On the Three-Dimensional Mechanical Behavior of Human Breast Tissue

Abstract

As the human breast undergoes complex, large-scale, fully three dimensional deformations in vivo, three-dimensional (3D) characterization of its mechanical behavior is fundamental to its diagnosis, treatment, and surgical modifications. Its anisotropic, heterogeneous fibrous structure results in complex behavior at both the tissue and organ levels. Mathematically modeling of this complex anisotropic behavior is thus critical to the proper simulation of the human breast. Yet, current breast tissue constitutive models do not account for these complexities, so that there is a pressing need for more detailed fully 3D analysis. To this end, we performed a full 3D kinematic mechanical evaluation of human fibroglandular and adipose breast tissues. We utilized our recently developed 3D kinematic numerical-experimental approach to acquire force-displacement data from both breast tissue subtypes. This was done by subjecting cuboidal test specimens, aligned to the anatomical axes,to both pure shear and simple compression loading paths. We then developed novel constitutive model that was able to simulate the unique anisotropic tension/compression behaviors observed. Constitutive model parameters were determined using a detailed finite element model of the experimental setup coupled to nonlinear optimization. We found that human breast tissues displayed complex anisotropic behavior, with strong, directionally dependent non-linearities. This was especially true for the fibroglandular tissue. The novel constitutive model was also able fully capture these behaviors, including states of combined tension and compression (i.e. in pure shear). The results of this study suggest that human breast tissue is complex in its mechanical response, exhibiting varying levels of anisotropy. Future studies will be required to link the observed anisotropy to the physical structure of the tissue, as well as mapping this heterogeneity and anisotropy across individuals.

Appendices

Appendix 1: Convexity Analysis

We examined the convexity of our fitted models by examining the structure of their extrema. We do this under the assumption of incompressibility by the method of Lagrange multipliers. In effect, we seek to solve the system

$$\begin{aligned} {\text {Grad}}\left( \psi -\lambda \phi \right) =0, \end{aligned}$$

(9)

where $\psi$ is our strain energy, $\lambda$ is a Lagrange multiplier, $\phi$ is the incompressibility constraint, and ${\text {Grad}}$ is taken over the variable set $\{E_{11},E_{22},E_{33},E_{12},E_{13},E_{23},\lambda \}$. Solutions to this system are stationary points of the energy, constrained to the hypersurface of isochoric strains. Once we know these extremal points, we can examine the structure of the energy near these points to determine if they are local minima, maxima, or saddle points.

Adipose

Examining this system for the fitted adipose parameters, we find that there is a unique solution lying at the point of zero strain. It is not difficult to see that this point is a minimum; the incompressibility constraint can be solved for $E_{11}$ for instance, and the energy constrained to isochoric motions by substituting this solution into the expression for the energy. Doing this, and taking the Hessian of this constrained energy density at the point of zero strain yields the Hessian matrix

$$\begin{aligned} \begin{bmatrix} 261.6 &{} -39.2 &{} 0 &{} 0 &{} 0\\ -39.2 &{} 255.6 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 473.6 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} 487.6 &{} 0\\ 0 &{} 0 &{} 0 &{} 0 &{} 743.6 \end{bmatrix}, \end{aligned}$$

(10)

which is positive definite, having a minimum eigenvalue 219.285. Therefore, the zero strain state in the incompressible model is a global minimum.

