Quantitative evaluation by measurement and modeling of the variations in dose distributions deposited in mobile targets

Abstract

The objective of this study is to quantitatively evaluate variations of dose distributions deposited in mobile target by measurement and modeling. The effects of variation in dose distribution induced by motion on tumor dose coverage and sparing of normal tissues were investigated quantitatively. The dose distributions with motion artifacts were modeled considering different motion patterns that include (a) motion with constant speed and (b) sinusoidal motion. The model predictions of the dose distributions with motion artifacts were verified with measurement where the dose distributions from various plans that included three-dimensional conformal and intensity-modulated fields were measured with a multiple-diode-array detector (MapCheck2), which was mounted on a mobile platform that moves with adjustable motion parameters. For each plan, the dose distributions were then measured with MapCHECK2 using different motion amplitudes from 0–25 mm. In addition, mathematical modeling was developed to predict the variations in the dose distributions and their dependence on the motion parameters that included amplitude, frequency and phase for sinusoidal motions. The dose distributions varied with motion and depended on the motion pattern particularly the sinusoidal motion, which spread out along the direction of motion. Study results showed that in the dose region between isocenter and the 50% isodose line, the dose profile decreased with increase of the motion amplitude. As the range of motion became larger than the field length along the direction of motion, the dose profiles changes overall including the central axis dose and 50% isodose line. If the total dose was delivered over a time much longer than the periodic time of motion, variations in motion frequency and phase do not affect the dose profiles. As a result, the motion dose modeling developed in this study provided quantitative characterization of variation in the dose distributions induced by motion, which can be employed in radiation therapy to quantitatively determine the margins needed for treatment planning considering dose spillage to normal tissue.

Keywords

Motion dose model motion artifacts respiratory motion spread-out dose distribution

1 Introduction

Patient motion causes considerable deviation of the dose coverage from the intended initial treatment plan in radiation therapy. Patients move in different patterns that include voluntary motions such as shifts or relaxations and involuntary motions such as respiration, heart beats or bowel motion. Discrepancies between the planned dose and the delivered dose may have serious consequences on the overall outcome of the treatment in radiation therapy, which include under-dosing of the tumor or overdosing of normal surrounding tissues. Intra-fractional dose discrepancies particularly in thoracic and abdominal cancer patients induced by respiratory motion have been investigated intensively in several studies [1, 2]. Large hypo-fractionated doses delivered by advanced stereotactic body radiation therapy techniques in few fractions impose stringent requirements on patient positioning, intra-fractional motion management and dose delivery. Various techniques have been used to reduce the effects of patient motion on the dose delivery that include breath-holding [3, 4], beam-gating [5 –8], motion tracking of implanted markers [9] and multi-leaf collimator tracking techniques [10, 11]. However, these techniques have limitations and may not be applicable for all patients. For example, a limited number of lung cancer patients can not hold breathing for long time periods to deliver dose from the different treatment beams because of compromised pulmonary functionality. Beam gating often uses external markers that are placed on the patient chest skin where the motion of these makers may not be correlated with internal tumor motion. Motion tracking of internally implanted markers provides a superior technique in tracking tumors where the robotic arm of the Cyberknife [9 , 13] follows the motion signal obtained from optical and radiographic imaging. However, invasive surgical procedures are required to implant the markers within the patient. In all previously mentioned techniques, the reduction of motion effects is performed during dose delivery.

Other techniques that consider motion effects include performing four-dimensional-CT (4D-CT) imaging, where the projections are acquired with a respiratory signal [14, 15]. The projections are then sorted in different respiratory phases and the corresponding CT image sets are reconstructed. These images are used to outline the internal target volume that includes motion blurring in the treatment planning process [16]. One limitation of 4D-CT treatment planning is the use of large margins to define the treatment planning volume which may result in irradiation of large proportions of normal tissue with high doses. In this work, the variations in the dose distributions for open or intensity-modulated fields were assessed quantitatively by both measurement and modeling using stationary and mobile phantoms. Two-dimensional dose distributions were measured using a two-dimensional multiple-diode-array detector that was placed in a solid water phantom, which was mounted on a mobile platform. The dose distributions based on convolution of the stationary dose distributions with a motion kernel were compared quantitatively with the measured dose distributions from open-conformal and intensity-modulated beams with the mobile MapCheck2 phantom.

