Analytic derivation of central axis percent depth dose calculations in transition zones with loss of electronic equilibrium

Purpose: The study of megavoltage photon dose distribution behind and near small areas of low and high density material is best understood with Monte Carlo (MC) dose calculation or direct measurements which may not be always be possible. This is especially true for air-tissue area where the replacement of soft tissue scattering material by air results in the loss of electronic equilibrium and changes in the lateral spread of the beam as well. Monte Carlo calculations are the standards to correctly evaluate in homogeneities in transition zones. If one could develop a model with sufficient accuracy to obtain similar results, this would be very helpful clinically. Methods: To this end, we have developed an exponential model and derive an explicit expression that accounts for the under dosage. The model is an extension of a much earlier work done with electrons and photons. Our analytic model is based on the experience of the underlying physics assuming exponential attenuation of photons in matter. Results: It differs from a similar work by solving the problem correctly and introducing parameters that can be traced to direct measurements without the need of extensive statistical data analysis. It combines the generation of free electrons through ionization and their attenuation to a simple differential equation for the central axis depth dose. It involves two parameters, which can be obtained from 1) direct beam measurements, 2) primary photon attenuation coefficients from physics tables and 3) iteration techniques. Conclusion: The simplicity of the model allows us to extend our derivation to situations such as transitions zones of different densities in areas such as head and neck and lung. A clinical example is illustrated to demonstrate the problems encountered in treating cancer of the larynx.


Introduction
High energy photons ionize matter indirectly; photon interactions in a medium release charged particles (electrons or positrons), which in turn deposit energy via Coulomb interactions with orbital electrons of the atoms. The intensity of a monoenergetic photon beam, Ip, incident on a medium attenuates according to the exponential law: Io is the initial photon intensity, E the energy of the photon, μ linear attenuation coefficient for the medium and x is the depth of interest. Eq. (1) represents the primary component of the photon beam. The linear attenuation coefficient μ is the sum of the attenuation coefficients of several interactions, μ = τ + σR + σC +κ, where τ denotes the photo-electric coefficient effect, σR for Rayleigh scattering, σC for Compton scattering and κ for pair productions. The most important interaction in the therapeutic range is Compton scattering, which dominates in the energy range of 6 to 18 MV X-rays. The absorbed dose is defined as the mean energy E deposited by ionizing radiation to a medium of mass m in a finite volume V. For monoenergetic photons traveling along a depth x, the absorbed dose can be written as where (1/ρ) (dE/dx) med, Av., also known as the Stopping Power (S/ρ)med, Av, is the average energy loss along the depth x and Φ(x) is the fluence or number of secondary electrons, Ie(x), generated by the incident photons. Taking into account the fact that the photon fluence is inversely proportional to the square of the distance from the source, Equ.(3) becomes The central axis depth dose is defined as the ratio of the central axis dose divided by the maximum dose on the central axis, that is; The purpose of this paper is to show that by means of a simple analysis of first order scattering of high energy photons in a an absorbing medium it is possible to compute, for practical purposes, the distribution of secondary radiation in an absorber containing a region of a different density and derive the way the total radiation is attenuated. Our solution can account for the radial distribution of the beam by multiplying the central axis by a simple empirical factor F(x,y) which takes into account the non-planarity of the field as well as the sidewise straggling and scattering of electrons. This will be the subject of a future investigation. In

Theory
High-energy photons incident on a homogeneous material of density ρ are gradually attenuated. Each centimeter of material attenuates a constant fraction of the initial intensity. As a consequence, their intensity follows an exponential decay law.
where, Ip(x) is the intensity left a depth x, Io is the incident intensity and μρ an effective linear attenuation coefficient for the photons in the given material. μp (cm -1 ) depends on the photon energy, type of medium and its value decreases with higher energies. Values for μp can be found in physics reference data tables or calculated from measured beam data. The primary effect of photons is to knock out electrons from the cell material. The energy lost by the photons is converted into ionization energy. These ionization electrons or electron fluence are proportional to absorbed dose from which the central axis % percentage depth dose is derived. This is an extension of our earlier work.
The surface dose is small but not zero. It is interesting to note that Eq. (7) agrees closely with the form obtained empirically by Johns et al. 5  2 × 2 to 40 × 40 cm 2 with an accuracy of ± 2% over a range of 0 to 30 cm depth. In Figure 4, we compared the calculated % central axis depth dose Eq. (7) against the Eclipse AAA for a 10 × 10 cm 2 and found excellent agreement.        Due to lack of measured data, we approximated values for xm, and μe. Computer Monte Carlo simulation 8 of a 6 MV photon beam in heterogeneous media containing bone, demonstrate that the absorbed dose is 11.1% lower in bone than in water for the same depth. Figure 5 shows the % central axis depth dose calculated for water and bone respectively.

