Calculation of Mass Stopping Power and Range of Protons as Well as Important Radiation Quantities in Some Biological Human Bodyparts (Water, Muscle, Skeletal and Bone, Cortical) ()

Ahlam S. Almutairi^{1}, Khalda T. Osman^{2}

^{1}Department of Physics, College of Science, Qassim University, AL-Rass, Saudi Arabia.

^{2}Department of Physics, College of Science, Qassim University, Buraydah, Saudi Arabia.

**DOI: **10.4236/ijmpcero.2022.112009
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In this work, the electronic mass stopping power and the range of protons in some biological human body parts (Water, Muscle, Skeletal and Bone, Cortical) were calculated in the energy range of protons 0.04 to 200 MeV using the theory of Bethe-Bloch formula as giving in the references. All these calculations were done using Matlab program. The data related to the densities, average atomic number to mass number and excitation energies for the present tissues and substances were collected from ICRU Report 44 (1989). The present results for electronic mass stopping powers and ranges were compared with the data of PSTAR and good agreements were found between them, especially at energies between 1 - 200 MeV for stopping power and 4 - 200 MeV for the range. Also in this study, several important quantities in the field of radiation, such as thickness, linear energy transfer (LET), absorbed dose, equivalent dose, and effective dose of the protons in the given biological human body parts were calculated at protons energy 0.04 - 200 MeV.

Keywords

Biological Human Bodyparts, Protons, Range, MatLab, PSTAR, Electronic Mass Stopping Power, LET, Absorbed Dose, Effective and Equivalent Dose

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Almutairi, A. and Osman, K. (2022) Calculation of Mass Stopping Power and Range of Protons as Well as Important Radiation Quantities in Some Biological Human Bodyparts (Water, Muscle, Skeletal and Bone, Cortical). *International Journal of Medical Physics, Clinical Engineering and Radiation Oncology*, **11**, 99-112. doi: 10.4236/ijmpcero.2022.112009.

1. Introduction

In many fields, such as radiation dosimetry, radiation biology, and many others, radiation chemistry, radiotherapy, and nuclear physics, the stopping power, energy loss, rangestraggling, and equivalent dose rate of ions in the air, tissue, and polymers are very important. The use of protons or heavier ions as an alternative to external photon beams in radiotherapy is increasing, with the reason being that they preserve the target dose, ensure a higher dose delivered to the tumor, and can transfer energy in the form of a point shot through diseased tissue due to the Bragg curve [1]. The stopping power of charged particles has been measured using a variety of ways, including direct energyloss measurements through films, backscattering from thick substrates with deposited absorbing layers, gamma resonance shift measurements, self-supporting methods, and indirect verification of the stopping power based on alpha energy losses in the air have all been reported as methods for measuring the stopping power of charged particles. [2] [3] [4].

In the present work, the electronic mass stopping power and range of proton in some biological human body parts (Water, Muscle, Skeletaland Bone, Cortical) are calculated using the Bethe-Bloch formula as reported in the references.

As it is known in any therapeutic unit with protons, it needs to calculate the absorbed dose, the equivalent dose to the tissue, and the effective dose according to the energy of the protons. Therefore, in this work, a variety of radiation quantities such as thickness, absorbed dose, equivalent dose, and effective dose of the protons in Water, Muscle, Skeletal and Bone, Cortical were also computed in proton energy range 0.04 - 200 MeV.

2. Methods

2.1. Calculations of Electronic Mass Stopping Power

Bethe was the first person to use quantum mechanical studies on stopping power. The Bethe theory of stopping power is valid when the projectile’s velocity surpasses the Bohr velocity. In Bethe’s theory, the goal is assumed to be charged particle. In Bethe’s approach to energy loss, the Born approximation is employed to represent inelastic collisions between heavy particles and atomic electrons. In this theory, the projectile heavy particle is as assumed to be structureless, whereas the target nucleus is assumed to be infinitely massive [4]. For the energy range 0.04 - 200 MeV, the Bethe mass stopping power equation [4] [5] [6] [7] was used:

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

where *β* is *v*/*c* where*v* is the proton velocity and *c* is light velocity,*I* is the mean excitation energy and *F*(*β*) is given by

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

*n* is calculated using the following relation:

$n={N}_{av}\rho \langle \frac{Z}{A}\rangle $ (3)

where *N _{a}* Avogadro number,

2.2. Calculations of Range

The range of a heavy particle is the straight distance it travels within the target. Light particles like electrons and positrons scatter widely throughout the path of targets due to their low mass, making it difficult to determine their journey duration. The path length of light particles has been calculated with remarkable success using Monte Carlo methods, which are based on a broad class of computational algorithms. On the other hand, heavy particles like protons have a practically straight line path length. The range of protons can be calculated using numerical integration methods. The Continuous Slowing Down Approximation (CSDA), on the other hand, is a straightforward and extensively used method for finding a variable’s range. This study used a simple and standard method for calculating the range of heavy particles such as protons in the targets. The CSDA approach uses incident particles to constantly lose energy in the route of the targets. neglects energy loss fluctuations. The range, R for an incident proton in the CSDA method is given as [1] [4] [7]:

$R=\underset{{E}_{0}}{\overset{{E}_{f}}{{\displaystyle \int}}}\frac{\text{d}E}{MS\left(E\right)}$ (4)

Table 1. Basic data for calculating mass stopping powers.

Table 2. Elemental composition of Water, Muscle, Skeletal and Bone, Cortical tissues.

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.

