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The role of structural viscoelasticity in deformable porous media with incompressible constituents: Applications in biomechanics

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  • The main goal of this work is to clarify and quantify, by means of mathematical analysis, the role of structural viscoelasticity in the biomechanical response of deformable porous media with incompressible constituents to sudden changes in external applied loads. Models of deformable porous media with incompressible constituents are often utilized to describe the behavior of biological tissues, such as cartilages, bones and engineered tissue scaffolds, where viscoelastic properties may change with age, disease or by design. Here, for the first time, we show that the fluid velocity within the medium could increase tremendously, even up to infinity, should the external applied load experience sudden changes in time and the structural viscoelasticity be too small. In particular, we consider a one-dimensional poro-visco-elastic model for which we derive explicit solutions in the cases where the external applied load is characterized by a step pulse or a trapezoidal pulse in time. By means of dimensional analysis, we identify some dimensionless parameters that can aid the design of structural properties and/or experimental conditions as to ensure that the fluid velocity within the medium remains bounded below a certain given threshold, thereby preventing potential tissue damage. The application to confined compression tests for biological tissues is discussed in detail. Interestingly, the loss of viscoelastic tissue properties has been associated with various disease conditions, such as atherosclerosis, Alzheimer’s disease and glaucoma. Thus, the findings of this work may be relevant to many applications in biology and medicine.

    Mathematics Subject Classification: Primary: 35Q74, 35B44; Secondary: 35C10.

    Citation:

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  • Figure 1.  Illustration of a novel hypothesis on potential causes of microstructure damage in deformable porous media. In the case that (ⅰ) the medium is subjected to a time-discontinuous mechanical load and (ⅱ) structural viscoelasticity is reduced or absent, then the fluid velocity within the porous medium will experience a blow-up, possibly leading to microstructural damage.

    Figure 2.  Schematic representation of the one-dimensional problem considered in this article. The deformable porous medium may be either poro-elastic or poro-visco-elastic. The forcing term $P(t)$ may have discontinuities in time.

    Figure 3.  Schematic representation of the dimensionless step pulse $\hat{P}(\hat{t}~)$ defined in (36). Here, the signal is discontinuous at the switch on time.

    Figure 4.  Dimensionless displacement $\hat{u}_{\hat{\eta}}$ as a function of $\hat{x}$ and $\hat{t}$. Left panel: $\hat{\eta}=0$. Right panel: $\hat{\eta}=1$.

    Figure 5.  Dimensionless discharge velocity $\hat{v}_{\hat{\eta}}$ as a function of $\hat{x}$ and $\hat{t}$. Left panel: $\hat{\eta}=0$. Right panel: $\hat{\eta}=1$. In order to highlight the velocity blow-up at $\hat{x}=1$, $\hat{t}=0$, the $\log_{10}$ plot of $|\hat{v}_{\hat{\eta}}|$ is plotted in both panels.

    Figure 6.  Dimensionless power density $\hat{\mathcal{P}}_{\hat{\eta}}$ as a function of $\hat{t}$. Left panel: $\hat{\eta}=0$. Right panel: $\hat{\eta}=1$. In order to highlight the power density blow-up at $\hat{t}=0$, the $\log_{10}$ plot of $\hat{\mathcal{P}}_{\hat{\eta}}$ is plotted in both panels.

    Figure 7.  Dimensionless discharge velocity $\hat{v}_{\hat{\eta}}(\hat{x},0)$ as a function of $\hat{x}$. In order to highlight the velocity blow-up at $\hat{x}=1$, $\hat{t}=0$, in the case $\hat{\eta}=0$, the $\log_{10}$ plot of $|\hat{v}_{\eta}|$ is plotted.

    Figure 8.  Left panel: maximal discharge velocity as a function of $\hat{\eta}$. Right panel: power density as a function of $\hat{\eta}$. Log$_{10}$-scale is used on the $y$-axis to better highlight blow-up of both quantities as $\hat{\eta} \rightarrow 0$.

    Figure 9.  Left panel: dimensional maximal discharge velocity as a function of $\hat{\eta}$. Right panel: dimensional power density as a function of $\hat{\eta}$. Log$_{10}$-scale is used on the $y$-axis to better highlight blow-up of both quantities as $\hat{\eta} \rightarrow 0$. We set $K_0/L = 1 ~ \mathrm{m^2 s Kg^{-1}}$. The black arrows indicate increasing values of $P_{\text{ref}}$ in the range $[10^{-3}, 10^3] ~ \mathrm{N m^{-2}}$.

    Figure 10.  Schematic representation of the dimensionless trapezoidal pulse $\hat{P}(\hat{t})$ defined in (41). Here, the signal switch on and switch off are characterized by linear ramps.

    Figure 11.  Dimensionless discharge velocity $\hat{V}_{\hat{\eta}}\left( \hat{x},\hat{t} \right) $ for $\hat{\eta}=0.1$, $\hat{\varepsilon}=0.2$, $\hat{\tau}=1$.

    Figure 12.  Dimensionless maximal discharge velocity $\hat{V}_{\max }\left( \hat{\eta},\hat{\varepsilon}\right)$ as a function of $\hat{\eta}$ and $\hat{\varepsilon}$.

    Figure 13.  Schematic representation of a confined compression chamber.

    Figure 14.  Comparison between the dimensionless maximum velocity $\hat{V}_{\max}(\hat{\eta},\hat{\varepsilon})$ obtained in the case of trapezoidal pulse using expression (44) and the threshold velocity $\hat{V}_{\text{th}} = 16.3$ typical of confined compression experiments [37].

    Figure 15.  Colormap of the difference $ \hat{V}_{\text{th}}-\hat{V}_{\max }$ as a function of $\hat{\eta}$ and $\hat{\varepsilon}$. As in Fig. 14, $\hat{V}_{\max}(\hat{\eta},\hat{\varepsilon})$ is obtained using expression (44) and $\hat{V}_{\text{th}} = 16.3$ is the typical velocity of confined compression experiments [37]. The curve in the parameter space for which $ \hat{V}_{\text{th}}=\hat{V}_{\max }$ is reported in a thick black mark.

    Table 1.  Numerical values of model parameters in the confined compression experiment for articular cartilage reported in [37].

    symbol value units
    $L$ $0.81 \cdot 10^{-3}$ $\mathrm{m}$
    $\mu$ $0.97 \cdot 10^6$ $\mathrm{N m^{-2}}$
    $\eta$ $0$ $\mathrm{N ~s ~m^{-2}}$
    $K_0$ $2.9 \cdot 10^{-16}$ $\mathrm{m^4 N^{-1} s^{-1}}$
    $P_{\text{ref}}$ $6 \cdot 10^{4}$ $\mathrm{N m^{-2}}$
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