Inferring simple but precise quantitative models of human oocyte and early embryo development

Abstract

Macroscopic, phenomenological models are useful as concise framings of our understandings in fields from statistical physics to finance to biology. Constructing a phenomenological model for development would provide a framework for understanding the complicated, regulatory nature of oogenesis and embryogenesis. Here, we use a data-driven approach to infer quantitative, precise models of human oocyte maturation and pre-implantation embryo development, by analysing clinical in-vitro fertilization (IVF) data on 7399 IVF cycles resulting in 57 827 embryos. Surprisingly, we find that both oocyte maturation and early embryo development are quantitatively described by simple models with minimal interactions. This simplicity suggests that oogenesis and embryogenesis are composed of modular processes that are relatively siloed from one another. In particular, our analysis provides strong evidence that (i) pre-antral follicles produce anti-Müllerian hormone independently of effects from other follicles, (ii) oocytes mature to metaphase-II independently of the woman's age, her BMI and other factors, (iii) early embryo development is memoryless for the variables assessed here, in that the probability of an embryo transitioning from its current developmental stage to the next is independent of its previous stage. Our results both provide insight into the fundamentals of oogenesis and embryogenesis and have implications for the clinical IVF.

Keywords: human embryos; in-vitro fertilization; pre-implantation development; quantitative biology.

Figures

Figure 1.

(a) Distributions and correlations of the four variables AMH (measured in International Units, IU), Eggs, MII and E2 (measured in pg ml−1). (b) Left: the number of metaphase-II oocytes retrieved (MII) is strongly correlated with the patient’s serum AMH. Grey dots: raw data, red circles: raw data binned into 20 separate bins with equal counts, green line and shaded region: nonlinear regression and errors. Centre: that correlation disappears after regressing against the total number of retrieved oocytes (Eggs). Grey dots: residuals after regressing against Eggs, red circles: residuals binned into 20 separate bins with equal counts, green line and shaded region: linear fit to the residuals, with slope and standard error shown at top of plot. The axes range of the plots is cropped to show details of the trends. Right: this conditional independency suggests a graph of the form AMH–Eggs–MII. (c) Left: the patient’s serum oestradiol concentration (E2) is strongly correlated with AMH. Centre: that correlation disappears after regressing against Eggs. Right: this conditional independency suggests a graph of the form AMH–Eggs–MII. (d) Left: MII is strongly correlated with E2. Centre: while regressing against Eggs greatly weakens that correlation, E2 and MII remain correlated after conditioning on Eggs. This suggests a fully connected graph is needed to describe these three variables (right). (e) A graphical model that is consistent with the data. Edge labels show the conditional correlation coefficients after conditioning on all other incoming edges; the data are consistent with the arrow marked with a * oriented in either direction. (f) The graphical model expected from prior knowledge of ovarian stimulation.

Figure 2.

MII versus the patient’s age (a), BMI (b), and the doses of stimulation drugs FSH (c) and HMG (d), after regressing against Eggs and E2. Grey dots show the residuals from the regressions, red circles and error bars show the mean and standard error of the data binned into 20 bins with equal number of points, and green lines, shaded regions and labelled slopes show the mean and standard error of the best linear fit to the residuals. (e) The fraction of MII oocytes (MII/Eggs) versus Eggs. (f) Histogram of observed MII (red circles) versus that expected from independently triggering follicles (green line and shaded region show the expected counts and their standard deviation), for the 116 cycles that have 17 eggs retrieved, which is where the discrepancy between the two histograms is the largest as measured by a χ2-test (p < 10−300, primarily due to the cycles with MII of 1–5). On the scale of the plot, the expected histograms from a binomial distribution where the probability for an oocyte being metaphase-II is constant is indistinguishable from one where the probability varies with E2.

Figure 3.

(a) A graphical model of oogenesis that is consistent with a mechanistic interpretation of the data. Labels show conditional correlation coefficients. For edges with two labels, the upper corresponds to FSH, the lower to HMG; dashes signify no dependence. The data are consistent with marked arrows (*) oriented in either direction. (b) Rank plots of p-values for the 99 conditional correlations corresponding to the conditional independencies predicted by the model. The green line shows that from the training data, the red line that from the test data. The black line and shaded regions show the median, 95% and 99.9% centred percentile of rank plots from 3000 datasets simulated according to the proposed model. (c) The same as (b), but showing the distribution of rank plots for the 99 conditional correlations from fully connected, linear Gaussian models.

Figure 4.

(a) AMH versus Eggs. Grey dots show raw data, red circles show the mean and standard error of AMH binned at each value of Eggs and green line shows the best linear fit to the data, with slope of 0.20 ± 0.01 and intercept 0.36 ± 0.11. A linear fit is the best-fit polynomial to the data, as determined by the model evidence. (b) Variance and error estimate of AMH versus Eggs, trimmed to the central 95% for each value of Eggs (red dots, errors calculated using the variance of the k-statistic). The green line shows the best linear fit to the variance, with a slope of 0.28 ± 0.01 and intercept −0.22 ± 0.03. (c) E2 versus Eggs. The green line shows the best linear fit to the data. A quadratic model (not shown) provides the best fit to E2 versus Eggs. (d) Variance and error estimate of E2 versus Eggs, trimmed to the central 95% for each value of Eggs.

Figure 5.

(a) Day 5 Stage versus Day 3 Cells. Grey dots show the raw data, red circles show the mean Day 5 Stage for each separate value of Day 3 Cells, the green line and shaded region show the nonlinear model with the highest evidence and its uncertainty. The Day 5 Stages are: (1) degenerate or arrested, (2) morula with incomplete compaction, (3) morula with complete compaction, (4) early blastocyst, (5) expanding blastocyst, (6) full blastocyst, (7) expanded blastocyst, (8) hatching blastocyst, and (9) hatched blastocyst (see electronic supplementary material, §2 for details). (b) Estimated probability of an embryo resulting in a fetal heartbeat (FH) as a function of Day 3 Cells alone, for embryos recorded on Day 3 and transferred. The red circles and error bars show the probability estimated by a model that fits an independent probability of implantation for each number of cells; the green line and shaded region shows the nonlinear model with the highest model evidence and its uncertainty. (c) The estimated probability of FH as a function of Day 5 Stage alone, for embryos recorded on Day 5 and transferred. (d) The logit of the estimated probability of FH as a function of Day 3 Cells, after regressing against Day 5 Stage. Red circles and error bars show the additional log probability estimated from a model that fits an independent logit for each value of Day 3 Cells; green line shows the best linear model and uncertainty for the logit. The data are consistent with Day 3 Cells having no additional predictive power on FH once Day 5 Stage is known. (e) The graph with the minimal number of edges that is consistent with the data and a mechanistic interpretation. Black nodes and arrows show the measured data; grey nodes and arrows show the data’s missingness. Only the arrow Age → BMI can be re-oriented without breaking consistency with the data or a mechanistic interpretation. Edge labels for continuous variables are conditional correlation coefficients (roman typeface). Edge labels for discrete variables are coefficients from logistic regression (italic typeface), after treating the effects of other variables with edges into the discrete variable and after normalizing the input variable by its mean and standard deviation. The missingness variables D5 Rec. and Trans. are 1 if the embryo is recorded on Day 5 or transferred, respectively, and 0 otherwise. The distribution of Trans. changes depending on whether Day 5 Stage was recorded; the two labels on edges into Trans correspond to Day 5 Stage missing or recorded. (f) The data are consistent with a picture where processes which control pre-implantation development are largely different from those which control post-implantation development.