A Novel Breast-Volume Self-Measurement Method with Improved Convenient and Accuracy

Featured Application

This self-measurement method for breast volume allows women to conveniently assess their body dimensions, promoting personal health management and improving body image. It aids lingerie manufacturers in designing better-fitting bras by providing accurate volume data, fostering comfort and support. Additionally, it can contribute to future research on body composition and psychological well-being, enhancing understanding of the relationship between physical measurements and women’s health.

Abstract

Breast volume is crucial for ensuring proper bra fit and comfort, significantly influencing women’s physiological and psychological well-being. This study aims to develop a novel method for breast-volume self-measurement, allowing women to accurately assess their breast volume without specialized equipment. We employed a geometric approximation of the breast as a combination of a partial elliptical cone and an irregular partial ellipsoid, leading to the formulation of a new volume equation. The method was validated against established standards, including the specimen drainage method and 3D scanning techniques. The findings revealed that our self-measurement approach achieved a relative error of only 3.8%, outperforming the 4.8% of 3D scanning and the 86.3% associated with traditional breast-volume equations. This innovative self-measurement technique enhances accuracy and serves as a practical solution for health and nutritional assessments, alongside body image evaluations. Its user-friendly nature positions it as a valuable tool for women’s health, particularly in personal fitness and ergonomic design.

1. Introduction

A well-fitting bra is essential for both physiological and psychological well-being, as it provides necessary support for the breasts [1]. However, research shows that over 85% of women wear the wrong bra size, largely due to inaccuracies in measuring breast volume [2]. Proper bra fit requires accurate breast-volume measurements to ensure that the cup size fully accommodates the breast shape [3,4,5]. This is not only important for women’s health management, but also critical for manufacturers to design bras that cater to diverse body types.

Accurately measuring breast volume has always been challenging due to the complexity of breast anatomy [6]. Current methods, including advanced imaging technologies, low-cost techniques, and formula-based approaches, show significant variability in results and often lack practicality for everyday use [7,8,9,10,11,12]. Techniques such as X-rays, MRI, and 3D scanning, while accurate, are costly, complex, and often involve radiation exposure, making them unsuitable for routine health assessments [13,14,15,16]. Low-cost methods, such as thermoplastic molding and water displacement, though simple, require assistance and lack precision, limiting their applicability for self-measurement [9,17,18,19,20]. Formula-based methods are more accessible but often oversimplify the breast’s complex geometry, leading to significant measurement errors [21,22,23].

To address these challenges, this study aims to develop a novel method for breast-volume self-measurement, allowing women to accurately assess their breast volume without specialized equipment.

2. Methodology

The overall framework of this study is illustrated in Figure 1, which consists of three main phases: (1) Selection of Measurement Parameters; (2) Breast-Volume Mathematical Modelling; (3) Breast-Volume Measurement Validation.

2.1. Selection of Measurement Parameters

When measuring breast volume, it is essential to first select the appropriate measurement parameters. Due to the varying shapes of women’s breasts, even breasts of the same volume may differ in shape and position. To ensure accuracy and relevance, this study selected parameters that effectively represent breast shape and position, including length, width, and height.

Following the Chinese national standards GB/T 16160-1996 Measurement Methods for Body Measurements for Garments [24] and GB/T 5703-2010 Basic Human Body Measurements for Technological Design [25], as well as measurement methods related to bra design from the Beijing Institute of Fashion Technology and Aimer Human Engineering Research Institute, this study identified five key breast measurement parameters. These include length measurements (upper-cup vertical distance, lower-cup arc distance), width measurements (anterior-cup arc distance, lateral-cup arc distance), and height measurement (breast height). These parameters provide essential references for subsequent breast shape fitting and volume calculation.

2.2. Breast-Volume Mathematical Modeling

Breast shape analysis is the most complex and critical aspect of breast-volume quantification. To achieve accurate breast-volume measurements, this study employed an approximate geometric method, constructing a mathematical model to represent the true shape of the breast. Geometric morphometrics, widely used in shape variation analysis for its flexibility and precision, was selected for breast shape modeling in this study [26]. Existing studies often use simple geometric forms, such as ellipsoids or cones, to approximate breast shapes [27,28,29]; however, these models struggle to accurately capture the complexity of breast morphology, particularly regarding the irregularity of the breast base and the fullness of the lower part. In this study, a combination of geometric assumptions was used to construct a mathematical model that closely approximates the true breast shape.

As shown in Figure 2, 3D scans reveal irregularities in the breast’s lower contour, and significant differences in curve length between the upper and lower sections. This indicates that simple cone or hemisphere assumptions are inadequate to describe the breast’s true shape. Therefore, we modeled the breast shape as a composite of irregular geometric forms to more accurately reflect its structure.

Figure 3 illustrates the composite geometric model of the breast developed in this study. In the model, the breast is divided into upper and lower sections along the Bi-Ai-Ci axis, with different geometric shapes representing the overall structure of the breast. To ensure the model’s accuracy, a three-dimensional reference coordinate system was established, with the coordinate axes following anatomical directions: +X from back to front, +Y from right to left, and +Z from bottom to top [30,31]. Within this coordinate system, the characteristic points of breast morphology were precisely marked, laying the foundation for further volume calculations.

Given the sagging of the breast under gravity and the distinct shape differences between the upper and lower parts, the model incorporates curves and lines that represent the actual breast shape, such as AC, AB, DC, DB, AD, and straight lines like AA′, CB, CG, and BF, with AA′ perpendicular to CB. These geometric features serve as key references for the shape fitting and the development of the volume calculation formula.

2.3. Breast-Volume Measurement Validation

To ensure objectivity and authoritative results, the breast specimen drainage method was adopted as the gold standard [32]. Two groups were defined in this study: the reference group, in which breast volumes were measured using the gold standard, and the comparison group, which included breast-volume measurements obtained through different methods (the proposed self-measurement method, existing breast-volume formulas, and 3D scanning techniques). By analyzing the error between each method in the comparison group and the gold standard, we systematically evaluated the accuracy of the new method.

