A modified Michaelis-Menten equation estimates growth from birth to 3 years in healthy babies in the US

Background and Objectives: Standard pediatric growth curves cannot be used to impute missing height or weight measurements in individual children. The Michaelis-Menten equation, used for characterizing substrate-enzyme saturation curves, has been shown to model growth in many organisms including nonhuman vertebrates. We investigated this equation could be used to interpolate missing growth data in children in the first three years of life. Methods: We developed a modified Michaelis-Menten equation and compared expected to actual growth, first in a local birth cohort (N=97) then in a large, outpatient, pediatric sample (N=14,695). Results: The modified Michaelis-Menten equation showed excellent fit for both infant weight (median RMSE: boys: 0.22kg [IQR:0.19; 90%<0.43]; girls: 0.20kg [IQR:0.17; 90%<0.39]) and height (median RMSE: boys: 0.93cm [IQR:0.53; 90%<1.0]; girls: 0.91cm [IQR:0.50;90%<1.0]). Growth data were modeled accurately with as few as four values from routine well-baby visits in year 1 and seven values in years 1-3; birth weight or length was essential for best fit. Conclusions: A modified Michaelis-Menten equation accurately describes growth in healthy babies aged 0-36 months, allowing interpolation of missing weight and height values in individual longitudinal measurement series. The growth pattern in healthy babies in resource-rich environments mirrors an enzymatic saturation curve.


Introduction
Height, weight, and growth are foundational indicators of child health. Growth charts, created by the World Health Organization 1 and the US Centers for Disease Control and Prevention 2 , serve as clinical references to evaluate individual pediatric physical sizes and growth rates against population means. These reference ranges represent cross-sectional information from tens to tens of thousands of children per age group. Longitudinal studies, however, demonstrate the unpredictability of individual patterns, with short growth spurts punctuating periods of minimal growth (i.e., a saltatory pattern) 3,4 . Thus, actual growth for an individual child is statistically unique and cannot be reconstructed from group data 5 .

Stanford's Outcome Research Kids (STORK) is a birth cohort recruited in the San Francisco Bay Area,
California, designed to evaluate the impact of infections on growth from birth to age 36 months 6 . In this project, some infants were missing necessary time-speci c weight measurements. We sought to identify an empirical longitudinal growth model that would provide the best interpolation of missing weight values given only the available weight values for that individual baby -in essence, a function that would smooth noisy existent data to t a line and that was simple, to avoid over tting.
The Michaelis-Menten equation was originally used in biochemistry to describe how substrate concentration affects the rate of enzyme catalysis 7 . The equation was subsequently slightly modi ed and applied to a wide range of chemical and biological processes, ranging from antibody development to soil microbial activity to tree growth [8][9][10] . The Michaelis-Menten equation also describes growth accurately in sh, birds and mammals of various sizes 11 . To date, however, the equation has not been used to model human growth.
We applied a modi ed Michaelis-Menten equation to each STORK baby's individual weight curve and evaluated its t. We then validated the use of this equation for weight and also height using a large longitudinal dataset from healthy babies (Stanford Medicine Research Data Repository (STARR)) and additionally identi ed those well-baby visit timepoint combinations essential for best model t. Finally, we evaluated the accuracy of this equation to predict weight and height during the second and/or third year of life when using growth measures from earlier timepoints.

Babies
Detailed methods for the STORK birth cohort have been described previously 6 . In brief, a multiethnic cohort of mothers and babies was followed from the second trimester of pregnancy to the babies' third birthday. Healthy women aged 18-42 years with a single-fetus pregnancy were enrolled. Households were visited every four months until the baby's third birthday (nine baby visits), with the weight of the baby at each visit recorded in pounds. Medical charts were abstracted for birth weight and length. All data were managed in REDCap 12 hosted at Stanford University. STARR (starr.stanford.edu) contains electronic medical record information from all pediatric and adult patients seen at Stanford Health Care (Stanford, CA). STARR staff provided anonymized information (weight, height and age in days for each visit through age three years; sex; race/ethnicity) for all babies during the period 03/2013-01/2022 followed from birth to at least 36 months of age with at least ve well-baby care visits over the rst year of life.

