J Clin Ultrasound 20:381388, JulylAugust 1992 CCC 0091-2751/92/060381-08$04.00 0 1992 by John Wiley & Sons, Inc.
Growth Standards for Anatomic Measurements and Growth Rates Derived from Longitudinal Studies of Normal Fetal Growth Russell L. Deter, MD,* and Ronald B. Harrist, MD,?
Abstract: A statistical procedure for deriving growth standards for anatomic measurements and their growth rates from longitudinal studies of fetal growth was evaluated using Rossavik growth models for the biparietal diameter (BPD), head circumference (HC), abdominal circumference (AC), and femur diaphysis length (FDL) determined in a previous study of normal fetal growth. For each anatomic parameter, the coefficients c and s of the model was used to define a set of growth curves that constituted the boundary growth curves of a region containing 95% of the growth curves of this data set. The set of boundary growth curves was used to specify the mean, lower limit, and upper limit values for the anatomic parameter and its growth rate at weekly intervals between 14 and 38 weeks, menstrual age. Comparison of these values to those determined from cross-sectional studies of fetal growth gave differences of -1.9% to 4.8% (SD: iO.9 to i 2 . 6 ) for mean vs. predicted value of the anatomic measurements. For the lower limit, similar values were 0.4%to 13.8%(SD: 51.7 to k8.8); for the upper limit the values were 8.3%to 18.0% (SD: 21.5 to i7.0). Comparisons of HC growth rates determined using polynomial and Rossavik growth models gave values of -3.4% (SD: 24.4)for mean vs. predicted value, -12.6% (SD: i10.6) for the lower limit and 5.2% (SD: i9.3) for the upper limit. The degree of agreement was similar for AC growth rates. These results indicate that reasonable growth standards for anatomic measurements and their growth rates can be determined from longitudinal studies of as few as 20 normal fetuses, although better estimates of normal variability could be obtained with a larger sample. Indexing Words: Rossavik growth model * Fetal growth Growth rates, fetal
The assessment of fetal growth during pregnancy has long utilized comparisons of individual measurements to growth curve standards derived primarily from cross-sectional studies of fetal growth.' In most instances anatomic measurements are used in these comparisons but data has been presented which claimed to provide the standards needed for making similar comparisons of growth rates for the biparietal diameter, head circumference (HC), abdominal circumference (AC), and femur diaphysis length From the *Department of ObstetricslGynecolog,Baylor College of Medicine, Houston, Texas, and the ?Department of Biometry, University of Texas School of Public Health, Houston, Texas. For reprints contact Russell L. Deter, MD, Department of Obstetrics/Gynecolog,Baylor College of Medicine, Houston, TX 77030.
(FDL).2-7 Growth curve standards derived from cross-sectional studies depend on the assumption that the sample studied is composed of a single group of normally growing fetuses having similar growth trajectories which show only random variation.8 However, rarely is data presented which justifies this assumption. As pointed out previously,' longitudinal studies of growth can provide the information needed to establish that one is studying a single group of normally growing fetuses. However, standard methods of analysis do not permit estimation of the normal variability for individual measurements unless all fetuses are studied at the same time points during pregnancy: a difficult task in the human fetus. Recently, it has been possible to remove this restriction through use of a modi38 1
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mean values reported in a previous publi~ation’~ fication of a statistical method for defining tolerwere used in the regression analyses. Thus the ance regions.“,” This approach requires only growth model for each parameter in each fetus that there be a single mathematical model that was specified by the values of c and s obtained by adequately describes the growth of the anatomic regression analysis. parameter in all fetuses studied. To obtain For each parameter the pairs of c and s values growth rate standards, the first derivative of this for the 20 fetuses studied were used to determine function with respect to fetal age must also be normal ranges at different time points for both known. As has been shown p r e v i o u ~ l y , ~ the ~-~~ anatomical measurements and growth rates. Rossavik growth model” is a mathematical Briefly, this procedure first specifies the location function which has these necessary properties. of each growth curve in “coefficient space,’’ the This investigation presents the results of apaxes of this space being the coefficients c and s, plying this method to Rossavik models derived as a point having coordinates given by the c and from longitudinal studies of one-dimensional pas values of its Rossavik growth model. The c and rameters. Growth standards for both anatomic s values for the mean growth curve were determeasurements and growth rates were determined from the sample of c’s and s’s as the avermined and compared to those obtained previage of the c and s coefficients, with each c and s ously by other methods. value weighted by the inverse of the appropriate variance- covariance matrix determined for that MATERIALS AND METHODS individual.21 This procedure takes into account differences in the number of time points studied Data Set and their distribution during pregnancy as well as measurement and fitting errors. The c and s This investigation was carried out using data values of the average growth curve specify the obtained by ultrasound from 20 normal fetuses coordinates of the centroid of the cloud of points delivered at term as described in detail previin “coefficient space” which represent the indi0us1y.’~Briefly, fetuses judged to be normal by vidual c-s pairs.lo7” Using knowledge of the detailed pediatric assessment at birth were studvariance and covariance of each c-s pair and the ied at 2 to 3 week intervals from 15 weeks, mengroup of c-s pairs, the 95% confidence ellipse strual age (MA), [initial scan: 15.2 (k1.4 SD) around the centroid can be determined using the weeks] to delivery [final scan: 38.2 (k1.5 SD) Hotellings T2 statistic, a multivariate form of weeks]. At each examination the biparietal dithe t-statistic.lo’l1Within this ellipse are 95% of ameter (BPD), head circumference (HC), abdominal circumference (AC) and femur diaphysis the points representing the growth curves of this length (FDL) were measured as described previgroup of normally growing fetuses. The perimeously6,20The fetal age at the time of each examter of this ellipse defines the boundary of the norination was determined from the first day of the mal range for the growth curves (not individual last menstrual period (LMP), confirmed by a measurements) of the parameter being studied. known date of ovulation or early ultrasound To determine the normal ranges for individual data, or by crown- rump length measurement^.'^ measurements, similar to those obtained in cross-sectional growth studies, the following procedure was carried out. Forty equally spaced Data Analysis points on the boundary of the 95% confidence ellipse were selected, and the c and s coordinate Regression analysis was carried out with the values for each point, together with the appropridata sets for each fetus to determine the optimal Rossavik growth models.16 The Rossavik model ate k value, were used to specify 40 Rossavik has the following general form’? growth models for each anatomic parameter. This set of models represents the extreme growth k + dt) curves for the parameter being studied. Two of P = c(t) these growth curves (but not necessarily the where P is the anatomical parameter studied, t same two) specify the maximum and minimum the duration of growth, and c, k , and s the model values for the anatomic measurement at any coefficients. Previous studies have indicated that given time point. These 40 models were used to the coefficient k is specified by the anatomic pagenerate 40 predicted values at weekly intervals rameter being studied and should be the same from 14 weeks to 38 weeks, MA. From these for all individuals. l2 The coefficient k was considdata, the mean, maximum, and minimum values ered a constant in these investigations, and the were determined. The minimum value was taken JOURNAL OF CLINICAL ULTRASOUND
GROWTH STANDARDS DERIVED FROM LONGITUDINAL STUDIES
as the lower limit of the normal range and the maximum value as the upper limit. The procedure for obtaining normal ranges for growth rates from longitudinal studies was essentially the same as that just described for individual measurements except that the following equation for growth rate, derived from the Rossavik model, was used in place of the Rossavik model itself:
_@
dt -
'
'7
s
+ s In ( t )P~
data were compared with that obtained in a previous longitudinal study using polynomial mode l ~ ~because , ~ , ~valid data from cross-sectional studies of growth rates are not available. Differences in values (longitudinal - cross-sectional) were expressed as a percentage of the value obtained in the cross-sectional study tor previous longitudinal study in the case of the head and abdominal circumference growth rates). These percentage differences were averaged and their variability assessed by calculation of standard deviations and ranges. It should be noted that the predicted values used in these calculations are not independent, being completely determined by the longitudinal and the cross-sectional regression models, respectively. For this reason, statistical testing of differences is not possible. The means, standard deviations, and ranges presented serve as summaries by which to judge the similarity of the growth curves derived from longitudinal and cross-sectional data.
