PERSONALIZED RISK PREDICTION FOR CANCER SURVIVORS: A GENERALIZED BAYESIAN SEMI-PARAMETRIC MODEL OF RECURRENT EVENTS WITH COMPETING OUTCOMES

Abstract

Multiple primary cancers are increasingly more frequent due to improved survival of cancer patients. Characteristics of the first primary cancer largely impact the risk of developing subsequent primary cancers. Hence, model-based risk characterization of cancer survivors that captures patient-specific variables is needed for healthcare policy making. We propose a Bayesian semi-parametric framework, where the occurrence processes of the competing cancer types follow independent non-homogeneous Poisson processes and adjust for covariates including the type and age at diagnosis of the first primary. Applying this framework to a historically collected cohort with families presenting a highly enriched history of multiple primary tumors and diverse cancer types, we have derived a suite of age-to-onset penetrance curves for cancer survivors. This includes penetrance estimates for second primary lung cancer, potentially impactful to ongoing cancer screening decisions. Using Receiver Operating Characteristic (ROC) curves, we have validated the good predictive performance of our models in predicting second primary lung cancer, sarcoma, breast cancer, and all other cancers combined, with areas under the curves (AUCs) at 0.89, 0.91, 0.76 and 0.68, respectively. In conclusion, our framework provides covariate-adjusted quantitative risk assessment for cancer survivors, hence moving a step closer to personalized health management for this unique population.

1. Introduction.

The most recent report from the National Cancer Institute’s Surveillance, Epidemiology, and End Results (SEER) program estimated approximately 20% of all incident cancer cases in the United States are second or later primary cancers, i.e., additional cancers occurring in cancer survivors [Forjaz et al. (2022)]. By 2026, the total number of cancer survivors living in the US will exceed 20 million [Hulvat (2020)]. Currently, the health care management of cancer survivors is largely indifferent from either at-risk healthy individuals or new cancer patients, thus not yet adequately configured to deliver the desirable high-quality care in this growing population [Mayer, Nasso and Earp (2017)]. Accurate characterization of risks among cancer survivors, however, is challenging because the cancer occurrences are not independent of one another: patients with bladder cancer present the highest risk of developing second cancer [Donin et al. (2016)]; and patients with breast cancer have a higher risk of developing a secondary lung than the general population [Bao et al. (2021)]. These results emphasize the importance of the cancer-type-specific component when modeling multiple primary cancers, given that our ultimate goal is to achieve personalized cancer risk prediction.

Multiple primary cancers can be regarded as recurrent events [Cook and Lawless (2007)]. Within each cancer recurrence, patients may encounter different cancer types and we can regard them as competing events since they preclude the occurrence of others at a given occurrence. Over the past 50 years, much work has been done on modeling recurrent events [Pena, Strawderman and Hollander (2001); Schaubel, Zeng and Cai (2006); Huang, Qin and Wang (2010)] and competing risks [Gray (1988); Fine and Gray (1999); Chatterjee, Hartge and Wacholder (2003); Gorfine and Hsu (2011)] separately. Several recurrent event models have been proposed to include a semi-competing risk [Ghosh and Lin (2002); Liu, Wolfe and Huang (2004); Kalbfleisch et al. (2013)] such as death, which censors the events of interest. However, these models do not allow for non-terminal competing risks within each recurrence, thus cannot model patients’ exposure to different cancer types at any given time point. Thus, a unified statistical framework that jointly models both recurrent events and competing risks remains a significant gap in knowledge. To address this gap, Sankaran and Anisha (2011) proposed a shared frailty model, in which the frailty term induces an unobserved correlation structure among the gap times between successive events. This frailty term, however, has the same multiplicative effect on all the cause-specific and gap-time-specific hazard functions, thus does not appropriately capture the complex interactions between events. Sankaran and Anisha (2012) proposed an additive hazards model, which, methodologically, has the advantage of providing the regression coefficient estimators in closed form. Nevertheless, the model relies on a strong assumption that, conditional on the covariates, the gap times are independent, hence making it unsuitable for many clinical applications, including ours. Another limitation of both models lies in the separate modeling of gap times, which makes it challenging to capture the dependence of the second primary cancer (SPC) on the timing of the first primary cancer (FPC).

In this paper, we fill in this gap by developing a generalized Bayesian model that allows for both terminal and non-terminal competing risks within each recurrence. To avoid the caveat of modeling the gap times separately, we use a non-homogeneous Poisson process to model the occurrence processes of the competing cancer types and, if applicable, competing mortality. This approach has been successfully employed by Shin et al. (2020) for the modeling of recurrent events without competing risks. By introducing covariates that capture the types of previous cancer occurrences into the model, we now construct a fully interpretable model that explicitly quantifies how the risk of a specific cancer type, at a specific recurrence, depends on the patients’ cancer history. Our generalized model works on both individual datasets, in which we have a collection of unrelated patients, as well as family datasets, in which family members are genetically correlated. The latter has not received major attention, possibly due to the unavailability of extensively collected family datasets and the methodological complexities that arise from the transmission of genetic information between family members. Choi et al. (2017) proposed a copula-based model for successive cancer risks in families with Lynch syndrome in the presence of competing mortality, but their method relies on empirically estimating the unknown genotypes from the data subset that has observed genetic information. This approach, however, could be potentially unreliable given the limited availability of genetic testing. In contrast, Shin et al. (2018) utilized the peeling algorithm [Elston and Stewart (1971)] to fully account for the pedigree structures of families in their likelihood computation. We adopt this approach in our model, augmenting it with a shared family-specific frailty term [Hougaard (1995)] to account for latent correlations among family members. Given appropriate data, our model can accommodate any number of recurrences and competing risks, and any set of patient-specific covariates, hence can be utilized to reveal important clinical insights.

Comprehensive data collection for cancer survivors, as needed for mathematical modeling, remains a major bottleneck. There exists a well-characterized inherited cancer syndrome, called Li-Fraumeni syndrome (LFS) [Li and Fraumeni (1969)], which is mainly caused by deleterious germline mutations in the TP53 tumor suppressor gene [Malkin et al. (1990)]. LFS presents a wide spectrum of cancer types, most notably soft-tissue sarcoma, osteosarcoma and breast cancer [Li and Fraumeni (1969)], as well as a much higher incidence rate of multiple primary cancers than the general population (50% as compared to 2–17%) [Mai et al. (2016); Vogt et al. (2017)]. Hence, LFS datasets offer an appropriate and available data source for the modeling of multiple primary cancers with competing risks. We therefore resort to a dataset that was prospectively collected from patients affected by LFS for real-data motivation, model training and model-based risk prediction application. The dataset consists of 429 families, each of which was recruited through a proband based on clinical LFS criteria [Li et al. (1988); Chompret et al. (2001); Bougeard et al. (2015)] at MD Anderson Cancer Center (MDACC) from year 2000 to year 2015. There are 138 families with at least one TP53 mutation carrier (i.e., positive families) and 291 families without any mutation carrier (i.e., negative families). Table 1 summarizes the dataset by gender, TP53 mutation status, type of FPC and type of SPC. We observe that over 90% of the individuals in the MDACC dataset have unknown mutation status. In fact, most people in the general population do not undergo genetic testing, most likely due to the cost and burden of such procedure. In addition, the LFS data is highly biased due to its focus on a high-risk population, which otherwise cannot be captured in practice. Therefore, ascertainment bias correction is required to generate model parameters that are valid for a general population.

Table 1.

Categorization of family members in the MDACC prospective dataset by gender, mutation status, type of first event, and type of second event. BR = breast cancer, SA = sarcoma, OT = other cancers, D = death, C = censored.

	First event						Second event
	BR	SA	OT	D	C	Total	BR	SA	OT	D	C	Total
Male
Wildtype	0	27	64	3	116	210	0	10	23	22	36	91
Mutation	0	41	48	1	40	130	0	15	24	25	25	89
Unknown	0	70	737	1,131	3,317	5,255	0	6	54	551	196	807
Subtotal	0	138	849	1,135	3,473	5,595	0	31	101	598	257	987
Female
Wildtype	86	47	94	2	142	371	42	22	50	23	90	227
Mutation	89	36	48	2	30	205	42	24	31	21	55	173
Unknown	350	55	494	881	3,235	5,015	34	8	59	522	276	899
Subtotal	525	138	636	885	3,407	5,591	118	54	140	566	421	1,299
Total	525	276	1,485	2,020	6,880	11,186	118	85	241	1,164	678	2,286

A natural and highly clinically meaningful question is to accurately predict the TP53 mutation status of individuals in the general population. Using our joint model, we will be able to fully utilize the available data to provide complete risk assessment for both healthy individuals and cancer survivors in a personalized way. Importantly, although this historical dataset has been analyzed in multiple studies over the years, the competing cancer outcomes among cancer survivors have never been quantified in a principled approach. Our newly constructed general model enables an evaluation of these real-life complex outcomes and allows us to utilize the estimated parameters for risk prediction among cancer survivors.

