The relevance of breast motions and gaits in running exercises

Abstract

The control of breast motions is a critical indicator to evaluate the comfort and function of sports bras. If the breast motions can be predicted based on the gait parameters detected by wearable sensors, it will more economical and convenient to evaluate the bras. Thirteen unmarried Chinese females with a breast cup of 75B were recruited in this study to investigate the regularity of breast motions and the relevance between breast motions and gaits during running exercises. The breast motion indicator is the distance alteration of breast regions. The gaits were described by the rotation angles of the hip, knee, ankle joints, and the foot height off the ground. Firstly, the Mann-Whitney U test and the Kruskal-Wallis H test were utilized to analyze the motion diversity among the eight breast regions. Then, the gray correlation analysis was applied to explore the relevance between breast motions and gaits. Finally, the back-propagation neural network, the genetic algorithm, and the particle swarm optimization algorithm were utilized to construct the prediction models for breast motions based on gait parameters. The results demonstrate that the same breast regions on the bilateral breasts and the different breast regions on the ipsilateral breasts present a significant motion diversity. There is a moderate correlation between breast motions and gait parameters, and the back-propagation neural network optimized by the particle swarm optimization algorithm performs better in breast motion prediction, which has a coefficient of determination of 84.58% and a mean absolute error of 0.2108.

Introduction

The female breasts primarily consist of external skins and internal structures (fibro-adipose and fibro-glandular tissues) (McGhee & Steele, 2020a). The thickness and elasticity of human skins decrease as age increases (Coltman et al., 2017; Den Tonkelaar et al., 2004). Besides, the configuration of the internal structure is affected by age, race, weight, hormones, and so on (Boyd et al., 2009; Huang et al., 2011; Lee et al., 1997; McGhee & Steele, 2020a). Specifically, the proportion of fibro-glandular tissue decreases while the fibro-adipose percentage increases by the aging process (Boyd et al., 2009; Huang et al., 2011; Lee et al., 1997). It was well-documented that the lack of breast support causes women to experience frequent breast pain during exercises (McGhee et al., 2007). Hence, enough external support for breasts is necessary for women. However, according to the relevant surveys, 56% of females have experienced breast pain during exercises (Lorentzen & Lawson, 1987). The larger breasts are, the more painful the sensations that women feel. As bras providing with sufficient support are capable of reducing the excessive breast motions and even the perceived pains (Scurr et al., 2011), numerous studies have been done to observe the characteristics of the breast motions among different exercises to improve bra support (McGhee et al., 2013; Scurr et al., 2011; Zhou et al., 2012).

Due to a lack of advanced measurement techniques and awareness of breast motions, Gerhlsen and Albohm only investigated the absolute displacement of the breasts in the coronal plane (Gehlsen & Albohm, 1980). The displacement trajectory tended to a figure-of-eight pattern. With the development of motion capture techniques, researchers began to investigate the breast motions in three dimensions (Scurr et al., 2009; Wood et al., 2012; Zhou et al., 2012). Awareness of the relative property of breast motions in three dimensions also improves the development of a series of new coordinate systems and breast motion indicators (Milligan et al., 2015; Scurr et al., 2009; Zhou et al., 2009, 2012). Zhou et. al. (2012) established a breast coordinate system based on four torso markers to eliminate the torso motions from the breast motions. The motions of six breast points presented a butterfly shape. In addition, it was speculated that relative breast motions were caused by the breast deformations and the stretching or contracting behavior of pectoralis major muscle (McGhee & Steele, 2020a). Lots of numerical models have also been constructed to study the constitutive material coefficients for breast tissues. Cai et. al. (2018) established a mass-spring-damper model for simulating breast motions. The abstract model verified the diversity of breast motions among difference regions. Most of the existing researches focused on a series of point motion for the breasts, whereas the point motion cannot effectively reflect the deformation of the breasts. Thus, this study aims to investigate the extension and contraction of the breast surface based on the distance alterations of different breast regions. This awareness may help bra designers to improve the bra support in terms of the bra structure and fabric elasticity.

During exercises, breast motions are influenced by the torso driving force, the viscoelastic mechanical properties, and the bra’s supporting ability (Haake & Scurr, 2010). Previous literature reported that the females may change their body posture to reduce the breast motions and pains (White et al., 2009). Milligan et. al. (2015) compared the motion patterns of human torso, pelvis, and upper arm during a 5 km running exercise. The results demonstrated that the high supportive bras perform better in terms of decreasing arm extension than the lower one. In addition, the breast displacements increase as the stride length exaggerates (Eden et al., 1992). Meanwhile, the rhythm of running or walking for females wearing sports bras was significantly lower than those baring breasts (Li et al., 2018). There might be a relationship between breast motions and gait patterns. Such experimental motion analyses are very restricted by their complicated setup, large workload of data cleaning and processing. If the breast motions can be predicted based on a wider breadth of gait patterns, the bra’s support will be evaluated, thus more information on bra performance is bale to be provided efficiently. Besides, by deeper perception of how lower limb movements influence breast motions, the bra designers can optimize the application-centered bra design. Therefore, it is necessary to explore the patterns of breast motions under different gaits. The results will provide a scientific basis for the design of sports bras.

