# Accuracy of Methods for Calculating Postnatal Growth Velocity for Extremely Low Birth Weight Infants

## Abstract

*Objective.* No uniform method for calculating growth velocity (GV) (grams per kilogram per day) among extremely low birth weight (ELBW) infants has been reported. Because the calculation of actual GV is so labor intensive, investigators have estimated GV with varying approaches, making comparisons across studies difficult. This study compares the accuracy of 3 mathematical methods used for estimating average GV, namely, 2-point models using the difference between weights at 2 time points divided by time and weight (either birth weight [BW] or average weight), linear regression models that are normalized for either BW or average weight, and an exponential model. The accuracy of all models was compared with actual GVs calculated from daily weight measures for a group of ELBW infants.

*Methods.* Actual GVs were calculated from daily weights for 83 ELBW infants admitted to the special care nursery and were compared with estimated GVs from each of the 5 models for the same infants.

*Results.* The exponential model, using weights from 2 time points, ie, GV = [1000 × ln(*W*_{n}/*W*_{1})]/(*D _{n}* −

*D*

_{1}), was extremely accurate, with mean absolute errors of 0.02% to 0.10%. The 2-point and linear models were highly inaccurate when BW was used in the denominator, with mean absolute errors of 50.3% to 96.4%. The 2-point and linear models were fairly accurate when average weight was used in the denominator, with mean absolute errors of 0.1% to 8.97%. Additional analyses showed that the accuracy of the 2-point and linear model estimates was affected significantly by the combination of BW, length of stay, and chronic lung disease, whereas the exponential model was not affected by these combined factors.

*Conclusions.* GV estimates calculated with 3 commonly used models varied widely, compared with actual GVs; however, the exponential model estimates were extremely accurate. The exponential model provides the accuracy and ease of use that are lacking in current methods applied to infant growth research.

In the United States, 22845 extremely low birth weight (ELBW) infants (birth weight [BW] of <1000 g) were born in the most recent year for which data are available.^{1} These infants account for a significant percentage of the deaths associated with prematurity^{2–6} and suffer the majority of short- and long-term morbidities, including poor postnatal growth. In fact, most of these infants are below the 10th percentile for weight at discharge, compared with intrauterine reference fetuses of similar gestational ages.^{4,7–9} Several investigators have demonstrated that postnatal growth is influenced by the severity of coexisting morbidities that affect infant nutritional and metabolic status.^{4,8–13} Both Ehrenkranz et al^{4} and Wright et al^{13} showed that infants without major morbidities grew at a significantly faster rate than did less-healthy infants. Radmacher et al^{8} reported a significantly faster rate of growth for infants without bronchopulmonary dysplasia, compared with those with bronchopulmonary dysplasia.

The measurement of postnatal growth is central to clinical care and research for ELBW infants, because it provides an indirect measure of overall infant health, nutritional adequacy, and care practices. Although in clinical practice postnatal growth measurements commonly include daily weight, weekly head circumference, and weekly length measures, growth velocity (GV) (grams per kilogram per day) is the most frequently reported measure in growth research among ELBW infants. Average GV is an attractive measure for research purposes because it summarizes the infant's weight gain over a desired time interval, often smoothing the variability seen in daily weight measures. However, the calculation of GV on a daily basis from actual weight measures, averaged over the desired time interval, is extremely labor intensive. Therefore, differing methods for estimating the actual average GV have been used in studies that have reported GV as a dependent variable.^{3,4,6–8,14–17} However, with such widely varying procedures for estimating average GV, the results of therapeutic practices cannot be compared across settings and studies.

Table 1 summarizes the various methods that previous investigators have reported for estimating average GV and demonstrates their use of different time intervals and starting points for weight measures. The frequency with which weight was actually measured varied from daily to weekly to other intervals specific to individual studies. The starting time point at which GV estimation began also varied among studies, with time points that included the following: from birth, from the nadir of early postnatal weight loss, from regaining BW, and from other points specific to individual studies.

