Associate editor’s commentary: Mathematical modelling and clinical prediction

Associate editor’s comment on ‘The implantation of every embryo facilitates the chances of the remaining embryos to implant in an IVF program: a mathematical model to predict pregnancy and multiple pregnancy rates’ by Matorras et al. Wider utilization of assisted reproductive technology and better success rates are good news, but with more live births there is an increasing need to reduce multiple births. Multiple birth is a public health problem because assisted reproductive technology twins are common (20–30% of registry births), they are associated with high rates of pre-term birth (7% <32 weeks and 48% <37 weeks), and they encounter more health problems during infancy and childhood (Scholz et al., 1999; Helmerhorst et al., 2004). Prevention is important because twins and especially higher order gestations encounter more pregnancy losses, perinatal mortality, and long-term morbidity, and give rise to psychological, social and financial problems for multiple birth families. Nevertheless, preventing multiple birth in individual cases is difficult because of our inability to identify accurately which patients or embryos are at high risk (ESHRE Campus, 2001). Until reproductive science discovers better predictors of successful implantation of transferred embryos, the only choice is to transfer fewer embryos. Single embryo transfer (SET) is a simple, sensible and available means of preventing most assisted reproductive technology twin births. The transfer of fewer embryos is, however, associated with lower overall live-birth rates. Three randomized controlled trials compared the transfer of one or two embryos in a single cycle, under generally optimal conditions that were determined by various inclusion criteria. The SET pregnancy rates per transfer were 35, 12 and 15% lower than the double embryo transfer (DET) groups, the smaller differences being in the more recent trials (Gerris et al., 1999; Martikainen et al., 2001; Gardner et al., 2004). If the reported rates in each trial arm are converted to live-birth rates using a standard 0.85 factor, the differences are slightly less impressive: 30, 10 and 13%. Of course, in each case, twins pushed up the average DET pregnancy rate, which involved 30–50% twin gestations. A larger study randomly allocated women <36 years of age who had at least two good quality embryos either to a fresh SET transfer and, if there was no live birth, subsequent transfer of a single frozen–thawed embryo, or to undergo a single transfer of two fresh embryos (Thurin et al., 2004). The live-birth rates were 43% (142/331 women) in the DET group and 39% (128/330 women) in the SET group. The difference was 4% (95% CI –3 to 12). Multiple birth rates were 33% in the DET group compared with 1% in the SET group (P < 0.001). Transferring one fresh embryo and, if needed, one frozen–thawed embryo virtually prevents multiple births without seriously compromising the live-birth rate in good prognosis assisted reproduction cycles. SET is a natural component of the trend toward less aggressive ovarian stimulation protocols, and the associated lower costs could lead to even higher utilization of assisted reproductive technology. One limitation of the trials is that the results are not relevant to couples with a less than optimal prognosis, because they would not be eligible for such trials. The reasons for a suboptimal prognosis could be female partner’s age, poor follicular development, low fertilization rates or below average embryo development. At best it can only be said that SET has not been evaluated in couples with a suboptimal prognosis. For these couples, what means are available to prevent multiple pregnancy, given that SET could seriously reduce their chance of conception? Clinicians and patients would be in a better position to make judgements about the number of embryos to transfer if there were reliable predictors of the prognosis for pregnancy and multiple birth gestation, but no such indicators are accurate enough to apply (ESHRE Campus, 2001). Nor is it possible to rely on guidelines because they are rarely detailed enough to direct each case. Legislation does govern the decisions in some countries, but the law tends to swing a rather broad axe. In similar clinical situations where information is lacking to help in specific judgements and decisions, attempts are often made to fill the gap with mathematical models. Recent models in reproductive medicine were developed to address whether assisted reproductive technology can compensate for the decline in fertility with age (Leridon, 2004), and to estimate the overall live birth rate with comprehensive treatment for infertility (Collins and Van Steirteghem, 2004). Mathematical models are also common in economic analysis because some trials do not incorporate cost information (Daya et al., 2001). Pertinent to SET decisions, a logistic regression model to select patients for SET was based on 642 first IVF cycles (Hunault et al., 2002). The cycles involved no more than two embryos transferred among women with median age 32 years (range 21–43). The terms in the pregnancy model were woman’s age, the number of retrieved oocytes, embryo morphology score and developmental score. In the twin pregnancy model the number of retrieved oocytes was not included. Receiver operating curve (ROC) analysis indicated, however, that the predictive ability of the models was only modestly better than chance: the areas under the