Abstract={In vitro cardiovascular disease models need to recapitulate tissue-scale function in order to provide in vivo relevance. We have developed a new method for measuring the contractility of engineered cardiovascular smooth and striated muscle in vitro during electrical and pharmacological stimulation. We present a growth theory-based finite elasticity analysis for calculating the contractile stresses of a 2D anisotropic muscle tissue cultured on a flexible synthetic polymer thin film. Cardiac muscle engineered with neonatal rat ventricular myocytes and paced at 0.5 Hz generated stresses of 9.2 ± 3.5 kPa at peak systole, similar to measurements of the contractility of papillary muscle from adult rats. Vascular tissue engineered with human umbilical arterial smooth muscle cells maintained a basal contractile tone of 13.1 ± 2.1 kPa and generated another 5.1 ± 0.8 kPa when stimulated with endothelin-1. These data suggest that this method may be useful in assessing the efficacy and safety of pharmacological agents on cardiovascular tissue.},

Doi={10.1016/j.biomaterials.2010.01.079},

pmid={20149449},

File={[Alford2010]_Biohybrid thin films for measuring contractility in engineered cardiovascular muscle.pdf:[Alford2010]_Biohybrid thin films for measuring contractility in engineered cardiovascular muscle.pdf:PDF},

Apart from $\sigma_x$, there are a number experimental parameters that may affect the curling behaviour of your films \cite{Alford2010}. In this chapter we attempt to quantify how substrate stiffness \param{Efilm}, substrate thickness \param{tfilm}, cell layer thickness \param{tcells}, substrate width \param{width}, and stress fibre distribution \param{fiblab} affect our read-out $\frac{l}{L}$ (in the simulations).

Apart from $\sigma_x$, there are a number experimental parameters that may affect the curling behaviour of your films \cite{Alford2010}. In this chapter we attempt to quantify how the geometries and material parameters of the substrate and cell layer, and how stress fibre distribution \param{fiblab} affect our read-out $\frac{l}{L}$ (in the simulations).

We investigate this parameter space with a multiple linear regression model and the results of 100 simulations with random combinations of input parameters, similar to Van Haaften \textsl{et al.}\cite{Haaften2019}. In addition to $\frac{l}{L}$, we also validate two assumptions in the analysis of `radius of curvature' $R$ from projected edge lengths $l$ (chapter \ref{estsigmax}).

Isotropic films and anisotropic films are treated separately, and we use an undeformed film length $L =4$\,mm (20 elements, 41 equidistant nodes) throughout this chapter.

We investigate this parameter space with a multiple linear regression model and the results of about 100 simulations with random combinations of input parameters, similar to Van Haaften \textsl{et al.}\cite{Haaften2019}. In addition to $\frac{l}{L}$, we also validate two assumptions in the analysis of `radius of curvature' $R$ from projected edge lengths $l$ (chapter \ref{estsigmax}).

Isotropic films and anisotropic films are treated separately, the number of variables is large enough already.

\subsection{Assumption of no contraction}

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...

@@ -39,7 +38,7 @@ That second assumption seems naive. Why would the average curvature occur at the

The first assumption strikes us as much less far fetched, and is implicitly used by Grosberg \textsl{et al.}\cite{Grosberg2011}: `Because the radius of curvature would no longer be constant throughout the MTF if the film comes into contact with itself, we did not cut the films long enough to make a full circle and designed the analysis code to eliminate any experiments in which the measured $x$-projection was $x < (L/2\piup)$.'

While the assumption of constant curvature may be valid for `free' curling films \cite{Alford2010}, it may not be for us as we introduce asymmetric boundary conditions. Also, for anisotropic films that curl `sideways' it is unlikely that the radii of curvature are equal for the two long edges of the film.\\

While the assumption of constant curvature may be valid for `free' curling films \cite{Alford2010}, it may not be for us as we introduce asymmetric boundary conditions (one short edge is fixed, the other one free). Also, for anisotropic films that curl `sideways' it is unlikely that the radii of curvature are equal for the two long edges of the film.\\

And again: not curious?\\

\begin{figure}[t!]

