include dictating the BI-RADS tissue composition reporting and for risk prediction purposes. Evidence indicates that many women may experience shifts in their composition classification due to operator variability [17]. Although there are many studies assessing various breast density methods for risk prediction, currently there is no clinical standard [11], and it is not included routinely in clinical risk assessments [16]. An accepted standardized measure of breast density is required to fully actualize the benefits of personalized breast screening and interventions. Future work includes evaluating these Fourier metrics in other study populations, including population-based samples, to enhance generalizability.