Electron Beam Processing of Biological Objects and Materials

Abstract

The research explores a wide range of applications for electron accelerators in industrial irradiation processing. It also compares the physical properties of electron beams, dose ranges, and methods used for irradiation of polymers, medical items, transplantology objects, pharmaceuticals, and foods. Moreover, the study discusses the depth dose non-uniformity in objects irradiated with accelerated electrons. The research also highlights the dependency of geometry, density, and chemical composition of the object on the dose distribution. Another focus of the study is computer simulation of electron irradiation method, encompassing all physical and technical parameters to assess the dose distribution throughout the irradiated objects, since without knowing the precise electron beam spectrum, it is impossible to accurately reconstruct the dose distribution throughout the objects. Considering that the beam spectrum cannot always be identified, especially for industrial accelerators, the study presents algorithm for reconstructing the dose distribution in irradiated objects. The final part of the research provides methods for increasing the dose uniformity throughout objects irradiated with electron beams.

1. Introduction

Irradiation with accelerated electron beams is a convenient all-purpose technology for the processing of various biological objects and materials since it offers undeniable advantages, such as the ability to vary the beam intensity and penetration depth of electrons, minimal changes in the temperature and pressure, as well as the absence of negative effects of chemical compounds, which allows to solve a wide range of tasks, from plant growth stimulation to increasing the wear resistance of metals [1]. Electron beam irradiation of objects is enabled by the transfer of the energy to molecules and atoms of substance of irradiated object. As electrons act on the substance, ionization, and excitation of atoms and molecules lead to physical and chemical reactions that change the properties of the object. One of the main factors that determine the effect of accelerated electrons on the substance is the irradiation dose, which is the ratio between the energy absorbed by the volume and its mass [2, 3]. The research discusses the application of electron beams for processing biological and non-biological objects and establishes the dose ranges and physical properties of electron beams to solve various tasks.

Since objects and materials have diverse properties, the dose distribution throughout irradiated objects differs considerably, even if the same irradiation method is applied, depending on the chemical composition, density, and shape of the objects. The research discusses the influence of physical and chemical properties as well as electron energy on the distribution of absorbed dose over the volume of object.

The dose distribution in various objects can be simulated using transport codes [4], provided that all physical and technical parameters of the irradiation method are accurately reproduced, taking into account the individual properties of biological objects and materials. The study compares different transport codes used for irradiation simulation and presents an algorithm for simulating irradiation exposure using the GEANT4 tool kit [5, 6, 7, 8], which is by far the most accurate tool for obtaining the absorbed dose distribution in objects of any geometry and chemical composition when irradiated with accelerated electrons.

Considering that it is not always possible to obtain the beam energy spectrum from manufacturers of electron accelerators, it is crucial to have algorithms for reconstructing the beam spectrum of accelerators used by irradiation facilities [9, 10, 11, 12, 13, 14]. The study presents an algorithm to reconstruct the electron beam spectrum and the depth dose distribution in an object of any chemical composition and shape, based on the experimentally measured absorbed dose distribution in any known substance.

Since biological objects have complex biochemical compositions and geometry, they are highly sensitive to the lack of dose uniformity [15, 16, 17]. The research presents a method of electron beam modification using aluminum modifier plates to significantly improve the irradiation dose uniformity, which makes it possible to increase the spectrum of biological objects irradiated with electron beams.

2. Electron beam processing in industrial irradiation

Electron beam irradiation is an environmentally friendly technology that uses electron energy to initiate a range of physical, chemical, and biological effects without the use of chemical compounds, high temperature, and pressure. The high kinetic energy and penetrating power of electrons offer significant advantages over the traditional methods used for processing biological objects and materials [1, 18, 19].

An important advantage of accelerated electrons over other radiation sources is their ability to vary the beam current and electron energy [20]. By varying the beam current, it is possible to change the intensity of radiation and, consequently, the dose rate absorbed by the treated object. Varying the energy of electrons allows to control the depth of electron penetration throughout the object, depending on the purpose of irradiation treatment.

Electron accelerators are classified according to the energy levels of electrons they generate [1, 21]. Low-energy electron accelerators with energies ranging from 0.08 to 0.8 MeV are commonly used for the treatment of surfaces with a mass thickness of up to 0.2 g/cm². Medium-energy electron accelerators, which generate 0.8–5 MeV electrons, are used for the treatment of objects at the depth not exceeding 0.5 g/cm². High-energy electron accelerators with energies of 5–10 MeV are applied for sterilization of medical supplies and instruments, biological materials used in transplantation, food processing, treatment of biowaste, decomposition of industrial effluents, as well as sewage and greywater treatment since the electron beams penetrate the object at the depth of up to 5 g/cm².

Flexibility of electron accelerators allows using electron beams in diverse areas, such as crosslinking, curing, and grafting of polymers and composites; modification of the material surface; water purification; improvement of material properties; sterilization of medical supplies and instruments; and irradiation of food products [3].

