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% (c) 2005 -- Harish Narayanan

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\usepackage{amsfonts}
\usepackage{psfrag}
\usepackage{amssymb}
\usepackage{multimedia}
\usepackage[english]{babel}
\usepackage[latin1]{inputenc}
\usepackage{chemarr}
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% Simple shorthand for bold faced symbols (for vectors, tensors...)
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% The preamble. Details of which are not used for evil reasons.
\title   {\textcolor[named]{BrickRed}{Coupled Mechanics and Transport
    in Growing Soft Tissue -- A Continuum Treatment}}
\author  {\small H. Narayanan, K. Garikipati, E. M. Arruda, K. Grosh
  \& S. Calve\\
  \small University of Michigan}
\subject {The mechanicsy talk at McMat.}
\institute[McMat]{\small McMat 2005 -- Baton Rouge, LA}
\date[BR 2005] {\small June $3^{\textsf{rd}}$, 2005}

% Document starts here
\begin{document}

  %00 --  Clean title slide.
  %\begin{frame}
  %\titlepage
  %\end{frame}
  
  %00 -- Actual title slide. Not clean, but what're you going to do?
  \begin{frame}
    \frametitle{\textcolor[named]{BrickRed}{Coupled Mechanics and
	Transport in Growing Soft Tissue}}
    \begin{columns}
      \begin{column}{6cm}
	\centering
	    {\movie[width=4.85cm,height=4.0cm,poster,loop]{}
	      {constrict_ref_conc_3.avi}}\\
      \end{column}
      \begin{column}{6cm}
	{\em Nutrient transport is pivotal}
      \end{column}
    \end{columns}
    \vspace{0.8cm}
    \centering{\small H. Narayanan, K. Garikipati, E. M. Arruda,
      K. Grosh \& S. Calve}\\
    \centering{\small University of Michigan}\\
    \centering{\small McMat 2005 -- Baton Rouge, LA}\\
    \centering{\small June $3^{\textsf{rd}}$, 2005}\\
  \end{frame}
  
  %01 -- Motivation and definitions
  \begin{frame}
    \frametitle{\textcolor[named]{BrickRed}{Motivation and
	definition}}
    {\sl Growth -- An addition or loss of mass}
    \begin{columns}
      \begin{column}{5cm}
	\begin{figure}[hbt]
	  \begin{center}
	    \includegraphics[width=4.02cm]{one-construct.eps}
			    {\\ \tiny Engineered tendon
			      constructs [Calve et al]}
	  \end{center}
	\end{figure}
      \end{column}
      \begin{column}{5cm}
	\begin{figure}[hbt]
	  \begin{center}%4.69cm
	    \includegraphics[width=4.69cm]{content.eps}
			    {\tiny Increasing collagen
			      concentration with age}
	  \end{center}
	\end{figure}
      \end{column}
    \end{columns}
    \vspace{0.5cm}
    \onslide+<2->Open system with multiple species
	       inter-converting and interacting
  \end{frame}

  %02 -- Issues that arise
  \begin{frame}
    \frametitle{\textcolor[named]{BrickRed}{Modelling approach}}
    Classical balance laws enhanced via fluxes and sources
    \pause
    \begin{itemize}
    \item<2-> Solid -- Collagen, proteoglycans, cells
    \item<2-> Extra cellular fluid\\
      -- diffuses relative to the solid phase\\
    \item<2-> Dissolved solutes (sugars, proteins, $\dots$)\\
      -- undergo transport relative to fluid
    \end{itemize}
    \tiny \onslide+<3->Brief subset of related literature:
    \begin{itemize}
    \item[$\circ$]<3-> Cowin and Hegedus [1976]
    \item[$\circ$]<3-> Kuhl and Steinmann [2002]
    \item[$\circ$]<3-> Sengers, Oomens and Baaijens [2004]
    \item[$\circ$]<3-> {\sl Garikipati et al. -- Journal of the
      Mechanics and Physics of Solids (52) 1595-1625 [2004]}
    \end{itemize}
  \end{frame}
  
