Expansion of the Human Embryonic Brain During Rapid Growth: Area Analysis

Authors


Abstract

This report focuses on growth of the brain of the early human embryo, Carnegie stages 12–23. Areas of median sections from 50 to 58 embryos were measured to determine the best mathematical model to describe growth of the three primary brain vesicles and to determine the change in the ratio of tissue to cavity areas (T/C). An exponential model best describes growth of the brain and head during this time period. The head expands 248-fold compared with a 171-fold growth of the brain. The whole brain, forebrain, and midbrain all exhibit larger cavities than tissue initially followed by a reversal of such at a critical time (stages 21–24). The presumptive cerebellar tissue which was twice the cavity initially grows to become more than six times the cavity. Boxplots of the T/C ratios for the head and brain plus its components reveal that initially the tissue is less than the cavity (10–20% and 40–60%, respectively) but eventually becomes larger (60–200%). Anat Rec, 292:472–480, 2009. © 2009 Wiley-Liss, Inc.

The brain and spinal cord or the central nervous system (CNS) first becomes apparent between the fourth and eighth weeks of development in the human (Desmond and O'Rahilly,1981; O'Rahilly and Muller,2006). Human embryology texts feature the elaborate changes in morphology that occur in the CNS during this time period but are remiss in conveying the tremendous growth of the embryonic CNS during these 4 weeks (Moore and Persaud,2008a,b; Sadler,2006). In fact, only one article exists in the literature to this day that documents such growth. This was a study reported more than 25 years ago by Desmond and O'Rahilly in which they measured the major axes of the three embryo brain vesicles. In that study, they compared the growth of the brain during the embryonic period, 4 to 8 weeks, with the fetal period, 8 weeks to birth. The study was limited to measuring one-dimensional lengths of the three major axes of the three embryo brain vesicles, namely, the bi-temporal (ear to ear) measurement, the frontal-occipital (forehead to back-of-head) measurement, and the dorsal-ventral (top-of-head to base) measurement. These findings, based on using one-dimensional axes, showed that the rates of growth of all three embryonic brain vesicles was much greater than the corresponding vesicles of the fetal period (Desmond and O'Rahilly,1981).

Interestingly, many of the current texts of human embryology draw attention to some of the more prominent abnormalities of the central nervous system such as hydrocephalus, spina bifida, and the Chiari type II malformation (Moore and Persaud,2008a,b; Sadler,2006). These same texts make no mention of the growth of the CNS and how mechanisms of growth might well play a role in these malformations.

Because of the lack of attention to growth as distinct from morphogenesis of the early embryonic brain in the human, we decided to elaborate upon our earlier study. We also thought that such an investigation was timely, because there has been so much interest in modeling head and brain growth during early fetal and postnatal years (Deter and Rossavik,1987; Deter and Harrist,1992; Biagiotti et al.,1994). Our analysis focuses on the embryo during the first 2 months of the first trimester and can thus provide a baseline for future fetal and postnatal modeling studies.

In the intervening years since this initial study, more data and more sophisticated, computer-based analyses have become available, enabling us to elaborate on our initial findings. In particular, we have been able to create two-dimensional models of this human embryonic brain growth. With our two-dimensional models, we were able to compare the rate of growth of the tissue to the rate of growth of the cavity for each of the three components of the brain, as well as for the whole brain. This greatly expands our understanding of brain growth during the embryonic period and thus provides unique data for the many fetal and natal studies to use in the future.

This new analysis demonstrated that an exponential model or a power function model is the most appropriate way to describe the growth of the whole brain and of its individual parts. Moreover, we found that initially, the cavities grow faster, but, at critical times, the tissue grows faster than the corresponding cavity.

The goals of our analysis were: (1) to determine mathematical models that describe the growth of the human embryo head, brain, and three components of the brain (forebrain, midbrain, hindbrain/presumptive cerebellum), including the tissue and cavity of each; and (2) to examine the changes in the ratio of the area of the tissue to the area of the cavity during this period of human embryo growth for the head, brain, and three components of the brain.