Table 1 shows the applications of electron beam processing, the combinations of the energy ranges and dose ranges that are used in each application, as well as the beneficial effects used in industry.

Application	Irradiated objects	Energy (MeV)	Doses (kGy)	Purpose
Modification of polymers [21, 28, 29, 30, 31, 32]	Polyolefins: polyethylene, polyvinylchloride, ethylene-propylene rubber, polyvinylidene fluoride, and ethylene tetrafluoroethylene copolymer	0.3–5	50–400	Wire and cable insulation
		0.5–3	50–150	Heat-shrinkable materials with a “memory” effect
	Elastomers (automobile tires); Rubbers (rubber gloves, roofing, and waterproofing materials)	0.8–1	1–200	Crosslinking
	Materials containing oligomers and monomers (adhesives, cellulose, varnishes, inks, films, concrete)	0.15–0.4	15–50	Polymerization and Grafting
	Composites (automobile and aircraft components)	1.2–1.5	150–250
	Hydrogels: polyvinyl alcohol, polyacrylamide, polyvinylpyrrolidone, polyethylene oxide, and methylcellulose	3–10	25–50
	Materials containing polytetrafluoroethylene; cellulosic waste (sawdust, shavings, straw, small chips, etc.); Teflon waste; Rubber, Cellulose, and Polypropylene materials	0.1–12	500–1500	Decomposition
Sterilized medical supplies [29, 33, 34]	Disposable medical equipment (syringes, needles, masks, surgical gloves, etc.); Packaging material (droppers, Petri dishes, pipette bottles, blood collection tubes); Pharmaceuticals (eye ointments and drops, ointments for burns, saline solutions)	5–10	10–35	Disinfection/sterilization
Preserving foods [15, 35, 36, 37, 38]	Potatoes, onions, garlic, carrots, etc.	up to 10 MeV	0.05–0.2	Inhibiting sprouting
	Seeds, spices, fruits		0.15–0.5	Sterilization for pest management
	Fruits, vegetables		0.5–1	Suppressing ripening
	Meat, fish, poultry, seafood, canned food		1–4	Extending shelf-life by suppressing mold, bacteria, and fungi growth
			3–7	Killing pathogenic bacteria
	Food additives, hospital diets, emergency rations, space food		10–70	Sterilization
Pollution control [39, 40]	Natural sources of water (rivers, lakes, reservoirs, artesian wells)	0.3–2	0.2–1	Cleaning of industrial wastewater, flue gases, and solid waste
	Wastewater and sewage sludge		0.4–4
	Gases containing SO2 and NOx, where x = 1, 2.		10–20
	Hospital and airport waste, activated carbon regeneration, and contaminated soil cleanup		1000–1500
Metals and mineral processing [41, 42]	Gems and minerals	2–50	25–1500	Crystalline and glass coloring/discoloring
	Semiconductors (diodes and thyristors)	9–13	0.1–100	Improvement of properties

Table 1.

Electron beam application and operating parameters [22, 23, 24, 25, 26, 27].

Irradiation with continuous and pulsed electron beams has become the main tool for obtaining various polymers in free-radical and ionic polymerization reactions. Since crosslinking enables the formation of chemical bonds between molecular chains and the creation of three-dimensional structures, it is commonly used to improve the physical characteristics of the polymer, including heat resistance [21].

Crosslinking at industrial accelerator with the power up to 350 kW is performed with electron energy ranging from 0.5 MeV to 3 MeV, and beam current from 50 mA to 100 mA. Modification of polymers uses the dose ranging from 1 kGy to 400 kGy depending on the desired effect [25]. Polymerization reaction of low molecular weight compounds with free radicals occurring in the process of curing due to indirect action of accelerated electrons in polymerizing monomers is enabled by electrons with the energy of up to 300 keV [31]. Graft polymerization, which involves grafting various monomers onto a polymer chain to give the polymer properties of the monomer, is found to be efficient at 0.3–0.5 MeV [32]. Irradiation with electron beams with energies of up to 0.3 MeV is used for the synthesis of nanocomposites for structural and magnetic applications, nanogels and hydrogels for drug delivery systems, and for the synthesis of membranes for medical and industrial applications [30, 31].

Electron-beam melting with 0.01–0.3 MeV electrons is utilized to produce complex, intricate geometries with excellent mechanical properties by selectively melting and fusing metal power particles [26]. This technique is used in airspace, medical, and automatic industries for manufacturing high-precision components since it allows to change the composition, morphology, and hardness of metals as well as improve wear resistance [27].

Irradiation to suppress microbial contamination in biological and non-biological objects requires a more complex approach since the desired effect is achieved with a precise combination of the absorbed dose, dose rate, and electron energy [37, 38]. Moreover, the dose range for biological objects is commonly narrower compared with metals, minerals, and other non-organic objects due to their chemical complexity. Sterilization of food products, medical items, and materials used in transplantology is carried out using electron beams with energies from 3 to 10 MeV and a power of up to 50 kW [34, 35]. In food irradiation, varying beam penetration depth allows to solve a range of tasks, from sprout inhibition with the doses of up to 0.2 kGy to food sterilization with the doses of up 50 kGy [32, 33, 34, 35]. The required absorbed dose increases to 35 kGy in the treatment of medical devices for inactivation of viruses and microorganisms [29, 33].

