Quantitative decision-making in preimplantation genetic (aneuploidy) screening (PGS)

Abstract

Purpose

To analyze using hypergeometric probability statistics the impact of performing preimplantation genetic screening (PGS) on a cohort of day 3 cleavage stage embryos.

Methods

Statistical mathematical modeling.

Results

We find the benefit of performing PGS is highly dependent on the number of day 3 embryos available for biopsy. Additional hidden variables that determine the outcome of PGS are the rates of aneuploidy and mosaicism, and the probability of a chromosomally mosaic embryo to test “normal”. If PGS is performed, our analysis shows that many combinations of the number of biopsiable embryos, and the rates of aneuploidy and mosaicism results in a marginal benefit from the intervention. Other combinations are detrimental if PGS is actually undertaken. Finally, increases in PGS error rates lead to a rapid loss in the ability of PGS to provide useful discriminatory information.

Conclusion

We set out the statistical framework to determine the limits of PGS when a specific number of day 3 preimplantation embryos are available for biopsy. In general, PGS cannot be recommended a priori for a specific clinical situation due to the statistical uncertainties associated with the different hidden variable quantitative parameters considered important to the clinical outcome.

Addendum

In the “Methods” section we outlined the basic parameters used to calculate the benefits of PGS. We also considered the impact of aneuploidy rates on day 3 embryo transfers without performing PGS. A more mathematical treatment will be presented in a separate publication. Below, we present in more detail the impact of PGS using day 3 biopsied embryos. The discussion that follows considers the intervention of PGS and its impact on the probability of transferring fully diploid embryos.

The calculations in this addendum form the basis for the data summarized in Tables 1, 2, 3, 4, 5, 6, 7, 8 and 9.

The loss rate (spontaneous abortion)

We define the spontaneous abortion rate σ, the rate at which chromosomally aneuploid embryos that survive to day five implant and cause a spontaneous miscarriage. (As most aneuploidies do not go to term, this is essentially equivalent to the implantation rate for aneuploid embryos.) In parallel to P _N , we define P _A the probability that exactly N _A of the N embryos are truly aneuploid. We can then calculate the probability P _SA to undergo a spontaneous miscarriage:

$$ {P_{SA}} = \sum\limits_{{n_a} = 1}^N {{P_A}\left( {\left. {{n_a}} \right|N,\alpha } \right)} \left( {\sum\limits_{i = 1}^{\min \left( {{t_3},{n_a}} \right)} {{P_H}\left( {\left. i \right|{n_a},N,\min \left( {{t_3},N} \right)} \right)\sum\limits_{j = 1}^i {{P_{ts}}\left( {i,j,{\rho_{uA}}} \right)\left( {1 - {{\left( {1 - \sigma } \right)}^j}} \right)} } } \right) $$

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Depending on the purpose of the calculation, one in principle should be careful to properly correlate the spontaneous miscarriage and successful transfer cases.

We have not done so here.

Day 5 transfer, without PGS

Day five transfer without PGS differs from the day three case due to the fact that some actionable filtering (based on the differential between ρ_vA and ρ_vN) of normal and abnormal embryos can take place. Formally, when written out fully one will find that the indices on the sums are entwined in a slightly different way. We reiterate that the value α is the aneuploidy rate observed at day 3; due to the ρ_vA-ρ_vN differential the aneuploidy rate is different on day five and must be calculated anew.

First we calculate the joint probability P _NA (N _N , N _A , N,α,ρ_vA,ρ_vN) that on day five there are exactly N _N normal and N _A aneuploid embryos. We find this by summing joint survival probabilities products P _ts over all possible values of N _N ³ and N _A ³ consistent with N, N _N , and N _A :

$$ {P_{NA}}\left( {\left. {{N_N},{N_A}} \right|N,\alpha, {\rho_{vA}},{\rho_{vN}}} \right) = \sum\limits_{N_N^3 = {N_N}}^{N - {N_A}} {{P_N}\left( {N_N^3,N,\alpha } \right){P_{ts}}\left( {N_N^3,{N_N},{\rho_{vN}}} \right){P_{ts}}\left( {N - N_N^3,{N_A},{\rho_{vA}}} \right)} $$

