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Could Effective Mechanical Properties of Soft Tissues and Biomaterials at Mesoscale be Obtained by Modal Analysis?

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Abstract

Background

The mechanical properties of biological tissues and soft biomaterials are difficult to explore even though they play an important role in mechanobiological responses and organ homeostasis. Limited availability of harvested tissue and careful handling must be considered as well as discrepancies in biomaterial development.

Objective

We hypothesized that a mixed analytical-experimental modal analysis could be used to determine effective mechanical properties at the mesoscale for hydrated and fragile, poorly available and small-sized biological tissue and biomaterials.

Methods

Young's modulus E, shear modulus G and Poisson's ratio \(\nu\) were obtained from the measurement of first two natural frequencies of a set-up associating tested specimen with a cantilever. Tangent modules are calculated using a set of two analytical governing equations in linear vibration framework. A complementary parametric sensitivity analysis was performed. The methodology was evaluated using materials known to be challenging, namely agarose for biomaterials and bone marrow for biological tissues.

Results

Frequencies were in the range of 350 Hz and acquisition time of few seconds. Linear responses was checked and solution triplets (E, G, \(\nu\)) were (99 ± 10 kPa, 43 ± 0.3 kPa, 0.16 ± 0.1) for agarose and (61 ± 12 kPa, 28 ± 7 kPa, 0.07 ± 0.03) for bone marrow.

Conclusion

Comparisons with literature when available, confirmed approach acceptability. Limited influences of boundary conditions, brief experiments and reproducibility can be considered for applications to fragile and rare biomaterials and biological tissues, in addition to conventional characterization methods.

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Abbreviations

v(x,t), w(x,t) :

Transverse displacement of the cantilever

Q(x,t), M(x,t) :

Shear force and bending moment in the cantilever

\({\omega }_{i}\), \({\phi }_{i}\) (x) :

Angular frequency and mode shape

EI, E c, \({\rho }_{c}\) , S, L :

Cantilever properties: bending stiffness per unit length, Young’s modulus, density, cross- section, length

E, G, \(\nu\) :

Specimen properties: effective Young’s modulus, shear modulus and Poisson’s ratio

\(\rho\), \(\alpha\) , s, l :

Specimen properties: effective density, shear coefficient, cross-section and length

k, m :

Specimen properties: translational stiffness and mass

\({k}_{\theta }\), \({i}_{\theta }\) :

Specimen properties: rotational stiffness and inertia

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Acknowledgements

The French Minister of Education and Research, the French National Center for research (CNRS) and the Region Occitanie (France) are acknowledged for their assistance and financial support.

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Authors and Affiliations

  1. IMFT UMR 5502, Toulouse University, 2 Allées C. Soula, 31400, Toulouse Cedex, France

    P. Swider, Y. Abidine & P. Assemat

  2. Transverse Laboratory Company, 2 allées C. Soula, 31400, Toulouse Cedex, France

    Y. Abidine

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  1. P. Swider
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Correspondence to P. Swider.

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Appendix

Appendix

Under separation of variables hypothesis, the displacement v(x,t) is written as the product of the mode shape ϕ(x) by the time function f(t). Verifying equation (1) gives conditions (A1) and (A2).

$$\begin{array}{ll}\frac{{\partial }^{4}\phi \left(x\right)}{\partial {x}^{4}}-\frac{{\rho }_{c}S}{{E}_{c}I}\cdot {\omega }^{2}\phi \left(x\right)=0& \left(\text{a}\right)\\\frac{{\partial }^{2}f\left(t\right)}{\partial {t}^{2}}+{\omega }^{2}f\left(t\right)=0& (\text{b})\end{array}$$
(A1)

Solutions of f(t) are harmonic functions with constants depending upon time initial conditions as expressed by equation (A2). The mode shape ϕ(x) is expressed by equation (A3a) with roots \(\beta\) depending upon angular frequencies as detailed in (A3b).

$$f\left(t\right)=A\,\mathrm{sin}\,\omega t+B\,\mathrm{cos}\,\omega t$$
(A2)
$$\begin{array}{ll}\phi \left(x\right)=C\;\mathrm{cos}\;\beta x+D\;\mathrm{cos}\;\beta x+E\;\mathrm{sinh}\;\beta x+F\;\mathrm{cosh}\;\beta x &\left(\text{a}\right)\\with &{}\\\beta ={\left({\rho }_{c}S{\omega }^{2}/{E}_{c}I\right)}^{1/4}& (\text{b})\end{array}$$
(A3)

Constants C, D, E and F depend on space boundary conditions. Clamping conditions in x = 0 give the two first equation (A4a, b) independent from f(t).

$$\begin{array}{ll}\phi \left(0\right)=0 &\left(\text{a}\right)\\\text{and}&{} \\\frac{\partial \phi \left(0\right)}{\partial x}=0 &(\text{b})\end{array}$$
(A4)

