@@ -520,11 +520,11 @@ 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 predominant orientation$\mu$, about half the size of the offset (95\,\% CI$=\left[-4.39\cdot10^{-1}\quad-3.71\cdot10^{-1}\right]$, SS/SS$_\text{tot}=0.86$, $p =0.000$). Contributions of peak standard deviation $\sigma$ and anisotropic fraction $P$ are each an order of magnitude smaller, which is also reflected in their SS/SS$_\text{tot}$ of $\mathcal{O}\left(10^{-3}\right)$.

There is unmistakeably an effect of predominant orientation$\mu$, about half the size of the offset (95\,\% CI$=\left[-4.39\cdot10^{-1}\quad-3.71\cdot10^{-1}\right]$, SS/SS$_\text{tot}=0.86$, $p =0.000$). Contributions of peak standard deviation $\sigma$ and anisotropic fraction $P$ 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 predominant orientation$\mu$ 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}).

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 predominant orientation$\mu$ 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 predominant orientation$\mu$ might be `real'.

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 predominant orientation$\mu$ might be `real'.

The 95\,\% confidence intervals of the other two coefficients, $a_2$ for peak standard deviation $\sigma$ (figure \ref{pspaceani_lL_a2}) and $a_3$ for anisotropic fraction $P$ (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 $\sigma_x$.\\

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@@ -595,7 +595,7 @@ Before we assess we look at the difference in curvature between the left and rig

\end{equation}

At $\sigma_x =0.015$\,MPa, the model (equation \ref{aregress}) predicts for this difference (table \ref{anispace_dk_tab}):

\input{anispace_dk_eq.tex}

The intercept of a difference of about 40\,\% seems to represent the modus of the highly skewed distribution of $\Deltaup\kappa$ over a wide range of $\sigma_x$ (figure \ref{anispace_dk_fig}), but the model does not fit well and fails to explain most of the variation in the data ($R_a^2=0.10$, SS$_\text{error}$/SS$_\text{tot}=0.86$). About two thirds of the remaining variation is attributed to predominant orientation$\mu$ (SS/SS$_\text{tot}=0.09$, $p =0.003$), and about one third to the anisotropic fraction $P$ (SS/SS$_\text{tot}=0.05$, $p =0.023$). Peak standard deviation seems to have no role of interest (SS/SS$_\text{tot}=0.01$, $p =0.445$).

The intercept of a difference of about 40\,\% seems to represent the modus of the highly skewed distribution of $\Deltaup\kappa$ over a wide range of $\sigma_x$ (figure \ref{anispace_dk_fig}), but the model does not fit well and fails to explain most of the variation in the data ($R_a^2=0.10$, SS$_\text{error}$/SS$_\text{tot}=0.86$). About two thirds of the remaining variation is attributed to predominant orientation$\mu$ (SS/SS$_\text{tot}=0.09$, $p =0.003$), and about one third to the anisotropic fraction $P$ (SS/SS$_\text{tot}=0.05$, $p =0.023$). Peak standard deviation seems to have no role of interest (SS/SS$_\text{tot}=0.01$, $p =0.445$).

\begin{table}[t!]

\center\small

\caption{Multiple linear regression model fitted to $\Deltaup\kappa$ for $\sigma_x =0.015$\,MPa 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_dk_tab}}

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@@ -760,7 +760,7 @@ Al in all, we can conclude that the analysis of film curvature $R$ from projecte

\noindent Alford \textsl{et al.} performed an analytical parameter space analysis of free curling constructs with $\nu_\text{film}=\nu_\text{cells}=0.5$\cite[figure 5]{Alford2010}. The results of our analysis qualitatively agree with theirs: film thickness and cell layer thickness have an opposite effect, cell layer stiffness has no effect. Alford \textsl{et al.} did not investigate film stiffness $E_\text{film}$. Our analysis suggests that this parameter may be the most important one (based on its coefficient), and that construct length $L$, also not investigated in \cite{Alford2010}, may be as important as cell layer thickness $t_\text{cells}$.

Within the limitations of our analysis we found that in order of decreasing effect, film stiffness $E_\text{film}$, film length $L$, cell layer stiffness $t_\text{cells}$, and film thickness $t_\text{film}$ affect the amount of construct curling (table \ref{isospace_lL_tab}). Of these parameters, construct length $L$ is probably accurately assessed from the image analysis. Proper experimental control or measurement of film stiffness $E_\text{film}$ therefore seems most the important factor to take into account.

