# Designing the Uniform Stochastic Photomatrix Therapeutic Systems
## 2 Theory  
The main advantage of the PMTS in comparison with other optical sources is the provision of the required irradiation intensity over the entire area of a sufficiently extended biological surface. However, as such, the spatial irradiation of a bio-object by a certain group of LEDs cannot be considered uniform if the LED sources are not located on the PMTS substrate in accordance with a strictly specified calculated location. Without this, irradiation of the body surface at different points of the biological object will occur with different and uncontrolled intensities. Many scientific methods have been proposed in the research articles on the issue of ensuring the uniform irradiation of an extended object using a group of LED sources or LED matrices [28; 29; 30; 31; 32; 33]. It should be noted that one of the promising modern areas in this field is the development of various lens solutions for LED systems [34; 35; 36; 37]. However, despite this, the task of uniform irradiating using various LED schematics is still more than relevant. Thus, let us initially consider here the theoretical aspects of the mathematical and engineering calculations of LED matrices, which allow designing the PMTS satisfying the properties of the spatial uniformity of the irradiation intensity over the entire surface of the body. In accordance with this, the geometrical dimensions and relief parameters of the biotissue as a basis determine the spatial formation of the PMTS substrate. The main parameters of the typical LED used in PMTS are the following:  
- wavelength of LED irradiation at maximum \(\lambda\), nm;
- half-width of irradiation spectrum \(\Delta\lambda\), nm;
- luminous intensity at given current through LED, mcd;
- double aperture angle \(2\alpha_{0}\), deg;
- general voltage and current characteristics;
- dependence of the luminous intensity on the direct current through LED;
- maximum permissible electrical features of LED;
- physical dimensions for particular LED and total amount of them;
general geometrical sizes of PMTS and shape of matrix surface. In the current case, particular for our tasks of biomedical application of PMTS, the following assumptions could be made:  
* LED irradiation is quasi-monochromatic;
* each LED is considered to be a point source radiating within the dimensional aperture angle;
* the wavelength of the LED irradiation belongs to visible and near-infrared ranges;
* the optical axis of LED is normal to the inner surface of PMTS and to irradiated biotissue. Now, let us determine the illuminance produced by a single LED. Consider orthogonal coordinates with LED at the origin of coordinates and \(Z\)-axis coinciding with the optical axis of the LED (i.e., the axis of the diode lens) (Figure 2). In this case, the illuminance at some physical point \((x_{0},\,y_{0},\,z_{0})\) of the irradiated surface is determined by the following equation:  
\[E=\frac{J(a)\cdot\cos\beta}{r_{0}^{2}} \tag{1}\]  
where \(r_{0}^{2}\) is derived from:  
\[r_{0}=\sqrt{x_{0}^{2}+y_{0}^{2}+z_{0}^{2}} \tag{2}\]  
and angle \(\alpha\) is determined by:  
\[\alpha=arccos\left(\sqrt{\frac{z_{0}^{2}}{x_{0}^{2}+y_{0}^{2}+z_{0}^{2}}}\right) \tag{3}\]  
Therefore, in the aforementioned equations, the key parameter \(J(\alpha)\) is the luminous intensity of the single LED, then \(\alpha\) is the angle between the direction of irradiation and the optical axis of the LED, and \(\beta\) is the angle between the direction of irradiation and the normal to the biological surface at a given point. Equations for the angle \(\beta\) depend on the frame shape of PMTS and the irradiated biotissue. Let us indicate here some typical cases of this interrelation:  
* flat PMTS irradiates a cylindrical surface;
* flat PMTS irradiates a spherical surface;
* cylindrical PMTS irradiates a cylindrical surface;  
Figure 2: Initial schematics for calculation of illuminance of a single LED point source.