The back-propagation (BP) neural network is a kind of feed-forward neural network with a simple structure and wide applications. The neural network can approximate any continuous function and square productive function with arbitrary accuracy (Zhou & Ma, 2019). Thus, this method has been widely applied to deal with non-linear problems such as girdle pressure prediction (Zhou & Ma, 2020), fabric surface detection (Jin et al., 2020), and flow stress prediction (Ding et al., 2020). However, the BP neural network has great defects in threshold and weight initialization. To avoid the BP neural network falling into the local optimum, researchers (Ding et al., 2020; Zhang & Guo, 2019; Zhou & Ma, 2020) usually adopted the genetic algorithm (GA) and the particle swarm optimization (PSO) to optimize the BP neural network. The GA is a computational model of biological evolution. This method searches for the optimal solution by simulating the natural evolutionary process (Zhang & Guo, 2019). The PSO is a kind of optimization algorithm based on iteration, and it can search the optimal threshold and weight of the BP neural network in a large space (Ding et al., 2020).

The distance alteration of the breast regions was applied to investigate the characteristics and diversities of breast motions in this study to address the above mentioned issues. Meanwhile, four gait parameters, including the hip, the knee, the ankle joint motion angles, and the foot height off the ground, were utilized to explore the correlation between breast motions and gaits. Moreover, the BP, the back-propagation neural network optimized by the genetic algorithm (GA-BP), and the back-propagation neural network optimized by the particle swarm optimization (PSO-BP) were adopted to establish the prediction models for breast motions based on gait parameters. Therefore, the following are the research questions to be answered. (1) What characteristics do the breast motions present in different parts during running exercise, and whether the motion differs in various breast regions? (2) What kind of correlation exists between breast motions and gait parameters? (3) If there is a desirable correlation between breast motions and gaits, the BP neural network optimized by the GA and PSO algorithms will be utilized to forecast breast motions. The mean absolute error and the coefficient of determination were determined to evaluate the accuracy of the models.

Methods

Subjects

Fifteen young Chinese females were recruited for measuring their breast sizes. B-cup breasts are the most prevalent in unmarried Chinese females (Liang et al., 2007; Mao et al., 2020). Thus, thirteen subjects with B-cup were chosen and two females with other cups were excluded in this study. All subjects were given consent form and had no pregnancy or breast disease experience. Ethical approval was obtained from the ethics committee of Xi'an Polytechnic University. The essential physical characteristics of the subjects are shown in Table 1.

Table 1 The essential physical characteristics of the subjects

Data capture

The experiment was carried out in a quiet, windless laboratory, with 25 ± 2 °C and relative humidity of 65 ± 3%. The subjects were asked to run on a treadmill (slope of 0°) at 7 km/h with nude upper body and tight-fitting athletic pants. Sixteen reflective markers were pasted on their bodies. The three-dimensional (3D) coordinates of the reflective markers were collected at 120 Hz frequency by a Vicon 612 (Oxford Metric, UK) motion analysis system. The system is comprised of 12 M-series cameras and its mean residual is 0.605 mm.

The positions of ten breast markers and six leg markers were defined to explore breast motions and gaits (Ren et al., 2015; Wang et al., 2019; Zhou et al., 2012). The definitions of the marker positions are shown in Table 2.

Table 2 The definitions of the marker positions

Firstly, the subjects carried out a 3-min exercise on the treadmill following their running habits. Then, the subjects were asked to stand naturally and a professional technician pasted the reflective markers on their body based on the Table 2. Finally, the Vicon workstation software (v5.2.9) was utilized to track and record the 3D coordinates of the reflective markers during natural standing and running. No less than ten seconds of natural standing and one minute of running were recorded for three times. The data without missing values for each marker at the same time were adopted for this study.

Data processing

The gait cycles were divided according to the vertical coordinate of the Ll₃. A complete gait cycle was defined as a nadir to another nadir in the vertical coordinate. Three complete consecutive gait cycles were extracted for each subject. As a result, a total of 39 gait cycles were obtained to analyze the breast motions and gaits.

