Dose rate and accelerator types

The ability to provide a dose of 2 Gy in a volume of 1 liter is a common clinical criterion for proton therapy. This equates to 1.9 × 10¹¹ protons for a cube of 10 cm × 10 cm × 10 cm at a depth of 10–20 cm in water at 2 Gy. For the FLASH treatment, 4 Gy (3.8× 10¹¹ protons) delivered within approximately 100 ms corresponds to a current of 600 nA or an average dose rate of 40 Gy/s [26]. With current medical accelerator technology, only isochronous cyclotrons can generate such high continuous beam currents.

Cyclotrons are the most often used accelerators in proton therapy. A cyclotron generates a high-intensity continuous beam at a fixed energy. To obtain the required energy, it must use a degrader. The degrader adjusts the physical thickness or the inserted degrader material. If there is a need for a quick shift in accelerator energy, the degrader is required to change beam energy in several tens of millisecond. But shorter than 0.1 ms/energy step is a significant challenge. At the same time, when the beam travels through the degrader, scattering and ionization losses occur, resulting in an increase in the beam emittance and energy spread, especially at low energies [27, 28]. For FLASH therapy, 3D ridge filter is used to avoid energy change [26].

The Linac accelerator exhibits the advantages of a higher peak current and smaller beam emittance. New linac designs have a sufficiently short time to modify the beam energy from pulse to pulse [29-31]. When it comes to true, it may realize complete spot scanning in the clinic. The enormous size and expensive price of linac are also factors to consider [1, 5].

A slow-cycling synchrotron can preselect the extracted energy, eliminating the need for a degrader. Its whole cycle comprising of the injection, ramping up, extraction and subsequent return of the magnetic field, takes a few seconds. Considering this duration, the typical synchrotron usually operates at a low injection energy, resulting in a small space-charge limit. Generally, slow-cycling synchrotrons deliver a lower average and instantaneous dose to patients [4, 32].

Several studies have been conducted on the laser-plasma acceleration of proton beams [25, 33]. Researchers have frequently combined ultrathin targets with beam optics to realize the transmission of several Gys in nanoseconds. In terms of clinical applications, some challenges, such as the maximum energy and intensity of the proton beams, low repetition rate, quality assurance, patient safety, control, and repeatability of the laser pulse output, remain to be addressed. Although proton laser plasma acceleration, to date, is not available for therapeutic use, it provides a new way to achieve FLASH therapy. Inspired by laser plasma researchers, we can use a very high instantaneous dose rate with a low average dose rate to realize the FLASH effect.

The rapid-cycling synchrotron shares similar beam characteristics with the laser plasma accelerator, but it exhibits much higher beam energy and intensity. We plan to design an RCMS capable of providing 10¹⁰ protons/pulse at a repetition rate of 25 Hz. This equates to 2.5×10¹¹ protons/s, slightly more than 60 Gy in 1 min in one liter of volume, with the mean dose rate of approximately 1 Gy/s. But the length of a beam pulse is only a few tens nano-seconds. If we can change beam energy from pulse to pulse and for scattering method that means several Gys for a energy slice, the instantaneous dose rate is more than 10⁷ Gy/s. For scanning method, this means a spot requires a pulse, and the instantaneous dose rate is up to 10⁹ Gy/s, but the dose control is very difficult.

The lattice design

Taking into account practical considerations, such as cost and spatial limitations, the rapid cycling medical synchrotron has a lattice with a short circumference of 27.2 m, composed by four arcs and four straight sections. One straight section is used for the injection, one for the acceleration, and others for the extraction. Eight dipoles have the same length of 1.5 m and their maximum magnetic field strength are less than 1.3  T. The edge angles of dipoles are 11.25. Eight quadrupoles, focusing in the horizontal and vertical plane, have the same length. The normalized strength of defocusing quadrupoles is -2.2 and the normalized strength of focusing quadrupoles is 2. Considering sextupolar component induced by the eddy current, two sextupole families are used to fit the chromaticity whose strengths are -15.01 and -28.42. Figure 1 shows the layout of the lattice, where B(n) represents a dipole, QD(n) and QF(n) represent a defocusing quadrupole and a focusing quadrupole separately, and SX(n) represents a sextupole. The beta-functions and the dispersion is presented in Fig. 2 and the primary physical parameters of the synchrotron ring are listed in Table 1.