We were able to show: (1) the exponential model or the power function model is appropriate to describe all features of head and brain growth in the human embryo that we measured; (2a) the tissue area appears to increase relative to the cavity area, but the tissue is always less than the cavity for the human embryo head, and (2b) the tissue appears to start out smaller than the cavity, but eventually it appears to become larger than the cavity for the brain and its three components.

MATERIALS AND METHODS

Area Measurements

Area tracings were made from median sections of 50–58 human embryos of the Carnegie Collection housed at the time (summer 1979) at the University of California, Davis. At the time the data for this study was collected, the Carnegie Collection was the largest collection of staged human embryos in the world and considered the “gold standard” of human embryo collections. Stages of the Carnegie embryos were based on the criteria published by O'Rahilly and Muller (1987). These same stages are now found online at virtualhumanembryo. lsuhsc.edu/HEIRLOOM/asp/SearchEmbryos.asp. However, it needs to be pointed out that there is overlap of stages based on morphological characters with real time. That is, a stage 12 can be considered 26 days old and a stage 13, 28 days old if using the criteria of Oliveir and Pineau, or a stage 12 according to Jirasek. However, according to Jirasek (1978), a stage 12 would be 26–30 days old and a stage 13 would be 28–32 days old. These criteria are cited in O'Rahilly (1973). It is important to realize that the benefit of staging is that embryos having identical morphological features are grouped together. This is particularly important when individuals are being grouped to make comparisons of size over time. It is equally important to realize, that within a given stage, the actual sizes of the embryos may vary.

Only embryos that exhibited good to excellent fixation were employed. The embryos ranged from stage 12 to stage 23 (approximately 24–57 postovulatory days or the first trimester), and measured from 3 to 30 mm in C.-R. length. This period of time is considered the embryo proper time and precedes the fetal time that begins at the third month. The age in days corresponding to the Carnegie stages are as follows: stage (s) 12: 26–30 days; s13: 29–32 days; s14: 31–35 days; s15: 35–38 days; s16: 37–42 days; s17: 42–44 days; s18: 44–48 days; s19: 48–51 days; s20: 51–53 days; s21: 53–54 days; s22: 54–56 days; s23: 56–60 days.

The sections were traced using a standard camera lucida attached to a Zeiss compound research microscope. No correction was made for shrinkage. As reported in an earlier study using the same embryos, the difference in shrinkage for 93% of the brains used in the study was only 1% (Desmond and O'Rahilly,1981). Finally, the areas were integrated using a Keuffel & Esser polar planimeter.

The median section was chosen as the basis for comparing growth of the embryonic brain over time. The median section is easily recognized for all stages at this time in development, and this is why it was chosen. Equally important is that it is easy to identify the morphological boundaries separating the three brain vesicles. The boundaries between the three primary brain vesicles were established by drawing a perpendicular line connecting the dorsal and ventral primary fissures separating the prosencephalon from the mesencephalon and the mesencephalon from the rhombencepalon. The rostral border of the otocyst was used as the caudal border for the hindbrain.

Consequently, the measurements reflect only two-dimensional growth and not three-dimensional growth. Granted, such measurements are not as ideal as three-dimensional, volume measurements, but they do give some idea as to the amount of increase in size of the brain during the early period of development. As the measurements were made at the site of the collection, time constraints prevented tracing more than the median sections.

The median areas were measured for the embryo head, brain, and three brain components: forebrain, midbrain, and hindbrain/presumptive cerebellum. For each, the whole organ and its tissue and cavity were measured. As the measurements were made from embryo sections of stage 12 through stage 23, it was necessary to trace the sections at different magnifications. Consequently, corrections were made to account for these differences.

We present the data for all measurements generically in the style: organ whole, organ cavity, and organ tissue. These data have the relationship: organ whole = organ tissue + organ cavity. Specifically, we use the following notation: head whole (hw), head cavity (hc), head tissue (ht), brain whole (bw), brain cavity (bc), brain tissue (bt), forebrain whole (fw), forebrain cavity (fc), forebrain tissue (ft), midbrain whole (mw), midbrain cavity (mc), midbrain tissue (mt), cerebellum whole (cw), cerebellum cavity (cc), and cerebellum tissue (ct). We apply this relationship to the two major organs, head and brain, as well as to the three brain subdivisions: forebrain, midbrain, and cerebellum. We also note that head cavity is the same as brain whole (hc = bw).