As shown in Figure 4, the experimental materials included both single and bilateral medical breast models, all at a 1:1 scale. To ensure consistency and comparability, bilateral breast models with identical parameters were used as test samples in other control experiments. The medical breast model parameters include the following: upper-cup arc length of 12.0 cm, lower-cup arc length of 7.5 cm, front-cup arc length of 9.0 cm, side-cup arc length of 8.5 cm, and breast height of 5.5 cm.

The breast specimen drainage method, based on the principle of buoyancy, accurately determines breast volume by measuring the volume of displaced liquid. This method is commonly used to validate the accuracy of other breast-volume measurement techniques.

In the existing breast-volume formula methods, mechanical tools such as tape measures and Martin calipers are typically used to measure the linear or curved distances between various parts of the breast. These measured values are then applied to fitting formulas to calculate breast volume, a process known as the linear calculation method. The most common formula was proposed by Qiao et al. [33], as shown in the Equation below:

V = π/3 × MP2 (MR + LR + IR).

In the formula, MP, MR, LR, and IR represent diameters obtained from surface measurements of the breast, corresponding to breast height, anterior arc length, lateral arc length, and inferior arc length, respectively.

For the 3D scanning method, this study employed the ATOS Q 12M 3D scanner (GOM, Braunschweig, Germany) to scan the medical breast model, as shown in Figure 5. The device uses triangulation and structured light projection to capture the 3D coordinates of the object. Since the scanned breast model cannot directly provide volume measurements, the final volume was automatically calculated using the GOM Inspect Professional module within the GOM Suite 2022 software.

The steps for comparing and validating breast-volume measurements are as follows: (1) Place the simulated breast model in a container filled with water, record the breast volume measured by the specimen drainage method, and use it as reference data. (2) Apply the linear breast-volume calculation method, compute the breast volume based on the model’s parameters, and record the results. (3) Use 3D scanning equipment to scan the breast model, acquire the 3D data, and calculate the volume. (4) Utilize the new breast-volume measurement method proposed in this study, measure the model’s volume, and record the results.

2.4. Statistical Analysis

To validate the effectiveness of the proposed breast-volume measurement method, we assessed the accuracy and consistency of each measurement method by calculating the relative error compared to the gold-standard (breast specimen drainage method).

Relative error is used to assess the deviation of the measured value from the standard value, and the calculation formula is as follows:

Relative error = (x0 − x)/x = (Δx)/x

where x0 represents the measured breast volume, and x refers to the volume obtained from the gold-standard measurement.

3. Results

3.1. Breast-Volume Mathematical Model

3.1.1. Breast Shape Formula Fitting

First, input the necessary parameters for breast-volume measurement, which include the following:

• x-axis parameter
  $\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e} Ai-{Ai}^{\prime }$, denoted as $H_{Ai-{Ai}^{\prime }}$.
• y-axis parameters
  $\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e} Bi-Ci$, denoted as $W_{Bi-Ci}$;
  $\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e} {Ai}^{\prime }-Ci$, denoted as $W_{{Ai}^{\prime }-Ci}$;
  $\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e} {Ai}^{\prime }-Bi$, denoted as $W_{{Ai}^{\prime }-Bi}$.
• z-axis parameters
  $Distance Di-Ai$, denoted as $L_{Di-Ai}$;
  $Distance Ei-Ai$, denoted as $L_{Ei-Ai}$.
• Anchor points Ai, Bi, Ci, Gi, and Fi
  The coordinate of $Ai$ in the breast shape measurement coordinate system $o-xyz$ is denoted as ($Aix,Aiy,Aiz$), where $Aix,Aiy,\mathrm{a}\mathrm{n}\mathrm{d} Aiz$ are the $x,y$, and $z$-coordinates of $Ai$, respectively. Similarly, the coordinate of $Bi$ is ($Bix,Biy,Biz),$ the coordinate of $Ci$ is ($Cix,Ciy,Ciz),$ the coordinate of $Gi$ is ($Gix,Giy,Giz)$, and the coordinate of $Fi$ is $(Fix,Fiy,Fiz)$.
• $\mathrm{\angle }GiCiAi$ and $\mathrm{\angle }FiBiAi$
  Calculate $\mathrm{\angle }GiCiAi$ and $\mathrm{\angle }FiBiAi$ in the projection plane $yoz$.

Through practical breast shape analysis, it was found that using a block ellipsoid to denote the lower breast can better reflect a realistic breast shape. Hence, with the right breast as an example, the shape equation was derived. The left and right regions are divided by the plane that is perpendicular to the y-axis and passes the line $A1D1$ projected on the $yoz$ plane, and both regions can be described by a 1/8 ellipsoid. For the complete ellipsoid, as shown in Figure 6, the local coordinate system $o-xyz$ is built with the center of the ellipsoid as the origin, which can be expressed by the following mathematical equations:

$$\mathrm{Complete} \mathrm{ellipsoid} \mathrm{in} \mathrm{the} \mathrm{left} \mathrm{region}: {\displaystyle \frac{x^2}{{H_{Ai-{Ai}^{\prime }}}^2}}+{\displaystyle \frac{y^2}{{W_{{Ai}^{\prime }-Bi}}^2}}+{\displaystyle \frac{z^2}{{L_{Di-{Ai}^{\prime }}}^2}}=1$$

(1)

$$\mathrm{Complete} \mathrm{ellipsoid} \mathrm{in} \mathrm{the} \mathrm{right} \mathrm{region}: {\displaystyle \frac{x^2}{{H_{Ai-{Ai}^{\prime }}}^2}}+{\displaystyle \frac{y^2}{{W_{{Ai}^{\prime }-Ci}}^2}}+{\displaystyle \frac{z^2}{{L_{Di-{Ai}^{\prime }}}^2}}=1$$

(2)

Since the practical breasts only account for 1/8 of the ellipsoids in the left and right regions respectively, only the 1/8 parts of Equations (1) and (2) are taken, as shown below:

1/8 ellipsoid in the left region:

$${\displaystyle \frac{x^2}{{H_{Ai-{Ai}^{\prime }} }^2}}+{\displaystyle \frac{y^2}{{W_{{Ai}^{\prime }-Bi} }^2}}+{\displaystyle \frac{z^2}{{L_{Di-{Ai}^{\prime }} }^2}}=1, x\ge 0, y\ge 0, z\ge 0$$

(3)

