Estimate of the penetrance of BRCA mutation and the COS software for the assessment of BRCA mutation probability

Abstract

We have designed the user-friendly COS software with the intent to improve estimation of the probability of a family carrying a deleterious BRCA gene mutation. The COS software is similar to the widely-used Bayesian-based BRCAPRO software, but it incorporates improved assumptions on cancer incidence in women with and without a deleterious mutation, takes into account relatives up to the fourth degree and allows researchers to consider an hypothetical third gene or a polygenic model of inheritance. Since breast cancer incidence and penetrance increase over generations, we estimated birth-cohort-specific incidence and penetrance curves. We estimated breast and ovarian cancer penetrance in 384 BRCA1 and 229 BRCA2 mutated families. We tested the COS performance in 436 Italian breast/ovarian cancer families including 79 with BRCA1 and 27 with BRCA2 mutations. The area under receiver operator curve (AUROC) was 84.4 %. The best probability threshold for offering the test was 22.9 %, with sensitivity 80.2 % and specificity 80.3 %. Notwithstanding very different assumptions, COS results were similar to BRCAPRO v6.0.

Appendices

Appendix 1

We have modified the original method of Parmigiani et al. [22] in three different points. First, we have included third degree relatives plus the grandparent sibling offspring (IV degree); second, we use age and birth cohort specific incidence and penetrance; third, we have incorporated the possibility of taking into account an hypothetical third gene. We present here the formulas respecting as far as possible the notation that Parmigiani used in his article.

For each family member b, the contribution to the likelihood function depends on the age a _b , the genetic status of BRCA1i, of BRCA2j, of BRCA3k, and on the disease status. As Parmigiani did, we factorize the conditional probability ρ ^ijk _b of observing the disease history of member b, given the genetic status ijk, into two disease-specific term, one for breast cancer and one for ovary cancer (notice that at this point we can add other phenotypes).

$$ \rho_{b}^{ijk} = \rho_{b}^{Oijk} \cdot \rho_{b}^{Bijk} $$

where the meaning of indexes is: family member b, O = ovary, B = breast and

$$ \begin{aligned} i & = 0,1,2\quad BRCA1\; mutated\; alleles \\ j & = 0,1,2\quad BRCA2\; mutated\; alleles \\ k & = 0,1,2\quad BRCA3\; mutated \;alleles \\ \end{aligned} $$

the ρ are defined in the following formula

$$ \rho_{b}^{Oijk} = \left\{ {\begin{array}{*{20}l} {r^{Oijk} \left( {a_{b}, g_{b} } \right) } \hfill & {if\;b\; has\;O\text{var} y \;cancer\; at\; age\; a_{b} } \hfill \\ {1 - R^{Oijk} \left( {a_{b}, g_{b} } \right)} \hfill & {if\; b\; hasnot\; O\text{var} y \;cancer\; at\; age a_{b} } \hfill \\ \end{array} } \right. $$

$$ \rho_{b}^{Bijk} = \left\{ {\begin{array}{*{20}l} {r^{Bijk} \left( {a_{b}, g_{b} } \right)} \hfill & {if\;b\;has\;Breast\;cancer\;at\;age \;a_{b} } \hfill \\ {1 - R^{Bijk} \left( {a_{b}, g_{b} } \right)} \hfill & {if\;b\;hasnot\;Breast\;cancer\;at\;age\;a_{b} } \hfill \\ \end{array} } \right. $$

where R ^ijk(a, g) is the cumulative risk at age a for the generation g and r ^ijk(a, g) is the age and birth cohort specific incidence.

To rewrite the formula used by Parmigiani in the case of 3 genes and for III degree relatives, we assign a label b to every family member and we define the following functions

m(b)	returns the label of the mother of b
f(b)	returns the label of the father of b
no(b)	returns the number of sons and daughters of b
o l (b) l = 1, …, no(b)	returns the label of person l belonging to the offspring of b
q(b)	returns the label of the mate of b
ns(b)	returns the number of brothers and sisters of b
s l (b)	l = 1, …, ns(b) returns the label of person l belonging to the sibling of b

For instance, the label of the paternal grandmother of b will be f(m(b)).

