Pocket formula for mass attenuaion coefficient, effective atomic number and electron density of human tissues

NUCLEAR CHEMISTRY, RADIOCHEMISTRY, NUCLEAR MEDICINE

Pocket formula for mass attenuaion coefficient, effective atomic number and electron density of human tissues

H.C. Manjunatha，

L. Seenappa

Nuclear Science and Techniques

Vol.30, No.3

Article number 36

Published in print 01 Mar 2019

Available online 09 Feb 2019

DOI：10.1007/s41365-019-0565-7

42401

We have proposed a pocket formulae for mass attenuation coefficient (μ/ρ), mass energy absorption coefficient (μ_en/ρ), and effective atomic number (Z_eff) in different tissues of human organs. We have also assigned a new chemical formula for all studied tissues based on their composition. We have introduced a new parameter called effective composition index (C_eff). Based on this we have introduced a new method to compute the effective atomic number. The evaluated photon interaction parameters are graphically represented. The evaluated average, maximum, minimum and standard deviations of effective atomic number are tabulated. The proposed formulae produces a mass attenuation coefficient, mass-energy absorption coefficient, and effective atomic number from their composition.

Mass attenuation coefficientsTissues

1. Introduction

These attenuation coefficients are extensively used in shielding and dosimetric computations which are strongly dependent on the energy of photon and composition of elements of interacting medium. The knowledge of attenuation in tissues is also useful in the mammographic examination which is the most effective method for early diagnosis of breast cancer. The attenuation coefficient and effective atomic number are fundamental parameters in radiology. Literature survey shows that there were several works on theoretical and measurements of mass attenuation coefficients of dosimetric interest [1-3].

Hubble and Seltzer [4] was given attenuation coefficients data for the elements and compounds. Berger and Hubbel [2] developed software called XCOM for calculating mass attenuation coefficients. Gerward et al.,[5] developed WinXCom programme to calculate mass attenuation coefficients. Hine [6] introduced the concept of Z_eff. This parameter is useful in selecting a tissue substitute. A literature survey shows that some researchers have measured or calculated effective atomic number in biological materials [7-9].

Kurudirek and Onaran [10] studied the Z_eff of biomolecules for electron, proton, alpha particle, and photon interactions. Kurudirek [11] also studied Z_eff and N_e of human tissues. The same workers [12] also studiedZ_eff of dosimetric materials for different interactions. Previous workers [13-18] measured the X-ray and gamma interaction parameters in some compounds of dosimetric interest. We also reported theoretical studies on the X-ray and gamma interaction parameters of biological samples [19-26].

Tissue equivalent materials are required for dose distribution studies in the radiotherapy and diagnoisis. The photon interaction parameters in the tissues of human organs are important for the preparation of tissue equivalent materials. The tissue equivalent materials and tissues should have a similar behaviour with photons.

In the present work, we have proposed a new semi empirical formulae for photon interaction parameters such as mass attenuation coefficient, mass energy absorption coefficient, and effective atomic number in different tissues of human organs (Kidney, Liver, Lung, Lymph, Muscle, Ovary, Pancreas, Cartilage, Red marrow, Spongiosa, Yellow marrow, Skin, Spleen, Testis, Thyroid, Skeleton-cortical bone, Skeleton-cranium Skeleton-femur, Skeleton-humerus, Skeleton-mandible, Skeleton-ribs (2nd, 6th), Skeleton-ribs (10th), Skeleton-sacrum, Skeleton-spongiosa, Skeleton-vertibral column (c4), and Skeleton-vertibral column (D6, L3)). In the second section of the paper, we have explained the proposed empirical formula. The third section of the paper describes the comparison of experimental results with present work.

2. Methodology

2.1 Semi empirical formula for effective atomic number (Z_eff) in terms of composition

To establish the exact relation between effective atomic number and composition, we have introduced a new parameter called effective composition index (C_eff). In general, it is the ratio of the sum of the product of composition and atomic weight of elements in large proportion to one plus the sum of the product of composition and atomic weight of remaining elements. The major elements present in the tissues of human organs are H, C, N, and O. Thus effective composition index (C_eff) for tissues of human organ is defined as the ratio between the sum of the product of composition and atomic weight of H, C, N, and O to one plus sum of the product of composition and atomic weight of remaing elements in the tissue.