The SEER program provides databases that allow researchers to query cancer-specific incidence rates, defined as the number of new cases per 100,000 people per year, for FPC (www.seer.cancer.gov/seerstat version 8.4.0.1). The program recently updated its SEER*Stat software to allow for queries from patients that developed a specific cancer type as SPC. However, these rates are not fully personalized, i.e., they do not account for important person-specific features beyond age group and gender. For example, the personal risk of developing cancer is significantly increased by the carrier status of a harmful genetic mutation [Parmigiani, Berry and Aguilar (1998); Chen et al. (2006)], which is not captured by SEER. We will demonstrate the significance of this shortcoming in Section 4.3, where we compare penetrance estimates from our model with those imputed from SEER’s incidence rates for patients with early-onset breast cancer.

The paper is organized as follows. In Section 2, we present the modeling and derivation of individual and family-wise likelihoods, techniques for addressing necessary ascertainment bias correction and incorporating frailty into family data, as well as simulation studies to evaluate the parameter estimation performances. In Section 3, we demonstrate how the individual likelihood can be used to construct effective risk models, with a special emphasis on lung cancer. In Section 4, we apply our family-wise model to the LFS dataset to obtain novel penetrance estimates for TP53 mutation carriers and non-carriers. Both sections include a cross-validation study. Section 5 concludes with future research directions.

2. Method.

2.1. Age-at-onset penetrance of multiple cancer types in multiple primaries.

We assume multiple cancer occurrences as recurrent events [Cook and Lawless (2007)]. To describe the model, we begin by introducing some notations. Let $N(t)$ be a non-homogeneous Poisson process that counts the number of primary cancers by age $t$, and $Z$ be the indicator of cancer type (i.e. $Z\in \{1,\dots ,K\}$) with $K$ being the number of cancer types. In addition, $\mathit{X}(t)$ denotes a vector of patient-specific covariates, which concurrently captures the historical information of $N(t)$ up to time $t$. The $k$th cancer-specific intensity function, ${\lambda }_k(t\mid \mathit{X}(t)),k=1,2,\cdots ,K$, is defined as

$${\lambda }_k(t\mid \mathit{X}(t))=\underset{\Delta t\to 0^{+}}{\text{lim}}\frac{P[N(t+\Delta t)-N(t)>0,Z=k\mid \mathit{X}(t)]}{\Delta t}.$$

We note that the overall intensity of $N(t)$ is $\lambda (t\mid \mathit{X}(t))=\sum _{k=1}^K{\lambda }_k(t\mid \mathit{X}(t))$. The notion of age-at-onset penetrance is defined as the probability that a patient develops the disease of interest by age $t$ given his or her characteristics [Langbehn et al. (2004)]. Since then, the definition has been extended to accommodate more complex settings. For example, the $k$th cancer-specific age-at-onset penetrance is defined as the conditional probability of having the $k$th type of cancer by age $t$ prior to developing other cancer types. Compared to our previous model that estimates age-at-onset cancer-specific penetrance among the first primary cancers [Shin et al. (2018)], here we focus on addressing the need of characterizing the $k$th cancer type among multiple primary cancers during the patient’s lifetime. The desired solution will add another layer of complexity to further extend the definition of penetrance. To this end, let $L$ be a non-negative integer that denotes the number of primary cancer occurrences. Then, we define the cancer-specific age-at-onset penetrance for multiple primary cancer as the probability of having the $k$th cancer type at the $l$th occurrence by age $t$ given the ($l-1$)th occurrence at time $u$ in the past, conditional on covariates representing patient characteristics and cancer history up to a time point $u$. Let $T_l$ be the time and $Z_l$ be the cancer type of the $l$th cancer diagnosis. The penetrance of interest, denoted by $q_{kl}(t\mid u,\mathit{X}(u)),k=1,\dots ,K;l=1,\dots ,L$, is then given by

$$q_{kl}\left(t∣u,\mathit{X}\left(u\right)\right)=P\left[T_l\le t,Z_l=k\mid T_{l-1}=u,\mathit{X}\left(u\right)\right]\text{for}t>u,$$

where $T_0=0$ for convenience. For $t>u$, we then have

$$q_{kl}\left(t∣u,\mathit{X}\left(u\right)\right)={\int }_u^t{\lambda }_k\left(v∣\mathit{X}\left(u\right)\right)\prod _{p=1}^K{S}_p\left(v∣u,\mathit{X}\left(u\right)\right)dv,$$ (1)

where $S_k(t\mid u,\mathit{X}(u))=\text{exp}\left[-{\int }_u^t{\lambda }_k(v\mid \mathit{X}(u))dv\right]$ for $k=1,\cdots ,K$.

For a patient with $l-1$ primary cancers, we are also interested in the probability of having the next cancer of any type by age $t$ given the ($l-1$)th occurrence at time $u$. This penetrance can be obtained by summing over all cancer types, i.e., $q_l(t\mid u,\mathit{X}(u))=\sum _{k=1}^K{q}_{kl}(t\mid u,\mathit{X}(u))$.

2.2. Intensity function modeling.

One of the challenges in modeling recurrent events with competing risks is accounting for the correlations between the events. To address this, we introduce a binary variable $D_k(t)$ that indicates whether the patient had the $k$th cancer type as the first primary by time $t$. That is, $D_k(t)=1$ if $T_1<t$ and $Z_1=k$, and $D_k(t)=0$ otherwise. The variable $D_k(t)$ allows the risk of later primary to depend on the type of the first one. The relationship between the first and second primary has been established in a number of epidemiological studies [Travis et al. (2005); Mariotto et al. (2007); Donin et al. (2016)]. We define $D_{k,l}(t),k=1,\dots ,K$, as indicators of the $l$th primary cancer. That is, $D_{k,l}(t)=1$ if $T_l<t$ and $Z_l=k$. By incorporating the set of covariates $\left\{D_{k,l}(t):k=1\dots ,K\right.$ and $\left.l=1,\dots ,L-1\right\}$ into the model specification, we allow the risk of the $l$th primary cancer to depend on the types of the previous $l-1$ occurrences. In our application, we consider only dependence on the type of the first primary cancer due to limited data availability. For each individual with an inherited cancer syndrome such as LFS, we let $G$ be the mutation status (0 for wildtype, 1 for mutated) for the gene of interest, and $S$ be the sex (0 for female, 1 for male). Therefore, the covariate vector is $\mathit{X}(t)={\left\{G,S,D_1(t),\dots ,D_K(t)\right\}}^T$.

We model the cancer-specific intensities using a proportional intensity model as follows:

$${\lambda }_k(t\mid \mathit{X}(t))={\lambda }_{0,k}(t)\text{exp}\left({\mathit{\beta }}_k^T\mathit{X}(t)\right),k=1,\dots ,K$$ (2)

where ${\mathit{\beta }}_k$ denotes the vector of coefficients, and ${\lambda }_{0,k}(t)$ is the baseline intensity function. Let ${\Lambda }_{0,k}(t)={\int }_0^t{\lambda }_{0,k}(v)dv$ be the cumulative baseline intensity function. We model ${\Lambda }_{0,k}(t)$ using Bernstein polynomials, which are often used to approximate functions with constraints such as monotonicity [Lorentz (2012)]. After rescaling $t$ to [0, 1], we can approximate ${\Lambda }_{0,k}(t)$ using the Bernstein polynomials as follows.

$${\Lambda }_{0,k}(t)\approx \sum _{m=1}^M{\Lambda }_{0,k}\left(\frac{m}{M}\right)\left(\genfrac{}{}{0pt}{}{M}{m}\right)t^m(1-t{)}^{M-m},$$

where $M$ denotes the order of the Bernstein polynomials. By reparameterizing ${\gamma }_{m,k}={\Lambda }_{0,k}\left(\frac{m}{M}\right)-{\Lambda }_{0,k}\left(\frac{m-1}{M}\right)$ with ${\Lambda }_{0,k}(0)=0$, it can be shown that

$${\lambda }_{0,k}\left(t\right)\approx \sum _{m=1}^M{\gamma }_{m,k}{f}_{B\left(m,M-m+1\right)}\left(t\right),$$

where $f_{B(m,M-m+1)}$ denotes the beta density with parameters $m$ and $M-m+1$ [McKay Curtis and Ghosh (2011)]. In practice, death from other causes is another competing risk that should be taken into account. One can model the hazard function for death in an identical way to the other cancer-specific intensities from (2).

2.3. Individual likelihood with known $G$.

We first derive the full likelihood of an individual’s observed cancer history, including the development of first, second, or more primary cancers over time, until censorship, when the genotype $G$ is known. For individual $j$, where $j=1,2,\cdots ,n$, let $L_j$ be the number of observed cancer occurrences. Although we focus on $L_j\le 2$ in the applications, we describe our model in a more general setting. We observe a dataset ${\left\{{\mathit{t}}_j,{\mathit{z}}_j,c_j,g_j,s_j\right\}}_{j=1}^n$, where ${\mathit{t}}_j$ and ${\mathit{z}}_j$ are $L_j$-dimensional vectors that indicate the times and cancer types at diagnoses; $c_j$ is the censoring time; and $g_j$ and $s_j$ denote TP53 mutation status and sex, respectively. Now, we introduce $d_{j,k}(t)=1$ if $t_{j,1}<t$ and $z_{j,1}=k$, and $d_{j,k}(t)=0$ otherwise. Finally, the vector of covariates is given by ${\mathit{x}}_j(t)={\left\{g_j,s_j,d_{j,1}(t),\dots ,d_{j,K}(t)\right\}}^T$.