2 Material and methods

2.1 Phantom setup and dose measurement

The dose distributions from various conformal open fields and intensity-modulated beams from a TrueBeam Stx Linear Accelerator (Varian Medical Systems, Inc., Palo Alto, CA) were measured using a multiple-diode-array detector (MapCheck2, Sun Nuclear, Melbourne, FL) [17, 18]. Different open fields were used to irradiate a phantom that ranged in size from 0.5×0.5 cm² to 20×20 cm². The MapCheck2 detector was mounted on a mobile platform phantom (Standard Imaging Inc., Middleton, WI) that moved with adjustable sinusoidal motion patterns in one-dimension along the superior-inferior direction (Y-Axis). Figure 1 shows the MapCheck2 phantom mounted on the mobile platform with adjustable motion amplitude and frequency that is actuated by a small electrical motor. The phantom was moved in one direction with different motion amplitudes from 0–25 mm using three frequencies 10, 20 and 30 cycles per minute (CPM) to simulate patient breathing. Two-dimensional (2D) dose distributions were measured with MapCheck2, which is made of 1,527 diode detectors arranged in a 2D-array with an effective area of 26×32 cm². The effective detection area of each diode is about 0.8 mm×0.8 mm, which allows high position resolution dose measurements locally at the site of the individual detectors. The vertical and horizontal distances between two consecutive diode detectors are 10 mm, while the minimum diagonal distance between two nearby detectors is 7 mm. The detectors are located at 12 mm depth in a phantom which corresponds to 20 mm of water-equivalent thickness. As shown in Fig. 1, an additional 50 mm thick solid water phantom is placed on the top of the diode-array detectors. Linear interpolation with the nearest neighboring points is used to improve position resolution of the measured 2D-dose distributions by MapCheck2.

Fig.1

Experimental setup with MapCheck2 phantom mounted on the mobile platform.

2.2 Motion dose modeling

A dose algorithm was introduced in this study to model the spread-out of dose distribution introduced by motion following an approach similar to a previous work that models the effects of motion on CT-number distributions [19]. Two motion patterns were considered in this work: (a) a target moving with a constant speed and (b) a target moving with sinusoidal motion along the superior-inferior direction during dose delivery. This model relates the variations in the dose level and elongation of the dose distributions with motion parameters such as speed, amplitude and frequency.

2.2.1 Motion model with constant speed

The dose distribution from a beam with a length L_o that irradiates a phantom moving with a constant speed V_P along the Y-direction spreads out over a length L in the forward direction as the phantom moves relative to the radiation isocenter. During dose delivery, each point in the mobile target shifts from its initial place where the end of the mobile target extends by V_PT, where T is the dose delivery time. Thus the dose distribution spreads out over a length, as given by the following:

$\begin{matrix} L = L_{o} + V_{P} T \\ L^{'} = V_{P} T - L_{o} \end{matrix}$ (1) where L_o is the length of the dose distribution deposited in the stationary target, L′ is the length of the high dose level. Figure 2a shows a simulation of the variations in the dose distribution of the mobile target with speed of the target where it spreads out with increased motion speed and the dose level received by the mobile target decreases.

Fig.2

(a) Simulation of the dose spread-out from an open field (40×40 mm²) deposited in a phantom moving with a constant speed equal to 1 mm/sec. (b-c) Simulation of the dose spread-out from an open beam (40×40 mm²) deposited in a phantom moving with a sinusoidal motion with amplitudes of 10 mm and 30 mm, respectively.

For a stationary target, the dose distribution is equal to D_S over the length L_o of the open field. As the target moves, the dose distribution spreads out over a longer distance (L) as given in equation (1) and the dose level (D_M) drops to a lower value than the level of the stationary target. The cumulative value of the dose that spreads out due to target motion will be preserved and equal to the integral of the dose level over the length of the open field as given by the following equation:

$D_{S} L_{o} = \frac{1}{2} D_{M} {L + L^{'}} = \frac{1}{2} D_{M} {L_{o} + V_{P} T + V_{P} T - L_{o}} = D_{M} V_{P} T$ (2)

Thus the relationship between the dose level for the mobile and stationary targets is given by the following equation: $D_{M} = D_{S} \frac{L_{o}}{V_{p} T}$ (3)

Thus the dose level D decreases with increasing V_P of the mobile target as shown in Fig. 2(a).