...

...

@@ -384,7 +383,7 @@ In isotropic films, the radius of curvature $R$ as estimated from projected edge

Anisotropic films cause films to (also) curve sideways out of the $\vec{y}\vec{z}$-plane (unless the axis of anisotropy are perfectly parallel to the films edges). To get an impression of what that looks like and what to expect, we ran 10 simulations with different orientations of the main axis of anisotropy ($90\leq\mu\,[\degree]\leq180$, figure \ref{lgmu}). The orientation of the anisotropy affects the amount of `sideways curling' (figure \ref{lfmu_def1}-\ref{lfmu_def2}), as expected. And because the projected edge length $l$ does not assess this out of plane curvature, its measure is also affected by this orientation (figure \ref{lfmu_sigma}).

Figure \ref{lgmu} suggests that when the axis of anisotropy is parallel to the long edges, all curvature is in the $\vec{y}\vec{z}$ plane and properly estimated by $\frac{l}{L}$, as with isotropic films. But that for an axis of anisotropy perpendicular to the long edges, curvature is in the $\vec{x}\vec{z}$ plane and therefore missed by $\frac{l}{L}$. Further, note that three of the films seem to show \textsl{a minimum} in their $\sigma_x$ -- $\frac{l}{L}$ relationship in figure \ref{lfmu_sigma} (110\,\degree, 120\,\degree, and 130\,\degree).\\

Figure \ref{lgmu} suggests that when the axis of anisotropy is parallel to the long edges, all curvature is in the $\vec{y}\vec{z}$ plane and properly estimated by $\frac{l}{L}$, as with isotropic films. But that for an axis of anisotropy perpendicular to the long edges, curvature is in the $\vec{x}\vec{z}$ plane and therefore missed by $\frac{l}{L}$. Further, note that three of the films seem to show \textsl{a minimum} in their $\frac{l}{L}\left(\sigma_x\right)$ relationship in figure \ref{lfmu_sigma} (110\,\degree, 120\,\degree, and 130\,\degree).\\

\begin{figure}[ht!]

\tiny\center

\subfloat[\label{lfmu_fibs}]{%

...

...

@@ -459,8 +458,11 @@ In addition, we attempt to assess to what degree the observation that $\sigma_x$

\subsection{Practical: convergence}

Since \param{FEMbuild}\texttt{ = 'matlab'} for these simulations, no python files could fail to be parsed, and all Abaqus input files `just worked' (in fact, we could have done with a single input file for all these 2500 simulations).

\begin{table}[t!]

\center\small

\caption{Multiple linear regression model fitted to the maximum value of $\sigma_x$ reached for the 100 random models. With the coefficients, their 95\,\% confidence intervals (CI), the amount of variation in the model explained by the parameter (SS/SS$_\text{tot}$), and the $p$-values for the coefficients. The last two columns contain the predicted range in the simulations based on the parameter space, and that predicted range divided by the complete range of the variable found in the simulations.\label{anispace_msig_tab}}

\input{anispace_msig_tab.tex}

\end{table}

\begin{figure}[p!]

\center

\subfloat[\label{pspaceani_n}]{%

...

...

@@ -469,13 +471,10 @@ Since \param{FEMbuild}\texttt{ = 'matlab'} for these simulations, no python file

\caption{Convergence for the 100 anisotropic films, with \textbf{(a)} the number of successfully converged simulations as a function of $\sigma_x$, and \textbf{(b)} the distribution of maximum cell traction stress $\sigma_x$.\label{aniconverge}}

\end{figure}

\begin{table}[b!]