Let $\mathit{\beta }=\left\{{\mathit{\beta }}_k:k=1,\dots ,K\right\}$ and $\mathit{\gamma }=\left\{{\gamma }_{m,k}:m=1\dots ,M;k=1\dots ,K\right\}$. For an individual with cancer history ${\mathit{h}}_j=\left\{{\mathit{t}}_j,{\mathit{z}}_j,c_j\right\}$, the likelihood can be expressed as a product of the observed cancer occurrences and the censoring event

$$P\left[{\mathit{h}}_j\mid {\mathit{x}}_j,\mathit{\beta },\mathit{\gamma }\right]=P\left[T_{j,L_j+1}>c_j\mid T_{j,L_j}=t_{j,L_j},{\mathit{x}}_{ij}\left(t_{ij,L_j}\right),\mathit{\beta },\mathit{\gamma }\right]\times \prod _{l=1}^{L_j}P\left[T_{j,l}=t_{j,l},Z_{j,l}=z_{j,l}\mid T_{j,l-1}=t_{j,l-1},{\mathit{x}}_j\left(t_{j,l-1}\right),\mathit{\beta },\mathit{\gamma }\right].$$

By noting that $d_k(t),k=1,\dots ,K$, are periodically fixed, we obtain the likelihood contribution from the $l$th cancer occurrence as follows.

$$P\left[T_{j,l}=t_{j,l},\right.\left.Z_{j,l}=z_{j,l}\mid T_{j,l-1}=t_{j,l-1},{\mathit{x}}_j\left(t_{j,l-1}\right),\mathit{\beta },\mathit{\gamma }\right]={\lambda }_{z_{j,l}}\left(t_{j,l}\mid {\mathit{x}}_j\left(t_{j,l}\right)\right)\prod _{k=1}^K{S}_k\left(t_{j,l}\mid t_{j,l-1},{\mathit{x}}_j\left(t_{j,l-1}\right)\right),l=1,\dots ,L_j,$$ (3)

with $t_{j,0}=0$. The likelihood contribution from the censored event is given by

$$P\left[T_{j,L_j+1}>c_j\mid T_{j,L_j}=t_{j,L_j},{\mathit{x}}_j\left(t_{j,L_j}\right),\mathit{\beta },\mathit{\gamma }\right]=\prod _{k=1}^K{S}_k\left(c_j\mid t_{j,L_j},{\mathit{x}}_j\left(t_{j,L_j}\right)\right)$$

Finally, the likelihood for the $j$th individual is

$$P\left[{\mathit{h}}_j\mid {\mathit{x}}_j,\mathit{\beta },\mathit{\gamma }\right]=\prod _{k=1}^K{S}_k\left(c_j\mid t_{j,L_j},{\mathit{x}}_j\left(t_{j,L_j}\right)\right)\times \left\{\prod _{l=1}^{L_j}{\lambda }_{z_{j,l}}\left(t_{j,l}\mid {\mathit{x}}_j\left(t_{j,l}\right)\right)\prod _{k=1}^K{S}_k\left(t_{j,l}\mid t_{j,l-1},{\mathit{x}}_j\left(t_{j,l-1}\right)\right)\right\}.$$ (4)

Since we assume the individuals are independent, the overall likelihood of the dataset is simply given by the product of all the individual likelihoods. The complete derivations of the equations above are relegated to Supplementary Materials [Nguyen et al. (2025)], Section 1.

2.4. Family-wise likelihood with unknown $G$.

For inherited cancer syndromes in the general population, the total number of carriers is very low, e.g., ~1 out of 1,300 for TP53 mutations [Gao et al. (2020)]. This rare condition means even one of the most extensively collected datasets would still have just a few hundred mutation carriers ($n=335$, see Table 1) to be used for estimating a series of penetrance curves, in order to capture dynamic cancer outcomes within the first and the second primary cancers (referred to as FPC and SPC from now on). On the other hand, data collection for inherited cancer syndromes, such as LFS, usually includes cancer history information from family members. Accordingly, we have data from additional tens of thousands of family members (e.g., $n=\mathrm{10,270}$, see Table 1), who have all the required information except for mutation status. Many of these family members can carry the mutation due to their genetic relationship with the carriers. Therefore, we introduce a model that includes all family members, including those with unknown genotypes. This missing information can be effectively employed in the inference since its distribution is given by the law of Mendelian inherencence. For example, the probability that a child is a mutation carrier is 0.5 if one of the parents is a carrier. This enables us to define the familywise likelihood as the likelihood of a family marginalized over missing genotypes, and this is a backbone of our method as successfully employed in Shin et al. (2018) and Shin et al. (2020).

2.4.1. Frailty modeling for the family units.

Let $i=\{1,\dots ,I\}$ denote the families. In order to account for often-observed non-genetic correlations between family members, after conditioning on $\mathit{X}(t)$, as a random effect, we introduce a frailty term [Hougaard (1995)], denoted by ${\xi }_{i,k}$, into the modeling of the cancer-specific intensities (2). Importantly, this family-wise random effect may vary across families and across different cancer types. Hence we have

$${\lambda }_k\left(t\mid {\xi }_{i,k},\mathit{X}(t)\right)={\xi }_{i,k}{\lambda }_{0,k}\left(t\right)\text{exp}\left({\mathit{\beta }}_k^T\mathit{X}\left(t\right)\right),k=1,\dots ,K.$$ (5)

For each cancer type $k$, we assume the frailty term to follow a gamma distribution with the same shape and scale parameters, i.e., ${\xi }_{1,k},\dots ,{\xi }_{I,k}\stackrel{\mathit{\text{iid}}}{~}\mathit{\text{Gamma}}\left({\varphi }_k,{\varphi }_k\right)$, a canonical choice in modeling frailty term [Duchateau and Janssen (2008)]. With the presence of the frailties, the age-at-onset penetrance defined in eq.(1) is further modified by marginalizing over the frailties $\mathit{\xi }=\left\{{\xi }_1,\dots ,{\xi }_K\right\}$. For $t>u$, we then have

$$q_{kl}(t\mid u,\mathit{X}(u))={\int }_u^t{\lambda }_{0,k}(v)\text{exp}\left({\mathit{\beta }}_k^T\mathit{X}\left(u\right)\right)R_k(v\mid u,\mathit{X}(u))\prod _{p=1}^K{R}_p(v\mid u,\mathit{X}\left(u\right){)}^{{\varphi }_p}dv,$$ (6)

where $R_k(t\mid u,\mathit{X}(u))={\varphi }_k/\left\{{\varphi }_k+{\int }_u^t{\lambda }_{0,k}(v)\text{exp}\left({\mathit{\beta }}_k^T\mathit{X}(u)\right)dv\right\}$.

2.4.2. Peeling algorithm to model the Mendelian inheritance.

Let $\mathit{\theta }=\{\mathit{\beta },\mathit{\gamma },\mathit{\varphi }\}$ be the full set of model parameters. The likelihood contribution of the $j$th individual in the $i$th family, which we call family-wise likelihood, can be derived similarly to (4), when the covariate vector ${\mathit{x}}_{\mathit{i}\mathit{j}}$ is fully observed for all family member $j$ in family $i$:

$$P\left[{\mathit{h}}_{ij}\mid {\mathit{x}}_{ij},\mathit{\theta },{\mathit{\xi }}_i\right]=\prod _{k=1}^K{S}_k\left(c_{ij}\mid {\xi }_{i,k},t_{ij,L_{ij}},{\mathit{x}}_{ij}\left(t_{ij,L_{ij}}\right)\right)\times \left\{\prod _{l=1}^{L_{ij}}{\lambda }_{z_{ij,l}}\left(t_{ij,l}\mid {\xi }_{i,z_{ij,l}},{\mathit{x}}_{ij}\left(t_{ij,l}\right)\right)\prod _{k=1}^K{S}_k\left(t_{ij,l}\mid {\xi }_{i,k},t_{ij,l-1},{\mathit{x}}_{ij}\left(t_{ij,l-1}\right)\right)\right\}.$$ (7)

Let ${\mathit{x}}_i=\left\{{\mathit{x}}_{ij}:j=1,\dots ,n_i\right\}$ and ${\mathit{h}}_i=\left\{{\mathit{h}}_{ij}:j=1,\dots ,n_i\right\}$ be the aggregated covariate vector and cancer history across all family members of the $i$th family, with ${\mathit{h}}_{ij}=\left\{{\mathit{t}}_{ij},{\mathit{z}}_{ij},c_{ij}\right\}$. Direct evaluation of (7), however, is not possible in our application since the TP53 mutation status is unknown for most family members.

