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Density functional theory to study stopping power of proton in water, lung, bladder, and intestine

  • Wejdan A. Jasim EMAIL logo and Rashid O. Kadhim
Published/Copyright: March 23, 2024
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Open Engineering
From the journal Open Engineering Volume 14 Issue 1

Abstract

Stopping power, range, and time of proton in water, lung, bladder, and intestinal human tissues are calculated using density functional theory and Beth’s relativistic equation in range of proton energy (0.01–1,000 MeV). The experimental data extracted from SRIM-2013 program were used to proton to the same human tissues applied in the MATLAB-2021 program, and the mean ionization potential of water and the studied tissues is calculated using Gaussian 09W program. A good agreement has been found between our calculations for stopping power, range, and time of protons in the studied human body tissues and SRIM-2013 program calculations.

Keywords: density functional theory; stopping power; range; stopping time; mean ionization potential; SRIM-2013 program; human body tissues

1 Introduction

Proton beams are used in radiotherapy to treat cancers effectively. The precise dosimetry of proton radiation is dependent on a detailed information regarding proton stopping power and range values in the substance of interest [1,2,3,4,5,6,7]. Calculating the stopping power and range of charged particles in matter has long been a topic of study and experimentation. In radiation physics, chemistry, medicine, microdosimetry, proton treatment, and biology, simple yet precise information on the stopping power and range values for protons is commonly required. Stopping power and range information are crucial for characterizing phantom and radiation detector materials [8]. Many scientists have investigated proton stopping power and range in various organic materials [9,10]. However, stopping power and range data encompassing tissues are uncommon; therefore, stopping power and range information of body tissues for protons are required, especially in proton treatment. The electronic stopping power, which is based on inelastic collisions with the target’s electrons, contributes the most to the total stopping power for protons. Nuclear stopping power, on the other hand, contributes the least to overall stopping power and is only important at extremely low energies due to elastic Coulomb collisions with target nucleons. For example, nuclear stopping power contributes to more than 1% of overall stopping power only at energies below 20 keV for protons in water [11].

The research aims at the possibility of adopting protons in the treatment and diagnosis of some defects in human tissues by calculating the stopping power, range, and stopping time of protons in these tissues and by studying the charge distribution based on the theory of density function.

2 Theory

The best extending formula for average energies, Bohr [12]. The effective charge Z 2 * of projectiles was provided. Yarlagadda et al. and Brandt [13,14] have detailed in full the Bohr criteria of abstraction. All outer shell electrons attached to the moving ion in orbitals with velocities less than u are removed off the target, according to the criteria. The criteria should, at the very least, be relevant to the target atom, taking into consideration the symmetry between the projectile and the target. Consider the effective atomic number Z 2 * and effective mean excitation energy I * for the target electrons, both of which are affected by the projectile’s energy. The summed corrections for Z 2 should be Z 2 * and I * at higher energies, which are nearly similar to the inner shell adjustments; nonetheless, the values of I * differ from those of I , which are considered energy independent. The Bethe modified formula is rewritten as [15]:

(1) S e = 4 π ( Z 1 * ) 2 e 4 N m v 2 Z 2 * ln 2 m v 2 I * ,

in which the effective atomic number Z * (Z stands for Z 1 and Z 2 ) relationship can be calculated as follows [16]:

(2) Z * = 4 π ∫ r b ∞ r 2 ρ ( r ) dr ,

where r is the distance from the nucleus, r b is determined from the Bohr stripping criterion v ≥ bv F (r b) = bh[3π2 ρ(r b)]1/3, where v F ( r ) is the velocity of the projectile or target atom’s Thomas-Fermi orbital, b is a constant of proportional about 1.26, and ρ ( r ) is the electronic charge density distribution in atom. The effective mean excitation energy I * is given by the Lindhard and Scharff hypothesis [17]:

(3) ln I * = 4 π Z 2 * ∫ r b ∞ ln [ γ ℏ ω p ( r ) ] r 2 ρ ( r ) dr ,

where γ = 2 for Z ≥ 30 , and ω p ( r ) is the local plasma frequency ( 4 π e 2 ρ ( r ) / m ) 1 / 2 A criterion was proposed by Lindhard and Scharff: 2 m v 2 ≥ γ ℏ ω p ( r ) . However, for the time being, the Bohr stripping criteria are preferred.