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Choosing now to transfer at most t _5f embryos, what is the probability P _noPGS5 (N,α,ρ_vA,ρ_vN, t _5f ) that at least T of them are normal? It is the probability to select at least T normal embryos from a set of N _N normal and N _A abnormal embryos, summed over all possible values of N _N and N _A weighted by their joint probabilities:

$$ {P_{noPGS - 5}}\left( {\left. {{N_{NX}} \geqslant T} \right|N,\alpha, {\rho_{vA}},{\rho_{vN}},{t_{5f}}} \right) = \sum\limits_{{N_N} = T}^N {\sum\limits_{{N_A} = 0}^{N - {N_N}} {{P_{NA}}\left( {\left. {{N_N},{N_A}} \right|N,\alpha, {\rho_{vA}},{\rho_{vN}}} \right)\sum\limits_{t = T}^{\min \left( {{N_N} + {N_A},{t_{5f}}} \right)} {{P_H}\left( {\left. t \right|{N_N},{N_N} + {N_A},\min \left( {{N_N} + {N_A},{t_{5f}}} \right)} \right)} } } $$

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Loss rate (spontaneous abortion) day 5 without PGS

Again, the spontaneous abortion rate σ is defined as the rate at which chromosomally aneuploid embryos that survive to day five implant and cause a spontaneous abortion. The probability of a spontaneous miscarriage is given by summing over all possible values of N _A and all possible values of the number of transferred aneuploid embryos:

$$ {P_{SA - 5f}}\left( {N,\alpha, {\rho_{vN}},{\rho_{vA}},{t_{5f}}} \right) = \sum\limits_{{N_A} = 1}^N {\sum\limits_{{N_{N = 0}}}^{N - {N_A}} {{P_{NA}}\left( {\left. {{N_N},{N_A}} \right|N,\alpha, {\rho_{vA}},{\rho_{vN}}} \right)\sum\limits_{{N_{AX}} = 1}^{\min \left( {{N_A},{t_{5f}}} \right)} {{P_H}\left( {\left. {{N_{AX}}} \right|{N_A},{N_N} + {N_A},\min \left( {{t_{5f}},{N_N} + {N_A}} \right)} \right)\left( {1 - {{\left( {1 - \sigma } \right)}^{{N_{AX}}}}} \right)} } } $$

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Day 3 PGS

The probability P_noPGS5(N _N,X  ≥ T| T, N, α, ρ_vN = 0) just calculated must be compared to the probability P _PGS that at least T normal embryos are transferred on day five following a day three PGS biopsy. In this case, the PGS analysis labels embryos as normal or aneuploid, but with some error rate due to the combined effects of laboratory uncertainties and embryonic mosaicism.

When undergoing PGS and transferring embryos on day 5 theoretically two separate effects favor successful transfer of normal embryos. The first, naturally, is the diagnostic information from PGS itself, separating normal from aneuploid embryos. The second is the filtering effect of culturing embryos from day 3 to day 5, as demonstrated in the previous section above. To preserve clarity in the strategy of calculation, we first go through the calculation of successful transfer in the case that all values of ρ are zero. We then introduce the complications of in vitro filtering in the following section, which are essentially the same as in the case discussed above in day 5 transfer without PGS.

Success probability, neglecting in vitro loss (mortality)

The strategy we take is to break the calculation into the following pieces:

• The probability to find a particular number of $N_N^N$and $N_N^A$

• For each set of $N_N^N$and $N_N^A$, the probability to select at least T of the truly normal (out of $N_N^N$) to transfer

The probability P _truth that the N embryos contain exactly N _N truly normal embryos is:

$$ {P_{truth}} = \left( {\left. {{N_N}} \right|N,\alpha } \right) = \left( {\begin{array}{*{20}{c}} N \\ {{N_N}} \\ \end{array} } \right){\left( {1 - \alpha } \right)^{{N_N}}}{\alpha^{\left( {N - {N_N}} \right)}} $$