Treatment of drinking water and wastewater is carried out using accelerated electrons with the energy of 0.3–1 MeV [39, 40]. While for drinking water treatment doses up to 1 kGy are applied, biowaste treatment for inactivation of a wide range of viruses and bacteria is carried out at doses up to 1 MGy, which is the maximum dose used to control contamination of organic objects [24].

3. Depth dose distribution in biological objects irradiated with accelerated electrons

Irradiation of biological objects is particularly sensitive to depth dose distribution due to the nature of such objects, different physical and chemical composition as well as a lack of homogeneity and complex geometry. All these factors have an impact on the irradiation uniformity and the efficiency of irradiation processing.

In treatment with accelerated electrons, non-uniformity of irradiation is inevitable due to the nature of the depth dose distribution throughout the irradiated object. The ratio of the minimum value of absorbed dose D_min to the maximum value of absorbed dose D_max in the object volume is commonly used as a criterion of irradiation dose uniformity [1]. Different categories of biological objects, such as transplantology objects, pharmaceuticals, and food products, require uniformity of at least 80% [1, 38]. Considering that achieving an irradiation uniformity of more than 80% for the objects with a mass thickness of more than 2 g/cm² is a challenging task, it is necessary to take into account such factors as packaging filling irregularity, as well as geometry, structure, chemical composition, and density of irradiated objects [15, 43].

This part will discuss the factors, which affect the depth dose distribution in the biological object irradiated with accelerated electrons.

3.1 Electron beam energy

One of the key factors that lead to the non-uniformity of electron beam irradiation is the character of depth dose distribution, which depends on the energy of electrons. Figure 1a and b shows the dependency of the relative absorbed dose on the penetration depth in water parallelepiped irradiated with 4 MeV, 6 MeV, 8 MeV, and 10 MeV, calculated using the GEANT4 toolkit [44]. Figure 1a shows that the higher the electron energy, the deeper the penetration of electrons into the object is and the lower maximum dose is achieved during irradiation.

Figure 1.

(a) Depth dose distribution in water parallelepiped irradiated with 4 MeV, 6 MeV, 8 MeV, and 10 MeV electrons; (b) Parameters of absorbed dose distribution in water irradiated with 10 MeV electrons.

To illustrate the dependency between the electron energy and depth dose distribution, let us introduce the following parameters of the absorbed depth dose distribution:

L_max is the distance from the surface of the object with the maximum absorbed dose;

L_opt is the optimal distance from the surface of the object at which the dose value is equal to the surface dose;

$K=\frac{D_{\mathit{min}}}{D_{\mathit{max}}}$ is the dose uniformity coefficient, which is the ratio of the minimum value of absorbed dose D_min to the maximum absorbed dose D_max in the object.

As can be seen from Figure 1b when the water parallelepiped is irradiated with 10 MeV electrons, L_max is 28.0 mm and L_opt is 38.75 mm, and the irradiation uniformity coefficient is 0.72. With an increase in electron energy from 4 MeV to 10 MeV, L_max increases from 10.25 mm to 27.5 mm, and L_opt increases from 15 mm to 38.75 mm (Figure 1a). Therefore, varying the beam energy allows to change the dose uniformity of the irradiated object.

3.2 Density of the irradiated object

Considering that biological objects vary in terms of density and density distribution, it is important to investigate the impact of the object density on the parameters of the absorbed depth dose distribution, such as the optimal distance L_opt and dose uniformity K. While the irradiation of biological objects is performed with 4–10 MeV electrons, it is feasible to estimate the depth dose distribution coefficients for the objects irradiated within this energy range having the density from 0.3 g/cm³ to 1.6 g/cm³, which is similar to that of biological objects at industrial irradiation facilities. Figure 2a shows that dose uniformity K varies from 0.62 to 0.72 and practically does not depend on the density of the irradiated object for the water parallelepipeds with densities ranging from 0.3 g/cm³ to 1.6 g/cm³.

Figure 2.

(a) Dependency of the dose uniformity K in water parallelepiped with the density ranging from 0.3 g/cm³ to 1.6 g/cm³ on the electron energy and the function approximating the calculated dependency (green line); (b) Dependency of L_opt on the electron energy for irradiated water parallelepiped with the different density and the functions approximating the calculated dependencies (solid lines).

Figure 2b shows the dependency of optimal distance L_opt on the electron energy for irradiation of parallelepipeds with densities of 0.3 g/cm³, 0.6 g/cm³, 1.0 g/cm³, and 1.6 g/cm³. As it can be seen, the higher the energy of electrons, the greater the value of L_opt, which means that at higher energies a greater dose uniformity can be achieved for the objects of greater thickness. At the same time, the lower the density of the irradiated object, the greater the growth rate of L_opt value with an increase in electron energy.