  %03 -- The balance of mass -- solid
  \begin{frame}
    \frametitle{\textcolor[named]{BrickRed}{The balance of mass}}
    \begin{columns}
      \begin{column}{7.5cm}
	\begin{figure}[ht]
	  \centering
	  \psfrag{D}{\tiny$\bX$}
	  \psfrag{F}{\tiny$\bx$}
	  \psfrag{E}{\renewcommand{\baselinestretch}{1.5}\tiny$\Pi^c$}
	  \psfrag{G}{\tiny$\pi^c$}
	  \psfrag{C}{\tiny$\Bvarphi$}
	  \psfrag{A}{\tiny$\Omega_0$}
	  \psfrag{B}{\tiny$\Omega_t$}
		 {\includegraphics[width=6.5cm]{bomass-solid.eps}}
	\end{figure}
      \end{column}
      \begin{column}{4.5cm}
	\small $\rho^c$ -- Collagen concentration\\
	\small $\Pi^c$ -- Collagen production
      \end{column}
    \end{columns}
    \begin{itemize}
    \item<1-> For collagen: \quad\quad
      $\frac{\partial\rho^c}{\partial t} = \Pi^c\nonumber$
      \pause
    \item<2-> No flux; No boundary conditions
    \end{itemize}
  \end{frame}

  %04 -- The balance of mass -- fluid
  \begin{frame}
    \frametitle{\textcolor[named]{BrickRed}{The balance of mass}}
    \begin{columns}
      \begin{column}{7.5cm}
	\begin{figure}[ht]
	  \centering
	  \psfrag{D}{\tiny$\bX$}
	  \psfrag{F}{\tiny$\bx$}
	  \psfrag{C}{\tiny$\Bvarphi$}
	  \psfrag{A}{\tiny$\Omega_0$}
	  \psfrag{B}{\tiny$\Omega_t$}
	  \psfrag{H}{\tiny$\bN\cdot\bM^f$}
	  \psfrag{I}{\tiny$\bn\cdot\bm^f$}
		 {\includegraphics[width=6.5cm]{bomass-fluid.eps}}
	\end{figure}
      \end{column}
      \begin{column}{4.5cm}
	\small $\rho^f$ -- Fluid concentration\\
	\small $\bM^f$ -- Fluid flux
      \end{column}
    \end{columns}
    \begin{itemize}
    \item<1-> For the fluid: \quad\quad
      $\frac{\partial\rho^f}{\partial t} = -\Bnabla\cdot\bM^f\nonumber$
      \pause
    \item<2-> No source; Concentration or flux boundary conditions --
      {\sl Tissue exposed to fluid in a bath, fluid injected in at the
	boundary}
    \end{itemize}
  \end{frame}
  
  %05 -- The balance of mass -- solute
  \begin{frame}
    \frametitle{\textcolor[named]{BrickRed}{The balance of mass}}
    \begin{columns}
      \begin{column}{7.5cm}
	\begin{figure}[ht]
	  \centering
	  \psfrag{D}{\tiny$\bX$}
	  \psfrag{F}{\tiny$\bx$}
	  \psfrag{C}{\tiny$\Bvarphi$}
	  \psfrag{A}{\tiny$\Omega_0$}
	  \psfrag{B}{\tiny$\Omega_t$}
	  \psfrag{H}{\tiny$\bN\cdot\bM^s$}
	  \psfrag{I}{\tiny$\bn\cdot\bm^s$}
	  \psfrag{E}{\renewcommand{\baselinestretch}{1.5}\tiny$\Pi^s$}
	  \psfrag{G}{\tiny$\pi^s$}
		 {\includegraphics[width=6.5cm]{bomass-solute.eps}}
	\end{figure}
      \end{column}
      \begin{column}{4.5cm}
	\small $\rho^s$ -- Solute concentration\\
	\small $\Pi^s$ -- Solute production\\
	\small $\bM^s$ -- Solute flux
      \end{column}
    \end{columns}
    \begin{itemize}
    \item<1-> For a solute: \quad\quad
      $\frac{\partial\rho^s}{\partial t} = \Pi^s - \Bnabla\cdot\bM^s$
      \pause
    \item<2-> Flux and source; Concentration boundary condition --
    {\sl Tissue exposed to solute in solution in a bath}
    \end{itemize}
  \end{frame}

