Mass Stopping Power and Range of Alpha Particles in Biological Human Body Tissues (Blood, Brain, Adipose and Bone)

Arwa Alshibel, Khalda T. Osman

Department of Physics, College of Science, Qassim University, Buraydah, Saudi Arabia.

DOI: 10.4236/oalib.1110775                

Abstract

The mass stopping power and range of alpha particles in biological human body tissues (blood, brain, adipose, and bone) were computed using the MATLAB Program at energies ranging from 0.2 MeV to 200 MeV. The Bethe Block formula was used to compute the mass stopping power. Furthermore, the alpha particle ranges at the tissues were calculated using the simple integration (continuous slowing down approximation) method. The results of mass-stopping power versus energy and range versus energy were graphed, and empirical equations for determining mass-stopping power and ranges were derived. The current results for mass-stopping powers and ranges of bone cortical and adipose tissues were compared to those produced by ASTAR results (Alpha Stopping Power and Range). They showed good agreement, especially for energies that extend from 1 to 200 MeV.

Keywords

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

1. Introduction

In the majority of studies and applications involving radiation science, including those involving blood cancer, cell modification, radiation mutations, nuclear physics, and so forth, the energy loss and range of particles or ions in the air, tissues, and polymers are of great importance. One of the most important calculations in reducing the risk of radiation damage to tissue near the sick tissue during radiotherapy is the stopping power calculation [1] .

In recent years, theoretical and experimental studies of the Stopping Power (SP) and range of charged particles have increased, especially in the field of radiation physics. This area has been the subject of numerous successful theoretical and experimental studies [2] .

Because the Bragg curve can transfer energy in the form of point shots through diseased tissue while maintaining the target dose and delivering a higher dose to the tumor (3), the use of heavier ions or alpha particles as an alternative to external photon beams in radiotherapy is increasing [3] .

Charged particles are commonly utilized in radiotherapy due to their precise tissue penetration and the fact that their depth relies on both the type of material being treated and incident particle energy. Charged particles, including as alpha particles, deuterons, and protons, are important in radiotherapy because of their ability to transmit energy to the target [4] . Therefore, exact knowledge of the stopping power and range of alpha particles is necessary for the precise dosimetry of alpha radiation. Researching the stopping power and range of alpha in biological tissues requires identifying or gathering information from research studies or ICRU reports.

Alpha particles, on the other hand, are the least invasive and hence lose more energy by ionizing the target’s atoms. This is why alpha particles are the most harmful to health [5] .

The stopping power is the rate of loss of energy per unit length (dE/dx). It depends on the speed and charge of both the fallen particles and the target material [1] .

An early study of the energy loss of charged particles traveling through materials yields the following general formula for stopping power:

$-\frac{\text{d}E}{\text{d}X}=\frac{4\mu e^{4}N{Z}_2}{m{v}^2}{Z}_1^{2}B$ (1)

where: N is the target density, Z₂ is the target atomic number, m is the electron mass, v and Z₁ are the projectile velocity and charge respectively and B is called the “stopping number” [6] .

The negative sign in Equation (1) signifies the fact that the particles lose energy as they pass through the material. For most practical purposes, the physics of the energy loss phenomena is complex and will be not covered in detail [7] .

When an ion beam enters a solid, it loses energy via collisions with target nuclei and electrons (electronic stopping and nuclear stopping, respectively) [8] . The stopping power owing to nuclear and electrical interactions is then simply added together to determine the total stopping power. It is also possible to ignore the nuclear component of the stopping power [9] .

Stopping power is the energy loss of charged particles in matter.

The charged particle suffers from loss of energy per unit of distance, which is

referred as $-\frac{\text{d}E}{\text{d}X}$ is conventionally known as the stopping power. This basic quantity is calculated using quantum and mechanical quantum theories. It is important to express $-\frac{\text{d}E}{\text{d}X}$ in terms of the properties specifying the incident

charged particle, such as its velocity v and charge and the properties pertaining to the atomic medium, the charge of the atomic nucleus, the density of atoms, and the average ionization potential. R.D. Evan and W.E. Meyerhof consider

only a crude, approximate derivation of the formula for $-\frac{\text{d}E}{\text{d}X}$ . They begin with

an estimate of the energy loss suffered by an incident charged particle when it interacts with a free and initially stationary electron. Referring to a collision cylinder whose radius is the impact parameter b and whose length is the small distance traveled dx [10] .