Fig. 1Full-screen View

Layout of the lattice design. B(n) represents a dipole. QD(n) and QF(n) represent a defocusing quadrupole and a focusing quadrupole, respectively. Furthermore, SX1 and SX2 are two sextupole families that fit the chromaticity.

Fig. 2Full-screen View

Twiss parameters of the RCS. The first solid line, dotted line, and the second solid line represent horizontal betatron function, vertical betatron function, and horizontal dispersion function, respectively.

The horizontal and vertical betatron functions were designed to be about 5 m. The horizontal dispersion function was smaller than 3.85 m. Small beta function means small beam envelope, requiring small aperture, and could realize large acceptance.

The injection design

Usually, there are three injection methods for synchrotron, namely the single-turn injection, the multi-turn injection and the stripping injection. Single-turn on-axis injection is simplest if could meet the particle numbers needed for the clinical treatment. A linac which can provide 7 MeV proton beams was chosen as the injector of RCS [34]. The revolution period of injection beam is 748 ns. The maximum time of single-turn on-axis injection depends on the descend time of injection element. If the injection beam current is 10 mA, this corresponds to 2.33×10¹⁰ protons within 374 ns, which could satisfy the requirement of 10¹⁰ protons per pulse. If higher particle number is required, higher injection beam current and shorter descend time of injection element is required.

The injection system is composed by a septum and a kicker. The septum is made up of a cutting plate that divides the field-free zone where the circulating beam is situated from the strong magnetic field where the injected beam is placed, therefore avoiding altering the circulating beam flow in the near area. The septum is thin enough that the injection beam stays as close to the reference orbit as feasible. The beam from the injection line is bent to a slight angle by the beam center after passing the septum. After some movement, the injection beam’s trace coincides with the closed orbit, and the direction of motion has an angle. The magnetic field produced by the kicker that installed here gives the particles a kick, forcing the beam to proceed in a direction same to the closed orbit, and the beams then move in the ring along the closed orbit. After the injection, the magnetic field strength of the kicker must be promptly decreased to zero to avoid interfering with the beams that return here (Figs. 3, 4).

Fig. 3Full-screen View

Single turn injection design.

Fig. 4Full-screen View

Fast extraction design.

Septal and kicker parameters are listed in Table 2.

Extraction design

In comparison to injection, the energy of the extraction beam is higher, and the strength of the extraction components is larger. The beam extraction in the ring accelerator is much more complex. The multi-turn slow extraction is generally used in medical synchrotron for long pulse extraction beam [35]. While employing the rapid cycling synchrotron with a high repetition frequency, the beam stays at a energy for very short time. The single-turn fast extraction method is widely utilized and used in our design.

Here we choose a septum and two same kickers which are easy to process in reality to kick out all the beams in one turn. Placing the kickers in front of the septum could make enough space and avoid beams hitting the extraction septum. The following is a basic outline of the process: in the ring, beams are deflected by a strong magnetic field of the kicker, pass the dipole and then enter the gap of the septum. The septum bents the beams in a much larger angle to get out of the way of other components in the ring and the beams are finally transmitted to the high-energy transport line to achieve ring separation. The ramping time of the kickers must be within the time gap of the single bunch between the turn and the last turn. In the single turn rapid extraction, the extraction platform time must be sufficient to kick all the particles and the requirement for descent time is relaxed. The revolution time of the beam to be extracted depends the required energy, varies from 0.15 μs to 0.3 μs. And the bunch length of circling beam is around 6 meters. So the rising time of kicker should be less than 0.076 μs.

The length of the septum is 0.8 m and two kickers are 0.6 m apart. The septum is installed in another straight section. The height of the center orbit at the septum’s entrance is 0.044 m and the separation height at the exit is 0.356 m. The parameters of the extraction kickers and septum are shown in Table 3.

Basic parameters

Eddy current effects are influenced by the hardware shape and size, material electrical conductivity, and magnetic field ramping rate. Low conductivity materials, thin pipe thicknesses, and appropriate pipe cross-section designs decrease eddy current effects in the vacuum chambers for the changing magnetic fields. According to the beam transverse envelope size, an elliptical cross-section of the vacuum chamber was chosen, where a is the vacuum chamber half-width, b is the chamber half-height, and e is the chamber thickness. To reduce the eddy current, the vacuum pipes should be made by small conductivity materials, such as ceramics. But the thickness of ceramic vacuum pipe is too high and increase the magnet gap and power very much. A low conductivity material Inconel 625 is used [38] (Table 4).