1/8 ellipsoid in the right region:

$${\displaystyle \frac{x^2}{{H_{Ai-{Ai}^{\prime }} }^2}}+{\displaystyle \frac{y^2}{{W_{{Ai}^{\prime }-Ci} }^2}}+{\displaystyle \frac{z^2}{{L_{Di-{Ai}^{\prime }} }^2}}=1, x\ge 0, y\le 0, z\ge 0$$

(4)

The anchor point $Ai$ is projected onto the $YOZ$ plane and the projected point $A11$ is obtained. The coordinate of $A11$ can be expressed as follows:

$$A11=Ai\times l(n, x)=(0, Aiy, Aiz)$$

(5)

By moving the origin $o$ of the local coordinate systems of 1/8 ellipsoids in the left and right regions to the point $A11$ of the global coordinate system, we can rewrite Equations (3) and (4) as

1/8 ellipsoid in the left region:

$${\displaystyle \frac{{\left(x\right)}^2}{{H_{Ai-{Ai}^{\prime }}}^2}}+{\displaystyle \frac{{\left(y+{Ai}_y\right)}^2}{{W_{{Ai}^{\prime }-Bi}}^2}}+{\displaystyle \frac{{\left(z-{Ai}_z\right)}^2}{{L_{Di-{Ai}^{\prime }}}^2}}=1, x\ge 0, y\ge 0, z\ge 0$$

(6)

1/8 ellipsoid in the right region:

$${\displaystyle \frac{{\left(x\right)}^2}{{H_{Ai-{Ai}^{\prime }}}^2}}+{\displaystyle \frac{{\left(y+{Ai}_y\right)}^2}{{W_{{Ai}^{\prime }-Ci}}^2}}+{\displaystyle \frac{{\left(z-{Ai}_z\right)}^2}{{L_{Di-{Ai}^{\prime }}}^2}}=1, x\ge 0, y\le 0, z\ge 0$$

(7)

When the ellipsoid rotates around the z axis, it can better fit the practical breast shape (Figure 7). The rotation angle is determined in the following way: determine the straight line connecting Bi and Ci, project the line onto the xoy plane, and the included angle between the projected line and the y axis is denoted as θ, the rotation angle.

By rotating the ellipsoid θ degrees around z axis clockwise, we can further convert Equations (6) and (7) to the following forms:

$$\begin{array}{c}{\displaystyle \frac{{\left(x·cos\theta -\left(y+{Ai}_y\right)·sin\theta \right)}^2}{{H_{Ai-{Ai}^{\prime }}}^2}}+{\displaystyle \frac{{\left(\left(y+{Ai}_y\right)·cos\theta +x·sin\theta \right)}^2}{{W_{{Ai}^{\prime }-Bi}}^2}}+{\displaystyle \frac{{\left(z-{Ai}_z\right)}^2}{{L_{Di-{Ai}^{\prime }}}^2}}=1\\ x\ge 0, y\ge 0, z\ge 0\end{array}$$

(8)

$$\begin{array}{c}{\displaystyle \frac{{\left(x·cos\theta -\left(y+{Ai}_y\right)·sin\theta \right)}^2}{{H_{Ai-{Ai}^{\prime }}}^2}}+{\displaystyle \frac{{\left(-\left(y+{Ai}_y\right)·cos\theta -x·sin\theta \right)}^2}{{W_{{Ai}^{\prime }-Bi}}^2}}+{\displaystyle \frac{{\left(z-{A1}_z\right)}^2}{{L_{Di-{Ai}^{\prime }}}^2}}=1\\ x\ge 0, y\le 0, z\ge 0\end{array}$$

(9)

Equations (8) and (9) describe the shape of the right region of the upper breast. To obtain the shape expression of the left region of the upper breast, the left region is reflected along the xoz plane. Mathematically speaking, $\left(y+{Ai}_y\right)·cos\theta +x·sin\theta$ in Equations (8) and (9) is replaced by $-\left(y+{Ai}_y\right)·cos\theta -x·sin\theta$. Then, the left breast shape can be expressed as follows:

1/8 ellipsoid in the left region:

$$\begin{array}{c}{\displaystyle \frac{{\left(x·cos\theta -\left(y+{Ai}_y\right)·sin\theta \right)}^2}{{H_{Ai-{Ai}^{\prime }}}^2}}+{\displaystyle \frac{{\left(-\left(y+{Ai}_y\right)·cos\theta -x·sin\theta \right)}^2}{{W_{{Ai}^{\prime }-C1}}^2}}+{\displaystyle \frac{{\left(z-{Ai}_z\right)}^2}{{L_{Di-Ai}}^2}}=1\\ x\ge 0, y\ge 0, z\ge 0\end{array}$$

(10)

1/8 ellipsoid in the right region:

$$\begin{array}{c}{\displaystyle \frac{{\left(x·cos\theta -\left(y+{Ai}_y\right)·sin\theta \right)}^2}{{H_{Ai-{Ai}^{\prime }}}^2}}+{\displaystyle \frac{{\left(-\left(y+{Ai}_y\right)·cos\theta -x·sin\theta \right)}^2}{{W_{{Ai}^{\prime }-Bi}}^2}}+{\displaystyle \frac{{\left(z-{Ai}_z\right)}^2}{{L_{Di-Ai}}^2}}=1\\ x\ge 0, y\le 0, z\ge 0\end{array}$$

(11)

The shape of the upper breast can be determined by Equations (12) and (13).

