Reference dosimetry in MRI-linacs: evaluation of available protocols and data to establish a Code of Practice

The Elekta Unity^™ MRI-linac (Elekta Instrument AB Stockholm, Sweden) was invented by the UMC Utrecht and developed in collaboration with Elekta and Philips (Raaijmakers et al 2005, Raaymakers et al 2017). The system incorporates daily verification of variation in patient anatomy, i.e. variation in tumour position and healthy surrounding tissues by creating a new on-line adapted treatment plan for every treatment. A first-in-man study to demonstrate the safe and accurate delivery was performed in 2017 on the clinical prototype MRI-linac (Raaymakers et al 2017). The Unity^™ system was clinically released in 2018 after CE-certification.

The device is a combination of a 1.5 T Philips 70 cm wide-bore MRI scanner, a 7 MV flattening-filter-free (FFF) beam generated with a standing-wave linear accelerator (linac) and an adaptive treatment software solution.

The active shielding method of the magnet provides a zone with a low magnetic field around the cryostat in which the magnetic-field sensitive linac equipment is located. The linac equipment is mounted on a gantry ring allowing for continuous rotation at 6 RPM. The source-to-isocenter-distance (SID) of the linac is 143.5 cm. The magnetic field inside the bore is directed towards −y_f, IEC61217 (2011), see figure 6, perpendicular to the radiation beam. The radiation beam travels through the beam portal region in the cryostat (approximately 15 cm wide), which is free of the super-conducting coils and contains lower attenuating construction. Radiation field size ranges from 1 × 1 cm² up to 22 × 57 cm². The MLC, an adapted Agility design being compatible to the high-magnetic field, consist of 160 leaves (projected leaf width at isocenter is 7.2 mm) and two backup jaws at a fixed collimator angle.

The helium level in the cryostat is maintained high and constant to keep gantry angle dependent dose variations due to this level to less than 1%. Dose output variation due to local inhomogeneities of the cryostat (e.g. welds) and couch are characterised and taken into account in the TPS. The maximum dose rate at the isocenter is approximately 4.25 Gy min⁻¹.

The ViewRay MRIdian^™ system (ViewRay Inc. Oakwood, USA) was the first commercially available MRI-integrated radiation device (Mutic and Dempsey 2014). It obtained its Food and Drug Administration (FDA) approval in 2010 and the first patient was treated with this system in 2014 at Washington University (St. Louis, USA) (Acharya et al 2016).

This device has been described previously by Mutic and Dempsey (2014) and consists of a split bore 0.35 T MRI with a bore size of 70 cm. The magnetic field is directed towards +y_f IEC61217:2011 (2011), see figure 6, perpendicular to the radiation beam. The MRI employs two superconducting solenoids, self-shielded gradient coils with gradient strengths and slew rates of up to 18 mT m⁻¹ and 200 T m⁻¹ s⁻¹, respectively, a 75 cm diameter radiofrequency transmitting birdcage coil and a surface receiver coil with only 0.75% beam attenuation. In addition, the MRI allows planar as well as volumetric imaging with a field of view of up to 50 cm.

In the initial version of this device, a ring-gantry with three equally-spaced ⁶⁰Co-sources (mounted within the gap between the two solenoids) was used, delivering a dose rate of about 5.5 Gy min⁻¹ at the isocenter (Mutic and Dempsey 2014). This beam does not pass through any of the MRI components before entering the bore. The beam is shaped by double-focused multi-leaf collimators and the device allows delivery of 3D-conformal as well as step-and shoot intensity modulated radiotherapy. Recently, the three ⁶⁰Co-cources have been replaced by a single standing-wave FFF 6 MV-linac (termed as MRIdian^™ linac). All new installations are equipped with the linac-version (Klüter 2019). SID and maximum field size at the isocenter are 105 cm/27.3 × 27.3 cm² for the ⁶⁰Co and 90 cm/24.1 × 27.4 cm² for the linac-version, respectively (ViewRay 2015, 2017). MR-imaging during treatment may be performed using either a single sagittal plane at 4 frames per second or three parallel sagittal planes at 2 frames per second and latency of image-based target tracking is specified to be 300 ms (Mutic and Dempsey 2014). The treatment planning system offers auto contouring, a fast Monte Carlo based dose calculation as well as inverse dose optimization and thus in principle the basic features required for adaptive radiotherapy treatments (Mutic and Dempsey 2014).

While this review mainly focusses on reference dosimetry for devices with a perpendicular orientation between the beam and the magnetic field, so-called orthogonal systems, for which two MRI-linac systems are commercially available (ViewRay MRIdian^™ and Elekta Unity^™), there are currently two projects studying MRI-linac designs in which the beam is parallel to the magnetic field, so-called inline systems. Both inline systems are under development and have not been clinically applied yet. For completeness, these systems are also briefly outlined here.

One project is the Australian Magnetic Resonance Imaging-linac program (Keall et al 2014), which investigated a prototype consisting of an ex-clinical, closed 1.5 T MRI scanner with an inline-oriented linac (with nominal beam energies of 4 MV and 6 MV) (Liney et al 2016). An advantage of this construction is the absence of attenuation and scatter radiation generated by the cryostat compared to orthogonal systems produced by a linac mounted outside a closed magnet. Apart from the general strategies of MR-guided radiotherapy, the project especially investigates potential advantages of an inline alignment over an orthogonal arrangement. A more recent prototype was realized as an actively shielded 1.0 T open-bore magnet (Jelen et al 2020). An obvious advantage of the inline design is the fixed position of the linac relative to the magnet, simplifying the linac operation, and the confinement of the secondary electrons along the beam direction by the magnetic field. This confinement leads to a reduced exit dose compared to the orthogonal systems. However, it may lead to a significantly increased in-field skin dose on the entrance side, which depends heavily on the fringe field outside of the magnet (Oborn et al 2012). A second advantage is the reduction of scattered electrons, because the beam does not pass through the cryostat. A further technical challenge is that the whole magnet-linac construction (or the patient) must be rotated to achieve different beam angles.