Statistical analysis
All observed weight and height values were evaluated in kilograms (kg) and centimeters (cm), respectively. Any values assessed beyond 1,125 days (roughly 36 months) and values for height and weight deemed implausible by at least two reviewers (e.g., signi cant losses in height, or marked outliers for weight and height) were excluded from the analysis. Additionally, weights assessed between birth and 19 days were excluded, as weight loss often occurs immediately after birth, and approximately 95% of babies return to their birth weight by 19 days 13 . At least ve observations across the 36-month period were required: babies with fewer than ve weight or height values after the previous criteria were excluded from analyses.

Model
We developed our weight model using values from STORK babies and then replicated it with values from the STARR babies. Height models were evaluated in STARR babies only because STORK data on height were scant.

The Michaelis-Menten equation is described as follows
where v is the rate of product formation, V max is the maximum rate of the system, [S] is the substrate concentration, and K m is a constant based upon the enzyme's a nity for the particular substrate.
For this study the equation became: where P was the predicted value of weight (kg) or height (cm), Age was the age of the infant in days, and c1 was an additional constant over the original Michaelis-Menten equation that accounted for the infant's non-zero weight or length at birth. Each of the parameters a1, b1 and c1 was unique to each child and was calculated using the nonlinear least squares (nls) method. In our case, weight data were tted to a model using the statistical language R (version 3.4.0) 14 , by calling the formula nls() with the following parameters: tted_model <-nls(weights~(c1+(a1*ages)/(b1+ages)), start = list(a1 = 5, b1 = 20, c1=2.5)) where weights and ages were vectors of each subject's weight in kg and age in days. The default Gauss-Newton algorithm was used. The optimization objective is not convex in the parameters, and can suffer from local optima and boundary conditions. In such cases good starting values are essential: the starting parameter values (a1=5, b1=20, c1=2.5) were adjusted manually using the STORK dataset to minimize model failures; these tended to occur when the parameter values, particularly a1 and b1, increased without bound during the iterative steps required to optimize the model. (Using higher starting a1 and b1 parameter values, i.e., closer to the mean/median values upon which the nls function previously converged, gave similar a1 and b1 parameter values, but also a higher rate of model failures due to more a1 and b1 values increasing without bound.) These same parameter values were used for the larger STARR dataset.
The starting height parameter values for height modeling were higher than those for weight modeling, due to the different units involved (cm vs. kg) (a1=60, b1=530, c1=50). Correlations between the c1 parameter and birth weight or birth length for all babies by sex and by study were evaluated using Spearman's rank coe cient.
Because this was a non-linear model, goodness of t was assessed primarily via root mean squared error (RMSE) for both weight and height 15 . (Lower RMSE values indicate better model t.) The values of RMSE are in the same units as those measured (kg or cm) and can be used as estimates of the deviation in values predicted by the model from the observed values. To evaluate the effect of age on the RMSE, we considered the RMSE for each timepoint and visualized all RMSE vs. age.

Imputation tests
To test for the in uence of speci c time points on the models, we limited our analysis to STARR babies with all recommended well-baby visits (12 over three years 16 ). Each scheduled visit except day 1 occurred in a time window around the expected well-baby visit (Visit1: Day 1, Visit2: days 20-44, Visit3: 46-90, Visit4: 95-148, Visit5: 158-225, Visit6: 250-298, Visit7: 310-399, Visit8: 410-490, Visit9: 500-600, Visit10: 640-800, Visit11: 842-982, Visit12: 1024-1125). We considered two different sets: infants with all scheduled visits in the rst year of life (seven total visits) and those with all scheduled visits over the full three-year timeframe (12 total visits). We t these two sets to the model, identifying baseline RMSE. Then, every visit, and every combination of two to ve visits were dropped, so that the RMSE or model failures for combination of visits could be compared to baseline.

