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\usepackage[utf8x]{inputenc} \usepackage{siunitx} \usepackage{todonotes} @@ -55,8 +57,6 @@ Nevertheless we include some microscopic properties. \item \textbf{Density} \end{itemize} -\cite{frantz2010} mentions Matrigel™ and collagen type I gels, so we will focus on these. - \subsection{Viscoelasticity} \todo{What is viscoelasticity? Show some graphs and \enquote{oral} explanation} @@ -65,7 +65,7 @@ Generally modeled using differential equations involving the elastic modulus $E$ \cite{roylance2001} mentions these constitutive models: \begin{itemize} - \item Maxwell: A Viscous flow on the long timescale, but additional elastic resistance to fast deformations (e.g. silly putty, warm tar). + \item Maxwell: Viscous flow on the long timescale, but additional elastic resistance to fast deformations (e.g. silly putty, warm tar). Does not describe creep or recovery. \item Kelvin-Voigt: Does not describe stress relaxation. \item Zener/Standard linear solid: Models creep and stress relexation. @@ -75,7 +75,32 @@ The Lethersich and Jeffreys models are models for viscoelasticity that specifica \subsection{Rheology and Materials Science of the \acrshort{ecm}} -E.g. \cite{sherman2015, gautieri2013} +\cite{frantz2010} mentions Matrigel and collagen type I gels, so we will focus on these. + +Great review with great figures: \cite{chaudhuri2020}. + +\begin{itemize} + \item \cite{sherman2015} lists the elastic modulus of collagen structures at different scales, see \autoref{fig:sherman2015-table1}. + \item \cite{puxkandl2002} defines a model for the viscoelasticity of collagen. + \todo{expand, give actual values} + \item \cite{slater2017} discusses properties of Corning® Matrigel®. + \begin{itemize} + \item Lists elastic moduli for different concentrations and mixtures involving collagen type I around $10^1$ to $10^3$ \si{\pascal}. + \todo{This seems very low; investigate sources} + \item This paper shows the viscuous component in the graphs but doesn't really go into it. + \end{itemize} + \item \cite{aisenbrey2020} discusses alternatives to Corning® Matrigel®. + \item \cite{sheu2001} experimentally investigate the elastic and viscous moduli of collagen gels. + They find that the Kelvin-Voigt model can be used to model their viscoelastic behavior. +\end{itemize} + +\begin{figure}[h] + \includegraphics[width=\textwidth]{figures/sherman2015-table1} + \caption{Comparison of Young's modulus of collagen at multiple hierarchical levels. From \cite{sherman2015}.}. + \label{fig:sherman2015-table1} +\end{figure} + +Since viscoelastic behavior is inherently time-dependent, it will be a challenge to choose a sensible time step resolution for the model. \section{\acrfull{cpm}} @@ -107,6 +132,8 @@ E.g. \cite{sherman2015, gautieri2013} \item Discretisation in space makes it possible to calculate \acrshort{lbm} time steps using stencil codes. \item Extensive literature exists including implementation details, e.g. \cite{krueger2017} \item Can be used to model viscoelasticity, e.g. \cite{giraud1998, malaspinas2010, ispolatov2002} + \item Probably not that simple to model matrix porosity. + \todo{Elaborate} \end{itemize} \section{\acrshort{ecm} Models in the \acrshort{cpm}}