The second project which uses an inline system is located at the University of Alberta (Edmonton, Canada) (Fallone 2014). Here, the prototype is composed of a biplanar high-temperature superconducting open-bore magnet and a 6 MV linac, which is aligned either perpendicular or inline to the magnetic field. A field strength of 0.6 T was selected as the optimal trade-off between good image quality and minimized dosimetric effects of the secondary electrons. To achieve different beam angles relative to the patient, the whole construction consisting of magnet and linac is rotated.

The most important photon–electron interaction types in radiotherapy are photoelectric effect, Compton scatter and pair production. Generally, it is assumed that the impact of the magnetic field on these interactions is insignificant for field strengths used in MRI-linacs (Szymanowski et al 2015). Therefore, the energy transferred to secondary charged particles (kerma) remains unaffected by the magnetic field for a constant photon fluence. However, the secondary charged particles (electrons and positrons) produced by the primary photon beam traverse through the magnetic field and experience the Lorentz force. Therefore, the transport of charged particles and consequently the absorbed dose (distribution) will be affected by the magnetic field.

For the electric field free case, the Lorentz force, $\overrightarrow {{F_L}}$, is given by

$$\begin{equation}\overrightarrow {{F_L}} = q\vec v \times \vec B\end{equation} \tag{ 1 }$$

where $q = \pm e$ (+ for positrons and − for electrons) with e the elementary charge and $\vec \upsilon$ the velocity of the electron moving in the magnetic field, $\vec B$. Since the proportion of secondary positrons is small compared to that of secondary electrons, in the remainder only secondary electrons will be considered.

According to equation (1), only the velocity component perpendicular to the magnetic field determines the Lorentz force, while the parallel component does not contribute. Therefore, in vacuum, an electron moving with an angle α between its velocity vector and the magnetic field vector will be forced to a helical trajectory with a gyroradius, r_g ,

$$\begin{equation}{r_g} = \frac{{{E_{{{tot}}}}\beta \sin \alpha }}{{e\left| {\vec B} \right|c}}\end{equation} \tag{ 2 }$$

with the total energy, E_tot , defined as the sum of the kinetic, E_k and the electron rest energy, E₀ , and $\beta = \left| {\vec v} \right|/c$ being the velocity relative to the speed of light $c$. With α equal to 90°, the trajectory reduces to a circle (figure 1). r_g increases with increasing kinetic energy and decreasing magnetic field strength (figure 2). This corresponds with the fact that the larger the r_g , the more the electron trajectory resembles the trajectory of the magnetic-field-free case.

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Since the secondary electrons generated in MV x-ray beams predominantly have the same direction as the x-rays, α will be small for inline MRI-linacs where the beam and magnetic field are parallel and hence the radius of the trajectory will be small. For the same reason in orthogonal MRI-linacs, r_g will be considerably larger because α is close to 90°.

The trajectory of the secondary electrons in media and with a magnetic field present depends on two interactions: firstly, on the electron interactions with the medium and secondly, on the interaction of the electron with the magnetic field by the Lorentz force. Neglecting energy loss by synchrotron radiation, the Lorentz force does not increase or decrease the electrons kinetic energy. Therefore, energy loss of electrons is completely dominated by the first interaction. Hence, electrons lose their energy over the same path length as in the magnetic-field-free case.

Their trajectory within the medium will however be different due to the Lorentz force, which depends on the electron velocity magnitude and direction relative to the magnetic field vector. Since the electron energy decreases due to interactions with the medium, the gyroradius of the electron trajectory also decreases along its trajectory. Therefore, in the continuous slowing down approximation (CSDA), in which scattering of the electron due to interactions with the medium is neglected, the circular or helical shaped electron trajectory in vacuum changes to a spiralling trajectory in a medium, as shown in figure 3.

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When accounting for scatter and realistic energy loss by electrons in media, the trajectories show a stochastic behaviour. In this more realistic approach, the concept of dose kernels, i.e. a 3D dose distribution in a homogeneous medium of a 1D beam around a single interaction point, is more useful to qualitatively evaluate the impact of magnetic fields on electron transport. For orthogonal MRI-linacs, electrons will be swept aside in the direction perpendicular to the magnetic field. The dose kernels will therefore be tilted and shortened in the forward direction (Raaijmakers et al 2008, Gargett et al 2015). This also results in the dose being deposited closer to the interaction point. Therefore, in a homogenous medium the dose distribution will be more similar to the kerma distribution than without a magnetic field (van Asselen et al 2018).

For inline MRI-linacs (section 2.3) the electrons will be focused in the direction of the photon field as they will spiral along the magnetic field lines and therefore travel further away from the interaction point in the photon direction. In this case, the dose kernel will be stretched relative to a situation without a magnetic field (Gargett et al 2015).

As mentioned before, the trajectory of electrons in the presence of magnetic fields is determined by the interactions with medium and the interaction with the Lorentz force. Since only the former depends on the medium mass density, it is useful to approach the characteristics of electron trajectories in magnetic fields as a function of medium mass density. A helpful concept for this are the so-called mass thickness trajectories, which are particle trajectories in a coordinate system linearly scaled with mass density (Bouchard et al 2015b). When neglecting the stopping power density effect in the CSDA approximation, mass thickness trajectories of an electron in a homogeneous medium are independent on mass density with no magnetic field present (figure 4). As can be seen from this figure, this is not the case for the situation with a magnetic field present. Where the mass thickness trajectory of a 1 MeV electron in a medium density close to water hardly reaches a quarter of a circle for 1 T magnetic field strengths, the trajectory reaches more than 2 revolutions in a medium density 10 times lower. For lower medium densities the number of revolutions further increases. Note that the gyroradius of trajectories in a normal coordinate system will be similar for all densities.

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This strong dependence of the electron trajectory characteristics on medium density is also demonstrated in the Monte Carlo simulation of figure 5. Therefore, interfaces between low- and high-density media will impact the electron trajectories considerably. This highlights the impact of the magnetic fields on the response of detectors especially for those with interfaces between low- and high-density media such as ionization chambers.

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An important aspect of Codes of Practice for reference dosimetry is the relation between the dose to the cavity and the dose to the (water) medium in the absence of the cavity. Codes of Practice base this relation on cavity theories, such as Bragg–Gray and Spencer–Attix cavity theory, which are valid under certain conditions. The condition for Bragg–Gray cavity theory states that the charged particle fluence, $\varphi$, in the cavity is not perturbed by the presence of the cavity. To account for departure from this condition perturbation correction factors are applied (Bielajew 1986, Andreo 1992, Andreo et al 2000).