The right region of the upper breast:

$$\left\{\begin{array}{c}{\displaystyle \frac{{\left(x·cos\theta -\left(y+{A1}_y\right)·sin\theta \right)}^2}{{H_{Ai-{Ai}^{\prime }}}^2}}+{\displaystyle \frac{{\left(\left(y+{Ai}_y\right)·cos\theta +x·sin\theta \right)}^2}{{W_{{Ai}^{\prime }-B1}}^2}}+{\displaystyle \frac{{\left(z-{A1}_z\right)}^2}{{L_{Di-Ai}}^2}}=1, \\ x\ge 0, y\ge 0, z\ge 0\\ {\displaystyle \frac{{\left(x·cos\theta -\left(y+{A1}_y\right)·sin\theta \right)}^2}{{H_{Ai-{Ai}^{\prime }}}^2}}+{\displaystyle \frac{{\left(\left(y+{Ai}_y\right)·cos\theta +x·sin\theta \right)}^2}{{W_{{Ai}^{\prime }-C1}}^2}}+{\displaystyle \frac{{\left(z-{A1}_z\right)}^2}{{L_{Di-Ai}}^2}}=1, \\ x\ge 0, y\le 0, z\le 0\end{array}\right.$$

(12)

The left region of the upper breast:

$$\begin{array}{c}\left\{\begin{array}{c}{\displaystyle \frac{{\left(x·cos\theta -\left(y+{Ai}_y\right)·sin\theta \right)}^2}{{H_{Ai-{Ai}^{\prime }}}^2}}+{\displaystyle \frac{{\left(-\left(y+{Ai}_y\right)·cos\theta -x·sin\theta \right)}^2}{{W_{{Ai}^{\prime }-Ci}}^2}}+{\displaystyle \frac{{\left(z-{Ai}_z\right)}^2}{{L_{Di-{Ai}^{\prime }}}^2}}=1, \\ x\ge 0, y\ge 0, z\ge 0\\ {\displaystyle \frac{{\left(x·cos\theta -\left(y+{Ai}_y\right)·sin\theta \right)}^2}{{H_{Ai-{Ai}^{\prime }}}^2}}+{\displaystyle \frac{{\left(-\left(y+{Ai}_y\right)·cos\theta -x·sin\theta \right)}^2}{{W_{{Ai}^{\prime }-Bi}}^2}}+{\displaystyle \frac{{\left(z-{Ai}_z\right)}^2}{{L_{Di-{Ai}^{\prime }}}^2}}=1, \\ x\ge 0, y\le 0, z\ge 0\end{array}\right.\\ x\ge 0, y\ge 0, z\ge 0\end{array}$$

(13)

According to preliminary breast shape analysis, the breast can be divided into upper and lower parts along the $z$ axis, and using a block elliptical cone to describe the upper breast shape can better reflect a realistic breast shape. The shape equation was derived using the right breast as an example. The left and right regions are divided by the plane that is perpendicular to the $y$ axis and passes the line $A1E1$ projected on the yoz plane, and both regions can be described by a part of the elliptical cone. For the complete elliptical cone (Figure 8) the local coordinate system $o-xyz$ is built with the center of the elliptical cone base as the origin and the height direction of the elliptical cone as the $z$ axis, which can be expressed by the following mathematical equations:

$$\mathrm{Complete} \mathrm{elliptical} \mathrm{cone} \mathrm{in} \mathrm{the} \mathrm{left} \mathrm{region}: {\displaystyle \frac{z^2}{{\left(h1\right) }^2}}={\displaystyle \frac{x^2}{{H_{Ai-{Ai}^{\prime }} }^2}}+{\displaystyle \frac{y^2}{{W_{{Ai}^{\prime }-Bi} }^2}}$$

(14)

$$\mathrm{Complete} \mathrm{elliptical} \mathrm{cone} \mathrm{in} \mathrm{the} \mathrm{right} \mathrm{region}: {\displaystyle \frac{z^2}{{\left(h2\right) }^2}}={\displaystyle \frac{x^2}{{H_{Ai-{Ai}^{\prime }} }^2}}+{\displaystyle \frac{y^2}{{W_{{Ai}^{\prime }-Ci} }^2}}$$

(15)

where $h1=\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathrm{\angle }FiBiAi\right)·W_{{A1}^{\prime }-Ci}, h2=\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathrm{\angle }GiCiAi\right)·W_{{Ai}^{\prime }-Ci}$. The practical upper breast only takes a part of the elliptical cones, as shown in Figure 9. Hence, based on Equations (14) and (15), we can obtain the elliptical cones in the left and right regions:

$${\displaystyle \frac{z^2}{{\left(h1\right) }^2}}={\displaystyle \frac{x^2}{{H_{Ai-{Ai}^{\prime }} }^2}}+{\displaystyle \frac{y^2}{{W_{{Ai}^{\prime }-Bi} }^2}}, x\ge 0, y\ge 0, z\ge 0$$

(16)

$${\displaystyle \frac{z^2}{{\left(h2\right) }^2}}={\displaystyle \frac{x^2}{{H_{Ai-{Ai}^{\prime }} }^2}}+{\displaystyle \frac{y^2}{{W_{{Ai}^{\prime }-Ci} }^2}}, x\ge 0, y\le 0, z\ge 0$$

(17)

By moving the elliptical cone base to the $xoy$ plane of the global coordinate system and taking a part of the elliptical cone based on the measured distance $Ei-Ai$ as the practical breast shape,

$${\displaystyle \frac{{\left(z-h1\right)}^2}{{\left(h1\right) }^2}}={\displaystyle \frac{x^2}{{H_{Ai-{Ai}^{\prime }} }^2}}+{\displaystyle \frac{y^2}{{W_{{Ai}^{\prime }-Bi} }^2}}, x\ge 0, y\ge 0, z\le {L }_{Ei-Ai}$$

(18)

$${\displaystyle \frac{{\left(z-h2\right)}^2}{{\left(h2\right) }^2}}={\displaystyle \frac{x^2}{{H_{Ai-{Ai}^{\prime }} }^2}}+{\displaystyle \frac{y^2}{{W_{{Ai}^{\prime }-Ci} }^2}}, x\ge 0, y\le 0, z\le {L }_{Ei-Ai}$$

(19)

By projecting the anchor point $Ai$ onto the $yoz$ plane, we can obtain the projected point $A11$, whose coordinate can be expressed as follows:

$$A11=Ai\times l(n, x)=(0, Ai y, Aiz)$$

(20)

By moving the base center of the elliptical cones in the left and right regions of the upper breast to the point $A11$ of the global coordinate system, we can rewrite Equations (18) and (19) as

$${\displaystyle \frac{{\left(z-h1-{Ai}_z\right)}^2}{{\left(h1\right) }^2}}={\displaystyle \frac{x^2}{{H_{Ai-{Ai}^{\prime }} }^2}}+{\displaystyle \frac{{\left(y+{Ai}_y\right)}^2}{{W_{{Ai}^{\prime }-Bi} }^2}}, x\ge 0, y\ge 0, z\le {L }_{Ei-Ai}$$

(21)

$${\displaystyle \frac{{\left(z-h2-{Ai}_z\right)}^2}{{\left(h2\right) }^2}}={\displaystyle \frac{x^2}{{H_{Ai-{Ai}^{\prime }} }^2}}+{\displaystyle \frac{{\left(y+{Ai}_y\right)}^2}{{W_{{Ai}^{\prime }-Ci} }^2}}, x\ge 0, y\le 0, z\le {L }_{Ei-Ai}$$

(22)

The complete shape of the upper and lower breasts can be obtained by Equations (23) and (24).

