Dose properties of a laser accelerated electron beam and. prospects for clinical application

Transcription

1 Dose properties of a laser accelerated electron beam and prospects for clinical application K. K. Kainz a), K. R. Hogstrom, J. A. Antolak, P. R. Almond, and C. D. Bloch, Department of Radiation Physics, The University of Texas M. D. Anderson Cancer Center, Houston, Texas C. Chiu, M. Fomytskyi, F. Raischel +, M. Downer, and T. Tajima ++, Department of Physics, The University of Texas at Austin, Austin, Texas a) Correspondence to: Kristofer K. Kainz, Ph.D., Department of Radiation Physics, Unit 0094, The University of Texas M.D. Anderson Cancer Center, Houston, TX Phone No.: (713) Fax No.: (713) kkainz@mdanderson.org. + Present address: Institut fur Theoretische Physik, Julius-Maximilians-Universitat at Wurzburg, Am Hubland, Wurzburg, Germany ++ Present address: Kansai Research Establishment, JAERI, 8-1 Umemidai, Kizu, Kyoto, , Japan (Received )

2 Abstract: Laser Wakefield Acceleration (LWFA) technology has evolved such that it now holds the potential as an alternative to existing technology that produces therapeutic electron and x-ray beams. The purpose of the present work is to investigate the dosimetric properties of an electron beam that should be achievable using existing LWFA technology. This paper first qualitatively reviews the fundamental principles of LWFA and describes a potential design for a 30-cm accelerator chamber containing a gas target. The energy spectrum of the electron beam from this potential LWFA system was calculated from one-dimensional particle-incell (1D PIC) simulations; these simulations showed the dependence of the energy distribution upon the plasma wavelength (λ p ). We studied only the high-energy electrons to determine their potential for clinical electron beams of central energy from 9 21 MeV. Each electron beam was broadened and flattened by designing a dual scattering foil system to produce a uniform beam (103% > off-axis ratio > 95%) over a cm 2 field. An energy window ( E) ranging from MeV was selected to study central-axis depth dose, beam flatness, and dose rate. Dose was calculated in water at a 100-cm source-to-surface distance using the EGS4/BEAM Monte Carlo algorithm. According to the 1D PIC simulations, only approximately % of the LWFA electrons are emitted with an energy greater than 1 MeV. As λ p decreased from µm, the maximum electron energy decreased from approximately MeV, the peak in the electron spectrum shifted to lower energies, and the number of electrons produced in the range of 9 21 MeV was as much as 10 times greater. Dose calculations showed that the beam flatness was fairly insensitive to E, and that a E as great as 2.5 MeV provided increased dose rates with little increase to the falloff of the depth-dose curve (R 10 R 90 ). As E increased above 2.5 MeV, dose rate increased; however, R 10 R 90 increased as well, resulting in a tradeoff between depth-dose curve falloff and dose rate. If a value of R 10 R 90 that is 0.5 cm above its minimum ( E = 0.5 MeV) is acceptable, the maximum practical dose rates are approximately 0.6 Gy min 1 for a 9-MeV beam and 0.2 Gy min 1 for a 21-MeV beam. Current LWFA technology should allow a table-top terawatt (T 3 ) laser to produce therapeutic electron beams that have acceptable flatness, penetration, and falloff of depth dose, although the dose rate is still below that which would be acceptable. Future progress in laser technology, e.g., 2

3 increasing the beam repetition rate from 10 to 1000 pulses s 1, should give acceptable dose rates for electron beams and possibly for LWFA-produced x-ray beams. Measurements confirming proof of principle and numerous engineering developments are needed for this technology to advance and to compete with existing electron accelerators. Keywords: electron radiotherapy, laser-electron acceleration, LWFA, dosimetry 3

4 I. INTRODUCTION In 1979, Tajima and Dawson 1 originally proposed a method by which the wake generated in a plasma by high-intensity laser pulses could accelerate charged particles to ultrarelativistic energies. This process, which has since been established experimentally, is referred to as Laser Wakefield Acceleration (LWFA). While the original theoretical and experimental efforts regarding the acceleration of charged particles using lasers have concentrated on electron beam formation, the acceleration of protons 2 4 and heavy ions 5 is also feasible. LWFA shows the promise as a method to perform the same applications that conventional linear accelerators (linacs) do today, including radiation therapy. The fundamental requirements of an accelerator system for electron radiotherapy are that it be able to produce beam energies in the range of approximately 6 20 MeV and deliver dose at a rate of at least 2 Gy min 1 to be effective. In other words, it should be able to reproduce the output from today s conventional linacs. The primary goal of the study performed by the group at The University of Texas M. D. Anderson Cancer Center is to determine whether a laser accelerator system can meet the above conditions for direct electron and x-ray therapies. Thus, a detailed analysis of the dose capabilities of this system, for energies in the range E (6, 20) MeV, is required. Part of this study includes calculations for the rate of total absorbed dose and the dependence of dose versus depth. The present investigation is based on particle-in-cell simulation work, conducted by the 4

5 team at The University of Texas at Austin (UTA), which predicts the phase space of the electron beam from LWFA. 6 An overview of LWFA physics and an introduction to some of the key hardware elements of a laser accelerator system are given in Section II. Section III discusses the conditions for which the particle-in-cell simulations were conducted, and describes how a therapeutic beam might be produced from the LWFA beam. This includes energywindow selection and beam broadening. There is then a description of the methods used by investigators at M. D. Anderson Cancer Center to design the dual foil scattering system and to calculate key properties of the dose distribution (e.g. off-axis profiles, central-axis percent depth dose, and dose rate). Section IV presents the results for the angular and energy distributions of these simulated electron beams, along with calculations of dosimetric properties of the therapeutic beams. The prospects for a clinical electron or x-ray therapy device, given these results, are discussed in Section V, along with a proposed experimental program related to this LWFA system. II. OVERVIEW OF LASER WAKEFIELD ACCELERATION FOR ELECTRON BEAMS II. A. LWFA physics A comprehensive overview of the LWFA and other plasma-based accelerator concepts has been given by Esarey et al. 7 Here, we briefly summarize the main features of the 5