  %06-- The balance of momentum -- solid
  \begin{frame}
    \frametitle{\textcolor[named]{BrickRed}{The balance of momentum}}
    \begin{columns}
      \begin{column}{7.5cm}
	\begin{figure}[ht]
	  \centering
	  \psfrag{D}{\tiny$\bX$} \psfrag{F}{\tiny$\bx$}
	  \psfrag{C}{\tiny$\Bvarphi$} \psfrag{A}{\tiny$\Omega_0$}
	  \psfrag{B}{\tiny$\Omega_t$} \psfrag{H}{\tiny$\bN\cdot\bM^c$}
	  \psfrag{I}{\tiny$\bn\cdot\bm^c$}
	  \psfrag{E}{\renewcommand{\baselinestretch}{1.5}\tiny$\Pi^c$}
	  \psfrag{G}{\tiny$\pi^c$} \psfrag{J}{\tiny$\bg$}
	  \psfrag{K}{\tiny$\bq^c$} \psfrag{L}{\tiny$\bP\bN$}
	  \psfrag{M}{\tiny$\Bsigma\bn$}
		 {\includegraphics[width=6.5cm]{bomom-solid.eps}}
	\end{figure}
      \end{column}
      \begin{column}{4.5cm}
	\small $\rho^c$ -- Collagen concentration\\
	\small $\bV$ -- Solid velocity\\
	\small $\bg$ -- Body force\\
	\small $\bq^c$ -- Interaction force\\
	\small $\bP^c$ -- Partial stress
      \end{column}
    \end{columns}
    \begin{itemize}
    \item For collagen: \quad\quad
      $\rho^c\frac{\partial \bV}{\partial t} =
      \rho^c\left(\bg+\bq^c\right) + \Bnabla\cdot\bP^c$
    \end{itemize}
  \end{frame}
  
  %07-- The balance of momentum -- fluid
  \begin{frame}
    \frametitle{\textcolor[named]{BrickRed}{The balance of momentum}}
    \begin{columns}
      \begin{column}{7.5cm}
	\begin{figure}[ht]
	  \centering
	  \psfrag{D}{\tiny$\bX$}
	  \psfrag{F}{\tiny$\bx$}
	  \psfrag{C}{\tiny$\Bvarphi$}
	  \psfrag{A}{\tiny$\Omega_0$}
	  \psfrag{B}{\tiny$\Omega_t$}
	  \psfrag{H}{\tiny$\bN\cdot\bM^f$}
	  \psfrag{I}{\tiny$\bn\cdot\bm^f$}
	  \psfrag{J}{\tiny$\bg$}
	  \psfrag{K}{\tiny$\bq^f$}
	  \psfrag{L}{\tiny$\bP\bN$}
	  \psfrag{M}{\tiny$\Bsigma\bn$}
		 {\includegraphics[width=6.5cm]{bomom-fluid.eps}}
	\end{figure}
      \end{column}
      \begin{column}{4.5cm}
	\small $\rho^f$ -- Fluid concentration\\
	\small $\bV$ -- Solid velocity\\
	\small $\bV^f$ -- Fluid relative velocity\\
	\small $\bg$ -- Body force\\
	\small $\bq^f$ -- Interaction force\\
	\small $\bP^f$ -- Partial stress
      \end{column}
    \end{columns}
    \begin{itemize}
    \item<1->  For the fluid, velocity relative to the solid: 
      $\bV^f = (1/\rho^f)\bF\bM^f$
      \pause
    \item[]<2-> $\rho^f\frac{\partial}{\partial
      t}\left(\bV+\bV^f\right) = \rho^f\left(\bg+\bq^f\right) +
      \Bnabla\cdot\bP^f  \textcolor{Lighter}{ - (\Bnabla (\bV+\bV^f))
      \bM^f}$
    \end{itemize}
  \end{frame}
  