Because the stopping power depends on the energy of particle, then it is possible to calculate the path of the particle which corresponds to the lower particle energy from the initial value E₀ to some smaller value E₁. Based on the definition of the stopping power, that path is equal to the integral:

$R={\displaystyle {\int }_{E_1}^{E_0}\frac{\text{d}E}{-\frac{\text{d}E}{\text{d}X}}}$ (2)

In general, the particle’s range does not correspond to the particle’s penetration depth. The penetration depth is defined as the greatest gap between the material’s surface and the particle as it goes into the substance. The penetration depth is determined by the geometry of the particle’s trajectory and is always less than the range. Because the interaction with the material’s atoms is random, various particles will be deflected differently, resulting in varied penetration depths (even if those particles are identical and have similar beginning energies). However, their material ranges will be quite comparable (practically equal). Because of the irregular trajectories of electrons, the discrepancy between range and penetration depth is extremely pronounced. Heavy charged particles, on the other hand, have a range that is nearly equal to their penetration depth (assuming the particle beam is normal to the material’s surface), since they move in straight lines [10] .

The objective of the present work is to calculate the mass stopping power and the range of alpha particles for the tissues under consideration at energies ranging from 0.2 MeV to 200 MeV. Also, we try to compare the obtained results of mass stopping power and range with available ASTAR data.

2. Method for Calculating Mass Stopping Power and Ranges

Bethe Block conducted the first quantum mechanical research of stopping power. When the projectile’s velocity exceeds the Bohr velocity, the Bethe Block theory of stopping is valid. The target is supposed to be an elemental material in Bethe theory. Bethe’s approach to energy loss is based on the born approximation, which is applied to inelastic collisions between heavy particles and atomic electrons. The projectile heavy particle is supposed to be structure less in this theory, while the target nucleus is assumed to be indefinitely massive [11] .

2.1. Calculations of Electronic Stopping Power

The following Bethe Block stopping power equation has been used for energy range 0.2 - 200 MeV [3] [12] [13] :

$-\frac{\text{d}E}{\text{d}X}=\frac{5.08\times {10}^{-31}{z}^{2}n}{{\beta }^2}\left[F\left(\beta \right)-\ln I\right]$ (3)

where: β is v/c where v is the velocity of alpha particles and c is light speed, I is the mean excitation energy and F(β) is given by:

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

n is calculated using the following relation:

$n=\frac{\rho z{N}_a}{A}$

where: N_a is Avogadro number, ρ is the density of substances and Ζ/A is the ratio of atomic number to the mass number of substances. The basic data for calculating mass stopping powers which is equal to the electronic stopping power divided by the density of the tissues and are tabulated in Table 1. The elemental composition of blood, Braine, Bone and Adipose tissues are given in Table 2. It is well known that human tissue chemical compositions play a significant role in the investigation of micro diametric distributions in irradiated human beings [1] . Then Equation (3) is used to find the values of stopping power and then we divided it by the density of the tissues to find mass stopping power for all the energy values for all tissues. All calculations were done using MATLAB program.

The percentage deviation error for the present results of mass stopping power and that of ASTAR results of bone cortical and adipose tissues results are calculating by the following relation

$\text{DeviationError}=\left(\frac{\text{PresentResult}-\text{ASTARResult}}{\text{ASTARResult}}\right)\times 100\%$ (4)

Table 1. Basic Data for calculating mass stopping powers.

Table 2. Elemental composition of blood, bone, Braine and Adipose tissues.

2.2. Calculations of Alpha Particles Range

A heavy particle’s range is defined as the straight path it takes inside the target. Due to their low weight and difficulty in calculating their journey length, light particles like electrons and positrons can scatter in the path of targets at enormous angles. For determining the path length of light particles in particular, Monte Carlo techniques built on a wide range of computing algorithms have been utilized successfully. In contrast, the travel length of heavier particles, such as alpha particles, is nearly linear. Some numerical integration techniques can be used to calculate the alpha particle range. But the Continuous Slowing Down Approximation (CSDA) is a simple and common method to calculate the range of the heavy particles like alpha particles in the targets and this method is employed in this study. Incident particles continuously lose their energy in the path of the targets and the CSDA method neglects energy loss fluctuations. to find range of a given tissues Equation (2) is used. All calculations were done using MATLAB program.