Breast motion parameters

The displacement, velocity, and acceleration were utilized to investigate the breast point’s motions (Scurr et al., 2010; Zhou et al., 2009), however, the motions of the breast areas were ignored in these reports. The distance alteration of the breast regions ($\Delta d$) was applied in this study to explore the extension and contraction of the breast surface during exercises as shown in Eq. (1). The eight breast regions are divided as shown in Fig. 1.

$$\Delta d_{ij} = d_{ij} \left( t \right) - d^{\prime}_{ij} ,$$

(1)

where d_ij(t) is the euclidean distance between breast point i and j at moment t, $d^{\prime}_{ij}$ is the mean euclidean distance between breast point i and j in a natural standing state.

Fig. 1

The breast regions

To improve the quality of the data, the normality and outliers of $\Delta d$ were checked. Measurements of $\Delta d$ that smaller than the first quartile—(1.5 × interquartile range) or larger than the third quartile + (1.5 × interquartile range) were considered as outliers (Shen et al., 2018). The change of the data before and after deleting the abnormal values is shown in Fig. 2.

Fig. 2

The outliers of the $\Delta d$

As shown in Fig. 2, the $\Delta d$ data maintain the same distribution after eliminating the outliers. Compared with the right breasts, there are more abnormal values in the left breasts, especially in the d₀₁ regions.

Gait parameters

The vertical displacement of the breasts is closely related to foot strikes during running (McGhee & Steele, 2020b). Therefore, the foot height off the ground ($\Delta h$), the hip, the knee, and the ankle joint rotation angles ($\alpha$, $\gamma$, $\beta$) were determined as the gait parameters in this study. And the illustration of four parameters (left leg) is shown in Fig. 3.

Fig. 3

Gait parameters (left leg)

The Ll₃′ denotes the coordinates of the ankle joint points in a natural standing state, and the four parameters were calculated in Eqs. (2) and (3).

$$\begin{aligned} a & = \sqrt {\left( {x_{1} - x_{2} } \right)^{2} + \left( {y_{1} - y_{2} } \right)^{2} + \left( {z_{1} - z_{2} } \right)^{2} } \\ b & = \sqrt {\left( {x_{1} - x_{3} } \right)^{2} + \left( {y_{1} - y_{3} } \right)^{2} + \left( {z_{1} - z_{3} } \right)^{2} } \\ c & = \sqrt {\left( {x_{2} - x_{3} } \right)^{2} + \left( {y_{2} - y_{3} } \right)^{2} + \left( {z_{2} - z_{3} } \right)^{2} } \\ \end{aligned}$$

(2)

where a is the euclidean distance between Ll₁ and Ll₂, b is the euclidean distance between Ll₁ and Ll₃, and c is the euclidean distance between Ll₂ and Ll₃.

$$\alpha = \arccos \;\frac{{\left| {y_{1} - y_{2} } \right|}}{a},\;\beta = \arccos \frac{{\left| {y_{3} - y_{2} } \right|}}{c},\;\gamma = 180^{^\circ } - \arccos \frac{{a^{2} + c^{2} - b^{2} }}{2ac},\;\Delta h = z_{3} - z_{3}^{\prime } .$$

(3)

Prediction models for breast motions

Numerous researches for breast motions are based only on running and they lack other exercises (McGhee & Steele, 2020a). With the development of wearable sensors, it is convenient and easy to detect the gaits outdoor (Prakash et al., 2018). If the breast motions are able to be predicted by gait parameters, the bra comfort and support will be evaluated in any situation. The BP neural network has excellent nonlinear mapping capability (Zhou & Ma, 2019), while the GA and the PSO algorithms can effectively optimize the BP neural network. The BP, the GA-BP, and the PSO-BP algorithm were applied in this study to establish the prediction models for breast motions.

The principle for BP neural network

Matlab 2017B version (MathWorks, USA) was utilized in this study to construct a BP neural network reflecting the nonlinear mapping relationship between breast motion indicators and gait parameters. Meanwhile, the gray correlation degree was adopted to calculate the weight of gait parameters ($\xi$) related to $\Delta d$. The $\xi$ was calculated in Eq. (4).

$$\xi_{ij} = \frac{{\delta_{ij} }}{{\sum\limits_{i = 1}^{8} {\delta_{ij} } }},$$

(4)

where i and j are the gait parameters and the $\Delta d$, respectively. $\delta_{ij}$ denotes the gray correlation degree between i and j. The results of $\xi$ are shown in Table 3.

Table 3 Weights of gait parameters related to $\Delta d$

There were nine nodes in the input layer, including eight weighted gait parameters and breast region numbers (the breast regions d_01(R), d_02(R), d_03(R), d_04(R), d_01(L), d_02(L), d_03(L), d_04(L) were set as 1, 2, 3, 4, 5, 6, 7, 8, respectively). The output layer was the value of $\Delta d$. The optimal number of hidden layer nodes was mainly obtained by the trial-and-error method and the mean absolute error (MAE) was utilized as the evaluation index. The number was first set to five and then gradually increased to 50. When the number was 35, the MAE reached the smallest and the BP neural network performed better. Thus, the topology of the BP neural network was finally determined to be 9-35-1, as shown in Fig. 4.