To tackle this, we define familywise likelihood by marginalizing (7) over missing genotypes given observed ones. Its computation is feasible since the distribution of missing genotypes given the observed are available from the law of Mendelian inheritance. Although its computational complexity is quite intensive especially when the family size is large, we can circumvent this issue by employing the peeling algorithm [Elston and Stewart (1971)] described in the following. Let ${\mathit{g}}_i=\left\{g_{ij}:j=1,\dots ,n_i\right\}$ be the set of genotype information. We then partition the covariate vector into genotype and other non-genetic covariates, hence ${\mathit{x}}_i={\mathit{g}}_i\cup {\mathit{g}}_i^C$. Let ${\mathit{g}}_{i,\mathit{\text{obs}}}=\left\{g_{ij}:g_{ij}\text{is}\text{known}\}\right.$ be the observed part and ${\mathit{g}}_{i,\mathit{\text{mis}}}$ the missing part of the genotype data for the $i$th family. We then have ${\mathit{g}}_i={\mathit{g}}_{i,\mathit{\text{mis}}}\cup {\mathit{g}}_{i,\mathit{\text{obs}}}$. We sum over all possible genotype configurations for the family members who have missing genotype information to calculate the family-wise likelihood as $P\left[{\mathit{h}}_i\mid {\mathit{g}}_{i,\mathit{\text{obs}}}\right]=\sum _{g_{i,\mathit{\text{mis}}}}P\left[{\mathit{h}}_i\mid {\mathit{g}}_{i,\mathit{\text{mis}}},{\mathit{g}}_{i,\mathit{\text{obs}}}\right]P\left[{\mathit{g}}_{i,\mathit{\text{mis}}}\mid {\mathit{g}}_{i,\mathit{\text{obs}}}\right]$. Here, the probability $P\left[{\mathit{g}}_{i,\mathit{\text{mis}}}\mid {\mathit{g}}_{i,\mathit{\text{obs}}}\right]$ can be sequentially computed by the law of Mendelian inheritance for a given pedigree structure. For example, if both parents have known genotype as ${\mathit{g}}_{\mathit{\text{obs}}}=\left(g_{\mathit{\text{father}}},g_{\mathit{\text{mother}}}\right)=(\mathrm{1,0})$ then the conditionl probability for a unknown genotype of their son, ${\mathit{g}}_{\mathit{\text{mis}}}=g_{\mathit{\text{son}}}$ is readily obtained by $P\left(g_{\mathit{\text{son}}}=1\mid g_{\mathit{\text{father}}}=1,g_{\mathit{\text{mother}}}=0\right)=0.5$ and $P\left(g_{\mathit{\text{son}}}=1\mid g_{\mathit{\text{father}}}=1,g_{\mathit{\text{mother}}}=\right.0)=0.5$. Although this summation procedure seems computationally intractable as the family size increases, the peeling algorithm can solve it very efficiently by taking advantages of the sequential strucutre of Mendelian inheritance. The details of the implementation of the peeling algorithm are provided in Supplementary Materials [Nguyen et al. (2025)], Section 2].

2.4.3. Ascertainment bias correction for families with LFS.

The ascertainment bias is inevitable and its correction is essential in studying rare diseases like LFS since the data can only be collected from high-risk populations. Introducing an ascertainment indicator for the $i$th family ${𝒜}_i$ that takes 1 if the $i$th family is ascertained and 0 otherwise, the ascertainment-corrected joint (ACJ) likelihood [Iversen Jr and Chen (2005)] is

$$P\left[{\mathit{h}}_i,{\mathit{g}}_{i,\mathit{\text{obs}}}\mid {\mathit{g}}_i^C,\mathit{\theta },{\mathit{\xi }}_i,{𝒜}_i=1\right]=\frac{P\left[{𝒜}_i=1\mid {\mathit{h}}_i,{\mathit{g}}_{i,\mathit{\text{obs}}},{\mathit{g}}_i^C,\mathit{\theta },{\mathit{\xi }}_i\right]P\left[{\mathit{h}}_i\mid {\mathit{g}}_{i,\mathit{\text{obs}}},{\mathit{g}}_i^C,\mathit{\theta },{\mathit{\xi }}_i\right]P\left[{\mathit{g}}_{i,\mathit{\text{obs}}}\mid {\mathit{g}}_i^C,\mathit{\theta },{\mathit{\xi }}_i\right]}{P\left[{𝒜}_i=1\mid {\mathit{g}}_i^C,\mathit{\theta },{\mathit{\xi }}_i\right]}.$$

In practice, the ascertainment decision is often made based on the phenotype of the proband in a deterministic way. If so, the first term in the numerator, $P\left[{𝒜}_i=1\mid {\mathit{h}}_i,{\mathit{g}}_{i,\mathit{\text{obs}}},{\mathit{g}}_i^C,\mathit{\theta },{\mathit{\xi }}_i\right]$ is reduced to $P\left[{𝒜}_i=1\mid {\mathit{h}}_{i1}\right]$ which is independent of the model parameters. In addition, it is not practical for any model to predict genotype data from non-genetic covariates, hence we can reasonably assume that $P\left[{\mathit{g}}_{i,\mathit{\text{obs}}}\mid {\mathit{g}}_i^C,\mathit{\theta },{\mathit{\xi }}_i\right]$ is simplified to $P\left[{\mathit{g}}_{i,\mathit{\text{obs}}}\right]$. We then have the following ACJ likelihood up to a constant of proportionality

$$P\left[{\mathit{h}}_i,{\mathit{g}}_{i,\mathit{\text{obs}}}\mid {\mathit{g}}_i^C,\mathit{\theta },{\mathit{\xi }}_i,{𝒜}_i=1\right]\propto \frac{P\left[{\mathit{h}}_i\mid {\mathit{g}}_{i,\mathit{\text{obs}}},{\mathit{g}}_i^C,\mathit{\theta },{\mathit{\xi }}_i\right]}{P\left[{𝒜}_i=1\mid {\mathit{g}}_i^C,\mathit{\theta },{\mathit{\xi }}_i\right]}.$$

Note that the numerator is the likelihood contribution of the $i$th family without considering the ascertainment bias, which can be obtained by the peeling algorithm as described in Section 2.4. For most LFS studies, the probands can be ascertained due to the diagnosis of a variety of cancer types, in particular those that belong to the LFS spectrum. The ascertainment probability for LFS studies, where the ascertainment decision is based on the first primary cancer diagnosis of the proband, is given by

$$P\left[{𝒜}_i=1\mid {\mathit{g}}_i^C,\mathit{\theta },{\mathit{\xi }}_i\right]=\sum _{k=1}^{K}P\left[{𝒜}_i\mid T_{ip,1}=t_{ip,1},Z_{ip,1}=k\right]P\left[T_{ip,1}=t_{ip,1},Z_{ip,1}=k\mid {\mathit{g}}_i^C,\mathit{\theta },{\mathit{\xi }}_i\right],$$

where the subscript $p$ denotes the proband. We are interested in the ACJ likelihood up to proportionality, hence only the ratio between the terms $P\left[{𝒜}_i\mid T_{ip,1}=t_{ip,1},Z_{ip,1}=k\right],k=1,\dots ,K$, matters. Denoting these terms by $a_1,\dots ,a_K$, where $a_K=1$, it follows that

$$P\left[{𝒜}_i=1\mid {\mathit{g}}_i^C,\mathit{\theta },{\mathit{\xi }}_i\right]\propto \sum _{k=1}^K{a}_{k}P\left[T_{ip,1}=t_{ip,1},Z_{ip,1}=k\mid {\mathit{g}}_i^C,\mathit{\theta },{\mathit{\xi }}_i\right]=\sum _{k=1}^K\sum _{g_{ip}\in \left\{0,1\right\}}{a}_{k}P\left[T_{ip,1}=t_{ip,1},Z_{ip,1}=k\mid g_{ip},{\mathit{g}}_i^C,\mathit{\theta },{\mathit{\xi }}_i\right]P\left[g_{ip}\mid {\mathit{g}}_i^C\right].$$

In our application, the TP53 mutation prevalence is known to be independent of the non-genetic covariates which yields $P\left[g_{ip}\mid {\mathit{g}}_i^C\right]=P\left[g_{ip}\right]$. This probability can be calculated readily from the mutated allele frequency ${\kappa }_A:P\left[G_{ip}=0\right]={\left(1-{\kappa }_A\right)}^2$, and $P\left[G_{ip}=1\right]=1-{\left(1-{\kappa }_A\right)}^2$. In the Western population, we have ${\kappa }_A=0.0006$ [Lalloo et al. (2003)]. In Section 4, we will show that our model is not sensitive to the choice of $a_k,k=1,\dots ,K$.