\center\small

\caption{Multiple linear regression model fitted to the maximum value of $\sigma_x$ reached for the 100 random models. With the coefficients, their 95\,\% confidence intervals (CI), the amount of variation in the model explained by the parameter (SS/SS$_\text{tot}$), and the $p$-values for the coefficients. The last two columns contain the predicted range in the simulations based on the parameter space, and that predicted range divided by the complete range of the variable found in the simulations.\label{anispace_msig_tab}}

\input{anispace_msig_tab.tex}

\end{table}

Since \param{FEMbuild}\texttt{ = 'matlab'} for these simulations, no python files could fail to be parsed, and all Abaqus input files `just worked' (in fact, we could have done with a single input file for all these 2500 simulations).

Converge, of course, is still an issue: only 18 models made it to $\sigma_x =0.025$\,MPa (figure \ref{aniconverge}), much in contrast to the isotropic films. When we fit equation \ref{aregress} for $\max\sigma_x$ we find (table \ref{anispace_msig_tab}):

\input{anispace_msig_eq.tex}

There seems to be a critical value of about $\sigma_x =0.015$\,MPa beyond which the number of successfully converged simulations collapses in figure \ref{pspaceani_n}, but this is not reflected in equation \ref{anispace_msig_eq}. The intercept is close to the to the maximum value of 0.025\,MPa (95\,\% CI$=\left[2.07\cdot10^{-2}\quad2.27\cdot10^{-2}\right]$, $p < 0.001$), and the effects are small. If anything, larger peaks seem to make for slightly more difficult simulations ($P$, 95\,\% CI$=\left[-8.08\cdot10^{-3}\quad-6.07\cdot10^{-3}\right]$, SS/SS$_\text{tot}=0.53$, $p < 0.001$).\\

...

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@@ -511,22 +510,36 @@ None of the films in these 100 models seem have to have completed `full circle c

When we fit the model of equation \ref{aregress} to $\frac{l}{L}$ at $\sigma_x =0.015$\,MPa, we find (table \ref{anispace_lL_tab}):

\input{anispace_lL_eq.tex}

There is unmistakeably an effect of $\mu$, about half the size of the offset (95\,\% CI$=\left[-4.39\cdot10^{-1}\quad-3.71\cdot10^{-1}\right]$, SS/SS$_\text{tot}=0.86$, $p < 0.001$). Contributions of $\sigma$ and $P$ are each an order of magnitude smaller, which is also reflected in their SS/SS$_\text{tot}$ of $\mathcal{O}\left(10^{-3}\right)$.

The evaluation of the coefficients of equation \ref{anispace_lL_eq} over the tested range of $\sigma_x$ (figure \ref{pspaceani_lL_as}) shows an intercept that is decreasing with increasing $\sigma_x$, as expected (figure \ref{pspaceani_lL_a0}). The coefficient for $\mu$ shows a non-linear response to $\sigma_x$, and our analysis at $\sigma_x =0.015$\,MPa is close to the minimum of this curve (figure \ref{pspaceani_lL_a1}).

It is hard to assess from this data whether this coefficient really increases after $\sigma_x =0.015$\,MPa, or whether we are looking at an effect of the decreasing numbers of observations beyond that value (figure \ref{pspaceani_n}). The consistent response of the intercept could be taken as a suggestion that the non-linear response for $\mu$ might be `real'.

The 95\,\% confidence intervals of the other two coefficients, $a_2$ for $\sigma$ (figure \ref{pspaceani_lL_a2}) and $a_3$ for $P$ (figure \ref{pspaceani_lL_a2}), oscillate around (and mostly include) zero, and the estimated sizes of the effects of these two parameters are (very) small over the entire range of $\sigma_x$.\\

\noindent\textbf{Round up} Stress fibre anisotropy, in particular its predominant orientation, can affect the measure of $\frac{l}{L}$ as much as $\sigma_x$ does.