Fibroglandular

We note that the unconstrained fibroglandular energy density can be written as

$$\begin{aligned} \psi =f\left( E_{11},E_{22},E_{33}\right) +158E_{23}^2+242 E_{13}^2+116 E_{12}^2, \end{aligned}$$

(11)

which is clearly convex in the shear terms, since for fixed $E_{11}$, $E_{22}$, $E_{33}$, this function takes its minimum value when these shear components vanish. Hence, we simply have to examine the restriction of the energy when these shear terms vanish, i.e. the function f. Applying the incompressibility constraint allows us to represent f as a function of two independent strain components, say $E_{11}$ and $E_{22}$. Plotting this constrained energy as a surface reveals two local minima, and a saddle point, the saddle point occurring at $\mathbf {E}=\mathbf {0}$ with the constrained energy taking the value $\psi =1.59$. The global minimum occurs at $\mathbf {E}={\text {diag}}\left( 0.460,-0.193,-0.075\right)$, with the energy taking a value of $\psi =-4.176$, and the other local minimum occurs at $\mathbf {E}={\text {diag}}\left( -0.204,0.175,0.126\right)$ with a value of $\psi =-0.665$. While this technically means that the fitted model is unstable at zero strain, the energy at zero strain is sufficiently near the global minimal energy. As a comparison, the model’s energy at the equibiaxial strain of state $\mathbf {E}={\text {diag}}\left( 0.2,0.2,-.245\right)$ is $\psi =45.069$, meaning that putting the tissue in this strain state from the zero strain state requires over 7.5 times the energy released by taking the tissue from zero strain to its global minimum. This lack of convexity causes the tissue to behave very softly in regions near there local extrema, while behaving stiffly at strains far from these extrema, replicating the observed softness at low strains and stiffness at higher strains.

Fibroglandular Correction

The lack of convexity in the fibroglandular model can be removed by adding the term

$$\begin{aligned} \psi _{add}=\begin{bmatrix} E_{11}&E_{22}&E_{33} \end{bmatrix} \begin{bmatrix} 32.85 &{} -21.93 &{} -12.75\\ -21.93 &{} 14.64 &{} 8.51\\ -12.75 &{} 8.51 &{} 4.945 \end{bmatrix} \begin{bmatrix} E_{11}\\ E_{22}\\ E_{33} \end{bmatrix}, \end{aligned}$$

(12)

to the total energy. This amounts to a correction on the fitted parameters $M_{11}$, $M_{22}$, $M_{33}$, $M_{12}$, $M_{13}$, and $M_{23}$, such that the final incompressible model has a global minimum at zero strain. This correction was found by expressing the matrix $\mathcal {M}$ in terms of its eigenbasis, and subsequently increasing the smallest eigenvalue until the three critical points merged into one. This modified matrix was then expressed back in terms of the original strain components, from which the correction above was computed. It is important to note that this does not make $\mathcal {M}$ positive definite, but it does make the restriction of the incompressible energy convex at the zero strain point. This correction term can be increased to completely remove the convexity from $\mathcal {M}$, which will guarantee a convex energy for all strains, though in practice, this is likely more than sufficient.

Appendix 2: Tension-Compression Asymmetry

Here we demonstrate how a neo-Hookean term coupled with incompressibility generates tension-compression asymmetry in pure shear.

For simplicity, consider an incompressible neo-Hookean material in pure shear in the $1-2$ plane. The energy is then

$$\begin{aligned} \psi =\frac{\mu }{2}\left( C_{11}+\frac{1}{C_{11}}-2\right) . \end{aligned}$$

(13)

We consider a displacement $\alpha$, with $C_{11}=1+\alpha$, and the energy becomes

$$\begin{aligned} \psi =\frac{\mu }{2}\left( \alpha +\frac{1}{1+\alpha }-1\right) =\frac{\mu }{2}\left( \frac{\alpha ^2}{1+\alpha }\right) . \end{aligned}$$

(14)

For positive $\mu$, this has a minimum at $\alpha =0$, hence our material is stable at zero strain, however, it is clear to see that

$$\begin{aligned} \psi \left( \alpha \right) =\frac{\mu }{2}\left( \frac{\alpha ^2}{1+\alpha }\right) \ne \psi \left( -\alpha \right) =\frac{\mu }{2}\left( \frac{\alpha ^2}{1-\alpha }\right) . \end{aligned}$$

(15)

This asymmetry is

$$\begin{aligned} \psi \left( \alpha \right) -\psi \left( -\alpha \right) =\frac{\mu \alpha ^2}{2}\left( \frac{1}{1+\alpha }-\frac{1}{1-\alpha }\right) =\frac{\mu \alpha ^3}{\alpha ^2-1}, \end{aligned}$$

(16)

therefore it is clear that the neo-Hookean term under the constraint of incompressibility generates the observed tension-compression asymmetry.