Mathematical Modeling and Statistical Analysis

Least squares regression is typically used to find an approximate model relating continuous or discrete independent variables with a continuous dependent variable (Draper and Smith,1981). For the given data set, the independent variable can only take on integer values and, hence, the derived models should only be evaluated in the domain of this discrete variable. The pattern of the data suggested exponential, power function, or logistic models. The analysis of variance (ANOVA) derived from the regression procedure generating the models determined how significantly these models explained the variation in the data. When the logistic model (three parameters) was compared with the exponential and power function models (two parameters each) for all data sets, the logistic model resulted in at most a negligible improvement at the cost of an additional parameter. Therefore, it was decided to only consider the exponential model, y = aebx and the power function model, y = cxd. The independent variable x was the Carnegie stage (12, … , 23), and the dependent variable y was the area of the median section of the organ, tissue or cavity being considered. All derived ANOVA tables had P < 0.0001, and all constants a, b and c, d associated with these models had P < 0.001. This suggested that these models and their associated parameters were statistically significant. An analysis of the residuals also supported the choice of these models. The square of the correlation coefficient r2 is a measure of how well these models explain the variation in the data, where a maximum value of 1.0 represents a perfect fit. (See Table 1 for a summary of results.)

Table 1. Models of human embryo head and brain growth, stages 12–23a
RegionnExponential: y = aebxPower: y = cxd
r2a × 10−2b × 10−1r2cd
  • a

    x = stage 12, …, 23; y = area (mm2) of the median section of the region; n = sample size; r2 = fit of model; significance of all models, P < 0.0001; significance of all estimated parameters a and b, c and d, P < 0.001.

Head whole (hw)520.972.854.630.988.66 × 10−98.11
Head cavity (hc) = bw520.963.334.370.982.26 × 10−87.68
Head tissue (ht)520.950.155.470.973.02 × 10−119.59
Brain whole (bw) = hc520.963.334.370.982.26 × 10−87.68
Brain cavity (bc)520.896.523.740.932.88 × 10−76.63
Brain tissue (bt)520.930.195.320.937.85 × 10−119.24
Forebrain whole (fw)580.950.744.940.979.63 × 10−108.61
Forebrain cavity (fc)580.881.774.190.921.98 × 10−87.39
Forebrain tissue (ft)580.930.026.150.929.61 × 10−1310.58
Midbrain whole (mw)580.912.923.840.941.22 × 10−76.71
Midbrain cavity (mc)580.804.213.370.846.71 × 10−75.95
Midbrain tissue (mt)580.870.324.440.872.44 × 10−97.67
Cerebellum whole (cw)520.840.034.950.843.74 × 10−118.63
Cerebellum cavity (cc)500.760.083.830.772.99 × 10−96.74
Cerebellum tissue (ct)520.830.015.220.836.56 × 10−129.08

As the corresponding exponential and power function models produce similar graphs over the given domain, the exponential function may be used to illustrate the behavior of both. There are three exponential growth models presented for each organ, whole, tissue and cavity (see Figs. 1a–2c). To determine if there were significant differences in the data producing these curves, a number of procedures were implemented. As the tissue and cavity measurements were from the same organ, these were not independent observations. Therefore, for each of the five organs (head, brain, forebrain, midbrain, and cerebellum), both the difference, tissue minus cavity, and the quotient, tissue divided by cavity, were analyzed. An ANOVA was performed on each difference and on each quotient to see if the means for each were the same across all stages. When considering the differences for the head, brain, forebrain, and cerebellum, all with P-values < 0.001, this suggested that the means of the differences were not all equal across stages. For the midbrain (P = 0.0924), equal means could not be rejected. The quotients for the five organs, all with P-values < 0.0076, suggested that the means of the quotients were not all equal.

Figure 1.

(a) Exponential models of the growth of regions of the head in the human embryo. The symbols with the structures they represent are: ▪ Head Whole Mean, ○ Head Cavity Mean, ▴ Head Tissue Mean, equation image Exponential Model for Head Whole, equation image Exponential Model for Head Cavity, equation image Exponential Model for Head Tissue. (b) Exponential models of the growth of regions of the brain in the human embryo. The symbols with the structures they represent are: ▪ Brain Whole Mean, ○ Brain Cavity Mean, ▴ Brain Tissue Mean, equation image Exponential Model for Brain Whole, equation image Exponential Model for Brain Cavity, equation image Exponential Model for Brain Tissue.

Figure 2.

(a) Exponential models of the growth of regions of the forebrain in the human embryo. The symbols with the structures they represent are: ▪ Forebrain Whole Mean, ○ Forebrain Cavity Mean, ▴ Forebrain Tissue Mean, equation image Exponential Model for Forebrain Whole, equation image Exponential Model for Forebrain Cavity, equation image Exponential Model for Forebrain Tissue. (b) Exponential models of the growth of regions of the midbrain in the human embryo. The symbols with the structures they represent are: ▪ Midbrain Whole Mean, ○ Midbrain Cavity Mean, ▴ Midbrain Tissue Mean, equation image Exponential Model for Midbrain Whole, equation image Exponential Model for Midbrain Cavity, equation image Exponential Model for Midbrain Tissue. (c) Exponential models of the growth of regions of the cerebellum in the human embryo. The symbols with the structures they represent are: ▪ Cerebellum Whole Mean, ○ Cerebellum Cavity Mean, ▴ Cerebellum Tissue Mean, equation image Exponential Model for Cerebellum Whole, equation image Exponential Model for Cerebellum Cavity, equation image Exponential Model for Cerebellum Tissue.

In addition, the nonparametric Jonckheere–Terpstra (J–T) test (Daniel,1990) was conducted on these differences and quotients. The J–T test is an extension of the Wilcoxon rank sum test and investigates the significance of trends for medians, in this case, across increasing stages. When the J–T test was conducted for differences, the head and midbrain (both P-values < 0.0003) revealed a significant falling trend for differences, while the cerebellum (P < 0.0001) revealed a significant rising trend for differences. There were no significant trends for differences associated with the brain or forebrain. However, when the J–T test was applied to quotients for the five organs, (all P-values < 0.0011), all indicated a significant rising trend. Furthermore, the J-T test applied to all organs for quotients, tissue/whole and cavity/whole, validated the previous trend results, namely, for all organs, there was a significant rising trend for tissue/whole (all P-values < 0.0042) and a significant falling trend for cavity/whole (all P-values < 0.0011). The results of these tests suggest that there are significant differences in growth between the tissue, cavity and whole during these stages.

RESULTS

Descriptions of Brain, Head, and Brain Component Growth

During the human embryonic period of development (weeks 4–8), the growth of the head, brain and three components of the brain, including the tissue and cavity of each, appear to follow an exponential or power function model (Tables 16, Figs. 1a–2c). From the models (Fig. 1a) and the mean area data (Table 2), the head cavity always appears to be larger than the head tissue. The approximate expansion of the head whole is 248-fold, the head cavity 171-fold, and the head tissue 1130-fold. However, the situation appears to be different for the brain whole and its three components.