Fig. 4

The topology of BP neural network

There are 21,827 sets of data in the sample set. In this study, 21,500 sets of data were randomly selected as the training set of the models, and the rest were the test set (327). In order to accelerate the convergence of the BP neural network, the input and output data were normalized to the range of [− 1, 1] based on the Eq. (5).

$$y = \frac{{\left( {y_{\max } - y_{\min } } \right)\left( {x - x_{\min } } \right)}}{{x_{\max } - x_{\min } }} + y_{\min } ,$$

(5)

where x and y represent the original data and normalized data, respectively.

The transfer functions of the BP neural network include linear function (purelin) and sigmoid function (logsig and tansig). The output of the purelin function is an arbitrary value. The logsig function maps the neuron from (− ∞, + ∞) to (0, 1), while the tansig function maps the neuron from (− ∞, + ∞) to (− 1, 1). Therefore, the tansig and the purelin were determined as the transfer function of the hidden and output layer, respectively. Meanwhile, the training function of the models is trainlm in this study. The maximum number of iteration steps, the training target, the learning rate, and the number of verification failures were set as 1000, 0.001, 0.01, and 6, respectively.

In order to avoid the BP neural network falling into the local optimum, we applied the GA and PSO algorithms to optimize the initial weights and thresholds for the BP neural network. Meanwhile, the topology of the BP neural network was maintained.

The principle for GA-BP neural network

The principle for the GA algorithm is to utilize individuals to represent the initial weights and thresholds of the BP neural network. Meanwhile, the GA algorithm takes the prediction error of the initial neural network as an individual fitness value to find the optimal initial parameters (Ding et al., 2020). The parameters of the GA algorithm were initialized firstly in this study. After repeated tests, the population size, the maximum number of iteration, the crossover probability, and the mutation probability were determined to be 8, 50, 0.4, and 0.2, respectively. Then, the real coding was utilized for individual coding, and the length of coding was calculated in Eq. (6).

$$S = S_{1} S_{2} + S_{1} S_{3} + S_{1} + S_{3} ,$$

(6)

where S₁ is the number of hidden layer nodes, S₂ and S₃ are the number of the input layer and output layer nodes, respectively.

The next step is to select the fitness function. The individual was utilized to represent the initial weights and thresholds of the neural network, and the absolute error of the neural network initialized by the individual was adopted as the fitness value F as shown in Eq. (7).

$$F = k\left( {\sum\limits_{i = 1}^{{S_{3} }} {\left| {y_{i} - o_{i} } \right|} } \right),$$

(7)

where k is the coefficient, y_i and o_i are the expected output and the predictive output, respectively.

The fitness ratio method was adopted to carry out the selection operation of the genetic algorithm. If an individual is set as X_i and its fitness is F_i, the selection probability is P_i as shown in Eq. (8).

$$P_{i} = \frac{{F_{i} }}{{\sum\limits_{i = 1}^{S} {F_{i} } }}.$$

(8)

The process of gene recombination is known as cross operation. The cross mode of the k chromosome a_k and the l chromosome a_l at the j position are shown in Eq. (9).

$$\left\{ {\begin{array}{*{20}ll} {a_{kj} = a_{kj} \left( {1 - b} \right) + a_{lj} b} \\ {a_{lj} = a_{lj} \left( {1 - b} \right) + a_{kj} b} \\ \end{array} } \right.,$$

(9)

where b is a random number between 0 and 1.

The j-th gene of the i-th individual is selected for mutation operation as shown in Eqs. (10) and (11).

$$a_{ij} = \left\{ {\begin{array}{*{20}ll} {a_{ij} + \left( {a_{ij} - a_{\max } } \right)f\left( g \right),\;r > 0.5} \\ {a_{ij} + \left( {a_{\min } - a_{ij} } \right)f\left( g \right),\;r \le 0.5} \\ \end{array} } \right.,$$

(10)

$$f\left( g \right) = r_{1} \left( {1 - \frac{g}{{G_{\max } }}} \right)^{2} ,$$

(11)

where a_max is the maximum value of a_ij and a_min is the minimum value of a_ij; g is the number of iterations; G_max is the maximum number of evolution; r₁ is a random number, and r is a random number between 0 and 1.

After the evolution of the GA algorithm, the optimal individuals were obtained. Finally, the prediction model was established by training the neural network. The specific process is shown in Fig. 5.

Fig. 5

The process of the GA-BP algorithm

The principle for PSO-BP neural network

The PSO algorithm is a parallel random search algorithm, which performs global searches under individual fitness conditions. In this process, the speed and position of each particle are constantly updated to accurately approach the target within a short period (Ding et al., 2020). The PSO algorithm also requires population initialization like the GA algorithm, but it has no selection, crossover, and mutation operations. Herein, the population size and the maximum number of iterations were also determined as 8 and 50, respectively.