C_{eff} = \frac{C_{H} A_{H} + C_{C} A_{C} + C_{N} A_{N} + C_{O} A_{O}}{1 + C_{Ca} A_{Ca} + C_{P} A_{P} + C_{Mg} A_{Mg} + C_{S} A_{S} + C_{Cl} A_{Cl} + C_{K} A_{K} + C_{Fe} A_{Fe} + C_{I} A_{I}}

(1)

In the above equation, C_H, C_C, C_N, C_O, C_Ca, C_P, C_Na, C_Mg, C_S, C_CI, C_K, C_Fe, and C_I are composition of the elements indicated in the corresponding subscripts. A_H, A_C, A_N, A_O, A_Ca, A_P, A_Na, A_Mg, A_S, A_Cl, A_K, A_Fe and A_I are atomic weight of the elements indicated in the corresponding subscripts. In the numerator of the above equation, we have considered the composition and atomic weight of H, C, N, and O because these elements are major elemental contents of tissue.

In the denominator, remaining elemental composition is considered.

The equation of effective composition index (C_eff) for a single element is reduced to

C_{eff} = C_{i} A_{i}

(2)

We have calculated the effective composition index (C_eff) for all tissues of human organs. A search was made for their best parametrization with the effective atomic number. Finally we have established relation between effective atomic number (Z_eff) and effective composition index (C_eff).

Z_{eff} = {\begin{cases} 1 / (2 .04541353 \times 10^{-2} ln (E C_{eff}) + 0 .2074035539) for 1-500 keV \\ 9 .301386273 \times 10^{-6} (E C_{eff}) + 4 .245546293                    for 0 .5-20 MeV \end{cases}

(3)

In the above Eq. (3), E represents photon energy in MeV. Effective atomic number can be calculated with the simple inputs of photon energy (E) and effective composition index (C_eff). The effective composition index can be calculated using their composition. Thus, Eq. (3) represents the simple semi empirical formula which produces the effective atomic number using their composition.

2.2 Semi empirical formula for of mass attenuation and energy absorption coefficients

Most of the tissues of human organs consisting of elements such as H, C, N, O, Ca, P, Na, Mg, S, Cl, K, Fe, and I are in their elemental composition. We have studied the variation of mass attenuation and energy absorption coefficients with atomic number at different energies for the elements which are constitutes of tissues of human organs. We have studied the variation of the mass attenuation coefficient (μ/ρ) with energy and atomic number. It is observed from this study that the mass attenuation coefficients (μ/ρ) do not vary linearly with energy and atomic number. To select the best fit for mass attenuation coefficients in the low energy region (1-100 keV), we have studied suitable functions such as $α_{1} E^{α_{2}} + α_{3} E^{α_{4}}$ , $α_{1} + \frac{α_{2}}{E} + \frac{α_{3}}{E^{2}} + \frac{α_{4}}{E^{3}},$ $\frac{1}{α_{1} E^{2} + α_{2} E + α_{3}},$ $\frac{α_{1} E^{α_{2}} + α_{3}}{E^{α_{4}} + α_{5}}$ , $(α_{1} E^{α_{2}} + α_{3}),$ $α_{1} \exp (α_{2} E^{α_{3}}) + α_{4},$ $α_{1} \exp (α_{2} {(\ln E + α_{3})}^{2}) + α_{4},$ $\frac{α_{1}}{{({(E + α_{2})}^{2} + 1)}^{α_{3}}} + α_{4},$ $α_{1} \exp (\frac{α_{2}}{E} + α_{3} \ln E),$ $α_{1} \exp (α_{2} E) + α_{3} \exp (α_{4} E),$ $α_{1} E + α_{2} \exp (α_{3} E) + α_{4}$ , $β_{1} E + β_{2} + \frac{β_{3}}{E} + β_{4} \ln E,$ $δ_{1} E + δ_{2} \exp (δ_{3} E) + δ_{4},$ and polynomial function $(α_{1} E^{4} + α_{2} E^{3} + α_{3} E^{2} + α_{4} E + α_{5})$ where α’ s are functions of atomic number [ $α = α (Z)$ ]. Among these functions the double exponential function such as $α_{1} E^{α_{2}} + α_{3}$ is the best suitable function. This function is also valid for the low energy region (1-100 keV) and for elements H, C, N, O, Ca, P, Na, Mg, S, Cl, K, Fe, and I. Hence we have fit this exponential function such as $α_{1} E^{α_{2}} + α_{3}$ to the mass attenuation coefficient data for the low energy region (1-100 keV) for elements H, C, N, O, Ca, P, Na, Mg, S, Cl, K, Fe and I.

\frac{μ}{ρ} = \sum_{i = 0}^{3} α_{i} Z^{i} E^{^{(\sum_{i = 0}^{3} β_{i} Z^{i})}} + \sum_{i = 0}^{4} δ_{i} Z^{i} for  1-100keV

(4)

here α_i, β_i and δ_i are fitting parametrs which are given in Table 1.

Fitting parametrs for mass attenuation coefficients (μ/ρ)

	i=0	i=1	i=2	i=3	i=4
α_i	10757.14872	- 3540.820395	398.4373505	-4.461736803	0
β_i	- 3.343463789	6.12684539·10^-2	- 2.825112754·10^-3	4.967048249·10^-5	0
δ_i	0.4320534583	- 8.406094275·10^-2	8.3504118·10^-3	- 3.691850298·10^-4	5.714514667·10^-6

We have also formulated the equation for the mass attenuation coefficient (μ/ρ) and the product of energy and atomic number in the energy region 100 keV to 20 MeV for elements H, C, N, O, Ca, P, Na, Mg, S, Cl, K, Fe, and I

\frac{μ}{ρ} = 1.591 {(Z E)}^{- 0.3865} f o r 0.1 - 20 M e V

(5)

We have also fit the following non linear function to mass attenuation coefficients (μ_en/ρ) in the low energy region (1-100 keV) for elements H, C, N, O, Ca, P, Na, Mg, S, Cl, K, Fe, and I;

\frac{μ_{e n}}{ρ} = \sum_{i = 0}^{7} φ_{i} Z^{i} E^{^{(\sum_{i = 0}^{7} ψ_{i} Z^{i})}} + \sum_{i = 0}^{7} χ_{i} Z^{i} f o r 1 - 100 k e V

(6)

here φ_i, ψ_i and χ_i are fitting parameters which are given in Table 2.

Fitting parametrs for mass attenuation coefficients (μ_en/ρ)

	Φ_i	Ψ_i	Χ_i
i=0	- 6183.518517	- 3.356499096	3.445436562·10^-1
i=1	9775.049268	+ 5.501861004·10^-2	7.017102505·10^-2
i=2	- 4419.256446	1.769339873·10^-2	- 7.455569824·10^-2
i=3	928.8872984	- 6.915457646·10^-3	1.881469124·10^-2
i=4	- 99.72245142	+ 9.55841425·10^-4	- 2.214979366·10^-3
i=5	+ 5.829884688	- 6.382510627·10^-5	1.346734218·10^-4
i=6	- 0.173724236	2.059213322·10^-6	- 4.080025291·10^-6
i=7	2.05344172·10^-3	-2.563187022·10^-8	4.856052612·10^-8

The proposed formula for the mass energy absorption coefficient (μ_en/ρ), and the product of the energy and atomic number in the energy region 100 keV to 20 MeV for elements H, C, N, O, Ca, P, Na, Mg, S, Cl, K, Fe and I is;

\frac{μ_{e n}}{ρ} = - 3.1228 \times 1 0^{- 3} l n (Z E) + 5.0891 \times 1 0^{- 2} f o r 0.1 - 20 M e V

(7)