(2)
+
The value of P at different time points was determined using the appropriate Rossavik model. The data obtained were the instantaneous growth rates at specific times in pregnancy. Evaluation of the mean, maximum, and minimum values obtained from the longitudinal data set was carried out by comparison with predicted, lower limit, and upper limit values (values that defined a range including 95% of the measurements) obtained in previous crosssectional growth st~dies.''-~~Head circumference and abdominal circumference growth rate
RESULTS
Comparisons of standard growth curves for BPD, HC, AC, and FDL, obtained from both the longi-
TABLE 1 Comparison of Predicted Values (Cross-Sectional Data Set) and Mean Values (Longitudinal Data Set) for Various Anatomic Parameters) Percentage Difference" Paramete?
NCsb
NLb
Nb
Mean
SD
YO
%
533 252 252 254
20 20 20 20
23 23 23 23
% -1.9 1.o 4.8 2.8
Y O
BPD HC AC FDL
k1.4 kO.9 kl.l k2.6
-0.6 1.8 5.9 5.1
-5.2 -1.9 1.4 -2.7
Maximum
Minimum
aBPD = biparietal diameter; HC = head circumference; AC = abdominal circumference; FDL = femur diaphysis length. bN,s = sample size in cross-sectional study; NL = sample size in longitudinal study; N = number of time Doints evaluated. (longitudinal prediction at t - cross-sectional prediction at t ) CPercentagedifference at age t= x 100. (cross-sectional prediction at t )
TABLE 2 Comparison of Lower Limits of Cross-Sectional and Longitudinal Standard Growth Curves for Various Anatomic Parameters Percentage Difference ~
~~
~
~
Parameter'
Ncs
N,
N
Mean
SD
Maximum
Minimum
BPD HC AC FDL
533 252 252 254
20 20 20 20
23 23 23 23
% -13.8 -5.1 0.4 -8.2
% kl.7 k2.6 14.3 18.8
% -11.9 -2.1 4.3 3.1
% -17.7 -8.4 -9.3 -22.8
Abbreviations the same as in Table 1. VOL. 20, NO. 6, JULYiAUGUST 1992
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TABLE 3 Comparison of Upper Limits of Cross-Sectional and Longitudinal Standard Growth Curves for Various Anatomic Parameters ~-
Percentage Difference Parameter
Ncs
N,
N
Mean %
YO
%
%
BPD HC AC FDL
533 252 252 254
20 20 20 20
23 23 23 23
12.0 8.3 11.6 18.0
21.5 22.1 k3.2 27.0
14.1 11.2 16.0 26.8
6.7 6.0 7.3 6.2
S.D.
Maximum
Minimum
Abbreviations the same as in Table 1.
cross-sectional data sets, this difference being greatest for AC. Similar comparisons of the lower limits of the normal range are presented in Table 2. On average (except for AC) the lower limits determined from the longitudinal data sets were smaller than those found using the cross-sectional data sets, these differences being approximately 10% or less. In some cases differences of 15% to 20% (BPD, FDL) were seen, but these occurred before 20 weeks where small differences (0.5 cm to 0.6 cm) were more important due to the small measurements (c4.0cm). Results of the comparisons of the upper limits (Table 3) were similar to those for the lower limits except that the values derived from the longi-
tudinal and cross-sectional data sets, are presented in Tables 1, 2, and 3. The age range was limited to 16 to 38 weeks because the crosssectional data sets had very few measurements before 16 weeks (520) and Rossavik models do not specify growth well after 38 weeks because of the growth cessation that occurs in normal fetuses.25 As can be seen in Table 1,there is good agreement between the predicted values specified by functions derived from the cross-sectional data sets and the mean values obtained from the longitudinal data sets. No percentage difference was greater than 6%. On average (except for BPD) the mean values from the longitudinal data sets were larger than the predicted values from the
TABLE 4 Growth Standards for Biparietal Diameter and Head Circumference Growth Rates Menstrual Age wk 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
38
Biparietal Diameter
Head Circumference
Lower Limit
Mean
Upper Limit
Lower Limit
Mean
Upper Limit
cmlwk 0.3
cmlwk 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
cm/wk 0.4 0.4 0.4 0.4
cmlwk 1.2 1.2 1.2 1.2 1.2
cmlwk 1.4 1.4 1.4 1.4 1.4
cmlwk
1.2
1.3
1.5
1.1 1.1 1.1 1.1 1.o 1.o 0.9 0.8 0.8 0.7 0.7 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2
1.3 1.3 1.2 1.2 1.2 1.1 1.1 1.o
1.5 1.4 1.4 1.4 1.3 1.3 1.3 1.2 1.2 1.2 1.1 1.1 1.1 1.o 1.o
0.3 0.3 0.3 0.3
0.3 0.3 0.3
0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.1 0.1 0.1
0.1 0.1 0.1
0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.3 0.3
0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3
1.o
0.9 0.9 0.8 0.8
0.8 0.7 0.7 0.6 0.6 0.5
1.7 1.7 1.6 1.6 1.6
1 .o
1.o 0.9 0.9
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tudinal data sets were larger than those derived from the cross-sectional data sets. The largest differences were found for FDL, particularly below 26 weeks where differences of 21% to 26% were seen. Also, the upper limit for AC derived from the longitudinal data set was, on average, 11.6% larger than that derived from the crosssectional data set, but the lower limits were almost the same. Growth rate standards for various parameters, derived from the longitudinal data sets, are given in Tables 4 and 5. The data are the instantaneous growth rates at various times in pregnancy. The mean values, together with minimum (lower limit) and maximum (upper limit) values, are presented. Differences in the mean value, lower limit, and upper limit for the head circumference growth rate were 0.1 cm to 0.2 cm/wk when the data obtained using polynomial growth models were compared to that obtained with Rossavik growth models. Expressed as percentages, these differences over the 12 to 38 week time period were -3.4% (k4.4 SD) for the predicted value, -12.6% (k10.6 SD) for the lower limit, and 5.2% (29.3 SD) for the upper limit. The larger percentage errors occurred toward the end of pregnancy (after 3 1 weeks) when the HC growth rates decreased.
