Mass Stopping Power and Range of Alpha Particles in Biological Human Body (Water and Eye Lens Tissue)

Arwa Alshibel, Khalda T. Osman

Department of Physics, College of Sciences, Qassim University, Buraidah, Saudi Arabia.

DOI: 10.4236/oalib.1111081                

Abstract

We used the MATLAB Program to calculate the mass stopping power and range of alpha particles in biological human body tissues (such as water and the eye lens tissues) at energies between 0.2 and 200 MeV. The Bethe Block formula was used to calculate the mass stopping power, and the simple integration (continuous slowing down approximation) method was used to calculate the alpha particle ranges. Empirical formulae were developed to determine mass-stopping power and ranges of water and eye lens tissues. The results of graphing the mass-stopping power versus energy and range versus energy are presented, and compared the ASTAR (Alpha Stopping Power and Range) results with the present results for the mass-stopping powers and ranges of water. The results indicated remarkable agreement, particularly for energies that extend from 2 to 200 MeV, with a percentage deviation error of 4.81% - 6.39%. A graphical representation of the mass stopping power of water and eye lens tissue with their compositions is also provided. In addition, various radiation parameters essential in cancer treatment, such as absorbed dose, equivalent dose, effective dose, alpha particle depth, and linear energy transfer, are computed in this study.

Keywords

Human Body Tissues, Alpha Particles, MATLAB, Mass Stopping Power, Range

1. Introduction

For the last century, scientists have been studying how charged particles interact with matter in order to understand their stopping power and energy dissipation. This research has a wide range of important applications, including ion implantation, fundamental particle physics, nuclear physics, radiation damage, and radiography [1] [2] [3] [4] . As heavy charged particles move through matter, they primarily lose energy through ionization and atomic excitation. This process has been extensively researched and is crucial to various scientific and technological advancements.

When heavy charged particles travel through matter, they tend to lose energy primarily through ionization and atomic excitation [4] [5] [6] .

The stopping power is defined as the mean energy loss per unit path length ?dE/dx. It depends on the charge and velocity of the projectile and, of course, the target material [7] [8] [9] . Early investigations of the energy loss of charged particles traversing matter arrive at a general stopping power formula. If an ion beam penetrates through matter, it loses energy due to collisions with electrons (electronic stopping) and target nuclei (nuclear stopping) [1] . The total stopping power is then just the sum of the stopping powers due to electronic and nuclear interactions [5] [6] . At low energies the total energy loss is usually described in terms of electronic stopping power [4] [5] [6] . The nuclear component of the stopping power can also be ignored [1] .

In this work, we will examine the interactions between alpha particles and matter. The mass stopping power of alpha particles will be studied at a range of energies, from 0.2 to 200 MeV. We will also investigate how these interactions affect water and eye lens tissues. Additionally, we will calculate the alpha particle ranges in water and lens tissues at energies between 0.2 and 200 MeV.

2. Stopping Power

The electronic mass stopping power for alpha particles on water and eye lens tissues is calculated by the Bethe-Bloch formula. The full expression for the Bethe-Bloch formula can be written as:

$-\frac{\text{d}E}{\text{d}x}=\frac{5.08\times {10}^{-31}{z}^{2}n}{{\beta }^2}\left[F\left(\beta \right)-\ln I\right]$ (1)

where: β is v/c where v is the velocity and c is light speed, I is the mean excitation energy and F(β) is given by [10] :

$F\left(\beta \right)=\ln \ln \frac{1.02\times {10}^6{\beta }^2}{1-{\beta }^2}-{\beta }^2$ (2)

The electron density n is calculated by:

$n=N_{av}\langle \frac{z}{A}\rangle =6.02\times {10}^{23}\rho \langle \frac{z}{A}\rangle$ (3)

2.1. Mass Stopping Power of Eye Lens

The mass stopping power of eye lens tissues is calculated after substituting the appropriate constant of electron density n and mean excitation energy I given in Table 1 in Equation (1) by the following equation:

Table 1. Basic data for calculating mass stopping powers.