The mass attenuation coefficient and mass energy absorption coefficient of tissues of human organs at different energies can be expressed by substituting Z=Z_eff in above Eqs. (1) and (2):

\frac{μ}{ρ} = (\begin{array}{l} (\sum_{i = 0}^{3} α_{i} Z_{e f f}^{i}) E^{^{(\sum_{i = 0}^{3} β_{i} Z_{e f f}^{i})}} + (\sum_{i = 0}^{4} δ_{i} Z_{e f f}^{i}) f o r 1 - 100 k e V \\ 1.591 {(Z_{e f f} E)}^{- 0.3865} f o r 0.1 - 20 M e V \end{array}

(8)

and

\frac{μ_{e n}}{ρ} = (\begin{array}{l} (\sum_{i = 0}^{7} φ_{i} Z_{e f f}^{i}) E^{^{(\sum_{i = 0}^{7} ψ_{i} Z_{e f f}^{i})}} + (\sum_{i = 0}^{7} χ_{i} Z_{e f f}^{i}) f o r 1 - 100 k e V \\ - 3.1228 \times 1 0^{- 3} l n (Z E) + 5.0891 \times 1 0^{- 2} f o r 0.1 - 20 M e V \end{array}

(9)

Above Eqs. (3) and (4) are simple semi empirical formulae which represent the mass attenuation coefficient (μ/ρ) and mass energy absorption coefficient (μ_en/ρ) in terms of the effective atomic number (Z_eff) of tissues. In the above Eqs. (3) and (4), E represents photon energy in keV. Both these coefficients can be calculated with the simple input of effective atomic number (Z_eff) at a given energy.

3. Results and discussions

Based on the composition [32] of elements in the tissues of human organs, we have formulated an equivalent chemical formula. The proposed equivalent chemical formula for tissues of human organs is shown in Table 3. We have calculated mass attenuation coefficients (μ/ρ), mass energy abosorption coefficients (μ_en/ρ), and effective atomic numbers using the formulae proposed in the present work. The variation of mass energy abosorption coefficients (μ_en/ρ) with energy for different tissues of human organs for a wide energy range 1 keV-20 MeV is shown in Figs. 1, 2, 3. It also observed a similar variation of mass attenuation coefficients (μ/ρ) with photon energy. The calculated effective atomic numbers of the tissues of human organs for a wide energy range 1 keV-20 MeV are also shown in Figs. 4, 5, 6. We have also highlighted the average value, maximum value, minimum value, and standard deviation of the calculated effective atomic number of tissues of human organs for a wide energy range 1 keV-20 MeV. These values are also presented in Table 4.