Stopping power of composites and tissues is given as sum of stopping powers of its constituent elements according to the Bragg equation [18]:

(4) S com ( E ) = ∑ i w i S i ( E ) ,

where w i is the weight ratio of each element in compound, and S i ( E ) is the stopping power in element.

The stopping range (R) is the distance traveled by the incident particle in the target material. In proton therapy, the range values are also determined as the distance between the starting point of the target’s surface and 80% of the Bragg peak. The particle range in the target material can be determined using the continuous slowing down approximation [19]:

(5) R ( E ) = ∫ E 0 − 1 S ( E ) d E .

The stopping time is defined as the time required to stop the charged particle in a medium, and it can be calculated from the following integral relationship [20]:

(6) t ( E ) = ∫ E 0 − 1 v S ( E ) d E ,

where v is the ion velocity.

3 Results and discussion

The mean ionization potential was calculated for the constituent elements of water and studied human tissues using Gaussian 09W program for a number of basis sets, which are 3–21G, 6–31G, 6–311G, LanL2DZ, LanL2MB, and SDD; the mean ionization potential calculations are listed in Table 1 in eV units.

Table 1

Mean ionization potential of water, lung, bladder, and intestine by different basis sets

Tissue Basis sets
3–21G 6–31G 6–311G LanL2DZ LanL2MB SDD
Water 88.19 88.98 89.05 88.84 86.75 88.83
Lung 137.82 139.60 139.56 129.24 125.01 137.79
Bladder 138.34 140.06 140.01 129.34 125.14 137.32
Intestine 135.56 137.18 137.14 131.69 127.32 136.29

Figure 2, Section a, shows the relationship between the charge density distribution of orbital electrons in the atoms of the elements that make up water and the studied tissues in ( e / a 0 ) unit, and the ratio of the radial distance to the Bohr radius a 0 , while Section b shows the relationship of the electron charge density distribution with the energy of the proton. The charge density distribution curves contain several peaks representing the centers of the electronic shell in each atom, and all curves start from zero where the nucleus of the atom is, and we also note that the hydrogen atom has the lowest value for the distribution of the electronic charge density because it contains only one electron, which makes its contribution to stopping the proton. The proton in the studied human tissues is very few, while the atoms of chlorine and potassium have the largest number of electrons among the constituent atoms, so they have the greatest values for the distribution of charge density, which makes their great contribution to stopping the proton.

The stopping power, range, and stopping time are calculated in Table 3 and Figures 1–3) using the Bethe formula and density functional theory (DFT) for water, lung, bladder, and intestine with a proton energy range from 10−2 to 103 MeV, whereas the Bethe equation (equation (1)) was used to calculate the stopping power of each of the elements that make up the water and lung, bladder, and intestine, then the stopping power for each of water and tissues was calculated from the Bragg equation (4), and the weight ratios are listed in Table 2.

Table 3

Stopping power calculations S total for proton in water and studied human body tissues

Proton energy (MeV) S total (MeV cm2/g)
Water Lung Bladder Intestine
0.01 −9844.7612 −11141.9148 −11162.6750 −11074.2823
0.02 −2148.9928 −3056.8217 −3067.0356 −3021.4691
0.05 606.9184 106.6873 102.6895 121.6411
0.07 818.1915 424.9167 422.0841 435.8114
0.1 858.1965 556.2108 554.2450 563.9954
0.2 706.5357 529.5924 528.6260 533.6390
0.5 429.4418 344.9190 344.5410 346.6196
0.7 345.3144 281.3256 281.0579 282.5620
1 270.3860 222.9037 222.7179 223.7854
2 163.1681 136.7943 136.7029 137.2512
5 80.2856 68.3110 68.2752 68.5028
7 61.3826 52.4420 52.4165 52.5815
10 46.0344 39.4787 39.4609 39.5784
20 26.1813 22.5899 22.5810 22.6421
50 12.4774 10.8319 10.8283 10.8546
70 9.5776 8.3298 8.3271 8.3467
100 7.3013 6.3611 6.3592 6.3736
200 4.4982 3.9303 3.9292 3.9376
500 2.7463 2.4073 2.4068 2.4116
600 2.5582 2.2439 2.2433 2.2478
700 2.4285 2.1312 2.1307 2.1349
800 2.3353 2.0503 2.0499 2.0538
900 2.2664 1.9906 1.9902 1.9940
1000 2.2143 1.9456 1.9452 1.9489
Figure 1 Charge density distribution of elements in water and studied human body tissues.
Figure 1

Charge density distribution of elements in water and studied human body tissues.

