A Novel and Stable Benchmark for Breast Measurement

Featured Application

The new breast measurement coordinate system proposed in this study offers a practical tool for accurately measuring breast dimensions in adolescent females, particularly during their dynamic growth stages. This system can be utilized in designing ergonomically optimized bras and other wearable products that provide a better fit and greater comfort. Additionally, it holds potential for use in medical assessments, sports science, and ergonomic studies, where precise breast measurements are essential for evaluating health and performance.

Abstract

Selecting an appropriate bra product has long been a challenge for adolescent girls, whose breasts are rapidly growing. This challenge arises due to the absence of a rational benchmark for breast measurement. Traditional benchmarks are based on ergonomic principles, using reference points located on the human skeleton. However, the breast lacks a bone structure for support, leading to highly variable measurement results. In this study, the jugular notch and the xiphoid process were selected as breast measurement points according to academic principles. Their accuracy, stability, and deviation were experimentally investigated through tactile assessment of adolescent girls at different stages of development. Based on the experimental results, a novel breast measurement coordinate system was established, and its accuracy was verified by conversion calculations using a geodetic coordinate system. The results indicated that the numerical calculation of the breast shape was more accurate using the newly established breast measurement coordinate system.

1. Introduction

Breasts are an important sexual characteristic of women. Female breast development starts at puberty, and during this period, the breasts go through different developmental stages [1]. Wearing a well-fitting and comfortable bra is essential for comfort, health, cardiorespiratory activity, and especially the normal development of adolescent girls’ breasts [2]. However, up to 100% of adolescent girls face difficulty in choosing the right bra [3]. This implies that, for non-protective vest bras and adult bras, there is a lack of not only products that can protect and be used by adolescent girls during their development stages [4,5,6] but also measurement techniques and methodological tools for selecting appropriate products [7]. This is due to the breast measurement benchmark problem, including the unstable nipple position and inaccurate upper and lower chest circumferences.

Anthropometric measurement is a quantitative measure of the body used to understand physical changes in an individual [8]. According to ergonomics academic principles, the first step in anthropometric measurement is to locate the measurement benchmark [9]. The International Society for the Advancement of Kinanthropometry defines anthropometric measurement benchmarks as identifiable skeletal points that can be found by palpation [10]. Some studies have also pointed out that rational and accurate ergonomics-based measurement benchmarks can be located at skeletal sites, such as bone protrusions, joint ends, incisions, and sutures, and have distinctive and fixed features [11].

However, according to the breast anatomy, it lacks a bone structure for support. As a result, the contact measurement tools adopted in the current breast shape measurement approach normally use human surface feature points, such as the nipple position and breast root circumference, rather than stable skeletal points as the benchmark. These feature points not only have a blurred boundary but may also shift with the growth of the breasts [12], especially during the rapid developmental stage of adolescent girls’ breasts. This hampers accurate comparison and referencing of breast measurement results. For non-contact measurement techniques such as three-dimensional (3D) body scanning, identifying the measurement benchmark is a very difficult process [13]. This is because skeletal points cannot be accurately identified from the surface shape scanned by the machine. As a result, the benchmark position obtained by scanning a human body is not always the same, resulting in variable breast measurements and poor measurement repeatability. This makes it difficult to accumulate measurement data and predict breast growth changes.

In addition to the measurement starting points, a data recording framework is required to provide the initial measurement reference to help accumulate measurement data. The human body is a 3D structure with directions and anatomical positions, which can be described using a coordinate system [14]. Such coordinate systems can be divided into geodetic coordinate systems and body coordinate systems [15]. In geodetic coordinate systems, the anatomical positions of the human body are measured based on the external reference points and with the surrounding environment as a scale. In contrast, body coordinate systems are defined by human bone landmarks and generally use Cartesian coordinate systems for measurement. Paxino’s body coordinate systems describe the original space of human physical tissues because they are directly imaged and the most similar to the original human form [16]. Between them, the geodetic coordinate systems are the most commonly used in existing breast shape measurements. However, geodetic coordinate systems are not suitable because they often consider the inclined plane of the human skeleton, the vertical plane, during the measurements, which is a simplification of the spatial dimensions of the human body and affects the accuracy of the measurement results [17]. Additionally, to obtain the final measurement values, the obtained data must be converted with its own specific mathematical model. Also, to save computational costs, spatial body dimensions are simplified into planar dimensions, which leads to deviations between measurement results and practical values.

Therefore, existing breast measurements are limited by the lack of measurement points that conform to the ergonomics characteristics and body coordinate systems that can accurately record breast shape data. In this study, to define new breast measurement points and establish a breast measurement coordinate system, the breast measurement points were first filtered based on the skeletal sites and distinctive and fixed features of the measurement benchmark in ergonomics. The incisura jugularis (IJ) and processus xiphoideus (PX) at the sternum were selected as new breast shape measurement points, and their accuracy and stability was verified during a touching check. Subsequently, adolescent girls at different developmental stages were included in practical movement experiments to examine the selected measurement points through a touching check. These studies aimed to verify whether all the adolescent girls at different developmental stages could successfully find the new breast measurement points during the touching check. A secondary aim was to evaluate the stability of the measurement points and the deviation between the new breast shape measurement points and the measurement points obtained using geodetic coordinate systems during the touching check. The measurement points with the smallest deviation were used as the origin for constructing the novel coordinate system for breast measurement. Finally, according to the experimental results, a novel coordinate system for breast measurement was established and its accuracy verified experimentally.

2. Methodology

The overall structure of this study is illustrated in Figure 1, comprising three main phases.

(1) Selection of Breast Measurement Points: A thorough literature review was conducted to identify anatomical landmarks that meet anthropometric standards in ergonomics. This involved selecting skeletal points, evaluating feature prominence, and ensuring fixation for measurement consistency. The optimal breast measurement points were determined based on this process.

(2) Establishment of the Measurement Coordinate System: Following the literature review, adolescent females at different developmental stages participated in practical experiments to assess the stability and palpation accuracy of the newly identified measurement points. The point with the least deviation was selected as the origin for the new breast measurement coordinate system. The parameters were defined in both geodetic and body coordinate systems, with a transformation matrix developed to align the two systems.

(3) Validation of the Measurement Coordinate System: Validation experiments were conducted to compare the accuracy of the new breast measurement coordinate system with the conventional terrestrial coordinate system. The breast volumes were calculated in both systems and compared to assess the precision of the new system in alignment with actual anatomical values.

Figure 1 provides a visual summary of the stages involved, from the selection of measurement points to the validation of the new coordinate system.

2.1. Breast Measurement Points Confirmation

This study began with a comprehensive review of the national and international literature, including scientific articles and encyclopedic entries, to identify the defining characteristics of anthropometric reference points.

The key breast measurement regions for bra design were identified on the chest, back, and shoulders, which are crucial for providing structural support, distributing weight evenly, and enhancing both lift and comfort. Consequently, we focused on selecting breast measurement points based on skeletal landmarks, as they offer greater stability and accuracy.

The scapula, located on the posterior thorax, includes key landmarks such as the acromial angle, acromial tip, and coracoid process [18]. The sternum, positioned anteriorly in the thoracic cavity [6], has the jugular notch and xiphoid process as its primary landmarks. The ribs form the thoracic cage, but breast tissue may obscure these landmarks in females [19]. The vertebrae, comprising 33 segments, have the spinous processes as the most palpable points [20].

After reviewing the skeletal landmarks for breast measurements, we initially identified four key regions: the scapula, sternum, ribs, and vertebrae, along with nine specific landmarks. However, the acromial angle, acromial tip, coracoid process, and C7 spinous process were excluded due to the instability caused by limb movement and difficulty in palpation [21,22]. We then focused on the sternum and ribs, which offer greater stability due to their immovable joints.

Given the requirements for stability and prominence, two landmarks on the sternum—the jugular notch and xiphoid process—were selected as the primary measurement points [23]. These landmarks will undergo further validation through experimental testing.

2.2. Breast Coordinate System Establishment

Three-dimensional human body data require a reference coordinate system, which includes both a primary and a secondary system. The primary system is the fixed terrestrial coordinate system, while the secondary system is the body coordinate system. In current chest measurement methods, the terrestrial coordinate system is commonly used, with the xiphoid process often serving as the origin (0,0,0).

The breast measurement coordinate system was developed after identifying the key skeletal landmarks. During the establishment of this new system, the most stable and reliable skeletal reference point was selected as the origin (0,0,0), based on palpation accuracy experiments involving the jugular notch and the xiphoid process.

The key measurement points for the breast morphology were marked within the new coordinate system. To determine the transformation equations for these points, we evaluated whether the accuracy of the breast measurement system based on the body coordinate system was superior to that of the terrestrial coordinate system. By defining the breast parameters in both systems and applying affine transformations, we derived equations mapping the terrestrial system to the breast measurement system. The accuracy of both systems was compared by calculating the volume of simplified breast models in each system and comparing them to the actual volume, thereby establishing the framework for future breast measurement and computational models.

2.3. Measurement Points Validation

2.3.1. Participants

In this study, 60 healthy adolescent females aged 8 to 18 years were recruited from the Guangzhou region and categorized into three developmental groups: Group A (8–12 years), Group B (13–15 years), and Group C (16–18 years), with 20 participants per group. All the participants were screened to ensure they had no skeletal disorders, such as scoliosis or shoulder asymmetry, ensuring the reliability of the results. Written informed consent was obtained from all the participants, and for those under 18, consent from their legal guardians was also secured, as required by the Institutional Review Board (IRB). The purpose, procedures, and potential risks of the study were clearly explained to both the participants and their guardians to ensure their full understanding. The descriptive statistics of the participants are summarized in

Table 1. Descriptive statistics of participants.

Variable	Group	Sample Size (n)	Mean (Range)	Standard Deviation (SD)
Age (years)	A	20	10.25 (8–12)	1.58
	B	20	14.10 (13–15)	0.83
	C	20	17.10 (15–18)	0.83
Height (cm)	A	20	141.54 (123.2–158.3)	10.13
	B	20	157.09 (150.0–162.5)	3.04
	C	20	159.21 (152.0–166.5)	4.02
Weight (kg)	A	20	36.68 (26.0–53.2)	7.49
	B	20	50.19 (42.7–62.2)	5.14
	C	20	50.74 (42.3–65.6)	5.62

.

2.3.2. Operational Definition

An operational definition specifies the detailed procedures and criteria that researchers use to measure or manipulate variables in a specific study. In this research, the xiphoid process and the jugular notch were selected as the new breast measurement points. The definitions of these bony landmarks and the palpation methods used for their identification are provided below. The specific palpation techniques are illustrated in Figure 2.

Xiphoid Process (PX): To identify the xiphoid process, place four fingers at the lower margin of the last costal cartilage. Slide the fingers upward along the edge of the costal cartilage and continue sliding along the seventh costal cartilage until a central depression is felt. The upper edge of the xiphoid process can be palpated above this depression. The central point of this upper margin is designated as the reference point.

Jugular Notch (JN): From the PX reference point, move the index finger vertically upwards until the sternal notch between the clavicles is palpated. The central point of this notch is selected as the reference point.

2.3.3. Experimental Procedure

(1) Measurement Method

Participants used their dominant hand to palpate the jugular notch and their non-dominant hand to palpate the xiphoid process. Once both points were located, data were recorded and images were captured. The process was then repeated with the hands switched. This sequence formed one complete measurement cycle. Each participant completed three cycles, with 10-second intervals between each. All the measurements were conducted by a certified level 1 anthropometrist with 5 years of experience to minimize the bias.

(2) Experimental Conditions

All the measurements were conducted under controlled conditions. The laboratory environment was maintained at a temperature of (25 ± 2) °C, with a relative humidity of (65 ± 5)% and air velocity below 0.1 m/s. Testing was performed at various times of the day under randomized lighting conditions. Participants stood in a pre-marked area, and a Canon EOS 800D digital camera (24.2 megapixels, 128 GB SD card) positioned 1.5 m away on a Selens T170 tripod was used to capture the images.

(3) Participant Preparation

Prior to testing, participants wore fitted clothing, stood barefoot, and had their hair tied back. Circular markers were placed between their index fingers for the image analysis. The palpation procedure and definitions were explained by the researchers, and participants were given three practice trials. The height and weight were measured and recorded before the experiment began.

(4) Data Processing

The z-axis of the body coordinate system was calculated by connecting the midpoints of the finger markers. A grid overlay was used to define the z-axis of the terrestrial coordinate system, passing through the brow, nose tip, chin, and navel. Deviations between the body’s z-axis and the z-axis of the terrestrial system were analyzed.

(5) Deviation Measurement

The deviations of the breast measurement points from the terrestrial coordinate system were measured using CorelDRAW 2023 software. The alignment and dynamic guide functions of the software ensured the precision, with the measurements recorded in millimeters, accurate to two decimal places.

2.3.4. Data Analysis

The statistical analysis for evaluating the stability and deviation in the objective assessments was conducted using SPSS version 25.0 (SPSS for Mac, Chicago, IL, USA). To assess the stability, a matched samples t-test was employed to compare the differences among the three repeated measurements within each group, focusing on paired quantitative data.

Intra-group comparisons of the deviations of the two new breast measurement points from the z-axis of the terrestrial coordinate system were performed using one-way analysis of variance (ANOVA).

2.4. Coordinate System Validation

2.4.1. Study Design

This experiment aimed to validate the new breast measurement coordinate system by comparing the volume measurement errors between this system and the conventional terrestrial coordinate system. According to the literature, the irregular shape of the breast makes it challenging to determine its exact volume, particularly in the early developmental stages of adolescent females, where the breast boundaries are more indistinct. This leads to inherent measurement uncertainties. To address these challenges, simple geometric objects with known volumes were used as reference standards for the validation. The objects were measured within both the terrestrial and the new breast measurement coordinate systems, and the errors between the measured volumes and the actual volumes were compared to provide objective data.

2.4.2. Materials

The materials used for the validation are shown in Figure 3. A scaled medical sternum model with a total height of 170 cm and a length of 16 cm from the jugular notch to the xiphoid process was used. The angle between the z-axis of the new breast measurement coordinate system and the z-axis of the terrestrial coordinate system was 17°. Additionally, four hemispheres with diameters of 3 cm, 5 cm, 8 cm, and 10 cm were selected as geometric models. These models were used to calculate and analyze the volume errors and error ranges in both the terrestrial and the breast measurement coordinate systems.

2.4.3. Procedure

Positioning of Models: Each geometric model (hemisphere) was placed within both the terrestrial coordinate system and the breast measurement coordinate system, as aligned with the reference points (jugular notch and xiphoid process) on the medical sternum model.

Volume Measurement: The volume of each hemisphere was measured in both coordinate systems. The volume calculations within the breast measurement coordinate system were based on the coordinate transformation methods previously established.

Error Calculation: Errors were defined as the absolute differences between the measured volumes and the known true volumes of the hemispheres. The volume errors and their ranges for each model were recorded for both coordinate systems.

2.4.4. Data Analysis

Statistical analysis was conducted to compare the measurement accuracy between the two coordinate systems. A paired t-test was performed to assess the differences in measurement errors for each geometric model. Additionally, the error ranges were analyzed to evaluate the consistency of the measurements obtained from each system.

3. Results

3.1. Point Measurement Accuracy

3.1.1. Stability Analysis Results

The stability analysis results for Groups A, B, and C are shown in

Table 2. Matched t-test results of Group A, Group B and Group C.

Group	Analyzed Item	Match (Mean ± SD)		D-Value	t	p
		Match 1	Match 2	(Match 1–Match 2)
A	DHO1-DHO2	17.16 ± 0.80	17.31 ± 0.78	−0.15	−0.421	0.678
	NHO1-NHO2	17.18 ± 0.57	17.31 ± 0.69	−0.14	−1.371	0.186
	DHO2-DHO3	17.36 ± 0.80	17.31 ± 0.78	0.05	0.608	0.550
	NHO2-NHO3	17.29 ± 0.72	17.31 ± 0.69	−0.02	−0.221	0.827
B	DHO1-DHO2	17.16 ± 0.80	17.31 ± 0.78	−0.15	−0.421	0.678
	NHO1-NHO2	17.18 ± 0.57	17.31 ± 0.69	−0.14	−1.371	0.186
	DHO2-DHO3	17.36 ± 0.80	17.31 ± 0.78	0.05	0.608	0.550
	NHO2-NHO3	17.29 ± 0.72	17.31 ± 0.69	−0.02	−0.221	0.827
C	DHO1-DHO2	17.43 ± 0.97	17.50 ± 1.00	−0.07	−1.211	0.241
	NHO1-NHO2	17.48 ± 0.93	17.52 ± 0.83	−0.04	−0.351	0.730
	DHO2-DHO3	17.46 ± 1.00	17.55 ± 1.07	−0.09	−0.930	0.364
	NHO2-NHO3	17.56 ± 0.83	17.59 ± 0.97	−0.03	−1.073	0.297

and Figure 4. In this analysis, “Dominant Hand Over Non-dominant Hand” is abbreviated as DHO, “Dominant Hand Under Non-dominant Hand” as DHU, “Non-dominant Hand Over Dominant Hand” as NHO, and “Non-dominant Hand Under Dominant Hand” as NHU. Furthermore, the first, second, and third measurements with the dominant hand on top are denoted as DHO1, DHO2, and DHO3, respectively, while the first, second, and third measurements with the non-dominant hand on top are denoted as NHO1, NHO2, and NHO3.

The results of the paired t-test analysis showed no significant differences in the action experiment data among adolescent females aged 8–12, 13–15, and 15–18 years (p > 0.05), as presented in Table 2 and Figure 5. The analysis of the four paired datasets across these three groups revealed consistent and stable length measurements between the jugular notch and the xiphoid process, regardless of whether the dominant hand palpated the jugular notch and the non-dominant hand palpated the xiphoid process (DHO), or the reverse (NHO). This consistency was maintained across three repeated measurements.

In Figure 4, ns indicates not significant, meaning that the t-test results did not reveal any statistically significant differences.

The colors and black/white dots in Figure 5 represent the distribution of the data points. The box in each box plot shows the interquartile range (IQR), with the central line representing the median. The whiskers indicate the range of the data, and the dots outside the whiskers represent outliers.

Additionally, the deviation analysis revealed that the distance measurements obtained with the non-dominant hand on top (values: −0.13, 0.10; −0.14, −0.02; −0.04, −0.03 for the three groups, respectively) were consistently smaller than those obtained with the dominant hand on top (values: −0.16, 0.19; −0.15, 0.05; −0.07, 0.09). This suggests that the distance measurements between the jugular notch and the xiphoid process were more stable when the non-dominant hand was used to palpate the jugular notch, and the dominant hand was used to palpate the xiphoid process.

In this figure, ns denotes not significant, indicating that no statistically significant differences were observed.

Table 3. Analysis results of the variance between groups.

Variable	Group (Mean ± SD)			F	p
	Group A (n = 20)	Group B (n = 20)	Group C (n = 20)
DHO	−15.41 ± 1.26	−17.24 ± 0.93	−17.39 ± 0.93	21.826	0.000 **
DHU	−15.56 ± 1.15	−17.87 ± 0.66	−17.96 ± 0.85	32.684	0.000 **

** p < 0.01.

demonstrates that for all three age groups, there were significant differences (p < 0.05) in the length measurements between the two points, depending on whether the dominant or non-dominant hand was used first for palpating the jugular notch and xiphoid process. This indicates that the length measurements between the two points varied significantly across the different groups and showed low consistency when using different hand positions for palpation.

The analysis of variance among the three groups for the length measurement deviations between the jugular notch and the xiphoid process—when the dominant hand was used to palpate the jugular notch and the non-dominant hand was used to palpate the xiphoid process (DHO)—revealed a statistically significant difference at the 0.01 level (F = 21.826, p = 0.000). Further pairwise comparisons showed that the mean deviation scores of Group A were significantly higher than those of Group B (Group A > Group B) and Group C (Group A > Group C).

3.1.2. Deviation Analysis Results

The results of the intergroup stability variance analysis for Groups A, B, and C are presented in Figure 5.

Figure 5 shows the intergroup deviations for the adolescent females from the three different age groups after three repeated measurements using both the dominant and non-dominant hands to palpate the jugular notch and xiphoid process. In this figure, “DHO” indicates the deviation from the z-axis when the dominant hand is used to palpate the jugular notch, “DHU” represents the deviation when the dominant hand is used to palpate the xiphoid process, “NHO” indicates the deviation when the non-dominant hand is used to palpate the jugular notch, and “NHU” represents the deviation when the non-dominant hand is used to palpate the xiphoid process.

The intergroup deviation analysis showed no significant differences (p > 0.05) in the deviations from the z-axis across the three groups when using both the dominant and non-dominant hands for palpation over three repetitions. This suggests the high consistency and stability in the measurement positions across all the repetitions.

For Group A, the deviations from the z-axis during the three palpation trials using the non-dominant hand on top to palpate the jugular notch (NHO) were 0.37, 0.32, and 0.31, respectively. For Group B, the deviations were 0.26, 0.26, and 0.23, respectively. For Group C, the deviations were 0.26, 0.21, and −0.22, respectively.

These results indicate that, across all three groups, the accuracy of the measurements is higher when using the non-dominant hand than the dominant hand, and it is higher when palpating the jugular notch compared to the xiphoid process.

3.2. System Establishment Accuracy

3.2.1. Coordinate System Transformation

The terrestrial coordinate system was first established to serve as the reference for the new breast measurement coordinate system in adolescent females during their developmental stages.

In Figure 6, the geodetic coordinate system O-XYZ is defined with PX as the origin, as aligned to the anatomical orientation of the body. The +X-axis extends from posterior to anterior, the +Y-axis from right to left, and the +Z-axis from lower to upper. The breast feature points Ai, Ai′, Bi, Ci, Di, and Ei, representing the nipple, inner edge, outer edge, lower edge, and upper edge of the breast, respectively, are defined in this system. Their coordinates are {AiX, AiY, AiZ}, {Ai′X, Ai′Y, Ai′Z}, {BiX, BiY, BiZ}, {CiX, CiY, CiZ}, {DiX, DiY, DiZ}, and {EiX, EiY, EiZ}.

Figure 7 introduces the breast measurement coordinate system O-XYZ, using IJ as the origin. This Cartesian system aligns similarly: +X from posterior to anterior, +Y from right to left, and +Z from upper to lower, with the z-axis connecting IJ and PX. In this system, the breast feature points Ai, Ai′, Bi, Ci, Di, and Ei maintain the same coordinate notation as in the geodetic system.

Define any point $\mathrm{Q}$. The coordinates for the point in the geodetic coordinate system and the breast measurement coordinate system can be expressed as follows:

Point $\mathrm{Q}$ in the geodetic coordinate system: $\{\mathrm{Q}\mathrm{X},\mathrm{Q}\mathrm{Y},\mathrm{Q}\mathrm{Z}\}$

Point ${\mathrm{Q}}^{\mathrm{’}}$ in the breast measurement coordinate system: $\{\mathrm{Q}\mathrm{X},\mathrm{Q}\mathrm{Y},\mathrm{Q}\mathrm{Z}\}$

Point $\mathrm{Q}$ in the geodetic coordinate system can be mapped and transformed into Point ${\mathrm{Q}}^{\mathrm{’}}$ in the breast measurement coordinate system through an affine transformation, that is, multiply linear matrix $\mathrm{T}$ and translate vector $\mathrm{b}$.

$$\left\{\begin{array}{c}\mathrm{Q}\mathrm{x}\\ \mathrm{Q}\mathrm{y}\\ \mathrm{Q}\mathrm{z}\end{array}\right\}=\left[\begin{array}{ccc}{\mathrm{T}}_{\mathrm{x}\mathrm{x}}& {\mathrm{T}}_{\mathrm{x}\mathrm{y}}& {\mathrm{T}}_{\mathrm{x}\mathrm{z}}\\ {\mathrm{T}}_{\mathrm{y}\mathrm{x}}& {\mathrm{T}}_{\mathrm{y}\mathrm{y}}& {\mathrm{T}}_{\mathrm{y}\mathrm{z}}\\ {\mathrm{T}}_{\mathrm{z}\mathrm{x}}& {\mathrm{T}}_{\mathrm{z}\mathrm{y}}& {\mathrm{T}}_{\mathrm{z}\mathrm{z}}\end{array}\right]\left\{\begin{array}{c}{\mathrm{Q}}_{\mathrm{X}}\\ {\mathrm{Q}}_{\mathrm{Y}}\\ {\mathrm{Q}}_{\mathrm{Z}}\end{array}\right\}+\left\{\begin{array}{c}{\mathrm{b}}_{\mathrm{X}}\\ {\mathrm{b}}_{\mathrm{Y}}\\ {\mathrm{b}}_{\mathrm{Z}}\end{array}\right\}$$

(1)

The transformation from the geodetic coordinate system to the breast measurement coordinate system is performed in line with the following processes:

(1) Inversely transform the $Z$-axis

Through the inverse transformation of the $z$-axis, we can obtain the position of the converted Point ${\mathrm{Q}}_1$ as follows:

$$\left\{\begin{array}{c}{\mathrm{Q}}_{1\mathrm{x}}\\ {\mathrm{Q}}_{1\mathrm{y}}\\ {\mathrm{Q}}_{1\mathrm{z}}\end{array}\right\}={\mathrm{T}}_1\left\{\begin{array}{c}{\mathrm{Q}}_{\mathrm{X}}\\ {\mathrm{Q}}_{\mathrm{Y}}\\ {\mathrm{Q}}_{\mathrm{Z}}\end{array}\right\}=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]\left\{\begin{array}{c}{\mathrm{Q}}_{\mathrm{X}}\\ {\mathrm{Q}}_{\mathrm{Y}}\\ {\mathrm{Q}}_{\mathrm{Z}}\end{array}\right\}$$

(2)

(2) Rotate $\mathrm{n}$ degrees clockwise around the y-axis

By rotating Point ${\mathrm{Q}}_1$ clockwise around the y-axis by $\mathrm{n}$ degrees, we can obtain the position of the converted Point ${\mathrm{Q}}_2$ as follows:

$$\left\{\begin{array}{c}{\mathrm{Q}}_{2\mathrm{x}}\\ {\mathrm{Q}}_{2\mathrm{y}}\\ {\mathrm{Q}}_{2\mathrm{z}}\end{array}\right\}=\mathrm{T}2\left\{\begin{array}{c}{\mathrm{Q}1}_{\mathrm{X}}\\ {\mathrm{Q}1}_{\mathrm{Y}}\\ {\mathrm{Q}1}_{\mathrm{Z}}\end{array}\right\}=\left[\begin{array}{ccc}\mathrm{c}\mathrm{o}\mathrm{s}(-{\mathrm{n}}^{\mathrm{o}})& 0& -\mathrm{s}\mathrm{i}\mathrm{n}(-{\mathrm{n}}^{\mathrm{o}})\\ 0& 1& 0\\ \mathrm{s}\mathrm{i}\mathrm{n}(-{\mathrm{n}}^{\mathrm{o}})& 0& \mathrm{c}\mathrm{o}\mathrm{s}(-{\mathrm{n}}^{\mathrm{o}})\end{array}\right]\left\{\begin{array}{c}{\mathrm{Q}1}_{\mathrm{X}}\\ {\mathrm{Q}1}_{\mathrm{Y}}\\ {\mathrm{Q}1}_{\mathrm{Z}}\end{array}\right\}$$

(3)

(3) Connect the origins of the two coordinate systems

By moving Point ${\mathrm{Q}}_2$ along the direction $\mathrm{O}-\mathrm{o}$ connecting the origins of the two coordinate systems by distance $\mathrm{b}$, the position of the translated Point ${\mathrm{Q}}_3$ can be calculated as follows:

$$\left\{\begin{array}{c}{\mathrm{Q}}_{3\mathrm{x}}\\ {\mathrm{Q}}_{3\mathrm{y}}\\ {\mathrm{Q}}_{3\mathrm{z}}\end{array}\right\}=\left\{\begin{array}{c}{\mathrm{Q}}_{2\mathrm{x}}\\ {\mathrm{Q}}_{2\mathrm{y}}\\ {\mathrm{Q}}_{2\mathrm{z}}\end{array}\right\}+\mathrm{b}=\left\{\begin{array}{c}{\mathrm{Q}}_{2\mathrm{x}}\\ {\mathrm{Q}}_{2\mathrm{y}}\\ {\mathrm{Q}}_{2\mathrm{z}}\end{array}\right\}+\left\{\begin{array}{c}{\mathrm{b}}_{\mathrm{X}}\\ {\mathrm{b}}_{\mathrm{Y}}\\ {\mathrm{b}}_{\mathrm{Z}}\end{array}\right\}$$

(4)

where $\left\{\begin{array}{c}{\mathrm{b}}_{\mathrm{X}}\\ {\mathrm{b}}_{\mathrm{Y}}\\ {\mathrm{b}}_{\mathrm{Z}}\end{array}\right\}$ is the translation distance from the origin of the geodetic coordinate system to the origin of the breast measurement coordinate system. Through a series of transformations and translations, the conversion formula is derived from the geodetic coordinate system to the breast measurement coordinate system as follows:

$$\left\{\begin{array}{c}\mathrm{Q}\mathrm{x}\\ \mathrm{Q}\mathrm{y}\\ \mathrm{Q}\mathrm{z}\end{array}\right\}=\left[\begin{array}{ccc}\mathrm{c}\mathrm{o}\mathrm{s}(-{\mathrm{n}}^{\mathrm{o}})& 0& -\mathrm{s}\mathrm{i}\mathrm{n}(-{\mathrm{n}}^{\mathrm{o}})\\ 0& 1& 0\\ \mathrm{s}\mathrm{i}\mathrm{n}(-{\mathrm{n}}^{\mathrm{o}})& 0& \mathrm{c}\mathrm{o}\mathrm{s}(-{\mathrm{n}}^{\mathrm{o}})\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& -1\end{array}\right]\left\{\begin{array}{c}{\mathrm{Q}}_{\mathrm{X}}\\ {\mathrm{Q}}_{\mathrm{Y}}\\ {\mathrm{Q}}_{\mathrm{Z}}\end{array}\right\}+\left\{\begin{array}{c}{\mathrm{b}}_{\mathrm{X}}\\ {\mathrm{b}}_{\mathrm{Y}}\\ {\mathrm{b}}_{\mathrm{Z}}\end{array}\right\}$$

(5)

3.2.2. Coordinate System Accuracy Verification

The practical volumes of the breasts simulated by the hemispheres with different diameters and the volumes calculated using both coordinate systems are shown in Figure 8.

The actual volumes of the hemispheres with diameters of 3 cm, 5 cm, 8 cm, and 10 cm are 7.07 cm³, 32.72 cm³, 134.04 cm³, and 261.80 cm³, respectively. As shown in Figure 8, a comparison reveals that the volumes calculated using the terrestrial coordinate system for hemispheres with diameters of 3 cm, 5 cm, 8 cm, and 10 cm were 2.09 cm³, 10.29 cm³, 45.98 cm³, and 97.72 cm³, with errors of 70.4%, 68.6%, 65.7%, and 62.7%, respectively. In contrast, the volumes calculated using the breast measurement coordinate system were 6.18 cm³, 28.62 cm³, 117.23 cm³, and 228.96 cm³, with errors of 12.6%, 12.5%, 12.5%, and 12.5%, respectively.

4. Discussion

This study uses developmental stages as a starting point for understanding the dynamic changes in female breast morphology, proposing a new breast measurement standard based on ergonomic principles. Selecting the correct reference and providing clear definitions in the initial step of human measurement procedures is crucial to establish a consistent reference point, minimizing measurement errors and ensuring accuracy [24]. Defining a new breast measurement standard allows for its conversion into a specific and operational measurement coordinate system. Due to the unique characteristics of the breast, the selection and definition of breast measurement standards have not been fully established [25]. Our study aims to address the inaccuracies associated with traditional methods that use the nipple as a reference point. By applying anthropometric principles that require measurement points to be “located on bones, distinct, and stable”, we identified the jugular notch and xiphoid process on the sternum as new reference points for breast measurement. The experimental results demonstrate that these new measurement points provide high accuracy and stability across different developmental stages and that the breast measurement coordinate system offers more precise calculations of the breast volume compared to the conventional terrestrial coordinate system.

The analysis results (Table 3) showed that the intragroup deviations in the length measurements relative to the z-axis for the three groups exhibited significant differences at the 0.01 level (F = 21.826, p = 0.000) when the dominant hand was used to palpate the jugular notch and the non-dominant hand to palpate the xiphoid process. Pairwise comparisons revealed that the mean deviation scores were significantly higher in Group A compared to Group B (Group A > Group B) and Group C (Group A > Group C). For the measurements taken with the non-dominant hand on top to palpate the jugular notch and the dominant hand on the xiphoid process, the deviations also showed significant differences at the 0.01 level (F = 32.684, p = 0.000), with Group A > Group B and Group A > Group C. These findings are likely due to growth-related changes in the sternum, which develops as the body grows [26]. Particularly, the participants in Group A (ages 8–12) are in a rapid growth phase with significant increases in height, while the participants in Groups B (ages 13–15) and C (ages 16–18) have slower skeletal growth, resulting in smaller deviations in the length measurements between the two points. This also reflects the slowdown in skeletal growth in females after age 12 [27], with the sternum becoming more stable and the positions of the jugular notch and xiphoid process showing high stability.

Furthermore, it is noteworthy that the distance measurements using the non-dominant hand on top (values: −0.13, 0.10; −0.14, −0.02; −0.04, −0.03) were consistently smaller than those using the dominant hand on top (values: −0.16, 0.19; −0.15, 0.05; −0.07, 0.09), indicating that the accuracy of the measurements with the non-dominant hand is higher than that with the dominant hand. Previous studies have shown that initial practice with the non-dominant hand leads to better learning outcomes for spatial accuracy tasks, while initial practice with the dominant hand is more effective for tasks requiring maximum force production [28].

The results also indicate that participants with a higher BMI could accurately locate the reference points. Different BMI levels affect the ease of palpating skeletal landmarks [29]. A high BMI generally indicates more fat tissue overlying bones, making skeletal landmarks harder to palpate and locate. However, consistent with previous studies, even in individuals with a higher body weight or BMI, fat accumulation around the sternum remains limited [30], as fat tends to accumulate more in areas such as the abdomen, hips, thighs, and upper arms [31]. Therefore, for women with a higher BMI, fat is primarily concentrated in the soft tissues around the breasts and chest, rather than directly over the sternum.