$-\frac{\text{d}E}{\text{d}x\rho }=\frac{0.717269584}{{\beta }^2\times 1.070}\left[F\left(\beta \right)-4.308110952\right]$ (4)

2.2. Mass Stopping Power of Water

The mass stopping power of water is calculated after substituting the appropriate constant of electron density n and mean excitation energy I given in Table 1 in Equation (1) by the following equation:

$-\frac{\text{d}E}{\text{d}x\rho }=\frac{0.680136816}{{\beta }^2\times 1.00}\left[F\left(\beta \right)-4.317488114\right]$ (5)

3. Calculation of Alpha Particles Range

As it passes through the material, the alpha particle’s range ionizes, reducing its energy slowly until it is almost zero. Because of the infinite range of Coulomb, the particle interacts with multiple electrons simultaneously and loses energy gradually but steadily as it moves. Following a predetermined distance, it then runs out of energy. The alpha particle range is the name given to this distance. The alpha particle’s range is therefore defined as the average distance covered before it loses all of its initial kinetic energy [11] .

The range of charged particle is computed by numerical integration of the stopping power [12] . The range of the alpha particles for the tissues under considerations is calculated using the following relation:

$R={\displaystyle \underset{E_0}{\overset{E_f}{\int }}\frac{\text{d}E}{MS\left(E\right)}}$ (6)

where, $E_0$ is the initial energy of incident charged particle in material, $E_f$ is the final energy of incident charged particle in material and $MS\left(E\right)$ is the mass stopping power.

The higher alpha particles energy, the stronger their penetration through a given substance because the more Colombian interactions between the particles of alpha and absorption electrons will be required to dissipate energy before coming to rest [12] . All calculations were done using MATLAB program.

4. Alpha Depths, Absorbed Dose, Equivalent Dose, and Effective Dose

The Depth

The depth of a charged particles in the tissue or substance is given by the following relation:

$T=\frac{R}{\rho }$ (7)

where R is the range and is ρ the density of the tissue or substance.

Absorbed Dose

Two different materials, if subjected to the same gamma-ray exposure, will in general absorb different amounts of energy. Because many important phenomena, including changes in physical properties or induced chemical reactions, would be expected to scale as the energy absorbed per unit mass of the material, a unit that measures this quantity is of fundamental interest. The mean energy absorbed from any type of radiation per unit mass of the absorber is defined as the absorbed dose. The historical unit of absorbed dose has been the rad, defined as 100 ergs/gram. As with other historical radiation units, the rad has been replaced by its SI equivalent, the gray (Gy) defined as 1 joule/kilogram. The two units are therefore simply related by: 1 Gy = 100 rad the absorbed dose is a reasonable measure of the chemical or physical effects created by a given radiation exposure in an absorbing material. The absorbed dose is calculated by the following relation:

$Ad=\frac{E}{1\text{grm}}\frac{1.6\times {10}^{-13}\text{J}}{1\text{MeV}}\frac{{10}^7\text{erg}}{1\text{J}}\frac{1\text{rad}}{\frac{100\text{erg}}{\text{gram}}}$ (8)

Equivalent Dose

The equivalent dose is calculated by the following relation:

$H_T={\displaystyle \underset{R}{\sum }{W}_R\times D}$ (9)

where $W_R$ = Weighting factor = 20 for Alpha particle and D is Absorbed dose.

Effective Dose

The tissues differ in their sensitivity to the late effects of radiation, which represent the biological responses to the tissue, which are delayed for a long period of time, often several years. The effective dose is used to estimate the incidence of delayed effects in the future. If a part of the body such as the lungs receives a radiation dose, it represents a risk factor of lung cancer. If the same dose is given to another organ, it causes a different risk factor and is called the dose measured by the effective dose, is designated E.

$E={\displaystyle \underset{T}{\sum }{W}_T{H}_T}$ (10)

where $W_T$ is the weighting factor of tissue.

And for water and eye lens $W_T=0.12$ .

5. Percentage Deviation Error

The percentage deviation error for the present results of mass stopping power and that of ASTAR results of water are calculating by the following relation:

$\text{DeviationError}=\left(\frac{\text{PresentResult}-\text{ASTARResult}}{\text{ASTARResult}}\right)\times 100\text{\%}$ (11)

the Percentage deviation error was calculated in a Table 13 for the tissues that were compared to the Aster.