We denote the overall likelihood of the dataset by $P\left[\mathit{h},{\mathit{g}}_{\mathit{\text{obs}}}\mid {\mathit{g}}^C,\mathit{\theta },\mathit{\xi },𝒜\right]$, where the family subscript $i$ is dropped to represent aggregation across all the $I$ families (e.g., $\mathit{h}=\left\{{\mathit{h}}_i:i=\right.1,\dots ,I\}$). Once the ACJ likelihood has been obtained for each family, the overall likelihood can easily be evaluated as a product as follows, since the families are independent

$$P\left[\mathit{h},{\mathit{g}}_{\mathit{\text{obs}}}\mid {\mathit{g}}^C,\mathit{\theta },\mathit{\xi },𝒜\right]\propto \prod _{i=1}^I\frac{P\left[{\mathit{h}}_i\mid {\mathit{g}}_{i,\mathit{\text{obs}}},{\mathit{g}}_i^C,\mathit{\theta },{\mathit{\xi }}_i\right]}{P\left[{𝒜}_i=1\mid {\mathit{g}}_i^C,\mathit{\theta },{\mathit{\xi }}_i\right]}.$$

2.5. Markov chain Monte Carlo (MCMC) estimation.

We assume a normal prior distribution for ${\mathit{\beta }}_k:{\mathit{\beta }}_k~N\left(0,{\sigma }^2\mathit{I}\right),k=1,\dots ,K$, where $0$ and $\mathit{I}$ denote the zero vector and the identity matrix respectively. For the baseline intensity function, we use a non-informative prior for ${\gamma }_{m,k},m=1,\dots ,M$ and $k=1,\dots ,K$. We set ${\varphi }_k~\mathit{\text{Gamma}}(\psi ,\psi ),k=1,\dots ,K$. This prior distribution is non-negative as required of ${\varphi }_k$. For the real data application in Section 4, we set $\sigma =100$ and $\psi =0.01$ to reflect little prior knowledge about the model parameters, and the simulation study in Section 2.6 shows that the estimates are not sensitive to the choice of hyperparameters. We denote the priors of $\mathit{\beta },\mathit{\gamma }$, and $\mathit{\varphi }$ by $P[\mathit{\beta }],P[\mathit{\gamma }]$, and $P[\mathit{\varphi }]$, respectively. Then, the joint posterior distribution of $\mathit{\theta }$ and $\mathit{\xi }$ is given by

$$P\left[\mathit{\theta },\mathit{\xi }\mid \mathit{h},{\mathit{g}}_{\mathit{\text{obs}}},{\mathit{g}}^C,𝒜\right]\propto P\left[\mathit{h},{\mathit{g}}_{\mathit{\text{obs}}}\mid {\mathit{g}}^C,\mathit{\theta },\mathit{\xi },𝒜\right]\cdot P[\mathit{\beta }].P[\mathit{\gamma }]\cdot P[\mathit{\xi }\mid \mathit{\varphi }]\cdot P[\mathit{\varphi }]$$

We use a random walk Metropolis-Hastings-within-Gibbs algorithm to generate posterior samples. That is, each of parameters $\mathit{\beta },\mathit{\gamma },\mathit{\xi },\varphi$ are updated one by one conditioned on all others fixed, by applying random walk Metropolis-Hastings algorithm (see Algorithm 1). We discarded the first 10% of samples as burn-in. The algorithm is coded in R with the peeling algorithm implemented in C++ and linked via the Rcpp library.

2.6. Simulation study.

We simulated 50 datasets, each consisting of $I$ families with 30 family members spanning three generations (See Fig. 3.1 in Supplementary Materials [Nguyen et al. (2025)], Section 3). In each family, the genotype of the proband is simulated from $G~\mathit{\text{Bernoulli}}(0.005)$, based on which the mutation genotypes of their family members are sequentially generated according to the Mendelian transmission. For simplicity, we consider two competing cancer types (i.e., $K=2$), and the covariate vector is $\mathit{X}(t)={\left\{G,D_1(t),D_2(t)\right\}}^T$, where $G$ denotes the genotype and $D_k\left(t\right),k=1,2$, indicates whether cancer of type $k$ occurred before time $t$ as the first primary cancer. We set the regression coefficients as ${\mathit{\beta }}_1={\{{\beta }_1^G,{\beta }_1^{D_1},{\beta }_1^{D_2}\}}^T=\{4,2,3{\}}^T$ and ${\mathit{\beta }}_2={\{{\beta }_2^G,{\beta }_2^{D_1},{\beta }_2^{D_2}\}}^T=\{3,3,2{\}}^T$. In addition, we specify constant baseline intensities ${\lambda }_{\mathrm{0,1}}(t)={\lambda }_{\mathrm{0,2}}(t)=0.001$ and frailties ${\xi }_1,{\xi }_2\stackrel{\text{iid}}{~}\mathit{\text{Gamma}}(1,1)$ (i.e., ${\varphi }_1={\varphi }_2=1$). Using these predefined model parameters, we generate cancer outcome data for all family members, and devised an ascertainment scheme to collect families that meet certain criteria until the total reaches the required number. To mimic unknown genotypes, we randomly select a percentage of family members in each family, and censor their mutation status. Estimation performance of the model parameters are evaluated via absolute bias and root mean square error. Since the end goal is to estimate age-at-onset penetrances, we also evaluate our penetrance estimation in terms of the absolute bias, mean credible interval width (MCIW) and coverage probability (CP). The exact definitions of the performance measures are described in Supplementary Materials, [Nguyen et al. (2025)] Section 3.

We consider the following penetrances

$$q_{k1}^{(1)}=q_{k1}(t\mid u=0,\mathit{X}(0)=\{0,0,0{\}}^T)$$

$$q_{k1}^{(2)}=q_{k1}(t\mid u=0,\mathit{X}(0)=\{1,0,0{\}}^T)$$

$$q_{k2}^{(1)}=q_{k2}(t\mid u=30,\mathit{X}(30)=\{0,1,0{\}}^T)$$

$$q_{k2}^{(2)}=q_{k2}(t\mid u=30,\mathit{X}(30)=\{1,1,0{\}}^T)$$

$$q_{k2}^{(3)}=q_{k2}(t\mid u=30,\mathit{X}(30)=\{0,0,1{\}}^T)$$

$$q_{k2}^{(4)}=q_{k2}(t\mid u=30,\mathit{X}(30)=\{1,0,1{\}}^T)$$

for $k=1,2$. For sensitivity analysis, we vary the sample size (i.e., the number of families), the censoring rate, as well as the hyperparameters of the prior distributions, resulting in a total of 11 simulation settings. We refer the reader to Supplementary Materials [Nguyen et al. (2025)], Section 3 for a detailed description of our simulation study.

Table 2 shows the estimation performance for one of these simulation settings, in which $I=200$, the censoring rate is 50%, and the hyperparameters are $\sigma =100$ and $\psi =0.01$. Supplementary Materials [Nguyen et al. (2025)], Section 3 provides the results for the other simulation settings. Overall, our MCMC estimates converge well with promising performance across all scenarios, and our model is not overly sensitive to the choice of hyperparameters. Considering the high complexity of our model, we are satisfied with its overall performance.

Table 2.

Performance of parameter and penetrance estimation under a simulation setting, in which $I=200$, the censoring rate is 50%, and the hyperparameters are $\sigma =100$ and $\psi =0.01$. Performance is measured via absolute bias (Bias) and root mean squared error (RMSE) for parameter estimation, and via absolute bias, mean credible interval width (MCIW) and coverage probability (CP) for penetrance estimation.

Parameter	Abs. Bias	RMSE	Penetrance	Abs. Bias	MCIW	CP
${\beta }_1^G$	0.111	0.135	$q_{11}^{(1)}$	0.004	0.018	0.915
${\beta }_1^{D_1}$	0.082	0.101	$q_{11}^{(2)}$	0.032	0.184	0.960
${\beta }_1^{D_2}$	0.061	0.073	$q_{12}^{(1)}$	0.012	0.060	0.960
${\varphi }_1$	0.249	0.300	$q_{12}^{(2)}$	0.034	0.179	0.941
${\lambda }_{0,1}$	0.00008	0.00010	$q_{12}^{(3)}$	0.021	0.104	0.931
			$q_{12}^{(4)}$	0.024	0.125	0.922
${\beta }_2^G$	0.146	0.189	$q_{21}^{(1)}$	0.004	0.017	0.874
${\beta }_2^{D_1}$	0.071	0.087	$q_{21}^{(2)}$	0.028	0.146	0.977
${\beta }_2^{D_2}$	0.102	0.122	$q_{22}^{(1)}$	0.019	0.099	0.941
${\varphi }_2$	0.236	0.280	$q_{22}^{(2)}$	0.034	0.178	0.942
${\lambda }_{0,2}$	0.00008	0.00010	$q_{22}^{(3)}$	0.014	0.060	0.946
			$q_{22}^{(4)}$	0.024	0.131	0.922

3. Application of the individual likelihood.

We apply our models to the MDACC patient cohort as introduced in Section 1. Since the positive families have characteristics that are distinct from the negative ones (e.g., more mutation carriers, more family members with SPC), we put these two types of families into two separate strata. For the cross-validation study, we split the full dataset, randomly with each strata, into two subsets of equal size to have training and validation datasets comparable in cancer-type distributions. See Table 4.1 (Supplementary Materials [Nguyen et al. (2025)], Section 4).

