#### 3.1. Physical Meaning of Opacity Curve

Simply put, opacity is a measure of incubated hatching egg absorbance. However, the unit is different from absorbance. Absorbance is widely used in optical sensors, but opacity is defined here by an electrical/optoelectrical sensor. Common spectroscopic techniques cannot be applied to incubated hatching eggs after day 10 of incubation because of high opaqueness. The concept of using opacity is as follows: as the embryonic components within the egg become larger during incubation, the amount of transmitted light is reduced. Consequently the opacity, that is, the ratio of input LED current over average output voltage, increases [

18]. The opacity of incubated hatching eggs during incubation increased more or less exponentially, peaking at day 15–16 before slowing (

Figure 2). During the first half of incubation (until day 10), the rate of increase was relatively slow compared to that in the second half, since the embryos are relatively small and immature during this stage, referred to as the formation phase. The opacity of an incubated hatching egg is largely influenced by embryo size, followed by allantois, yolk and yolk sac. During this formation phase, the growth rate of the embryo is much lower than that during the last half of incubation, referred to as the growth phase. Subsequent rapid growth of the embryo from day 12 to 16 of incubation is thought to be responsible for the rapid changes seen in opacity during this period.

On the other hand, embryo weight gain follows an exponential or sigmoid curve, where the rate of growth is much slower during the first half of incubation, followed by rapid growth in the second half of incubation (

Figure 2). Finally, at the very end of the incubation period, the rate of growth begins to slow down. This is a common growth phenomenon in vertebrates. The embryo weights separately observed by Byerly and Romanoff for precocial birds are very similar [

17,

19].

#### 3.3. Curve Fitting and Mathematical Growth Models

When performing curve fitting, several important points need to be considered. Firstly, the opacity value reflects growth information from all embryonic components (embryo size, yolk sac, allantois). Therefore, fitting a mathematical function may eliminate the influence of yolk sac and allantois on the opacity curve. This is because embryo growth generally follows an exponential, sigmoid or power function. Secondly, embryo growth needs to be monitored non-invasively if it is to be more precise. However, a clear deviation was seen between opacity points and fitted curve, especially from day 7 to day 12 of incubation, due to the influence of allantois.

The smaller size egg group had a lower average value of opacity over the incubation period than the larger egg group in all mathematical models (

Figure 5). This makes sense, as chick embryos rapidly gain body weight in the second half of incubation and thus the differences gradually increase over time. After curve fitting with various mathematical growth models, the calculated opacity as a function of time (

y =

f(

t)) was correlated with the embryo weight reported by Byerly [

17]. The calculated values of opacity for various models showed a high correlation with embryo weight (

Figure 6).

Therefore, the embryo weights could be calculated from the calculated opacity value using exponential, sigmoid, power and Gompertz growth models (Equations (5)–(8)).

where

x is embryo weight, and

y is calculated opacity obtained by fitting various mathematical models.

The performances of the various equations are shown in

Table 2. The exponential and power function models had higher performance in explaining embryo growth in terms of the coefficient of determination, R

^{2} = 0.991 and 0.994, respectively.

The above equations came after optimization by minimizing the error between actual opacity and calculated opacity at certain days of incubation (Equations (5)–(8)). Coefficients (parameters)

a and

b are calculated based on minimization of the error with multiple iterations. The Gompertz function is defined by M

$=A{e}^{-b{e}^{-ct}}$, where M is embryo mass,

A is the upper asymptote and

a and

b are the growth rates. The Gompertz function is a kind of sigmoid function. It is a type of time series mathematical model, where growth is slowest at the start and end of a time period (

Figure 5d). The upper asymptote was 70 mA/V, the opacity value of the eggs.

All mathematical models were almost equally acceptable in terms of performance parameters such as RMSE and coefficient of determination (

Table 2). The lowest RMSE was for the power function, since the trend in the data was more similar to that of the power function curve. The Extra Large (XL) eggs group had higher variations in opacity and a greater number of samples. Calculated opacity values fitted with exponential and power functions performed better in terms of the coefficient of determination with embryo weight (

Figure 6). Although the Gompertz function had a lower performance in fitting opacity data, it provided a clearer idea of the embryo weight as this value was very close to embryo weight values (Equation (8)).

Although embryo size is highly correlated with egg size, if graded eggs are used, egg size might not always provide precise information about embryo weight or size during incubation. This is because two eggs of the same size can produce different sizes or weights of hatchling chicks. This variation may come from different conversion ratio, hypoxia, gender, or resources allocated inside the eggs (e.g., ratio of egg yolk and albumen). Thus, the technique developed for embryonic growth observation in this research may solve this issue.

One limitation of this study is that it estimated growth only until day 16 of incubation. This technique or wavelength may not be suitable for studying growth in the last few days of incubation. Researchers may need to look for another approach to study this range in the future. During this very late stage of incubation, the internal structure of the egg is very different—it is relatively dry, feathers are fully developed and shell thickness is becoming much thinner, and thus much higher transmission results in lower opacity values. Another concern is regarding embryo exposure to high voltage or current. Near-infrared light is relatively safe for living organisms, especially if the exposure time is short. No adverse effects were observed regarding hatchability, since the hatch rate was more than 90%, including infertile eggs. Generally, about 5–7% of incubated eggs are found to be infertile. However, it is not recommended to apply a higher input current or light intensity for longer than 10 s for the inspection of hatching eggs, especially before day 10, due to the tiny embryo size and higher sensitivity to external stresses.