2.2.2 Motion model with sinusoidal motion

Considering a sinusoidal motion of the phantom platform, the location of a given point on the multiple-diode-array detector varies along the superior-inferior direction (Y-axis) with time according to the following equation: $y (t) = y_{o} + A sin (ω t + φ)$ (4) where y_o is the position at t = 0, A is the motion amplitude, ω is the motion angular frequency and φ is the motion phase. The dose distribution spreads out by 2A from [L₀/2 + A, L₀/2 + A] along the Y-axis. The dose distribution, D_M (y), can be calculated as follows: $D_{M} (y) = \int_{- \infty}^{+ \infty} D_{S} (x, y - y^{'}, z) P (y^{'}) {dy}^{'}$ (5) where D_S (y) is the dose distribution received by the stationary phantom. P (y) represents the probability distribution function for the sinusoidal motion along the superior-inferior direction (Y-axis) and is given by the following equation: $P (y) = \frac{1}{π} \frac{1}{\sqrt{1 - (\frac{y - y_{o}}{A})^{2}}}$ (6)P (y) is independent of motion frequency and phase and thus the dose distribution calculated by equation (5) is only dependent on motion amplitude. In the same way for a target moving with constant speed, the integral of the dose over the length of the stationary and mobile targets is preserved for a target with sinusoidal motion as given by the following equation: $D_{S} L_{o} = \int_{y_{o} - L_{o} - A / 2}^{y_{o} + L_{o} + A / 2} D_{M} (y) dy$ (7)

Figures 2(b-c) show the dose spread-out due to sinusoidal motion of the target where the cumulative dose integral over the length of the target is constant for the stationary target and mobile target regardless of the motion amplitude.

3 Results

3.1 Dose variation of open conformal beams with motion

Figures 3(a,c,e) show a comparison of the measured dose profiles deposited in the stationary and mobile phantom along the Y-axis. Figures 3(b,d,f) show the ratio dose profiles from mobile phantom relative to the stationary phantom for the indicated range-of-motion (ROM). The central dose remained unchanged and agreed within 3% with the static phantom dose as long as the ROM is smaller than the length of the open field (L) along the direction of motion. However, in the penumbra region and beyond larger dose deviations were obtained with increase in motion amplitude. In Figs. 3(a,c,e), cold regions were obtained between the center of the field and the 50% isodose line that was considered the dose needed to cover the PTV. While hot regions occurred beyond the 50% isodose line and progressed toward the periphery of the field and beyond that was considered regions of normal tissue. All motion dose profiles intersected the stationary dose profile at the 50% isodose line for ROM’s less than the length of the stationary beam. The dose level in the profiles in the cold spots within the open fields decreased with increasing ROMs. For example, in a 10×10 cm² open field with a ROM = 25 mm, the dose profile decreased by nearly 30% compared with that for the static phantom as shown in Fig. 3d. At the edges of the fields beyond 50% dose level, the ratio of dose profiles (motion relative to stationary) in the regions of hot spots outside the open fields increased significantly as the ROM increased by nearly 30% for a 10×10 cm² open field and ROM = 25 mm as shown in Fig. 3d. This represented dose from the treatment fields that will invade into normal tissue. Additionally, the dose distributions of the central axis dose profile in the direction perpendicular to the phantom motion did not change for all ROM’s that are smaller than the length of the open stationary field. The right and left asymmetry in the dose difference between the measured doses of the mobile relative to the stationary phantom was due to the limited spatial resolution of MapCheck detector by the spacing between the diode-array detectors.

Fig.3

(a,c,e) represent the stationary and mobile measured dose profiles along the motion direction (Y-axis) from an open 5×5, 10×10, 20×20 cm² fields, respectively, for the indicated ROM’s. (b,d,f) represent the ratio of the measured dose profile from mobile phantom relative to that from static phantom (ROM = 0 mm) for the same open field.

3.2 Dose variations with intensity-modulated beams

Figure 4 shows IMRT fields for head-and-neck and lung patients that were measured with a stationary and a mobile MapCheck2 phantom with various motion amplitudes as indicated and a frequency of 15 cycles-per-minute (CPM). The mobile IMRT dose distributions varied from the stationary distributions where cold spots were created in the regions that were covered by the stationary distribution and hot spots were generated outside the regions covered by the stationary fields. The percentage variation of the dose distribution in the H&N plans were shown in Fig. 4b and the lung plan Fig. 4d. The deviations between stationary and mobile distributions increased systematically with motion range. For large ROM’s, the dose variations were as high as 25% for lung and 13 % for H&N.