\subsection{Assumption of no contraction}

We can do the maths (table \ref{anispace_dL_tab}):

\input{anispace_dLl_eq.tex}

\input{anispace_dLr_eq.tex}

And we can plot the deformations on a scale that makes it look like something is going on (figure \ref{anispace_dL_fig}), but any discussion on `effects' would be a waste of time.

\begin{table}

\caption{Multiple linear regression model fitted to $\Deltaup L$ for $\sigma_x =0.015$\,MPa for the 100 random models for \textbf{(a)} the left edge, and \textbf{(b)} the right edge. With the coefficients, their 95\,\% confidence intervals (CI), the amount of variation in the model explained by the parameter (SS/SS$_\text{tot}$), and the $p$-values for the coefficients. The last two columns contain the predicted range in the simulations based on the parameter space, and that predicted range divided by the complete range of the variable found in the simulations.\label{anispace_dL_tab}}

Nothing happens here.

Anisotropic sideways curling seems even more a purely geometrical affair (no lengthening/shortening of the constructs) than isotropic curling.

\begin{table}[h!]

\caption{Multiple linear regression model fitted to $\lambda_L$ for $\sigma_x =0.015$\,MPa for the 100 random models for \textbf{(a)} the left edge, and \textbf{(b)} the right edge. With the coefficients, their 95\,\% confidence intervals (CI), the amount of variation in the model explained by the parameter (SS/SS$_\text{tot}$), and the $p$-values for the coefficients. The last two columns contain the predicted range in the simulations based on the parameter space, and that predicted range divided by the complete range of the variable found in the simulations.\label{anispace_dL_tab}}

@@ -534,29 +547,32 @@ When we fit the model of equation \ref{aregress} to $\frac{l}{L}$ at $\sigma_x =

\caption{Distribution of $\lambda_L$ (equation \ref{eq:lambdal}) for the 100 anisotropic models for \textbf{(a)} the left edge, and \textbf{(b)} the right edge.\label{anispace_dL_fig}}

\caption{Coefficients of the equations for $\lambda_L$ (red) with the left edge on the left (equation \ref{anispace_dLl_eq}) and the right edge on the right (equation \ref{anispace_dLr_eq}), and from top to bottom the coefficients for: the intercept $(a_0)$, the mean of the Gaussian distribution $\mu$ ($a_1$), the standard deviation of the Gaussian distribution $\sigma$ ($a_2$) and the anisotropic fraction $P$ ($a_3$). \label{pspaceani_dL_as}}

%\caption{Coefficients of the equations for $\lambda_L$ (red) with the left edge on the left (equation \ref{anispace_dLl_eq}) and the right edge on the right (equation \ref{anispace_dLr_eq}), and from top to bottom the coefficients for: the intercept $(a_0)$, the mean of the Gaussian distribution $\mu$ ($a_1$), the standard deviation of the Gaussian distribution $\sigma$ ($a_2$) and the anisotropic fraction $P$ ($a_3$). \label{pspaceani_dL_as}}

%\end{figure}

\subsection{Assumption of constant curvature}

\input{anispace_dk_eq.tex}

\begin{table}

...

...

@@ -663,7 +679,7 @@ When we fit the model of equation \ref{aregress} to $\frac{l}{L}$ at $\sigma_x =

\caption{Coefficients of the equations for $R_\varepsilon$ (red) with the left edge on the left (equation \ref{anispace_Rel_eq}) and the right edge on the right (equation \ref{anispace_Rer_eq}), and from top to bottom the coefficients for: the intercept $(a_0)$, the mean of the Gaussian distribution $\mu$ ($a_1$), the standard deviation of the Gaussian distribution $\sigma$ ($a_2$) and the anisotropic fraction $P$ ($a_3$). \label{pspaceani_Re_as}}

\end{figure}

\subsection{Minimum for $\sigma_x$--$\frac{l}{L}$}\label{sec_md}

\subsection{Minimum for $\frac{l}{L}\left(\sigma_x\right)$}\label{sec_md}