Mass Stopping Power and Range of Protons in Biological Human Body Tissues (Ovary, Lung and Breast)

In this work, the mass stopping power and range of protons in biological human body tissues (ovary, lung and breast) were calculated at the energ y rang ing from 0.04 MeV to 200 MeV using the MATLAB Program. The data relat ing to the densities, average atomic number to mass number and excitation energ y for the present tissues were collected from ICRU Report 46. The mass stopping power was calculated by the Bethe formula . Moreover, the simple integration (continuous slowing down approximation) method was employed for calculating protons range at the tissues. The results of the mass stopping power versus energy and the range versus energy were presented gra phically and the empirical formulae for calculating the mass stopping power and the ranges were obtained. The present results for mass stopping powers and ranges were compared with the results obtained by others. Good agreements were found between them, especially at the energ y ranging from 3 to 200 MeV.

• Human Body Tissues
• Protons
• Range
• MATLAB
• Mass Stopping Power

1. Introduction

The stopping power and energy dissipation of charged particles through matter has been a subject of great interest for 100 years, because of its wide areas of application, such as ion implantation, fundamental particle physics, nuclear physics, radiation damage, radiology [1].

In recent years, the theoretical and experimental studies of the Stopping Power (SP) and range of charged particles have been increasing, especially in radiation physics. Many theoretical, as well as experimental studies have been established on this topic quite efficiently [2]. In particular, the use of protons or heavier ions are increasing as an alternative to external photon beams in radiotherapy, the reason is attributed to preserving the target dose, ensuring a higher dose delivered to the tumor and being able to transfer energy in the form of point shot through the diseased tissue due to the Bragg curve [3].

Charged particles are often used for radiotherapy because they have a well-defined penetration in tissues, the depth is dependent on the incident energy of particles and the nature of the irradiated material. Charged particles (protons, deuterons, alpha particles) have an important effect in radiotherapy as they have the ability to deliver their energy to the target [4].

For protons, the main contribution to the total stopping power is provided by the electronic stopping power which is based on inelastic collisions with the target’s electrons. Conversely, the least contributing to the total stopping power comes from nuclear stopping power which arises from elastic collisions with the target’s nucleons and is only significant at very low energy. For instance, the nuclear stopping power conduces more than one percent to the total stopping power only at energy below 20 KeV for protons [5]. Hence, precise knowledge of stopping power and range of protons are needed for the exact dosimetry of proton radiation. When studying the stopping power and range of protons in biological tissues, one must determine or collect data from experimentally or ICRU reports.

Heavy charged particles traversing matter lose energy primarily through the ionization and excitation of atoms. 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 [6] [7].

Early investigation of the energy loss of charged particles traversing matter arrives at a general stopping power formula:

− d E d x = 4 π e 4 N Z 2 m e v 2 Z 1 2 B (1)

where: N is the target density, Z₂ is the target atomic number, m_e is the electron mass, v and Z₁ are the projectile velocity and charge respectively and B is called the “stopping number’’ [7].

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 are complex, and will be not covered in detail [8].

If an ion beam penetrates through matter it loses energy due to collisions with electrons (electronic stopping) and target nuclei (nuclear stopping) [9].

Then, the total stopping power is just the sum of the stopping powers due to electronic and nuclear interactions. The nuclear component of the stopping power can also be ignored [10].

2. Methods

The first quantum mechanical study of stopping power was done by Bethe. Bethe theory of stopping is valid when the projectile’s velocity is higher than the Bohr velocity. In Bethe theory, the target is assumed as an elemental material. Bethe’s treatment of the energy loss is based on the born approximation applied to the inelastic collisions between the heavy particle and the atomic electrons. In this theory, the projectile heavy particle is assumed to be structure less, and the target nucleus is assumed infinitely massive [11].

1) Calculations of electronic stopping power

The following Bethe mass stopping power equation has been used for energy range 0.04 - 200 MeV [12]:

− d E ρ d x = 5.08 × 10 − 31 z 2 n β 2 ρ [ F ( β ) − ln I ] (2)

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 ( β ) = ln 1.02 × 10 6 β 2 1 − β 2 − β 2 (3)

n is calculated using the following relation:

n = N a v ρ 〈 Z A 〉 (4)

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 human body tissues are given in Table 1. The calculated mass stopping power of protons for lung, breast and ovary are based on Bethe equation after substituting the constants from Table 1. In Table 2, the compositions of the human tissues are given.