Prediction
We sought to predict weight or height at 36 months (Y3) from growth measures assessed only up to 12 months (Y1) or to 24 months (Y1+Y2), utilizing the "last value" approach 17 . In brief, the last observation for each child (here, growth measures at 36 months) is used to assess overall model t, by focusing on how accurately the model can extrapolate the measure at this time point. We identi ed all STARR infants with at least ve time points in Y1 and at least two time points in both Y2 and Y3, with the selection of these time points based on maximizing the number of later time points within the constraints of the wellbaby visit schedule for Y2 and Y3. The per-subject set of time points (Y1-Y3) was tted using the modi ed Michaelis-Menten equation and the mean squared error was calculated, acting as the "baseline" error. The model was then run on the subset of Y1 only and of Y1+Y2 only. To test predictive accuracy of these subsets, the RMSE was calculated using the actual weights or heights versus the predicted weights or heights of the three time series.
All analyses were performed in R 3.4.0 14 (data available in Supplemental Data Tables). The STORK study and the extended anonymized STARR dataset were approved by the Stanford IRB (protocol 17756).

Results
A total of 126 STORK and 14,817 STARR babies were considered for this analysis (Figure 1). After excluding values per protocol, 97 (77.0%) STORK and 14,695 (99.2%) STARR babies had su cient measurements to be included in the weight analyses. For height, examined only in STARR, 11,655 (78.7%) babies were included.
The sex of infants was similar in both cohorts but STORK babies were slightly heavier than STARR babies (Table 1). For STORK babies, weight values were spread fairly consistently across the 36 months by design; for STARR babies, the number of weight and height timepoints per subject was variable (range: weight: 5-15; height: 5-13).

Weight models
The Michaelis-Menten model was successfully tted to 94 STORK babies (95.9%) and 14,596 STARR babies (99.3%). The c1 parameter followed a normal distribution and approximated birthweight (Spearman Rho correlation: 0.79, 0.84 and 0.87 for STORK boys, STORK girls and both STARR boys and girls, respectively; difference between mean c1 values and mean birth weight: 0.30, 0.14, 0.06 and 0.05 kg in STORK boys, STORK girls, STARR boys and STARR girls, respectively) (Table 2, Figure 2 -gure supplement 1). Distributions of the model parameters a1 and b1 were right-skewed; extremely high a1 and b1 parameters indicated linear growth, and a higher b1 to a1 ratio indicated both less rapid early growth in the infants and a more linear curve overall. The parameter values for a1 and b1 were weakly correlated with the c1 parameter value, indicating that birth weight might play a role in predicting these values (Spearman's Rho correlation ~0.30). Apart from the shape of the growth curve and the location of the in exion point, however, we did not discern a physiological meaning for either a1 or b1.
Visual inspection of plots of infant weights over time indicated a good t with this model for all babies (Figure 2, A-D). Model t was high, as measured by low RMSE (Figure 3A-B, Table 2 Visual inspection of the tted data for height indicated excellent model t (Fig 1, E-F) and RMSE values were low, with both median and 90% values under 1 cm. Only ve subjects (0.043%) had RMSE over 3 cm

Imputation Tests
Considering growth only in the rst year, the removal of visit1 (birth weight or length) increased RMSE more than the removal of any other recommended well-baby visit (Table 3); the visit at approximately 12 months had the second largest impact on model t. Considering growth over three years, while removal of birth weight had a large impact on RMSE, removal of any other individual well-baby visit alone had a far more modest effect. Many combinations of up to three visits in year 1 and up to ve visits in years 1-3 could be dropped with only a small increase in RMSE, leaving as few as four visit timepoints needed in year 1, and as few as seven visit timepoints needed in years 1-3, with exceptions: removal of combinations of visit1 with other visits, particularly during year 1, led to a sizable increase in RMSE, as did removal of consecutive visits at the nal time points (visits 5-7 for the year 1 subset; visits 10-12 for the years 1-3 subset). The RMSE could be rescued partly for missing visit1 data by increasing the initial a1 and b1 parameters to higher values (e.g., a1=15, b1=500).