385
Comparison of AC growth rates derived from the Rossavik models with those derived from polynomial modeling of individual growth curves was affected by selection of the linear model in the latter investigations. This model specifies a constant expected growth rate and growth rate variability. As seen in Table 5, the mean AC growth rate was 1.2 cm/wk through 29 weeks then became 1.1cm/wk to 38 weeks. Estimates for the expected AC growth rates were 1.1 cm/wk5 and 1.2 cm/wk7 in the two studies published to date. The lower limit in both these latter studies was 1.0 cmlwk; the upper limit was 1.3 cmiwk in the first study and 1.4 cm/wk in the second. These values are in good agreement with the data in Table 5 except for the lower limit after 33 weeks. These lower values may be indicative of the fact that the AC growth reaches a plateau toward the end of the third trimester in some f e t ~ s e s . ~ DISCUSSION
To evaluate the validity of this approach to growth curve specification, we have compared our results to empirical cross-sectional growth curves for the anatomic parameters studied. Such growth curves provide the most direct information on the variability seen in anatomic parameters during pregnancy. The growth curves
TABLE 5 Growth Standards for Abdominal Circumference and Femur Diaphysis Length Growth Rates Menstrual Age wk 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
Abdominal Circumference
Femur Diaphysis Length
Lower Limit
Mean
Upper Limit
Lower Limit
Mean
Upper Limit
cmlwk 1.o 1.o 1.o 1.o 1.o 1.o 1.o 1.o 1.o
cmlwk 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1
cmlwk 1.5 1.5 1.5 1.5 1.5 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.5 1.5 1.5 1.5 1.5 1.6 1.6 1.6
cmlwk 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.0 0.0 0.0
cmlwk 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.2 0.2 0.2
cmlwk 0.5 0.5 0.5 0.5 0.4 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3
1.o 1.o 1.o 1.0 1.o 0.9 0.9 0.9 0.8 0.8 0.8 0.7 0.7 0.7 0.6 0.6
VOL. 20, NO. 6, JULYIAUGUST 1992
0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.1
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DETER AND HARRIST
used in these comparisons were obtained from valid cross-sectional investigations.' As shown in Table 1, there was a remarkable degree of agreement between the mean growth curves derived from the longitudinal data sets and the expected growth curves derived from the crosssectional data sets. Such findings are consistent with our previous work that has shown great similarities in expected growth curves derived from cross-sectional and longitudinal studies, the latter determined from functions whose coefficients were averages of the coefficients obtained in longitudinal studies of individual These results constitute strong empiric support for the validity of the method used in this investigation. A more rigorous test of this method is its ability to predict normal variability. In view of the fact that only 20 fetuses were studied longitudinally, one would not expect the degree of agreement between estimates of the upper and lower limits of normal variability to be as good as that found between the mean and predicted growth curves. As seen in Tables 2 and 3, this was indeed the case, but the degree of agreement was still good except for the upper limit of the FDL. In the latter situation, most of the problems were with the values for the early part of pregnancy where, because of the short length of the FDL, a 0.5 cm difference represented a significant percentage difference. In almost all cases, the values derived from the longitudinal studies were more conservative, being lower for the lower limit and higher for the upper limit. These results indicate that, although more accurate estimates of normal variability may be obtained from a larger sample of fetuses studied longitudinally, the estimates provided by the current sample are quite reasonable. This again supports the validity of this method for determining standard growth curves. Although there are advantages associated with being able to determine standard growth curves for various anatomic parameters using only a small number of fetuses studied longitudinally, the primary importance of this approach is its ability to provide standards for growth rates. Growth rates are direct indicators of fetal growth between time points and are more sensitive detectors of growth abnormalities because they can be abnormal even when individual values are within the normal range. Establishing growth rate standards using cross-sectional methods requires data sets with the same characteristics as those used for determining growth standards for the anatomic parameters themselves. These in-
clude (1) only one pair of measurements (one growth rate calculation) per patient and (2) uniform distribution of occasions of measurement over the time interval of interest. This is to assure independence among measurements and representativeness of the sample so that valid estimates of normal growth rate variability can be obtained.26 The assumptions concerning differences among individuals made in studies of the anatomic parameters themselves must also be valid for growth rates. To date, a cross-sectional data set with these characteristics has never been studied and acquiring such a data set would be difficult. Most previous publications purporting to give values for average growth rates and normal growth rate variability have used multiple pairs of measurements from the same fetus, and the same measurements have been used to calculate more than one growth rate Such data sets do not provide growth rate measurements that are independent; thus variability estimators derived from these data sets are biased.26 In the approach used in this investigation, these problems are avoided by applying the statistical analysis to the entire longitudinal growth curve (as represented by the coefficients c and s), rather than individual measurements or even pairs of measurements. Since there is only one curve per fetus, independence is assured. When the set of extreme curves is identified, the normal variability of any parameter (including growth rate) can be evaluated using this set. At any given time point, parameter values derived from two (but not necessarily the same two) curves of this set will be the extreme values. Because the set of extreme curves defined in this investigation were determined from individual growth curves from normally growing fetuses, these extreme values can logically be considered to represent the range of normal variability. The reasonably good agreement between upper and lower limit values derived from cross-sectional and longitudinal growth studies of anatomic parameters supports this conclusion. The similarity of upper and lower limits for HC and AC growth rates derived from polynomial and Rossavik growth models is additional support. It should also be noted that despite their questionable validity, the normal growth rate ranges published p r e v i ~ u s l y ~are - ~ very similar to those given in Table 4 and 5, except for the data of F e ~ c i n a . ~ In using the data in Tables 4 and 5, it is important to realize that the values given are instantaneous growth rates at the menstrual ages JOURNAL OF CLINICAL ULTRASOUND
GROWTH STANDARDS DERIVED FROM LONGITUDINAL STUDIES indicated. However, a growth rate estimate calculated from measurements at two time points is the average growth rate for the time interval between scans. To obtain appropriate values for evaluating such average growth rates, it is necessary t o first determine the upper and lower limit values at the time points defining the time interval. Since the change in values over weekly intervals is small, w e use linear extrapolation for time points that do n o t fall o n an exact week. T h e average of e a c h p a i r of values (e.g., lower limit) is then determined and these averages taken as the upper and lower limits of the normal range against which the measured average growth rate for the time interval is compared. The results of this s t u d y have demonstrated the validity of this statistical method for obtaining important clinical information a b o u t fetal growth from longitudinal data sets. Preliminary studies have indicated that this method c a n be applied to two- and three-dimensional paramet e r and ~ ~ should ~ be applicable t o a variety of mathematical models of growth including those with more than two coefficients. In the c u r r e n t investigation, this method has provided specific information o n the normal variability of growth rates for four anatomic p a r a m e t e r s that are important in the evaluation of fetal growth. S u c h information should make a valuable contribution t o the detection and characterization of growth
abnormalities.