Figure 2 Stopping power calculations for proton in (a) water, (b) lung, (c) bladder, and (d) intestine.
Figure 2

Stopping power calculations for proton in (a) water, (b) lung, (c) bladder, and (d) intestine.

Figure 3 Range calculations of electron in (a) water, (b) lung, (c) bladder, and (d) intestine.
Figure 3

Range calculations of electron in (a) water, (b) lung, (c) bladder, and (d) intestine.

Table 2

Weight ratios of elements in water and studied human body tissues

Tissue H C N O Na P S Cl K
Water 0.112 — — 0.888 — — — — —
Lung 0.007 0.089 0.031 0.846 0.003 0.004 0.007 0.008 0.006
Bladder 0.007 0.081 0.026 0.858 0.003 0.004 0.005 0.007 0.008
Intestine 0.008 0.099 0.022 0.858 0.002 0.002 0.002 0.005 0.003

The calculations of the stopping power of water, lung, bladder, and intestine are represented in Figure 2 in Sections a–d respectively, the black curve represents the calculations of the global program SRIM-2013, the blue curve represents the calculations of the relative Beth equation using the DFT, and the red curve represents the calculations of the curve-fitting formula. All calculations for the blue and red curves have been multiplied by coefficients to show the difference between the curves because they are very similar. The calculations of the stopping power are directly proportional to the energy of the projectile (proton), as it increases with the increase of energy up to the maximum value of the stopping power. Else, then, the stopping power decreases with increasing proton energy. The correlation coefficient between the practical calculations of the SRIM-2013 program and the theoretical calculations are listed in each section of Figure 1. The values of the correlation coefficient show a good agreement between the calculations of the SRIM-2013 program and the calculations of the current study, especially in the high-energy region (1–1,000 MeV).

The range and stopping time calculations calculated by equations (5) and (6) for water, lung, bladder, and intestine are represented in Figures 2 and 3 in Sections a–d, respectively, and in the same order and colors as Figure 1 for stopping power. The range and stopping time are directly proportional to the energy of proton, as they increase with the increase in energy. The correlation coefficient between the practical calculations of SRIM-2013 program and the theoretical calculations is listed at the bottom of each section of Figures 2 and 3, and they show the complete agreement between the calculations of the SRIM-2013 program and the calculations of the current study.

The ratio of the stopping range to the stopping time is directly proportional to the energy and at the same time is inversely proportional to the stopping power, which is included in the theoretical relations to calculate the range (equation (5)) and the stopping time (equation (6)).

And at the energies in which the process of ionization and irritation occurs, the effect of the denominator (stopping power) is greater than that of the numerator (energy), and at high energies, the effect of energy is clear in the results (Figure 4).

Figure 4 Stopping time calculations of electron in (a) water, (b) lung, (c) bladder, and (d) intestine.
Figure 4

Stopping time calculations of electron in (a) water, (b) lung, (c) bladder, and (d) intestine.

4 Conclusions

Through the research results that we reached to calculate the total stopping power using the Bethe equation by DFT and with an energy range (0.01–1,000) MeV, it was found that the stopping power increases at low energies less than 0.1 MeV and decreases with the increase in energy greater than 0.1 MeV, where it was found through calculations that they depend on the speed projectile. By calculating the Bethe equation to calculate the total stopping power in the studies tissue, it was found that results of the curve match are close to the practical results of SRIM-2013 with approximately equal correlation coefficients R c = 0.9 .

  1. Conflict of interest: Authors state no conflict of interest.

  2. Data availability statement: The most datasets generated and/or analysed in this study are comprised in this submitted manuscript. The other datasets are available on reasonable request from the corresponding author with the attached information.