2) Calculations of range

Range, R the range for a heavy particle is the straight distance traveled by the particle within the target. Light particles like electrons, positrons can be scattered in the path of the targets with large angles due to their low weight and it is difficult to calculate their path length. Monte Carlo methods that are based on a broad class of computational algorithms have been used in a successful manner specially for calculating the path length of light particles. On the other hand, the path length of heavy particles like protons is almost straight line. The range of protons can be calculated by some numerical integration methods. But the Continuous Slowing Down Approximation (CSDA) is a simple and common method to calculate the range of the heavy particles like protons 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.

Table 1

Basic Data for calculating mass stopping powers

I (eV)	n (electrons/m3)	〈 Z / A 〉	Density ρ (g/m3)	Substances
75.2	3.476 × 1029	0.550	1050,000	Lung
70.3	3.383 × 1029	0.551	1020,000	Breast
75	3.482 × 1029	0.551	1050,000	Ovary

Table 2

Elemental composition of Lung, Breast and Ovary tissues

Composition	Substances
C (0.105000), N (0.031000), H (0.103000), O (0.749000), Na (0.002000), P (0.002000), S (0.003000)	Lung
C (0.332000), N (0.030000), H (0.106000), O (0.527000), Na (0.001000), P (0.001000), S (0.002000), Cl (0.001000)	Breast
C (0.093000), N (0.024000), H (0.105000), O (0.768000), Na (0.002000), P (0.002000), S (0.002000), Cl (0.002000), K (0.002000)	Ovary

The range, R for an incident proton in the CSDA method is given as:

R = ∫ E 0 E f d E M S ( E ) (5)

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 M S ( E ) is the mass stopping power [1].

3. Results and Discussion

The results of mass stopping powers for the present tissues are given in Table 3. In Figures 1-3, the mass stopping power of tissues and their compositions are plotted and it is noted that the mass stopping power of all tissues represent the average values of its compositions. The comparison between the present calculated mass stopping powers and that of Ref. [5] are shown in Figures 4-6, and good agreements between two results are observed in energy range 3 - 200 MeV.

In Table 4, the ranges of all present human tissues are given. In Figures 7-9, a comparison between present results of ranges in all tissues and that of Ref. [5] are shown in energy range 0.04 - 200 MeV and good agreements are observed. In Table 5 and Table 6 the empirical formulae for calculating mass stopping powers and ranges for all present human tissues are given. Also percentage difference between the present results of mass stopping powers and ranges and that of Ref. [5] are calculated for all tissues and the range of differences of percentage errors are given in Table 7.

4. Conclusion

In this work, calculations of mass stopping power and range of protons incident on the three different human body tissues (lung, breast and ovary) have been

Table 3

Values of mass stopping power (in MeV cm²/g) of lung, breast and ovary tissues

Mass stopping power (in MeV cm2/g)				Proton energy (MeV
	Ovary	Breast	Lung
	288.6849	417.5179	282.2618		0.04
	727.9282	813.8908	722.7380		0.06
	830.8763	895.3875	826.5003		0.08
	841.5006	893.1341	837.7000		0.1
	695.3133	721.1679	692.9473		0.2
	484.8960	497.8423	483.4803		0.4
	376.7606	358.3989	375.7261		0.6
	311.0256	317.5082	310.2014		0.8
	266.4701	271.6586	265.7808		1.0
	160.6155	163.2136	160.2245		2.0
	93.9559	95.2568	93.7372		4.0
	67.9364	68.8044	67.7816		6.0
	53.7599	54.4113	53.6391		8.0
	44.7426	45.2639	44.6429		10.0
	25.0335	25.2946	24.9792		20.0
	17.7039	17.8781	17.6660		30.0
	13.8058	13.9365	13.7764		40.0
	11.3646	11.4692	11.3406		50.0
	9.6832	9.7704	9.6628		60.0
	8.4504	8.5251	8.4326		70.0
	7.5053	7.5707	7.4896		80.0
	6.7563	6.8145	6.7423		90.0
	6.1474	6.1997	6.1346		100.0
	5.6419	5.6895	5.6301		110.0
	5.2151	5.2588	5.2043		120.0
	4.8498	4.8901	4.8397		130.0
	4.5333	4.5707	4.5239		140.0
	4.2563	4.2912	4.2475		150.0
	4.0117	4.0445	4.0034		160.0
	3.7941	3.8249	3.7863		170.0
	3.5991	3.6282	3.5917		180.0
	3.4234	3.4510	3.4163		190.0
	3.2641	3.2903	3.2574		200.0