Elliptical cone in the left region of the upper breast:

$${\displaystyle \frac{{\left(z-h1-{Ai}_z\right)}^2}{{\left(h1\right) }^2}}={\displaystyle \frac{x^2}{{H_{Ai-{Ai}^{\prime }} }^2}}+{\displaystyle \frac{{\left(y+{Ai}_y\right)}^2}{{W_{{Ai}^{\prime }-Bi} }^2}}, x\ge 0, y\ge 0, z\le {L }_{Ei-Ai}$$

(23)

Elliptical cone in the right region of the upper breast:

$${\displaystyle \frac{{\left(z-h2-{Ai}_z\right)}^2}{{\left(h2\right) }^2}}={\displaystyle \frac{x^2}{{H_{Ai-{Ai}^{\prime }} }^2}}+{\displaystyle \frac{{\left(y+{Ai}_y\right)}^2}{{W_{{Ai}^{\prime }-Ci} }^2}}, x\ge 0, y\le 0, z\le {L }_{Ei-Ai}$$

(24)

Elliptical cone in the left region of the lower breast:

$$\begin{array}{c}{\displaystyle \frac{{\left(x·cos\theta -\left(y+{Ai}_y\right)·sin\theta \right)}^2}{{H_{Ai-{Ai}^{\prime }}}^2}}+{\displaystyle \frac{{\left(\left(y+{Ai}_y\right)·cos\theta +x·sin\theta \right)}^2}{{W_{{Ai}^{\prime }-B1}}^2}}+{\displaystyle \frac{{\left(z-{Ai}_z\right)}^2}{{L_{Di-Ai}}^2}}=1\\ x\ge 0, y\ge 0, z\ge 0\end{array}$$

(25)

Elliptical cone in the right region of the lower breast:

$$\begin{array}{c}{\displaystyle \frac{{\left(x·cos\theta -\left(y+{Ai}_y\right)·sin\theta \right)}^2}{{H_{Ai-{Ai}^{\prime }}}^2}}+{\displaystyle \frac{{\left(\left(y+{Ai}_y\right)·cos\theta +x·sin\theta \right)}^2}{{W_{{Ai}^{\prime }-Ci}}^2}}+{\displaystyle \frac{{\left(z-{Ai}_z\right)}^2}{{L_{Di-Ai}}^2}}=1\\ x\ge 0, y\le 0, z\ge 0\end{array}$$

(26)

Then, we reflect the above equation along the $xoz$ plane, that is, change $\left(y+{Ai}_y\right)·cos\theta +x·sin\theta$ to $-\left(y+{Ai}_y\right)·cos\theta -x·sin\theta$ and then to $-y-{Ai}_y$. Hence, the shape of the left breast can be determined by Equations (27)–(30).

Ellipsoid in the left region of the upper breast:

$${\displaystyle \frac{{\left(z-h2-{Ai}_z\right)}^2}{{\left(h2\right)}^2}}={\displaystyle \frac{x^2}{{H_{Ai-{Ai}^{\prime }}}^2}}+{\displaystyle \frac{{\left(y+{Ai}_y\right)}^2}{{W_{{Ai}^{\prime }-Ci}}^2}}, x\ge 0, y\le 0, z\le L_{Ei-Ai}$$

(27)

Ellipsoid in the right region of the upper breast:

$${\displaystyle \frac{{\left(z-h1-{Ai}_z\right)}^2}{{\left(h1\right)}^2}}={\displaystyle \frac{x^2}{{H_{Ai-{Ai}^{\prime }}}^2}}+{\displaystyle \frac{{\left(-y-{Ai}_y\right)}^2}{{W_{{Ai}^{\prime }-Bi}}^2}}, x\ge 0, y\ge 0, z\le L_{Ei-Ai}$$

(28)

Ellipsoid in the left region of the lower breast:

$$\begin{array}{c}{\displaystyle \frac{{\left(x·cos\theta -\left(y+{Ai}_y\right)·sin\theta \right)}^2}{{H_{Ai-{Ai}^{\prime }}}^2}}+{\displaystyle \frac{{\left(-\left(y+{Ai}_y\right)·cos\theta -x·sin\theta \right)}^2}{{W_{{Ai}^{\prime }-Ci}}^2}}+{\displaystyle \frac{{\left(z-{Ai}_z\right)}^2}{{L_{Di-Ai}}^2}}=1\\ x\ge 0, y\le 0, z\ge 0\end{array}$$

(29)

Ellipsoid in the right region of the lower breast:

$$\begin{array}{c}{\displaystyle \frac{{\left(x·cos\theta -\left(y+{Ai}_y\right)·sin\theta \right)}^2}{{H_{Ai-{Ai}^{\prime }}}^2}}+{\displaystyle \frac{{\left(-\left(y+{Ai}_y\right)·cos\theta -x·sin\theta \right)}^2}{{W_{{Ai}^{\prime }-Bi}}^2}}+{\displaystyle \frac{{\left(z-{Ai}_z\right)}^2}{{L_{Di-Ai}}^2}}=1\\ x\ge 0, y\ge 0, z\ge 0\end{array}$$

(30)