6 LWFA. Fig. 1 illustrates the basic interactions between a laser pulse and the plasma that the pulse creates. When a laser pulse of sufficiently high intensity travels through a neutral gas, the leading edge of the pulse ionizes the gas. Because this ionization process occurs rapidly, the remainder of the laser pulse does not interact with neutral particles, but rather with a plasma. The remainder of the laser pulse, in propagating through the plasma, applies a force upon the electrons that is proportional to the gradient of the pulse intensity. This force, referred to as the ponderomotive force, has a significant component in the direction of propagation of the laser pulse. Thus, the electrons in the plasma will be longitudinally displaced relative to the heavier positive ions. This space charge displacement pulls the electrons back and forth, setting up a longitudinally oscillating plasma wave. The oscillation frequency ω p of the plasma wave, in a plasma with electron density n e, is as follows 8 (given in meter-kilogram-second units): ω p nq ε m 2 = e e, (1) 0 e where m e is the electron mass, q e is the electron charge, and ε 0 = C 2 N 1 m 2 is the permittivity of free space. The phase velocity v p of the plasma wave matches the group velocity v g EM of the laser pulse: ω p EM v p = c 1 = vg < 2 ω 2 c, (2) where ω is the oscillation frequency of the photons in the laser pulse. 6

7 Under certain circumstances, the travelling plasma wave, which is oscillating longitudinally, will trap and accelerate charged particles, such as electrons, to velocities that match and then exceed that of the travelling wave. The charged particles to be accelerated could be injected into the plasma wave from an external source. If the plasma wave amplitude is sufficiently high, they could also be drawn from the thermal background charged particles that are trapped in the plasma wave. For high plasma wave amplitude, there could also be a supply of charged particles due to so-called wave breaking phenomena. To bring about plasma wave formation using a single laser source, a high-intensity pulse (on the order of W cm 2 or higher) is necessary. In the standard LWFA scheme, a pulse length on the order of the plasma wavelength λ p provides the most efficient wakefield excitation: L pulse 2π vp λp =. (3) ω p An optimum resonance condition occurs when the plasma-wave oscillation period τ p is twice the temporal width τ of the laser pulse (or, equivalently, when the plasma wavelength λ p is approximately twice the laser pulse length L pulse ). 7 For a hydrogen or helium gas at standard temperature and pressure, the electron density n e is about cm 3 and ω p is about s 1, making τ = 2 π / ω about 14 fs. As the widths of pulses typically generated by terawatt laser systems are on the order of 100 fs, the optimum resonance condition would be difficult to meet unless the plasma density was p p 7

8 below that for gas at STP. Thus, lower-density plasmas on the order n e ~ cm 3 are used for this standard LWFA scheme. Fig. 2 illustrates an inelastic scattering mechanism called Raman scattering, which is used for generating plasma waves at a much higher plasma density, when L pulse >> λ p. When Raman scattering occurs in atoms or molecules, incident photons at ω 0 excite a bound level of intermediate frequency ω 1 < ω 0, such as a molecular vibration, then emerge with reduced frequency ω scatter = ω 0 ω 1 (see [A]). When Raman scattering occurs in plasmas, incident photons at ω 0 excite a plasma wave at ω p, and emerge at the Stokesshifted frequency ω scatter = ω 0 ω p. The scattered light can emerge in any direction, but forward Raman scattering 9 is of greatest interest for acceleration because it excites a plasma wave that travels at nearly the speed of light. In this case, the Raman-scattered light co-propagates and beats with the incident light, as shown in [B] and [C]. The beating modulates the incident light pulse, breaking it up into a train of shorter pulses, as shown in [D]. Because the beat frequency is at ω p, the pulse train reinforces the growth of the plasma wave at ω p, which in turn deepens the modulation of the incident pulse. In the simple one-dimensional (1D) limit shown in Fig. 2, this positive feedback mechanism is often called the forward Raman instability. 9 When its 2D or 3D aspects are included, it is called the self-modulation instability. 10 The use of this laser-plasma instability to create an accelerating structure for charged particles has come to be known as the selfmodulated LWFA (SM-LWFA). 11 8

9 Of the various schemes for accelerating charged particles via laser-plasma interactions, the SM-LWFA so far yielded electron bunches of the highest energy, charge (> 1 nc per bunch), and collimation (transverse emittance ε < 0.1π mm mrad). In an experiment conducted at the University of Michigan, the generation of electron beams with energy E up to 20 MeV from a plasma of density n e = cm 3 was reported. 12 Another group at the Naval Research Laboratory reported an electron beam with energy E max = 30 MeV from a plasma of density n e = cm A maximum electron beam energy E max = 90 MeV, achieved using a plasma density n e = cm 3, was reported by a collaboration at the Rutherford Appleton Laboratory. 14 Leemans et al. 15 at the Lawrence Berkeley Laboratory have demonstrated SM-LWFA that produces nc bunches with E max = MeV at repetition rates as high as 10 Hz. This represents about the upper limit of repetition rate achievable with the SM-LWFA with current laser technology because of the high peak power required to drive the modulation instability. Clearly, laser-plasma acceleration can produce electron beam energies and charge per bunch adequate for therapy, although repetition rate (and therefore average current) needs to be improved. Recent simulations 16, 17 have suggested that the pulse energy needed to drive the forward Raman scattering process can be reduced significantly by the method of Raman seeding, in which the primary pulse co-propagates into the plasma with a much lower intensity secondary pulse that is offset in frequency by ω p from the primary pulse. The Raman-seeded LWFA superficially resembles another laser-plasma acceleration scheme, the plasma beat-wave accelerator (PBWA), 18 in which the primary and Raman-shifted secondary pulses are comparable in energy. However, according to simulations, it 9