We use the following notation to express multi-summation symbol

$$ \mathop \sum \limits_{i = 0}^{2} \mathop \sum \limits_{j = 0}^{2} \mathop \sum \limits_{k = 0}^{2} \rho_{{}}^{ijk} = \mathop \sum \limits_{ijk \in [0,1,2]} \rho_{{}}^{ijk} $$

We rewrite the Parmigiani function d(m ₁, m ₂, f ₁, f ₂, n) in the case of three genes

$$ d_{b} ( m_{1}, m_{2}, m_{3},f_{1}, f_{2}, f_{3} ) = \mathop \prod \limits_{l = 1}^{no(b)} \mathop \sum \limits_{ijk \in [0,1,2]} \rho_{{o_{l} (b)}}^{ijk} \cdot P[i|m_{1}, f_{1} ] \cdot P[j|m_{2}, f_{2} ] \cdot P[k|m_{3}, f_{3} ] $$

It represents the probability of observing a determined clinical history in the offspring of b [formed by no(b) individuals] given the allele configuration (m ₁, m ₂, m ₃) in the mother and (f ₁, f ₂, f ₃) in the father (the temporary proband b could be either the mother or the father label). The function P[i|m, f] represents the probability to inherit i mutated alleles given m mutated alleles in the mother and f in the father following mendelian rules of gene transmission.

We can extend the d function in order to take into account two generations (sons and nephew) by defining

$$ \begin{aligned} & d2_{b} ( m_{1}, m_{2}, m_{3}, f_{1}, f_{2}, f_{3} ) \\ & \quad = \mathop \prod \limits_{l = 1}^{no(b)} \mathop \sum \limits_{ijk \in [0,1,2]} \rho_{{o_{l} (b)}}^{ijk} \cdot P[i|m_{1}, f_{1} ] \cdot P[j|m_{2}, f_{2} ] \cdot P[k|m_{3}, f_{3} ] \cdot \mathop \sum \limits_{{i_{q} j_{q} k_{q} \in [0,1,2]}} \rho_{{q(o_{l} \left( b \right))}}^{{i_{q} j_{q} k_{q} }} \cdot P[i_{q}, j_{q}, k_{q} ] \cdot d_{{o_{l} \left( b \right)}} (i,j,k,i_{q}, j_{q}, k_{q} ) \\ \end{aligned} $$

where $\rho_{{q(o_{l} (b))}}^{{i_{q} j_{q} k_{q} }}$ is related to offspring mates (i.e. sons and daughter in law), if no information is available we consider ρ ^ijk _q()  = 1 for every ijk; the function P[i _q , j _q , k _q ] represents the probability of the mate to have a certain allele configuration and it is directly related to the prevalence of the three mutations, that is

$$ P\left[ {i,j,k} \right] = P_{1} [i] \cdot P_{2} [j] \cdot P_{3} [k] $$

with

$$ P_{a} \left[ i \right] = \left\{ {\begin{array}{*{20}l} {(1 - f_{a} )^{2} } \hfill & {for\;i = 0} \hfill \\ {2 \cdot \left( {1 - f_{a} } \right) \cdot f_{a} } \hfill & {for\;i = 1} \hfill \\ {f_{a}^{2} } \hfill & {for\;i = 2} \hfill \\ \end{array} } \right. $$

where f _a is the allele frequency for gene BRCA _a .

In case of three generations we can define the function

$$ \begin{aligned} & d3_{b} ( m_{1}, m_{2}, m_{3}, f_{1}, f_{2}, f_{3} ) \\ & \quad = \mathop \prod \limits_{l = 1}^{no(b)} \mathop \sum \limits_{ijk \in [0,1,2]} \rho_{{o_{l} (b)}}^{ijk} \cdot P[i|m_{1}, f_{1} ] \cdot P[j|m_{2}, f_{2} ] \cdot P[k|m_{3}, f_{3} ] \cdot \mathop \sum \limits_{{i_{q} j_{q} k_{q} \in [0,1,2]}} \rho_{{q(o_{l} \left( b \right))}}^{{i_{q} j_{q} k_{q} }} \cdot P[i_{q}, j_{q}, k_{q} ] \cdot d2_{{o_{l} \left( b \right)}} (i,j,k,i_{q}, j_{q}, k_{q} ) \\ \end{aligned} $$

which incorporates the d2 function, and so on for further generations. Now we have solved how to “move down” in the family.