An important concept related to the conditions for cavity theories is charged particle equilibrium (CPE). Full CPE is defined as

$$\begin{equation}\vec \nabla \varphi = 0.\end{equation} \tag{ 3 }$$

Considering energy conservation under the condition of CPE, absorbed dose equals collision kerma. In addition, based on the Boltzmann transport equation (BTE), Fano (1954) has shown that CPE remains unaffected by material density variations in a medium with uniform atomic properties (condition 1) and a uniform source scaled with mass density (condition 2). Hence, CPE in the absence of the ionization chamber assures that for ionization chambers of radiologically water equivalent materials, the Bragg–Gray condition is fulfilled and that perturbation corrections are unity (Bouchard et al 2015a).

Due to photon attenuation and scattering, CPE cannot be achieved in practice for reference fields. Only transient CPE (TCPE), which is defined as a constant (i.e. depth independent) non-unity ratio between dose and kerma, is achievable. Under TCPE conditions, perturbation corrections of ionization chambers are small and can be treated independently (Palmans et al 2017) which is the basis of the treatment of perturbations in CoPs (Almond et al 1999, Andreo et al 2000). For example, TRS-398 assumes the perturbation correction for the presence of the cavity, p_cav, to be unity with an uncertainty of 0.1%.

Bouchard and Bielajew (2015) derived a modified BTE for charged particle transport in the presence of magnetic fields. Based on this modified BTE, Bouchard et al (2015a) and de Pooter et al (2015) have proven that the two Fano conditions, described above, are inadequate to establish CPE in a medium with mass density variations in the presence of magnetic fields. To achieve CPE, an isotropic angular distribution of charged particles is required. Because the production of secondary electrons in MV photon beams is dominated by Compton interactions, which produces an an-isotropic distribution of secondary electrons, this condition cannot be fulfilled in realistic radiation fields in the presence of magnetic fields.

Therefore, this condition of isotropy for the secondary electron field highlights a more fundamental aspect of the applicability of cavity theory for ionization chambers in the presence of magnetic fields. It shows that CPE in the absence of the chamber is insufficient to guarantee, that for ionization chambers of radiologically water equivalent materials, the Bragg–Gray condition is fulfilled and that perturbation corrections can be considered unity. To make this point clear, consider the following example with a uniform photon fluence in a medium of uniform atomic composition and mass density. When a small cavity of a different mass density but with the same atomic composition is introduced, the secondary electron fluence will not be perturbed in the absence of a magnetic field. In the presence of a magnetic field, however, this in general will not be the case because the secondary electron fluence will not be isotropic.

This point is supported by Monte Carlo simulations for a model with a water vapour cavity in a water medium and a uniform density scaled source of electrons (de Pooter et al 2015). Results for realistic angular distributions based on Compton electrons show a >10% deviation from unity for the ratio between dose to the cavity and kerma in the presence of a magnetic field (B = 1.5 T) compared to no deviation from unity without a magnetic field present. For non-realistic isotropic angular distributions, the deviation of the ratio from unity are insignificant and similar to those without a magnetic field. This suggests that for realistic radiation fields much larger perturbations can be expected for ionization chambers in the presence of magnetic fields than for radiation fields without a magnetic field present.

In commercially available MRI-linacs (Mutic and Dempsey 2014, Raaymakers et al 2017), the static magnetic field is parallel to the bore and the y_f-axis (IEC61217). The linear accelerator is mounted on a gantry ring which can rotate around the y_f-axis, as shown in figure 6. The direction of the magnetic field is different for the two commercially available MRI-linacs: +y_f for the ViewRay MRIdian^™ and −y_f for the Elekta Unity^™).

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As described in section 3.1, the direction of the Lorentz force is perpendicular to the magnetic field direction and the velocity of the electrons, and thus to the direction of the photon beam. Since the direction of the magnetic field is fixed and the photon beam axis rotates in the x_f-z_f plane, the Lorentz force vector is also rotating in the same plane orthogonal to the beam axis.

Detectors are usually placed along one of the three main axes of the facility (x_f, y_f or z_f). Except for the axis aligned with the beam axis, the detector can be placed in two opposing directions: with the tip in the positive direction or with the tip in the negative direction. Therefore, there are five possible orientations for the detector relative to the beam and the magnetic field (represented by (a)–(e) in figure 7). In the most commonly used setup, the detector points in the direction of $\vec B$ (orientation b in figure 7). Most publications refer to this as the parallel orientation. For symmetry reasons there is no difference in response between the parallel orientation and the opposing orientation (orientation a in figure 7) for axisymmetric detectors. Also, the setup with the detector placed orthogonal to the magnetic field and in the direction of the Lorentz force, the perpendicular orientation, has been widely investigated. For this orientation the two opposing orientations are not symmetrically equivalent. Most papers refer to orientation d (figure 7) as the perpendicular orientation. In some papers the opposite orientation e is also investigated and referred to as the counter-clockwise perpendicular orientation.

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Orientation c with the detector aligned along the beam axis has also been investigated but is not commonly used. Note that relating the orientation of the chamber with the direction of $\vec B$ requires relating the direction of $\vec B$ to the machine coordinate system which is different for the two commercially available MRI-linacs (+y_f and −y_f).

Generally, reference fields with field sizes of 10 × 10 cm² are recommended in CoPs. In the presence of a magnetic field and for orthogonal MRI-linac facilities (figure 6), the following characteristics of a reference field change:

• build-up region;
• beam exit region;
• depth dose distribution; and
• penumbra.

The dose distribution in the build-up region changes due to the change in contaminant electron fluence and the change of electron trajectories in the phantom due to the Lorentz force. The contaminant electrons (generated in air, in the accelerator and in the cryostat) travelling towards the surface will be captured by the magnetic field and spiral around the magnetic field lines. As a result they will be swept out of the radiation field which reduces the surface dose in the field (Oborn et al 2009, Hackett et al 2018, Malkov et al 2019). Furthermore, electrons resulting from photon–electron interactions inside the phantom or patient will deposit their energy closer to the surface since their path is curved by the Lorentz force (section 3.2), which will increase the surface dose. As a result the build-up region will be shorter compared to no magnetic field (Woodings et al 2018), depending on the magnetic field strength (Oborn et al 2009, 2010), see figure 8. In addition, the electron streaming effect (Malkov et al 2019) contributes to out-of-field dose. Here, electrons backscattered from the build-up region into the air will spiral along the magnetic field and travel out-of-field. The effect depends on the curvature of the surface.