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If there are N _A (=N-N _N ) truly aneuploid embryos, the probability P _NA to observe $N_A^N$ of them labeled as normal is:

$$ {P_{NA}}\left( {\left. {N_A^N} \right|{N_A},\mu, \eta } \right) = \left( {\begin{array}{*{20}{c}} {{N_A}} \\ {N_A^N} \\ \end{array} } \right){\left( {{\mu_A} + {\eta_A}} \right)^{N_A^N}}{\left( {1 - {\mu_A} - {\eta_A}} \right)^{\left( {{N_A} - N_A^N} \right)}} $$

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Similarly, if there are N _N truly normal embryos, the probability P _NN to observe $N_N^N$of them labeled as normal is:

$$ {P_{NN}}\left( {\left. {N_N^N} \right|{N_N},\mu, \eta } \right) = \left( {\begin{array}{*{20}{c}} {{N_N}} \\ {N_N^N} \\ \end{array} } \right){\left( {{\mu_N} + {\eta_N}} \right)^{N_N^N}}{\left( {1 - {\mu_N} - {\eta_N}} \right)^{\left( {{N_N} - N_N^N} \right)}} $$

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So we can write that the probability P _label that the final normally-labeled sample consists of $N_N^N$truly normal embryos and$N_A^N$ aneuploid embryos is:

$$ {P_{label}}\left( {\left. {N_N^N,N_A^N} \right|N,\alpha, \mu, \eta } \right) = \sum\limits_{n = 1}^N {{P_{truth}}\left( {\left. n \right|N,\alpha } \right){P_{NN}}\left( {\left. {N_N^N} \right|n,\mu, \eta } \right){P_{NA}}\left( {\left. {N_A^N} \right|N - n,\mu, \eta } \right)} $$

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Once again, we must use the hypergeometric distribution to assign the probability

P _X to select at least T truly normal embryos out of the collection of normally-labeled embryos:

$$ {P_X}\left( {\left. {{N_{X,N}} \geqslant T} \right|T,N_N^N,N_A^N} \right) = \sum\limits_{i = T}^{\min \left( {{t_{5p}},N_N^N + N_A^N} \right)} {{P_H}\left( {\left. i \right|N_N^N,N_N^N + N_A^N,\min \left( {{t_{5p}},N_N^N + N_A^N} \right)} \right)} $$

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We now have all the expressions needed to evaluate the desired quantity P _PGS :

$$ {P_{PGS}}\left( {\left. {{N_{X,N}} \geqslant T} \right|N,\alpha, \mu, \eta, \rho = 0} \right) = \sum\limits_{N_N^N = T}^N {\sum\limits_{N_A^N = 0}^{N - N_N^N} {{P_{label}}\left( {\left. {N_N^N,N_A^N} \right|N,\mu, \eta } \right){P_X}\left( {\left. {{N_{X,N}}} \right|N_N^N,N_A^N} \right)} } $$

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Spontaneous miscarriage rate, neglecting in vitro loss (mortality), day 5 transfer with PGS

We may straightforwardly define the probability to select for transfer a number of aneuploid embryos P _XA in parallel to P _X . We then calculate the probability for a spontaneous abortion P _SAPGS (ρ = 0) while undergoing PGS:

$$ {P_{SAPGS}}\left( {N,T,\sigma, \rho = 0} \right) = \sum\limits_{{n_a} = 1}^{\max \left( {{t_{5p}},N} \right)} {\sum\limits_{{n_n} = 0}^{\max \left( {{t_{5p}},N} \right) - {n_a}} {{P_{label}}\left( {{n_n},{n_a}} \right)\sum\limits_{n = 0}^{{n_a}} {{P_{XA}}\left( {n,T,{n_n},{n_a}} \right)\left( {1 - {{\left( {1 - \sigma } \right)}^n}} \right)} } } $$

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Successful transfer probability accounting for in vitro loss (mortality), day 5 transfer with PGS