REFERENCES 1. Deter, RL. Evaluation of studies of normal growth. In: Quantatiue Obstetrical Ultrasonography. New York, Wiley, 1986, pp 65-112. 2. Campbell S, Neuman GB: Growth of the fetal biparietal diameter during normal pregnancy. J Obstet Gynaecol Br Commonw 78:513-519, 1971. 3. O’Brien GD, Queenan JT: Growth of the ultrasound fetal femur length during normal pregnancy. Am J Obstet Gynecol 141:833-837, 1981. 4. Fescina RH, Ucieda FJ, Cordano MC, Neito F, Fenzer SM, Lopez R Ultrasonic patterns of intrauterine fetal growth in a Latin American country. Early H u m Deu 6:239-248, 1982. 5. Deter RL, Harrist RB, Hadlock FP, Poindexter AN: Longitudinal studies of fetal growth with the use of dynamic image ultrasonography. Am J Obstet Gynecol 143:545-554, 1982. 6. Deter RL, Hadlock FP, Harrist RB: Evaluation of fetal growth and the detection of intrauterine growth retardation. In Callen PW (ed.): Ultrasonography in Obstetrics and Gynecology. Philadelphia, WB Saunders, 1983, pp 113-140. 7. Deter RL, Harrist RB, Hadlock FP, Cortissoz CM, VOL. 20, NO. 6, JULY/AUGUST 1992
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Batten GW: Longitudinal studies of fetal growth using volume parameters determined with ultrasound. J Clin Ultrasound 12:313-32, 1984. 8. Deter RL, Harrist RB, Hadlock FP, Carpenter RB: The use of ultrasound in the assessment of normal fetal growth. J Clin Ultrasound 9:481-493, 1981. 9. Laird NM, Ware JH: Random-effect models for longitudinal data. Biometries 38:963-974, 1982. 10. Guttman I: Statistical tolerance regions: Classical and Bayesian. In: A Stuart (ed): Griffin’s Monograph #26, Darien, Connecticut, Hafner Publishing Co, 1970, pp 3-149. 11. Guttman I: Statistical tolerance regions. In: Kotz S, Johnson NL (eds): Encyclopedia of Statistical Sciences, vol 9, New York, Wiley 1988, pp. 272287. 12. Deter RL, Rossavik IK, Harrist RB, Hadlock FP. Mathematical modeling of fetal growth: Development of individual growth curve standards. Obstet Gynecol68:156-161, 1986. 13. Rossavik IK, Deter RL, Hadlock FP: Mathematical modeling of fetal growth: 111. Evaluation of head growth using the head profile area. J Clin Ultrasound 15:23-30, 1987. 14. Rossavik IK, Deter RL, Hadlock FP: Mathematical modeling of fetal growth: IV. Evaluation of trunk growth using the abdominal profile area. J Clin Ultrasound 15:31-36, 1987. 15. Deter RL, Rossavik IK, Cortissoz C, Hill RM, Hadlock FP: Longitudinal studies of thigh circumference growth in normal fetuses. J Clin Ultrasound 15:388-393, 1987. 16. Deter RL, Rossavik I K A simplified method for determining individual growth curve standards. Obstet Gynecol 705301-805, 1987. 17. Deter RL, Rossavik IK, Harrist RB: Development of individual growth curve standards for estimated fetal weight. I. Weight estimation procedure. J Clin Ultrasound 16:215-226, 1988. 18. Rossavik IK, Deter R L Mathematical modeling of fetal growth. I. Basic principles. J Clin Ultrasound 12:529-533, 1984. 19. Deter RL, Rossavik IK, Hill RM, Cortissoz CM, Hadlock F P Longitudinal studies of femur growth in normal pregnancies. J Clin Ultrasound 15:299305, 1987. 20. Hadlock FP, Deter RL, Harrist RB, Park SK: Fetal biparietal diameter: Rational choice of the plane of section for ultrasonic measurement. A J R 138:871-874, 1982. 21. Gumpertz M, Pantula SG A simple approach to inference in random coefficient models. American Statistician, 43:203- 210, 1989. 22. Hadlock FP, Deter RL, Harrist RB: Fetal biparieta1 diameter: A critical re-evaluation of the relation to menstrual age using realtime ultrasound. J Ultrasound Med 10:97- 104, 1982. 23. Deter RL, Harrist RB, Hadlock FP, Carpenter W: Fetal head and abdominal circumferences: 11. A critical re-evaluation of the relationship to menstrual age. J Clin Ultrasound 10:365-372, 1982.
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24. Warda.A, Deter RL, Rossavik IK, Carpenter RJ, Hadlock FP. Fetal femur length: A critical reevaluation of the relationship to menstrual age. Obstet Gynecol 66:69-75, 1985. 25. Deter RL, Hill RM, and Tennyson LM: Predicting the birth characteristics of normal fetuses 14 weeks before delivery. J Clin Ultrasound 17:89-93, 1989.
26. Elston RC, Grizzle JE: Estimation of timeresponse curves and their confidence bands. Biometrics 18:148- 159, 1962. 27. Deter RL, Harrist RB: Assessment of normal fetal growth. In: Chervenak FA, Isaacson G, Campbell S teds): Textbook of Ultrasound in Obstetrics and Gynecology, Boston, Little, Brown 1992 (in press).
