H.C.Manjunatha·L.Seenappa

Abstract We have proposed a pocket formula for mass attenuation coefficient(μ/ρ),mass energy absorption coefficient(μen/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 formula produces a mass attenuation coefficient,mass energy absorption coefficient,and effective atomic number from their composition

Keywords Effective atomic number·Mass attenuation coefficient·Tissues

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 measurements of mass attenuation coefficients of dosimetric interest[1—3].

Hubble and Seltzer[4]gave attenuation coefficients data for the elements and compounds.Berger and Hubbel[2]developed the 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 Zeff.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 Zeffof biomolecules for electron,proton,alpha particle,and photon interactions.Kurudirek[11]also studied Zeffand Neof human tissues.The sameworkers[12]also studied Zeffofdosimetric 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 diagnosis.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-empiricalformula forphoton interaction parameterssuch asmass 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(second,sixth),skeleton ribs(tenth),skeleton sacrum,skeleton spongiosa,skeleton vertebral column(c4),and skeleton vertebralcolumn(D6,L3)].In the second section of the paper,we have explained the proposed empirical formula.The third section ofthe paper describes the comparison of experimental results with the 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(Ceff).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(Ceff)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 reaming elements in the tissue.

In the above equation,CH,CC,CN,CO,CCa,CP,CNa,CMg,CS,CCI,CK,CFe,and CIare compositions of the elements indicated in the corresponding subscripts.AH,AC,AN,AO,ACa,AP,ANa,AMg,AS,ACl,AK,AFe,and AIare atomic weights 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,the remaining elemental composition is considered.

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

We have calculated the effective composition index(Ceff)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(Zeff)and effective composition index(Ceff).

In 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(Ceff).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 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 constituents 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 asδ2exp（δ3E）+ δ4， and polynomial function（α1E4+ α2E3+ α3E2+ α4E+ α5）,where α's are functions of atomic number[α = α（Z）].Among these functions,the double exponential function such as α1Eα2+ α3is 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 α1Eα2+ α3to 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.

here αi,βi,and δiare fitting parameters which are given in Table 1.

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

We have also fit the following nonlinear function to mass attenuation coefficients(μen/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:

here φi,ψi,and χiare fitting parameters which are given in Table 2.

The proposed formula for the mass energy absorption coefficient(μen/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:

for 0.1-20 MeV.

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

and

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

Table 1 Fitting parameters for mass attenuation coefficients(μ/ρ)

Table 2 Fitting parameters for mass attenuation coefficients(μen/ρ)

Table 3 Equivalent chemical formula for tissues of human organs

3 Results and discussions

Fig.1(Colour online)Variation of mass energy absorption coefficient of some tissues with photon energy(lung,lymph,muscle,ovary,pancreas,cartilage,red marrow,spongiosa)

Fig.2(Colour online)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

Fig.3(Colour online)Variation of mass energy absorption coefficient of some tissues[skeleton ribs(second,sixth),skeleton ribs(tenth),skeleton sacrum,skeleton spongiosa]with photon energy

Fig.4(Colour online)Variation of effective atomic number of some tissues with photon energy(adipose,blood,brain,breast,cartilage,cell nucleus,eye lens,GI tract,heart)

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 absorption coefficients(μen/ρ),and effective atomic numbers using the formulae proposed in the present work.The variation of mass energy absorption coefficients(μen/ρ)with energy for different tissues of human organs for a wide energy range 1 keV—20 MeV is shown in Figs.1,2,and 3.A similar variation of mass attenuation coefficients(μ/ρ)with photon energy is also observed.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,and 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.

Fig.5(Colour online)Variation of effective atomic number of some tissues(kidney,liver,lung,lymph,muscle,ovary,pancreas,red marrow,cortical bone)with photon energy

Fig.6(Colour online)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 formula,we have compared the values produced by the present work with the experimental values available in the literature.The comparison of mass attenuation coefficients produced by the present formula with that of experimental values is shown in Table 5.From this table,it is clear that the values produced by the present formula 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(Ceff)helps in achieving an accurate relation between effective atomic number and elemental composition of the tissue at a given energy.Inthe second stage,we have established the exact relation between effective atomic number,mass attenuation coefficients(μ/ρ),and mass energy absorption coefficients(μen/ρ).Hence,this set of simple formulae produces mass attenuation coefficients(μ/ρ)and mass energy absorption coefficients(μen/ρ)from the elemental composition at a given energy.

Table 4 Statistics of computed effective atomic numbers

Table 5 Comparison of the present work with experiments

4 Conclusion

Theproposed semi-empiricalformula ofmassattenuation coefficients(μ/ρ),mass energy absorption coefficients(μen/ρ),and effective atomic number for tissues of human organs in the energy range 1 keV—20 MeV produces values which agree well with experiments.This formula is the first of its kind,and it is useful in radiotherapy and medical physics.