10 requires less laser energy, is more compatible with plasmas of nonuniform density, and can generate multiple MeV electrons as efficiently as the SM-LWFA. Although not yet demonstrated in the laboratory, the Raman-seeded LWFA is a promising approach for increasing the repetition rate produced by laser-plasma accelerators. II. B. LWFA accelerator design In this section, we describe the key hardware elements that comprise a prospective electron LWFA system. Included are the laser and pulse amplification systems that are common to most LWFA experiments as well as a proposed option for a plasma-gas 6, 21 cylinder. High intensities are achieved by amplifying the laser pulse and focusing the laser pulse within the gas target. The laser source must be capable of generating pulses with intensity (irradiance) on the order of W cm 2. The components of such a table-top terawatt (T 3 ) system include a pump laser (such as a Nd:YAG laser) to provide an initial laser pulse and a series of typically 3 or 4 amplifiers (such as Ti:sapphire crystals) to provide the additional gain in pulse energy. The output terawatt laser pulse typically has a pulse length of fs and a wavelength of µm. To achieve the required pulse intensity without damaging the amplifier elements within the T 3 laser system, a technique known as chirped pulse amplification (CPA) is used. 19 The CPA scheme is illustrated in Fig. 3. First, an optical configuration consisting of a 10

11 pair of diffraction gratings and a telescope is used to broaden the pulse, in time, into its constituent frequencies. This stretched pulse is amplified, and the amplified stretched pulse is then cast upon another pair of diffraction gratings that recompresses the pulse in time. Initial experimental tests of CPA in increased the intensity of laser pulses by a factor of 10 6 ; since then, gain factors of have been achieved. 20 CPA pulses with energy approaching 1 joule and repetition rates as high as 10 Hz have been used for all SM-LWFA experiments to date. Details of an experiment in progress to implement the Raman-seeded LWFA have been illustrated by Downer et al. 21 The scheme being investigated generates a Raman-shifted seed pulse 1% as intense as the primary pulse in a barium nitrate crystal internal to a CPA system. Simulations suggest that this has the potential for acceleration equivalent to the SM-LWFA with lower pulse energy and higher repetition rate. A typical accelerator chamber consists of an evacuated vessel enclosing a gas jet or cell 1 2 mm thick and focusing mirrors, as shown in Fig. 4. A curved mirror on the input side focuses the incident laser pulse, or laser pulse combination, typically at f/35 to a spot radius on the order of 18 µm. To reach a peak intensity on the order of W/cm 2 needed for LWFA, the incident pulse should therefore have a peak power of 1 TW, e.g., an energy of 0.1 J for a 100 fs pulse. Somewhat higher intensity and energy is typically needed for SM-LWFA, and potentially somewhat less for the Raman-seeded scheme. At 10 Hz repetition rate, the average power of this beam is on the order of 1 W. The laser spot on the incident mirror should be 0.46 cm in diameter to keep the fluence below a 11

12 safe damage threshold of approximately 1 J/cm 2 for dielectric mirrors. Thus, a typical focal length of about 15 cm, or total chamber length of about 30 cm, is needed. The cylinder would be evacuated except for the gas cell. This is necessary because the pulse would not focus well if the entire cylinder were filled with the plasma gas. Small orifices would be placed on each side of the cell, along the chamber axis. This is required because the laser pulses would eventually damage the gas-cell walls. Because some of the gas within the cell will stream out of the orifices, the flow of gas into the cell must be maintained. The pressure difference between the gas cell and the vacuum would be maintained using differential pumping. To produce the plasma wave in this accelerator via the seeded-pulse scheme, the helium gas in the plasma cell would have a density of at least n e = cm 3, which is slightly above the STP density of cm 3. Using a higher-density plasma gas leads to a larger number of thermal electrons that are trapped and accelerated. This in turn allows a sufficiently large dose delivery without the need for an injected external beam. A low-z gas such as hydrogen or helium is used because a higher-z gas attenuates the laser pulse and increases the distortion of the pulse within the plasma, thus inhibiting the ability to focus the pulse. 12

13 III. METHODOLOGY III. A. Particle-in-cell simulations of LWFA Given the promise of medical application for LWFA, the UTA investigators conducted particle-in-cell (PIC) simulations of a laser accelerator apparatus similar to those described above that might fulfill the requirements for direct electron beam therapy. The output electron phase space distributions that are predicted by these simulations are used to model a therapeutic electron beam and to assess the dosimetric properties of such a beam. An iteration of the PIC-simulation algorithm proceeds as follows. Temporal and spatial grids are defined over the region of interaction between a collection of charged particles and the electric and magnetic fields with which they interact. These electric and magnetic fields arise internally from interactions among the charged particles and externally from the incident laser pulse. From the positions and velocities of the charged particles at a given moment in time, the charge density and current density at each spatial grid point is obtained. Maxwell s equations are then used to determine, from these charge and current densities, the electric and magnetic fields, and thus the Lorentz force, at the grid points. Incorporating these Lorentz forces into equations of motion, and then integrating these equations, gives the updated positions and velocities of the charges (i.e., the updated charge and current densities) to use in the next iteration. After a sufficient 13