In addition, methods for estimating average GV from these weight measures have not been standardized. Some investigators used absolute weight gain over a specified time interval, not normalizing data for either infant weight or time.^{10} Others used average daily weight gain calculated from weight measures over a specified time interval, without normalizing data for the infant weight.^{14} Although this method permits comparisons among infant groups of similar weights, the limitations are obvious when infants with very different weights are compared. Although some investigators attempted to correct for this limitation by dividing the net weight gain by the specified time interval and weight, the actual weight used to normalize average GV varied among investigators, with some using BW and others using averaged weight from the beginning to the end of the period.^{3,7,16} The weight chosen as the denominator in these GV estimations affects the value obtained for GV significantly; therefore, the use of different weights makes comparisons of average GV across studies impossible. Other researchers estimated average GV through the use of linear regression equations that predicted weight as a function of time.^{14,15,17} Still others reported GV but did not specify the formula used for estimations.^{4,6,8} Finally, some investigators used *z* scores to compare BW and postnatal weight at specified times with in utero weight at similar gestational ages.^{9,18,19}

Despite the variety of reported methods for estimating average GV in the literature, no previous research compared the accuracy of these methods. We hypothesized that different methods of estimating GV would yield discrepant results, making comparisons across studies difficult to perform. The purpose of this study was to compare the accuracy of 3 mathematical methods used for estimation of average GV, ie, 2-point models using the difference between weights at 2 time points divided by time and weight (either BW or average weight), linear regression models normalized for either BW or average weight, and an exponential model. The accuracy of all models was compared with actual GV calculated from daily weight measures for a group of ELBW infants.

## METHODS

### Subjects

One hundred fifty infants with BW of <1000 g were admitted to Rush University Medical Center Special Care Nursery between January 1, 1997, and December 31, 1998. Daily weight measures from admission to discharge or transfer were accessed retrospectively from a dataset of infants from this time period who survived to discharge.^{20} Exclusion criteria were (1) infants of mothers with positive drug screen results, (2) infants to be placed for adoption, (3) infants admitted after day 3 of life, and (4) infants for whom GV for the first month could not be calculated, including those with a length of stay (LOS) of <30 days or significantly incomplete weight data. Therefore, the weight measures for the remaining 83 infants served as the accurate standard for this study. This study was approved by the Rush University institutional review board.

### GV

The weights recorded in the dataset reflected the following weighing procedures. Staff nurses weighed infants daily. Infants in stable condition were weighed outside the Isolette, with a Scale-Tronix pediatric scale (model 4800; Scale-Tronix, Wheaton, IL). Infants receiving assisted ventilation were weighed by 2 nurses, with 1 nurse holding the attached tubes, lines, and sensors. Infants in unstable condition were weighed with in-bed scales (model 4004; Scale-Tronix), and weighing was not performed on days when infants were deemed to be in extremely unstable condition. The treatment of these missing data is described below.

The daily weights from the infants' medical records were collected and entered into the database by research assistants during the initial study.^{20} The weights were used as recorded in the medical records, with no editing or deletion of data. In this study, for each infant, we identified the age to regain BW, defined as the first of 3 successive days on which the weight was greater than or equal to the BW.^{15} Actual or accurate standard GV (grams per kilogram per day) was calculated as [(*W _{n}*

_{+1}−

*W*) × 1000]/ [(

_{n}*W*+

_{n}*W*

_{n}_{+1})/2] in daily increments until discharge, where

*W*is the weight in grams on day

_{n}*n*and

*W*

_{n}_{+1}is the weight in grams on the following day. If the weight had not been recorded for a period of 1 to 3 days, then the accurate standard GV was estimated by using the weights from the day before and the day after the missing day(s) and averaging the weights over the appropriate time interval. For the majority of cases, weights were missing for a single day at a time; however, weights were not available for 2 and 3 consecutive days for 5 and 2 infants, respectively. The 83 infants were hospitalized for a total of 6443 days; of the total stay, only 77 days (1.2%) for 44 infants reflect these estimated accurate standard GVs.