In the searching space, the PSO algorithm randomly initializes a group of particles. Each particle represents a potential optimal solution to the extreme value optimization problem. The fitness value of each particle is calculated according to the Eq. (7). The particle updates the individual position in the searching space by tracking the individual extreme value (Pbest) and the group extreme value (Gbest). In each iteration, the particle updates its speed and position through individual extreme value and group extreme value, as shown in Eq. (12).

$$\left\{ \begin{aligned} v_{ij} \left( {t + 1} \right) & = \omega v_{ij} \left( t \right) + c_{1} r_{2} \left[ {P_{best} \left( t \right) - x_{ij} \left( t \right)} \right] + c_{2} r_{3} \left[ {G_{best} \left( t \right) - x_{ij} \left( t \right)} \right] \\ x_{ij} \left( {t + 1} \right) & = x_{ij} \left( t \right) + v_{ij} \left( {t + 1} \right) \\ \end{aligned} \right.,$$

(12)

where v_ij represents the j-th dimension component of the i-th particle velocity. The particle velocity was limited between [− 1, 1] in this study. t is the current iteration number, and $\omega$ is the inertia weight, herein, its range was set in [0.4, 09]. c₁ and c₂ are acceleration constants, herein, c₁ = c₂ = 1.5; r₂ and r₃ are random numbers between 0 and 1; x_ij is the j-th dimension component of the position of particle i. The motion range of the particles was set in [− 5, 5].

At the end of the PSO algorithm, the optimal initial weights and thresholds were output, and the prediction model was acquired by training the network. The specific process is shown in Fig. 6.

Fig. 6

The process of the PSO-BP algorithm

Results

Breast motions

Some reports have found that there is a significant difference between the left and right breasts in terms of morphology and kinematics (Pei et al., 2019; Ren et al., 2016). The breast motions also exhibit considerable variability in different breast parts (Mason et al., 1999). The study aims to investigate the difference of breast extension and contraction between breast regions, SPSS 20.0 version (IBM, USA) was applied for the statistical analysis of $\Delta d$. Since the $\Delta d$ were detected to be non-parametric data by the test for normal distribution, the Mann–Whitney U test was applied to analyze the motion diversity between the same regions on the bilateral breasts. Meanwhile, the Kruskal–Wallis H test was utilized to analyze the motion difference between different regions (d₀₁, d₀₂, d₀₃, d₀₄) on the ipsilateral breasts. All significance levels were predetermined at p ≤ 0.05. The results are shown in Table 4.

Table 4 Statistical characteristics of $\Delta d$ (unit: mm)

As for the $\Delta d$, the positive and negative values represent the extension and contraction of breast regions, respectively. Combined with the $\Delta d$ distribution in Fig. 2, most breast regions present both extension and contraction, except for the d_01(L) region. According to the average values, the motion pattern of the d_01(L), the d_01(R), the d_02(L), the d_03(R), the d_04(L), and the d_04(R) regions are mainly extension, while the d_02(R) and d_03(L) regions take contraction as the primary motion. In the d₀₁ and d₀₂ areas, the breast motions are more distinct in the left than in the right, and the reverse phenomenon occurs in the d₀₃ and d₀₄ breast regions. The phenomenon demonstrates that the motions of breast regions vary between the left and right breasts. As for the upper and outer breast regions, the motions are more intense in the left breasts, whereas the lower and inner breast regions move more intensely in the right breasts.

In terms of the range values, the four regions on the right breasts have a wider range of stretch than the left breasts, and the maximum value (4.07 mm) occurs in the d_04(R). The result indicates that the breast motions are more active in the right breasts. The squared deviation (SD) indicates the dispersion degree of the breast motion indicators. Except for the d₀₃ areas where the SD values are similar in bilateral breasts, the other regions have different SD values between the right breasts and the left breasts. The SD values in the right breast regions were much greater than those in the left breasts. The result indicates that the right breast motions are more scattered and irregular than the left breasts. As shown in Table 4, there are significant motion differences in the same regions on the bilateral breasts or the different regions on the ipsilateral breasts according to the two non-parametric tests. Numerous reports have verified the morphological asymmetry of bilateral breasts (Losken et al., 2005; Pei et al., 2019). Based on the investigation of local breast motions, the asymmetry of bilateral breast motions was found in this study. Therefore, the bra producers and designers are advised to improve the bra comfort and support based on the differences in both breast morphology and motions.