To apply the individual likelihood model to individuals with known $G$ in the training set, we focused on all family members who were tested for TP53, plus those with a predicted mutation probability greater than 0.9 (inferred mutation carriers) or less than 0.01 (inferred non-carriers). These probabilities were calculated using the peeling algorithm as described in Section 2.4. To utilize the individual likelihood-based penetrance estimates in the validation set, we expanded our validation subjects to include family members with a predicted mutation probability greater than 0.9 or less than 0.01. Related summaries of the training and validation sets for the individual likelihood model are provided in Tables 4.2 and 4.3 (Supplementary Materials [Nguyen et al. (2025)], Section 4).

We also compare our penetrance estimates with those that are imputed from the most recent SEER Research Plus database (National Cancer Institute, April 2022). The database provides access to incidence data across eight registries in the US from 1975 to 2019. See Table 4.4 (Supplementary Materials [Nguyen et al. (2025)], Section 4) for summary of the SEER data.

3.1. Model specifications.

We choose three individual cancer-type models to demonstrate utility, with each focused on a different cancer type group among the second primary cancers: i) breast cancer, ii) sarcoma, and iii) lung cancer. Here we group all the other cancer types in contrast with the chosen one to maximize sample size and reduce the number of parameters to be estimated. Our model design is based on the frequency of cancer types among SPC as well as the potential relevance to public health (e.g., lung cancer screening). In each model, we categorize the outcomes as breast cancer (or sarcoma, lung cancer), other cancers, and death. For clarity of notations, we let $k\in \{br/sa/lu,ot,d\}$, instead of integers, where $br,sa,lu,ot$, and $d$ refer to breast cancer, sarcoma, lung cancer, other cancers, and death respectively. For the latter two groups, the intensity/hazard functions take the form given by Equation (4), where $\mathit{X}(t)=\{G,S,D(t){\}}^T$, and $D(t)$ is an indicator of previous cancer occurrences (i.e., $D(t)=1$ if the person has had at least one cancer in the past, and $D(t)=0$ otherwise). Here we use a common indicator to alleviate limited data availability, but cancer-specific indicators, as used in the simulation study, is still applicable to obtain insights on interactions between cancer types given enough data. Given this covariate vector $\mathit{X}(t)$, we write ${\mathit{\beta }}_k={\{{\beta }_k^G,{\beta }_k^S,{\beta }_k^D\}}^T$ for the regression coefficients in the intensity of cancer type $k$. Our dataset has no male patients diagnosed with breast cancer. Hence we restrict the intensity of breast cancer to 0 for male as ${\lambda }_{br}(t\mid \mathit{X}(t))={\lambda }_{0,br}(t)\text{exp}\left({\mathit{\beta }}_{br}^T\mathit{X}(t)\right)I\{S=0\}$. The sarcoma model categorizes the outcomes into sarcoma, other cancers, and death (i.e., $k\in \{sa,ot,d\}$). Lastly, the lung cancer model considers lung cancer, other cancers, and death (i.e., $k\in \{lu,ot,d\}$). For both models, we do not impose any restrictions such as the indicator function above on their intensity functions.

3.2. Model estimation.

Using the random-walk Metropolis-Hastings-within-Gibbs algorithm, we obtain 10,000 posterior samples, of which the first 5,000 are discarded as burn-in. While the overall sample size in the individual likelihood is smaller due to the exclusion of subjects with uncertain mutation status, all subjects are processed as one single group, resulting in larger matrix operations and, therefore, more intensive computation than the family-wise model. Each MCMC iteration takes about two minutes on a high-performance computing cluster at MDACC (407 nodes, each of which has 28 CPU cores with Intel Skylake 2.6Ghz). Posterior samples for all model parameters converged well within the first 10,000 iterations (Figures 5.1–5.3, Supplementary Materials [Nguyen et al. (2025)], Section 5).

Table 5.1 (Supplementary Materials [Nguyen et al. (2025)], Section 5) shows the estimated parameters from three individual models for breast, sarcoma and lung cancers, respectively. We observe that ${\beta }_{br}^G,{\beta }_{sa}^G$, and ${\beta }_{lu}^G$ are significantly positive across all models as expected, since breast cancer, sarcoma, and lung cancer all belong to the LFS spectrum. The parameter ${\beta }_{de}^D$ is also significantly positive in all three models as patients with one cancer in the past are at higher risk of mortality. In the lung cancer model, it is interesting to see that ${\hat{\beta }}_{lu}^D=1.41$, suggesting a higher chance of observing a second primary lung cancer. This is consistent with recent reports on second primary lung cancer cases among breast cancer patients [Bao et al. (2021)]. Similar observation can be made about sarcoma with ${\hat{\beta }}_{sa}^D=1.08$. This is consistent with recent reports on increased chance of developing sarcoma among patients who receive radiotherapy [Patel (2000)].

3.3. Age-at-onset penetrance.

For each of the three models, our primary goal is to estimate the cancer-specific age-at-onset penetrances. Using the notations introduced in Section 2, for a patient with TP53 mutation status $g$ and sex $s$, the set of cancer-specific penetrances to the first primary is denoted by $q_{k1}(t\mid 0,\mathit{X}(0)),k=1,2,3$, where $\mathit{X}(0)=\{g,s,0{\}}^T$. For a patient with a first primary of any type at age 10, the set of cancer-specific penetrances to the second primary is given by $q_{k2}(t\mid 10,\mathit{X}(10)),k=1,2,3$, where $\mathit{X}(10)=\{g,s,1{\}}^T$. While we can construct penetrance curves for other cancers and death, the primary purpose of the Breast Cancer model is to estimate penetrance for breast cancer (e.g., $q_{br,1}(t\mid 0,\mathit{X}(0))$ and $(q_{br,2}(t\mid 10,\mathit{X}(10))$), which is of clinical relevance. Similarly, we estimate the age-at-onset penetrances to the first and second primary for sarcoma, and lung cancer, respectively.

The estimated age-at-onset penetrance to the first primary is presented in Figure 6.1 (Supplementary Materials [Nguyen et al. (2025)], Section 6). For comparison, we also impute probability density curves using the SEER incidence rates (National Cancer Institute, April 2022). Given the rare occurrence of TP53 mutations, one would expect the SEER estimates, which are computed from the general population, to closely match our model-based penetrance estimates for noncarriers.

The major and novel contribution of our approach, however, is the set of covariate-adjusted penetrance estimates for the second primary cancer. Figure 1 shows the corresponding cancer-specific penetrance estimates to the second primary, given a first primary cancer of any type at age 10, from the three individual likelihood models. In contrast, to produce the SEER estimates that match the covariate values, we have to perform the following steps manually. We first select SEER individuals who were diagnosed with first primary cancers of any type at age 10–14 during the period 1975–1989. We then prospectively follow these individuals until 2019 to monitor the occurrences of second primary cancers. The cancer-specific incidence rates at gap times of 5, 10, 15, 20, 25, and 30 years are calculated as the proportions of these individuals who further developed the corresponding cancer type at ages 15–19, 20–24, 25–29, 30–34, 35–39, and 40–44 respectively. In Figure 1, our penetrance estimates for individuals with wildtype TP53 approximates the SEER estimates reasonably well in breast cancer and sarcoma, less well in lung cancer, which has much lower penetrance among SPCs as compared to the other two cancer types. The penetrance estimates of breast cancer and sarcoma are much higher for mutation carriers compared to wildtypes as expected. The penetrance estimates of lung cancer are also higher for mutation carriers, but the difference is much smaller. Although lung cancer belongs to the LFS spectrum, this small difference is not surprising given the very strong competing risks from breast cancer and sarcoma (i.e., mutation carriers are much more likely to develop these two cancer types before lung cancer). While lung cancer has been extensively studied by medical researchers, our study is the first to report age-at-onset penetrance estimates for this cancer type among SPCs, hence potentially impactful on lung cancer screening strategies. In Figures 6.2–6.3 (Supplementary Materials [Nguyen et al. (2025)], Section 6), we provide additional penetrance estimates for two hypothetical patients: one has a first primary cancer at age 20, and the other has a first primary cancer at age 30. These penetrance curves collectively describe how the risk landscapes of patients depend on the ages at diagnosis of their first primary cancers, and demonstrate the utility of our novel statistical framework that formally accounts for such covariates.

Fig 1:

Individual likelihood models: estimates of cancer-specific age-at-onset penetrance to the second primary cancer, given the first primary at age 10. The 95% credible intervals are given at gap times of 5, 15, and 25 years. Cumulative cancer-specific incidence rates from SEER are shown for comparison.