14 number of iterations, the phase space distribution of an output electron beam is determined. 23 The UTA PIC simulations modeled the evolution of a plasma generated by a seeded laser pulse described in Section II.B. The UTA group had optimized several key parameters of the laser-plasma system, such as primary pulse and seed pulse intensity and plasma density. The guiding principle of this optimization is to maximize the electron current for therapeutic electron energies (6 20 MeV) while keeping the laser intensity to a minimum. Although adjusting the pulse intensity should increase the current (and thus the dose rate) from the system, keeping the intensity as low as possible yields reductions in the size, complexity, and operating cost of the LWFA system. A list of the key parameters for the proposed accelerator system is given in Table I. Among these parameters are the density and dimensions of the plasma cell, as well as the power, size, and repetition rate of the oscillation-generating laser pulse. The sets of simulation results presented below, from which the dose-related results in Section IV are derived, were obtained using 1D PIC simulations. Eventually, a full assessment of LWFA-generated electron-beam dose properties must be performed on 2D PIC-simulation results, so as to make sound quantitative judgments in the laser-plasma parameter range of practical interest. However, use of the more mature 1D PIC results can, in the meantime, give a suitable qualitative understanding of the effects resulting from changes in the laser-plasma parameters in the simulation. 14

15 The electron beams from the simulation were obtained for 3 different plasma densities, corresponding to A, B, and C in Table I. Increasing the plasma density n e yields more background electrons that can be trapped within the plasma wave to comprise an eventual beam. However, increasing n e can also attenuate the incident laser pulse, thereby reducing the amplitude of the plasma wave and thus the maximum-achievable beam energy. The maximum energies of accelerated electrons, for the 3 different plasma densities considered, are also shown in Table I. In the PIC simulations, changes in plasma density were made by adjusting the plasma frequency (or, equivalently, the plasma wavelength). For each of the plasma density settings, Table I also lists the quantity N total per pulse. This is the total number of electrons irradiated by the pulse at the focal point and along an effective interaction length of 184 µm, and is thus the number of electrons that could potentially comprise the output beam following each laser pulse. The quantity N E > 1 MeV is the number of electrons accelerated per pulse with energy E > 1 MeV. Of the total number of plasma-cell electrons irradiated by a laser pulse, only about % of them are accelerated to energies suitable for therapy. III. B. Producing a therapeutic beam from simulated data As will be observed from the UTA 1D PIC simulation results presented in Section IV, the energy distributions for LWFA electron beams are quite broad, on the order of tens of MeV. Thus, tasks to perform within the M. D. Anderson Cancer Center analyses of the UTA PIC simulation output include the selection of subsets of the output beam that have 15

16 narrower energy spreads, and for each subset, the design of hardware to broaden this beam into the desired field area. The beam exiting a conventional linac has a diameter on the order of 1 mm, which must be transformed into a beam with a uniform region of at least cm 2 to treat the patient. Two standard techniques are used to broaden beams from conventional linacs. In one method, the beam is scanned across the treatment field using orthogonal, timevarying magnetic fields 24 ; in the other method, a dual scattering foil system is utilized. The former method has become less popular because of the risk of a malfunction of the scanning magnet that can be hazardous to the patient. 25 Broadening the beam using a dual scattering foil system is more widely used; therefore, we will restrict our analyses to that method for beam broadening. In the dual scattering foil system, a uniformly thick primary foil is used to provide an initial Gaussian spread for the beam profile. A Gaussian-shaped secondary foil is then used to scatter the central part of the primary-foil output farther from the beam axis. With appropriate foil dimensions, the resulting beam profile at the phantom surface is reasonably flat over the treatment field area, and falls off outside the edges of the field area. To select narrow ranges in energy from the raw beam distribution, one can implement an achromatic bending magnet to spatially separate the beam s component energies and a variable collimating slit that transmits only those electrons within the selected energy range. At this stage in the M. D. Anderson studies, details of energy window selection are not being modeled, but this will eventually be done using a particle transport code. It 16

17 may also be possible to perform some degree of beam energy selection without a magnet and collimators. High-energy components might be removed from the output beam by appropriately adjusting the plasma density. Also, some portion of the low-energy component might be scattered out of the treatment field by the flattening foils. Raischel 17 has investigated an alternative method that transports the broad beam within a defined energy spread through achromatic magnet systems. His results produced beams too narrow for most clinical situations, but which might be useful in abutting multiple fields. This offers the potential for intensity and energy-modulated electron therapy. 26 III. C. Dose simulation and calculation From the electron-beam phase-space distributions predicted by the UTA PIC simulations, various subsets of the beam energy distribution were used to assess the following attributes of an eventual therapeutic beam: (1) beam flatness at the phantom surface; (2) percent depth dose; and (3) peak dose rate. Calculating these three quantities was done using the EGS4/BEAM Monte Carlo program, 27, 28 which models the transport of electrons and photons through a given beamline geometry. The user provides the composition and dimensions of the beamline elements (for these analyses, the scattering foils and the water phantom), as well as the position, momentum, and energy spectrum of the incident electron beam (in this case, a subset of the output beam generated by the UTA PIC simulations). 17