Relevance of breast motions and gaits

Due to the influence of trunk displacements, breast motions also present a periodic pattern (McGhee et al. 2013). Besides, the intensity of breast motions may lead to gait changes during exercises (Eden et al., 1992; Li et al., 2018). Thus, whether a correlation exists between breast motions and gaits is speculated. The mean values of parameters were calculated for correlation analysis. The duration of three cycles varies with individual subjects for the unavoidable changes in the arm swing amplitude (Zhou et al., 2012). Therefore, the gait cycles were reshaped by the reshape function in Pandas of Python. The maximum duration of cycles was determined as the new duration, so the linear interpolation method was applied to increase the samples. The average gait and breast motion parameters for 13 subjects over time are shown in Figs. 7 and 8, respectively.

Fig. 7

Variations of gait parameters with time (left leg)

Fig. 8

Variations of $\Delta d$ with time

As shown in Fig. 7, the curve of $\Delta h$ with time is similar to the sine curve. The wave shapes of the lower limb rotation angles including the $\beta$, the $\gamma$, and the $\alpha$ are similar to each other, and the curves all present a ‘W’ shape with a single peak and two troughs. Although the $\Delta h$ curve is different from three other gait parameters, the four curves all present a stable periodic structure. Their peaks and the troughs also exhibit a period of synchronicity. On top of this, the hip rotation angle $\alpha$ has the smallest value, and the ankle rotation angle $\beta$ has the relatively minimum range of change (26.89°). The knee rotation angle $\gamma$ has the largest value and the maximum change range (42.98°). The result reveals that the knee rotation joint has the maximum motion amplitude while the ankle rotation joint has the minimum motion amplitude.

Figure 8 shows that the $\Delta d$ exhibits a continuous periodicity with time and the curves approximately present a periodic ‘V’ shape. There are two breast motion cycles in a gait cycle according to the gait cycle division. Thus, the period of breast motions is half of the gaits. The phenomenon demonstrates that breast motions are more frequent than trunk movements. The event is perhaps a critical factor to cause breast pain. Compared to the gait parameters, the $\Delta d$ values include more noise and present a relatively large fluctuation. As a result, the main features of breast motions are obscured and it may be detrimental to the correlation analysis between breast motion and gait. In order to reduce the noise of $\Delta d$, the wavelet threshold de-noising method was applied in this study to process the breast motion indicators. The result is shown in Fig. 8. In a single gait cycle, the patterns of the $\beta$, the $\gamma$, and the $\alpha$ are similar to the wave shape of the breast motion indicators. They all show a ‘W’ shape. Thus, the knee rotation angle $\gamma$ (left leg) having the largest amplitude was determined to compare with the breast motions. The results are shown in Fig. 6. Meanwhile, the Mann–Whitney U test was applied to analyze the variability of two breast motion cycles in a gait cycle. The significance level was predetermined at p ≤ 0.05.

According to Fig. 9, although there are differences in the peaks and troughs of $\Delta d$ and $\gamma$, the curves both show a ‘W’ shape within the same gait cycle. For the knee rotation angle $\gamma$, the peaks are more bulging while the troughs are a little concave. Meanwhile, the peaks and troughs have similar extents in the breast motion curves. In terms of the appearance times of the wave peaks and troughs, the peaks of $\Delta d$ for the d_03(R), the d_01(L), the d_02(L), and the d_04(L) regions are consistent to the $\gamma$, respectively. It is interesting to observe that the peaks of $\Delta d$ for the d_01(R), the d_04(R), and the d_03(L) regions lag behind the $\gamma$, while the crests of $\Delta d$ for the d_02(R) are ahead of the $\gamma$. The appearance times of troughs within a gait cycle show another characteristic. The first valley of $\Delta d$ always lags behind the $\gamma$, whereas the second valley of $\Delta d$ advances or synchronizes with the $\gamma$. The time lag of peaks and troughs may be due to the lack of bone and muscle in the breasts or the viscoelasticity variability between different breast parts (Cai et al., 2018; McGhee & Steele, 2020a).

Fig. 9

Variations of $\Delta d$ and $\gamma$ with time

Although the expansion and contraction of the breasts show a V-shaped cycle in general, there are some phenomena of 'rebound' in the wave troughs. The phenomena illustrate that breast motions have a non-linear and complex character. The breast motion curves were divided into two cycles in a gait cycle, and the proportion of each breast motion cycle (breast motion cycle/gait cycle $\times$ 100%) was calculated. The results find that the first cycle of $\Delta d$ is longer than the second cycle except for the d_02(R) and the d_03(R) regions. According to the results of the Mann–Whitney U test, breast regions have significant differences between the two cycles except for d_02(L) and more significant differences exist for d_03(R), d_01(L), and d_04(L) regions. The results mean that the breast motions are more intense and complex than the gaits and they lead to the poor stability of the $\Delta d$ curves.

