The implantation of every embryo facilitates the chances of the remaining embryos to implant in an IVF programme: a mathematical model to predict pregnancy and multiple pregnancy rates.

BACKGROUND We aimed to assess the validity of a theoretical mathematical model to predict the pregnancy rate and the multiple pregnancy rate in IVF/oocyte donation programmes on the basis of the implantation rate and the number of transferred embryos. METHODS A total of 1835 embryo transfers corresponding to three different programmes in two centres with different implantation rates were analysed. Pregnancy and multiple pregnancy rates observed in the aforementioned programmes were compared with those obtained following different mathematical models. Four models were tested: binomial model, ground model, maternal variability model and collaborative model. The goodness of fit was performed by means of the maximum likelihood fit method. RESULTS The binomial model could not predict the pregnancy rate, and especially the multiple pregnancy rate. The multiple pregnancy rate predicted following the binomial model was much lower than observed, up to 40-fold reduced. Ground model and maternal variability model adjusted to the data with more precision, but were still not accurate. Finally, the collaborative model reproduced with very great accuracy both pregnancy rate and the multiple pregnancy rate. A collaborative parameter of 22% was found, implying that the implantation probability of each embryo is increased by 22% for every embryo previously implanted. CONCLUSIONS Embryonic implantation does not follow a binomial law, showing that the implantation is not independent from the number of embryos implanted. The best fit to the data is obtained following a collaborative model by which the implantation of one embryo is facilitated by the implantation of other embryo(s). The mathematical formula of the collaborative model predicts very accurately the pregnancy rate and the multiple pregnancy rate in IVF/oocyte donation programmes, based on the implantation rate of this specific programme and the number of embryos transferred up to five embryos. We recommend using the aforementioned formula to quantify the pregnancy rate and the risk of multiple pregnancy in the counselling of the infertile couple at embryo transfer. Such a formula is freely available at www.ifca.unican.es/matorras/mathpreg/.