JOURNAL
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CLINICAL ULTRASOUND
Growth Standards for Anatomic Measurements and Growth Rates Derived from Longitudinal Studies of Normal Fetal Growth Russell L. Deter, MD,* and Ronald B. Harrist, MD,?
Abstract: A statistical procedure for deriving growth standards for anatomic measurements and their growth rates from longitudinal studies of fetal growth was evaluated using Rossavik growth models for the biparietal diameter (BPD), head circumference (HC), abdominal circumference (AC), and femur diaphysis length (FDL) determined in a previous study of normal fetal growth. For each anatomic parameter, the coefficients c and s of the model was used to define a set of growth curves that constituted the boundary growth curves of a region containing 95% of the growth curves of this data set. The set of boundary growth curves was used to specify the mean, lower limit, and upper limit values for the anatomic parameter and its growth rate at weekly intervals between 14 and 38 weeks, menstrual age. Comparison of these values to those determined from cross-sectional studies of fetal growth gave differences of -1.9% to 4.8% (SD: iO.9 to i 2 . 6 ) for mean vs. predicted value of the anatomic measurements. For the lower limit, similar values were 0.4%to 13.8%(SD: 51.7 to k8.8); for the upper limit the values were 8.3%to 18.0% (SD: 21.5 to i7.0). Comparisons of HC growth rates determined using polynomial and Rossavik growth models gave values of -3.4% (SD: 24.4)for mean vs. predicted value, -12.6% (SD: i10.6) for the lower limit and 5.2% (SD: i9.3) for the upper limit. The degree of agreement was similar for AC growth rates. These results indicate that reasonable growth standards for anatomic measurements and their growth rates can be determined from longitudinal studies of as few as 20 normal fetuses, although better estimates of normal variability could be obtained with a larger sample. Indexing Words: Rossavik growth model * Fetal growth Growth rates, fetal
The assessment of fetal growth during pregnancy has long utilized comparisons of individual measurements to growth curve standards derived primarily from cross-sectional studies of fetal growth.' In most instances anatomic measurements are used in these comparisons but data has been presented which claimed to provide the standards needed for making similar comparisons of growth rates for the biparietal diameter, head circumference (HC), abdominal circumference (AC), and femur diaphysis length From the *Department of ObstetricslGynecolog,Baylor College of Medicine, Houston, Texas, and the ?Department of Biometry, University of Texas School of Public Health, Houston, Texas. For reprints contact Russell L. Deter, MD, Department of Obstetrics/Gynecolog,Baylor College of Medicine, Houston, TX 77030.
(FDL).2-7 Growth curve standards derived from cross-sectional studies depend on the assumption that the sample studied is composed of a single group of normally growing fetuses having similar growth trajectories which show only random variation.8 However, rarely is data presented which justifies this assumption. As pointed out previously,' longitudinal studies of growth can provide the information needed to establish that one is studying a single group of normally growing fetuses. However, standard methods of analysis do not permit estimation of the normal variability for individual measurements unless all fetuses are studied at the same time points during pregnancy: a difficult task in the human fetus. Recently, it has been possible to remove this restriction through use of a modi38 1
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mean values reported in a previous publi~ation’~ fication of a statistical method for defining tolerwere used in the regression analyses. Thus the ance regions.“,” This approach requires only growth model for each parameter in each fetus that there be a single mathematical model that was specified by the values of c and s obtained by adequately describes the growth of the anatomic regression analysis. parameter in all fetuses studied. To obtain For each parameter the pairs of c and s values growth rate standards, the first derivative of this for the 20 fetuses studied were used to determine function with respect to fetal age must also be normal ranges at different time points for both known. As has been shown p r e v i o u ~ l y , ~ the ~-~~ anatomical measurements and growth rates. Rossavik growth model” is a mathematical Briefly, this procedure first specifies the location function which has these necessary properties. of each growth curve in “coefficient space,’’ the This investigation presents the results of apaxes of this space being the coefficients c and s, plying this method to Rossavik models derived as a point having coordinates given by the c and from longitudinal studies of one-dimensional pas values of its Rossavik growth model. The c and rameters. Growth standards for both anatomic s values for the mean growth curve were determeasurements and growth rates were determined from the sample of c’s and s’s as the avermined and compared to those obtained previage of the c and s coefficients, with each c and s ously by other methods. value weighted by the inverse of the appropriate variance- covariance matrix determined for that MATERIALS AND METHODS individual.21 This procedure takes into account differences in the number of time points studied Data Set and their distribution during pregnancy as well as measurement and fitting errors. The c and s This investigation was carried out using data values of the average growth curve specify the obtained by ultrasound from 20 normal fetuses coordinates of the centroid of the cloud of points delivered at term as described in detail previin “coefficient space” which represent the indi0us1y.’~Briefly, fetuses judged to be normal by vidual c-s pairs.lo7” Using knowledge of the detailed pediatric assessment at birth were studvariance and covariance of each c-s pair and the ied at 2 to 3 week intervals from 15 weeks, mengroup of c-s pairs, the 95% confidence ellipse strual age (MA), [initial scan: 15.2 (k1.4 SD) around the centroid can be determined using the weeks] to delivery [final scan: 38.2 (k1.5 SD) Hotellings T2 statistic, a multivariate form of weeks]. At each examination the biparietal dithe t-statistic.lo’l1Within this ellipse are 95% of ameter (BPD), head circumference (HC), abdominal circumference (AC) and femur diaphysis the points representing the growth curves of this length (FDL) were measured as described previgroup of normally growing fetuses. The perimeously6,20The fetal age at the time of each examter of this ellipse defines the boundary of the norination was determined from the first day of the mal range for the growth curves (not individual last menstrual period (LMP), confirmed by a measurements) of the parameter being studied. known date of ovulation or early ultrasound To determine the normal ranges for individual data, or by crown- rump length measurement^.'^ measurements, similar to those obtained in cross-sectional growth studies, the following procedure was carried out. Forty equally spaced Data Analysis points on the boundary of the 95% confidence ellipse were selected, and the c and s coordinate Regression analysis was carried out with the values for each point, together with the appropridata sets for each fetus to determine the optimal Rossavik growth models.16 The Rossavik model ate k value, were used to specify 40 Rossavik has the following general form’? growth models for each anatomic parameter. This set of models represents the extreme growth k + dt) curves for the parameter being studied. Two of P = c(t) these growth curves (but not necessarily the where P is the anatomical parameter studied, t same two) specify the maximum and minimum the duration of growth, and c, k , and s the model values for the anatomic measurement at any coefficients. Previous studies have indicated that given time point. These 40 models were used to the coefficient k is specified by the anatomic pagenerate 40 predicted values at weekly intervals rameter being studied and should be the same from 14 weeks to 38 weeks, MA. From these for all individuals. l2 The coefficient k was considdata, the mean, maximum, and minimum values ered a constant in these investigations, and the were determined. The minimum value was taken JOURNAL OF CLINICAL ULTRASOUND
GROWTH STANDARDS DERIVED FROM LONGITUDINAL STUDIES
as the lower limit of the normal range and the maximum value as the upper limit. The procedure for obtaining normal ranges for growth rates from longitudinal studies was essentially the same as that just described for individual measurements except that the following equation for growth rate, derived from the Rossavik model, was used in place of the Rossavik model itself:
_@
dt -
'
'7
s
+ s In ( t )P~
data were compared with that obtained in a previous longitudinal study using polynomial mode l ~ ~because , ~ , ~valid data from cross-sectional studies of growth rates are not available. Differences in values (longitudinal - cross-sectional) were expressed as a percentage of the value obtained in the cross-sectional study tor previous longitudinal study in the case of the head and abdominal circumference growth rates). These percentage differences were averaged and their variability assessed by calculation of standard deviations and ranges. It should be noted that the predicted values used in these calculations are not independent, being completely determined by the longitudinal and the cross-sectional regression models, respectively. For this reason, statistical testing of differences is not possible. The means, standard deviations, and ranges presented serve as summaries by which to judge the similarity of the growth curves derived from longitudinal and cross-sectional data.