18 III. C. 1. Beam Flatness Prior to running the BEAM simulations, the appropriate dimensions and material composition for the scattering foils were determined for each canonical beam energy under consideration. For a given beam energy, the dimensions of the foils were determined by calculating the phantom-surface planar fluence for a given foil configuration, and adjusting the foil dimensions within the calculation until the desired shape for the planar fluence distribution is achieved. 29 The initial design parameters for the foil system presumed a smooth-gaussian shape for the secondary foil. A corresponding stepped-foil approximation to the Gaussian shape was then obtained for the secondary foil; this consisted of 3 beveled-edged disks of equal thickness stacked on top of each other. This was done to more easily represent the secondary foil geometry within the EGS4 simulation. The separation along the beam axis between the foils was set to 10 cm, and the distance between the primary-foil plane and the phantom surface was taken to be 100 cm (standard source-to-axis distance SAD). The foils were designed to scatter the beam into a circular treatment field of radius cm, to achieve flatness within a cm 2 square field. At the edges of the circular field, the goal for the relative planar fluence of the scattered beam was 95% of the central-axis planar fluence, with no value greater than 103% of the central-axis planar fluence. The materials used for the scattering foils were gold for the primary and aluminum for the secondary. 18

19 III. C. 2. Percent depth dose Having determined an acceptable dual scattering foil system, EGS4 was then used to propagate subsets of the LWFA electron-beam energy spectrum through the foils and onto a water phantom. BEAM 27 was used to model the beamline and a water phantom. The CHAMBER component module was used to model a water phantom and to score the central-axis depth dose. The following parameters were set within BEAM to halt further transport of electrons and photons whose kinetic energy fell below 10 kev: AE = MeV; ECUT = MeV; AP = MeV; and PCUT = MeV. The following quantities were extracted from each of the depth-dose curves generated using EGS4/BEAM. The therapeutic depth R 90 is the penetration depth along the beam axis at which the distal end of the 90% relative-dose contour resides. R 10 is the depth where the central-axis relative dose is reduced to 10%. The difference R 10 R 90 roughly characterizes the minimum distance between the target volume and critical structures for which the target receives the full prescribed dose and the critical structures receive minimal dose. A larger R 10 R 90 indicates a higher unwanted dose to regions beyond R 90. The depth-dose curves from EGS4 also indicate the bremsstrahlung dose expected from these dual-foil beam-scattering systems. 19

20 III. C. 3. Dose rate A determination of the rate at which dose may be absorbed from a LWFA-generated therapy electron beam is also of interest, given that such a device should be capable of delivering dose at a rate of at least 2 Gy min 1. EGS4 Monte Carlo simulations were used to determine the dose rate from a realistic beam i.e., one that emerges from the dual scattering foil system such that the electrons may lose energy and/or be scattered outside the treatment field. Of interest is the dose rate evaluated at R 100, the point on the central axis where the maximum dose is deposited, and this was calculated in the following manner. First, the depth-dose curves obtained via EGS4/BEAM, in the manner described above and for the same canonical values for E cent and E, were output in units of dose per incident electron. (This is in fact the default output format for central-axis dose results from a BEAM simulation.) This quantity was then multiplied by the number of electrons per minute (assuming a 10-Hz pulse repetition rate) predicted by the UTA PIC simulation for the selected E cent and E values. These rates of electron production were obtained for subsets of the electron energy distributions for λ p = 5.8, 5.2, and 5.0 µm. This prescription may be summarized by the equation: dd dt dose per incident electron = at R on central axis. (4) number of electrons per 10 pulses pulse within ( Ecent, E) second R from EGS4/BEAM from UTA PIC 20

21 The objective of these calculations of absorbed dose rate is to study the dependence of ( dd / dt ) R100 on the central beam energy E cent and the energy spread E for beams predicted using the UTA system with the current 1D PIC simulation parameters. IV. RESULTS AND DISCUSSION IV. A. Phase-space properties of LWFA electron beams Before presenting the energy distributions for the E > 1-MeV electrons (that comprise % of the output beam) predicted by the UTA PIC simulations, some comments regarding the beam angle distribution from 1D PIC and 2D PIC should be made. The polar angle θ is defined by Fig. 5 as the angle between the momentum vector of the electron and the z-axis; the azimuthal angle φ denotes the direction of the momentum s transverse component in the plane perpendicular to the z-axis. The z-axis is defined by the direction of the incident laser pulse, and coincides with the axis of the accelerator cylinder. One-dimensional PIC can predict the trends of the electron energy distribution reasonably well, since 1D PIC models the key effects of forward Raman scattering and the trapping and acceleration of electrons within the wakefield. However, only 2D PIC can model the transverse interactions that influence the wakefield amplitude, accelerated electron yield, and electron beam angle distribution. Thus, the 1D PIC predicts much narrower θ distributions (i.e., FWHM less than 0.2 degrees deviation from the z-axis) than what has 21

22 been observed in previous experiments (e.g. FWHM ~ 5 degrees). 12, 15 The current approach for our subsequent analyses, then, is to restrict our use of the 1D PIC results to determine the effect of the central beam energy and beam energy spread upon the shape of the depth-dose curves and the magnitude of the dose rate at R 100. Fig. 6[A] shows the distribution of dn/de versus E for the E > 1-MeV electrons that were generated via UTA PIC simulations, with λ p = 5.8 µm. The energy distribution is quite broad; for a therapy device, a means to select a narrower subset of beam energies from this distribution will be required. Clearly, the UTA LWFA system is capable of generating copious numbers of electrons in the therapeutically useful range of E (6, 20) MeV. To smooth this distribution, in preparation for the calculations of central-axis depth-dose curves and total dose rate, a fit of a fourth-degree polynomial has been made. The fit result is indicated by the dotted lines in Fig. 6. For the λ p = 5.8 µm energy distribution (corresponding to a plasma electron density of cm 3 ), the peak of the electron distribution occurs at approximately 35 MeV. Beam energies in excess of about 20 MeV are usually too high for conventional electron therapy because the steepness of decrease in dose with penetration depth is no longer sufficient. 30 However, the UTA group has demonstrated that increasing the plasma density n e (decreasing the plasma wavelength λ p in the PIC simulations) shifts the energy distribution toward values more favorable for conventional therapy. The plots in Fig. 6[B] and [C] show the energy distribution for λ p = 5.2 µm and λ p = 5.0 µm, respectively. More significantly, increasing the plasma gas density increases the yield of therapeutic 22