3.4. Validation of the penetrance.

Let $r_{kl}(t\mid u,\mathit{X}(u),\mathit{H})$, where $\mathit{X}(u)=\{G,S,D(u){\}}^T$ and $\mathit{H}$ is the family history, be the risk of developing cancer of type $k$ as the $l$th primary by age $t$ given the ($l-1$)th cancer occurrence at age $u$. That is, $r_{kl}(t\mid u,\mathit{X}(u),\mathit{H})=P\left(T_l\le \right.\left.t,Z_l=k\mid T_{l-1}=u,\mathit{X}(u),\mathit{H}\right)$. For patients with known TP53 mutation status, $\mathit{H}$ can be removed based on the conditional independence assumption. Since we select only family members with measured genotypes, $r_{kl}(t\mid u,\mathit{X}(u),\mathit{H})=P\left(T_l\le t,Z_l=k\mid T_{l-1}=u,\mathit{X}(u)\right)$, is equal to the age-at-onset penetrance $q_{kl}(t\mid u,\mathit{X}(u))$. For each cancer model, we evaluate its prediction performance for (1) the first primary, and (2) the second primary for individuals who already had one primary cancer. Receiver Operating Characteristic (ROC) curves for the first evaluation are given in Figure 7.1 (Supplementary Materials [Nguyen et al. (2025)], Section 7). The second evaluation is our focus, and the results are shown in Figure 2. All models displayed good predictive performances, with areas under the ROC curve (AUCs) being between 0.82 and 0.93, and further presenting small 95% bootstrap confidence intervals.

Fig 2:

ROC curves, along with the AUCs and their 95% confidence intervals, for cancer-specific prediction of the second primary based on the individual likelihood models. SA = sarcoma, BR = breast cancer, LU = lung cancer, OT = other cancer. Sample size: (a) $n(BR)=20,n(OT)=70$; (b) $n(SA)=16,n(OT)=74$; (c) $n(LU)=7,n(OT)=83$

4. Application of the family-wise likelihood.

4.1. Model specifications.

Aiming to address the data sparsity issue presented in the application of our individual model to the real dataset, our family-wise model systematically infers missing genotypes and correspondingly includes all family members in the likelihood calculation. Therefore, we stratify the full dataset into two subsets of equal sizes for training and cross-validation as before, then fit a single unified model to the training set across everyone reported. Motivated by both the biologically meaningful outcomes as well as the need for a sufficient sample size for each category, we feature three groups of cancer diagnosis in this model: (1) sarcoma, (2) breast cancer, and (3) all other cancers combined, which include both LFS-related (i.e., more frequently observed in LFS) and non-LFS malignancies. We include death from other causes as an additional competing risk. For clarity of notations, we set $k\in \{sa,br,ot,de\}$, for sarcoma, breast cancer, other cancers, and death respectively. We have $\mathit{X}(t)={\left\{G,S,D_{sa}(t),D_{br}(t),D_{ot}(t)\right\}}^T$ and ${\mathit{\beta }}_k={\{{\beta }_k^G,{\beta }_k^S,{\beta }_k^{D_{sa}},{\beta }_k^{D_{br}},{\beta }_k^{D_{ot}}\}}^T$ for the regression coefficients in the intensity of cancer type $k$. Note in the unified model, all combinations of cancer group outcomes in the first and the second primary are parameterized. For sarcoma, other cancers, and death, the intensity functions take the form given by Equation (5). In this dataset, no male patients were diagnosed with breast cancer. Hence, we impose the following restriction on the intensity function of breast cancer: ${\lambda }_{br}\left(t\mid {\xi }_{i,br},\mathit{X}(t)\right)={\xi }_{i,br}{\lambda }_{0,br}(t)\text{exp}\left({\mathit{\beta }}_{br}^T\mathit{X}(t)\right)I\{S=0\}$. For the ascertainment bias correction, we set $a_{sa}=10,a_{br}=2$ and $a_{ot}=1$ as we observe sarcoma to be more indicative of TP53 mutation, hence contributing to the patient ascertainment more often.

4.2. Model estimation.

In addition to the settings described in Section 2.5, we set $M=5$ for the degree of the Bernstein polynomial to ensure sufficient flexibility while keeping the computational cost reasonable. Using the random walk Metropolis-Hastings-within-Gibbs algorithm, we generate 30,000 posterior samples, of which the first 5,000 are discarded. Overall, the estimation is computationally intensive due to a large number of parameters, with each MCMC iteration taking about 45 seconds on the data when both frailty and ascertainment bias correction is applied. Trace plots of the posterior samples confirm good convergence within 30,000 iterations (Figures 8.1–8.4, Supplementary Materials [Nguyen et al. (2025)], Section 8).

Table 3 shows that all estimates of the regression coefficients are significant at 95% significance level. The ${\hat{\beta }}_{sa}^G,{\hat{\beta }}_{br}^G$ and ${\hat{\beta }}_{ot}^G$ are positive as expected since mutation carriers are much more susceptible to LFS-related cancer types. Parameters of the form ${\beta }_p^{D_q},p\in \{sa,br,ot\}$ and $q\in \{sa,br,ot\}$, are positive with ${\beta }_{br}^{D_{br}}$ being the sole exception, indicating that patients with a first primary are more likely to develop a second primary later in their lives. A major contributor for this phenomenon is the method of treatment used for the first primary. Radiotherapy and chemotherapy have long been linked to increased risk in cancer survivors [Boice Jr et al. (1985)]. Bhatia and Sklar (2002) found a strong relationship between radiation-related tumors and the dosage of radiation, and that the second cancers typically develop within the radiation field after a latency period. They also reported some chemotherapeutic agents, such as alkylating agents, that increase the risk of developing second cancer. Sarcoma is particularly likely to occur in patients who were treated with radiotherapy [Patel (2000)], which may explain why the largest effect for the onset of second primary cancers is presented by ${\beta }_{sa}^{D_{br}}$. Although data on treatment methods were not collected for our cohort due to its historical nature, they can readily be incorporated into our model as a covariate in future studies where such information has been reliable collected.

Table 3.

Summary of estimated ${\mathit{\beta }}_k,k\in \{sa,br,ot,de\}$, based on the last 25,000 posterior samples. The weights $a_{sa}$ for sarcoma, $a_{br}$ for breast cancer, and $a_{ot}$ for other cancers are set to 10, 2, and 1 respectively

Event	Parameter	Mean	Median	SD	2.5%	97.5%
Sarcoma	${\beta }_{\mathit{\text{sa}}}^G$	4.063	4.063	0.223	3.645	4.494
	${\beta }_{sa}^S$	0.426	0.426	0.182	0.073	0.789
	${\beta }_{\mathit{\text{sa}}}^{D_{sa}}$	1.416	1.417	0.327	0.753	2.043
	${\beta }_{\mathit{\text{sa}}}^{D_{br}}$	1.491	1.493	0.316	0.877	2.110
	${\beta }_{\mathit{\text{sa}}}^{D_{ot}}$	1.716	1.717	0.267	1.181	2.232
Breast	${\beta }_{br}^G$	3.516	3.514	0.201	3.118	3.908
	${\beta }_{\mathit{\text{br}}}^{D_{sa}}$	1.222	1.235	0.408	0.365	1.988
	${\beta }_{br}^{D_{br}}$	−0.785	−0.781	0.252	−1.300	−0.314
	${\beta }_{br}^{D_{ot}}$	0.861	0.860	0.197	0.472	1.238
Other	${\beta }_{\mathit{\text{ot}}}^G$	2.410	2.413	0.137	2.139	2.668
	${\beta }_{ot}^S$	0.392	0.392	0.080	0.239	0.545
	${\beta }_{ot}^{D_{sa}}$	1.183	1.186	0.216	0.762	1.596
	${\beta }_{ot}^{D_{br}}$	0.641	0.644	0.169	0.295	0.964
	${\beta }_{ot}^{D_{ot}}$	0.783	0.783	0.113	0.557	0.993
Death	${\beta }_{\mathit{\text{de}}}^G$	0.841	0.836	0.132	0.578	1.102
	${\beta }_{de}^S$	0.327	0.328	0.056	0.219	0.435
	${\beta }_{de}^{D_{sa}}$	2.309	2.314	0.157	1.987	2.614
	${\beta }_{de}^{D_{br}}$	1.573	1.576	0.112	1.345	1.790
	${\beta }_{de}^{D_{ot}}$	2.139	2.138	0.061	2.016	2.259

Estimates of $\gamma$ and $\varphi$ are provided in Tables 8.1–8.2 (Supplementary Materials [Nguyen et al. (2025)], Section 8). To evaluate the sensitivity of our model to the weights $a_k$, we also report the estimates of the regression coefficients when $a_{sa},a_{br}$ and $a_{ot}$ are set to 4, 2 and 1 respectively (Table 8.3, Supplementary Materials [Nguyen et al. (2025)], Section 8), i.e., reduced contribution to ascertainment from sarcoma. The parameter estimates are highly consistent with those in Table 2, e.g., ${\beta }_{sa}^{D_{br}}$ changed from 1.491 to 1.444, ${\beta }_{br}^{D_{sa}}$ changed from 1.222 to 1.246. hence our model is not overly sensitive to the choice of $a_k$.