According to the computer simulation using the GEANT4 toolkit and MATLAB, the analytical interdependencies of electron energy, dose uniformity K, optimal distance L_opt, and the object density can be expressed as follows [45]:

$$L_{\mathit{opt}}\left[\mathrm{cm}\right]=4\left[\frac{{\mathrm{cm}}^4}{\mathrm{MeV}\ast \mathrm{g}}\right]\times {\mathrm{\rho }}^{-0.96}\times E\left[\mathrm{MeV}\right]-1.59\left[\frac{{\mathrm{cm}}^4}{\mathrm{g}}\right]\times {\mathrm{\rho }}^{-0.46},$$E1

$$K=0.01\left[{\mathrm{MeV}}^{-1}\right]\times E\left[\mathrm{MeV}\right]+0.57,$$E2

where $\mathrm{\rho }$ is the object density $\left[\frac{\mathrm{g}}{{\mathrm{cm}}^3}\right]$, E is electron energy. These dependencies were obtained with a maximum interpolation error of no more than 2%.

3.3 The material of irradiated object

The material of irradiated object has an impact on the distribution of absorbed doses throughout the object. This can be explained by the difference in the number of electrons in electron shells of various atoms, which determines the nature of interaction between the accelerated electrons and matter. Figure 3 shows the dependencies of the relative absorbed dose distribution on the penetration depth of 10 MeV electrons in aluminum, graphite, water, and polypropylene, which have different effective charge Z_eff.

Figure 3.

Depth dose distribution in aluminum (Al, Z_eff = 13), graphite (C, Z_eff = 6), water (H₂O, Z_eff = 7.4), and polypropylene (C₂H₄, Z_eff = 5.4) irradiated with 10 MeV electrons.

Table 2 presents the optimal distances L_opt for the object made of aluminum, graphite, water, and polypropylene irradiated with 10 MeV electrons, as well as the irradiation uniformity throughout the object made of different materials, which is determined for the optimal distance L_opt.

Material	ρ (g/cm^3)	Zeff	ρ·Zeff (g/cm^3)	K (rel.un.)	Lopt (mm)
Aluminum	2.7	13	35.1	0.65	14
Graphite	1.7	6	10.2	0.72	19
Water	1.0	7.4	7.4	0.72	37
Polypropylene	0.9	5.4	4.9	0.75	42

Table 2.

The parameters of absorbed dose distribution in different materials.

Electrons with the energies up to 10 MeV, which is the maximum energy used in radiation treatment of biological objects, lose their energy due to ionization losses. It can be seen from Figure 3, the higher its effective charge Z_eff and density, the more energy is lost by an electron penetrating an equal depth in different materials, according to the Bethe-Bloch formula [46]. Moreover, with an increase in the effective charge Z_eff the maximum penetration depth of electrons throughout the object decreases, at the same time causing the surface absorbed dose to decrease compared to the maximum value. Therefore, the material with a higher electron density tends to have a lower irradiation uniformity compared with the material with a lower electron density, as Table 2 shows.

3.4 The geometry of irradiated object

Biological objects exposed to accelerated electrons, such as foodstuffs and materials used in transplantology may have complex geometry that can be dramatically different from parallelepiped. A parallelepiped is regarded as a perfect shape for irradiation because its simple geometry allows electrons to penetrate the object perpendicularly to its surface. Exposure of more complex geometries, such as a sphere, an ellipsoid, or a cylinder, to electron beams, however, does not permit the perpendicular penetration of electrons into the surface layer, making the depth dose distribution irregular and less predictable. Figure 4 is a 3D-color representation of the absorbed dose distribution over a parallelepiped with the edge of 6 cm, a ∅ 6 cm sphere, and a ∅ 6 cm cylinder simulated as water phantoms as they are unilaterally irradiated with 10 MeV electrons.

Figure 4.

3D-color map of absorbed dose distribution throughout the water phantoms: (a) parallelepiped with the edge of 6 cm, (b) ∅ 6 cm sphere, and (с)∅ 6 cm cylinder with the height of 6 cm during unilateral 10 MeV electron irradiation.

As it can be seen from Figure 4, the distance at which the absorbed dose reaches its maximum changes with the change in the object shape. In the case of the parallelepiped, as the electrons start penetrating the volume, the dose increases slightly but when the depth exceeds 2 cm the dose drops to zero at the distance of 5 cm from the surface (Figure 4a). It is interesting that the layers of the parallelepiped close to the edges are underexposed. When a sphere is irradiated with 10 MeV electrons the layers of the sphere close at the equator are clearly overexposed (Figure 4b). In the cylinder, overexposure can be observed on the lateral sides of the surface while the character of depth dose distribution is similar to that of the parallelepiped (Figure 4c). When irradiating from one side, all the shapes are partly underexposed the maximum electron penetration depth is lower compared to the linear dimensions of the objects.