23 electrons per pulse. For an eventual LWFA accelerator, perhaps the gas density could be adjusted to maximize the production of electrons for a specific central energy. Fig. 6 shows the energy distributions that are utilized in the dose calculations presented below. These figures indicate a broad energy distribution for the output beam. The dose calculations to follow will presume the ability to select narrower portions of this distribution for therapy. Energy spreads ( E) of 0.5, 2.5, 4.5, and 6.5 MeV will be examined; these spreads will be centered about the nominal central beam energies E cent under evaluation: 9, 12, 15, 18, and 21 MeV. IV. B. Dosimetric properties of LWFA beams IV. B. 1. Flatness A dual-foil system was designed for each of the canonical central energies E cent. The foil parameters were optimized for a mono-energetic beam ( E = 0), and then the same foil geometry was used to simulate the scatter of beams with a nonzero E centered about E cent. The resulting dimensions of the primary and secondary foils, for each beam energy, are summarized in Table II. For each foil configuration, the effective Gaussian width of the 3-layer secondary foil was (0.95) (1.4) cm. Before presenting results for the depth-dose curves from EGS4, we established that a beam profile distribution predicted by EGS4 agrees with the analytically calculated 23

24 profile distribution for the same scattering foil geometry. Fig. 7 compares the calculated profile distribution for a 9-MeV beam with the prediction from EGS4. The foil configuration was optimized to scatter a 9-MeV electron beam into a circular field of diameter 25 2 cm. Over the treatment field range (r cm), the calculation and the EGS4 prediction agree well. The dimensions of the scattering foils to broaden the electron distributions from 1D PIC were obtained using the assumption that the initial angular spread of the electrons is negligible, i.e. the experimentally-observed θ distribution of about 5 degree width was not taken into account 12. Given an initial Gaussian beam-angle distribution of the electrons from the LWFA device, the likely revision to the results that are shown below would be to simply reduce the thickness of the uniform primary scattering foil. Incorporating an initial angular spread into the foil-design algorithm will be a focus of future studies. It is necessary to evaluate the effect of increasing the beam energy width upon the flatness of the beam at the phantom surface. This effect has been estimated using the analytically calculated beam profile distributions, given that they are consistent with the predictions by EGS4 according to Fig. 7. Plots of the calculated profile distributions for beam energy widths up to 6.5 MeV are shown in Fig. 8 for [A] E cent = 9 MeV and [B] E cent = 21 MeV. At each energy, the same dual-foil geometry was used for each of the beam energy widths. The beam energy distributions in each case are subsets of the λ p = 5.8 µm energy distribution shown in Fig. 6[A]. For both the 9-MeV and 21-MeV 24

25 scattered beams, the beam profile changes minimally as the energy spread is increased. Thus, a broad central beam energy has only a small effect on the flatness of the beam. If unacceptable, the primary foil thickness could be re-optimized. The shoulders that appear in the vicinity of approximately 22 cm from the central axis in Fig. 8[B] are due to the radius of the bottom layer of the stepped-foil configuration being limited to 2.3 cm (projected to 23 cm at isocenter). IV. B. 2. Percent depth dose Subsets of the PIC energy spectra in Fig. 6 with E ranging from MeV served as the input energy spectra for EGS4/BEAM. Central-axis depth-dose curves calculated by EGS4/BEAM with the previously described scattering foils are shown in Figs. 9[A] and [C] for E cent = 9 MeV and Figs. 9[B] and [D] for E cent = 21 MeV. Figs. 9[A] and [B] used the energy spectrum generated for λ p = 5.0 µm, and Figs. 9[C] and [D] used the energy spectrum generated for λ p = 5.8 µm. The effect of increasing the width of the energy distribution is demonstrated in these plots. For the 9-MeV case, as the energy spread increases, R 90 decreases and R 10 increases. For the 21-MeV beam and λ p = 5.8 µm, as the energy spread increases, R 90 is relatively constant while R 10 increases. For the 21-MeV beam and λ p = 5.0 µm, R 90 decreases and R 10 remains almost constant. This latter anomaly is due to the presence of few electrons with energy above 22 MeV generated in the PIC simulations (c.f. Fig. 6[C]). 25

