2.1 Relevant parameters used in Monte Carlo simulation

As we know, Monte Carlo particle transport code is widely applied in dose calculation [9-12], and the calculation results of these codes (e.g. MCNP, EGS4 and GEANT4) have been accepted as benchmark data for other methods. Consequently, GEANT4 Monte Carlo code (version.4.9.2) was selected to calculate the local dose distributions of hip prostheses. The Monte Carlo (MC) simulations of this study were performed on a computing server with two Xeon E5504 processors.

2.2 Elemental composition of typical hip prostheses

Three typical prosthesis materials (titanium alloy, Co-Cr-Mo alloy, and ceramic) were selected in MC simulations. The elemental composition of the hip joint (bone, mass density = 1.4 g/cm^-3) was extracted from the No. 23 report of the ICRP (International Commission on Radiological Protection) [14], while that of hip prostheses were extracted from the reference data supplied by manufacturers. The compositions data listed in Table 1 was used as benchmark data and incorporated into the material functions of the GEANT4 detector-construction file.

2.3 Modeling of the linac treatment head

The aim of the modeling is to obtain the photon spectrum of ELEKTA medical linac. For ELEKTA linac, the main components of the treatment head are a target, a primary collimator, two flattening filters, and a secondary collimator system. As shown in Fig. 1, flattening filter No. 1 (Difference filter) was located at the basis of the primary collimator only for 15 MV X-ray mode, while flattening filter No. 2 (Low Energy X-ray filter) appears at the 4, 8, and 15 MV modes.

Fig. 1.Full-screen View

Geometry setup of the treatment head.

2.4 Energy spectra of clinical photon beams

For obtaining photon spectra of ELEKTA linac, the interactions between various electron beams and the target- collimator-filter combined system were simulated. Then, the calculated spectra of the resultant photon beams (Fig. 2) were used as reference data for the 4, 8, and 15 MV virtual sources.

Fig. 2.Full-screen View

Calculated photon spectra for 4, 8 and 15 MV clinical beams.

2.5 Modeling of the Monte Carlo calculation

The source-phantom and the prosthesis-detector configuration are illustrated in Fig. 3. As showed in Fig. 3(a), virtual photon sources (4, 8, and 15 MV) were placed above a cubic water phantom, and the spheroid hip prosthesis with a 4 cm diameter is located at the phantom center. Figure 3(b) (generated by OPENGL) depicts the interaction between the primary ray and the hip prosthesis. It also shows a sketch of the prosthesis-detector configuration. Several sensitive detectors were placed at an interval of 5 surrounding the prosthesis. As illustrated in Fig. 3(c), the geometric parameter θ is termed as detector angle, and it is defined as the angle between the beam axis and the vector connecting the prosthesis and the detector center. The parameter d (detector distance) quantify the distance between the prosthesis surface and detector centers. In this study, several simulations were accomplished using various d (5, 10, 15, and 20 mm) to reveal the relation between prosthesis dose perturbations and detector distance.

Fig. 3.Full-screen View

The MC calculation model: (a) Source-phantom system; (b) Prosthesis-detector system; (c) Illustration of two geometry setup parameters for prosthesis-detector system.

2.6 The dose differences between hip prostheses and hip joint

In this study, the dose differences between prostheses and the hip joint are quantified by a variable RDP (relative dose perturbation), which can be expressed as follows:

where mi is the abbreviation of prosthesis material, θ is the detector angle, d is the detector distance, $D{(\theta \mathrm{,}d)}_{m_i}$ is the calculated dose for mi, and D(θ, d)_bone is the calculated dose for the hip joint.

3.1 Local dose distributions of typical hip prostheses

In this section, the calculated dose distributions of typical prostheses were illustrated in Fig. 4. In most cases, about 2 billion events were simulated in each GEANT4 Run-Action, while the relative errors are smaller than 0.05. To simplify notations, the titanium alloy prosthesis, Co-Cr-Mo alloy prosthesis, and ceramic prostheses were respectively abbreviated as TC4, CCM, and ceramic.

Fig. 4.Full-screen View

Local dose distributions for 4, 8 and 15 MV photon beams.

As shown in Fig. 4, the calculated dose for the hip joint increases approx linearly with θ (detector angle). For hip prostheses, an inflection point appears at each dose-angle curve. The inflection point becomes the critical point that divides the entire angle range into two regions with different dose-angle variation laws. Compared to the hip joint, the dose distributions of prostheses show several obvious characteristics. Firstly, the inflection angles (correspondent θ of inflection points) of these prostheses are approximately the same for a specified d (detector distance), and it decreases with d. Secondly, the calculated dose increases with θ in accordance with a power function, while θ is smaller than the inflection angles (shadow region). Thirdly, the calculated dose for the prostheses is approximately the same as that for the hip joint for 10, 15 and 20 mm detector distances, while θ is larger than the inflection angles (scattering region).

Overall, the calculated dose is smallest for CCM and largest for ceramic in the prosthesis shadow region. In the prosthesis scattering region, the largest calculated dose for the prostheses appears for a 15 MV beam, while the smallest dose was observed for a 4 MV beam. In this region, it increases about 1% to 7% for 8 and 15 MV beams at a 5 mm detector distance.

