Longitudinal breast density as a marker of breast...

Longitudinal breast density as a marker ofbreast cancer risk

C. Armero(1), M. Rué(2), A. Forte(1), C. Forné(2),H. Perpiñán(1), M. Baré(3), and G. Gómez(4)

(1)BIOstatnet and Universitat de València,(2)BIOstatnet and Universitat de Lleida,

(3)Epidemiology and Assessment Unit UDIAT-Diagnostic Centre, CorporacióSanitària Parc Taulí, Sabadell,

(4)BIOstatnet and Universitat Politècnica de Catalunya

Joint Modeling and BeyondUniversiteit Hasselt, April 14-15, 2016

Background Scientific objective and material Statistical methods Results Conclusions

Outline

Background

Scientific objective and study population

Statistical methods

Results

EstimationPrediction

Conclusions

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Sección 1 Background

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Background

Breast density (BD) is a characteristic of the breast tissue that isreflected in mammograms. For most women, breast density decreaseswith age.

The ordinal BI-RADS scale (American College of Radiology, 2013) is acommonly tool used to measure breast density. It categorizes breastdensity in four groups

a (1): almost entirely fatty (low density),b (2): scattered fibroglandular densities (medium density),c (3): heterogeneously dense (high density), andd (4): extremely dense (very high density)

Several studies have shown that high breast density is associated with anincreased breast cancer (BC) risk.

Some research (Huo et al., 2014) concluded that longitudinalmeasurement of mammographic density might be used to personalizebreast cancer screening (Biomarker).

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Sección 2 Scientific objective and material

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Objective and material

Objective: To study the association structure and intensity betweenmammographic longitudinal measures of breast density and breast cancerrisk.

Study population:

13,685 women attending a population-based screening program inthe BC early-detection program in the Vallès Occidental East areain Catalonia (Spain), between October 1995 and June 1998.Women were followed for vital status or possible diagnosis of BCuntil December, 2013 (Baré et al., 2003, 2006, 2008).At study entry

50-69 years old.No history of BC.Invited for a biennial mammographic exam.

6 / 26


Study variables and scales: Breast density and breast cancer data

Data from the 13,685 women attending our population-based screeningprogram.

81,621 mammographic measures of breast density with thesubsequent identification and age of the woman.Survival time: Time from the initial age at entrance to thescreening program until BC-diagnosis (Administrative censoring).Time scale: Age (50-69 years).Time zero: 50 years. Women who entered the programme after thisage produced left truncated data.Baseline covariates:

First-degree relative with BC: Yes-No.Prior breast procedures (breast biopsy, fine needle aspiration, cystaspiration, breast reconstruction, lumpectomy and surgicaltreatment): Yes-No.

7 / 26


Data description

13 685 women and 81 621 mammograms

0

500

1000

1500

2000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Number of screenings per women

Cou

nt

1

2

3

4

50 55 60 65 70Years

BI−

RA

DS

Prior breast procedures: 7.1% (no diagnosis BC) and 13.2% (diagnosisBC)

First-degree relative with BC: 5.2% (no diagnosis BC) and 9.8%(diagnosis BC)

431 women diagnosed with BC (3.15%)

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Sección 3 Statistical methods

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Statistical methodology

Elements:

Longitudinal data: Breast density dataSurvival data: Time from the initial age of the screening programto BC detection. Left truncation and administrative censoring.

Objective: We want to construct a survival model to assess the risk of adiagnosis of breast cancer by means of baseline information and breastdensity temporal information.

Put a joint model in your problem ,

10 / 26


Objective and material and methodsJoint models

Different approaches: conditional models, shared-parameter models,random-effects models, and joint latent class models.Shared-parameter models:

1 Longitudinal model for the longitudinal data y .2 Survival model for the survival data s.3 Connection between both models by means of a common vector of

random-effects, b, which endows them with conditionalindependence.Frequentist

f (y , s, b) = f (y , s | b) f (b)= f (y | b) f (s | b) f (b)

Bayesian

f (y , s, b, θ,φ) = f (y , s | b, θ,φ) f (b, θ,φ)= f (y | b, θ,φ) f (s | b, θ,φ) f (b | θ,φ)π(θ, φ)= f (y | b, θ) f (s | b, θ) f (b | φ)π(θ, φ)

where θ and φ are the vector of parameters and hyperparameters,respectively.