6. Results and Discussion

The Bethe Block theory and the MATLAB application were used in the current work to determine the mass stopping power using Equation (4) and Equation (5). The range of alpha particles in water and eye lens tissues is calculated using Equation (6). Table 2 shows the elemental composition of water and eye lens. The results of mass stopping powers and range of alpha particles in water, and eye lens respectively are given in Table 3 and Table 4. In Table 5 a comparison between the present results and that of ASTAR results of mass stopping powers of water are given while a comparison between the present results and that of ASTAR result of alpha particles range for water are given in Table 6. Table 7 is Values of mass stopping power of Water and its compositions and Table 8 for the Values of mass stopping power of Eye lens and its some of compositions.

To study the relation between the mass stopping power and the alpha particles energy the graphical method is used as shown in Figure 1 and Figure 2 for the tissues under consideration. The empirical formula for calculating mass stopping powers knowing alpha particles energy are obtained for all tissues under study as given in Table 9. Figure 3 depicts a comparison of the mass stopping power of an alpha particle in water for the current results and those of ASTAR. In Figure 4 and Figure 5 the relationship between the range and alpha particles for the tissues under study are shown while the empirical formula for calculating alpha particles range of all studied tissues is given in Table 10. In Table 11 and Table 12 the calculated depth of alpha particles, LET, absorbed dose, equivalent dose and effective dose of alpha particle for both water and eye lens are given. The percentage deviation errors of mass stopping power for water are given in Table 13.

Table 2. Elemental composition of water and eye lens.

Table 3. Values of mass stopping power (MeV∙cm²/g) of water and eye lens.

Table 4. Values of range (in g/cm²) of water and eye lens.

Table 5. Comparison of mass stopping power (MeV∙cm²/g) of this work and that of ASTAR results for water.

Table 6. Comparison of range (in g/cm²) of water of this work and ASTAR results.

Table 7. Values of mass stopping power (in MeV∙cm²/g) of water and its compositions.

Table 8. Values of mass stopping power (in MeV∙cm²/g) of Eye lens and its some of compositions.

Table 9. The empirical formulae for calculating mass stopping powers.

Table 10. The empirical formulae for calculating range.

Table 11. Alpha depth in the tissue, LET, absorbed dose, equivalent dose and effective dose of Alpha particle in water.

Table 12. Alpha depth in the tissue, LET, absorbed dose, equivalent dose and effective dose of Alpha particle in eye lens.

Table 13. Percentage of deviation error of stopping power for water.

Figure 1. Mass stopping power of water and its composition.

Figure 2. Mass stopping power of eye lens tissues and its some of composition.

Figure 3. Mass stopping power of water measured in this work and that of ASTAR results versus energy of Alpha particles.

Figure 4. Range of water versus energy of Alpha particles.

Figure 5. Range of eye lens versus energy of Alpha particles.

7. Conclusions

Our research aims to evaluate how much energy alpha particles deposit in water and eye lens tissues. We used energy levels between 0.2 and 200 MeV and calculated the mass stopping power for each beam with the Bethe-Bloch formula. Based on our findings, we can draw the following conclusions:

1) The mass stopping power is proportional to Z (the charge of the incident particles), Z/A (the ratio of atomic and mass’ number of the tissue) and I (the ionization energy of the tissues).

2) The mass stopping power increases rapidly at low energies reaches a maximum and decreases gradually with increasing energy.

3) The mass stopping power allows us to calculate the range of the heavy particles in the absorber material.

4) The current fitting curves for mass stopping power and alpha particle range in water are excellent and accord with ASTAR data in the alpha energy rage of 2 to 200 MeV, with a deviation error of 4.81% - 6.39%.

5) Semi-empirical formulas for mass stopping power and range of alpha particles in water and eye lens tissues were developed for alpha particle energies ranging from 0.2 to 200 MeV.

6) The mass stopping power of alpha particles in water and eye lens tissues is equal to their average composition values.

Conflicts of Interest

The authors declare no conflicts of interest.

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