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At the beam exit of the phantom, ejected electrons can return to the phantom and deposit their dose. This electron return effect (ERE) can result in a high dose area at the beam exit (Raaijmakers et al 2007, Ahmad et al 2016, Woodings et al 2018). The effect depends on magnetic field strength, field size and curvature of the exit surface (Raaijmakers et al 2008, Oborn et al 2009, 2010, Keyvanloo et al 2012).

The shape of lateral dose profiles becomes asymmetric (see figure 9) in the direction of the Lorenz force, i.e. perpendicular to the magnetic field direction (figure 6) (Raaymakers et al 2004, Ahmad et al 2016, Woodings et al 2018). The asymmetry depends on the magnetic field strength. In the direction of the magnetic field the lateral dose profiles will remain symmetric.

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In reference dosimetry, the absorbed dose is usually measured for a reference field on the central axis downstream from the dose maximum. In this region the aforementioned changes in dose distribution will not play a role. At this point the dose changes slightly due to the curved path of the electrons. Since the energy is deposited closer to the photon interaction point, the absorbed dose will be closer to the kerma. As a result, the absorbed dose will be reduced in the presence of a magnetic field orthogonal to the photon field (see figure 8). For 1.5 T this reduction is around 0.5% in a 10 × 10 cm² field for an Elekta Unity^™ beam (O'Brien et al 2016, van Asselen et al 2018). For 0.35 T with the same beam this reduction is around 0.1%. For beams of experimental facilities, consisting of a clinical linear accelerator and an electromagnet, different values have been reported, 0.5% and 0.8% for B= 0.35 T and 1.5 T respectively (Delfs et al 2018).

In current Codes of Practice such as TRS-398 (Andreo et al 2000), TG-51 (Almond et al 1999) and NCS-18 (Aalbers et al 2008) the magnetic field is not part of the reference conditions and the determination of the absorbed dose to water is based on the k_Q formalism:

$$\begin{equation}{D_{{\text{w}},Q}} = {M_Q}{N_{D,{\text{w}},{Q_0}}}{k_{Q,{Q_0}}}\end{equation} \tag{ 4 }$$

with ${D_{{\text{w}},Q}}$ the absorbed-dose-to-water for beam quality $Q$, ${M_Q}$ the corrected reading of the dosimeter in beam quality Q and ${N_{D,{\text{w}}, {Q_0}}}$ the absorbed dose-to-water calibration coefficient at reference beam quality ${Q_0}$, usually ⁶⁰Co. To correct for the difference in calibration coefficient between the reference beam quality ${Q_0}$ and the beam quality $Q$, the beam quality correction factor ${k_{Q,{Q_0}}}$, is used, with

$$\begin{equation}{k_{Q,{Q_0}}} = \frac{{{N_{D,{\text{w}},Q}}}}{{{N_{D,{\text{w}},{Q_0}}}}}.\end{equation} \tag{ 5 }$$

The corrected reading, ${M_Q}$, is given by

$$\begin{equation}{M_Q} = {M_{\text{raw}}} {k_{p,T}} {k_{\text{h}}} {k_\mathrm{pol}} {k_{\text{s}}}\end{equation} \tag{ 6 }$$

where ${M_{\text{raw}}}$ is the raw dosimeter reading corrected for leakage, ${k_{p,T}}$ the factor to correct the reading for temperature and pressure, ${k_{\text{h}}}$ for relative humidity, ${k_{\text{pol}}}$ for polarity effect and ${k_{\text{s}}}$ for incomplete charge collection due to recombination.

The magnetic field affects the calibration coefficients of ionization chambers used in the CoPs (Meijsing et al 2009, Reynolds et al 2013, O'Brien et al 2016). To account for the difference in reference conditions (i.e. the presence of the magnetic field) between the beam used for the calibration of the chamber and the MRI-linac beam, the magnetic field correction factor is introduced by O'Brien et al (2016), van Asselen et al (2018) and Malkov and Rogers (2018), using slightly different notations of the measurement equation with different terminology and symbols. Van Asselen et al (2018) uses the following formulation:

$$\begin{equation}{D_{{\text{w}},Q}} = {M_Q}{N_{D,{\text{w}},{Q_0}}}{k_{Q,{Q_0}}}{k_{\vec B,Q}}\end{equation} \tag{ 7 }$$

with

$$\begin{equation}{k_{\vec B,Q}} = \frac{{N_{D,{\text{w}},Q}^{\vec B}}}{{{N_{D,{\text{w}},Q}}}}\end{equation} \tag{ 8 }$$

which enables compatibility with the existing k_Q,_Q0 formalism in CoPs. Note that $\vec B$ is used as a vector, since the correction factor depends on the orientation of the magnetic field relative to the photon beam and the orientation of the detector. The ${k_{\vec B,Q}}$ directly corrects the calibration coefficient in beam quality Q for the presence of the magnetic field.

The reference field in MRI-linacs in the absence of the magnetic field deviates from that in conventional linacs, because, among others, the MRI-linac FFF beam is filtered by the MRI-cryostat (Woodings et al 2018). For this purpose O'Brien et al (2016) have proposed to add the ${k_{\vec B,Q,{Q_0}}}$ factor to the formalism from Alfonso et al (2008) which is the basis of TRS-483 (Palmans et al 2017). This formalism adds a correction factor, $k_{{Q_{\text{msr}}},Q}^{{f_{\text{msr}}},{f_{\textrm{ref}}}}$, to the k_Q,_Q0 formalism, which accounts for the difference between reference fields in conventional beams and machine specific reference fields (msr). The complete measurement equation of O'Brien et al (2016) reads

$$\begin{equation}{D_{{\text{w}},Q}} = {M_Q}{N_{D,{\text{w}},{Q_0}}}{k_{Q,{Q_0}}}k_{{Q_{\text{msr}}},Q}^{{f_{\text{msr}}},{f_{\textrm{ref}}}}k_B^{{Q_{\text{msr}}}}.\end{equation} \tag{ 9 }$$