As in the earlier section, we must now consider all possible values of $N_{N,5}^A$ and $N_{N,5}^N$, the number of PGS-labeled "normal" embryos that are truly normal and aneuploid, respectively, surviving to day five. We find the probability P _NA,5 ($N_{N,5}^A$, $N_{N,5}^N$| N, N _N , N _A , α, ρ, μ,η) (where μ, ρ, and η are a notational shorthand for all relevant members of the parameter families with the various subscripts):

$$ {P_{NA,5}}\left( {\left. {N_{N,5}^A,N_{N,5}^N} \right|N,\alpha, \mu, \eta, \rho } \right) = \sum\limits_{N_N^A = N_{N,5}^A}^{N - N_{N,5}^N} {\sum\limits_{N_N^N = N_{N,5}^N}^{N - N_N^A} {{P_{label}}\left( {\left. {N_N^N,N_A^N} \right|N,\alpha, \mu, \eta } \right){P_{ts}}\left( {N_{N,5}^A,N_N^A,{\rho_{vA}}} \right){P_{ts}}\left( {N_{N,5}^N,N_N^N,{\rho_{vN}}} \right)} } $$

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We now sum over values of the transferred number of truly normal embryos for the probability to transfer that number, to find:

$$ {P_{PGS}}\left( {\left. {{N_{X,N}} \geqslant T} \right|N,\alpha, \mu, \eta, \rho } \right) = \sum\limits_{N_{N,5}^N = T}^N {\sum\limits_{N_{A,5}^N = 0}^{N - N_{N,5}^N} {{P_{NA,5}}\left( {\left. {N_{N,5}^N,N_{A,5}^N} \right|N,\alpha, \mu, \eta, \rho } \right){P_X}\left( {\left. {{N_{X,N}}} \right|N_{N,5}^N,N_{A,5}^N} \right)} } $$

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Miscarriage rate accounting for in vitro loss (mortality)

Now accounting properly for the loss rates (mortality) we find:

$$ \begin{array}{*{20}{c}} {{P_{SAPGS}}\left( {N,T,\sigma, \rho, \mu, \eta } \right) = \sum\limits_{{n_a} = 1}^{\min \left( {{t_{5p}},N} \right)} {\sum\limits_{{n_n} = 1}^{\min \left( {{t_{5p}},N} \right) - {n_a}} {{P_{label}}\left( {\left. {{n_n},{n_a}} \right|N,\alpha, \mu, \eta, \rho } \right) \times } } } \\ {\left[ {\sum\limits_{N_{A,3}^N = 1}^N {\sum\limits_{N_{N,3}^N = 1}^{N - N_{A,3}^N} {{P_{ts}}\left( {N_{A,3}^N,{n_a},{\rho_{vA}}} \right){P_{ts}}\left( {N_{N,3}^N,{n_n},{\rho_{vN}}} \right)} } \left( {\sum\limits_{n = 1}^{\min \left( {{n_a},{t_{5p}}} \right)} {{P_{XA}}\left( {\left. n \right|{n_a},{n_n} + {n_a},\min \left( {{n_a},{t_{5p}}} \right)} \right)\left[ {1 - {{\left( {1 - \sigma } \right)}^n}} \right]} } \right)} \right]} \\ \end{array} $$

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where for clarity the internal summands N _N and N _A stand for $N_{N,5}^N$and $N_{N,5}^A$, respectively.

ΔP and ΔC

Displaying now the full functional dependence, we have described the proper framework for calculating the increase in the probability of success:

$$ \Delta P = {P_{PGS}}\left( {{N_{X,N}} \geqslant T\left| {N,T,\alpha, {\mu_A},{\mu_N},{\eta_A},{\eta_N},{t_5},\rho } \right.} \right) - {P_{noPGS}}\left( {{N_{X,N}} \geqslant T\left| {N,T,\alpha, {t_3},\rho } \right.} \right) $$

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We can also calculate the increase in the cost function accounting for embryo loss:

$$ \begin{gathered} \Delta C = \left[ {{P_{PGS}}\left( {{N_{X,N}} \geqslant T\left| {N,T,\alpha, {\mu_A},{\mu_N},{\eta_A},{\eta_N},{t_5},\rho } \right.} \right) - {P_{noPGS}}\left( {{N_{X,N}} \geqslant T\left| {N,T,\alpha, {t_3},\rho } \right.} \right)} \right] \hfill \\ - \lambda \left[ {{P_{SA - PGS}}\left( {\sigma, N,T,\alpha, {\mu_A},{\mu_N},{\eta_A},{\eta_N},{t_5},\rho } \right) - {P_{SA - noPGS}}\left( {\sigma, N,T,\alpha, {t_3},\rho } \right)} \right] \hfill \\ \end{gathered} $$

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