The equivalent chemical formula for tissues of human organs

Tissue	Chemical formula
Adipose tissue	H₄₀₁₀C₁₇₆₅N₁₈O₆₁₆Na₂S₁Cl₁
Blood	H₅₆₅₂C₅₁₁N₁₃₂O₂₆₀₀P₂Na₂S₃Cl₅K₃Fe₁
Brain	H₁₇₀₂C₁₉₄N₂₅O₇₁₃P₂Na₂S₃Cl₅K₃
Breast	H₃₇₂₉C₉₈₀N₇₆O₁₁₆₈P₁Na₂Cl₁
Cell Nucleus	H₈₄₃C₆₀N₁₈O₃₇₂P₇S₁
Eye Lens	H₃₃₇₇C₅₇₆N₁₄₄O₁₄₃₁P₁Na₂S₃Cl₁
GI tract	H₄₁₁₂C₃₇₄N₆₁O₁₈₃₅P₁Na₂S₁Cl₂K₁
Heart	H₅₇₀₇C₅₆₃N₁₂₈O₂₅₆₂P₂Na₂S₃Cl₅K₃Fe₁
Kidney	H₄₀₉₆C₄₄₀N₈₆O₁₈₁₄Ca₁P₃Na₃S₂Cl₂K₂
Liver	H₁₇₆₄C₂₀₅N₃₈O₇₉₃P₂Na₂S₂Cl₁K₁
Lung	H₁₉₉₈C₁₇₁N₄₃O₉₁₅P₂Na₂S₂Cl₂K₁
Lymph	H₃₄₃₆C₁₀₉N₂₅O₁₆₆₇Na₄S₁Cl₄
Muscle	H₃₅₈₈C₄₂₂N₈₆O₁₅₇₃P₂Na₂S₃Cl₁K₄
Ovary	H₂₀₃₇C₁₅₁N₃₃O₉₃₈P₁Na₂S₁Cl₁K₁
Pancreas	H₃₃₇₂C₄₅₁N₅₀O₁₃₉₁P₂Na₃S₁Cl₂K₂
Cartilage	H₁₁₂₆C₉₇N₁₉O₅₅₀P₈Na₃S₃Cl₁
Red marrow	H₅₈₁₈C₁₉₂₅N₁₃₆O₁₅₃₂
Spongiosa	H₄₇₁₀C₁₈₇₈N₁₁₂O₁₂₈₁Ca₁₀₃P₆₁Na₂Mg₂S₃Cl₃K₁Fe₁
Yellow marrow	H₄₀₄₅C₁₉₀₁N₁₈O₅₁₂Na₂S₁Cl₁
Skin	H₃₈₇₉C₆₆₄N₁₁₇O₁₅₇₆P₁Na₃S₂Cl₃K₁
Spleen	H₂₃₄₉C₂₁₆N₅₃O₁₀₆₅P₂Na₁Cl₁K₂
Testis	H₃₂₅₇C₂₅₅N₄₄O₁₄₈₃P₁Na₃S₂Cl₂K₂
Thyroid	H₁₃₀₉₅C₁₂₅₇N₂₁₇O₅₉₀₉P₄Na₁₁S₄Cl₇K₃I₁
Skeleton cortical bone	H₇₇₆C₂₉₇N₆₉O₆₂₅Ca₁₂₉P₇₆Na₁Mg₂S₂
Skeleton cranium	H₁₁₄₀C₄₀₆N₆₆O₆₂₅Ca₁₀₁P₆₀Na₁Mg₂
Skeleton femur	H₂₄₆₂C₁₀₁₈N₇₁O₈₁₅Ca₁₁₄P₆₃Na₂Mg₁S₁Cl₂l₁
Skeleton humerus	H₁₄₄₇C₆₃₅N₅₄O₅₆₁Ca₉₂P₅₅Na₁Mg₁S₂
Skeleton mandible	H₁₀₄₉C₃₈₁N₆₇O₆₂₅Ca₁₀₇P₆₄Na₁Mg₂S₂
Skeleton ribs (2^nd,6^th)	H₂₄₈₃C₈₅₆N₁₀₉O₁₀₆₅Ca₁₂₈P₇₆Na₂Mg₂S₄Cl₁l₁
Skeleton ribs (10^th)	H₂₁₇₂C₇₆₅N₁₁₂O₁₀₆₁Ca₁₅₂P₉₁Na₂Mg₂S₄Cl₁K₁
Skeleton sacrum	H₄₁₀₀C₁₄₀₄N₁₄₈O₁₅₂₉Ca₁₃₇P₈₁Mg₂S₃Cl₂K₁Fe₁
Skeleton spongiosa	H₄₇₁₀C₁₈₇₈N₁₁₂O₁₂₈₁Ca₁₀₃P₆₁Na₂Mg₂S₃Cl₃K₁Fe₁
Skeleton vertebral column (C4)	H₃₄₉₁C₁₂₁₄N₁₅₅O₁₅₂₂Ca₁₈₅P₁₁₀Na₂Mg₂S₅Cl₂K₁Fe₁
Skeleton vertebral column (D6, L3)	H₃₈₇₉C₁₃₃₄N₁₅₂O₁₅₂₅Ca₁₅₅P₉₂Mg₂S₃Cl₂K₁Fe₁