  %08 -- Kinematics of growth
  \begin{frame}
    \frametitle{\textcolor[named]{BrickRed}{Growth kinematics}}
    \begin{figure}[ht]
      \psfrag{A}{\small $\Omega_0$}
      \psfrag{B}{\small $\Omega^\ast$}
      \psfrag{C}{\small $\Omega_t$}
      \psfrag{D}{\small $\bX$}
      \psfrag{E}{\small $\bX^\ast$}
      \psfrag{F}{\small $\bx$}
      \psfrag{G}{\small $\bF^{\mathrm{g}^{\mathrm{\iota}}}$}
      \psfrag{H}{\small $\widetilde{\bF}^{\mathrm{e}^{\mathrm{\iota}}}$}
      \psfrag{I}{\small $\widetilde{\bF}$}
      \psfrag{J}{\small $\bar{\bF}^\mathrm{e}$}
      \psfrag{K}{\small $\bF$}
      \psfrag{L}{\small $\Bkappa$}
      \psfrag{M}{\small $\bu^\ast$}
      \psfrag{N}{\small $\Bvarphi$}
	     {\includegraphics[width=7cm]{kinematics.eps}}
    \end{figure}
    \begin{itemize}
    \item <1->$\bF =
      \bar{\bF}^\mathrm{e}\widetilde{\bF}^{\mathrm{e}^{\mathrm{\iota}}}\bF^{
	\mathrm{g}^{\mathrm{\iota}}}$
      \pause
    \item <2->Internal stress due to
      $\widetilde{\bF}^{\mathrm{e}^{\mathrm{\iota}}}$
    \end{itemize}
  \end{frame}

  %09 -- Constitutive relations -- flux
  \begin{frame}
    \frametitle{\textcolor[named]{BrickRed}{Constitutive relations for
	fluxes}}
    \begin{itemize}
    \item<1-> Combine first and second laws to get dissipation
    inequality 
    \item<1-> Constitutive hypothesis $e^\iota =
            \hat{e}^\iota(\bF^{\mathrm{e}^\iota},\rho^\iota,\eta^\iota)$\\
            $\Rightarrow$ consistent constitutive relations
	    \pause
	  \item<2-> Fluid flux relative to collagen\\
            \quad$\bM^f = \bD^f\left(\rho^f\bF^T\bg +
            \bF^T\Bnabla\cdot\bP^f - \Bnabla(e^f - \theta\eta^f)\right)$
	    \pause
	  \item<3-> Solute flux (proteins, sugars, nutrients, $\dots$)
            relative to fluid\\
            \quad$\widetilde{\bV}^s=\bV^s-\bV^f$\\
            \quad$\widetilde{\bM}^s = \bD^s\left( - \Bnabla(e^s -
            \theta\eta^s)\right)$ %\rho^s \bF^\mathrm{T}\bg
	  \item<3-> $\bD^f$ and $\bD^s$ -- Positive semi-definite
            mobility tensors\\
            Magnitudes from literature, e.g. Mauck et al. [2003]
    \end{itemize}
\end{frame}


  %10 -- Constitutive relations -- WLCM
  \begin{frame}
    \frametitle{\textcolor[named]{BrickRed}{Worm-like chain model for
    collagen}}
    \begin{columns}
      \begin{column}{2cm}
	\begin{figure}[ht]
	  \psfrag{A}{\tiny $a$}
	  \psfrag{B}{\tiny $b$}
	  \psfrag{C}{\tiny $c$}
	  \centering {\includegraphics[width=2cm]{wlcm.eps}}
	\end{figure}
      \end{column}
      \begin{column}{10cm}
	{\small
	$\widetilde{\rho}^\mathrm{c}\hat{e}^\mathrm{c}(\bF^{\mathrm{e}^
	\mathrm{c}},\rho^\mathrm{c})$\\
	  \begin{eqnarray*}
	    &=& \frac{N k \theta}{4 A}\left(\frac{r^2}{2L} +
	    \frac{L}{4(1-r/L)} -
	    \frac{r}{4}\right)\\
	    &-&\frac{N k
	    \theta}{4\sqrt{2L/A}}\left(\sqrt{\frac{2A}{L}} + 
	    \frac{1}{4(1 - \sqrt{2A/L})} -\frac{1}{4}
	    \right)\log(\lambda_1^{a^2}\lambda_2^{b^2}\lambda_3^{c^2})
	    \\
	    &+& \frac{\gamma}{\beta}({J^{\mathrm{e}^\iota}}^{-2\beta}
	    -1) + 2\gamma{\bf 1}\colon\bE^{\mathrm{e}^\mathrm{c}}
	\end{eqnarray*}}
      \end{column}
    \end{columns}
    \begin{itemize}
    \item Embed in multi chain model [Bischoff et al.]\\
      \tiny\quad\small $r =
      \frac{1}{2}\sqrt{a^2\lambda_1^{\mathrm{e}^2} +
	b^2\lambda_2^{\mathrm{e}^2} + c^2\lambda_3^{\mathrm{e}^2}}$
    \item $\lambda_I^{\mathrm{e}}$ -- elastic stretches along a, b, c\\
      \tiny\quad\small $\lambda_I^{\mathrm{e}} =
      \sqrt{\bN_I\cdot\bC^{\mathrm{e}}\bN_I}$
    \end{itemize}    
  \end{frame}