Table 2. Head regions: mean areas (mm2), standard deviation areas (mm2)
nStageHead wholeHead cavity (brain whole)Head tissue
MeanSDMeanSDMeanSD
1123.870.003.560.000.310.00
21310.831.359.110.691.720.66
71417.933.5114.353.583.581.43
61527.275.1421.693.705.572.02
31651.814.6040.024.3911.792.54
51788.2316.0269.059.4319.177.18
618139.5624.93103.4816.6336.0811.28
719232.5970.56171.9459.3460.6515.85
420345.3471.63243.1254.23102.2219.56
321461.7475.68326.6372.39135.1133.59
522589.7630.57395.9817.65193.7916.42
323962.03192.08608.97198.85353.0629.43
Table 3. Brain regions: mean areas (mm2), standard deviation areas (mm2)
nStageBrain whole (head cavity)Brain cavityBrain tissue
MeanSDMeanSDMeanSD
1123.560.002.270.001.290.00
2139.110.696.610.192.500.88
71414.353.5810.793.723.561.67
61521.693.7016.503.065.202.33
31640.024.3928.169.5311.865.33
51769.059.4355.926.6213.135.46
618103.4816.6371.815.7331.6711.35
719171.9459.34106.4224.5965.5145.20
420243.1254.23147.4963.1195.6420.66
321326.6372.39157.4063.86169.2335.19
522395.9817.65204.3272.21191.6671.73
323608.97198.85186.6957.08422.29189.69
Table 4. Forebrain regions: mean areas (mm2), standard deviation areas (mm2)
nStageForebrain wholeForebrain cavityForebrain tissue
MeanSDMeanSDMeanSD
2121.400.770.920.440.480.32
2135.211.273.851.381.350.12
7147.082.055.702.041.390.67
61510.921.708.871.472.051.03
31624.772.6918.455.906.313.35
51744.489.1837.226.857.263.50
71869.0713.0950.646.4618.439.67
819116.1640.7072.3418.5043.8226.12
520151.9530.4188.5240.0163.4317.84
521238.8845.12101.3020.54137.5734.27
522290.3027.30143.0864.85147.2261.46
323433.23118.37130.9724.18302.27120.76
Table 5. Midbrain regions: mean areas (mm2), standard deviation areas (mm2)
nStageMidbrain wholeMidbrain cavityMidbrain tissue
MeanSDMeanSDMeanSD
2121.270.270.820.190.450.07
2133.691.882.681.221.010.66
7146.961.704.991.911.971.05
61510.172.047.362.112.811.11
31614.031.839.213.734.821.90
51722.743.9517.823.974.922.18
71830.234.8121.793.628.454.07
81958.8418.4836.919.7921.9315.54
52079.3521.7947.5718.9131.7815.05
521115.3868.4260.5345.3754.8533.52
52293.6111.2457.518.7336.0916.56
323141.3266.8249.8030.7591.5259.25
Table 6. Cerebellum regions: mean areas (mm2), standard deviation areas (mm2)
nStageCerebellum wholeCerebellum cavityCerebellum tissue
MeanSDMeanSDMeanSD
  • a

    For cerebellum tissue, n = 5.

1120.170.000.080.000.080.00
2130.220.070.080.030.140.10
7a140.300.200.140.040.200.13
6150.600.480.270.170.330.32
3161.220.450.500.080.720.37
5171.831.240.870.700.960.68
6183.060.521.190.541.880.78
7195.876.521.901.733.974.80
4208.362.972.221.096.142.03
3218.975.501.181.027.794.52
52212.075.193.731.988.344.20
32334.4216.215.923.2928.5013.79

The brain cavity starts out larger than the brain tissue for both the data and the models (Fig. 1b, Table 3). According to our data, the brain tissue becomes larger than the brain cavity between stages 21 and 23, and the models predict this change at approximately stage 22. The approximate expansion of the brain whole is 171-fold, the brain cavity 82-fold, and the brain tissue 327-fold.

The growth of the forebrain follows the same pattern as the whole brain. The tissue becomes larger at stage 21 while the models predict this change at approximately stage 22 (Table 4, Fig. 2a). The approximate expansion of the forebrain whole is 309-fold, the forebrain cavity 142-fold, and the forebrain tissue 630-fold.

The growth of the midbrain is also similar. The tissue exceeds that of the cavity at stage 23 and the models extrapolate this change at stage 24 (Table 5, Fig. 2b). The approximate expansion of the midbrain whole is 111-fold, the midbrain cavity 61-fold, and the midbrain tissue 203-fold. (Note that in Figs. 1b, 2a,b, the means for cavity appear to follow a sigmoid curve starting at stage 18. The logistic models did not noticeably follow this shape but continued in an upward direction.)