The models were applied over different time intervals, to determine whether the accuracy varied on the basis of the chosen interval. From the actual GVs calculated in daily increments, we calculated the average weekly GV for each infant by first averaging the daily GVs in sequential 7-night increments for the entire hospital stay and then calculating the mean of these weekly values. We then calculated the average monthly GV for each infant by first averaging the daily GVs in sequential 30-night increments for the entire hospital stay and then calculating the mean of these monthly values. We calculated until-discharge GV for each infant by averaging all daily accurate standard GVs for the hospital stay. For each infant, this entire process was applied from 2 different starting points, ie, day of birth and day of regaining BW. These actual GV data served as the accurate standard for our comparisons of estimated GVs from the different mathematical models. The 2-point and exponential models were applied to the same infants for identical time periods and starting points. The linear models were applied to the same infants but only for the until-discharge time interval, because the strength of these models is the ability to calculate the average rate of change by using all of the data points, theoretically minimizing error.

For each infant, we estimated the average GV with 3 methods described in the research literature, as follows: (1) 2-point BW model: net weight gain over the time interval divided by the time interval and BW, or estimated GV = [1000 × (*W _{n}* −

*W*

_{1})]/ [(

*D*−

_{n}*D*

_{1}) × BW]

^{16}; (2) 2-point average weight model: net weight gain over the time interval divided by the time interval and average weight, or estimated GV = [1000 × (

*W*−

_{n}*W*

_{1})]/ {(

*D*−

_{n}*D*

_{1}) × [(

*W*

_{1}+

*W*)/2]}, where

_{n}*W*is the weight in grams,

*D*is day, 1 indicates the beginning of the time interval, and

*n*is the end of the time interval, in days

^{3}; (3) linear BW model: linear regression of weight (in grams) versus time (in days), where GV = [1000 × (slope of regression line)/BW]

^{17}; (4) linear average weight model: linear regression of weight (in grams) versus time (in days), where estimated GV = [1000 × (slope of regression line)]/[(

*W*

_{1}+

*W*)/2] (although it is not described in the literature, we evaluated this model because we speculated that the weight used in the denominator might affect the accuracy of the model); (5) exponential model: estimated GV = [1000 × ln(

_{n}*W*/

_{n}*W*

_{1})]/ (

*D*−

_{n}*D*

_{1}) (see Appendix for derivation of formula). Although it was not described previously in the literature, an exponential model was used to estimate average GV because growth in biological systems is generally nonlinear; indeed, growth is often described as following an exponential pattern.

^{21}An exponential model assumes that growth occurs at a fraction of the previous weight, otherwise known as first-order kinetics. Although first-order kinetics are used commonly to describe phenomena with a constant rate of change, we speculated that they might be useful to describe weight gain. Although growth does not occur at a constant rate of change, the average GV smoothes the variability in weight gain observed clinically, thereby providing an estimate of a theoretically constant rate of change (Fig 1).

### Data Analysis

The accurate standard GVs and the estimated GVs from the 5 models (2-point BW, 2-point average weight, linear BW, linear average weight, and exponential) are reported as mean ± SD for the 83 infants. The GVs estimated with the 5 models were compared with the accurate standard GVs through calculation of the magnitude of error, as reflected by the percentage absolute difference, ie, difference = {[absolute(estimated GV − accurate standard GV)]/accurate standard GV} × 100%, and are reported as the mean percentage absolute difference, SD, maximal percentage absolute difference, and percentage of infants with values that were ≤5% and ≤20% different from the accurate standard GVs. We decided a priori that if the estimated GVs from a model were within 20% of the accurate standard GVs, then the model would be useful as a clinical tool. The mean estimated GVs from the models were also correlated with the mean accurate standard GV for each time interval and starting point with Pearson's correlation coefficients. Spearman correlation coefficients (*r*_{s}) and Mann-Whitney *U* tests were used to examine the relationship between the absolute magnitude of error in the estimated GVs and factors likely to influence the accuracy of GV estimates for the until-discharge time interval, such as necrotizing enterocolitis, BW, LOS, and chronic lung disease. Then, analysis of covariance (ANCOVA) was used to examine the interaction between the factors that were found to affect the models significantly and consistently. Results were considered significant at *P* < .05. Data were analyzed with SPSS-PC software, version 12.0 (SPSS, Chicago, IL).

