M1 model

variable		represents		depends on
t		time
\(t_e\)		exposure time
z		depth in tissue
\(\rho_s\)		sensitizer concentration
\(\rho_i\)		ink concentration
a		penetration depth		\(\rho_i\)
f		hardened fraction		z, \(i_s\), t
\(i_s\)		surface light intensity
i		light intensity		z, \(\rho_i\)
\(x_s\)		exposure		\(i_s\), \(t_e\)
r		hardening rate per unit intensity		\(\rho_s\)
s		solidified fraction (able to survive development)		f

The time dependence of the hardened fraction \(f\) (naturally constrained to \(0 < f < 1\)) at depth \(z\) is given by

\[ \frac{d f(z,i,t)}{dt} = r i (1-f) \]

with initial condition \(f(z, i, t=0) = f_0\) (\(f_0\) representing a possible pre-exposure hardening; normally, \(f_0 = 0\)). Thus, at each unit of time, the remaining unhardened fraction \(1-f\) gets converted to hardened at a rate \(r i\) where \(r\) is a rate per unit intensity and \(i\) is the (local) light intensity at depth \(z\).

For the hardening rate, we will assume it to be proportional to the concentration of light-sensitive chemicals (e.g. DAS to gelatine weight ratio),

\[ r = \rho_s \]

where we set the proportionality constant to one by an appropriate choice of units.

Given a light intensity \(i_s\) (assumed independent of time during the exposure) at the surface \(z=0\), and assuming that light is collimated in the \(\hat{z}\) direction, the intensity at depth \(z\) is given by

\[ i(z) = i_s e^{-z/a} \]

where \(a\) is the light penetration depth. We will assume that gelatine is purely transparent (hardened or not) and that penetration depth is inversely proportional to ink concentration, namely

\[ a = \frac{1}{\rho_i} \]

where the proportionality constant is set to \(1\) by choosing appropriate measurement units.

We thus get

\[ \frac{df}{dt} = \rho_s i_s e^{-\rho_i z} (1-f) \rightarrow \frac{d}{dt} ln (1-f) = -\rho_s i_s e^{-\rho_i z} \]

Integrating over time and using our initial condition yields

\[ \left. \ln (1-f) \right|_0^{t_e} = -\rho_s e^{-\rho_i z} i_s t_e \]

in which \(t_e\) is the exposure time. Namely,

\[ f(z, i_s, t_e) = 1 - (1-f_0) e^{-\rho_s e^{-\rho_i z} i_s t_e} \]

(as \(t_e \rightarrow \infty\), we of course get \(f \rightarrow 1\) as expected).

We can simplify notations by introducing the exposure \(x_s \equiv i_s t_e\):

\[ f(z, x_s) = 1 - (1-f_0) e^{-\rho_s x_s e^{-\rho_i z}} \]

The hardened fraction will lead, under development and transfer to the temporary support, to a solidified fraction \(s\) which will depend on the hardened fraction in a to-be-specified way.

During exposure, gelatine is hardened through the process of cross-linking. The transition between non-hardened and hardened gelatine is assumed to be similar to a percolation transition at some critical value \(f_c\) of the hardened fraction. The solidity of the gelatine will thus rapidly increase as \(f\) crosses this value.

Our interest is in the double transfer process, and we will seek to model the first transfer onto a temporary support (modeling the final transfer is simpler, see later). Our development process thus (for now) consists of the following steps:

• mating the tissue to a temporary support
• warm water development
• drydown of the temporary support

Here, we need to make some assumptions.

Assumption D1: during warm water development, all regions with \(f < f_c\) are washed away; any region with \(f > f_c\) is fully transferred.

Expressed in equations, Assumption D1 is (using the Heaviside function)

\[ s(f) = \theta (f - f_c) \]

Under assumption D1, the (post-drydown) height \(h_1\) for a given exposure is given by the solution to

\[ f_c = f(h_1, x_s) \hspace{5mm} \rightarrow \hspace{5mm} h_1 (x_s) = \theta (\rho_s x_s - g) ~\frac{1}{\rho_i} \ln \frac{\rho_s x_s}{g}, \hspace{5mm} g \equiv \ln \frac{1-f_0}{1-f_c} \]

in which \(g\) is a characteristic of the gelatine (and its eventual pre-hardening), and the height exists only for exposures above a threshold

\[ x_t = \frac{g}{\rho_s} \]

Summarizing, under Assumption D1, the post-drydown height on the temporary support:

• is zero for exposures below the threshold \(x_t\) (which is inversely proportional to the sensitizer concentration)
• for exposure above the threshold, grows logarithmically with exposure

An alternative assumption is

Assumption D2: during warm water development, all regions with \(f < f_c\) are completely washed away; regions with \(f > f_c\) are transferred, but have their non-hardened fractions washed away.