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The joint model, I

Longitudinal submodel: A cumulative logit model for the longitudinal ordinalmeasures of breast density based on the idea of a continuous latent variable(Lunn et al., 2001; Luo et al., 2014).

yij breast density in the BI-RADS scale, yij ∈ {1, 2, 3, 4}, of woman i ,i = 1, ..., n, at time tij (age 50 + tij ), j = 1, ..., ni .

We assume an underlying continuous latent variable y∗ij that determinesthe breast density BI-RADS category of woman i at time tij .

Relationship between yij and y∗ij

yij = k ↔ y∗ij ∈ (γk−1, γk ],

y∗ij ∼ Logistic(mij , s = 1)

logitP(yij > k) = mij − γk ,

mit = β0 + bi0 + (β1 + bi1)t,

with −∞ = γ0 < γ1 < γ2 = 0 < γ3 < γ4 =∞ unknown cutpoints, β0, β1unknown parameters and (b0i , b1i subject specific random effects).

12 / 26


The joint model, II

Survival submodel: A left-truncated Cox proportional hazard model for thetime-to-BC detection that incorporates information from the longitudinalprocess.

Ti , i = 1, . . . , n observed BC detection time for the i-th woman,Ti = min(T ∗i ,Ci ), T ∗i is the true failure time and Ci the right-censoringtime. The event indicator, δi = I(T ∗i ≤ Ci ).Hazard function of T ∗i in terms of the left truncated Cox proportionalhazard model

hi (t|x i , θis , t∗i > ai ) = h0(t | λ, η0) exp{η1Famhisti +η2Brstproci +αmit}, t > ai

h0(t | λ, η0) = λtλ−1 eη0 : baseline risk function of a Weibull distribution;x i : baseline covariates, First-degree relative with BC (Famhist) and priorbreast procedures (Brstproc) with regression coefficients η1 and η2;α: effect of the individual trajectory of breast density of a woman overtheir BC risk in terms of the latent BD mean;ai : age over 50 at which woman i enters the screening program thusproviding a left truncated time (Uzunogullari et al., 1992);θis = (λ, η0, η1, η2, α, θil )T , with θil = (β0, β1, b0i , b1i )T .

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The joint model, III

1 Longitudinal submodel: A cumulative logit model for the longitudinalordinal measures of breast density based on the idea of a continuouslatent variable (Lunn et al., 2001; Luo et al., 2014), and

yij = k ↔ y∗ij ∈ (γk−1, γk ],logitP(yij > k) = mij − γk ,

mit = β0 + bi0 + (β1 + bi1)t,

2 Survival submodel: A left-truncated Weibull proportional hazard modelfor the time-to-BC detection, that incorporates information from thelongitudinal process.

hi (t|x i , θis , t∗i > ai ) = h0(t | λ, η0) exp{η1Famhisti + η2Brstproci + αmi (t)}= h0(t | λ, η0) exp{η1Famhisti + η2Brstproci + α(β0 + bi0 + (β1 + bi1)t)}, t > ai

3 Both processes are connected through a shared vector of randomeffects, which, in the presence of covariates and parameters, endows bothprocesses with conditional independence (Rizopoulos, 2012).

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Bayes Inference

Bayesian Inference: Complete specification of the joint model needs theelicitation of a prior distribution for the subsequent parameters andhyperparameters.

Prior independence as a default specification and wide proper priordistributions with the aim of giving all inferential prominence to thedata.Normal distributions for the regression coefficients of thelongitudinal and survival submodels and the association coefficient.Truncated Normal distributions for the cutpoints of the latentscale, −∞ = γ0 < γ1 < γ2 = 0 < γ3 < γ4 =∞ selected to providethe same prior probability to each response category.Gamma distribution, Ga(1, 1), for the parameter of the baselinehazard function because it mimics a constant baseline hazardfunction (Guo et al., )Uniform hyperdistribution distribution Un(0, 10) for the standarddeviations σ0 and σ1 of the normal random effects bi0 ∼ N(0, σ0)and bi1 ∼ N(0, σ1).

