**6.2 Linear combination of modifier plates for improved dose uniformity**

It is possible to increase maximum thickness of the irradiated objects while maintaining the dose uniformity at a high level by using the combination of modifier plates of different thicknesses in such a way that the absorbed dose distribution in the object is as close as possible to a constant value, or, equivalently, and by minimizing the following function to calculate the weight coefficients *ai* :

$$\sum\_{i=1}^{N} \left( \sum\_{j=1}^{M} a^i d^{i j} - \text{const} \right)^2 \to \text{Min},\tag{15}$$

where *di*,*<sup>j</sup>* is the dose at the depth *xj* of the object irradiated with electrons by adding aluminum modifier plate with a thickness of *di*. Summation is carried out over *i* ranging from 1 to *N,* where *N* is the number of aluminum plates of different thicknesses, and over *j* ranging from 1 to *M*, where *M* is the number of points in the object at which the absorbed dose is determined.

**Figure 9a** shows the absorbed dose distribution in the water phantom irradiated with 10 MeV electrons without modifier plates, with one plate, and with a combination of plates with the thickness varying from 0.1 mm to 9.5 mm. **Figure 9b** shows the absorbed dose distributions in the water phantom irradiated with electron beams with the energy ranging from 5 MeV to 10 MeV by adding a combination of modifier plates. It can be seen while the phantom irradiated without adding plates has an optimal distance 3.8 g/cm2 and the dose uniformity 0.71, adding one modifier plate increases the dose uniformity to 0.97 and decreases the optimal distance to 1.2 g/cm2 . In contrast, using the combination of plates maintains the optimal distance at 3 g/cm2 and the dose uniformity at 0.98 (**Figure 9a**). **Figure 9b** shows that varying the electron energy and using a combination of modifier plates allows to increase the dose uniformity up to 0.95 for objects with the mass thickness ranging from 1.8 g/cm2 to 3.8 g/cm2 .

It should be noted that the proposed algorithm makes it possible to select the combination of plates that results in maximum dose uniformity for any beam spectrum. The maximum possible dose uniformity of 0.98 is achieved for objects with a mass thickness of up to 3.5 g/cm<sup>2</sup> irradiated with 10 MeV electrons.

**Figure 9.**

*(a) Absorbed dose distribution during 10 MeV irradiation with and without plates; (b) Absorbed dose distributions in the water phantom during irradiation with 5–10 MeV electrons with the combination plates [66].*