## RESULTS

### Subjects

The characteristics of the infants studied are summarized in Table 2. The gestational age and BW (mean ± SD) were 26.2 ± 2 weeks and 785 ± 140.5 g, respectively, and BW was regained at an average of 15.7 ± 5.2 days. Mean LOS was 77.6 ± 25.2 days, with a mean weight at discharge of 2087.5 ± 538.3 g. Major infant morbidities included chronic lung disease, defined as oxygen requirement at postconceptional age of 36 weeks (41.5%), necrotizing enterocolitis (6.0%), grade 3 or 4 intraventricular hemorrhage (6.0%),^{22} and periventricular leukomalacia (1.2%). Twenty-three (27.7%) infants were discharged from the hospital receiving oxygen.

### GV

For the 83 infants, mean accurate standard GVs and GVs estimated with the 5 models are summarized in Table 3. All accurate standard and estimated GVs were calculated from both starting points of birth and regaining BW and for 3 time intervals, ie, weekly, monthly, and until-discharge. For the birth starting point, the mean accurate standard GVs varied from 12.415 to 12.674 g/kg per day for the different time intervals; values varied from 15.699 to 16.615 g/kg per day for the regaining BW starting point.

The differences between the accurate standard and estimated GVs are summarized in Table 4. The exponential model approximated closely the accurate standard GVs, regardless of time period used, whereas there were large percentage differences with the 2-point BW and linear BW models and moderate differences with the 2-point average weight and linear average weight models. The differences between the accurate standard GVs and GVs estimated with the 2-point BW model were large, with 83.1% to 100% of the differences being >20%. For the linear BW model, there was a >20% difference between the accurate standard and estimated GVs for all 83 infants with the birth starting point and for 82 of 83 infants with the regaining BW starting point.

### Factors Affecting the Accuracy of GV Estimates

Figure 2 depicts the relationship between LOS and the magnitude of error, expressed as the percentage difference between accurate standard GVs and estimated GVs, for each of the models and with both starting points. As LOS increased, the 2-point BW and linear BW models overestimated the accurate standard GVs, whereas the 2-point average weight and linear average weight models underestimated the accurate standard GVs slightly. Increasing LOS was highly correlated with increasing absolute magnitudes of error for the 2-point BW (*r*_{s} = 0.87 and *r*_{s} = 0.86 for birth and regaining BW, respectively; *P* < .01 for both), 2-point average weight (*r*_{s} = 0.85 and *r*_{s} = 0.85 for birth and regaining BW, respectively; *P* < .01 for both), and linear BW (*r*_{s} = 0.83 and *r*_{s} = 0.82 for birth and regaining BW, respectively; *P* < .01 for both) models. In contrast, the linear average weight (*r*_{s} = −0.41; *P* < .01; and *r*_{s} = 0.28; *P* = .01; for birth and regaining BW, respectively) and exponential (*r*_{s} = 0.17; *P* = .12; and *r*_{s} = 0.30; *P* = .01; for birth and regaining BW, respectively) model estimates revealed weak linear correlations between increasing LOS and absolute magnitude of error.

Lower BW was associated with a greater magnitude of error for all of the models except the linear average weight model from birth, for which higher BW was associated with a greater magnitude of error. The correlations were statistically significant for the 2-point BW (*r*_{s} = −0.57 and *r*_{s} = −0.57 for birth and regaining BW, respectively; *P* < .01 for both), 2-point average weight (*r*_{s} = −0.55 and *r*_{s} = −0.55 for birth and regaining BW, respectively; *P* < .01 for both), and linear BW (*r*_{s} = −0.49 and *r*_{s} = −0.54 for birth and regaining BW, respectively; *P* < .01 for both) models and the linear average weight model from birth (*r*_{s} = 0.43; *P* < .01). The exponential model estimates were also statistically but weakly correlated (*r*_{s} = −0.22; *P* = .05; for birth and *r*_{s} = −0.38; *P* < .01; for regaining BW). Only the errors made with the linear average weight model from regaining BW were not statistically associated with BW. When BW was analyzed categorically as <750 g or ≥750 g, lower BW (<750 g) was associated with a larger and statistically significant percentage absolute difference for all of the models except the exponential model from birth. However, the mean percentage absolute differences in the accuracy of the exponential model were not clinically appreciable (0.015–0.025%).