(2)
+
The value of P at different time points was determined using the appropriate Rossavik model. The data obtained were the instantaneous growth rates at specific times in pregnancy. Evaluation of the mean, maximum, and minimum values obtained from the longitudinal data set was carried out by comparison with predicted, lower limit, and upper limit values (values that defined a range including 95% of the measurements) obtained in previous crosssectional growth st~dies.''-~~Head circumference and abdominal circumference growth rate
RESULTS
Comparisons of standard growth curves for BPD, HC, AC, and FDL, obtained from both the longi-
TABLE 1 Comparison of Predicted Values (Cross-Sectional Data Set) and Mean Values (Longitudinal Data Set) for Various Anatomic Parameters) Percentage Difference" Paramete?
NCsb
NLb
Nb
Mean
SD
YO
%
533 252 252 254
20 20 20 20
23 23 23 23
% -1.9 1.o 4.8 2.8
Y O
BPD HC AC FDL
k1.4 kO.9 kl.l k2.6
-0.6 1.8 5.9 5.1
-5.2 -1.9 1.4 -2.7
Maximum
Minimum
aBPD = biparietal diameter; HC = head circumference; AC = abdominal circumference; FDL = femur diaphysis length. bN,s = sample size in cross-sectional study; NL = sample size in longitudinal study; N = number of time Doints evaluated. (longitudinal prediction at t - cross-sectional prediction at t ) CPercentagedifference at age t= x 100. (cross-sectional prediction at t )
TABLE 2 Comparison of Lower Limits of Cross-Sectional and Longitudinal Standard Growth Curves for Various Anatomic Parameters Percentage Difference ~
~~
~
~
Parameter'
Ncs
N,
N
Mean
SD
Maximum
Minimum
BPD HC AC FDL
533 252 252 254
20 20 20 20
23 23 23 23
% -13.8 -5.1 0.4 -8.2
% kl.7 k2.6 14.3 18.8
% -11.9 -2.1 4.3 3.1
% -17.7 -8.4 -9.3 -22.8
Abbreviations the same as in Table 1. VOL. 20, NO. 6, JULYiAUGUST 1992
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TABLE 3 Comparison of Upper Limits of Cross-Sectional and Longitudinal Standard Growth Curves for Various Anatomic Parameters ~-
Percentage Difference Parameter
Ncs
N,
N
Mean %
YO
%
%
BPD HC AC FDL
533 252 252 254
20 20 20 20
23 23 23 23
12.0 8.3 11.6 18.0
21.5 22.1 k3.2 27.0
14.1 11.2 16.0 26.8
6.7 6.0 7.3 6.2
S.D.
Maximum
Minimum
Abbreviations the same as in Table 1.
cross-sectional data sets, this difference being greatest for AC. Similar comparisons of the lower limits of the normal range are presented in Table 2. On average (except for AC) the lower limits determined from the longitudinal data sets were smaller than those found using the cross-sectional data sets, these differences being approximately 10% or less. In some cases differences of 15% to 20% (BPD, FDL) were seen, but these occurred before 20 weeks where small differences (0.5 cm to 0.6 cm) were more important due to the small measurements (c4.0cm). Results of the comparisons of the upper limits (Table 3) were similar to those for the lower limits except that the values derived from the longi-
tudinal and cross-sectional data sets, are presented in Tables 1, 2, and 3. The age range was limited to 16 to 38 weeks because the crosssectional data sets had very few measurements before 16 weeks (520) and Rossavik models do not specify growth well after 38 weeks because of the growth cessation that occurs in normal fetuses.25 As can be seen in Table 1,there is good agreement between the predicted values specified by functions derived from the cross-sectional data sets and the mean values obtained from the longitudinal data sets. No percentage difference was greater than 6%. On average (except for BPD) the mean values from the longitudinal data sets were larger than the predicted values from the
TABLE 4 Growth Standards for Biparietal Diameter and Head Circumference Growth Rates Menstrual Age wk 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
38
Biparietal Diameter
Head Circumference
Lower Limit
Mean
Upper Limit
Lower Limit
Mean
Upper Limit
cmlwk 0.3
cmlwk 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2
cm/wk 0.4 0.4 0.4 0.4
cmlwk 1.2 1.2 1.2 1.2 1.2
cmlwk 1.4 1.4 1.4 1.4 1.4
cmlwk
1.2
1.3
1.5
1.1 1.1 1.1 1.1 1.o 1.o 0.9 0.8 0.8 0.7 0.7 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2
1.3 1.3 1.2 1.2 1.2 1.1 1.1 1.o
1.5 1.4 1.4 1.4 1.3 1.3 1.3 1.2 1.2 1.2 1.1 1.1 1.1 1.o 1.o
0.3 0.3 0.3 0.3
0.3 0.3 0.3
0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.1 0.1 0.1
0.1 0.1 0.1
0.4 0.4 0.4 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.3 0.3
0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3
1.o
0.9 0.9 0.8 0.8
0.8 0.7 0.7 0.6 0.6 0.5
1.7 1.7 1.6 1.6 1.6
1 .o
1.o 0.9 0.9
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tudinal data sets were larger than those derived from the cross-sectional data sets. The largest differences were found for FDL, particularly below 26 weeks where differences of 21% to 26% were seen. Also, the upper limit for AC derived from the longitudinal data set was, on average, 11.6% larger than that derived from the crosssectional data set, but the lower limits were almost the same. Growth rate standards for various parameters, derived from the longitudinal data sets, are given in Tables 4 and 5. The data are the instantaneous growth rates at various times in pregnancy. The mean values, together with minimum (lower limit) and maximum (upper limit) values, are presented. Differences in the mean value, lower limit, and upper limit for the head circumference growth rate were 0.1 cm to 0.2 cm/wk when the data obtained using polynomial growth models were compared to that obtained with Rossavik growth models. Expressed as percentages, these differences over the 12 to 38 week time period were -3.4% (k4.4 SD) for the predicted value, -12.6% (k10.6 SD) for the lower limit, and 5.2% (29.3 SD) for the upper limit. The larger percentage errors occurred toward the end of pregnancy (after 3 1 weeks) when the HC growth rates decreased.