4.3. Age-at-onset penetrance.

We present penetrance estimates for FPC and those for SPC with example onsets of the first primary in sarcoma and other cancers in Tables 9.1–9.5 (Supplementary Materials [Nguyen et al. (2025)], Section 9). For comparison, we also show the penetrances that are imputed from the incidence rates provided by SEER (National Cancer Institute, April 2022). Here we focus on discussing the penetrance estimates for SPC, which have never been reported in the literature. For a female patient with a first primary of breast cancer at age 20, the set of cancer-specific penetrances to the second primary cancer is given by $q_{k2}(t\mid 20,\mathit{X}(20))$, where $\mathit{X}(20)=\{g,0,0,1,0{\}}^T$. Figure 3A shows the penetrance curves for such a patient with $g=0$ and $g=1$. The SEER curve was constructed by the following steps. First we selected individuals who were diagnosed with first primary breast cancers at age 20–24 during the period 1975–1989, and prospectively followed them until 2019. We then computed incidence rates at gap times of 5, 10, 15, 20, 25, 30 years as the proportions of individuals who developed second primary cancers of any type during the corresponding age intervals. Similar to Figure 3B, we changed the onset of the first primary breast cancer to age 30. With the females who do not carry TP53 mutations, the SEER estimate approximates closely for those with FPC at age 30 but is much higher than those with FPC at age 20. One potential explanation for observing the high SEER estimate in Figure 3A is that the population that contributes data to SEER in this category mostly consists of germline mutation carriers of TP53 or BRCA1/2, as these women have a substantially higher chance of developing breast cancer at age 20 than wildtype carriers. However, SEER does not collect the individual’s mutation information. This highlights an advancement in our model to explicitly adjust for mutation status as a covariate, whereas curves derived from the SEER registry cannot differentiate those with or without deleterious germline mutations. For individuals with FPC of breast at age 20, their chances of carrying germline mutations are high. Hence the corresponding SEER curve should approximate more closely to the penetrance estimate for mutation carriers (right panel) than noncarriers (left panel) in Figure 3A.

Fig 3:

Estimates of cancer-specific age-at-onset penetrance to the second primary cancer, given that the first primary is breast cancer. The 95% credible intervals at gap times of 5, 15, and 25 years are shown. Cumulative incidence rates across all cancer types from SEER are shown for comparison.

4.4. Validation of the penetrance.

A majority of family members of patients with LFS do not undergo genetic testing for various reasons. Our family-wise likelihood is particularly advantageous in this case because it incorporates all family members’ cancer history into the computation of the time-to-event coefficients. Let ${\mathit{X}}_G(u)$ be the set of covariates without the mutation status (i.e., ${\mathit{X}}_G(u)=\mathit{X}(u)\setminus \left\{G\right\}$). By conditioning on the unobserved $G$, the risk $r_{kl}$ is given by a weighted average

$$r_{kl}\left(t\mid u,{\mathit{X}}_G(u),\mathit{H}\right)=\sum _{G\in \{0,1\}}{q}_{kl}(t\mid u,\mathit{X}(u))P(G\mid \mathit{H})$$

where the mutation probability $P(G\mid \mathit{H})$ is calculated via the peeling algorithm [Elston and Stewart (1971)].

We first evaluate the predictive performance of our joint model in predicting outcomes, such as the TP53 mutation status, number of primaries, or cancer types within the first primary, which are achievable by previous models [Shin et al. (2018, 2020)]. The joint model achieved comparable performances (Supplementary Materials [Nguyen et al. (2025)], Section 10).

We then move further to predict a unique outcome made available by the joint model, the type of cancer within the second primary. This evaluation is particularly noteworthy as it sets our model apart from others in the literature. Given the limited sample size, we include individuals with mutation probabilities greater than 0.99 as inferred carriers, and smaller than 0.01 as inferred wildtypes. For comparison, we consider a naïve model that assumes independence and identical distributions of the first and second primary. Under this assumption, we can show that the cancer-specific age-at-onset penetrances to the second primary is given by

$$q_{k2}(t\mid u,\mathit{X}(u))=\frac{P\left(T_2\le t\mid T_1=u,\mathit{X}(u)\right)P\left(T_1\le t-u,Z_1=k\mid \mathit{X}(u)\right)}{P\left(T_1\le t-u\mid \mathit{X}(u)\right)}.$$

The second term in the numerator is given by a cancer-specific model [Shin et al. (2018)], while the other two are given by an MPC model [Shin et al. (2020)]. We use the penetrance estimates produced by this naive model to perform risk prediction on the validation cohort. Figure 4 compares the validation results of our model with the naive model. For our model, the AUCs are 0.91, 0.76 and 0.68 for sarcoma, breast cancer, and other cancers, respectively. Furthermore, the ROC curves of the naive model hover around the diagonal line, with AUC not significantly different from 0.5 for sarcoma and other cancer types. This indicates poor predictive performance and further confirms the importance of accounting for the type and age-at-onset for the first primary when predicting the second primary.

Fig 4:

ROC curve, along with the AUCs and their 95% confidence intervals, for cancer-specific prediction of the second primary in the test dataset. SA = sarcoma, BR = breast cancer, OT = other cancer. Sample size: $n(SA)=12,n(BR)=16,n(OT)=36$.

5. Conclusion.

With the advancements in oncology, cancer patients are living longer, resulting in a sharp increase in the number of patients diagnosed with multiple primary cancer. Our Bayesian semiparametric framework that jointly models both multiple primary cancers and multiple cancer types represent one of the first attempts to build a quantitative foundation for characterizing cancer risk trajectories among cancer survivors. We provide explicit expressions of both an individual likelihood and a family-wise likelihood, allowing us to evaluate cancer risk trajectories among both independent individuals and family members. The individual likelihood can be used for risk characterization of a general population whose cancer history is captured by registries, while family-wise likelihood can be used for a given inherited disease where family history data are routinely collected. With estimated coefficients by the likelihoods, a suite of penetrance curves can be built, which are cancer-type-specific age-at-onset probabilities for both first and second primary cancers. These probabilities serve as the basis for calculating personalized risk estimates, some of which, for first primary cancer only, are already used in clinical practice to facilitate decision-making [Parmigiani, Berry and Aguilar (1998); Nguyen et al. (2024, 2023)]. Our joint model has the potential to move the needle in clinical management by expanding the current practice to the population of cancer survivors.

We have validated the performance of our models using both simulation studies and cross-validation for risk prediction in a real patient cohort that was extensively and meticulously collected over 40 years encompassing a geneticist’s entire research career. Although this cohort was previously used to address various questions, its complex outcome presented by cancer survivors could not be properly evaluated until we propose this new method. Using the individual likelihood approach, we accurately characterized the risk of second primary lung cancer (SPLC) among cancer survivors in our study population. Our estimated penetrance curves for SPLC is important for the future update of the United States Preventive Services Task Force (USPSTF) recommendation for lung cancer screening, which currently does not account for the cancer survivor status [Krist et al. (2021)]. Using the family-wise likelihood approach, we have obtained a new set of penetrance estimates for the genetic and risk counseling with families with LFS. A repeated and heart-wrenching question during counseling sessions with these families is when my child will develop next cancer and what it will be. Our new set of penetrance estimates have passed an initial validation, hence holds the promise to help provide more informed counseling experience to these families. Next steps include external validations using multiple patient cohorts collected under more heterogeneous ascertainment settings, in order to further evaluate the robustness and accuracy of these estimates. Our framework is general and can accommodate any number of cancer types and the number of primary cancers, as well as additional covariates such as treatment information, given additional data availability. Given enriched data with a long-term follow-up in the future, our unified model can be used to reveal a complete picture of the risk landscape in cancer patients.

One limitation of our method is the relatively high computation cost. Training the family-wise model required about three weeks on a high-performance computing cluster, and training individual likelihood models took even longer. To tackle this issue, we may consider using a block Gibbs sampler with appropriately selected proposal densities for the Metropolis-Hastings algorithm to simultaneously sample from multiple parameters, thus reducing the number of times the family-wise likelihoods are evaluated. One may also consider a data augmentation approach for the familywise likelihood calculation itself, i.e., to formulate the MCMC to estimate both the unknown genotype configurations and all parameters, removing the peeling algorithm calculation for each evaluation. Another limitation is that all TP53 mutations are treated as functionally equal in our model, which is a coarse approximation of TP53 biology. With suitable datasets and a well-validated grouping scheme of TP53 mutations in future when they become available, our model can incorporate this additional information to account for differences in the pathogenicity of these mutations [Neskey et al. (2015)], hence potentially further improving the predictive performance. Finally, we would also like to point out that our model assumes proportionality of the intensity, which is common in standard survival analysis. It is thus important to check the assumption in practice. To the best of our knowledge, however, there is no possible way to check it under the recurrence model with competing events. We believe this is one interesting research direction that warrants further investigation.

Supplementary Material

Supplement (DOI:; .pdf). Derivation of the likelihood, description of the peeling algorithm, full details of the simulation study, MCMC diagnostics, additional data analyses and additional validation exercises are included.