From the paper of van Asselen et al (2018) it is clear that ${k_{\vec B,Q}}$ is implicitly defined for the MRI-linac machine specific reference field. Therefore, despite the apparent differences in the measurement equations (8) and (9) both equations have the same meaning, and ${k_{\vec B,Q}}$ in the paper of van Asselen et al (2018) equals $k_B^{{Q_{\text{msr}}}}$ in the paper of O'Brien et al (2016). In addition, in a similar way, the product of ${k_{Q,{Q_0}}}$ and $k_{{Q_{\text{msr}}},Q}^{{f_{\text{msr}}},{f_{\textrm{ref}}}}$ in O'Brien et al (2016) equals ${k_{Q,{Q_0}}}$ in van Asselen et al (2018). Furthermore, in O'Brien et al (2016) the correction factor $k_{{Q_{\text{msr}}}}^{B, {f_{\text{msr}}}}$ is defined as the product of ${k_{Q,{Q_0}}}$, $k_{{Q_{\text{msr}}},Q}^{{f_{\text{msr}}},{f_{\textrm{ref}}}}$ and $k_B^{{Q_{\text{msr}}}}$. This can be written as a ratio of calibration coefficients by

$$\begin{equation}k_{{Q_{\text{msr}}}}^{B, {f_{\text{msr}}}}{\text{ }} = \frac{{N_{D,{\text{w}},{Q_{\text{msr}}}}^{\vec B}}}{{{N_{D,{\text{w}},{Q_0}}}}}.\end{equation} \tag{ 10 }$$

This factor is similar to the $k_Q^{mag}$ used in Malkov and Rogers (2018).

To avoid implicit definitions in the formalism and to make it generalizable, in the remainder of this paper the formalism of TRS-483 is used for the machine specific reference field of MRI linacs in the absence of the magnetic field. The formulation of van Asselen et al (2018) is used to make the step for this msr field in the presence of a magnetic field. To make clear that this correction applies for an msr field the subscript msr is added to Q, yielding ${k_{\vec B,{Q_{\text{msr}}}}}$. This leads to the following measurement equation:

$$\begin{equation}{D_{{\text{w}},Q}} = {M_Q}{N_{D,{\text{w}},{Q_0}}}k_{{Q_{\text{msr}}},{Q_0}}^{{f_{\text{msr}}},{f_{\textrm{ref}}}}{k_{\vec B,{Q_{\text{msr}}}}}.\end{equation} \tag{ 11 }$$

The relations between the used symbols in the three discussed papers and this paper is summarized in table 1.

Table 1. Relations between the used symbols in the various formulations of the measurement equation for reference dosimetry in an MRI-linac.

O'Brien et al (2016)	Malkov and Rogers (2018)	van Asselen et al (2018)	This paper
${k_{Q,{Q_0}}}\, \cdot \,k_{{Q_{\text{msr}}},Q}^{{f_{\text{msr}}},{f_{\textrm{ref}}}}$	${k_Q}$	${k_{Q,{Q_0}}}$	$k_{{Q_{\text{msr}}},{Q_0}}^{{f_{\text{msr}}},{f_{\textrm{ref}}}}$
$k_B^{{Q_{\text{msr}}}}$	${k_B}$	${k_{\vec B,Q}}$	${k_{\vec B,{Q_{\text{msr}}}}}$
$k_{{Q_{\text{msr}}}}^{B, {f_{\text{msr}}}}$	$k_Q^{mag}$	${k_{Q,{Q_0}}}\, \cdot \,{k_{\vec B,Q}}$	$k_{{Q_{\text{msr}}},{Q_0}}^{{f_{\text{msr}}},{f_{\textrm{ref}}}}\, \cdot \,{k_{\vec B,{Q_{\text{msr}}}}}$

For clinical reference dosimetry the definition of a beam quality specifier for conventional MV photon beams is usually based on either %dd(10)_x (AAPM TG-51) or TPR_20,10 (TRS-398 and most national CoPs). %dd(10)_x is defined as the percentage depth dose at 10 cm depth, %dd(10), in a 10 × 10 cm² radiation field and at a source to surface distance (SSD) of 100 cm, for photons only (i.e. excluding the contribution from contaminant electrons). Therefore, the determination of %dd(10)_x requires the application of a lead foil in the beam when measuring %dd(10), to exclude the effect of contaminant electrons (Almond et al 1999, McEwen et al 2014). This is not required for beams with nominal energies <10 MV and with a flattening filter for which %dd(10)_x and %dd(10) are considered similar.

TPR_20,10 is defined from the ratio of the measured absorbed doses (or corrected ionization chamber signals) at 20 cm and 10 cm water depth at a fixed source to detector distance (SDD), in a 10 × 10 cm² radiation field. The determination of TPR_20,10 requires only measurements at depths far beyond the build-up region, therefore TPR_20,10 is not affected by electron contamination. Although the definition of TPR_20,10 specifies an SDD of 100 cm, it turns out that the value of TPR_20,10 is essentially independent of SDD provided that the field size 10 × 10 cm² is maintained. Using the formalisms of section 4.3, and to avoid ramp-down of the MRI magnet, the beam quality specifier must be measured in the presence of a magnetic field. Therefore, it should be independent of the magnetic field for reference dosimetry in MRI-linacs.

The strong magnetic field in MRI-linac systems naturally tends to remove electron contamination from the beam. This has been demonstrated by O'Brien et al (2016), who calculated %dd(10) by Monte Carlo simulation of an Elekta Unity^™ MRI-linac, based on a full head model, which includes electron contamination, and based on a point source model without contaminant electrons, for no magnetic field and for a magnetic field of 1.5 T. This study found that the d_max and %dd(10) is similar on both models with magnetic field present, confirming that most of the contaminant electrons in the full model never reach the point of d_max. When no magnetic field is present, the difference in %dd(10) between the full head model and the point source model is much larger (2.0%). Therefore, this data suggests that %dd(10)_x and %dd(10) cannot be considered similar for MRI linacs when no magnetic field is present. Potentially this difference can be attributed to contaminant electrons from the part of the bore cover through which the beam passes.