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Sección 4 Results

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Posterior distributionLongitudinal submodel: A cumulative logit model for the longitudinal ordinal measures of breast density based onthe idea of a continuous latent variable (Lunn et al., 2001; Luo et al., 2014).

yij = k ↔ y∗ij ∈ (γk−1, γk ],

logit P(yij > Dk ) = mij − γk ,

mit = β0 + bi0 + (β1 + bi1)t,

Survival submodel: A left-truncated Weibull proportional hazard model for the time-to-BC detection, thatincorporates information from the longitudinal process.

hi (t|x i , θis , t∗i > ai ) = h0(t | λ, η0) exp{η1Famhisti + η2Brstproci + αmit}= h0(t | λ, η0) exp{η1Famhisti + η2Brstproci + α(β0 + bi0 + (β1 + bi1)t)}, t > ai

Posterior distribution: Approximated by Markov Chain Monte Carlo methods.

Mean Sd 2.5% Median 97.5% P(· > 0 | D)β0 -1.4262 0.0346 -1.4964 -1.4251 -1.3608 0.0000β1 -0.0524 0.0018 -0.0560 -0.0524 -0.0489 0.0000σ0 2.6067 0.0227 2.5643 2.6059 2.6534σ1 0.0053 0.0018 0.0015 0.0053 0.0087γ1 -4.6994 0.0269 -4.7521 -4.6998 -4.6489γ3 1.7362 0.0156 1.7060 1.7364 1.7675η0 -7.6066 0.3369 -8.2476 -7.6011 -6.9337 0.0000η1 0.6227 0.1716 0.2747 0.6308 0.9517 0.9984η2 0.4535 0.1440 0.1644 0.4600 0.7210 1.0000α 0.1490 0.0207 0.1089 0.1496 0.1887 1.0000λ 1.5366 0.1044 1.3287 1.5387 1.7386

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Posterior distribution for the probabilities associated to breastdensity categories

P(yit = a, b, c, d | θ, bi ,φ) depends on θ and bi .Posterior distribution for the probability associated to each breast densitycategories for a generic woman i of the same population at time t,

π(P(yit = a, b, c, d | θ, bi ,φ) | Data)

approximated from f (bi | φ) and π(θ, φ | Data).Posterior mean 95% credible interval of π(P(yit = a, b, c, d | θ, bi ,φ) | Data)

0.0

0.2

0.4

0.6

50 55 60 65 70Age (years)

Pro

babi

lity

a

b

c

d

18 / 26


Posterior predictive distribution for the latent breast density

Conditional latent breast density (y∗it | θ, bi ,φ) ∼ Logistic(mit , s = 1).Posterior predictive distribution of the latent breast density of a generic womani at t years over 50, (y∗it | Data), (50, 55, 60, 65 and 70 in the graphic).

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−10

−5

0

5

10

50 55 60 65 70Years

y*

0.0

0.2

0.4

0.6

50 55 60 65 70Age (years)

Pro

babi

lity

a

b

c

d

19 / 26


Posterior distribution for the hazard ratios, I

Posterior distribution for the hazard ratio (HR):� HR of BC diagnosis in women with prior breast procedures versuswomen without prior breast procedures.� HR of BC diagnosis in women with family history of BC versus womenwithout family history of BC.� HR of BC diagnosis in women with family history of BC and priorbreast procedures versus women without family history of BC and withoutprior breast procedures.

0.0

0.5

1.0

1.5

0 2 4 6Hazard Ratio

var

Brstproc

Famhist

Famhist and Brstproc

20 / 26


Posterior distribution for the hazard ratios, II

Posterior distribution of the HR of a BC diagnosis for a generic woman i withbreast density in category k at time t versus this generic woman with breastdensity in category k′ at t is NOT POSSIBLE because

hi (t|x i , θis , t∗i > ai ) = h0(t | λ, η0) exp{η1Famhisti + η2Brstproci + αmit}, YEShi (t|x i , θis , t∗i > ai ) = h0(t | λ, η0) exp{η1Famhisti + η2Brstproci + α yit}, NO