There were no statistically significant relationships between any model's absolute percentage difference and the presence of necrotizing enterocolitis. Oxygen requirement at discharge was associated statistically with the absolute percentage difference only for the linear average weight model from birth. Gender was associated statistically with the absolute percentage difference only for the exponential model, but these differences were not clinically appreciable (0.003–0.008%). However, the presence or absence of chronic lung disease was associated significantly with the mean absolute percentage difference for the 2-point BW, 2-point average weight, linear BW, and linear average weight models for both starting points. There was no effect of chronic lung disease in the exponential model when applied from the birth starting point. Although the effect of chronic lung disease in the exponential model from the regaining BW starting point was statistically significant (*P* = .02), the actual percentage difference (0.008%) was not clinically appreciable

ANCOVA was performed to examine the potential interactions of the 3 variables that were associated significantly and consistently with the magnitude of error in estimated GVs, ie, BW, LOS, and the presence or absence of chronic lung disease. The analysis demonstrated that the combination of these factors explained a statistically significant percentage of the absolute differences for the 2-point BW model (from birth, *R*^{2} = 0.885; from regaining BW, *R*^{2} = 0.878), 2-point average weight model (from birth, *R*^{2} = 0.850; from regaining BW, *R*^{2} = 0.683), linear BW model (from birth, *R*^{2} = 0.829; from regaining BW, *R*^{2} = 0.824), and linear average weight model (from birth, *R*^{2} = 0.209; from regaining BW, *R*^{2} = 0.137). In contrast, neither the full statistical model nor the individual factors were statistically significant in the ANCOVA for the exponential model (from birth, *R*^{2} = 0.046; from regaining BW, *R*^{2} = 0.089), indicating that GVs estimated with the exponential model were unaffected by these variables.

## DISCUSSION

This study is the first to report the comparison of actual and estimated GVs from several mathematical models for a sample of ELBW infants, for whom these measures are especially important for clinical practice and research. Postnatal GV is used to guide day-to-day decisions in the care of ELBW infants in the NICU, such as determining the feeding regimen. Postnatal GV is also used frequently in research as a dependent variable to assess the safety and efficacy of interventions, particularly nutritional regimens, as well as an independent variable to predict important outcomes such as neurocognitive development, IQ, and the risk of adult-onset diseases (such as cardiovascular diseases).^{23,24}

These data demonstrate that different mathematical models yield varying estimates of average GVs, compared with actual GVs, for a group of ELBW infants. Although variability from the accurate standard was a function of the starting point and the measurement interval for all of the models, the magnitude of error in GV estimates was dependent on the specific model. The 2 models that used BW to normalize the average GV demonstrated the greatest error, which was magnified by increasing LOS; this suggests that these models have limited utility in clinical practice or research (Table 4).

The models that used average weight as the denominator provided more accurate estimates of the average GV. GV estimates for the 2-point average weight^{3} and linear average weight models were close to the accurate standard GVs, with a magnitude of error of <20% for the majority of infants (Table 4). However, our analysis showed that the accuracy of these models was affected by factors observed commonly among ELBW infants, such as chronic lung disease and longer hospital stays. For hospital stays of >160 days, the error exceeded 20% for the 2-point average weight model (Fig 2). This model would thus be acceptable to use for the majority of ELBW infants, because recent studies reported mean LOS for this population of 86 and 100 days and median LOS of 87.5 days.^{25–27} However, the reported range of LOS for ELBW infants is quite broad, ranging from 39 to 365 days.^{27} Therefore, researchers and clinicians should take care to use this model only for infants with LOS of <160 days.