It can be concluded that not only the electron energy, but also the properties of the object—density, chemical composition, and shape—have an impact on the distribution of absorbed dose throughout the object. Since biological objects have non-homogeneous density, complex shapes, and chemical composition, it is necessary to take an individual approach to establish the dose distribution for various objects.

4. Computer simulation of depth dose distributions

The non-uniformity of the absorbed dose distribution throughout the irradiated object entails the non-uniformity of the irradiation effect: overexposure can cause material destruction or undesirable changes of physical and chemical material properties, while underexposure may not lead to the desired effect [15, 16, 17]. Therefore, the distribution of the absorbed dose throughout the irradiated object should be strictly controlled.

Modern dosimetry methods for determining the integral dose absorbed by an object and its dose distribution during industrial irradiation are time-consuming and expensive [47, 48]. Dosimetry requires regular repetition and calibration due to possible changes in the electron beam spectrum during operation of the industrial accelerator [44, 49].

Computer simulation is highly accurate method that allows to calculate 2D and 3D depth dose distributions, taking into account all the physical and technical parameters of the irradiation method, such as the geometry, density, and chemical composition of the irradiated object [50, 51, 52, 53, 54, 55].

4.1 Monte Carlo radiation transport codes

The Monte Carlo method, which is used to establish transport codes, has proved to be efficient for simulating different types of irradiation for objects of various shapes and geometries [4, 52, 53, 54, 55]. Table 3 provides transport codes for simulating the interaction of particles with the matter as well as energy ranges and materials that can be used in simulation [56].

Code	Developer	Particle type	Particle energy	Material
EGSnrc [5]	National Research Council of Canada, Canada	electrons, photons	1 keV–10 GeV	Homogeneous materials; Simple material geometry (parallelepiped, cylinder, sphere)
PENELOPE [6]	Nuclear Energy Agency, France	electrons, photons	1 keV–1 GeV
MCNPX [7]	Los Alamos National Laboratory, USA	electrons, photons neutrons	1 keV–100 GeV (photons); 1 keV–1 GeV (electrons)	Homogeneous and inhomogeneous materials; Complex material geometry
GEANT4 [8]	CERN, Switzerland	electrons, photons neutrons protons, muons	∼10 eV–100 TeV

Table 3.

Capabilities of transport modeling codes.

While the EGSnrc and PENELOPE codes are limited in their application to the objects of simple geometry [5, 6], transport codes MCNPX and GEANT4 are more widely used in the industry as these are more comprehensive instruments for simulating a wider variety of physical processes representing the interactions of electrons, photons, protons, and neutrons [7, 8]. The MCNPX code is used in nuclear medicine, radiation safety, accelerator development, and modeling industrial irradiation of biological objects and materials [52]. GEANT4 is currently the most complete set of tools for simulating the passage of particles through matter [8]. Its fields of application include high energy physics, nuclear and accelerator physics, and medical and space science research. Unlike other codes, GEANT4 simulates any geometry of the objects and the radiation source with any energy spectrum and traces physical processes selected for a particular irradiation method and objects, which makes GEANT4 the most flexible transport code with a wide scope of uses [57].

4.2 Computer simulation of the absorbed dose distribution in biological objects using GEANT4

During computer simulation biological objects irradiated with electron beams are usually represented by water phantoms since water is close in its properties to the range of biological objects involved in irradiation processing. To calculate the absorbed dose distribution using GEANT4, it is necessary to describe the geometry of the object and determine the radiation source and the volume for detecting the absorbed energy.

GEANT4 contains more than 20 standard shapes that can be used to simulate the geometry of biological objects. To increase the accuracy of depth dose distributions in biological objects, it is necessary to determine the geometry of the object as a set of volumes of different densities and chemical compositions that correspond to the different parts of the object. Electron beam is determined by specifying its size and shape, the coordinates of its center, electron energy spectrum, spatial distribution, and number of electrons emitted per second to simulate the irradiation method used. Detection is performed using virtual volumes, which divide the water phantom with the linear dimensions of X × Y × Z into the number of N_x × N_y × N_z cubic cells. In each cell, the total energy absorbed during interactions of electrons with matter is recorded using the following formula:

$$E_{i,j,k}=\sum _{n=0}^{N_{i,j,k}}{\mathrm{\Delta E}}_{i,j,k,n},$$E3

where i,j,k is the cell index, ${\mathrm{\Delta E}}_{i,j,k,n}$ is the energy absorbed by the i,j,k cells during the n interaction, $N_{i,j,k}$ is the number of interactions occurring in the i,j,k cell.