Appendix 3: Robustness Analysis

Due to the relatively large number of parameters appearing in our model, there may be concerns about parameter identifiability. We seek to establish the functional independence of these parameters and provide an a posteriori justification of their use. Recall that our proposed strain energy density takes the form

$$\begin{aligned} \psi =\mu {\text {Tr}}\left( \mathbf {E}\right) +\mathbf {E}\cdot \mathcal {M}\left[ \mathbf {E}\right] +a\exp \left( \frac{b}{2}E_{33}^2\right) +K\left( J-1\right) ^2. \end{aligned}$$

Ordering the material parameters as $\alpha =\{\mu ,M_{11},M_{22},M_{33},M_{12},M_{13},M_{23},M_{44},M_{55},M_{66},a,b,K \}$, the corresponding sensitivity functions take the form

$$\begin{aligned} \begin{bmatrix} {\text {Tr}}\left( \mathbf {E}\right) ,E_{11}^2,E_{22}^2,E_{33}^2,2E_{11}E_{22},2E_{11}E_{33},2E_{22}E_{33},E_{23}^2,E_{13}^2,E_{12}^2,\exp \left( \frac{b}{2}E_{33}^2\right) ,\frac{a}{2}\exp \left( \frac{b}{2}E_{33}^2\right) E_{33}^2,\left( J-1\right) ^2 \end{bmatrix} \end{aligned}$$

Note that these functions are independent of the parameters except for a and b, and the only sensitivities depending on a and b are the sensitivities with respect to a and b. Therefore, we can evaluate these functions at the fitted values for the fibroglandular tissues, and simply remove the two sensitivities corresponding to a and b when we want to analyze the adipose tissue.

We seek to quantify the independence of these functions over the strain range considered in this study. To do this, we consider the region of strain space containing our experimental data, namely we bound the normal components of the strain to be between $-0.2$ and 0.345, corresponding to the normal displacements $-2.25$ mm and 3 mm respectively. Likewise, we bound the shear components to be between $\pm 0.265$ corresponding to the maximum shear strain encountered in our study. We further restrict this region of strain space to those strains satisfying $J>0$ so that only physical strains are considered. We denote this domain $\mathcal {D}$. This domain then defines a Hilbert space, where the inner product is taken to be

$$\begin{aligned} \left\langle f,g\right\rangle = \int _{\mathcal {D}}f \bar{g} d\mathbf {E}, \end{aligned}$$

where the overbar denotes complex conjugation, though all functions we will consider are real-valued. Like any other inner product, this allows us to compute the magnitude of functions on this space as

$$\begin{aligned} \Vert f\Vert ^2=\left\langle f,f\right\rangle , \end{aligned}$$

and allows the computation of angles between functions as

$$\begin{aligned} \cos \theta _{f,g} = \frac{\left\langle f,g \right\rangle }{\Vert f\Vert \Vert g\Vert }. \end{aligned}$$