The ramping is realized by improving the strength of the dipoles and quadrupoles.

For a sinusoidal ramp, the time variation of the dipole magnetic field is provided by Eq. 2 [39] $B\left(t\right)=B_0\left[a_1-\cos \left(2\pi f_{\text{r}}t\right)\right]\mathrm{,}$ (2) where $B_0=\left(B_{\max }-B_{\min }\right)/2$, $a_1=\left(B_{\max }+B_{\min }\right)/\left(B_{\max }-B_{\min }\right)$, and $f_{\text{r}}$ is the repetition rate.

The minimum dipole strength is 0.20038 T when injected, and the maximum is 1.2294 T when extracted. Figure 5 illustrates the ramping curve of the dipole field.

Fig. 5Full-screen View

Sinusoidal ramping curve for the dipole field.

Induced sextupolar component in the dipoles chambers

The eddy current magnetic field within the vacuum chamber of the dipoles has a significant impact on the dipole component, leading to magnetic field attenuation and delay. Concurrently, the sextupolar component of the eddy-current magnetic field in the dipole vacuum chamber induces changes in chromaticities. Additionally, the quadrupole component of the eddy-current magnetic field in the vacuum chamber of the quadrupole magnet section results in a lateral tune shift.

The formula for the sextupolar component in elliptical vacuum chambers can be expressed by Eq. 3[39] as follows: $m_3=\frac{{\mu }_0\sigma e}{h}\frac{\dot{B}}{B\rho }J,$ (3) where $\sigma =0.8\times {10}^6\text{ S}/\text{m}$ denotes the electrical conductivity of Inconel 625. Furthermore, ${\mu }_0=4\pi \times {10}^{-7}$ denotes vacuum permeability, ρ denotes the dipole bending radius, ≈B denotes the rate of change in the field, and J denotes the geometric factor of the elliptical vacuum chamber. $J={\displaystyle {\int }_0^{\pi /2}}\sin \phi \sqrt{{{\displaystyle \cos }}^2\phi +{\left(\frac{b}{a}\right)}^2{{\displaystyle \sin }}^2\phi }d\phi ,$ where a denotes the vacuum chamber half-width and b denotes the half-height of the chamber.

The sextupole strength introduced during the sinusoidal ramping is shown in Fig. 6. When t=5.0 ms, the sextupolar exhibits a maximum value of 1.143 m^-3. This is also verified in Opera software’s ELEKTRA module as shown in Fig. 7. The geometrical factors and B’ / B are related to the strength of the eddy current magnetic field’s sextupole component.

Fig. 6Full-screen View

Sextupole strength introduced by the sinusoidal ramping curve.

Fig. 7Full-screen View

When t=0.01 s, the magnetic center plane is distributed along line x=-23 mm to x=23 mm.

The sextupolar component induced by eddy currents is calculated as the sextupole component errors of dipoles, and causes horizontal and vertical chromaticity modification described by ε_{x, co} and ε_{y, co}. $\{\begin{array}{l}{\epsilon }_{x\mathrm{,}\text{co}}={\epsilon }_{x\mathrm{,0}}+\Delta {\epsilon }_z=-0.73+31.03m_3\hfill \\ {\epsilon }_{y\mathrm{,}\text{co}}={\epsilon }_{y\mathrm{,0}}+\Delta {\epsilon }_y=-0.78-27.27m_3\hfill \end{array}$ (4) The maximum horizontal chromaticity is ε_{x, co} = 34.74, and the minimum vertical chromaticity is ε_{y, co} = -31.95. Because of chromaticity changes caused by the sextupole component of the eddy current effects, the transverse working tune may change and cross the resonance line. When designing beam optics, it is necessary to consider the errors introduced by the sextupole component, chromaticity correction should be done in the ramping. Two sextupole families are used to fit the chromaticity to the value of 0.5 and 0.5 in the horizontal and vertical direction.

Induced dipole component in the dipoles chambers

The dipole component can also be induced by eddy currents in the dipole vacuum chamber. This can be described by Eq. 5 [40] as follows: $\begin{array}{c}{m}_1=\frac{-{\mu }_0\sigma e\dot{B}F}{2\pi \rho Bh}\mathrm{,}\\ F=\pi \left(\frac{a_2^2{b}_2}{a_2+b_2}-\frac{a_1^2{b}_1}{a_1+b_1}\right)\mathrm{,}\end{array}$ (5) where a₁, a₂, b₁, and b₂ denote the vacuum chambers inside the half-width, outside half-width, inside half-height, and outside half-height, respectively. Maybe the presence of the vacuum box could cause the effect of magnetic stranded for the RCS, with a time constant of 3.5254 μs. This is acceptable. Figure 8 shows the acceptable effects of magnetic stranded for the sinusoidal ramping within 0.04 s.