26 Fig. 10 plots the quantity R 10 R 90 from the EGS4 depth-dose curves against R 90 for each of the central energies and energy spreads under consideration. In general, as the energy (and hence therapeutic depth) increases, so also does the separation between R 90 and R 10. At the lower values of R 90, corresponding to lower central beam energies, R 10 R 90 ranges from approximately cm. At the higher values of R 90, R 10 R 90 ranges from approximately cm. In Fig. 10[C], corresponding to λ p = 5.0 µm, one observes that the E = 6.5 MeV curve does not follow the trend set by the other curves at high R 90. Again, this is because there are relatively few electrons with energies above 22 MeV for the λ p = 5.0-µm beam energy distribution. IV. B. 3. Dose rate Fig. 11 shows a plot of the absorbed dose rate at R 100 versus energy spread for each of the central energies under consideration. As indicated in the energy distributions in Fig. 6, an increase in the plasma density (represented by reducing the plasma wavelength parameter in the PIC simulations) yields an increase in the rate of production of electrons with clinically useful energies. Thus, the absorbed dose rates shown in Fig. 11[A] for beams sampled from the λ p = 5.8 µm energy distribution are less than the dose rates shown in Fig. 11[B] and [C] for λ p = 5.2 µm and λ p = 5.0 µm, respectively. The dose rate increase with energy spread ( E) is approximately linear, with the exception of the 21-MeV data for λ p = 5.0 µm. Its nonlinear increase is again due to there being few electrons in the spectrum with energies greater than 22 MeV. 26

27 For λ p = 5.8 µm, the slope of dose rate varies little with E central, as its energy spectrum is relatively constant from 9 20 MeV. By contrast, for λ p = 5.0 and 5.2 µm, the slope of dose rate versus E is greater for the lower energy beams as their energy spectra curve decreases from 9 20 MeV. For a particular beam energy, there can be a significant range in dose rate as λ p and E vary. For example, at 9 MeV, the dose rate varies from approximately Gy min 1 (λ p = 5.8 µm, E = 0.5 MeV) to 0.95 Gy min 1 (λ p = 5.0 µm, E = 6.5 MeV). At 21 MeV, values range from approximately Gy min 1 (λ p = 5.8 µm, E = 0.5 MeV) to 0.20 Gy min 1 (λ p = 5.2 µm, E = 6.5 MeV). The greater dose rates require E = 6.5 MeV; however, this results in likely too great of a R 10 R 90 value. To better appreciate that relationship, Fig. 12 plots the dose rate versus R 10 R 90 for the 9- and 12-MeV beams (λ p = 5.0 µm) and for the 15-, 18-, and 21-MeV beams (λ p = 5.2 µm). If we were to restrict the increase in R 10 R 90 to 0.5 cm greater than its minimum value (at E = 0.5 MeV), then the maximum dose rates are approximately 0.6, 0.4, 0.3, 0.2, and 0.15 Gy min 1 for 9-, 12-, 15-, 18-, and 21-MeV beams, respectively. IV. C. Influence of 2D PIC upon dose properties Although progress on 2D PIC simulation of LWFA is beginning, results from initial 2D simulations might be compared with the 1D results above to estimate the beam flatness, depth dose, and dose rates we can expect once analysis of 2D data begins. Initial 2D PIC results have been obtained assuming a laser pulse power of 1.7 TW, a spot radius of 10 27

28 µm, and an electron density of cm 3 (corresponding to λ p = 4.8 µm). From this spectrum, the polar angle distribution appears to have a half width at half maximum of about 5 degrees. We anticipate that this divergence would be accommodated first by designing beam optics to focus the beam to a point at the location of the primary scattering foil. Second, it would most likely be necessary to decrease the thickness of the primary scattering foil derived earlier, to flatten the beam to the desired profile. The initial 2D PIC results also indicate that the falling edge of the energy distribution is located at about 16 MeV. This is similar to the general trend, observed within the λ p = 5.0 µm, 1D PIC results shown previously, i.e. decreasing maximum electron energy with increasing electron density. The shapes of the depth-dose curves for the 2D PIC results can be expected to change accordingly. Finally, initial 2D PIC results also indicate the number of beam electrons per pulse with E > 1 MeV to be about , for which the dose-rate calculations above could decrease by approximately a factor of 10. The reduced yield of electrons per pulse with E > 1 MeV is not inconsistent with the measured values of about 0.5 nc reported by Wagner et al. 12 V. CONCLUSIONS V. A. Electron beams According to the output from the 1D PIC simulations, the LWFA-generated electrons that are potentially clinically useful (i.e. with E > 1 MeV) exhibit energy spectra that are quite broad, on the order of tens of MeV. Whereas these spectra show the proposed LWFA 28

29 system to be capable of producing a clinically useful range of electron beam energies (6 20 MeV), the widths of the spectra require a means to select subsets with narrower beam energy widths. For example, a standard 270-degree achromatic magnet with collimating blocks inside could be used to select the range of energies for the therapy beam. The quality of the beam energy distribution can be improved by adjusting the plasma density. Increasing the plasma density (i.e., decreasing the plasma wavelength) can simultaneously reduce the peak of the electron energy distribution to a favorable level and increase the output electron beam current. Subsets of the beam energy distributions predicted by the 1D PIC simulations were implemented into EGS4, which propagated this distribution through a dual-foil beam scattering system and into a water phantom. Results from these simulations demonstrated the flatness of the beam, the shape of the central-axis depth-dose curve, and the absorbed dose rate at R 100 on the central axis. The dependence of these quantities on the central beam energy and beam energy width was investigated. The gradient of the depth-dose curves (R 10 R 90 ) increases as beam energy and beam energy width increase. From near-monochromatic beams ( E ~ 0.5 MeV) to beams with widths ~ 6.5 MeV, R 10 R 90 ranges from about 1.5 cm to about 5 cm. The absorbed dose rate tends to increase with increasing beam energy width, although the dose rate is below 1 Gy min 1 even for energy widths as great as 6.5 MeV. Shifting the central beam energy toward the peak of the distribution increases the dose rate; the direction of shift depends upon the shape of the distribution, which in turn depends upon the plasma density. The results for depth-dose shape and dose rate illustrate the trade-off involved when adjusting the beam 29