curve were 0.68 and 0.71, respectively, for the pregnancy and twin pregnancy models. This issue of the Journal includes a modelling approach which takes the reverse direction, by testing which of several mathematical models best approximates the authors’ clinical experience with pregnancy and multiple pregnancy (Matorras et al., 2005). The paper comes from the Institute of Physics at the University of Cantabria and two collaborating IVF clinics, where the mathematical models were referenced to 1835 IVF embryo transfer procedures. The authors evaluated four models: (i) a simple binomial model assumes that each embryo implants independently of other embryos and independently of maternal factors; (ii) a ground model also assumes that embryos implant independently of other embryos, but not independently of maternal factors, meaning that the model incorporates a constant reduction factor or barrier; (iii) a collaborative model makes the appealing assumption that once an embryo has implanted, the implantation of each further embryo is enhanced in a linear fashion; and (iv) a maternal variability model assumes that different sets of maternal factors affect the binomial model, a condition that may be more realistic, but seems to defy modelling. The ground and the collaborative models came close to fitting the observed data for pregnancy and multiple pregnancy rates, but only the collaborative model gave a non-significant χ2 for departures from goodness of fit (0.36). This model conformed within reason to both singleton and multiple implantation rates if the embryo assistance factor or collaborative parameter was 22%. Of course, a linear model can apply only to small numbers of embryos transferred because, with enough embryos transferred, eventually the implantation rate would be >100%. The binomial model overestimated pregnancy rate with 3–5 embryos transferred and underestimated multiple pregnancies, yielding a highly significant lack of fit. The maternal variability model, under the authors’ best assumptions, predicted multiple implantations fairly accurately, but gave higher rates for singleton implantations, yielding a significant departure from goodness of fit (P < 0.00001). Unfortunately, the authors were unable to test a combination of the ground model and the collaborative model because of data limitations. What do these results mean? The authors cite biological studies to support the implication that embryos assist one another to implant, and imply that the failure of the maternal variability model argues against an important uterine factor. The biological implications need to be tested under appropriate experimental conditions. The question is whether embryo implantation facilitates successive embryos; or do the higher implantation rates mean only that the first embryo has chanced upon a favourable endometrial environment, which contributes to the success of additional embryos. Should clinicians use the model in practice? The authors conclude that their model is useful for decisions about the best number of embryos to be transferred, using the pregnancy rate and implantation rate of the relevant centre to forecast the multiple pregnancy risk. It is interesting to download the models and enter relevant pregnancy rates, but a mathematical model by itself does not have the validity needed to counsel couples about clinical decisions such as the number of embryos to transfer. To establish the validity of a prediction model usually involves testing the model in randomly generated sets of the original data to confirm internal validity, and in data from other clinics to test external validity (Diamond, 1992; Harrell et al., 1996). In the Cantabria model, the analysis for goodness of fit had to involve all three clinic samples to have sufficient data and there is no formal test of validity within the samples, nor is there an external validation. This underlines the need for doing validation studies in larger data sets such as those in assisted reproductive technology registries. The accuracy of a prediction model is usually expressed by means of ROC analysis: the area under the curve from ROC analysis should be >50% (the area expected by chance), and the 95% confidence interval conveys the precision of the estimate. Even if there is a striking and significant correlation between models and observed experience, the area under the curve may indicate that the model is not useful in clinical practice. One reason is that modelling cannot reflect the uncertainties and exigencies of clinical practice. For example, in the Cantabria clinic non-donor IVF programme with the best results, the implantation rate was bi-modal, rising from 11% with one embryo transferred to 24% with three and falling to 16% with five embryos transferred. The Cantabria model is an interesting methodology for those who wish to explore the biology of implantation and multiple implantation. As other investigators and data bases yield similar or different results, knowledge of the best implantation model could form the rationale for clinical studies. For the present, in couples with a suboptimal prognosis, it would be prudent to base decisions about the number of embryos to transfer on the merits of each case, rather than on theoretical models that have not undergone testing for validity and accuracy. The clinical judgement might be assisted by the validated if somewhat limited prediction evidence that is available (Hunault et al., 2002).

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