Influence of mammographic density and compressed breast thickness on true mammographic sensitivity: a cohort study

Understanding the detectability of breast cancer using mammography is important when considering nation-wide screening programmes. Although the role of imaging settings on image quality has been studied extensively, their role in detectability of cancer at a population level is less well studied. We wish to quantify the association between mammographic screening sensitivity and various imaging parameters. Using a novel approach applied to a population-based breast cancer screening cohort, we specifically focus on sensitivity as defined in the classical diagnostic testing literature, as opposed to the screen-detected cancer rate, which is often used as a measure of sensitivity for monitoring and evaluating breast cancer screening. We use a natural history approach to model the presence and size of latent tumors at risk of detection at mammography screening, and the screening sensitivity is modeled as a logistic function of tumor size. With this approach we study the influence of compressed breast thickness, x-ray exposure, and compression pressure, in addition to (percent) breast density, on the screening test sensitivity. When adjusting for all screening parameters in addition to latent tumor size, we find that percent breast density and compressed breast thickness are statistically significant factors for the detectability of breast cancer. A change in breast density from 6.6 to 33.5% (the inter-quartile range) reduced the odds of detection by 61% (95% CI 48–71). Similarly, a change in compressed breast thickness from 46 to 66 mm reduced the odds by 42% (95% CI 21–57). The true sensitivity of mammography, defined as the probability that an examination leads to a positive result if a tumour is present in the breast, is associated with compressed breast thickness after accounting for mammographic density and tumour size. This can be used to guide studies of setups aimed at improving lesion detection. Compressed breast thickness—in addition to breast density—should be considered when assigning complementary screening modalities and personalized screening intervals.


Sub-model 1: Age at onset
We start with the Moolgavkar-Venson-Knudson (MVK) two-stage model of carcinogenesis [1,2] to describe the time from birth to the age at onset T of an invasive breast cancer tumor.The survival function for this model is given by with the parameters A < 0 and B, δ > 0. The corresponding probability density function (PDF) is For practical reasons, we assume that the tumor is a sphere with a starting diameter of 0.5mm at age T = t.From that point, we assume that the tumor will grow deterministically, and will be detectable with non-zero probability.

Sub-model 2: Tumor growth
Once onset occurs, we assume that the tumor is a sphere that grows exponentially.Given that onset occurred at age T = t, and given an inverse growth rate r, the tumor volume (mm 3 ) and diameter (mm) at age x > t are given by where d 0 = 0.5(mm), and v 0 is the volume of the sphere with diameter d 0 , i.e. the starting tumor diameter and volume, respectively.
To represent the heterogeneity in the tumor growth rates between tumors, we assume that the inverse growth rate of each tumor is drawn from a gamma-distributed random variable R = r, with for the parameters µϕ > 0, where E [R] = µ and V ar (R) = ϕµ 2 .While the individual inverse growth rates are not observed, the available data can be used to estimate the parameters µ and ϕ, and thus the inverse growth rate distribution of the study population.

Sub-model 3: Symptomatic detection
As the tumor progresses in size, the breast cancer has an increasing risk of displaying symptoms, such as a palpable lump, redness, or swelling.These symptoms then lead to detection of the cancer.
We assume that the continuous hazard rate of symptomatic detection at age u is proportional to the latent tumor volume V (u), i.e. for the age at symptomatic detection U : for the parameter η > 0. In this study, we let η depend on TBV through Since TBV is only measured at screening, we use each woman's average TBV across screenings as a constant effect on the symptomatic detection rate.

Sub-model 4: Screen-detection
If a woman attends mammography screening before a tumor is symptomatically detected, there is opportunity to detect it early.We assume that the screening test sensitivity (STS) for the mammography screening undertaken at age ω follows a logistic function of the latent tumor diameter for parameters β 0 , β s .The STS can be extended to depend on other factors than just size.In this study, we include the possible dependence on PD, CBT, EXP, CP.The STS function is then Over her lifetime (and up to her end of follow-up), a woman will be invited to attend mammography screening at certain ages, each of which she can choose to attend or not.This creates an individual screening history for each woman in the study, consisting of the ages when screened, and the results of each screening (positive or negative).For most women, this will be a sequence of negative screenings.Only the screen-detected cases will have one positive result-their final screening.If a woman attended screening at ages τ 1 , τ 2 , . . ., τ k , the probability of her screening history is where ST S * (ω j ) is the STS when the tumor size at age ω j is unknown.Since tumor size is only measured at diagnosis, then-for all but the screen-detected cases-these latent tumor sizes must be inferred by combining all four sub-models.For screen-detected cases, tumor sizes at all previous negative screenings need to be inferred in the same way.
This process of screen-detection becomes a competing risk to symptomatic detection, where only the first detection mode is observed.A cancer which is symptomatically detected between two mammography screenings is commonly referred to as an interval cancer.

Likelihood function
The four sub-models are combined into an individual timeline for each woman in the study.This constructed timeline is used to calculate the likelihood of the observed outcome.Thus, the unknown parameters in the sub-models can be estimated.The formulas for these individual likelihood contributions have been derived and presented previously [3,4].The observed outcome for each BC case in the study is the age at detection, tumor size at detection, and the mode of detection.The likelihood of being detected at age x with a tumor of volume v is , if screen-detected .

drdt.
For the derivations of these likelihood functions, and how they are adjusted for left truncation, see [3].
Each individual's likelihood contribution is then combined into a total log-likelihood.For N study participants, the total log-likelihood is This total log-likelihood is then maximized with respect to the parameters of the four submodels.
marginalizing over the age at onset, we can use the start point (age at onset of a 0.5mm tumor) and end point (detection at age x with a vmm 3 tumor) to calculate the probability of the specific observed outcome (mode, age, and tumor size).For censored individuals we do not have information on the final tumor size (nor of its existence).We therefore need to compound over the inverse growth rate R as well.Then the same idea lets us calculate the likelihood of not being detected by either mode up to end of follow-up: