1 Department of Obstetrics and Gynecology, Hospital de Cruces, Baracaldo, Vizcaya, País Vasco University, 2 Institute of Physics, University of Cantabria-CSIC, 3 Instituto Valenciano de Infertilidad, Department of Pediatrics, Obstetrics and Gynecology, Valencia, University of Valencia, Spain
4 To whom correspondence should be addressed at: María Diaz de Haro, 7, 6 i., 48013 Bilbao. Spain. E-mail: rmatorras{at}hcru.osakidetza.net
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Abstract |
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Key words: collaborative model/embryo implantation/mathematical model/multiple pregnancy/prediction formula
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Introduction |
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While in some countries there are legal regulations regarding the maximum number of embryos to be transferred, in others each case is evaluated on an individual basis. In recent reports, embryo scores have been developed to predict the probability of pregnancy (Terriou et al., 2001; De Placido et al., 2002
). However, the prediction of multiple pregnancy has been addressed in few studies (Martin and Welch, 1998
; Wheeler et al., 1998
).
In IVF programmes there is a remarkable variation in the pregnancy rate (PR) reported by the different groups. Thus, some groups have a 3-fold PR when compared with others (SART, 1997). While some groups advocate not transferring more than three or two embryos or even one, others advocate managing each case individually, especially taking into account the age of the woman and previous IVF attempts, as well as embryo quality (Martikainen et al., 2001
; Hunault et al., 2002
).
Hellins formula, conceived in 1895, calculates the frequency of multiple births N = 1/90(n 1), n being the number of fetuses and 1/90 the observed frequency of twins. This rule is moderately accurate for spontaneous conceptions, but of no predictive value in assisted reproduction procedures.
Both PR and multiple PR depend on the basal implantation rate (IR) of the specific IVF programme and on the number of embryos transferred. Bearing this in mind, we have tested the hypothesis that PR and multiple PR could be predicted by testing four different mathematical models corresponding to four possible implantation mechanisms in which the two aforementioned parameters (IR and number of embryos transferred) act as independent variables: binomial model, ground model, maternal variability model and collaborative model. These models have been tested in three different programmes corresponding to two centres. If such a formula could accurately predict the probability of pregnancy and the risk of the different orders of multiple pregnancy, it would be of great help when deciding the number of embryos to be transferred. This way it may be possible to maximize the PR, minimizing the multiple pregnancy rate.
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Materials and methods |
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In the 10% IVF programme the standard policy was to transfer four embryos when available, whereas in cases of poor embryo quality or previous implantation failures five embryos were transferred. In the 20% IR IVF programme, the general policy was to transfer three embryos except in cases of poor embryo quality or previous implantation failures, where four or five were transferred. The same policy was followed in the oocyte donation programme. In the three groups, patients receiving one or two embryos corresponded to those where no more available embryos were present. IVF general management and luteal phase support has been previously reported (Ruiz et al., 1997; Caligara et al., 2001
; Matorras et al., 2002
).
IR was defined as the number of gestational sacs observed at vaginal ultrasound 35 weeks post-transfer divided by the number of transferred embryos. In the three aforementioned programmes the following parameters were calculated according to the number of transferred embryos: PR, single PR, twin PR, triplet PR, quadruplet PR and quintuplet PR.
In order to assess the possibility of predicting the PR and the multiple PR based on the number of transferred embryos and the IR, four different mathematical models corresponding to four possible implantation mechanisms were tested. Other parameters which have a well-known influence on IVF outcome (female age, embryo quality, infertility duration, previous pregnancies, previous IVF failures, associated conditions) were not analysed, for two reasons: (i) the sample size does not allow for analysis of additional parameters; and (ii) since all the aforementioned parameters influence the IR, for the analysis it would seem enough to study the resulting variables and not also all the predictive initial variables.
The first mathematical theoretical model tested was the classical binomial distribution. In this concept, IR will depend exclusively on the embryo and the possibility of implantation of each embryo is independent. The probability A of having m successes in n events, assuming each of the events is totally independent from the others, is given by a binomial probability law (Altman, 1991). If each of the events has a probability p of success, the probability is given by:
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where p = IR, n = number of transferred embryos, and m = number of implanted embryos. A is the probability of obtaining the implantation of m embryos when n embryos are transferred. The pregnancy rate (PR) or probability of obtaining success in at least one (one or more) can be derived from the above equation and takes the form:
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According to the aforementioned formulae, we calculated the expected IR, PR and multiple PR in the three programmes under study and compared them with the observed rates.
The second model tested was the ground model, based on the hypothesis by Speirs et al. (1996). The observed success rate is the superposition of two random effects: one is purely binomial (the embryo) with a probability p; as before, an initial barrier (the endometrium or anything intrinsic to the IVF procedure) has to be broken with probability b.
This barrier represents a situation that may impede the attachment of any of the embryos. In this case the probability of m successes in n trials is given by the binomial expression described above multiplied by b:
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The probability of obtaining at least one success is given by:
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The collaborative model tested the hypothesis that there is some positive reinforcement or help (collaboration) between the embryos in such a way that when one embryo has successfully implanted, the implantation for the second is easier; when there are two it is easier still, and so on. This mathematical model is based on two parameters: the probability of the first success p and the increase of probability of success after having a success d. We assume a linear dependence such that the probability of obtaining a success after m successes is given by:
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Of course care has to be taken so that the probability is never >1. This was achieved with two approaches. First, the probability is filtered through a linear-truncated logistic function, bounding the probabilities to 1. The probabilities then take the form:
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Alternatively, we have filtered it with a smooth logistic function. In particular, the best choice to preserve linearity if the probability is significantly smaller than 1 and to naturally bind the result below 1 without introducing more free parameters is the hyperbolic tangent, which is defined by:
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The probabilities then take the form:
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The probability of obtaining m successes if n embryos are transferred can be calculated according to the following recursion rule, as a function of the probabilities that m or m 1 successes occur when n 1 embryos are transferred:
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with and
With this model, the probability of obtaining at least one success follows the same expression as the binomial case (here p is not equal to the IR).
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The maternal variability model considered the hypothesis that the probability
of implantation is not constant but depends on the ground (endometrium), but for a given probability the behaviour is purely binomial. This would mean that the observed rates are the superposition of the rates for different probabilities, each one following a binomial law. For instance, there could be a significant fraction of extremely receptive mothers or extremely low implanters. In this case an average of different binomial law is observed, which is no longer a binomial distribution.
To test these models, an a priori assumption on how the aforementioned probabilities are distributed is needed, although it can be shown mathematically that essentially the final law depends only on the average probability and the spread measured with the standard deviation. To illustrate these models we have first assumed that the probability is continuously distributed between two given values either uniformly or with a linear dependence such that the probability is 0 for the highest value. Second, we have used the extreme case, the one that mathematically gives the maximum prediction for multiplets, where higher rates of multiples are obtained, assuming that there are two samples with a fixed (different) probability.
Any of these models depends on a set of parameters, usually two, that were obtained from the data using a maximum likelihood fit. This technique provides mathematically the best parameters for a given model and observed data. These parameters are calculated by maximizing the probability that the observed data are produced by that probability distribution, as a function of the parameters. The fit is applied separately to each of the samples and for a given number of transferred embryos, because their quality is different in each case and therefore the fit parameters might vary from one case to another. A Pearson 2-test is applied to the resulting parameters to provide an indication of the goodness of the fit as well as which of the models reproduces the data better. This test is applied to all the available data, because we expect all the data to follow the same model (although not necessarily with the same parameters as stated above), providing a single value for each model. For a correct model we must expect a
2 value close to Npoint Npar, the difference between the number of data points and the number of parameters used in the fit. For this check, only the cases of three or more transferred embryos are used, making a total of 36 data points (3+4+5 x 3 hospitals). The number of parameters in the fit is nine for the binomial model and 18 for the rest (because two parameters in each set are needed). Therefore one should expect value of
27 for the binomial model and
18 for all the remaining cases. A value much greater indicates that the model does not reproduce the data correctly. It is important to remember that the opposite implication is not always truea wrong model can give a small value.
Since the observed IR varies according to the number of transferred embryos, in our mathematical models the IR value considered was that corresponding to the specific number of transferred embryos under study.
In the Tables, values in parentheses correspond to the percentage of cases where the number of implanted embryos was exactly that indicated in the heading of the column and no more. Values outside parentheses correspond to the implantation of at least the number of the heading. This means that, for instance when twins are considered, triplets or more are included outside parentheses, whereas inside parentheses only twins are reported. And when one embryo implantation is analysed, all the cases of multiple pregnancy are taken into account outside parentheses while inside parentheses only single implantations are given.
Theoretical data corresponding to the four models tested compared to the observed data are shown in Figures 1, 2 and 3. We omit the cases for one and two transferred embryos, because the prediction of the models is exact by definition, since we fit two parameters from two data points at most (except in the binomial model, with only one parameter).
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Results |
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Accuracy of the ground model
The estimations calculated according to the formula described above are shown in Figures 1, 2 and 3. Despite improving remarkably the prediction on the rate for multiplets, this model is not totally satisfactory in some cases, with a global tendency to predict higher rates of twins and lower rates for singletons and triplets. This was reflected in a better 2 value, 44 but still higher than the 18 expected in this case.
Accuracy of the collaborative model
The predictions of the collaborative model are shown in Tables I, II and III and Figures 1, 2 and 3. The predictions were almost identical when using linear-truncated or hyperbolic tangent logistic functions, but slightly better in the first case. All the results and subsequent discussion will be referred to this case. We found that this model reproduces very accurately the cases with two, three and four transferred embryos in the three centres. The number of cases of five transferred embryos is very small and, as a consequence, the statistical fluctuations are considerable and therefore the discrimination between models less clear. However, if we express observation and prediction in terms of total number of cases (Table IV) rather than in probabilities, we can establish that at least this model is not incompatible with the data. The 2 is 19.6, close to the 18 expected, confirming the compatibility of this model with the data.
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We have to highlight that irrespective of the number of transferred embryos and for all three populations, the collaboration parameter d (how the probability increases with the number of successes) is statistically compatible to 22%. Despite some fluctuations due to the poor statistical sensitivity of some of the samples to this parameter, the fit can be repeated by fixing the collaboration parameter to 22%, without significantly degrading the quality of the fit. Apparently this collaboration is an intrinsic effect of the embryo on the endometrium and does not strongly depend on the procedure in different centres.
Accuracy of the maternal variability model
This model also increases the rate of multiplets, but this increase is not enough to explain the data (Figures 1, 2, 3). This was found to be true independently of the assumption intrinsic to the model as described above. We show here only results for the simplest case, and the one with best fit, in which the probability distribution of the population is constant between two different values, the remaining cases being very similar. The expected PR was higher than observed (up to 50% increase), whereas multiplets were not accurately predicted. The lowest 2 value according the different assumptions was 127 (18 expected).
Comparison of the collaborative model and the ground model
The comparison of the goodness of the fit was made by calculating the upper tail probability (the probability of obtaining a given 2 value or higher). For the ground model, a
2 of 43 with 18 degrees of freedom corresponds to a probability 1 P < 0.001 (meaning that there are significant differences between the proposed model and the observed data). Although a quick look at the tables and plots could lead to the conclusion that the model is reasonably correct, the
2 value clearly rules out the model at 99.9% confidence level.
For the collaborative model, a 2 of 19 with 18 degrees of freedom corresponds to a probability 1 P = 0.36 (meaning that there are no significant differences between the proposed model and the observed data).
Collaborative and ground combined model
Since the ground hypothesis seems reasonable from the biological point of view, an attempt was made with a mixed model in which the described collaboration was superimposed on a ground probability. The drawback is that such a model has a minimum of three free parameters and therefore does not have predictive power for cases with fewer than four transferred embryos, where our sample is small. The fit performed in this way slightly improved the agreement with data, but not enough to compensate the loss of degrees of freedom. Mathematically a 2 of 17 is obtained with only 9 degrees of freedom, with a probability of 5%, smaller but still compatible at 95% confidence level.
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Discussion |
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It has been highlighted that since the IR could be different because of transfer policy reasons (in transfers of one and two embryos, the transferred embryos were the only ones available and a low IR should be expected, whereas in transfers of three and four embryos in a number of cases it was possible to select the embryosthus a higher PR could be expected), in all of our calculations we worked with the specific IR for the number of transferred embryos under study.
First of all we tried the binomial model, assuming that IR was independent of the number of embryos transferred. The binomial model we used predicted with reasonable accuracy the PR when the implantation rate was 10%, as well as when the implantation rate was close to 20% and only one or two embryos were transferred. When implantation rate was close to 20% and three or more embryos were transferred, the PR was lower than expected (5570%). We found a striking increase in the observed multiple PR, especially regarding the frequency of triplets and quadruplets, whose frequencies were 1040-fold higher than expected according the binomial model.
The binomial model we used was very similar to the classical rule of Hellin, but we used two variables instead of only one. Concerning Hellins rule, it should be borne in mind that it was based only on newborn data, whereas abortions, immature deliveries, fetal deaths and spontaneous reductions (which are calculated to be 1643%) (Seoud et al., 1992; Bollen et al., 1993
; Yaron et al., 1999
) were not considered. Thus triplets and quadruplets (in the pre-assisted reproduction era) were also more frequent than expected following an exponential model. Thus perhaps in spontaneous conceptions embryo implantation is also not an independent factor either.
The multiple PR we observed, much higher than expected, suggests that the implantation probability of each embryo is not independent of the others. In our opinion the most plausible hypothesis was that the implantation of one embryo facilitates that of the remaining embryos. In a previous preliminary study on the implantation rate, no evidence was found that embryos helped (or hindered) each other (Speirs et al., 1996), but no mention was made regarding the multiple PR.
In a recent report, Trimarchi (2001) observed that the obtained PR for having at least one success was compatible with the binomial model and concluded that it proved that the embryos implanted independently. However, the fact that a mathematical model that presupposes embryo independence is compatible with the observed data, does not imply that models based on embryo dependency are incorrect. This uncertainty can only be elucidated through the multiple pregnancy analysis, which was not addressed in the aforementioned article.
Therefore, we have developed a collaborative mathematical model to test the hypothesis that embryos during implantation help each other. Our collaborative model reproduced very accurately both pregnancy rate and the different types of multiple pregnancy rates up to the time five embryos were transferred.
The possibility that the embryo is able to modulate endometrial molecules, self-controlling the implantation process, is a challenging and obvious idea demonstrated in several species including the human. In humans, data from a number of studies provide convincing evidence of a molecular dialogue between the developing embryo and the maternal endometrial epithelium (De los Santos et al., 1996; Simon et al., 1997
; Meseguer et al., 2001
; Galan et al., 2000
; González et al., 2000
; Caballero-Campo et al., 2002
). This embryonicendometrial crosstalk is beneficial for the activation of specific endometrial molecules such as the interleukin-1 system (De los Santos et al., 1996), integrins
5,
3,
1, (Simon et al., 1997
), mucin MUC1 (Meseguer et al., 2001
), leptin (González et al., 2000
), induction of endometrial epithelial apoptosis (Galan et al., 2000
) and chemokines (Caballero-Campo et al., 2002
) in a timely manner that may improve the chances of implantation of the rest of the embryos.
In rodents, it has been demonstrated that previous adhesion of embryos increases the implantation rate of the other embryos (Shiotani et al., 1993). In fact, Wakuda et al. (1999)
have demonstrated that intraoviductal embryos exert a biological effect, by sending a signal to the endometrial epithelium and stroma, thus facilitating endometrial receptivity and improving implantation rates. This crucial concept has also been demonstrated in the rabbit (Harper et al., 1989
) and in the non-human primate (Fazleabas et al., 1999
). In this in vivo study a physiological effect of chorionic gonadotrophin was found on the uterine endometrium, suggesting that the primate blastocyst signals modulate the uterine environment prior to and during implantation (Fazleabas et al., 1999
). Therefore, these basic studies strongly agreed with the collaborative model presented herein, suggesting that there is a third dimension to be considered in implantation, namely the effect of the embryo on the regulation of endometrial receptivity and implantation.
However, from a theoretical biological point of view, there are three additional possibilities that could explain the overimplantation we found: (i) when endometrium is highly receptive the implantation rate could be less embryo dependent (ground model); (ii) embryos with high implantation potential could occur more frequently in the same couples (average binomial model); and (iii) some mothers could have a high implantation potential and pregnancy and multiple pregnancy could be more frequent (maternal variability model). In a report where a higher than expected rate of multiple implantations also was noted, it was speculated that it could be the result of some couples producing better-quality embryos, combined with some characteristics of the mothers associated with higher endometrial receptivity (Baker et al., 2000). However, such hypothetical models were not tested.
When we tested the ground model we found that, although the predictions were considerably more accurate compared with the binomial model, the fitting with the observed data was poorer than with the collaborative model. The second alternative hypothesis was not tested because, from a mathematical point of view, having different samples following a binomial law with different PR would also lead to a binomial law for the global sample (with a PR that is the average of the different PR). The third alternative hypothesis we tested (maternal variability) does not fit accurately with the observed data. The PR calculated with this last method were considerably higher than that observed (up to 50% increase), whereas multiplets were not predicted with accuracy.
Concerning our model, two limitations must be considered. First, our model was tested when up to five embryos were transferred and only with inadequate statistics here. It has been shown that the linearity assumption in the increase of probability is reasonable up to this number. This conclusion was also supported by the fact that including a logistic function to distort the linearity does not improve the results. With six or more embryos there must exist some competition which would be reflected in the failure of the linearity when more embryos are transferred. This point could not be investigated in our population, since such transfers, as in the majority of teams, are no longer performed in our centres. On the other hand it seems biologically sound that, in addition to embryo collaboration, ground plays a role. However, to test such a combined model (ground plus collaborative) needs at least three free mathematical parameters (IR, collaborative parameter, ground parameter), therefore being difficult to test without making strong assumptions. Indeed that model would be valid only for transfers of four or five embryos. When such a mixed model was applied, the accuracy of the predictions increased only slightly. The barrier probability was very close to 1, meaning that ground conditions were good in most cases. In our opinion this means that in most cases the diagnostic work-up effectively excluded the conditions that could interfere with implantation (myoma, polyps, sinechiae), luteal phase support was adequate and embryo transfer was performed atraumatically.
In summary, it is concluded that the multiple pregnancy rate in IVF cycles is much higher than that calculated following a binomial model, indicating that the implantation of each embryo is not an independent parameter. The collaborative mathematical model developed fitted accurately and simultaneously both PR and multiple PR. Thus, from a theoretical mathematical point of view, our data are compatible with the assumption that during the implantation the embryos help each other. This is in agreement with biological data reported in different animal species including the human. It is of interest that the probability of implantation of each embryo is increased on average by 22% for every previously implanted embryo. This collaboration parameter does not strongly depend on the number of transferred embryos nor on the population studied. Thus it seems it is an intrinsic property of the human embryo implantation. We believe that our model represents a useful tool in the decision-making process in order to ascertain the best number of embryos to be transferred, according to the IR of each centre and the multiple pregnancy risk assumed by the couples. Such a formula is freely available at www.ifca.unican.es/matorras/mathpreg/.
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Submitted on July 20, 2004; resubmitted on February 22, 2005; accepted on April 8, 2005.
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