Statistics of computed effective atomic numbers

Tissue	Z_eff
	Average	Max.	Min.	SD
Adipose	3.520	5.301	3.074	0.497
Blood	4.054	6.224	3.448	0.648
Brain	3.956	6.168	3.354	0.652
Breast	3.946	6.455	3.310	0.736
Cartilage	4.227	6.511	3.578	0.692
Cell nucleus	4.037	6.339	3.401	0.687
Eye lens	4.069	6.094	3.506	0.598
Gi tract	3.972	6.121	3.379	0.635
Heart	4.032	6.217	3.427	0.651
Kidney	4.021	6.185	3.422	0.643
Liver	4.037	6.203	3.437	0.644
Lung deflated	4.041	6.215	3.435	0.649
Lymph	3.996	6.200	3.381	0.617
Muscle	4.031	6.200	3.434	0.642
Ovary	4.012	6.190	3.407	0.648
Pancreas	3.941	6.095	3.357	0.635
Red marrow	3.789	5.819	3.269	0.576
Skeleton cortical bone	7.283	10.949	5.994	1.335
Skeleton cranium	6.196	9.896	5.009	1.289
Skeleton femur	5.182	8.793	4.159	1.181
Skeleton humerus	5.618	9.295	4.512	1.244
Skeleton mandible	6.432	10.132	5.216	1.306
Skeleton ribs(2nd,6th)	5.446	9.035	4.391	1.193
Skeleton ribs(10th)	5.854	9.496	4.721	1.250
Skeleton sacrum	4.979	8.352	4.039	1.088
Skeleton spongiosa	4.540	7.751	3.705	0.995
Skeleton vertibral column	5.468	9.076	4.426	1.165
Skeleton vertibral column	8.009	10.230	7.350	0.771
Skin	4.008	6.095	3.437	0.613
Spleen	4.032	6.194	3.430	0.644
Spongiosa	4.540	7.751	3.705	0.995
Testis	4.076	6.553	3.405	0.742
Thyroid	4.002	6.096	3.411	0.590
Yellow marrow	3.476	5.227	3.045	0.483

Figure 1.

Variation of mass energy absorption coefficient of some tissues with photon energy (Lung, Lymph, Muscle, Ovary, Pancreas, Cartilage, Redmarrow, Spongiosa)

Figure 2.

Variation of mass energy absorption coefficient of some tissues (Yellow marrow, Skin, Spleen, Testis, Thyroid, Skeleton-cortical bone, Skeleton-cranium Skeleton-femur, Skeleton-humerus) with photon energy

Figure 3.

Variation of mass energy absorption coefficient of some tissues (Skeleton-ribs (2nd,6th), Skeleton-ribs (10th), Skeleton-sacrum, Skeleton-spongiosa) with photon energy

Figure 4.

Variation of effective atomic number of some tissues with photon energy (Adipose, Blood, Brain, Breast, Cartilage, Cell nucleus, Eyelens, GI tract, Heart)

Figure 5.

Variation of effective atomic number of some tissues (Kidney, Liver, Lung, Lymph, Muscle, Ovary, Pancreas, Redmarrow, Cortical bone) with photon energy

Figure 6.

Variation of effective atomic number of some tissues (SK bone, Skeleton-femur, Skeleton-humerus, Skin, Spleen, Spongiosa, Testis, Thyroid, Yellow marrow) with photon energy

To verify the validity of the proposed formulae, we have compared the values produced by present work with that of experimental values available in literature. The comparison of mass attenuation coefficients produced by the present formulae with that of experimental values is shown in the Table 5. From this table, it is clear that the values produced by the present formulae agree well with the experiments. In the first stage of the work, we have established the relation between effective atomic number and elemental composition of tissues. The proposed new parameter effective composition index (C_eff) helps in achieving an accurate relation between effective atomic number and elemental composition of the tissue at a given energy. In the second stage, we have established the exact relation between effective atomic number, mass attenuation coefficients (μ/ρ) and mass energy abosorption coefficients (μ_en/ρ). Hence, this set of simple formulae produces mass attenuation coefficients (μ/ρ) and mass energy abosorption coefficients (μ_en/ρ) from the elemental composition at a given energy.