2.3. Calculations of Thickness, Absorbed Dose, Equivalent Dose and Effective Dose

Thickness of proton in the tissue or substance is given by

$T=\frac{R}{\rho}$ (5)

where *R* (in g/cm^{2}) is the range and is *ρ* the density of the tissue or substance [9]

Absorbed dose in (in rad): It is the transfer of an amount of energy of 100 erg per gram of the absorbent material when protons pass over it and is given by the following relation: [9]

$D=\frac{E}{1\text{\hspace{0.17em}}\text{grm}}\frac{1.6\times {10}^{-13}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{J}}{1\text{\hspace{0.17em}}\text{MeV}}\frac{{10}^{7}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{erg}}{1\text{\hspace{0.17em}}\text{J}}\frac{1\text{\hspace{0.17em}}\text{rad}}{\frac{100\text{\hspace{0.17em}}\text{erg}}{\text{gram}}}$ (6)

where *E* is the proton energy.

Equivalent dose: is given by

${H}_{T}={\displaystyle \underset{R}{\sum}{W}_{R}}\times D$ (7)

where *D *is the absorbed dose and
${W}_{R}$ is the weighting factor of radiation (proton) [9]

Effective dose: is given by

$E={\displaystyle \underset{T}{\sum}{W}_{T}{H}_{T}}$ (8)

where ${W}_{T}$ is the weighting factor of tissue or substance [9];

where ${W}_{T}=0.12$ for Water, Muscle, Skeletal and Bone, Cortical and ${W}_{R}=5$ for protons [10].

2.4. The Percentage Error of Difference

The percentage error of difference for the mass stopping power and range of protons in given biological human body parts is calculated by

$\%\text{error}=\frac{\text{PSTARresult}-\text{Presentresult}}{\text{PSTARresult}}\times 100$ (9)

3. Results and Discussion

The results of mass stopping power and range of protons in some biological human body parts (Water, Muscle, Skeletal and Bone, Cortical) are given in Table 3. In Figures 1-3 the mass stopping power of biological human body parts and their compositions are plotted using MathLab program and it is noted that the mass stopping power of all substances and tissues is approximately equal to t the average values of its compositions. The comparison between the present calculated electronic mass stopping powers and that of PSTAR program [8] are

Table 3. Values of electronicmass stopping power (in MeV cm^{2}/g) of Water, Muscle, Skeletal and Bone, Cortical tissues.

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

Figure 2. Mass stopping power of Muscle, Skeletal and its composition.

Figure 3. Mass stopping power of Bone, Cortical and its composition.

shown in Figures 4-6 angood agreements between two results are observed in energy range 1 - 200 MeV.

In Table 4 the ranges of protons in some Biological human body parts (Water, Muscle, Skeletal and Bone, Cortical) are given. In Figures 7-9 a comparison between present results of ranges and that of PSTAR data are shown in energy

Figure 4. Mass stopping power of Water versus $\langle \frac{Z}{A}\rangle {E}^{0.05}$.

Figure 5. Mass stopping power of Muscle, Skeletal versus $\langle \frac{Z}{A}\rangle {E}^{0.05}$.

range 0.1 - 200 MeV and good agreements are observed. In Table 5 and Table 6 the thickness, LET, absorbed dose, effective and equivalent dose are given. In Table 7 and Table 8, the empirical formulae for calculating mass stopping powers and ranges for protons in some biological human body parts (Water, Muscle Skeletal and Bone, Cortical) are given with the percentage difference error.

4. Conclusion

In this work, calculations of mass stopping power and range of protons incident

Figure 6. Mass stopping power of Bone, Cortical versus $\langle \frac{Z}{A}\rangle {E}^{0.05}$.

Figure 7. Range of Water versus $\langle \frac{Z}{A}\rangle {E}^{0.05}$.

on the three different biological human parts (Water, Muscle, Skeletal and Bone, Cortical) have been done and the following conclusions are drawn:

1) The mass stopping power of the Water, Muscle, Skeletal and Bone, and Cortical is equal to the average value of mass stopping power of their compositions in energy range 0.04 - 200 MeV.

2) Values for mass stopping power and ranges of protons in Water, Muscle, Skeletal and Bone, Cortical are in good agreement with the data of PSTAR program. The percentage value of error difference was between 0.09% - 28.88% for

Figure 8. Range of Muscle, Skeletal versus $\langle \frac{Z}{A}\rangle {E}^{0.05}$.

Figure 9. Range of Bone, Cortical versus $\langle \frac{Z}{A}\rangle {E}^{0.05}$

mass stopping power and was between 3.99% - 79.47% for range at proton energy ranging 0.04 - 200 MeV.

3) It was also observed that the maximum value of mass stopping power was at 0.1 MeV and after that, the mass stopping power start to decrease with increasing proton energy as expected.

Table 4. Values of range (in g/cm^{2}) of Water, Muscle, Skeletal and Bone, Cortical tissues.

Table 5. Thickness, LET, absorbed dose, equivalent dose and effective dose of proton in Water and Muscle, Skeletal.

Table 6. Thickness, LET, absorbed dose, equivalent dose and effective dose of proton in Bone, Cortical.

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

Table 8. The empirical formulae for calculating range.

4) The empirical formulae suggested for mass stopping power are simple and accurate at proton energy (1 - 200 MeV) while the empirical formulae suggested for proton range give a good result compared to calculated values at proton energy (4 - 200 MeV).

5) Also in this study, some radiation quantities were calculated, such as linear energy transfer, adsorbed dose, and effective and equivalent dose that give good information to those interested in proton therapy.

Acknowledgements

The authors thank the Department of Physics, Qassim University, for their support and encouragement of this research.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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