The standard deviation of absorbed energy $S_{i,j,k}$in the i,j,k cell is determined by the sum of the squares of the absorbed energy ${\sum }_{n=1}^{N_{i,j,k}}{E}_{i,j,k}^2$ and is calculated using the formula:

$$S_{i,j,k}=\sqrt{\frac{1}{N_{i,j,k}}{\sum }_{n=1}^{N_{i,j,k}}{E}_{i,j,k}^2-{\left(\frac{1}{N_{i,j,k}}{\sum }_{n=1}^{N_{i,j,k}}{E}_{i,j,k}\right)}^2.}$$E4

Then, knowing the mass of the i,j,k cell m_i,j,k, the absorbed dose is determined by the formula

$$D_{i,j,k}=\frac{E_{i,j,k}}{m_{i,j,k}}.$$E5

To calculate a 3D-color map representing the relative absorbed dose distribution throughout the biological object the dose absorbed by each i,j,k cell of water phantom is color-coded, and each cell is marked with the color corresponding to the value $D_{i,j,k}^{\mathit{rel}}$, which is the ratio between $D_{i,j,k}$ and the maximum dose value in the water phantom.

5. Reconstruction of electron beam energy spectra

A necessary condition for successful irradiation with electron beams is complete information about the spatial distribution of the absorbed dose in the irradiated object, which is determined both by the properties of the irradiated object (i.e., geometry, elemental composition, and density) and by the source parameters, primarily the energy beam spectrum [13, 14]. Modern approaches to determine the energy spectra of accelerators are based on direct measurement of electron energy using special equipment [12] and on indirect methods based on the reconstruction of the spectra using experimentally measured data [9, 10, 11].

The indirect method of reconstructing the energy spectrum is based on the solution of the Fredholm integral equation of the first kind, which can be formulated as follows:

$$D\left(x\right)=\underset{0}{\overset{E_{\mathit{max}}}{\int }}f\left(E\right)d\left(x,E\right)\mathit{dE},$$E6

where $D\left(x\right)$ is the distribution of some parameters, such as the absorbed charge, dose, fluence, and flux density, along the parameter x (depth, angle, etc.); $d\left(x,E\right)$ is the distribution of the same parameters for a monoenergetic beam with the energy E; $f\left(E\right)$ is the energy spectrum.

Usually, this equation is reduced to a system of linear algebraic equations by approximating a continuous spectrum $f\left(E\right)$ linear combination of basic functions $F_j\left(E\right)$ with $a^j$ acting as weights and by approximating distribution $D\left(x\right)$ with a discrete set of values $D^i$ at points $x_i$. The corresponding system of linear algebraic equations takes the form:

$$D^i=\sum _{j=1}^N{d}^{i,j}{a}^j.$$E7

Here $d^{i,j}$ is the parameter at the point $x_i$ created by an electron beam with an energy spectrum $F_j\left(E\right)$.

A common method for solving the system (7) is the least squares method. Fredholm equation of the first kind in the general case is an incorrectly posed problem, that is, the solution of the equation may not exist or there may be several of them. Also, the solution of the system (7) can change greatly with small changes of $D\left(x\right)$. These properties of the integral equation are transferred to its discrete counterpart, which leads to non-physical sharply oscillating solutions that have little to do with the true spectrum of the beam, as demonstrated in Figure 5a.

Figure 5.

(a) The appearance of non-physical oscillations in the reconstructed spectrum; (b) Peak smoothing due to over regularization.

To address this phenomenon, the regularization procedure is used [11, 58] that involves the modification of the original problem, which turns an incorrectly posed problem into a correctly posed one. There are two types of regularization: the first type modifies the equation; the second type modifies allowed solutions. The simplest example of regularization of the second type is the imposition of a non-negativity condition. Such regularization is used in problems in which the non-negativity of an unknown quantity is guaranteed by its physical nature, for example, mass, spectrum, energy, etc. The first method usually consists of the introduction of regularizing operators. One of the most popular methods is $L_2$ regularization, better known as Tikhonov regularization [58, 59, 60]:

$$L_2={\sum }_i{\left(D^i-{\sum }_i{d}^{i,j}{a}^j\right)}^2+\alpha \bullet \theta \left[a^j\right],$$E8

where $\alpha$ is the regularization parameter, $\theta \left[a^j\right]$ is the regularizing operator.

There is a wide range of regularizing operators [61, 62] and values of regularization parameters. In general, the higher the value of the regularization parameter, the less the regularized problem will be related to the nonregularized problem, the smaller the value, the less noticeable the regularization effect will be. In the simplest case, the regularizing operator is a Euclidean norm $L_2$ . From practice, it is known [63] that for the simplest regularizing operator, the best results are obtained by the residual method, which prescribes to choose such values of the regularization parameter that:

$${\sum }_i{\left(D^i-{\sum }_i{d}^{i,j}{a}^j\right)}^2\approx {\sigma }^2{‖D^i‖}^2,$$E9

where $\sigma$ is the relative error of $D^i$.