Comparing %dd(10) with and without a magnetic field present shows differences of 0.7% and 1.7% for the full head model and the point source model, respectively. Because the point source model does not include contaminant electrons, the %dd(10) calculated with this model can be considered similar to %dd(10)_x. The difference of 1.7% translates into a change of 0.3% in k_Q,_Q0 factors for typical ionization chambers. Therefore, it was concluded that the change in %dd(10) is resulting from the changing impact of the Lorentz force on secondary electrons between d_max and 10 cm depth and from the reduction in electron contamination. The same study showed that TPR_20,10 remains unaffected by the presence of the magnetic field.

Malkov and Rogers (2018) have investigated the behaviour of TPR_20,10 and %dd(10)_x as a function of magnetic field strength (B = 0 T − 2.0 T) for 6 difference beam qualities (⁶⁰Co, 7 MV (Elekta Unity^™ beam), 6 MV and 25 MV (Elekta SL25 beams), 6 MV and 10 MV (Varian beams)). All models did not include contaminant electrons. They demonstrate that %dd(10)_x reduces by 1.89% for the Elekta Unity^™ beam quality at 1.5 T. Which is in agreement with the results of O'Brien et al (2016). For higher beam energies, the difference can be even higher than 6%. For the range B = 0 T − 0.35 T, the curve for %dd(10)_x does not show differences larger than 0.2%. For TPR_20,10 the difference with no magnetic field present is within 0.36% for all investigated beam qualities.

Therefore, the existing data suggests that TPR_20,10 is the preferred beam quality specifier for reference dosimetry in MRI-linac facilities. Moreover, a technical difficulty in measuring %dd(10)_x in an MRI-linac is the restricted range of possible SSD values, which would already make the direct use of TG-51 for reference dosimetry not feasible. Both O'Brien et al (2016) and Malkov and Rogers (2018) suggest to use a functional relationship between TPR_20,10 and %dd(10)_x (Kalach and Rogers 2003), to convert TPR_20,10 to %dd(10)_x for use with the TG-51 protocol.

Equation (6) describes the general use of corrections for polarity, recombination, air density and humidity in the k_Q,_Q0 formalism. The magnetic field dependency of these corrections has been investigated by several papers (Smit et al 2013, de Prez et al 2019b). For two ionization chamber types (PTW 30013 and IBA FC65-G) k_s and k_pol were determined in anti-parallel and perpendicular orientation to a magnetic field of 0 T and at 1.5 T in an Elekta Unity^™ (de Prez et al 2019b) using the same methods recommended by existing Codes of Practice (Weinhous and Meli 1984). A larger value for k_s could be expected with a magnetic field, compared to no magnetic field due to the longer (curved) path length of the ions in the chamber cavity. However, even though k_s at 1.5 T was slightly larger (0.06%), this was still within the expanded Type A uncertainty of 0.1%.

The application of corrections for volume averaging, k_vol, were first explicitly mentioned in the addendum to TG-51 (Almond et al 1999) and adopted later on in TRS-483 (Palmans et al 2017). In TRS-483, k_vol is included in $k_{{Q_{\text{msr}}},Q}^{{f_{\text{msr}}},{f_{\textrm{ref}}}}$, in the TG-51 addendum it is included as a separate correction, P_rp , in the formalism. Andreo et al (2020) recently proposed k_vol as a separate correction from k_Q,Q0 , for the update of TRS-398. k_vol is dependent on the beam profile over the chamber cavity and can be obtained by a simple integration of the beam profile over the length of the ionization chamber cavity or, for commonly applied FFF photon beams, by the equations given by TRS-483 Palmans et al (2017):

$$\begin{equation}{k_{{\text{vol}}}} = 1 + \left( {6.2 \times {{10}^{ - 3}}\,\cdot\,{{TP}}{{{R}}_{20,10}} - 3.57 \times {{10}^{ - 3}}} \right)\,\cdot\,{\left( {\frac{{100}}{{{{SDD}}}}} \right)^2}\,\cdot\,{L^2}.\end{equation} \tag{ 12 }$$

Here, k_vol is dependent on beam quality Q (specified by TPR_20,10), SDD and length, L, of the ionization chamber cavity (distances in cm). For the Elekta Unity^™ the results of this equation were compared to those of the integration of its beam profile over the length of the ionization chamber and shown to agree within 0.02(5)% (de Prez et al 2019b).

This suggests that the impact of the magnetic field on k_vol is small for reference fields and it does not affect ${k_{\vec B,{Q_{\text{msr}}}}}$. Consequently, similar methods as those described in TRS-483 and the TG-51 addendum are applicable to implement this correction in the formalism. This can be done either by the TRS-483 approach, where k_vol is considered as a contribution to the beam quality correction factor (or $k_{{Q_{\text{msr}}},Q}^{{f_{\text{msr}}},{f_{\textrm{ref}}}}$), or by the TG-51 approach in which it is included as a separate correction factor to the ionization chamber reading.

For reference dosimetry at commercially available MRI-linacs (sections 2.1 and 2.2) the knowledge of the ionization chamber's response is only needed for a few discrete magnetic field strengths (0.35 T for ViewRay MRIdian^™ or 1.5 T for Elekta Unity^™). Nevertheless, the investigation of the chamber response as a function of magnetic field strength has gained considerable interest to better understand the behaviour and characteristics of ionization chambers in magnetic fields.

For the experimental investigation of chamber response curves an adjustable magnetic field is needed, which is not available on commercially available MRI-linacs. Therefore, several papers investigate chamber response as a function of magnetic field strength experimentally using large electromagnets, or by applying Monte Carlo methods (see section 6.1). In most of these papers the relative response (i.e. the corrected ionization chamber reading, M_Q (equation (6), normalized to M_Q without magnetic field present) is determined as a function of magnetic field strength. Some papers calculate or measure ${k_{\vec B,{Q_{\text{msr}}}}}$ as a function of the magnetic field strength (e.g. Pojtinger et al (2018)).