Namely, Assumption D2 takes the solidified fraction to be

\[ s(f) = f \theta(f - f_c) \]

The post-drydown height under Assumption D2 is given by the integral

\[ h_2 (x_s) = \int_0^{h_1} dz f(z, x_s) \]

in which the upper bound of the integral is the largest \(z\) with a hardened fraction above the critical one, namely the height according to Assumption D1.

This integral is calculated as follows:

\[ h_2 = \int_0^{h_1} dz \left[ 1 - (1-f_0) e^{-\rho_s x_s e^{-\rho_i z}} \right] = h_1 - \frac{1 - f_0}{\rho_i} \int_{\rho_s x_s e^{-\rho_i h_1}}^{\rho_s x_s} d\eta \frac{e^{-\eta}}{\eta} \]

Substituting for \(h_1\) gives

\[ h_2 (x_s) = \frac{\theta (\rho_s x_s - g)}{\rho_i} \left[ \ln \frac{\rho_s x_s}{g} - (1 - f_0) \int_{g}^{\rho_s x_s} d\eta \frac{e^{-\eta}}{\eta} \right] \]

Using the definition of the exponential integral \(E_1 (x) \equiv \int_z^\infty dt \frac{e^{-t}}{t}\), we obtain

\[ h_2 (x_s) = \frac{\theta (\rho_s x_s - g)}{\rho_i} \left[ \ln \frac{\rho_s x_s}{g} - (1 - f_0) \left[ E_1 (g) - E_1 (\rho_s x_s) \right] \right] \] (since \(E_1\) is a decreasing function, the square brackets represent a positive value, so \(h_2 < h_1\) as expected).

Approximations

Using the simplest approximation of \(E_1\) to first order, \(E_1 (x) = -\gamma - \ln x + x + O(x^2)\), we get a first approximation

\[ h_2^{a_1} = h_1 - \frac{1-f_0}{\rho_i} \left[\rho_i h_1 + \rho_s x_s (e^{-\rho_i h_1} - 1) \right] = f_0 h_1 + (1-f_0) \frac{\rho_s}{\rho_i} (1 - e^{-\rho_i h_1}) x_s \]

which we can rewrite

\[ h_2^{a_1} = \frac{f_0}{\rho_i} \ln \frac{x_s}{x_0} + (1-f_0) \frac{\rho_s}{\rho_i} (x_s - x_0) \]

This approximation is not good for us: we expect the exposure to be such that \(\rho_s x_s \gg 1\), so a small argument cannot be assumed.

Using the (upper bound) approximation for \(E_1 (x)\), i.e. \(E_1 (x) < e^{-x} \ln (1 + 1/x)\), we get the approximation

\[ h_2^a = h_1 - \frac{1-f_0}{\rho_i} \left[ e^{-\rho_s x_s e^{-\rho_i h_1}} \ln \left( 1 + \frac{e^{\rho_i h_1}}{\rho_s x_s}\right) - e^{-\rho_s x_s} \ln \left(1 + \frac{1}{\rho_s x_s}\right) \right] \]

which is perhaps too complicated to be useful.

We define the (local) contrast as the slope of the (log-log) density (on temporary support, post-drydown) - exposure graph, namely (assuming ink concentration is constant)

\[ c(x) = \rho_i \frac{d}{d \ln x} h \]

Under Assumption D1, this yields

\[ c_1(x) = 1, \hspace{10mm} x > x_0 \]

Namely, Assumption D1 fails to account for contrast variation as a function of ink or sensitizer concentration.

Under Assumption D2, using approximation \(a_1\), we get

\[ c (x) = f_0 + (1-f_0) \rho_s x_s \]

Problem: contrast would increase with increasing sensitizer concentration. Precisely the contrary is required. Again, this approximation is not good.

To compute the contrast, it is easiest to take the derivative of the integral expression for \(h_2\),

\[ c_2 (x) = \rho_i \frac{d}{d \ln x} \left[ h_1 - \frac{1 - f_0}{\rho_i} \int_{g}^{\rho_s x} d\eta \frac{e^{-\eta}}{\eta} \right] \]

\[ = c_1 - (1-f_0) e^{-\rho_s x} \]

Problem: this means that contrast increases with increasing \(\rho_s\), the contrary to what we want.

Given a transmission density for the temporary support, a simple model can be given for the reflection density after transfer to the final support.

Let \(R_s\) be the reflection coefficient of the final support. The reflection coefficient of the image is given by \(R_i = T^2 R\) where \(T\) is the transmission coefficient (single pass) through the gelatine of the image (we need \(T^2\) because light has to go in, bounce and go through again).

We have

\[ R_i = R_s e^{-2h/a} \]