Gray relational analysis is a quantitative method to describe and compare the developmental dynamics of a system. The larger the gray correlation degree ($\delta$), the more similar the variation trend of the two variables, and vice versa (Xiao, 1997). The strength of the correlation is defined as weak $\delta \le {0.49}$, moderate $\delta \ge {0.50}$ to $\delta \le {0.74}$, and strong $\delta \ge {0.75}$ (Bridgman et al., 2010). The gray correlation degrees between parameters are shown in Fig. 10.

Fig. 10

Grey correlation degrees between breast motion indicators and gait parameters

From Fig. 10, the gait parameters have the smallest $\delta$ value between the left and right leg, and the maximum value among them is only 0.62. However, the $\delta$ values between the ipsilateral gait parameters are all above 0.75. The gray correlation degree between $\alpha$ and $\gamma$ for the right leg reached the maximum (0.92). Moderate $\delta$ values are observed between the breast motion indicators, with a minimum value of 0.68. The $\delta$ values between d_04(L) and other breast regions are the highest, while the gray correlation degree between d_03(L), d_04(R) and other breast regions are the lowest. The result indicates that the d_03(L) and d_04(R) areas have more complex and irregular motions. As the $\delta$ values between gait parameters and breast motion indicators are in the range of [0.6, 0.75], there is a moderate correlation between these two types of parameters. Therefore, the prediction models for breast motions based on gait parameters can be established in this study.

Forecast of breast motions

To compare the accuracy of the PSO-BP, the GA-BP, and the BP algorithms, the input and output data of the models were set up consistently in this study. The predictive values of the test set were output after training the three prediction models for breast motions. The predictive values and the experimental values of $\Delta d$ are shown in Fig. 11.

Fig. 11

The predictive results of the neural network models

Figure 11 shows that the values of $\Delta d$ predicted by the PSO-BP, the GA-BP, and the BP neural network models differ lightly from the experimental data. Thus, the breast motion indicators in this study can be effectively predicted by the above three models based on the gait parameters. To quantify the performance of the models, we adopted the coefficient of determination (R²) and the MAE to evaluate the prediction accuracy of the algorithms, as shown in Eqs. (13) and (14). The results are shown in Fig. 12 and Table 5, respectively.

$$R^{2} = 1 - \frac{{\sum\limits_{i = 1}^{m} {\left( {\Delta d^{(i)} - \Delta d^{\prime(i)}} \right)^{2} } }}{{\sum\limits_{i = 1}^{m} {\left( {\Delta d^{\prime(i)} - \overline{{\Delta d^{\prime}}} } \right)^{2} } }},$$

(13)

$$MAE = \frac{1}{m}\sum\limits_{i = 1}^{m} {\left| {\Delta d^{(i)} - \Delta d^{\prime(i)}} \right|} ,$$

(14)

where $\Delta d$ and $\Delta d^{\prime}$ are the experimental and predictive value of the models respectively; $\overline{{\Delta d^{\prime}}}$ is the average of the predictive values, and m indicates the sample size of the test set.

Fig. 12

The R² between experimental and predictive values: a PSO-BP, b GA-BP, c BP

Table 5 The MAE of PSO-BP, GA-BP, and BP models

Figure 12 shows that the R² of the PSO-BP, the GA-BP, and the BP neural network models reaches more than 80%. The R² of the GA-BP and the BP neural network models are similar to each other, while the R² of the PSO-BP model has the maximum value of 84.58%. The result indicates that these three neural network models have the ability to predict the breast motions, and the PSO-BP model is more suitable for breast motion prediction in this study.

As shown in Table 5, the overall MAE of the PSO-BP model is 0.2108, which is 11% lower than that of the BP neural network. Meanwhile, the overall MAE of the GA-BP model is 0.2289, which is 3% lower than that of the BP neural network. The phenomenon indicates that both the PSO and GA algorithms have the ability to improve the prediction accuracy of the BP neural network for breast motions. The PSO-BP model has the best prediction accuracy. Among the three prediction models for breast motions, the MAE of d_02(R) and d_04(R) are larger than other breast regions. The result demonstrates that the motion prediction in these two breast regions has poor performance. According to the MAE of the PSO-BP model, the left breasts have better prediction accuracy than the right breasts. Therefore, it is more effective to predict the left breast motions based on the gait parameters. The result might be related to the stretch ranges and the dispersion of the breast motions.