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Received: 2023-06-10
Revised: 2023-08-02
Accepted: 2023-08-08
Published Online: 2024-03-23

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  152. Possibility of production high strength lightweight concrete containing organic waste aggregate and recycled steel fibers
  153. An investigation of self-sensing and mechanical properties of smart engineered cementitious composites reinforced with functional materials
  154. Forecasting changes in precipitation and temperatures of a regional watershed in Northern Iraq using LARS-WG model
  155. Experimental investigation of dynamic soil properties for modeling energy-absorbing layers
  156. Numerical investigation of the effect of longitudinal steel reinforcement ratio on the ductility of concrete beams
  157. An experimental study on the tensile properties of reinforced asphalt pavement
  158. Self-sensing behavior of hot asphalt mixture with steel fiber-based additive
  159. Behavior of ultra-high-performance concrete deep beams reinforced by basalt fibers
  160. Optimizing asphalt binder performance with various PET types
  161. Investigation of the hydraulic characteristics and homogeneity of the microstructure of the air voids in the sustainable rigid pavement
  162. Enhanced biogas production from municipal solid waste via digestion with cow manure: A case study
  163. Special Issue: AESMT-7 - Part I
  164. Preparation and investigation of cobalt nanoparticles by laser ablation: Structure, linear, and nonlinear optical properties
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  166. The effect of urban environment on large-scale path loss model’s main parameters for mmWave 5G mobile network in Iraq
  167. Formatting a questionnaire for the quality control of river bank roads
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  169. Machine learning-based compressive strength estimation in nanomaterial-modified lightweight concrete
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  171. Behavior of container berth structure under the influence of environmental and operational loads
  172. Energy absorption and impact response of ballistic resistance laminate
  173. Effect of water-absorbent polymer balls in internal curing on punching shear behavior of bubble slabs
  174. Effect of surface roughness on interface shear strength parameters of sandy soils
  175. Evaluating the interaction for embedded H-steel section in normal concrete under monotonic and repeated loads
  176. Estimation of the settlement of pile head using ANN and multivariate linear regression based on the results of load transfer method
  177. Enhancing communication: Deep learning for Arabic sign language translation
  178. A review of recent studies of both heat pipe and evaporative cooling in passive heat recovery
  179. Effect of nano-silica on the mechanical properties of LWC
  180. An experimental study of some mechanical properties and absorption for polymer-modified cement mortar modified with superplasticizer
  181. Digital beamforming enhancement with LSTM-based deep learning for millimeter wave transmission
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  183. Design and optimization of two-stage controller for three-phase multi-converter/multi-machine electric vehicle
  184. Evaluation of microstructure and mechanical properties of Al1050/Al2O3/Gr composite processed by forming operation ECAP
  185. Calculations of mass stopping power and range of protons in organic compounds (CH3OH, CH2O, and CO2) at energy range of 0.01–1,000 MeV
  186. Investigation of in vitro behavior of composite coating hydroxyapatite-nano silver on 316L stainless steel substrate by electrophoretic technic for biomedical tools
  187. A review: Enhancing tribological properties of journal bearings composite materials
  188. Improvements in the randomness and security of digital currency using the photon sponge hash function through Maiorana–McFarland S-box replacement
  189. Design a new scheme for image security using a deep learning technique of hierarchical parameters
  190. Special Issue: ICES 2023
  191. Comparative geotechnical analysis for ultimate bearing capacity of precast concrete piles using cone resistance measurements
  192. Visualizing sustainable rainwater harvesting: A case study of Karbala Province
  193. Geogrid reinforcement for improving bearing capacity and stability of square foundations
  194. Evaluation of the effluent concentrations of Karbala wastewater treatment plant using reliability analysis
  195. Adsorbent made with inexpensive, local resources
  196. Effect of drain pipes on seepage and slope stability through a zoned earth dam
  197. Sediment accumulation in an 8 inch sewer pipe for a sample of various particles obtained from the streets of Karbala city, Iraq
  198. Special Issue: IETAS 2024 - Part I
  199. Analyzing the impact of transfer learning on explanation accuracy in deep learning-based ECG recognition systems
  200. Effect of scale factor on the dynamic response of frame foundations
  201. Improving multi-object detection and tracking with deep learning, DeepSORT, and frame cancellation techniques
  202. The impact of using prestressed CFRP bars on the development of flexural strength
  203. Assessment of surface hardness and impact strength of denture base resins reinforced with silver–titanium dioxide and silver–zirconium dioxide nanoparticles: In vitro study