Fig.4

Dose profiles for head-and-neck (a) and lung (c) IMRT plans delivered with motion at 15 CPM and different ranges of motion of 0, 5, 10, 15, 20 and 25 mm as indicated. The ratio of dose profiles for the head-and-neck (b) and lung (d) from mobile phantom relative to profile of the stationary phantom.

3.3 Comparison of measured and calculated dose profiles

Figures 5(a,c,e) show a comparison between the dose profiles measured with MapCheck2 and those calculated with the motion model (equation (5)) for the 5×5, 10×10, 20×20 cm² open fields, respectively, for different ROM’s. Figures 5(b,d,f) show the ratio of the measured relative to the calculated dose profiles. The dose profiles calculated by the motion model agreed within 3% with the corresponding measured dose profiles for the different open fields and ROM’s in the flat dose region within 80% isodose lines as shown in Figs. 5(b,d,and f). In the penumbra and dose spread-out regions, the calculated and measured dose profiles agreed within 10% which was due to high dose gradients and limited spatial resolution of MapCheck2. Larger deviations between the measured and calculated dose profiles were obtained outside the penumbra in the low dose region from scattered radiation. This resulted from lack of accuracy of dose measured by MapCheck2 due to increased signal noise and spatial resolution of MapCheck2.

Fig.5

Comparison between the measured and calculated dose profiles along the direction of motion (Y-axis) for the different field sizes (a) 5×5 cm², (c) 10×10 cm², and (d) 20×20 cm² with different motion amplitudes as indicated. (b,d,f) show the ratio of the measured relative to the calculated dose profiles for the corresponding fields in (a,c,e) and different ROM.

3.4 Variation of dose distributions with motion frequency and phase

Figures 6(a-d) show a comparison of the dose profiles measured with mobile MapCheck2 phantom for the open fields of 5×5, 2×2, 1×1 and 0.5×0.5 cm², respectively, using motion frequencies of 10, 15, 20 and 30 CPM. The measured dose profiles did not change with variations in motion frequency and phase of the mobile MapCheck2 phantom for a ROM = 10 mm. This agreed with the predictions of the motion model (equation (5)) that indicated that for a certain motion amplitude and lower dose rate where the dose was delivered over several respiratory cycles, the dose profiles were independent of motion frequency and phase. The deviation in the dose profile for the small fields of 1×1 cm and 0.5×0.5 cm² and in the high dose gradient regions were due to limitations in the spatial resolution and positioning accuracy of the mobile MapCheck2 phantom.

Fig.6

Dose profiles (a-d) are measured with mobile MapCheck2 phantom for the open fields of 5×5, 2×2, 1×1 and 0.5×0.5 cm², respectively, using motion frequencies of 10, 15, 20 and 30 CPM (cycles-per-minute) and a ROM = 10 mm.

3.5 Simulation of the dose distribution spread-out

Figures 7(a-c) show simulations of the spread-out of the dose distributions for three open beams (10×10 mm², 20×20 mm² and 40×40 mm²) deposited in MapCheck2 that was moving with different speeds along the Y-axis. The length of the dose distributions increased linearly with the motion speed as predicted by equation (1) which led to spilling of dose to regions beyond the PTV compromising normal tissue sparing. The dose level deposited in the mobile detector system decreases inversely with the speed of the target as given by equation (2) which led to compromise of the dose required to achieve control of the tumor. This simulation demonstrates the variations in the dose distributions due to motion when dose was delivered over a short period of time much smaller than the respiratory period using high dose rates where an average constant speed could be assumed for the moving target. The speed of the target in this case depended on the motion frequency and phase at the time of dose delivery.

Fig.7

Simulation of the spread-out of the dose distributions for three open beams (a) 10×10 mm², (b) 20×20 mm², and (c) 40×40 mm² deposited in a multiple-diode-array detector system (MapCheck2) moving with different constant speeds along the Y-axis.