The exponential model was extremely accurate regardless of starting point or time interval, with mean magnitudes of error of <0.1% and no errors exceeding 5% of the accurate standard GV for any of the 83 infants (Table 4). Although LOS, BW, and chronic lung disease affected the accuracy of the exponential model inconsistently, statistical significance was most likely a function of the large sample size, because the absolute differences between magnitudes of error were negligible. This explanation was verified with the ANCOVA, which showed that the exponential model was affected minimally by any of these factors and remained extremely accurate, with only 4.6% to 8.5% of the variance in the percentage absolute difference between the accurate standard and estimated GVs being attributable to LOS, BW, and the presence of chronic lung disease. Another point that indicates that the exponential model should be applicable to all infants, regardless of disease severity, is that it estimated accurately a wide range of average GVs (6.37–24.21 g/kg per day) and was accurate for individual infants during periods of weight loss. We speculate that the superior performance of the exponential model is a function of its nonlinearity, which provides a better fit for the growth of biological species.

Ease of application is another desirable characteristic of any model used to estimate average GVs. The simple mathematics of the 2-point models allow easy clinical application at the bedside. The linear models require linear regression techniques and thus are somewhat more labor intensive than the other models, which limits routine clinical use. However, the additional labor may be an acceptable exchange for the theoretical improvement in accuracy provided by this technique, which integrates all of the data points in its result. From a practical perspective, the exponential model is very simple to use, requiring only weight and day of life at 2 time points, but requires computerized calculations. A limitation of all 2-point models is that the accuracy is dependent on the accuracy of the weight measures used. Therefore, careful review of the weights should be performed to single out clearly aberrant values.

A limitation of this study may be that we studied only infants who survived to discharge or transfer, who do not reflect the entire range of illness severity seen in most NICUs. However, our range of neonatal morbidities was similar to that reported by Hack et al^{28} for a group of ELBW survivors. Prospective application of the exponential model to all infants would facilitate evaluation of its accuracy for the subgroup of infants who do not survive and presumably have poor postnatal growth.

There is a great need for uniformity in measurement of GV, as shown by our review of the multiple methods found in the literature. Although consistently applying the same starting point and time interval may facilitate comparisons, it may not be appropriate, because the use of each starting point has its own merits. Evaluation of growth starting from birth includes a gross assessment of disease severity, because sicker infants grow slowly and regain BW later.^{11} In contrast, evaluating growth starting from regaining BW measures the adequacy of the nutritional regimen, with the caveat that chronic and late-onset illnesses (eg, sepsis or chronic lung disease) may affect this growth rate. Therefore, it may be difficult to standardize the starting point and time interval, because these would vary with the particular research question.

One standardized method of calculating average GV should be adopted. Four of the models that we evaluated make the assumption of linear growth, although the concept of GV is expressed mathematically as an exponential function. This analysis has demonstrated that the current discrepancies in calculation methods found in the growth literature may be resolved by using an exponential model for estimating GV normalized for weight. The exponential model has numerous features that make it a desirable and powerful model, ie, (1) it is extremely accurate; (2) it is simple to use, requiring only weight and day of life at 2 time points; (3) it is robust under various testing conditions; and (4) it is unaffected by clinical factors found commonly among ELBW infants, which allows its broad application for the study of infant growth.

## APPENDIX

Average GV can be derived from the exponential relationship between initial weight (*W*_{1}) and weight at the second time point (*W _{n}*) and time, with

*D*representing day of life. An exponential model assumes that growth occurs at a constant fraction (

*k*) of the previous weight, such that weight changes over time by some fraction of the previous weight, (1) which can be rearranged to (2) with integration of both sides (3) to yield (4) If

*k*= GV and Δ

*t*= (

*D*−

_{n}*D*

_{1}) and if 1000 is used to correct units (grams per kilogram per day), then this can be restated as (5) Exponentiating both sides yields the exponential relationship between weights at times 1 and

*n*(6) Rearranging Eq 5 for GV yields (7)

## Footnotes

- Accepted March 24, 2005.
- Address correspondence to Aloka L. Patel, MD, Rush University Medical Center, 1653 W. Congress Pkwy, Murdock 622, Chicago, IL 60612. E-mail: aloka_patel{at}rush.edu
Conflict of interest: The exponential formula described in this publication is copyrighted © 2005 by Rush University Medical Center; all rights are reserved. Methods of using the formula are patent pending. Permission is granted to use the exponential formula for research purposes. No commercial use is permitted without a license from Rush University Medical Center.

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- Copyright © 2005 by the American Academy of Pediatrics