3.1.2. Breast-Volume Calculation Formula

To calculate the lower breast volume, we need to calculate the volume of the complete ellipsoid. The left region of the right breast is taken as an example. Let $f\left(x,y,z\right)={\displaystyle \frac{x^2}{{H_{Ai-{Ai}^{\prime }}}^2}}+{\displaystyle \frac{y^2}{{W_{{Ai}^{\prime }-Bi}}^2}}+{\displaystyle \frac{z^2}{{L_{Di-Ai}}^2}}$. According to the features of triple integrals, the ellipsoid volume (V) can be expressed as:

$$\begin{array}{l}V=\underset{\Omega }{\iiint }f\left(x, y, z\right)dxdydz\\ \Omega :{\displaystyle \frac{x^2}{{H_{Ai-{Ai}^{\prime }}}^2}}+{\displaystyle \frac{y^2}{{W_{{Ai}^{\prime }-Bi}}^2}}+{\displaystyle \frac{z^2}{{L_{Di-Ai}}^2}}\le 1\end{array}$$

(31)

Let ${\displaystyle \frac{x}{H_{Ai-{Ai}^{\prime }}}}=X,{\displaystyle \frac{y}{W_{{Ai}^{\prime }-Bi}}}=Y,{\displaystyle \frac{z}{L_{Di-Ai}}}=Z$, then Equation (31) can be rewritten as:

$$\begin{array}{ll}V& =\underset{\Omega }{\iiint }f\left(H_{Ai-{Ai}^{\prime }}X, W_{{Ai}^{\prime }-Bi}Y, L_{Di-Ai}Z\right)H_{Ai-{Ai}^{\prime }}·dX{·W}_{{Ai}^{\prime }-Bi}·dY·L_{Di-Ai}·dZ\\ & =H_{Ai-A{i}^{\prime }}{·W}_{{Ai}^{\prime }-Bi}·L_{Di-Ai}\iiint \left(X^2+Y^2+Z^2\right)dXdYdZ\end{array}$$

(32)

Then, the spherical coordinate system is used. Let

$$\left\{\begin{array}{l}X=rsin\theta sint\\ Y=rsin\theta cost\\ Z=rcos\theta \end{array}\right.$$

(33)

where $r\in \left[0,1\right],t\in \left[0,2\pi \right],\theta \in [0,\pi ]$. According to Equation (33),

$$dXdYdZ=r^{2}sin\theta drd\theta dt$$

(34)

By substituting Equation (34) into Equation (32),

$$\begin{array}{ll}V& =H_{Ai-A{i}^{\prime }}{·W}_{{A1}^{\prime }-B1}·L_{Di-Ai}\underset{\Omega }{\iiint }\left(X^2+Y^2+Z^2\right)dXdYdZ\\ & =\underset{\Omega }{\iiint }{r}^{2}sin\theta drd\theta dt=H_{Ai-A{i}^{\prime }}{·W}_{{Ai}^{\prime }-Bi}·L_{Di-Ai}{\int }_0^{2\pi }dt{\int }_0^{\pi }sin\theta d\theta {\int }_0^1{r}^{2}dt\\ & ={\displaystyle \frac{4\pi }{3}}{H}_{Ai-{Ai}^{\prime }}{·W}_{{Ai}^{\prime }-Bi}·L_{Di-Ai}\end{array}$$

(35)

Hence, the complete ellipsoid volume of the left region of the right breast can be determined by $V={\displaystyle \frac{4\pi }{3}}{H}_{Ai-{Ai}^{\prime }}{·W}_{{Ai}^{\prime }-Bi}·L_{Di-Ai}$. Likewise, the complete ellipsoid volume of the right region of the right breast can be determined by $V={\displaystyle \frac{4\pi }{3}}{H}_{Ai-{Ai}^{\prime }}{·W}_{{Ai}^{\prime }-Ci}·L_{Di-Ai}$; the complete ellipsoid volume of the right region of the left breast can be determined by $V={\displaystyle \frac{4\pi }{3}}{H}_{Ai-{Ai}^{\prime }}{·W}_{{Ai}^{\prime }-Bi}·L_{Di-{Ai}^{\prime }}$; and the complete ellipsoid volume of the left region of the left breast can be determined by $V={\displaystyle \frac{4\pi }{3}}{H}_{Ai-{Ai}^{\prime }}{·W}_{{Ai}^{\prime }-Ci}·L_{Di-Ai}$.

Based on the above derived equations of the complete ellipsoid volume, we can obtain an equation for the practical volume of the lower breast: left /right breast volume

$$\begin{array}{l}V={\displaystyle \frac{1}{8}}·{\displaystyle \frac{4\pi }{3}}{H}_{Ai-A{i}^{\prime }}{·W}_{{Ai}^{\prime }-Bi}·L_{Di-Ai}+{\displaystyle \frac{1}{8}}·{\displaystyle \frac{4\pi }{3}}{H}_{Ai-A{i}^{\prime }}{·W}_{{Ai}^{\prime }-Ci}·L_{Di-Ai}\\ ={\displaystyle \frac{\pi }{6}}{H}_{Ai-{Ai}^{\prime }}·L_{Di-Ai}\left(W_{A{i}^{\prime }-Bi}+W_{{Ai}^{\prime }-Ci}\right)\end{array}$$

(36)

The volume of the upper breast is the partial elliptical-cone volume. Hence, to obtain the volume of the upper breast, we need to first derive the equation of the complete elliptical-cone volume. It is assumed that the semi-major axis of the elliptical-cone base is $a$, the semi-minor axis of the elliptical-cone base is $b$, and the height of the elliptical cone is $h$. According to the triangle similarity theorems, we have

$${\displaystyle \frac{A}{a}}={\displaystyle \frac{h-z}{h}}$$

(37)

According to Equation (37),

$$A={\displaystyle \frac{\left(h-z\right)}{h}}a$$

(38)

According to the triangle similarity theorems,

$${\displaystyle \frac{B}{b}}={\displaystyle \frac{h-z}{h}}$$

(39)

According to Equation (39),

$$B={\displaystyle \frac{\left(h-z\right)}{h}}b$$

(40)

The complete elliptical-cone volume can thus be calculated as

$$\begin{array}{c}V={\int }_0^h\pi ABdz={\int }_0^h\pi ·{\displaystyle \frac{\left(h-z\right)}{h}}a·{\displaystyle \frac{\left(h-z\right)}{h}}bdz\\ ={\displaystyle \frac{\pi ab}{h^2}}{\int }_0^h{\left(h-z\right)}^{2}dz={\displaystyle \frac{1}{3}}\pi abh\end{array}$$

(41)