  204. A data augmentation approach to enhance breast cancer detection using generative adversarial and artificial neural networks
  205. Modification of the 5D Lorenz chaotic map with fuzzy numbers for video encryption in cloud computing
  206. Special Issue: 51st KKBN - Part I
  207. Evaluation of static bending caused damage of glass-fiber composite structure using terahertz inspection
https://doi.org/10.1515/eng-2022-0518
Keywords for this article
density functional theory; stopping power; range; stopping time; mean ionization potential; SRIM-2013 program; human body tissues
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  157. An experimental study on the tensile properties of reinforced asphalt pavement
  158. Self-sensing behavior of hot asphalt mixture with steel fiber-based additive
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  161. Investigation of the hydraulic characteristics and homogeneity of the microstructure of the air voids in the sustainable rigid pavement
  162. Enhanced biogas production from municipal solid waste via digestion with cow manure: A case study
  163. Special Issue: AESMT-7 - Part I
  164. Preparation and investigation of cobalt nanoparticles by laser ablation: Structure, linear, and nonlinear optical properties
  165. Seismic analysis of RC building with plan irregularity in Baghdad/Iraq to obtain the optimal behavior
  166. The effect of urban environment on large-scale path loss model’s main parameters for mmWave 5G mobile network in Iraq
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  170. In-depth analysis of critical factors affecting Iraqi construction projects performance
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  172. Energy absorption and impact response of ballistic resistance laminate
  173. Effect of water-absorbent polymer balls in internal curing on punching shear behavior of bubble slabs
  174. Effect of surface roughness on interface shear strength parameters of sandy soils
  175. Evaluating the interaction for embedded H-steel section in normal concrete under monotonic and repeated loads
  176. Estimation of the settlement of pile head using ANN and multivariate linear regression based on the results of load transfer method
  177. Enhancing communication: Deep learning for Arabic sign language translation
  178. A review of recent studies of both heat pipe and evaporative cooling in passive heat recovery
  179. Effect of nano-silica on the mechanical properties of LWC
  180. An experimental study of some mechanical properties and absorption for polymer-modified cement mortar modified with superplasticizer
  181. Digital beamforming enhancement with LSTM-based deep learning for millimeter wave transmission
  182. Developing an efficient planning process for heritage buildings maintenance in Iraq
  183. Design and optimization of two-stage controller for three-phase multi-converter/multi-machine electric vehicle
  184. Evaluation of microstructure and mechanical properties of Al1050/Al2O3/Gr composite processed by forming operation ECAP
  185. Calculations of mass stopping power and range of protons in organic compounds (CH3OH, CH2O, and CO2) at energy range of 0.01–1,000 MeV
  186. Investigation of in vitro behavior of composite coating hydroxyapatite-nano silver on 316L stainless steel substrate by electrophoretic technic for biomedical tools
  187. A review: Enhancing tribological properties of journal bearings composite materials
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  189. Design a new scheme for image security using a deep learning technique of hierarchical parameters
  190. Special Issue: ICES 2023
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  192. Visualizing sustainable rainwater harvesting: A case study of Karbala Province
  193. Geogrid reinforcement for improving bearing capacity and stability of square foundations
  194. Evaluation of the effluent concentrations of Karbala wastewater treatment plant using reliability analysis
  195. Adsorbent made with inexpensive, local resources
  196. Effect of drain pipes on seepage and slope stability through a zoned earth dam
  197. Sediment accumulation in an 8 inch sewer pipe for a sample of various particles obtained from the streets of Karbala city, Iraq
  198. Special Issue: IETAS 2024 - Part I
  199. Analyzing the impact of transfer learning on explanation accuracy in deep learning-based ECG recognition systems
  200. Effect of scale factor on the dynamic response of frame foundations
  201. Improving multi-object detection and tracking with deep learning, DeepSORT, and frame cancellation techniques
  202. The impact of using prestressed CFRP bars on the development of flexural strength
  203. Assessment of surface hardness and impact strength of denture base resins reinforced with silver–titanium dioxide and silver–zirconium dioxide nanoparticles: In vitro study
  204. A data augmentation approach to enhance breast cancer detection using generative adversarial and artificial neural networks
  205. Modification of the 5D Lorenz chaotic map with fuzzy numbers for video encryption in cloud computing
  206. Special Issue: 51st KKBN - Part I
  207. Evaluation of static bending caused damage of glass-fiber composite structure using terahertz inspection
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