Generally these response curves do not monotonically increase with magnetic field strength but may show a maximum or minimum depending on the orientation between radiation beam, chamber axis and direction of the magnetic field, see for example figure 10. Furthermore, the shape of the curves depends strongly on the chamber type and the curves are typically asymmetric for the perpendicular orientation. Most probably this is a results from the direction of deflection of the secondary electrons which is either to the stem (orientation e, figure 7) or to the tip of the ionization chamber (orientation d, figure 7), see also section 5.3. Table 2 gives an overview of the maximum change of response for different types of ionization chambers and for different orientations between the radiation beam, chamber axis and direction of the magnetic field. This data is extracted from publications discussed in the following.

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Meijsing et al (2009) were the first to investigate the impact of a magnetic field on the response of a Farmer NE2571 chamber experimentally and by means of Monte Carlo simulations for two orientations of the chamber axis with respect to the radiation beam. The Monte Carlo simulations (using the Geant4 toolkit) of the response of the Farmer chamber in the magnetic field match the measurements within the simulation precision of 2.5% standard deviation after a small misalignment of about 3° between the incident beam direction and the magnetic field has been considered. Furthermore, the paper directed the attention to the fact that the orientation of the chamber with respect to the magnetic field and radiation beam plays a crucial role for the response of ionization chambers in magnetic fields.

Reynolds et al (2013) extended these investigations by using the Monte Carlo code PENELOPE to model the responses of NE2571 and PR-06 C Farmer-type ionization chambers in the presence of orthogonal and inline magnetic fields (with respect to the photon beam) of varying magnitude. The simulated response curves for the NE2571 chamber agree very well with the curves obtained by Meijsing et al (2009).

This work was re-examined and extended in Reynolds et al (2015, 2017) who investigated the response of a Capintec PR-06 C ionization chamber in magnetic fields oriented either parallel or perpendicular to an incident photon beam. The ionization chamber response was calculated as a function of a number of polar and azimuthal angles. It was found that in a magnetic field perpendicular to the beam direction small angular deviations of the chamber axis of no more than 3° from its nominal directions either parallel or perpendicular to the beam can change the response by more than 1%—an effect which has also been observed unintentionally by Meijsing et al (2009). Malkov and Rogers (2016) calculated the response curve of an NE2571 ionization chamber as a function of the magnetic field strength. Their results largely agreed with previous results by Reynolds et al (2013) and Meijsing et al (2009).

A few papers have investigated response changes for orientations of chamber axis parallel to the beam (orientation c in figure 7) (Meijsing et al 2009, Reynolds et al 2017). The results show that the response changes in opposite direction compared to orientations a, b, d and e (i.e. response ratio < 1) with values down to 0.9.

It was found by Agnew et al (2017) that both the change of the chamber's response as well as the magnetic field strength at which the maximum change of response occurs depend on the chamber type. In their study measurements have been done in a ⁶⁰Co beam, which might be a reason for the difference in maximum response change with other studies for the same chamber types in table 2. Malkov and Rogers (2017) used the EGSnrc Monte Carlo system to calculate the change of response in magnetic fields for the ionization chambers investigated experimentally by Agnew et al (2017). Similar to Spindeldreier et al (2017), they found that the experimental results could only be reproduced by Monte Carlo simulations when a dead volume close to the chamber stem is considered.

From table 2 it can be seen that the response changes for cylindrical chambers in a perpendicular orientation are significantly larger than for a parallel orientation. This is supported by the dataset (including a large set of different chamber types) of Malkov and Rogers (2018) which shows that for a parallel orientation at B = 1.5 T, ${k_{\vec B,{Q_{\text{msr}}}}}$ does not deviate from unity by more than 2% for reference type chambers. Note that this does not imply that the maximum response for any magnetic field strength is below 2%.

The previous overview and table 2 only focus on cylindrical chambers. Both Pojtinger et al (2018) and Malkov and Rogers (2018) investigated response changes for several plane-parallel chambers. It is clear from these studies that response changes larger than 10% can be found with significant variations between chamber types.

From table 2 it can be seen that the most data is available for the chamber types NE2571 and PTW 30013. The variation in the data set NE2571 chamber can be attributed to the fact that the measurements are strongly affected by small air gaps around the cavity. This effect is further discussed in section 5.2. Another reason is that not all simulation results for both chamber types have included the dead volume, which improves the consistency between measurements and simulations. The variation for the data set of the PTW 31010 chamber can be attributed to the fact that data set includes measurement and simulation results from both ⁶⁰Co (Agnew et al 2017, Malkov and Rogers 2017) and linac beams (Malkov and Rogers 2017). In summary: response curves are affected by chamber type, chamber/magnetic field orientation, air gaps around the chamber and beam quality (⁶⁰Co vs. linac beam).

In reference dosimetry for conventional radiotherapy, non-waterproof chambers are often used in a water phantom in combination with a waterproof sleeve as recommended by several CoPs (Almond et al 1999, Andreo et al 2000, Aalbers et al 2008, McEwen et al 2014). Furthermore, solid phantoms are frequently used for QA measurements and sometimes even for absolute dose measurements. In both configurations, inserting of the ionization chamber requires that the insert or sleeve internal diameter is slightly larger (∼0.1 mm) than the ionization chamber's outer diameter resulting in thin air gaps between the ionization chamber and phantom.

While in dosimetry for conventional radiotherapy the influence of such thin air gaps on the response of the ionization chamber is negligible, it has been shown by several authors that in the presence of magnetic fields the response of ionization chambers is strongly influenced by these submillimetre air gaps (Hackett et al 2016, Agnew et al 2017, 2017a, O'Brien et al 2017b).

Hacket et al (2016) were the first to draw attention on this effect by measuring the responses of three waterproof Farmer-type chambers placed in a waterproof sleeve in a water phantom with the chamber axis, the radiation beam and the magnetic field pairwise perpendicular. A thin air gap of about 0.1 mm–0.2 mm existed between the chamber and the sleeve. On an MRI-linac (Elekta Unity^™ prototype) the response of the chambers increased by 0.7%–1.2% when the thin air gap was removed by filling the sleeve with water. The experiment was replicated on a conventional linac without a magnetic field resulting in an insignificant response change (<0.2%).