Within the limitations of our analysis we find that in order of decreasing effect, film stiffness $E_\text{film}$, film length $L$, cell layer stiffness $t_\text{cells}$, and film thickness $t_\text{film}$ affect the amount of construct curling (table \ref{isospace_lL_tab}). Of these parameters, construct length $L$ is probably accurately assessed from the image analysis. Proper experimental control or measurement of film stiffness $E_\text{film}$ therefore seems most the important factor to take into account.

Cell layer thickness $t_\text{cells}$ is not under experimental control and is probably also the hardest parameter to assess from these four. Yet, our results suggest that its effect on the accuracy of the analysis should not be neglected.

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@@ -769,13 +769,29 @@ Cell layer thickness $t_\text{cells}$ is not under experimental control and is p

Things get more hazy in our anisotropic analysis.

For starters, our analysis is hampered by convergence issues for larger values of $\sigma_x$

For starters, the analysis is hampered by convergence issues for larger values of $\sigma_x$(figure \ref{aniconverge}). Second, 3D `sideways' curling is not properly assessed with the projected edge length (figures \ref{lkaniexam}\&\ref{fig:anikappaxyz}). Both edges deform differently in 3D curling (figures \ref{anispace_dk_fig}\&\ref{anispace_ke_fig}) and even if the assessed 2D curvature is between that of the 3D curvatures of the edges, it seems naive to assume that it is the mean of those two (3D) curvatures (figure \ref{anispace_Re_fig}). In fact, for anisotropic cell layers, predominant fibre orientation $\mu$ affects the projected edge length $\frac{l}{L}$ as much as cell traction stress $\sigma_x$ does (table \ref{anispace_lL_tab}).

Projected edge length only assesses curvature in 2D. For anisotropic constructs it therefore becomes a non-linear proxy for the actual cell traction stresses that deforms the construct in 3D. And because the predominant fibre orientation affects the amount of out-of-plane deformation, comparison of projected edge lengths between constructs can only be valid for (near) equal values of $\mu$.\\

But it gets worse.\\

\noindent Even for a given predominant fibre orientation $\mu$, projected edge length $\frac{l}{L}$ may not be unique for a given value of cell traction stress $\sigma_x$ (figure \ref{lfmu_sigma}). We found a minimum in the $\frac{l}{L}\left(\sigma_x\right)$ relationship \textsl{for more than half of our random models} (figure \ref{anispace_md_fig}). The measurement of $\frac{l}{L}$ is ambiguous around these minima.\warning

Our current analysis does not suffice to predict the occurrence or severity of these minima in terms of the fibre parameters $\mu$, $\sigma$, and $P$. Figure \ref{lfmu_sigma} suggests that the occurrence this minimum is most prevalent around a predominant orientation $\mu$ that is about 30\,\degree{} off-axis $\left(90\,\degree\pm30\,\degree\right)$, but $\sigma$ and $P$ will also have their influence.\\

\noindent Our proposed analysis may be affected by convergence issues, but two conclusions seems inescapable:

\begin{enumerate}

\item 2D projected edge length $\frac{l}{L}$ is an ambiguous proxy for comparison of (actual 3D) cell traction stresses $\sigma_x$ between cell layers with different predominant fibre orientations $\mu$; and

\item the $\frac{l}{L}\left(\sigma_x\right)$ relationship may not be unique for a given (anisotropic) cell layer.

\end{enumerate}

The diligent researcher that aims to use this thin film platform for anisotropic studies, would therefore do well to first investigate this $\frac{l}{L}\left(\mu, \sigma_x\right)$ relationship in more detail.

\vfill\subsubsection{Final note to self}

The FibLab documentation reads \cite[original emphasis]{fiblab2018}: ``When you describe anisotropy with two parameters, $P_\text{ani}$ and $\sigma_\text{p}$, the \textsl{description} of the `amount of alignment' in your distribution may be unambiguous, what to put into your statistical comparisons is \textsl{not}.''\\

Figure \ref{lfmu_sigma} suggests that the occurrence this minimum is most prevalent around a predominant orientation $\mu$ of $90\,\degree\pm30\,\degree$.

\noindent The current analysis suggests, tentatively, that, for these thin films at least, size matters\\more than width.

The effects of the proportion of the angles that goes into the (Gaussian) peak, peak size $P$ (or $P_\text{ani}$) are in general (much) larger than those of peak standard deviation $\sigma$ (or $\sigma_\text{p}$, tables \ref{anispace_msig_tab}, \ref{anispace_lL_tab}, \ref{anispace_dk_tab}, and \ref{anispace_Re_tab}; table \ref{anispace_ke_tab} seems to be an exception).

effects of $\sigma$ and $P$ best investigated with $\mu=90$\,\degree

\ No newline at end of file

This is perhaps best investigated with a given $\mu=\frac{\piup}{2}$, which you neglected to do.