Prediction
Su cient data for weight prediction modeling was available for 4,829 STARR infants ( Figure 1); of these, 1.8% were dropped due to model failure to t their growth curve. RMSE values for the full models with these babies were similar to models using all STARR babies. In modeling data from Y1+Y2 to predict growth in Y3, RMSE increased by approximately 1.1kg for weight and 2cm for height, equivalent to 7.5% and 2.1% of sample mean weight and height at 36 months (Table 4; Figures 6 and 7; Table 1). Similarly, in modeling data from Y1 to predict growth in Y2+Y3, RMSE increased to approximately 1.3kg and 5.6cm (8.8% and 5.8% of mean weight and height at 36 months, respectively).

Discussion
Using longitudinal weight data rst in a small birth cohort and subsequently in a large healthcare database, we found that a modi ed Michaelis-Menten equation described individual babies' non-linear growth in weight and height from birth to age 36 months with minimal error. Although certain time points were essential for best model t (birth weight or length, and, for year 1 growth, the measure at approximately 12 months), the loss of most other data points had only modest effects on RMSE, indicating that our model can correctly interpolate weights and heights for a majority of infants, even when information from multiple well-baby visits is missing. Given routine baby follow-up, this equation provides an excellent method to estimate weight or height at any time point within the rst three years of life.
The modi ed Michaelis-Menten equation has been shown previously to describe growth in a wide array of living organisms and in particular mammals, including primates 11 . We believe our study is the rst to demonstrate its applicability in humans. This equation has the distinct advantage of being conceptually simple: although childhood height and weight are clearly in uenced by a multitude of factors, normal growth over time with su cient resources mirrors an elementary chemical reaction on consumable substrates. Whether this equation is valid for growth in premature babies, babies with severe illness or health conditions or babies in resource-poor environments remains to be determined.
We examined how well the modi ed Michaelis-Menten equation predicted growth at 36 months and found that estimates based on data from ages 0-24 months were within approximately 2.1% of actual height and 7.5% of actual weight. This difference in precision between height and weight may be because height measurements are less subject to intrinsic variation than weight measurements 18 ; additionally, height might be less prone to measurement error than weight, as children may be weighed with or without clothes. Using measures from only the rst year of life to predict height and weight to 36 months was more imprecise (within 5.8% and 8.8% of actual height and weight, respectively). To date, we have found no models designed speci cally to predict growth at three years of life; this equation may provide an interesting approach for identifying unexpectedly low or high growth within an individual child up to this age, without focusing on standardized growth curves. Of course, our model includes only the initial hyperbolic growth before age three years; different models should be used when considering other time frames when the growth rate changes signi cantly (i.e., at puberty).
Limitations of the Michaelis-Menten equation include failure of the model to t growth in children with linear (vs. non-linear) growth; the proportion of such babies in our study, however, was small (~0.7% overall) and these babies could potentially be t to a standard linear growth model. We were also unable to determine a physiologic interpretation for two of the three model parameters, although together they are important for shaping the growth curve. In this study, we limited our time frame from birth to 36 months; an evaluation of how far along the age spectrum this equation remains reliable would be of interest. It is important to note that body mass index (BMI), a function of height and weight, does not follow a similar curve. Finally, although weight and height have been considered useful measures of growth, growth trajectories -their derivatives -are perhaps of greater importance 19-21 .
In summary, a modi ed Michaelis-Menten equation is a useful tool to accurately describe weight and height in individual, ethnically diverse infants aged 0-36 months in California. Whether this equation can similarly explain growth in premature babies, sick children in resource-poor environments and those in older age categories has yet to be evaluated. Growth over time in an individual baby, like that of many known organisms, mirrors the saturation curve of a basic enzymatic reaction.

DATA AVAILABILITY
All information used in this study are provided in the attached source data.
Con ict of interest disclosures: The authors have no con icts of interest relevant to this article to disclose.
Funding/support: WW and RL are supported by the Max Planck Society. Both the STORK study and the data abstraction from STARR were funded by the National Institutes of Health (NIH) grant R01 5R01HD063142-02. Neither the Max Plank Institute nor the NIH had any role in the design or conduct of the study, or the decision to submit the work for publication.        Height prediction: Fitted curves for 10 randomly selected children (50% boys) using models t from early time frames (STARR only). Black dots indicate actual heights. Models were t for the full data (black), years 1 and 2 (blue), and year 1 alone (orange).

Supplementary Files
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