Secondary Attenuation Coefficient (μe)
The electron absorption coefficient depends not only on field size but on depth as well. The effects of electronic equilibrium require that two separate values be used for μe. For x < xm , in the buildup region, secondary electrons will be attenuated 1 more quickly than beyond the buildup region and will have a larger numerical value. The parameter μe for x < xm and x > xm cannot be solved analytically but using an iteration technique, an approximate value can be obtained for a given depth and field size. The following method is used to determine the best values for μe: We start with an arbitrary value of μe =2.60 cm -1 Inserting the value of 2.60 cm 1 , we calculate %D(x,f) with Eq (7) %DD(1,100) = 98.96 which is close to the measured value of 97.4. Since the agreement is within 2%, we accept the value of μe=2.60cm -1 and proceed to calculate the central axis depth dose for x < xm.
We have assumed that the average secondary electron energy of Ie(x) is proportional to the average photon energy and will vary with field size and depth. The effects of electronic equilibrium require two separate values for μe, that is, for x < xm and x > xm. (See Table 2)

Central axis percent depth dose derivation for inhomogeneous case
The introduction of a region with density other than the medium causes a reduction or increase of the dose along the central axis. It will be shown that the dose at x = a will be reduce by a factor proportional to the ratio of the stopping power of air to that of tissue. At x=b, the depth dose distribution will be more complicated. However, the problem is simplified by solving the dose in each specific region.

Central axis % Depth Dose Equations for each Region
Region S2 and S1 are the stopping power for medium 2 and 1.
The effective/average energy for the % depth dose in water, is obtained from photon interaction coefficients tables. 4 This also allowed us to extrapolate the average photon energies and stopping power ratio for different medium. The dependent parameters as a function of field size,μe ,μp,α, xm and S were calculated and obtained from measured data. (See Table 3) Figure 6 shows the central axis % depth dose for 6 and 16 MV photons with and without an air gap.
The results agree with findings from Monte Carlo studies. 7,8,9,10,11 For higher energies, the changes will be more pronounced. Based on our model, the calculations show that small fields have a greater reduction at the air junction (interface) zone than larger field sizes. Higher energies also exhibit higher dose reductions near air-tissue interface zone.
The primary photon linear attenuation coefficient in low density areas such air cavities is much less than values of tissue equivalent materials such as water and therefore cause a decrease in the dose distribution due to reduced generation of scattered electrons as reflected by the coefficient α in Eq.(31). This causes electronic disequilibrium and loss of dose in the region. Beyond the air cavity, there is an increase of dose. The primary reason for this is the increase in the production of electrons and reflected by the coefficient α.
For densities > 1 gm/cm 3 such as bone, the electron density and the linear attenuation is higher than water but the mass attenuation per gram of bone is less than water and causes a decrease of dose. At the interface of bone and soft tissue, there is an increase of dose due to backscatter of electrons from the bone surface and a buildup region of a few millimeters occurs. (Figure 7)

Clinical Application
As a single modality, radiation therapy provides excellent local regional control and survival for early T1 vocal cord lesions. 12, 13 There are several ways of treating a T1 larynx lesion with radiation but for simplicity we examine a simple case where the treatment plan consist of two opposing lateral fields using 6 MV photons. Figure  9 describes a typical geometry of a T1 larynx lesion.
For our calculation model, we assume a neck separation of 10 cm with an air gap of 2.0 cm. The lesion has a diameter of 0.5 cm. Patient setup is very crucial here and any deviation from the center of the tumor target can cause a further dose reduction. From radiobiology data and cell kinetic studies, we know that a 0.5 cm tumor mass has approximately 10 7 viable cells and a dose of 45-50 Gy is sufficient to kill more than 95% of the cells but in clinical practice we find that doses greater than 60 Gy are needed to control a T1 lesions of the larynx. Using our model we show that the dose reduction can be up to 20% when treated with parallel oppose fields. See Figure 10.
We feel that the high dose needed to control early lesions of the larynx can only be explained by the dose inhomogeneity occurring in the transition zone between air and tissue. This has been studied by several authors. 14,15,16,17, 18 We recommend these simple guidelines to improve the control of early T1 lesions of the larynx: a) use field sizes > 6 × 6 cm 2 with 6 MV only; and b) use daily Cone Beam for daily beam set up. The smaller the field size, the greater the under dosage. Higher photon energies > 6 MV exhibit higher under dosage as well.

Conclusion
Our analytic model accurately calculates the central axis percentage depth dose of high-energy photons in both homogeneous and non-homogeneous cases. Although our model is simplistic in that an average energy is used to represent the scattering of electrons and photons. The energy loss by the photons scattering into electrons is converted into ionization energy. These ionization electrons represent our electron fluence created by the primary photons and are related to the dose in the medium. The assumptions and meaning of the coefficients in regions 1-3 have been stated precisely. By applying boundary conditions for situation with inhomogeneities, we have derived an explicit expression for the central axis percent depth dose. In the case of treating cancer of the larynx, there can be a significant reduction of dose at the tumor site. We compared our calculations with data from Monte Carlo calculations and found good agreement.