In order to write the functions to “move up” in the family we need to define the following function U _b (o1, o2, o3) that represents the probability of observing the history of the parents and of the siblings of family member b given his gene status (o1, o2, o3)

$$ \begin{aligned} & U_{b} \left( {o1,o2,o3} \right) \\ & \quad = \mathop \sum \limits_{{i_{f}, j_{f}, k_{f}, i_{m}, j_{m}, k_{m} \in \left[ {0,1,2} \right]}} \rho_{f\left( b \right)}^{{i_{f} j_{f} k_{f} }} \cdot \rho_{m\left( b \right)}^{{i_{m} j_{m} k_{m} }} \cdot P\left[ {i_{f}, i_{m} |O1} \right] \cdot P\left[ {j_{f}, j_{m} |O2} \right] \cdot P\left[ {k_{f}, k_{m} |O3} \right] \cdot ds_{b} \left( {i_{f}, j_{f}, k_{f}, i_{m}, j_{m}, k_{m} } \right) \\ \end{aligned} $$

The function P[i _f , i _m |o], used in the latter, represents the probability to find i _f mutated alleles in the father and i _m in the mother given o mutated alleles in the son or daughter. It is defined as follow using the Bayes theorem.

$$ P\left[ {i_{f}, i_{m} |o} \right] = \frac{{P\left[ {i_{f} } \right] \cdot P\left[ {i_{m} } \right] \cdot P\left[ {o|i_{f}, i_{m} } \right]}}{{\mathop \sum \nolimits_{{i,j \in \left[ {01,2} \right]}} P\left[ i \right] \cdot P\left[ j \right] \cdot P\left[ {o|i,j} \right]}} $$

We have used the function ds where only the siblings of b (temporary proband) are taken into account;

$$ \begin{aligned} & ds_{b} ( m_{1}, m_{2}, m_{3}, f_{1}, f_{2}, f_{3} ) \\ & \quad = \mathop \prod \limits_{l = 1}^{ns(b)} \mathop \sum \limits_{ijk \in [0,1,2]} \rho_{{s_{l} (b)}}^{ijk} \cdot P[i|m_{1}, f_{1} ] \cdot P[j|m_{2}, f_{2} ] \cdot P[k|m_{3}, f_{3} ] \cdot \mathop \sum \limits_{{i_{q} j_{q} k_{q} \in [0,1,2]}} \rho_{{q(s_{l} \left( b \right))}}^{{i_{q} j_{q} k_{q} }} \cdot P[i_{q}, j_{q}, k_{q} ] \cdot d2_{{s_{l} \left( b \right)}} (i,j,k,i_{q}, j_{q}, k_{q} ) \\ \end{aligned} $$

It represents the probability of observing the history of siblings of b and of their offspring (up to two generations) given the genetic status of parents (m ₁, m ₂, m ₃) and (f ₁, f ₂, f ₃)

Note that in this case, even if the structure is the same of function d3, the label b is not referred to the parents as it was before but to temporary proband.

With the U function we can “move up” of one generation, but we want reach the great-grandparents so we need functions to do two more steps. We use U function to move from grandparents to great-grandparents and grandparents siblings; The U2 function defined by

$$ \begin{aligned} & U2_{b} \left( {o1,o2,o3} \right) \\ & \quad = \mathop \sum \limits_{{i_{f}, j_{f}, k_{f}, i_{m}, j_{m}, k_{m} \in \left[ {0,1,2} \right]}} \rho_{f\left( b \right)}^{{i_{f} j_{f} k_{f} }} \cdot \rho_{m\left( b \right)}^{{i_{m} j_{m} k_{m} }} \cdot P\left[ {i_{f}, i_{m} |O1} \right] \cdot P\left[ {j_{f}, j_{m} |O2} \right] \cdot P\left[ {k_{f}, k_{m} |O3} \right] \cdot U_{{f\left( {f\left( b \right)} \right)}} \left( {i_{f}, j_{f}, k_{f} } \right) \cdot U_{{m\left( {m\left( b \right)} \right)}} \left( {i_{m}, j_{m}, k_{m} } \right) \cdot ds_{b} \left( {i_{f}, j_{f}, k_{f}, i_{m}, j_{m}, k_{m} } \right) \\ \end{aligned} $$

is used to move from parents to grandparents, aunts and uncles and the function U3 defined by

$$ \begin{aligned} & U3_{b} \left( {o1,o2,o3} \right) \\ & \quad = \mathop \sum \limits_{{i_{f}, j_{f}, k_{f}, i_{m}, j_{m}, k_{m} \in \left[ {0,1,2} \right]}} \rho_{f\left( b \right)}^{{i_{f} j_{f} k_{f} }} \cdot \rho_{m\left( b \right)}^{{i_{m} j_{m} k_{m} }} \cdot P\left[ {i_{f}, i_{m} |O1} \right] \cdot P\left[ {j_{f}, j_{m} |O2} \right] \cdot P\left[ {k_{f}, k_{m} |O3} \right] \cdot U2_{f\left( b \right)} \left( {i_{f}, j_{f}, k_{f} } \right) \cdot U2_{m\left( b \right)} \left( {i_{m}, j_{m}, k_{m} } \right) \cdot ds_{b} \left( {i_{f}, j_{f}, k_{f}, i_{m}, j_{m}, k_{m} } \right) \\ \end{aligned} $$

is used to move from the proband to parents and proband’ s sibling.