  %11 -- Description of the example
  \begin{frame}
    \frametitle{\textcolor[named]{BrickRed}{Example of coupled computation}}
    \begin{columns}
      \begin{column}{4.5cm}
	\begin{figure}[ht]
	  \centering
	  \psfrag{M}{\tiny $\bN\cdot\bM^f$}
	  \psfrag{P}{\tiny $\bP \bN$}
		 {\includegraphics[width=4cm]{cylinder-in-bath.eps}}
	\end{figure}
      \end{column}
      \begin{column}{7.5cm}
	\begin{itemize}
	\item<1-> Simulating a tendon immersed in a bath
	\item<1-> Constrict it to force fluid and dissolved nutrient
	  flow $\Rightarrow$ Guided tendon growth
	  \pause
	\item<2-> Biphasic model
	  \begin{itemize}
	  \item<2-> worm-like chain model for collagen
	  \item<2-> ideal nearly incompressible fluid\\
            $\rho^f \hat{e}^f =
            \frac{1}{2}\kappa(\mbox{det}(\bF^{\mathrm{e}^f})-\bone)^2$
            \\
	  \end{itemize}
	\item<2-> Fluid mobility $D^f_{ij} = 1\times 10^{-8}
	  \delta_{ij}$,\\
	  Han et al. [2000]
	  \pause
	\item<3-> First order rate law:
	  $\Pi^\mathrm{f} = -k^\mathrm{f}(\rho^\mathrm{f} -
	  \rho_{0_\mathrm{ini}}^\mathrm{f}), \quad\Pi^\mathrm{c} =
	  -\Pi^\mathrm{f} $
	\end{itemize}
      \end{column}
    \end{columns}
  \end{frame}

  %12 -- Results -- 1
  \begin{frame}
    \frametitle{\textcolor[named]{BrickRed}{Results and inferences}}
    \begin{columns}
      \begin{column}{5cm}
	\movie[width=4.85cm,height=4.0cm,poster,loop]{}
	      {constrict_total_flux_3_4.avi}{}
      \end{column}
      \begin{column}{7cm}
	\begin{itemize}
	\item Total flux in the vertical direction
	\item Stress driven diffusion
	\end{itemize}
      \end{column}
    \end{columns}
  \end{frame}
  
  %13 -- Results -- 2
  \begin{frame}
    \frametitle{\textcolor[named]{BrickRed}{Results and inferences}}
    \begin{columns}
      \begin{column}{5cm}
	\movie[width=4.85cm,height=4.0cm,poster,loop]{}
	      {constrict_ref_conc_3.avi}{}
      \end{column}
      \begin{column}{7.5cm}
	\begin{itemize}
	\item Regions of high fluid concentration\\
          $\Rightarrow$ faster growth
	\item Relaxation after constriction concludes
	\end{itemize}
      \end{column}
    \end{columns}
  \end{frame}

  %14 -- Summary and things
  \begin{frame}
    \frametitle{\textcolor[named]{BrickRed}{Summary and further work}}
    \begin{itemize}
    \item<1-> Physiologically relevant continuum formulation
      describing growth in an open system -- consistent with mixture theory
    \item<1-> Relevant driving forces arise from thermodynamics\\ --
    coupling with mechanics
  \item<1-> Gained insights into the problem
    \pause
    \begin{itemize}
    \item<2-> Issues of saturation and growth
    \item<2-> Saturation and Fickian diffusion
    \item<2-> Configurations and physical boundary conditions
    \end{itemize}
    \end{itemize}
    \pause
    \begin{itemize}
    \item<3-> More careful treatment of biochemistry -- nature of
    sources
    \item<3-> Formulated a theoretical framework for remodelling
    \item<3-> Engineering and characterization of growing, functional
    biological tissue to drive and validate modelling
    \end{itemize}
  \end{frame}

\end{document}