Comparison of present work with experiments

Tissue	Energy (keV)	Mass attenuation coefficient (cm²/g)			Tissue	Energy (keV)	Mass attenuation coefficient(cm²/g)		Winxcom[5]
		Experimental values[10]	Present work	Winxcom[5]			Experimental values[10]	Present work
Adipose	8	6.0000[27]	5.560	5.501	Pancreas	27	0.4192[29]	0.4156	0.410
	11	2.4000[27]	2.596	4.90		60	0.20673[29]	0.2027	0.203
	15	1.0295[27]	1.118	1.10		122	0.1586[29]	0.1601	0.162
	20	0.5200 [27]	0.569	0.510		279	0.1211[29]	0.1175	0.110
	30	0.2947[27]	0.387	0.318		662	0.08615[29]	0.0749	0.079
Liver	30	0.3868[28]	0.429	0.410	Lung	27	0.32285[29]	0.3347	0.330
	40	0.2821[28]	0.218	0.250		60	0.15904[29]	0.1548	0.150
	50	0.2415[28]	0.203	0.250		122	0.11714[29]	0.1352	0.131
	60	0.2198[28]	0.199	0.210		279	0.08667[29]	0.089	0.080
	70	0.2145[28]	0.181	0.190		662	0.06267[29]	0.066	0.065
	80	0.1962[28]	0.171	0.180	Kidney	27	0.4323[29]	0.4369	0.421
	90	0.19056[28]	0.166	0.170		60	0.208[29]	0.2062	0.210
	100	0.1887[28]	0.161	0.160		122	0.157[29]	0.1601	0.159
	110	0.1557[28]	0.159	0.151		279	0.121[29]	0.1370	0.135
						662	0.085[29]	0.0909	0.089
Kidney	30	0.3705[28]	0.388	0.380	Breast	8	9.157[30]	8.708	8.501
	40	0.2657[28]	0.245	0.250		10	5.392[30]	4.454	4.100
	50	0.2343[28]	0.221	0.231		12	3.382[30]	2.91	3.103
	60	0.2076[28]	0.197	0.205		14	2.098[30]	1.811	2.103
	70	0.1895[28]	0.196	0.191	Bone	140	0.1255[31]	0.130	0.129
	80	0.19142[28]	0.190	0.190		364	0.10104[31]	0.118	0.101
	90	0.1838[28]	0.171	0.170		662	0.0667[31]	0.083	0.070
	100	0.1667[28]	0.177	0.175	Muscle	140	0.1495[31]	0.142	0.141
	110	0.1876[28]	0.173	0.172		364	0.09143[31]	0.102	0.090
						662	0.0771[31]	0.091	0.082
Brain	27	0.43365[28]	0.421	0.410	Liver	27	0.416[29]	0.410	0.410
	60	0.2096[28]	0.224	0.210		60	0.2085[29]	0.209	0.210
	122	0.1644[28]	0.154	0.145		122	0.1557[29]	0.1607	0.154
	279	0.1231[28]	0.122	0.132		279	0.1188[29]	0.1276	0.112
	662	0.0872[28]	0.103	0.080		662	0.0849[29]	0.0782	0.078

4. Conclusion

The proposed semi empirical formulae of mass attenuation coefficients (μ/ρ), mass energy abosorption coefficients (μ_en/ρ), and effective atomic number for tissues of human organs in the energy range 1 keV-20 MeV produce values which agree well with experiments. This formulae is the first of its kind and it is useful in radiotherapy and medical physics.

References

D.A Bradley, C.S Chong, A.M. Ghose,

Photon absorptiometric studies of elements, mixtures and substances of biomedical interest

. Phys. Med. Biol. 31,267-273 (1986),.doi: 10.1088/0031-9155/31/3/005

Pocket formula for mass attenuaion coefficient, effective atomic number and electron density of human tissues

1. Introduction

2. Methodology

2.1 Semi empirical formula for effective atomic number (Zeff) in terms of composition

2.2 Semi empirical formula for of mass attenuation and energy absorption coefficients

3. Results and discussions

4. Conclusion

2.1 Semi empirical formula for effective atomic number (Z_eff) in terms of composition