Based on the derived calculation equation of the elliptical-cone volume, we can derive the calculation equation of the practical volume of the upper breast using a similar method. The volume of the left region of the upper breast can be calculated by

$$\begin{array}{ll}V1& ={\displaystyle \frac{V}{4}}={\displaystyle \frac{1}{4}}{\int }_0^{L_{Ei-Ai}}\pi ABdz\\ & ={\displaystyle \frac{1}{4}}{\int }_0^{L_{Ei-Ai}}\pi ·{\displaystyle \frac{\left(h_1-z\right)}{h_1}}·H_{Ai-{Ai}^{\prime }}·{\displaystyle \frac{\left(h_1-z\right)}{h_1}}·W_{{Ai}^{\prime }-Bi}d\\ & ={\displaystyle \frac{\pi ·H_{Ai-{Ai}^{\prime }}·W_{{Ai}^{\prime }-Bi}}{4{h_1}^2}}{\int }_0^{L_{Ei-Ai}}{\left(h_1-z\right)}^{2}dz\\ & ={\displaystyle \frac{\pi ·H_{Ai-{Ai}^{\prime }}·W_{{Ai}^{\prime }-Bi}}{12h_1^2}}\left[h_1^3-{\left(h_1-L_{Ei-Ai}\right)}^3\right]\end{array}$$

(42)

Likewise, the volume of the right region of the upper breast can be calculated by

$$V2={\displaystyle \frac{V}{4}}={\displaystyle \frac{\pi ·H_{Ai-A{i}^{\prime }}·W_{{Ai}^{\prime }-Ci}}{12h_2^2}}\left[h_2^3-{\left(h_2-L_{Ei-Ai}\right)}^3\right]$$

(43)

Therefore, the volume of the upper breast can be calculated by

$$\begin{array}{c}{V}_{\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{e}\mathrm{r}}={\displaystyle \frac{\pi ·H_{Ai-{Ai}^{\prime }}·W_{{Ai}^{\prime }-Bi}}{12h_1^2}}\left[h_1^3-{\left(h_1-L_{Ei-Ai}\right)}^3\right]\\ +{\displaystyle \frac{\pi ·H_{Ai-{Ai}^{\prime }}·W_{{Ai}^{\prime }-Ci}}{12h_2^2}}·\left[h_2^3-{\left(h_2-L_{Ei-Ai}\right)}^3\right]\end{array}$$

(44)

Combined with Equation (36), the complete breast volume can be calculated by

$$\begin{array}{c}V={\displaystyle \frac{\pi }{6}}{H}_{Ai-{Ai}^{\prime }}·L_{Di-Ai}\left(W_{{Ai}^{\prime }-Bi}+W_{{Ai}^{\prime }-Ci}\right)+{\displaystyle \frac{\pi ·H_{Ai-{Ai}^{\prime }}·W_{{Ai}^{\prime }-Bi}}{12h_1^2}}\\ ·\left[h_1^3-{\left(h_1-L_{Ei-Ai}\right)}^3\right]+{\displaystyle \frac{\pi ·H_{Ai-{Ai}^{\prime }}·W_{{Ai}^{\prime }-Ci}}{12h_2^2}}·\left[h_2^3-{\left(h_2-L_{Ei-Ai}\right)}^3\right]\end{array}$$

(45)

3.2. Breast-Volume Measurement Results

In the operation of the breast specimen drainage-volume measurement method, the volume of water displaced by the unilateral breast simulation model was used to replace the volume of the breast model. The resulting volume of the unilateral medical breast model was 425 mL. To standardize the measurement units, the breast volume was converted to 425 cm³.

In the proposed measurement method for breast volume (Figure 10), the breast medical-simulation model showed that the coordinate of the nipple point $A$ in the proposed breast shape-measurement coordinate system was $(x, y, z)=(5.5,-10,7.5)$, the arc distance of the upper breast border $AE=12$, the arc distance of the lower breast border $AD=7.5$, the arc distance of the front breast border $AB=9$, the arc distance of the lateral breast border $AC=8.5$, and the breast height ${AA}^{\prime }=5.5$. It was assumed that $C1G1=F1B1, E1G1=A1C1, E1F1=A1B1, \mathrm{a}\mathrm{n}\mathrm{d} h_1=L_{Ei-Ai}$. Based on Equation (45), we replaced the measurement parameters in the equations and obtained

$$\begin{array}{l}{H}_{Ai-{Ai}^{\prime }}={AA}^{\prime }=5.5,L_{Di-Ai}=\sqrt{{AD}^2-{{AA}^{\prime }}^2}=5.099\\ W_{{A1}^{\prime }-B1}=\sqrt{{AB}^2-{{AA}^{\prime }}^2}=7.1239,W_{{A1}^{\prime }-C1}=\sqrt{{AC}^2-{{AA}^{\prime }}^2}=6.4807\\ h1=L_{Ei-Ai}=\sqrt{{AE}^2-{{AA}^{\prime }}^2}=10.6654\\ \begin{array}{cc}V=& {\displaystyle \frac{\pi }{6}}{H}_{Ai-{Ai}^{’}}·L_{Di-Ai}\left(W_{{A1}^{\prime }-B1}+W_{{A1}^{\prime }-C1}\right)+{\displaystyle \frac{\pi ·H_{Ai-{Ai}^{\prime }}·W_{{A1}^{\prime }-B1}}{12h_1^2}}\hfill \\ & ·\left[h_1^3-{\left(h_1-L_{Ei-Ai}\right)}^3\right]+{\displaystyle \frac{\pi ·H_{Ai-{Ai}^{\prime }}·W_{{A1}^{\prime }-C1}}{12h_2^2}}·\left[h_2^3-{\left(h_2-L_{Ei-Ai}\right)}^3\right]=408.7\end{array}\end{array}$$

Given that the breast volume of the reference group, measured by the gold standard, is 425 cm³, the breast-volume results calculated using different methods for the comparison group were summarized. The calculated breast volumes and relative error data for each method in the comparison group are presented in Figure 11.

As shown in Figure 11, a comparison reveals that the new breast-volume measurement method proposed in this study has the smallest relative error compared to the reference group, at only 3.8%, demonstrating the high accuracy of this method. The 3D scanning method follows, with a relative error of 4.8%. The breast-volume equation method yielded the largest relative error, reaching 86.3%.