Our analysis also demonstrated that the breast volume calculations based on the terrestrial coordinate system tend to be underestimated, with larger errors compared to the actual volume. While the breast volume calculated using the new breast measurement coordinate system is also slightly underestimated, it is much closer to the actual volume. As the true diameter of the hemisphere increases, the volume calculation error using the breast measurement coordinate system remains almost constant at approximately 12.5%. This suggests that the traditional breast volume calculation method based on the terrestrial coordinate system has larger errors. When obtaining or converting anthropometric data, the human coordinate system is often employed; however, the inclined skeletal planes of the body are typically treated as vertical planes, which simplifies the spatial dimensions and affects the measurement accuracy [32]. Thus, using a breast measurement coordinate system based on anatomical features provides a more accurate calculation of the breast dimensions.

To address the physiological and psychological changes in breast shape during adolescence, this study developed a novel breast measurement tool based on the new breast measurement coordinate system, which has been granted a Chinese invention patent (Patent No. ZL 2022 1 0831873.1). The new wearable breast measurement tool uses the breast measurement reference points as a starting framework. It allows for self-measurement and fitting after locating the origin of the new breast measurement coordinate system at the jugular notch. The tool measures the breast circumference (underbust circumference), length (vertical distance of the upper cup, arc distance of the lower cup), width (arc distance of the front cup, arc distance of the side cup), and height (breast height) to obtain the dimensional parameters of various breast features. These parameters are used to calculate the breast volume and assess the breast asymmetry, with standard operating procedures provided to ensure the measurement accuracy.

The strength of this study lies in using the developmental stages as the basis for understanding the dynamic changes in female breast morphology. The challenge that 90% of women face in finding a suitable bra that fits, coupled with the significant risk posed to breast health by wearing improperly fitting bras, remains an unresolved issue in both academic and industrial settings. The core problem lies in the lack of reasonable, accurate, and ergonomically compliant measurement standards for breast morphology, making it difficult to accurately measure the changing female breast shape and lacking a stable framework for recording measurements. This study challenges the century-old traditional breast measurement methods used in the industry by defining appropriate measurement standards, developing new measurement methods and tools, establishing standard operating procedures, and proposing new measurement theories and technical systems. This provides robust methods and tools for data collection and product development in the lingerie industry.

Although this study has made significant progress, certain limitations in terms of the study design must be acknowledged. Given that puberty marks the onset of dynamic changes in the female breast morphology, this study focused on a specific age group. Although it has been shown that the size of the sternum stabilizes after adolescence, certain special populations with different sternum shapes and sizes will need to be considered in future studies to make the new breast measurement standard more universally applicable. Additionally, due to the irregular shape of the breast, its true volume is difficult to determine and its boundaries are hard to define. The most accurate current method for measuring the breast volume is the specimen displacement method [33]. Thus, simple geometric objects with known volumes were used as reference standards for the validation. Furthermore, since the human body is dynamic, measurement data are influenced by factors such as muscle hydration, skin expansion due to heat, limb movements, and breathing patterns. Given that the female breast volume may fluctuate by approximately 10–20% each month, this study only focused on establishing measurement standards. Future research will incorporate measurements of actual breasts and the development of mathematical models.

Future research will further refine the breast measurement coordinate system by developing methods tailored to the dynamic changes of different groups. This includes not only the rapid developmental stages of adolescent girls but also other specific populations such as breastfeeding women, post-surgical recovery patients, breast cancer survivors, and elderly women experiencing gradual breast shape degeneration. For these groups, the physiological changes in their breast morphology present unique health and medical challenges. In particular, the breast size and shape can fluctuate significantly during breastfeeding and post-surgical recovery. The measurement system will derive shape equations specific to these developmental stages, ensuring accuracy in data collection. Additionally, future research will focus on creating more universally applicable measurement tools to meet the needs of these different populations. This will enable individuals to accurately measure their breast shape and select appropriate bras, enhancing their health and quality of life. These improvements are expected to benefit not only ergonomic applications such as bra design but also medical assessments, including breast health monitoring and post-surgical rehabilitation.

5. Conclusions

This study established and validated a novel breast measurement standard using stable skeletal landmarks to enhance the accuracy. The proposed breast measurement coordinate system represents a significant improvement over traditional methods by focusing on fixed anatomical landmarks such as the jugular notch and xiphoid process. These points demonstrated high stability and accuracy across different developmental stages of adolescent females, allowing for more reliable breast shape measurements.

The system effectively minimizes the errors commonly found in traditional terrestrial coordinate systems, particularly in calculating the breast volume. By aligning with skeletal landmarks, this method provides a reproducible and standardized approach to breast measurement. This research lays the groundwork for more precise assessments, which are critical for ergonomic bra design, improved comfort, and better fit during dynamic growth phases in adolescents.

6. Patents

China Invention Patent. A breast measurement device and system: ZL 2022 1 0831873.1.