Finally, the probability of observing the clinical family history given the genetic status (i, j, k) of the proband is

$$ \begin{aligned} & P\left[ {family\; history|BRCA1 = i,BRCA2 = j,BRCA3 = k} \right] \\ & \quad = \rho_{p}^{ijk} \cdot \mathop \sum \limits_{{i_{m}, j_{m}, k_{m} \in \left[ {0,1,2} \right]}} \rho_{q\left( p \right)}^{{i_{m} j_{m} k_{m} }} \cdot P[i_{m}, j_{m}, k_{m} ] \cdot d3_{p} \left( {i,j,k,i_{m}, j_{m}, k_{m} } \right) \\ \end{aligned} $$

where ρ ^ijk _p is refers to the proband personal clinical history, i _m j _m k _m are refer to the genetic status of the proband’ s mate and ρ ^ijk _q(p) to his history.

Finally according to the Bayes theorem the probability of observing mutation status [BRCA1, BRCA2, BRCA3] given the family history can be written

$$ \begin{aligned} & P\left[ {BRCA1,BRCA2,BRCA3|family \;history} \right] \\ & \quad = \frac{{P[BRCA1,BRCA2,BRCA3] \cdot P\left[ {family \;history|BRCA1,BRCA2,BRCA3} \right]}}{{\mathop \sum \nolimits_{{i,j,k \in \left[ {0,1,2} \right]}} P[i,j,k] \cdot P\left[ {family \;history|i,j,k} \right]}} \\ \end{aligned} $$

Appendix 2

The observed incidence rates in the general population ir _all can be written as the sum of two or terms: the incidence rates for women carrying wild-type BRCA gene ir ₀ time the prevalence of wild-type and the incidence rates ir ₁ for women carrying BRCA mutation times the prevalence of mutation

$$ IR_{All} = IR_{0} \cdot \left( {1 - f} \right)^{2} + IR_{1} \cdot \left( {f^{2} + 2f\left( {1 - f} \right)} \right) = IR_{0} \cdot \left( {1 - f} \right)^{2} + IR_{1} \cdot \left( {2f - f^{2} } \right) $$

where f is the allele frequency, (1 − f)² is the wild-type prevalence and (f ² + 2f(1 − f)) is the mutation prevalence. Because of the very low allele frequency we can approximate the latter by omitting the second order terms f ².

$$ IR_{All} \approx IR_{0} \cdot \left( {1 - 2f} \right) + IR_{1} \cdot 2f $$

In the case of three genes the latter becomes

$$ IR_{All} \approx IR_{0} \cdot \left( {1 - 2f_{1} - 2f_{2} - 2f_{3} } \right) + IR_{1} \cdot 2f_{1} + IR_{2} \cdot 2f_{2} + IR_{3} \cdot 2f_{3} $$

The incidence rates for the wild-type BRCA is then

$$ IR_{0} \approx \frac{{IR_{All} - IR_{1} \cdot 2f_{1} + IR_{2} \cdot 2f_{2} + IR_{3} \cdot 2f_{3} }}{{1 - 2f_{1} - 2f_{2} - 2f_{3} }} $$

For instance we have simulated how mutation probability changes in a proband with breast cancer at different ages using respectively ir _all and ir ₀

Age	Mut. prob. using ir all	Mut. prob. using ir 0
20	0.452	0.837
25	0.243	0.323
30	0.141	0.166
40	0.071	0.077
50	0.033	0.035
60	0.028	0.030
70	0.015	0.016

We can note that the correction is important for younger ages when there is a considerable proportion of genetic cases.

Appendix 3

Software input

To estimate BRCA mutation probability, the COS software requires sex, current age or age at death, age at BC or OC diagnosis, age at diagnosis of second or contralateral BC, and age at any bilateral prophylactic mastectomy or oophorectomy in each family member.

The results of genetic testing can also be included (allowing researchers to calculate the mutation probability in untested family members).

These data are also required by BRCAPRO. Unlike BRCAPRO, however, the COS software additionally requires the date of birth of each family member (to allow for changing incidence and penetrance over generations).

Since data on the distant relatives of a proband are frequently incomplete, the COS software incorporates routines for the automatic imputation of missing information. These routines incorporate the following assumptions: that the interval between generations is 25 years, that BC and OC deaths occur 5 and 2 years, respectively, after diagnosis, and that life expectancy has increased from 65 years for people born before 1920 to 75 years for those born afterwards. A routine to automatically import family data from Progeny is available at http://www.progenygenetics.com/.