# Designing the Uniform Stochastic Photomatrix Therapeutic Systems
## 2 Theory
To calculate the illuminance resulting from a flat LED matrix, for example, at a point \(K\) lying on the optical axis of one of the diode sources, the following equation could be used:  
\[ \begin{array}{c}E_{k}=\frac{I_{0}}{Z^{2}}\bigg{|}z>0+\frac{4J_{0}Z}{\left(4 \delta^{2}+Z^{2}\right)^{3/2}}\bigg{|}z>2I_{0}+\frac{4J_{0}Z}{\left(8\delta^{2} +Z^{2}\right)^{3/2}}\bigg{|}z>2\sqrt{2}I_{0}+\frac{4J_{0}Z}{\left(16\delta^{2} +Z^{2}\right)^{3/2}}\bigg{|}z>4I_{0}\cdots\\ \qquad\qquad\qquad+\frac{4J_{0}Z}{\left(32\delta^{2}+Z^{2}\right)^{3/2}}\bigg{|} z>4\sqrt{2}I_{0}+\ldots\end{array}  \tag{13}\]  
where \(2d\) is the distance between the adjacent LED point sources, and \(I_{0}\) is derived trigonometrically as \(l_{0}=2d\cdot\cos\alpha_{0}\) (Figure 3). The first term of Series (13) describes the LED irradiation nearest to the point \(K\). As the distance from the irradiating surface increases, new terms occur in the series due to obvious overlapping of light beams. This equation shows that theoretical determination of illuminance at an arbitrary point requires complicated calculations and unjustified simplifications. It is more appropriate to determine the LED light distribution over a surface of a certain type generated by one of the aforementioned types of PMTS or combinations of them. Also, it is better to assume that the shape of the irradiated biotissue can be approximated with a flat, cylindrical, or spherical surface. As a result, due to such assumptions, the general task of calculating the LED intensity created by the matrix structure can be divided into several special cases. The illuminance at each point of the irradiated surface is calculated as the summarization of illuminances produced by each LED. In the process of calculation of the contribution of each LED the coordinate system is translated so that the origin of coordinates coincides with the LED under consideration, whereas the \(Z\)-axis is directed along the optical axis of the diode source (in the direction of light distribution). Thus, if coordinates \((x_{0},\ y_{0},\ z_{0})\) of the point \(K\) in the new coordinate system are determined, then the illuminance is calculated using Formulas (1), (2) and (3). Such a basic coordinate system was selected to simplify the specification of parameters of the irradiated surface. Now, based on all of the aforementioned assumptions, let us consider several main methods of illuminance calculation produced by PMTS of various types. _Flat PMTS_ (Figure 4, left). For the flat type of PMTS the calculation begins from the upper left LED. In this case, the coordinates of the irradiated point are:  
\[\left\{ \begin{array}{c}x_{1,1}=x_{0}+0.5(N_{x}-1)dx\\ y_{1,1}=y_{0}+0.5\big{(}N_{y}-1\big{)}dy\\ z_{1,1}=z_{0}\end{array} \right. \tag{14}\]  
where \(dx\) and \(dy\) are the distances between adjacent LEDs along the \(X\)- and \(Y\)-axes of the basic coordinate system, respectively; \(N_{x}\) and \(N_{y}\) represent the numbers of LEDs along these axes. Figure 3: Simple schematics for calculation of illuminance produced by flat PMTS. The illuminance is calculated sequentially for LED rows parallel to the \(X\)-axis. As the coordinate system is translated along the \(X\)-axis, the coordinates are transformed as follows:  
\[\left\{ \begin{array}{c}x_{i+1,j}=x_{i,j}-dx\\ y_{i+1,j}=y_{i,j}\\ z_{i+1,j}=z_{i,j}\end{array} \right. \tag{15}\]  
Furthermore, as the coordinate system is translated along the \(Y\)-axis, the following transformation of coordinates is used as well:  
\[\left\{ \begin{array}{c}x_{i,j+1}=x_{i,j}\\ y_{i,j+1}=y_{i,j}-dy\\ z_{i,j+1}=z_{i,j}\end{array} \right. \tag{16}\]  
_Cylindrical PMTS_ (Figure 4, center; Figure 5). Prior to the calculations of the cylindrical type of PMTS, the following parameters have to be specified: quantities \(N_{g}\) and \(N_{\mathrm{a}}\) of LEDs along the generating line and the arc of the cylinder, respectively; distance \(dx_{c}\) between adjacent LEDs along the generating line; angle \(\gamma\) between adjacent LEDs lying on the same arc; PMTS cylindrical radius \(R_{c}\).