We do this for each pair of functions $f_i,f_j$, and obtain a matrix of cosines $\mathbf {A}$ where

$$\begin{aligned} A_{ij} = \cos \theta _{f_i,f_j}. \end{aligned}$$

This matrix of cosines was numerically computed at the fitted fibroglandular parameters to be

$$\begin{aligned} \begin{bmatrix} 1, 0.664, 0.453, 0.453, 0.664, 0.453, 0.664, 0.466, 0.466, 0.466, 0.655, 0.654, 0.566\\ 0.664, 1, 0.320, 0.320, 0.471, 0.121, 0.471, 0.512, 0.512, 0.512, 0.683, 0.445, 0.557\\ 0.453, 0.320, 1, 0.176, 0.320, 0.176, 0.121, 0.131, 0.131, 0.131, 0.175, 0.114, 0.500\\ 0.453, 0.320, 0.176, 1, 0.121, 0.176, 0.320, 0.131, 0.131, 0.131, 0.202, 0.322, 0.500\\ 0.664, 0.471, 0.320, 0.121, 1, 0.320, 0.471, 0.512, 0.512, 0.512, 0.683, 0.445, 0.557\\ 0.453, 0.121, 0.176, 0.176, 0.320, 1, 0.320, 0.131, 0.131, 0.131, 0.202, 0.322, 0.500\\ 0.664, 0.471, 0.121, 0.320, 0.471, 0.320, 1, 0.512, 0.512, 0.512, 0.756, 0.996, 0.557\\ 0.466, 0.512, 0.131, 0.131, 0.512, 0.131, 0.512, 1, 0.556, 0.556, 0.737, 0.481, 0.446\\ 0.466, 0.512, 0.131, 0.131, 0.512, 0.131, 0.512, 0.556, 1, 0.556, 0.737, 0.481, 0.446\\ 0.466, 0.512, 0.131, 0.131, 0.512, 0.131, 0.512, 0.556, 0.556, 1, 0.737, 0.481, 0.446\\ 0.655, 0.683, 0.175, 0.202, 0.683, 0.202, 0.756, 0.737, 0.737, 0.737, 1, 0.723, 0.608\\ 0.654, 0.445, 0.114, 0.322, 0.445, 0.322, 0.996, 0.481, 0.481, 0.481, 0.723, 1, 0.541\\ 0.566, 0.557, 0.500, 0.500, 0.557, 0.500, 0.557, 0.446, 0.446, 0.446, 0.608, 0.541, 1 \end{bmatrix} \end{aligned}$$

The fact that none of the off-diagonal values are 1 or $-1$ indicates that our sensitivities are functionally independent pairwise, but we must consider their independence as a group to determine if they are linearly independent.

To establish the independence of the sensitivity functions on $\mathcal {D}$, we can compute the volume of the hyper-parallelepiped spanned by the normalized functions. This value will be 0 if the functions are linearly dependent, and will be 1 if the functions are mutually orthogonal, but interpreting intermediate values is more difficult. Due to the relationship between hyper-volumes and dimension, for a large number of functions, this value will rapidly become small if these functions are not perfectly orthogonal. Therefore, to eliminate the effect of dimensional scaling on this measure, we take the nth root of this volume to get the side length of a hyper-cube with the same volume as this parallelepiped, and we use this effective length ratio as the relevant measure of independence.

The volume of the hyper-parallelepiped can be computed as the square root of the determinant of the matrix of cosines. The volume of the unit hyper-parallelepiped with these directional cosines is the same as that of a hyper-cube with side length 0.592, indicating that these functions, while not perfectly independent, are sufficiently independent to justify their use. Removing the rows and columns corresponding to the parameters a and b, we obtain an equivalent hyper-cube length of 0.752, indicating that the underlying sensitivities for adipose tissue are more independent than those for the fibroglandular tissue. Alternatively, one could consider a unit-parallelepiped spanning the same volume, but with uniform angles between its edges. This equivalent parallelepiped gives an effective “average angle” between the sensitivities. Doing this yields an effective angle of $42.5^\circ$ between the fibroglandular sensitivities, and $48.6^\circ$ between the adipose sensitivities.

By either of these metrics, we see that at the fitted values we obtained, and over the strain range we have considered, our model depends on our parameters independently. Therefore, while there is some covariance between our parameters, this covariance is not so severe as to justify a reparameterization of our model, or a reduction of the number of parameters.