30 energy width; whereas greater widths result in high dose rates, smaller widths result in sharper falloff of the depth-dose curves. For the prospective electron LWFA device with the above-proposed operating parameters, the maximum practical dose rate (assuming a value of R 10 R 90 that is 0.5 cm above the value for a monoenergetic beam) ranges from approximately Gy min 1. To achieve dose rates of at least 2 Gy min 1, the beam current will need to be increased by a factor ranging from V. B. Potential for x-ray beams For LWFA to be useful in a clinical setting, it is likely that this technology must also be capable of producing suitable x-ray beams. Although not the subject of the present work, studies of x-ray beams produced using LWFA electrons are important and should be the focus of future studies. The present study showed that dose rates of about 2 Gy min 1 for electrons should be achievable with a modest increase in beam current ( times greater) using the current set of UTA 1D PIC simulation parameters. However, the beam currents will need to be an additional 2 3 orders of magnitude greater to achieve similar dose rates for 6 20 MV x-ray therapy. 24 A key shortcoming of LWFA at this stage is the low pulse repetition rate of the current terawatt laser systems. Increasing the pulse repetition rate would help to achieve the higher beam currents necessary for x-ray therapy. The damage thresholds of the optical 30

31 elements in the laser cavity restrict the pulse repetition rate to the 10-Hz value currently used. However, advances are being made that would considerably increase these damage thresholds. A recent development 31 suggests that pulse rates up to 1 khz should be achievable for pumped-laser systems. Additionally, one could envision adjusting some combination of the plasma density, the intensity gain of the laser pulse amplification system, and the diameter of the laser pulse at its focus within the plasma gas. Also, for x- ray beams, it should be possible to relax the requirement for a narrow spread in the energy of the electron beam upon the x-ray target, given that the energy distribution is already broad for x-rays utilized in conventional treatments. We are planning studies similar to the present one in the near future to evaluate whether LWFA can produce clinically acceptable x-ray beams as well. V. C. Future work In this work, our analysis is based on the UTA group s 1D PIC simulation results, and this approach is justified to some extent. Our analysis needs to be repeated once the 2D simulation results become available. Eventually, experimental verification is intended to be carried out by two experiments. One experiment would construct the hardware to generate the seeded laser pulse, and establish that the plasma wave amplitude and the subsequent trapping and acceleration of electrons within the plasma are indeed enhanced relative to the SM-LWFA regime. Also, it must be demonstrated that the accelerator cylinder and gas cell apparatus are viable. Measurements verifying our calculations of dose rate, depth-dose, and the ability to flatten the beam would follow. 31

32 Another measurement of considerable interest would be the relative biological effectiveness of the output beam from an experimental LWFA electron accelerator. This is because the instantaneous dose rate (the dose delivered per beam bunch or per pulse) from the LWFA system would be substantially higher than that from conventional radio frequency linacs. Using electron beams from conventional linacs, Meyn et al. 32 demonstrated a measurable decrease in the survival fraction versus dose for mouse jejunum crypt cells if the instantaneous dose rate was increased from about Gy min 1 to about an order of magnitude higher. Using the data from Fig. 6 and assuming a pulse repetition rate of 10 Hz and a pulse width of 100 fs (for the laser and the electron beam bunch), one could estimate the instantaneous dose rates from LWFA to be as great as Gy min 1. The relative biological effectiveness for such an electron beam relative to those from conventional linacs is unknown. Additionally, dosimetry of beams with such high instantaneous dose rates will be challenging, due to dose rate effects upon traditional dosimeters, particularly ionization chambers 33. The response from thermoluminescent dosimeters, film, or solid state dosimeters (such as diamond or silicon) may be less affected by such high instantaneous dose rates. Along with beam broadening, other design issues for an eventual clinical LWFA device must be examined. For example, it is necessary to adequately shield the substantial number of electrons (> 99.7 %) produced by LWFA that have energies below those 32

33 suitable for therapy. The heating of an eventual low-energy electron shield must be estimated along with any possible leakage dose from such a shield. 33

34 References: 1 T. Tajima and J. M. Dawson, Laser electron accelerator, Phys. Rev. Lett. 43, (1979). 2 E. L. Clark, K. Krushelnick, J. R. Davies, M. Zepf, M. Tatarkis, F. N. Beg, A. Machacek, P. A. Norreys, M. I. K. Santala, I. Watts, and A. E. Dangor, Measurements of energetic proton transport through magnetized plasma from intense laser interactions with solids, Phys. Rev. Lett. 84, (2000). 3 A. Maksimchuk, S. Gu, K. Flippo, D. Umstadter, and V. Yu. Bychenkov, Forward ion acceleration in thin films driven by a high-intensity laser, Phys. Rev. Lett. 84, (2000). 4 T. E. Cowan, M. Roth, J. Johnson, C. Brown, M. Cristl, W. Fountain, S. Hatchett, E. A. Henry, A. W. Hunt, M. H. Key, A. MacKinnon, T. Parnell, D. M. Pennington, M. D. Perry, T. W. Phillips, T. C. Sangster, M. Singh, R. Snavely, M. Stoyer, Y. Takahashi, S. C. Wilks, and K. Yasuike, Intense electron and proton beams from PetaWatt lasermatter interactions, Nucl. Instrum. Meth. A 455, (2000). 5 M. Hegelich, S. Karsch, G. Pretzler, D. Habs, K. Witte, W. Guenther, M. Allen, A. Blazevic, J. Fuchs, J. C. Gauthier, M. Geissel, P. Audebert, T. Cowan, and M. Roth, MeV ion jets from short-pulse-laser interaction with thin foils, Phys. Rev. Lett. 89, (2002). 34