It is worth noting that the Tikhonov regularization with the simplest regularizing operator leads to a smoothing of the peaks of the spectrum (Figure 5b), which may be undesirable. This can be addressed by modifying the regularizing operator, for example [64]:

$$\theta \left[a^j\right]={\sum }_{j}log{\left(a^j\right)}^2.$$E10

Also, one can decompose the spectrum into a regular and singular part [64]:

$$f\left(E\right)=f_s\left(E\right)+f_r\left(E\right),$$E11

where $f_r\left(E\right)$ is regular and $f_s\left(E\right)$ is singular component:

$$f_s\left(E\right)=\beta e^{-\frac{{\left(E-\mu \right)}^2}{{∆}^2}}.$$E12

Such decomposition allows to treat two parts of spectrum separately and can lead to improved results.

6. Methods to increase dose uniformity in the objects irradiated with accelerated electrons

Sterilization of biological and non-biological objects using accelerated electrons involves the reduction of microbial growth to the required level due to high energy values absorbed by microorganisms. As electrons penetrate biological objects, they lose energy in direct interaction with atomic electrons of bacteria cell structures, which results in DNA breaks and the destruction of cell membrane [1, 3]. Reactive oxygen species appearing during radiolysis of water in biological objects destroy chemical bonds in cell molecules causing inactivation of microorganisms. Moreover, the higher the dose absorbed by the object, the greater number of microorganisms inhibited during irradiation. However, the increase in irradiation dose is limited by the irreversible physical and chemical changes occurring in the biological object. Considering that each object has its specific chemical composition and physical properties, it is necessary to select the dose range individually to ensure that effective irradiation dose does not lead to irreversible changes in the object. The dose range for the treatment of biological objects is narrower compared to non-biological objects due to a great complexity of biochemical composition of such objects, and even a small change in the dose can lead to a significant alteration in the structure and functionality of cells.

Treatment of biological objects with accelerated electrons is characterized by the non-uniformity of dose distribution over the volume of irradiated objects due to the nature of dose distribution, non-homogeneous density of biological objects, complex geometry, and chemical composition. The distribution of radiation effect on both microbiological parameters and properties of the object is non-homogeneous, which reduces the efficiency of irradiation treatment and makes it difficult to maintain the microbial values throughout the object at the required level. Thus, it is highly important to develop feasible methods to increase uniformity of absorbed dose distribution over the volume of irradiated objects.

Currently, the following methods are used in industrial irradiation facilities to increase the uniformity of dose distribution:

• Two-side irradiation for effective treatment of the objects with a high thickness;

• Application of special polymer absorbers with a density close to that of the irradiated object to fill the empty space in the package, making the complex shape of the object close to a parallelepiped;

• Varying the energy of accelerated electrons in several irradiation sessions.

When the thickness of the object exceeds the maximum path of electrons with energies up to 10 MeV, it is reasonable to carry out double-side irradiation [17]. However, this is not applicable to objects with a shape different from that of a parallelepiped because it does not provide consistent irradiation uniformity.

The second way to improve the uniformity of irradiation is the use of polymer absorbers designed to fill the empty space in the package or irregularities of the irradiated object, imitating the shape of a parallelepiped to enable the use of the previous method [17]. However, this way of tackling the problem of irradiation non-uniformity is not cost-effective since it requires the replacement of polymers that are destroyed during irradiation.

Another way to increase the irradiation uniformity is to vary the energy of accelerated electrons during several irradiation sessions. Figure 6 shows the dependencies of the absorbed dose on the depth of a parallelepiped irradiated with 3 MeV and 10 MeV electrons, as well as the distribution of the absorbed dose when the parallelepiped is irradiated by a combination of 3 MeV and 10 MeV electron, beams with weighted coefficients 0.1 and 0.9, respectively.

Figure 6.

Distribution of absorbed dose in the water parallelepiped during irradiation with 3 MeV (blue line) and 10 MeV (red line) electrons, as well as irradiation with the combination of 3 MeV and 10 MeV electron beams (yellow line).

As can be seen from Figure 6, the combination of two irradiation energies allows to increase the dose uniformity. However, the use of multiple sessions increases treatment time, which makes it difficult to apply to a wide range of categories of biological objects.

6.1 Electron beam modification method for improving dose uniformity

This study proposes a method of modifying the beam spectrum using aluminum plates, which allows to increase the dose uniformity during irradiation with accelerated electrons with the energy up to 10 MeV [44]. The main idea of increasing the uniformity of irradiation consists in additional placement of modifier plates between the beam output and the irradiated object. The energy and angular distributions of the directed monoenergetic electron beam are blurred after passing through the plate, which leads to the appearance of electrons with lower energies in the beam, improving the dose uniformity throughout the object during one irradiation session.

Figure 7 shows that the addition of aluminum modifier plates changes the dose distribution throughout the irradiated object. The dose value increases in the surface layers at a distance ranging from 0 to 1.5 g/cm², while the maximum electron path in the substance decreases. Figure 8 shows that the uniformity coefficient K grows linearly with increasing thickness of the aluminum modifier plate d.

Figure 7.