For mode shapes describing motion in the vertical plane, i.e. v(x,t), mass and stiffness coefficients used in system (3) are detailed as follows:

$$\begin{aligned}&E_{c}\,I=E_{c}\,bh^{3}/12,\;m=\rho sl/3,\;i_{\theta}=\rho sl^{3}/105,\\&k=Es/l,\;l_{\theta}=(4+a)\,Ei/ (2+a)\,l\end{aligned}$$

with i = s2 / 4\(\pi\) and a = 3es /\(\alpha \pi\)l 2 G with \(\alpha\): shear coefficient

For mode shapes describing motion in the horizontal plane, i.e. w(x,t), mass and stiffness coefficients used in system (3) are detailed as follows:

$$\begin{aligned}&E_{c}I=E_{c}.hb^{3}/12,\;m=13\rho sl/35,\;i_{\theta}=\rho s^{2}l/6\pi,\\& k=12\;Ei/(1+a)\,l^{3},\;k_{\theta}=Gs^{2}/2\pi l\end{aligned}$$

with i = s2 / 2\(\pi\) and a = 3Es /\(\alpha \pi\)l 2 G

In those equations, b and h are respectively the width and height of cantilever cross-section, Ec the cantilever Young’s modulus. The parameters \(\rho ,\) s and l are respectively the effective density, cross-section surface and length of the specimen whereas E, G and v are the effective Young’s modulus, the shear modulus and the Poisson’s ratio of the specimen.

Coefficients of equation 4a, b are detailed as follows:

$$\begin{aligned}&{a}_{1}=945{s}^{4}\left(1-\mathrm{cos}\;\beta L\;\mathrm{ch}\;\beta L\right) \\&{a}_{2}=1260{s}^{3}\;\alpha \pi {l}^{2}\left(1-\mathrm{cos}\;\beta L\;\mathrm{ch}\;\beta L\right)\end{aligned}$$
$$\begin{aligned}{a}_{3}=&-{s}^{3}\rho {l}^{2}{\omega }^{2}\left(315s+36\pi {l}^{2}\right)\left(1-\mathrm{cos}\;\beta L\;\mathrm{ch}\;\beta L\right)\\&+945EI\,{\beta }^{3}l{s}^{3}\left(\mathrm{cos}\;\beta L\;\mathrm{sh}\;\beta L+\mathrm{sin}\;\beta L\;\mathrm{ch}\;\beta L\right)\\& +3780EI\,\beta \pi l{s}^{2}\left(\mathrm{cos}\;\beta L\;\mathrm{sh}\;\beta L+\mathrm{sin}\;\beta L\;\mathrm{ch}\;\beta L\right)\end{aligned}$$
$$\begin{aligned}{a}_{4}=&\;-{s}^{3}\rho {l}^{2}{\omega }^{2}\left(315s+36\pi {l}^{2}\right)\left(1-\mathrm{cos}\;\beta L\;\mathrm{ch}\;\beta L\right)\\&+945EI\;{\beta }^{3}l{s}^{3}\left(\mathrm{cos}\;\beta L\;\mathrm{sh}\;\beta L+\mathrm{sin}\;\beta L\;\mathrm{ch}\;\beta L\right)\\& +3780EI\;\beta \pi l{s}^{2}\left(\mathrm{cos}\;\beta L\;\mathrm{sh}\;\beta L+\mathrm{sin}\;\beta L\;\mathrm{ch}\;\beta L\right)\end{aligned}$$
$$\begin{aligned}{a}_{5}=&\;-36\pi EI\;{\beta }^{3}{l}^{5}{s}^{2}\rho {\omega }^{2}\left(\mathrm{cos}\;\beta L\;\mathrm{sh}\;\beta L+\mathrm{sin}\;\beta L\;\mathrm{ch}\;\beta L\right)\\&+1260\pi EI\;\beta {l}^{3}{s}^{2}\rho {\omega }^{2}\left(\mathrm{cos}\;\beta L\;\mathrm{sh}\;\beta L-\mathrm{sin}\;\beta L\;\mathrm{ch}\;\beta L\right)\\& +3780\pi E{I}^{2}\;{\beta }^{4}{l}^{2}s\,\left(1+\mathrm{cos}\;\beta L\;\mathrm{ch}\;\beta L\right)\\&+12\pi {l}^{6}{s}^{3}{\rho }^{2}{\omega }^{4}\left(1-\mathrm{cos}\;\beta L\;\mathrm{ch}\;\beta L\right)\end{aligned}$$
$$\begin{aligned}{a}_{6}=&\;-12{\pi }^{2}EI\,{\beta }^{3}{l}^{7}\alpha s\rho {\omega }^{2}\left(\mathrm{cos}\;\beta L\;\mathrm{sh}\;\beta L+\mathrm{sin}\;\beta L\;\mathrm{ch}\;\beta L\right)\\&+420{\pi }^{2}EI\,\beta {l}^{5}\alpha s\rho {\omega }^{2}\left(\mathrm{cos}\;\beta L\;\mathrm{sh}\;\beta L-\mathrm{sin}\;\beta L\;\mathrm{ch}\;\beta L\right)\\& +1260{\pi }^{2}E{I}^{2}\,{\beta }^{4}{l}^{4}\alpha \,\left(1+\mathrm{cos}\;\beta L\;\mathrm{ch}\;\beta L\right)\\&+4{\pi }^{2}{l}^{3}\alpha {s}^{2}{\rho }^{2}{\omega }^{4}\left(1-\mathrm{cos}\;\beta L\;\mathrm{ch}\;\beta L\right)\end{aligned}$$
$$\begin{aligned}&{a}_{7}=630\alpha {s}^{4}\left(1-\mathrm{cos}\;\beta L\;\mathrm{ch}\;\beta L\right)\\&{a}_{8}=-39\pi \alpha {s}^{3}{l}^{4}\rho {\omega }^{2}\left(1-\mathrm{cos}\;\beta L\;\mathrm{ch}\;\beta L\right)\\&\quad\quad+105\pi \alpha {s}^{2}EI\,{\beta }^{3}\left(\mathrm{cos}\;\beta L\;\mathrm{sh}\;\beta L+\mathrm{sin}\;\beta L\;\mathrm{ch}\;\beta L\right)\end{aligned}$$
$$\begin{aligned}{a}_{9}=&\;-3\rho {\omega }^{2}{l}^{2}{s}^{4}\left(39+70\alpha \right)\left(1-\mathrm{cos}\;\beta L\;\mathrm{ch}\;\beta L\right)\\&+315EI\;{\beta }^{3}l{s}^{3}\left(\mathrm{cos}\;\beta L\;\mathrm{sh}\;\beta L+\mathrm{sin}\;\beta L\;\mathrm{ch}\;\beta L\right)\\& -1260EI\;\beta \pi l\alpha {s}^{2}\left(\mathrm{cos}\;\beta L\;\mathrm{sh}\;\beta L-\mathrm{sin}\;\beta L\;\mathrm{ch}\;\beta L\right)\end{aligned}$$
$$\begin{aligned}{a}_{10}=&\;-35\pi EI\,{\beta }^{3}{l}^{5}\alpha {s}^{2}\rho {\omega }^{2}\left(\mathrm{cos}\;\beta L\;\mathrm{sh}\;\beta L+\mathrm{sin}\;\beta L\;\mathrm{ch}\;\beta L\right)\\&+78{\pi }^{2}EI\,\beta {l}^{5}\alpha s\rho {\omega }^{2}\left(\mathrm{cos}\;\beta L\;\mathrm{sh}\;\beta L-\mathrm{sin}\;\beta L\;\mathrm{ch}\;\beta L\right)\\& +210{\pi }^{2}E{I}^{2}\,{\beta }^{4}{l}^{4}\alpha \left(1+\mathrm{cos}\;\beta L\;\mathrm{ch}\;\beta L\right)\\&+13\pi {l}^{6}{s}^{3}{\rho }^{2}{\omega }^{4}\alpha \left(1-\mathrm{cos}\;\beta L\;\mathrm{ch}\;\beta L\right)\end{aligned}$$
$$\begin{aligned}{a}_{11}=&\;-105EI\,{\beta }^{3}{l}^{3}{s}^{3}\rho {\omega }^{2}\left(\mathrm{cos}\;\beta L\;\mathrm{sh}\;\beta L+\mathrm{sin}\;\beta L\;\mathrm{ch}\;\beta L\right)\\&+234\pi EI\,\beta {l}^{3}{s}^{2}\rho {\omega }^{2}\left(\mathrm{cos}\;\beta L\;\mathrm{sh}\;\beta L-\mathrm{sin}\;\beta L\;\mathrm{ch}\;\beta L\right)\\& +630\pi E{I}^{2}\,{\beta }^{4}{l}^{2}s\,\left(1+\mathrm{cos}\;\beta L\;\mathrm{ch}\;\beta L\right)\\&+39{l}^{4}{s}^{4}{\rho }^{2}{\omega }^{4}\,\left(1-\mathrm{cos}\;\beta L\;\mathrm{ch}\;\beta L\right)\end{aligned}$$

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Swider, P., Abidine, Y. & Assemat, P. Could Effective Mechanical Properties of Soft Tissues and Biomaterials at Mesoscale be Obtained by Modal Analysis?. Exp Mech 63, 1055–1065 (2023). https://doi.org/10.1007/s11340-023-00974-7

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