The data for the cerebellum does not follow the same pattern as the rest of the brain. After stage 12, the cerebellum tissue is always larger than the cerebellum cavity, and the models suggest that this occurs after stage 14 (Table 6, Fig. 2c). The approximate expansion of the cerebellum whole is 202-fold, the cerebellum cavity 74-fold, and the cerebellum tissue 356-fold.

The head and brain growth is initiated by a rapid increase in the size of the cavity which is later directed by a rapid increase in the size of the tissue. Summarizing the descriptions of the data and the models, the critical time at which the tissue appears to surpass the cavity occurs at approximately stage 22 for the whole brain and the forebrain, at approximately stage 24 for the midbrain, and at approximately stage 14 for the cerebellum. The data and the models for the developmental stages examined indicate that the degree of tissue growth for the head, brain, and its components was always many-fold greater than that for the cavity.

Brain Growth Assessed by Examining Tissue to Cavity Growth

One can get a different perspective of the growth of the median area of the tissue relative to the median area of the cavity by examining the boxplots of the ratios: tissue/cavity. The ratio of head tissue to head cavity (Fig. 3a) shows that the head tissue grows somewhat faster than the head cavity, with the tissue starting at 10–20% of the cavity and appearing to increase linearly to ∼60% of the cavity. (The ANOVA derived from the regression showed that a linear model was significant (P < 0.0001) in fitting this data, with r2 = 0.54.) The ratio of brain tissue to brain cavity (Fig. 3b) and the ratio of forebrain tissue to forebrain cavity (Fig. 4a) appear to have a similar pattern with one another. In the early stages of both, the tissue is ∼40%–50% of the cavity. This tissue to cavity ratio appears to drop to 20%–30% by stage 17 and then grows to ∼200% by stage 23. Both the brain and forebrain tissues grow to double the size of their respective cavities. The ratio of midbrain tissue to midbrain cavity (Fig. 4b) and the ratio of cerebellum tissue to cerebellum cavity (Fig. 4c) appear to start out with small variation until stage 19. After that, both start to increase with large variation until the ratios are more than twice what they were early on.

Figure 3.

(a) Boxplots of the ratio, head tissue (ht) to head cavity (hc), from stages 12 through 23 in the human embryo. The head tissue appears to grow faster than the head cavity (brain whole) during this sequence of stages. Each box contains the middle 50% of the distribution for each stage, and a horizontal line interior to a box represents the median. A circle ○ represents an outlier, and an asterisk * represents an extreme value. (b) Boxplots of the ratio, brain tissue (bt) to brain cavity (bc), from stages 12 through 23 in the human embryo. From stage 18, it appears that the brain tissue grows faster than the brain cavity until it appears to be about double the brain cavity by stage 23. It appears that the median value of bt/bc is greater than 1 starting at stage 22, so brain tissue is now larger than brain cavity.

Figure 4.

(a) Boxplots of the ratio, forebrain tissue (ft) to forebrain cavity (fc), from stages 12 through 23 in the human embryo. The pattern of growth of the forebrain tissue relative to the forebrain cavity appears to follow the pattern of growth of the brain tissue relative to the brain cavity. It appears that the median value of ft/fc is greater than 1 starting at stage 21, so forebrain tissue is now larger than forebrain cavity. (b) Boxplots of the ratio, midbrain tissue (mt) to midbrain cavity (mc), from stages 12 through 23 in the human embryo. It appears that the median value of mt/mc is greater than 1 starting at stage 23, so midbrain tissue is now larger than midbrain cavity. (c) Boxplots of the ratio, cerebellum tissue (ct) to cerebellum cavity (cc), from stages 12 through 23 in the human embryo. It appears that the median value of ct/cc is approximately 2 at stage 13, eventually growing to approximately 6 at stage 23. For this organ, tissue always appears larger than cavity.

The transition, when tissue overtakes cavity, can be visualized by seeing the position of the boxplots relative to the horizontal line with value 1 (Figs. 3b, 4a–c).