Fig. 8Full-screen View

Dipole strength errors introduced by the sinusoidal ramping curve.

Induced quadrupolar component in the quadrupole vacuum chamber

Similar to the dipole vacuum chamber, eddy current effects exist, and only the quadrupolar component would be generated. For a circular vacuum chamber of $\overline{r}$ medium radius, the quadrupolar strength is given by Eq. 6[39] $m_2=-\frac{7}{16}{\mu }_0\sigma e\overline{r}\times \frac{\dot{B}}{B\rho }\mathrm{.}$ (6) Figure 9 shows that the magnetic hysteresis induced by the quadrupolar component in the vacuum chamber is small. As shown in Fig. 10, the maximum tune shift introduced by the quadrupolar component is within 4×10^-4.

Fig. 9Full-screen View

Quadrupole strength errors introduced by the sinusoidal ramping curve.

Fig. 10Full-screen View

Horizontal and vertical tune shifts introduced by quadrupolar component.

Compared to the initial transverse tune, the transverse tune shift due to the quadrupole component of the eddy-current magnetic field is significantly less than the error introduced by the quadrupole magnet (Fig. 11).

Fig. 11Full-screen View

Resonance line diagram.

Power loss introduced by eddy currents in the vacuum chamber

The existence of electrical resistance would lead to heating effects. When the vacuum chamber is heated to a high temperature for a long time, the vacuum inside the chamber would decrease. In an extremely high vacuum environment, the gas molecules adsorbed on the vacuum chamber wall absorb heat and diffuse into the vacuum chamber. Then the vacuum would decrease and the beam lifetime would shorten. From an engineering point of view, overheated tube temperatures can cause aging and cracking of the epoxy resin in the dipole magnet core. Generally speaking, the temperature of the vacuum chamber does not exceed 80 ℃. The eddy current power loss per unit length is given by Eq. 7 and is shown in Fig. 12[41]. $\frac{P_{\text{eddy}}}{L}=4\sigma {\dot{B}}^{2}e{a}^3\times H,$ (7) where $H={\displaystyle {\int }_0^{\pi /2}}{{\displaystyle \sin }}^2\phi \sqrt{{{\displaystyle \cos }}^2\phi +{\left(\frac{b}{a}\right)}^2{{\displaystyle \sin }}^2\phi }\text{d}\phi ,$

Fig. 12Full-screen View

Eddy current power loss per unit length.

and a, b, h are the vacuum chamber half-width, vacuum chamber half-height and magnet gap half-height. To visualize the effect of power loss on the vacuum chamber we perform a transient EMF research using the ELEKTRA TR module in Opera software and calculated eddy current loss distribution. The eddy current loss is proportional to the magnetic field rate of change squared. The rate of change of the magnetic field in a single cycle is not constant. In the eddy current loss thermal analysis, the average eddy current loss in a single cycle is used and the average eddy current loss factor is 0.6434 by averaging and integrating the values over a time period. The we create a static thermal analysis model in Opera software’s TEMPO ST module. The thermal conductivities of the magnet core and Inconel 625 is shown in Table 5. The exterior surface of the vacuum chamber is configured for heat dissipation by air convection and the heat transfer coefficient is 14 $\text{W}/\left({\text{m}}^2\cdot \text{K}\right)$. The initial ambient temperature is 28 ℃.

We multiplied the average eddy current loss factor of 0.6434 by the eddy current loss distribution of the vacuum chamber under changing magnetic fields derived from the ELEKTRA TR module’s eddy current effect simulation. We imported it as a steady-state heat source distribution into the vacuum chamber model in the TEMPO ST module. After simulation the temperature distribution of the chamber after the eddy current loss is depicted in Fig. 13.

Fig. 13Full-screen View

(Color online) Temperature distribution of the thick vacuum chamber after eddy current loss temperature rise.

The power losses at vacuum chamber was 23.63 W, and we can observe from the picture that the highest temperature is 75.165 ℃ within acceptable limits for engineering applications. This result is very close to the theoretical result.