We have constructed an ad hoc procedure to represent each value of the breastdensity at time t in terms of a particular value of the true latent breast densityat t

ordinal yit latent y∗it true latent mit (y∗it = mit + εit)

t = 0BI-RADS b c d

a 1.667 (1.450, 1.904) 2.531 (1.959, 3.207) 3.434 (2.437, 4.684)b - 1.512 (1.351, 1.685) 2.046 (1.681, 2.460)c - - 1.350 (1.244, 1.460)

t = 10BI-RADS b c d

a 1.690 (1.464, 1.937) 2.704 (2.054, 3.483) 3.698 (2.569, 5.132)b - 1.593 (1.403, 1.798) 2.171 (1.754, 2.649)c - - 1.359 (1.251, 1.473)

21 / 26


Dynamic estimation and prediction, I

Bayesian reasoningEstimation of the conditional survival probability that a woman i withlongitudinal information Yi at ti,ni and free of a diagnosis of breast cancerat ti,ni is also free at t, P(Ti > t | θ, b,φ,Yi , x i ,Ti > ti,ni ), by means ofits posterior distribution

π(P(Ti > t | θ, b,φ,Yi , x i ,Ti > ti,ni ) | Data), t > ti,ni

that we approach from the posterior distributionπ(θ, b,φ | Yi ,Ti > ti,ni ,Data).Prediction of the longitudinal breast cancer measurement of a givenwoman in a scheduled time tni +1, as the predictive distribution

P(yi,tni +1 = a, b, c, d | Yi , x i ,Data), k = 1, 2, 3, 4

computed fromThe conditional probabilityP(yi,tni +1 = a, b, c, d | θ, b,φ,Yi , x i ,Ti > ti,ni ), andThe posterior distribution π(θ, b,φ | Yi ,Ti > ti,ni ,Data)

22 / 26


Dynamic prediction and prediction, II

Both posterior distributionsπ(P(Ti > t | θ, b,φ,Yi , x i ,Ti > ti,ni ) | Data), t > ti,niP(yi,tni +1 = a, b, c, d | Yi , x i ,Data), k = 1, 2, 3, 4

based on π(θ, b,φ | Yi ,Ti > ti,ni ,Data) and could be applied to womenin the study as well as women not in the study. But we only have anapproximate sample from π(θ, b,φ | Data). Consequently, if woman i is

In the study and Data includes Yi , x i ,Ti > ti,ni then

π(θ, b,φ | Data) = π(θ, b,φ | Yi ,Ti > ti,ni ,Data)

In the study but with new longitudinal and survival information notincluded in Data then Yi , x i ,Ti > ti,ni then

π(θ, b,φ | Data) 6= π(θ, b,φ | Yi ,Ti > ti,ni ,Data)

Update the posterior distribution with the new information. SequentialMCMC methods in progress.Not in the study with sequential interest we need to update

π(θ, b,φ, bi | Yi ,Ti > ti,ni ,Data)

Update the posterior distribution with the new information. SequentialMCMC methods in progress.

23 / 26


Dynamic estimation and prediction, III

Posterior distribution:π(P(Ti > t | θ, b,φ,Yi , x i ,Ti > ti,ni ) | D), t > ti,nif (yi,tni +1 | Yi , x i ,D)

Posterior mean and 95% credible interval of the survival probabilitydistribution and posterior mean of the predictive mammogram breastdensity of woman id17540 (No family history of BC and no prior breastprocedures) with a given follow-up which includes all her historicallongitudinal mammogram and covariates.

24 / 26


Sección 5 Conclusions

25 / 26


Conclusions

Joint models of longitudinal and survival data are a suitable tool foranalyzing the relationship between mammographic breast density andbreast cancer risk.

Our joint model is a good starting point for learning about the problembut we need to introduce more flexibility and work more on the problem:

Non-linear trajectories in the longitudinal mammogram BDmodeling.Leave the Weibull baseline risk function and explore other proposalswhich allow for multimodal or heavy tailed patterns.Assess the performance of the biomarker: Sensitivity, specificity andtime-dependent receiver operating characteristic (ROC) curves.Sensitivity of the prior distributions, in particular the hyperpriors forthe random effects.Improve our knowledge about latent variables in ordinal models.

26 / 26

Longitudinal breast density as a marker of breast...

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