Discussion

The awareness of breast motion patterns is significant to the optimization of bra comfort and support. A scientific and reasonable parameter utilized to quantify breast motions makes the results more comprehensive and reliable. Numerous reports have described the breast motions based on the points (Scurr et al., 2010; Zhou et al., 2009), but these studies were unable to reflect the motions of the breast regions. Herein, the distance alteration between breast points ($\Delta d$) was utilized to analyze the breast motions. The indicator aims to explore the line variation of different breast regions and it reflects the extension and contraction of breast regions. In this study, the results reveal that the main breast motions are the extension. Whether the same breast regions on the different sides of the breasts, or the various breast regions on the same side of the breasts, they all present a significant diversity in breast motions. The phenomenon may be related to the asymmetrical mass of the breasts. There are no consistent results for the motion intensity in different breast regions, and the motions in the right breasts show relatively larger fluctuation than that in the left breasts. The breast displacement of females varies mainly in the vertical direction during the two-step jumping exercise, and the vertical displacement accounts for 58% of the breast displacements (Bridgman et al., 2010). When females exercise on a treadmill at 10 km/h, the vertical, mediolateral, and anteroposterior components account for an average of 56%, 22%, and 19% of breast motions, respectively (Zhou et al., 2012). However, there is a dissimilar result for the breast motions in this study. The breast motions obtain maximum and minimum values in the d₀₁ and d₀₃ regions respectively, which demonstrates that the motions are more active in the lateral breast areas, followed by the vertical breast areas. Meanwhile, the medial breast areas present the slightest motions. The discrepancies with previous reports may be due to the different indicators of breast motions.

As for the breast motions and the trunk movements, some reports claimed that they all present a pattern of the sinusoidal curve (Haake & Scurr, 2010). However, the foot height off the ground is the only parameter founded in this study that shows a sinusoidal curve with time. The rotation angles of the lower limb joints and the distance alterations of the breast regions all show a ‘W’ shape during a gait cycle. It is interesting to note that the breast regions exhibit a ‘rebound’ phenomenon during exercises. In research of dynamic model for the breasts, the ‘double bounce’ phenomenon was also observed at the minimum value of the breast motion curves (Haake & Scurr, 2010), and the researchers suggested that this phenomenon might be related to the dynamic forces of heel and forefoot strike during contact. Besides, there is a time delay between the curves of certain breast areas and that of limb rotation angles. The phenomenon is also discovered in the study investigating the absolute and relative displacement of breasts during walking and running (Scurr et al., 2009). In the report, both the vertical and resultant breast displacements show a ‘bimodal distribution’ within a single gait cycle and the peaks and troughs of breast motion curves lag slightly behind the temporal division of the gaits. Once the subject's heel hits the ground during running, that is, at the lowest point where the foot is high off the ground, the trunk suddenly decelerates and falls vertically. Nevertheless, the soft tissue of the breasts continues to move downward due to the inertia, and the breasts only begin to decelerate when the trunk starts to move upward (McGhee et al., 2010). The time delay in trunk-breast motions causes the breasts to slap down against the chest wall and thus leads to the breast pain (Scurr et al., 2009; McGhee et al. 2013).

According to this analysis, there is a moderate correlation between breast motions and gaits. Thus, the BP neural network, the PSO, and the GA algorithms were applied in this study to develop three prediction models for breast motions. The R² of these three models reaches more than 80%, and the PSO-BP model obtains the maximum R² of 84.58%. Meanwhile, the PSO-BP model also has the smallest overall MAE of 0.2108. Therefore, the PSO-BP algorithm is considered to be more appropriate for breast motion prediction in this study. The validity of the prediction model allows researchers to predict breast motions with gait parameters, and the method can facilitate the investigation of breast motions in different environments lacking the access to measure the breast motions.

There are some limitations in this study, such as the correlation between breast motions and human gaits is only investigated under the bare breasts. For further research, it is vital to explore the difference of breast motions between the bare breasts and the bra-wearing breasts. Furthermore, the influence of bra structures, fabrics, and styles for the breast motions should also be analyzed to reduce breast expansion and contraction during exercises. Besides, the methods in this study are also applicable to investigate the breast motions for other subjects. The awareness of breast motions among different females in a series of exercises is helpful to improve bra support and comfort.

Conclusions

In this study, the distance alteration of the breast regions ($\Delta d$) was determined to evaluate breast motions. According to the analysis of the breast motions, a significant motion difference appears in the same regions on the bilateral breasts and the different regions on the ipsilateral breasts. Meanwhile, the $\Delta d$-time curves of eight breast regions show the same continuous ‘V’ pattern. Within a gait cycle, four gait parameters (the hip, the knee, the ankle joint rotation angles, and the foot height off the ground) demonstrate one period while breast motions present two periods. At the same time, the curves of the hip, the knee, the ankle joint rotation angles, and the $\Delta d$ perform one peak and two troughs respectively. However, there is a slight difference in the occurred time of peaks and troughs. In this case, there is a moderate correlation between breast motion indicators and gait parameters in this study. Therefore, the BP neural network combined with optimization algorithms (the GA and PSO algorithms) is valid to predict breast motions based on gait parameters. Among these three models, the PSO-BP algorithm has the highest prediction accuracy which has an R² of 84.58% and an MAE of 0.2108.