4. Discussion

The primary finding of this study is that the proposed self-measurement method for breast volume achieved a relative error of only 3.8%, demonstrating superior accuracy compared to traditional methods, including 3D scanning (4.8%) and conventional volume equations (86.3%). This high accuracy indicates that the method provides reliable breast-volume data, making it a practical solution for applications in bra sizing, personal health monitoring, and body image assessments.

This study introduces an innovative approach for self-measuring breast volume, utilizing a new volume calculation formula based on a geometric approximation model that combines irregular ellipsoids and elliptical cones. Accurate breast measurements are essential for proper bra sizing, as they form a foundational basis for lingerie design and production. To optimize the model’s accuracy, five key measurement parameters were selected, including upper-cup vertical distance, lower-cup vertical distance, front-cup arc distance, side-cup arc distance, and breast height. The proposed method thus stands out in terms of accuracy, ease of use, and applicability.

The analysis shows that the specimen drainage method, considered the “gold standard” for breast-volume measurement, involves immersing breast specimens in a water-filled container and calculating the displaced water volume to determine breast volume. Due to its fixed sample nature, this method offers high repeatability, but is impractical for daily measurements, making it somewhat limited [34]. Additionally, breast morphology changes are equally important for breast shape analysis, not just absolute volume data [35]. In this study, the defined breast-shape characteristics primarily assess both breast volume (size) and breast shape features (length, width, height, position, and other shape parameters). The combined measurement of breast volume and shape features is also a key indicator for assessing bilateral breast asymmetry [36,37], allowing for a complete evaluation of the overall breast shape. These findings align with previous studies that evaluated breast volume, shape, and surface area [38]. Although 3D scanning provides accurate measurements, it is expensive and cannot directly capture breast volume, with limitations in detecting finer details [39]. Furthermore, most scanning devices used for human measurement rely on laser technology, which may pose potential risks if used improperly [40], and each scan can be costly [41]. These factors restrict the use of 3D scanning in routine human measurements. Unlike the complex and expensive 3D scanning technology, this new method simplifies breast-volume measurement, reducing the need for specialized equipment and enhancing the feasibility of self-assessment.

This study developed a novel breast shape-measurement tool based on the newly proposed breast-volume measurement method, which has been granted a Chinese patent (Patent No.: ZL 2022 1 0831873.1). The tool is wearable, and its measurement parameters are derived from the method proposed in this research, focusing on key breast dimensions such as circumference (underbust), length (upper-bust vertical distance, lower-bust arc distance), width (anterior-bust arc distance, lateral-bust arc distance), and height (breast height). These parameters are used to calculate breast volume and assess bilateral breast asymmetry. To ensure the accuracy of self-measurement, the tool includes a standard operating procedure. Additionally, a WeChat mini-program (v. 8.0.53) has been developed to enable efficient data recording, conversion, and analysis when using the new breast shape-measurement tool.

This study also highlights the relationship between breast volume and women’s psychological health and body image. Previous research has shown that body measurements and body shape perception significantly impact women’s self-awareness and psychological well-being [42]. Therefore, accurate breast-volume measurements not only aid in improving bra selection, but also help women develop a positive body image. The self-measurement method developed in this study supports daily health monitoring and lays the foundation for future research on body image and psychological health. This method allows for long-term tracking of changes in breast volume, helping women better understand their physical condition, thereby enhancing self-confidence and self-acceptance.

While this study has made significant progress in the development and preliminary validation of a breast-volume measurement method, we acknowledge certain limitations in the research design. First, the sample size was relatively limited, as we only used one unilateral and one bilateral standardized medical breast model. Although these initial results provided a solid foundation for verifying the accuracy of the formula, future studies should expand the sample size to further validate the applicability of this method across different populations. Additionally, a formal power analysis to determine the required sample size has not yet been conducted. However, we plan to include power calculations in future studies to optimize the research design and ensure that the selected sample size can robustly verify the reliability and validity of the method.

Second, while the geometric model demonstrated high accuracy in this study, the substantial variation in breast morphology among individuals suggests that further refinement is necessary. Future research could optimize the geometric model to accommodate more complex or larger breast shapes, thus improving the method’s general applicability.

Accurate breast-volume measurement plays a crucial role not only in health management and bra design but also in breast reconstruction and cosmetic procedures, particularly for correcting breast asymmetry [43]. Precise volume estimation is essential for surgical planning, as it helps ensure symmetrical outcomes by accurately assessing the volume of each breast [44]. Although the self-measurement method proposed in this study was preliminarily validated using medical models, it has shown great potential in providing accurate volume data, and could be applied in preoperative evaluations. In future studies, we plan to further validate the applicability of this method in real-world scenarios.

Furthermore, future research should explore the broader applications of this method in health and nutritional monitoring, especially its role in promoting long-term physiological and psychological well-being. With further refinement and widespread adoption, the proposed method could become a valuable tool for monitoring women’s health and assessing psychological wellness.

5. Conclusions

This study successfully proposes a novel breast-volume self-measurement method that significantly enhances measurement accuracy and convenience through an innovative application of geometric models. The newly developed volume-calculation formula, based on a combination of irregular-ellipsoid and elliptical cone models, was validated experimentally, and demonstrated a relative error of only 3.8%, substantially outperforming existing methods, such as 3D scanning (4.8%) and traditional volume equations (86.3%). This high accuracy suggests that the proposed method provides reliable breast-volume data, which can improve bra sizing accuracy, fit, and comfort.

Compared to current methods, this approach offers ease of operation and reduces reliance on specialized equipment, making it feasible for routine use in personal health monitoring, bra design, and body image assessment. Additionally, the method’s simplicity enhances its applicability for a variety of health-related scenarios without the need for extensive technical expertise or expensive tools.

Future research can build on this foundation by further refining the geometric model to accommodate different breast shapes, thereby broadening the method’s applicability. Development of corresponding digital measurement systems could also expand its usability. With these optimizations, the proposed method holds promise for supporting nutritional assessments, anthropometry, and studies on psychological health, thus making it a valuable tool for comprehensive health and wellness monitoring.

6. Patents

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