Figures 8(a-c) show simulation of the dose distributions for three open beams (10×10 mm², 20×20 mm² and 40×40 mm²) deposited in MapCheck2 that was moving with a sinusoidal pattern along the Y-axis. The spread-out of the dose distributions increased linearly with motion amplitude such as obtained by equation (5) where it elongated by the range of motion (ROM = 2A). The dose level deposited in the mobile detector system decreased with the increase in the motion amplitude of the detector system as given by equation (7). If the ROM was smaller than the length of the beam (L) along the direction of motion, then the central dose level was equal to that of the stationary detector. However, when the ROM motion was larger than L then the central axis dose decreased and showed a dip in the middle with maximal dose deposited outside the region covered by the stationary beam as shown in Fig. 8.

Fig.8

Simulations of the dose distributions from three open beams (a) 10×10 mm², (b) 20×20 mm², and (c) 40×40 mm² deposited in the multiple-diode array detector system (MapCheck2) moving with sinusoidal motion with different motion amplitudes along the Y-axis.

4 Discussion

The motion model developed in this work predicts accurately the spread-out of dose distributions measured with MapCheck2 mounted on a mobile phantom. The dose profiles calculated with the model were verified with measurements for simple open fields and complicated intensity-modulated fields used in the treatment of lung and head and neck tumors. One characteristic of the dose distributions in open fields is that the central axis dose and the dose at 50% isodose line in the penumbra regions do not change from the static dose distributions as long as the ROM is smaller than the length of the open field along the motion direction. However, in the region between the central-axis and the 50% isodose line, the dose levels decrease with increase in the ROM which causes lack of dose coverage for the planning target volume. Beyond the 50% isodose line in the penumbra and umbra regions, the dose level increases with increase in the ROM of the phantom which leads to dose hot spots in the surrounding normal tissue. Motion frequency and phase do not affect the dose distributions where several motion cycles are required to deliver the total dose or low dose rates are used. However, motion frequency and phase become important parameters that affect the dose distributions where low dose levels or high dose rates are used with the total dose delivered over a time interval that is shorter than one period of the sinusoidal motion.

The spread-out of the dose distributions from IMRT-fields is dependent mainly on motion amplitude similar to open fields as explained previously. However, the locations of hot and cold spots change significantly with motion because the beamlets in an IMRT-field are delivered with different dose rates. When the total dose from an IMRT field is delivered over few respiratory cycles, the dose profiles do not vary with motion frequency and phase. However, with high dose rates where the total dose from an intensity-modulated beam is delivered in a sub-cycle of motion, then the dose painting varied with motion’s frequency and phase. Although, an analytical formula for the spread-out of the dose distributions is derived that considers one-dimensional cyclic motion of the phantom, its application can be extended generally to complicated tumor motion patterns in patients. For example, simulations of the spread-out of dose distributions deposited in mobile targets moving with constant speed and cyclic motion patterns are shown in Figs. 7 and 8, respectively. Constant speed motion can be used to represent tumor shifts due to patient relaxation or variation in the filling of certain organs such as rectum or bladder. Cyclic motion represents an approximation of respiratory motion in regularly breathing patients. In addition, the largest dose discrepancies in dose delivery to mobile tumors results from breathing motion which mostly moves tumors in a certain direction back and forth that can be approximated as one-dimensional motion as simulated in the model introduced in this study. One interesting clinical application of the motion model and data analysis developed in this work is that the results of the dose spread-out by different motion patterns obtained from the dose profile simulation can be used to determine the margins needed for treatment planning. These margins need to cover the range of motion of the tumor which depends directly on the motion amplitude of the mobile tumor that is specific for each patient.

5 Conclusion

Motion causes considerable variations in the delivered dose distributions deposited in mobile tumors treated both with conformal or intensity-modulated fields. The motion artifacts are pronounced for large tumor motion where the motion amplitude is comparable to the length of tumor and margins used for the planning-target-volume along the direction of motion. The spread-out of the dose distributions and discrepancies with initial dose planned on stationary tumor targets increases with the ROM of cyclic motions. Maximal and minimal dose variations, spatial position of maximal dose and dose drop rate in the dose distributions of the mobile phantom are simulated using the motion model introduced in this work. The modeling introduced in this study provides quantitative characterization of dose distribution variations induced by motion which might be employed in radiation therapy to determine margins used for the ITV considering dose spillage to normal tissues.

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