385
Comparison of AC growth rates derived from the Rossavik models with those derived from polynomial modeling of individual growth curves was affected by selection of the linear model in the latter investigations. This model specifies a constant expected growth rate and growth rate variability. As seen in Table 5, the mean AC growth rate was 1.2 cm/wk through 29 weeks then became 1.1cm/wk to 38 weeks. Estimates for the expected AC growth rates were 1.1 cm/wk5 and 1.2 cm/wk7 in the two studies published to date. The lower limit in both these latter studies was 1.0 cmlwk; the upper limit was 1.3 cmiwk in the first study and 1.4 cm/wk in the second. These values are in good agreement with the data in Table 5 except for the lower limit after 33 weeks. These lower values may be indicative of the fact that the AC growth reaches a plateau toward the end of the third trimester in some f e t ~ s e s . ~ DISCUSSION
To evaluate the validity of this approach to growth curve specification, we have compared our results to empirical cross-sectional growth curves for the anatomic parameters studied. Such growth curves provide the most direct information on the variability seen in anatomic parameters during pregnancy. The growth curves
TABLE 5 Growth Standards for Abdominal Circumference and Femur Diaphysis Length Growth Rates Menstrual Age wk 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38
Abdominal Circumference
Femur Diaphysis Length
Lower Limit
Mean
Upper Limit
Lower Limit
Mean
Upper Limit
cmlwk 1.o 1.o 1.o 1.o 1.o 1.o 1.o 1.o 1.o
cmlwk 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1 1.1
cmlwk 1.5 1.5 1.5 1.5 1.5 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.4 1.5 1.5 1.5 1.5 1.5 1.6 1.6 1.6
cmlwk 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0.0 0.0 0.0
cmlwk 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.2 0.2 0.2
cmlwk 0.5 0.5 0.5 0.5 0.4 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3
1.o 1.o 1.o 1.0 1.o 0.9 0.9 0.9 0.8 0.8 0.8 0.7 0.7 0.7 0.6 0.6
VOL. 20, NO. 6, JULYIAUGUST 1992
0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.1
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DETER AND HARRIST
used in these comparisons were obtained from valid cross-sectional investigations.' As shown in Table 1, there was a remarkable degree of agreement between the mean growth curves derived from the longitudinal data sets and the expected growth curves derived from the crosssectional data sets. Such findings are consistent with our previous work that has shown great similarities in expected growth curves derived from cross-sectional and longitudinal studies, the latter determined from functions whose coefficients were averages of the coefficients obtained in longitudinal studies of individual These results constitute strong empiric support for the validity of the method used in this investigation. A more rigorous test of this method is its ability to predict normal variability. In view of the fact that only 20 fetuses were studied longitudinally, one would not expect the degree of agreement between estimates of the upper and lower limits of normal variability to be as good as that found between the mean and predicted growth curves. As seen in Tables 2 and 3, this was indeed the case, but the degree of agreement was still good except for the upper limit of the FDL. In the latter situation, most of the problems were with the values for the early part of pregnancy where, because of the short length of the FDL, a 0.5 cm difference represented a significant percentage difference. In almost all cases, the values derived from the longitudinal studies were more conservative, being lower for the lower limit and higher for the upper limit. These results indicate that, although more accurate estimates of normal variability may be obtained from a larger sample of fetuses studied longitudinally, the estimates provided by the current sample are quite reasonable. This again supports the validity of this method for determining standard growth curves. Although there are advantages associated with being able to determine standard growth curves for various anatomic parameters using only a small number of fetuses studied longitudinally, the primary importance of this approach is its ability to provide standards for growth rates. Growth rates are direct indicators of fetal growth between time points and are more sensitive detectors of growth abnormalities because they can be abnormal even when individual values are within the normal range. Establishing growth rate standards using cross-sectional methods requires data sets with the same characteristics as those used for determining growth standards for the anatomic parameters themselves. These in-
clude (1) only one pair of measurements (one growth rate calculation) per patient and (2) uniform distribution of occasions of measurement over the time interval of interest. This is to assure independence among measurements and representativeness of the sample so that valid estimates of normal growth rate variability can be obtained.26 The assumptions concerning differences among individuals made in studies of the anatomic parameters themselves must also be valid for growth rates. To date, a cross-sectional data set with these characteristics has never been studied and acquiring such a data set would be difficult. Most previous publications purporting to give values for average growth rates and normal growth rate variability have used multiple pairs of measurements from the same fetus, and the same measurements have been used to calculate more than one growth rate Such data sets do not provide growth rate measurements that are independent; thus variability estimators derived from these data sets are biased.26 In the approach used in this investigation, these problems are avoided by applying the statistical analysis to the entire longitudinal growth curve (as represented by the coefficients c and s), rather than individual measurements or even pairs of measurements. Since there is only one curve per fetus, independence is assured. When the set of extreme curves is identified, the normal variability of any parameter (including growth rate) can be evaluated using this set. At any given time point, parameter values derived from two (but not necessarily the same two) curves of this set will be the extreme values. Because the set of extreme curves defined in this investigation were determined from individual growth curves from normally growing fetuses, these extreme values can logically be considered to represent the range of normal variability. The reasonably good agreement between upper and lower limit values derived from cross-sectional and longitudinal growth studies of anatomic parameters supports this conclusion. The similarity of upper and lower limits for HC and AC growth rates derived from polynomial and Rossavik growth models is additional support. It should also be noted that despite their questionable validity, the normal growth rate ranges published p r e v i ~ u s l y ~are - ~ very similar to those given in Table 4 and 5, except for the data of F e ~ c i n a . ~ In using the data in Tables 4 and 5, it is important to realize that the values given are instantaneous growth rates at the menstrual ages JOURNAL OF CLINICAL ULTRASOUND
GROWTH STANDARDS DERIVED FROM LONGITUDINAL STUDIES indicated. However, a growth rate estimate calculated from measurements at two time points is the average growth rate for the time interval between scans. To obtain appropriate values for evaluating such average growth rates, it is necessary t o first determine the upper and lower limit values at the time points defining the time interval. Since the change in values over weekly intervals is small, w e use linear extrapolation for time points that do n o t fall o n an exact week. T h e average of e a c h p a i r of values (e.g., lower limit) is then determined and these averages taken as the upper and lower limits of the normal range against which the measured average growth rate for the time interval is compared. The results of this s t u d y have demonstrated the validity of this statistical method for obtaining important clinical information a b o u t fetal growth from longitudinal data sets. Preliminary studies have indicated that this method c a n be applied to two- and three-dimensional paramet e r and ~ ~ should ~ be applicable t o a variety of mathematical models of growth including those with more than two coefficients. In the c u r r e n t investigation, this method has provided specific information o n the normal variability of growth rates for four anatomic p a r a m e t e r s that are important in the evaluation of fetal growth. S u c h information should make a valuable contribution t o the detection and characterization of growth
abnormalities.