3.2 Local dose differences between hip prostheses and hip joint

In this section, the relative dose perturbation (RDP) is calculated and illustrated in Fig. 5.

Fig. 5.Full-screen View

Calculated RDP values for 4, 8 and 15 MV photon beams.

As shown in Fig. 5, each RDP-angle curve has an inflection point. The inflection angles are approx 60, 50, 45, and 40 for 5, 10, 15, and 20 mm, respectively. The expression of RDP implies that the positive RDPs quantify the relative dose increases of the prostheses compared to the hip joint, while the negative RDPs quantify the relative dose decreases. In the prosthesis shadow region, CCM and ceramic show the smallest and largest RDPs, respectively. For a specified hip prosthesis, RDP is minimized at the minimum θ, and then increases with the θ according to the power function law. Finally, it nearly approaches zero at the inflection point. In the prosthesis scattering region, the RDPs are very sensitive to the nominal energy and are nearly zero for the 4, 8 and 15 MV beams, while the detector distances are 10, 15 or 20 mm. However, the RDPs are slightly positive (1%-7%) for the 8 and 15 MV beams at a 5 mm detector distance, which indicates the calculated dose for the prostheses is slightly larger than that of the hip joint.

Generally, the differences between this study and previous reports [6-8] mainly originate from the differences in geometric setup parameters, radiation source characteristics, and calculation codes. Firstly, a spheroid configuration was used in our Monte Carlo modeling, but cubic or thin slice configuration was used in most previous works. Besides, the deviations of geometric setup parameters (e.g. source-surface-distance, source-axis-distance) can also introduce differences of calculated dose perturbations. Secondly, the differences of linac X-ray characteristic (e.g. photon spectrum and photon intensity) probably influence the calculation results. Thirdly, the Geant4 Monte Carlo toolkit (standard EM physics process) was applied in evaluating the RDP of hip prosthesis in this study, but the EGS4 or EGSnrc systems were mainly utilized in earlier works. Taking together, the combination of above factors has important dosimetric implications when calculation results of different investigations were compared.

3.3 Impact of nominal energy on relative dose perturbations

In the prosthesis shadow region, the RDP values for the Co-Cr-Mo alloy, titanium alloy, and ceramic hip prostheses are almost 37%, 22%, and 16% for the 4 MV beam; 35%, 21%, and 14% for the 8 MV beam; 35%, 21%, and 13% for the 15 MV beam. Obviously, the differences of maximum RDP values between 4, 8, and 15 MV beams are smaller than 2% for specified material type and detector distance. In other words, nominal energy is not the dominant factor influencing prosthesis perturbations in this region. That is because the beam with higher nominal energy also shows stronger penetrating power towards the water above the prostheses, when prostheses show stronger absorption and scattering compared to the hip joint. For this region, material type greatly influences RDPs, which is due to substantial attenuations by these prostheses. Thus, the prostheses with a higher atomic number or density are more likely to introduce stronger perturbations, owing to its stronger absorption or scattering.

In the prosthesis scattering region, however, the RDPs are greatly influenced by nominal energy or detector distance. Except for the 5 mm detector distance, almost no dose perturbations were observed for 4, 8, and 15 MV beams. The primary ray was not attenuated by the prostheses before reaching these detectors belonging to this region; therefore, the water-generated electrons and prosthesis-generated Compton electrons mainly influence prosthesis dose perturbations. The water-generated electrons remain approx constant after the replacement of the hip joint by prosthesis. Thus, the RDPs mainly depend on the Compton electrons generated by the prostheses. In other words, the calculated dose will increase with the yield of prosthesis-generated electrons, while detector distance is shorter than the range of these electrons. On the contrary, an increased electron yield hardly introduces any dose perturbations, while detector distances are larger than the Compton electron range. In most cases, photon beams with higher nominal energy are inclined to generate Compton electrons with a relatively higher kinetic energy. Thus, the dose perturbations for a 15 MV beam are stronger than that of 4 and 8 MV beams in this region, and it reaches up to 7% for a 5 mm detector distance.

3.4 Depth dose distribution beneath prostheses

In this section, percentage depth dose (Fig. 6) beneath the prosthesis was calculated for 4, 8, and 15 MV beams. The depth dose differences (Fig. 7) between the prostheses and hip joint were introduced to reveal the relation between material type and prosthesis attenuations.

Fig. 6.Full-screen View

Percentage depth dose beneath prostheses (Normalized by the maximum dose of hip joint).

Fig. 7.Full-screen View

Depth dose differences between prostheses and hip joint.

Obviously, 4, 8, and 15 MV beams were intensively attenuated by these prostheses, while the dose decrease closely depends on the atomic number or density of the prostheses. In detail, CCM illustrated the largest dose decrease; on the contrary ceramic shows the smallest decrease, owing to its smaller atomic number or density. To quantify the decrease of the depth dose, the authors calculated the dose differences between the prostheses and hip joint, and plotted these calculated differences in Fig. 7. As shown in Fig. 7, depth dose differences increase with depth and were determined by the atomic number or density. So far, we have evaluated the dosimetric perturbations of three typical prostheses and offered useful data for medical physicists and radiation oncologists.