A piecewise mass-spring-damper model of the human breast
Introduction
Previous research into breast movement has been limited to subjective measures (Risius et al., 2016) of comfort (Chen et al., 2016) and objective measures of kinematic data of breast displacements (Risius et al., 2015) during physical activities such as running (Scurr et al., 2010) and jumping (Nolte et al., 2016). The differences between breast displacements during bra-supported and unsupported activities (Lu et al., 2016) were obvious. However, the breast viscoelastic properties that affect the breast movement still remain unclear. Chen et al. (2013) conducted a finite element analysis to simulate a breast’s free vibration and ascertained its mean damping ratio to be 0.215 ± 0.013, yet a spring constant for the breast was not derived. McGhee et al. (2013) assumed the breast force to be the total breast mass multiplied by the superior-inferior nipple acceleration but gave no consideration to the elasticity and viscous damping within the breast. Haake and Scurr (2010) used a mass-spring-damper model to predict the superior-inferior nipple displacement. The estimated spring constants, damping coefficient and damping ratio were 245 ± 15 N m−1, 8.73 ± 0.88 N s m−1, 0.475 ± 0.062 respectively. They assumed that the breast (size 32C in their study) could be modeled as a single mass-spring-damper located at the nipple and its motion was regularly harmonic. However, in reality, the viscoelastic properties of the breasts vary with the dynamic change of distributions of densities of the fatty tissues, glandular tissues, and skin.
Therefore, this study used both experimental real-time motion analysis of a breast’s free vibration and new piecewise theoretical equations to create a mass-spring-damper system to represent the breast. Using the waveform of nipple displacements obtained from free-falling breast experiments (free vibration), the in-vivo material properties of individual breasts were derived. They were then used to predict the nipple displacements during bare-breasted running (forced vibration) that were validated with experimental data. This newly-developed model will serve as a foundation for future research into the simulation of 3D, nonlinear breast movement and the design of external support to prevent breast discomfort during different types of activities.
Section snippets
Experiments
To model the breast displacement during free and forced vibration, it is necessary to determine the effective mass (derived from the breast volume obtained from 3D body scanning), spring constants, and damping coefficients for the breast, as well as the thoracic displacement (obtained from the motion analysis).
Theoretical equations of breast damping
Based on the waveform of the nipple displacement during the free vibrations (Fig. 4), a Kelvin-Voigt model (Eldred et al., 1995) with two springs and two viscous dampers was created to describe both the elasticity and viscosity of the breast materials in a single degree of freedom as shown in Fig. 5.
This mass-spring-damper model assumed the breast to be a fixed point mass located at the nipple, which adheres to the thorax through a set of viscous damper and elastic spring acting in parallel
Results and discussions
As Fig. 7 shows, the period of phase 1 is Tb/2 = 0.05 s, the period of phase 2 is Ta/2 = 0.143 s, thus the overall periodSubstituting the known valuesinto Eq. (15) gives the damping ratio .
Substituting the known valuesinto Eq. (16) gives the damping ratio .
Model validation
Once the damping coefficients and spring constants were derived, they were used to predict the breast movement during bare-breasted running. The experimental breast displacements against time were compared with the theoretically derived data. The equilibrium equation of forced breast vibration during running is shown as Eq. (20) and rearranged as Eq. (21)where m is the effective breast mass, ka and k
Conclusion
The analysis of the experimental data of the free vibration of the human breast in the superior-inferior direction showed that the vibration behavior above the static equilibrium position was obviously different from that below it. Therefore, in order to characterize the visco-elastic nature of a human breast, the use of the piecewise mass-spring-damper model proposed in this study is advocated. The spring constants ka, kb, and damping coefficients ca, cb can be derived from the experimental
Acknowledgements
We would like to thank the Hong Kong Research Grants Council for funding the research through project PolyU 5306/12E and PolyU 5304/13E. We also thank Dr. Simon Harlock for his professional editing and the subject involved in the experiments in this study.
Conflict of interest statement
All the authors have no conflicts of interest in this publication.
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