Dependency of the relative absorbed dose on the depth in water phantom irradiated with 6 MeV (a) and 10 MeV (b) electron beams with the aluminum modifier plates with thicknesses ranging from 1 mm to 5 mm.

Figure 8.

Dependency of irradiation uniformity K of a water parallelepiped on the thickness of the aluminum modifier plate after irradiation with 4 MeV, 6 MeV, 8 MeV, and 10 MeV electron beams [44].

For mass thicknesses of the object L ranging from 1.025 to 3.125 g/cm² with an error of no more than 5% it is possible to select the thickness of the aluminum modifier plate d in the range from 0.5 to 5 mm, at which the thickness of the object corresponds to the optimal distance when the object is irradiated with electron beams with the energy ranging from 4 MeV to 10 MeV using the following formula:

$$d\left[\mathrm{cm}\right]=\begin{array}{l}-0.060\left[\mathrm{cm}\right]-0.199\left[\frac{{\mathrm{cm}}^3}{\mathrm{g}}\right]\times L\left[\frac{\mathrm{g}}{{\mathrm{cm}}^2}\right]+0.093\left[\frac{\mathrm{cm}}{\mathrm{MeV}}\right]\times E_0\left[\mathrm{MeV}\right]\\ --0.002\left[\frac{{\mathrm{cm}}^3}{\mathrm{MeV}\bullet \mathrm{g}}\right]\times E_0\left[\mathrm{MeV}\right]\times L\left[\frac{\mathrm{g}}{{\mathrm{cm}}^2}\right],\end{array}$$E13

and to calculate with an error not exceeding 5% the irradiation uniformity K for different combinations of modifier thicknesses d in the range from 0.5 mm to 5 mm and initial electron energies $E_0$:

$$K=\begin{array}{l}0.603-1.830\left[\frac{1}{\mathrm{cm}}\right]\times d\left[\mathrm{cm}\right]+0.012\left[\frac{1}{\mathrm{MeV}}\right]\times E_0\left[\mathrm{MeV}\right]--0.135\left[\frac{1}{\mathrm{MeV}\bullet \mathrm{cm}}\right]\\ \times E_0\left[\mathrm{MeV}\right]\times d\left[\mathrm{cm}\right].\end{array}$$E14

Thus, knowing the required minimum value of the coefficient K_min, it is possible to select combinations of electron beam energies and thicknesses of modifier plates, at which for a parallelepiped-shaped object the irradiation uniformity is achieved at a level that is not less than the required K ≥ K_min over its volume.

The proposed method allows to increase the irradiation uniformity up to 0.97 in parallelepiped-shaped objects irradiated with electron beam energy between 4 MeV and 10 MeV when using aluminum modifier plates with a thickness ranging from 0.5 mm to 5.5 mm. The optimal distance from the surface of the object at which the dose value is equal to the surface dose ranges from 1.025 g/cm² to 3.125 g/cm² and decreases with an increase in modifier plate thickness [65].

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 $a^i$:

$$\sum _{i=1}^N{\left(\sum _{j=1}^M{a}^i{d}^{i,j}-\mathit{\text{const}}\right)}^2\to \mathit{Min},$$E15

where $d^{i,j}$ is the dose at the depth $x_j$ of the object irradiated with electrons by adding aluminum modifier plate with a thickness of $d_i$. 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.

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].

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/cm² irradiated with 10 MeV electrons.

7. Conclusion

The analysis of various applications of electron beams shows that the dose ranges, electron energy, and beam power vary greatly depending on the purpose of irradiation, which determines the electron penetration depth, absorbed dose distribution, and the desired effect. The study uses the GEANT4 toolkit to simulate the impact of electron beam irradiation on the absorbed dose distribution depending on electron energy, density, chemical composition, and geometry of the irradiated object.

To increase the accuracy of computer simulation in the absence of open access to energy beam spectrum of industrial accelerators our team developed an algorithm to reconstruct the electron beam spectrum which would allow to calculate depth dose distribution in an object of any geometry and chemical composition with an accuracy of up to 95%. Using our extensive collection of GEANT4 data on irradiation of biological objects with electrons having the energy of up to 10 MeV we developed a method to increase the irradiation uniformity up to 0.98 for the objects with a mass thickness of up to 3.5 g/cm² when one-side irradiated with 10 MeV electrons.

Since computer simulation found that the edges of the object with the shape close to a parallelepiped are underexposed, our further research will be aimed at increasing irradiation uniformity throughout the object irradiated with electron beams. Another area of interest for research is to create a database of absorbed dose distribution for irradiation of biological objects with photons to develop an algorithm for reconstructing bremsstrahlung irradiation energy spectra, which is commonly used in irradiation facilities.

Acknowledgments

We thank researchers at Scobeltsyn Scientific Research Institute of Nuclear Physics, Lomonosov Moscow State University for conducting experiments using electron accelerator.

This research was funded by the Russian Science Foundation, grant number 22-63-00075.