DISCUSSION

Our study shows that the exponential model or the power function model is appropriate to describe all features of head and brain growth in the human embryo that we measured. With respect to the relative increase in tissue area compared with the cavity, the tissue area appears to increase relative to the cavity area, but the tissue is always less than the cavity for the human embryo head. In contrast, the tissue appears to become larger than the cavity for the brain and its three components.

Three Methods Used to Determine Stages of Growth Transition

We presented three methods for determining the stages of transition, where tissue changes from being less than cavity to being more than cavity. Analysis using these methods: (1) averages of the areas for each stage, (2) theoretical models created with the area data, and (3) boxplots of the ratio of area of tissue to cavity, show that the actual interval of transition using the three methods differs, but the difference is small, namely, 21–24 for the brain, the forebrain, and the midbrain, and 13–14 for the cerebellum.

Relative Growth of the Three Brain Vesicles

If we compare the size of the various parts of the three sections of the brain using mean area data, the forebrain is the largest for every stage (12–23), and it grows the fastest comparing stage 23 to stage 12. The fact that the forebrain grows fastest at the outset of brain growth most likely contributes to the fact that the human cerebrum is the largest among the vertebrates along with the odontocete whales and elephants (Pearson and Pearson,1976). Others have shown for the human fetus that growth of the forebrain is greater than for the rest of the brain (Dunn,1921,1926; Jenkins,1921; Grenell and Scammon,1943). Moreover, this growth trend holds true throughout the postnatal development until the adult-sized brain is established (Hesdorffer and Scammon1935; Blinkov and Glezer,1968).

As we discussed in our earlier article (Desmond and O'Rahilly,1981), the fact that the forebrain at the onset of its growth grows at a higher rate compared with the midbrain and hindbrain makes it appear as if the brain has been programmed to set aside a large mass of tissue that will become the cerebral cortex. Such precocious enlargement of the forebrain could also be necessary to set aside cell groups for future specialization in function. On the basis of this assumption, a significant change in the embryonic growth rate of the cell primordial of the cortex could have pronounced postnatal physiological and behavioral effects.

Choice of Models and Relation to Models Used by Others

Among the various models considered for the data in Tables 26, the principal of parsimony led us to consider models with two parameters. Using the value of r2 to measure the fit of the model led us to prefer the exponential and power function models over the linear model for each data set. Many other models for growth of the human embryo and fetus reported in the literature seem to be based on the model of Rossavik and Deter (1984). They considered a model for a “basic growth parameter (P)” of the “head cube” (assessing head volume). They assumed that P = c tk + st, where c, k and s are constants and t is the time variable. Among the family of models discussed was the case where s = 0, representing a power function, P = c tk. This growth model for volume may be converted into one representing area by raising it to the two-thirds power (as alluded to in their paper), resulting in a model that is also a power function.

There is one report from Wosilait et al., (1992) which analyzes embryonic and fetal growth data over a broad range of 25–300 days. They found that polynomial and Gompertz equations provided the best fit for growth of the normal human embryo and fetus over this period. Their data include a fast growth phase followed by a slow growth phase. In contrast, our data represent the embryonic growth prior to the data that they present. Thus, an exponential model describing this earlier rapid growth phase is somewhat similar to what they did by using the Gompertz equation.

Invariably, there are always many models to consider when approximating experimental data. As a model is, in general, an attempt to understand patterns presented by the data, there may be little value in choosing one model over another when both convey similar information with basically the same degree of accuracy. One way to compare the exponential model with the power function model is to observe that the corresponding values of r2, as found in Table 1, slightly favor the power function model. Another method is to apply the Bayesian Information Criterion which reaffirms this slight preference. However, given the variability of the data, one need not really question which of these two models is better since their r2 values are so close. If one examines the pattern of the parameters in the exponential model, the coefficients are essentially the same order of magnitude. In comparison, the order of magnitude for the coefficients in the power function model varies dramatically from 10−7 to 10−13. It is this consistency of the constants found in the exponential model that leads us to prefer the exponential model over the power function model.

Acknowledgements

The authors thank Ms. Anne Kearney Spratt who compiled most of the original data for this manuscript while working on her Masters' degrees in biology and statistics at Villanova University.

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