REFERENCES 1. Deter, RL. Evaluation of studies of normal growth. In: Quantatiue Obstetrical Ultrasonography. New York, Wiley, 1986, pp 65-112. 2. Campbell S, Neuman GB: Growth of the fetal biparietal diameter during normal pregnancy. J Obstet Gynaecol Br Commonw 78:513-519, 1971. 3. O’Brien GD, Queenan JT: Growth of the ultrasound fetal femur length during normal pregnancy. Am J Obstet Gynecol 141:833-837, 1981. 4. Fescina RH, Ucieda FJ, Cordano MC, Neito F, Fenzer SM, Lopez R Ultrasonic patterns of intrauterine fetal growth in a Latin American country. Early H u m Deu 6:239-248, 1982. 5. Deter RL, Harrist RB, Hadlock FP, Poindexter AN: Longitudinal studies of fetal growth with the use of dynamic image ultrasonography. Am J Obstet Gynecol 143:545-554, 1982. 6. Deter RL, Hadlock FP, Harrist RB: Evaluation of fetal growth and the detection of intrauterine growth retardation. In Callen PW (ed.): Ultrasonography in Obstetrics and Gynecology. Philadelphia, WB Saunders, 1983, pp 113-140. 7. Deter RL, Harrist RB, Hadlock FP, Cortissoz CM, VOL. 20, NO. 6, JULY/AUGUST 1992
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Batten GW: Longitudinal studies of fetal growth using volume parameters determined with ultrasound. J Clin Ultrasound 12:313-32, 1984. 8. Deter RL, Harrist RB, Hadlock FP, Carpenter RB: The use of ultrasound in the assessment of normal fetal growth. J Clin Ultrasound 9:481-493, 1981. 9. Laird NM, Ware JH: Random-effect models for longitudinal data. Biometries 38:963-974, 1982. 10. Guttman I: Statistical tolerance regions: Classical and Bayesian. In: A Stuart (ed): Griffin’s Monograph #26, Darien, Connecticut, Hafner Publishing Co, 1970, pp 3-149. 11. Guttman I: Statistical tolerance regions. In: Kotz S, Johnson NL (eds): Encyclopedia of Statistical Sciences, vol 9, New York, Wiley 1988, pp. 272287. 12. Deter RL, Rossavik IK, Harrist RB, Hadlock FP. Mathematical modeling of fetal growth: Development of individual growth curve standards. Obstet Gynecol68:156-161, 1986. 13. Rossavik IK, Deter RL, Hadlock FP: Mathematical modeling of fetal growth: 111. Evaluation of head growth using the head profile area. J Clin Ultrasound 15:23-30, 1987. 14. Rossavik IK, Deter RL, Hadlock FP: Mathematical modeling of fetal growth: IV. Evaluation of trunk growth using the abdominal profile area. J Clin Ultrasound 15:31-36, 1987. 15. Deter RL, Rossavik IK, Cortissoz C, Hill RM, Hadlock FP: Longitudinal studies of thigh circumference growth in normal fetuses. J Clin Ultrasound 15:388-393, 1987. 16. Deter RL, Rossavik I K A simplified method for determining individual growth curve standards. Obstet Gynecol 705301-805, 1987. 17. Deter RL, Rossavik IK, Harrist RB: Development of individual growth curve standards for estimated fetal weight. I. Weight estimation procedure. J Clin Ultrasound 16:215-226, 1988. 18. Rossavik IK, Deter R L Mathematical modeling of fetal growth. I. Basic principles. J Clin Ultrasound 12:529-533, 1984. 19. Deter RL, Rossavik IK, Hill RM, Cortissoz CM, Hadlock F P Longitudinal studies of femur growth in normal pregnancies. J Clin Ultrasound 15:299305, 1987. 20. Hadlock FP, Deter RL, Harrist RB, Park SK: Fetal biparietal diameter: Rational choice of the plane of section for ultrasonic measurement. A J R 138:871-874, 1982. 21. Gumpertz M, Pantula SG A simple approach to inference in random coefficient models. American Statistician, 43:203- 210, 1989. 22. Hadlock FP, Deter RL, Harrist RB: Fetal biparieta1 diameter: A critical re-evaluation of the relation to menstrual age using realtime ultrasound. J Ultrasound Med 10:97- 104, 1982. 23. Deter RL, Harrist RB, Hadlock FP, Carpenter W: Fetal head and abdominal circumferences: 11. A critical re-evaluation of the relationship to menstrual age. J Clin Ultrasound 10:365-372, 1982.
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24. Warda.A, Deter RL, Rossavik IK, Carpenter RJ, Hadlock FP. Fetal femur length: A critical reevaluation of the relationship to menstrual age. Obstet Gynecol 66:69-75, 1985. 25. Deter RL, Hill RM, and Tennyson LM: Predicting the birth characteristics of normal fetuses 14 weeks before delivery. J Clin Ultrasound 17:89-93, 1989.
26. Elston RC, Grizzle JE: Estimation of timeresponse curves and their confidence bands. Biometrics 18:148- 159, 1962. 27. Deter RL, Harrist RB: Assessment of normal fetal growth. In: Chervenak FA, Isaacson G, Campbell S teds): Textbook of Ultrasound in Obstetrics and Gynecology, Boston, Little, Brown 1992 (in press).
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