3. Results and Discussion

The Bethe Block theory and the MATLAB application were used in the current work to determine the mass stopping power and range of alpha particles in four tissues of the human body (brain, bone, adipose and blood tissues). The results of mass stopping powers of brain, bone, adipose and blood tissues respectively are given in Table 3. In Table 4 the range of alpha particles for bone, brain, adipose and blood tissues respectively are given. In Table 5 a comparison between the present results and that of ASTAR results of mass stopping powers of adipose and bone cortical tissues are given while a comparison between the present results and that of ASTAR result of alpha particles range for bone cortical and adipose tissue are given in Table 6. To study the relation between the mass stopping power and the alpha particles energy the graphical method is used as shown in Figures 1-4 for the tissues under consideration and the empirical formula for calculating mass stopping powers knowing alpha particles energy are obtained for all tissues under study as given in Table 7. In Figures 5-7 the relation between the range and alpha particles for the tissues under study are shown while the empirical formula for calculating alpha particles range of all

Figure 1. Mass stopping power of blood versus energy of Alpha particles.

Table 3. Values of mass stopping power (MeV cm²/g) of brain, bone cortical, Adipose and blood tissues.

Table 4. Values of range (in g/cm²) of Bone, brain, Adipose and Blood tissues.

Table 5. Comparison of mass stopping power (MeV cm²/g) of This work and That of ASTAR results for Adipose tissue and Bone Cortical (ICRU-44).

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

Table 7. The empirical formulae for calculating mass stopping powers.

Figure 2. Mass stopping power of bone versus energy of Alpha particles.

Figure 3. Mass stopping power of brain versus energy of Alpha particles.

Figure 4. Mass stopping power of Adipose tissue versus energy of Alpha particles.

Figure 5. Range of blood versus energy of Alpha particles.

Figure 6. Range of bone versus energy of Alpha particles.

studied tissues is given in Table 8. The percentage deviation errors calculated by Equation (4) of mass stopping power for bone cortical and adipose tissues are given in Table 9 and shown in Figure 8, Figure 9.

Figure 7. Range of brain and Adipose tissues versus energy of Alpha particles.

Table 8. The empirical formulae for calculating range.

Table 9. Percentage of deviation error of stopping power of for Adipose tissue and bone.

Figure 8. Deviation Error% for Adipose tissue versus energy of Alpha particles.

Figure 9. Deviation Error% for bone versus energy of Alpha particles.

4. Conclusions

In this study, estimates of the mass-stopping power and range of alpha particles incident on the four different human body tissues (brain, bone, adipose, and blood) in the energy range of 0.2 - 200 MeV have been made. The following findings are made:

1) As can be seen in Figures 1-4, all tissues under study have lower mass stopping powers as the energy of alpha particles rises. This can be explained by the fact that the rapid motion of charged particles at high energies reduces their survival time near the electron of the target material and decreases the probability of collision with it. The outcome is a slower energy transfer process, which reduces the mass-stopping power.

2) It was shown that for all investigated tissues, the alpha particle energy at 0.6 MeV had the highest mass-stopping power values.

3) As shown in Figure 7 the range of Brain and Adipose tissues are nearly equal.

4) The ASTAR results and the current results of the mass-stopping powers of bones cortical and adipose tissues are compared, and it is found that there is good agreement between the two results. The percentage deviation error ranged between 1.2% and −8.10% for bone cortical tissue and between 13% and −7.19% for adipose tissue at the energy range of 1 - 200 MeV.

5) The empirical equations for mass-stopping power and alpha particle ranges are straightforward and cover an extensive spectrum of alpha particle energies between 0.2 and 200 MeV for all tissues under consideration. They also provide important data to people researching alpha particle therapy and shielding.

Conflicts of Interest

The authors declare no conflicts of interest.

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