Table 4

Values of range (in g/cm 2 ) of lung, breast and ovary

Range (g/cm2)			Proton energy (MeV)
Ovary	Breast	Lung
1.499 × 10−⁵	1.345 × 10−⁵	1.512 × 10−5	0.04
2.983 × 10−5	2.692 × 10−5	3.008 × 10−5	0.06
4.860 × 10−5	4.403 × 10−5	4.900 × 10−5	0.08
7.098 × 10−5	6.449 × 10−5	7.154 × 10−5	0.1
2.301 × 10−⁴	2.109 × 10−⁴	2.318 × 10−⁴	0.2
7.462 × 10−⁴	6.902 × 10−⁴	7.510 × 10−⁴	0.4
0.0015	0.0014	0.0015	0.6
0.0024	0.0023	0.0024	0.8
0.0035	0.0033	0.0036	1.0
0.0115	0.0108	0.0115	2.0
0.0371	0.035	0.0373	4.0
0.0739	0.070	0.0742	6.0
0.120	0.115	0.120	8.0
0.175	0.169	0.176	10.0
0.570	0.555	0.571	20.0
1.134	1.110	1.137	30.0
1.848	1.815	1.852	40.0
2.699	2.659	2.704	50.0
3.678	3.631	3.684	60.0
4.778	4.727	4.784	70.0
5.994	5.939	6.001	80.0
7.320	7.265	7.327	90.0
8.753	8.699	8.761	100.0
10.290	10.239	10.298	110.0
11.927	11.882	11.936	120.0
13.662	13.625	13.671	130.0
15.493	15.465	15.502	140.0
17.418	17.402	17.427	150.0
19.434	19.433	19.442	160.0
21.540	21.555	21.548	170.0
23.734	23.769	23.741	180.0
26.014	26.071	26.021	190.0
28.380	28.461	28.386	200.0

Table 5

The empirical formulae for calculating mass stopping powers

The empirical formulae for calculating mass stopping powers		Substances
y = A 1 e x / t 1 + y o
Ref. [5]	This work
A1 = 1.0140 × 108, t1 = −0.04235 yo = −2.17377 R2 = 0.99982	A1 = 6.4723 × 108, t1 = −0.0374 yo = 0.10311 R2 = 0.99999	Lung
A1 = 1.3298 × 108, t1 = −0.04166 yₒ = −1.74685 R2 = 0.99988	A1 = 7.424 × 108, ,t1 = −0.03717 yo = 0.33231 R2 = 0.99999	Breast
A1 = 1.05463 × 108, t1 = −0.0423 yo = −1.998 R2 = 0.99984	A1 = 6.7234 × 108, t1 = −0.0374 yo = 0.25388 R2 = 1	Ovary

Table 6

The empirical formulae for calculating range

The empirical formulae for calculating range		Substances
y = A 1 e x / t 1 + y o
Ref. [5]	This work
A1 = 5.6893 × 10−16, t1 = 0.01861 yo = −0.0191 R2 = 0.99992	A1 = 2.0524 × 10−14, t1 = 0.02054 yo = −0.04347 R2 = 0.99999	Lung
A1 = 1.5038 × 10−15, t1 = 0.01912 yo = −0.08055 R2 = 0.99988	A1 = 3.1976 × 10−14, t1 = 0.02058 yo = −0.09494 R2 = 1	Breast
A1 = 1.5704 × 10−15, t1 = 0.01914 yo = −0.08739 R2 = 0.99984	A1 = 4.1072 × 10−14, t1 = 0.02102 yo = −0.09538 R2 = 1	Ovary

Table 7

Percentage difference between present result and that of Ref. [ 5 ]

Tissue	Range of percentage difference between present result and Ref. [5] of mass stopping power	Range of percentage difference between present result and Ref. [5] of range
Breast	−6.07% - 12.82%	−7.06% - 21.29%
Lung	−6.46% - 13.95%	−3.20% - 26.95%
Ovary	−6.44% - 13.91%	−8.13% - 23.47%

done and the following conclusions are drawn:

1) The mass stopping power of each tissue gives the average value of mass stopping power of its compositions in energy range of 0.04 - 200 MeV;

2) It is also noted that the empirical formulae for both mass stopping power and range for each human tissue studied in the present work are simple and widely applicable;

3) Values for mass stopping power and ranges of protons for breast, lung and ovary tissues are in good agreement with the data proposed by other researchers. This comparison is very important in terms of reliability of present work;

4) It also observed that the maximum value of mass stopping power was at 0.1 MeV and after that, the mass stopping power started to decrease with increasing energy;

5) The empirical formulae suggested for mass stopping power and ranges of protons are simple and they covered a large range of proton energy (1 - 200 MeV) as well as gave important information for those interesting in proton therapy.

Acknowledgements

The authors thank the Department of Physics, Qassim University, for its 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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