Malkov and Rogers (2016) investigated the influence of 0.5 mm and 1.0 mm air gaps on the response of a NE2571 Farmer-type chamber in magnetic fields using the EGSnrc Monte Carlo code. In the simulations, two orientations of the chamber axis with respect to the radiation beam were considered (perpendicular and parallel) with the chamber axis always perpendicular to the magnetic field. In both configurations, the chamber response without a magnetic field changed negligibly (<0.2%) when a uniform air gap of 1 mm was introduced around the walls of the chamber. In the perpendicular orientation, an increase of the chamber response was found for magnetic fields below 1.0 T, whereas for higher magnetic fields a reduction of the response in the order of 1% was observed for a 0.5 mm air gap in good agreement with the findings by Hackett et al (2016). For the orientation with chamber parallel to the beam axis the situation is reversed; with a reduction in response for B < 1 T and an increase for B > 1 T.

The previous results were confirmed by O'Brien et al (2017b) who placed three Farmer-type ionization chambers in plastic phantoms with machine drilled holes congruent with the chamber shape. Each chamber was rotated axially, and the response was recorded as a function of rotation angle both with and without a magnetic field in an Elekta Unity^™ system. Without a magnetic field a small response variation (<0.1%) was observed for all chambers. When a 1.5 T magnetic field was applied, variations of the response between ±0.2% and ±1.3% were observed depending on the chamber type. The larger variation of the response in the presence of a magnetic field was attributed to submillimetre air gaps whose thickness around the chamber changes during a rotation due to small misalignment of the rotation axis (wobbling).

Agnew et al (2017) investigated the influence of the location of the air gap around the chamber in greater detail. Several routinely used cylindrical ionization chambers were placed in custom-made PMMA (Poly(methyl methacrylate)) phantoms in a ⁶⁰Co beam between the poles of a strong electromagnet, the chamber axis, beam direction and magnetic field being pairwise perpendicular. The phantom was designed to be tightly fitting and rotational symmetric around the ionization chamber except for a small recessed region 0.3 mm in depth comprising one quarter of the chamber's circumference. This recessed region, which provides an asymmetric air gap, could be positioned at different angles around the ionization chamber. Depending on the chamber type, variations of the response between 1.1% ± 0.1% (Exradin A1SL) and 8.5% ± 0.2% (PTW 31006) were observed in an applied magnetic field of 1.5 T when the asymmetric air gap was rotated around the chamber. For a Farmer-type chamber PTW 30013 a variation of the response of 3.8% ± 0.2% was observed. After water was added to the phantom cavity to eliminate all air gaps, the variation of response with position of the cavity recess was reduced to 0.20% ± 0.01%. By changing the magnetic field strength used for the measurements, it was found that the observed variation of response increases with increasing field strength. For a PTW 31010 chamber a variation of the response with the position of the recessed region of 0.25% was observed for B = 0.25 T, whereas it increased to 2.9% at 2 T. However, for both field strengths used in commercially available systems, 0.35 T and 1.5 T, the response with asymmetric air gaps and the response with water-filled air gap differed by 1.0% or more. Additionally, it was found that also the magnetic field strength at which the maximum of the response curve occurs (see section 5.1) as well as the slope of this curve depend on the size of the air gap and—to a lesser extent—on the location of the air gap around the chamber.

In order to find an explanation for the observed changes of response in the presence of small air gaps, O'Brien et al (2017a) investigated the effect of submillimetre gaps between a Farmer-type chamber and a solid phantom by means of Monte Carlo simulations. For symmetrical air gaps, calculated chamber responses were all within 0.5% for air gap thickness up to 1.4 mm and magnetic field strength up to 1.5 T. A similar result was obtained for asymmetric ('one-sided') air gaps without a magnetic field; here the response varied by less than 0.5% depending on the orientation of the air gap with respect to the chamber and radiation source. With B = 1.5 T, a variation of the response of more than 2.5% was found for 0.2 mm air gaps depending on its position around the chamber. By investigating the contributions to the total dose from regions outside, adjacent and inside the ionization chamber it was found that the observed variation of the response is not simply caused by the electron return effect (Raaijmakers et al 2005, 2007, 2008), as the gyroradius of most electrons is too large to be responsible for the observed effects. Instead, the change of response is attributed to the loss of dose contributions from electrons originating inside the air gap volume, which is not completely compensated by more distant electrons owing to their reduced range in the magnetic field.

In summary, the presence of small air gaps around ionization chambers increase the variation in response change because of small positioning variations that affect the size and symmetry of the air gaps. In addition, the air gap has a significant impact on the response change compared to no (or a water-filled air gap). While the former effect seems to be mainly relevant for 1.5 T MRI-linacs, the latter has a significant impact for both field strengths used in commercially available systems, 0.35 T and 1.5 T. Apart from the aforementioned factors, the influence of air gaps on chamber response depends on the orientation of the chamber axis, beam axis and magnetic field. Generally, it is concluded that the use of solid phantoms or waterproof sleeves is not adequate for reference dosimetry or QA measurements in MRI-linacs, due to the unavoidable presence of submillimetre air gaps and the unknown distribution of the air gap thickness. Instead, the sensitive volume of the ionization chamber should be wholly immersed in water in order to obtain reproducible results and the non-waterproof chamber seems less suited for reference dosimetry in MRI-linacs.

McNiven et al (2008) have combined micro-CT images of ionization chambers with calibrations of these chambers. From this data it was possible to correlate the geometrical volume (from the micro-CT images) of the cavity with the 'electrical collecting volume' which is the volume derived from the produced charge and delivered dose to the cavity. The authors demonstrate that the volume of the cavity is significantly larger than the electrical collecting volume. Based on these findings, Ross (2009) suggested that the collecting volume could be obtained from finite element method (FEM) simulations of the electric field inside the ionization chamber and by excluding the volume in which the electric field lines from the chamber wall reach the guard ring instead of the electrode. In a more recent publication, Butler et al (2015) investigated the spatial response of ionization chambers with micro-beams. They demonstrate that the response of the ionization chamber cavity is not homogeneous and that Farmer-type ionization chambers exhibit a volume with a low sensitivity close to the chamber guard electrodes.

Malkov and Rogers (2017) used Monte Carlo simulations to calculate ${k_{\vec B,{Q_{\text{msr}}}}}$ for ionization chambers, for which the correction factors have been previously determined experimentally by Agnew et al (2017). In their simulations, part of the cavity close to the guard electrode was modelled as a cylindrically shaped volume of varying dimensions. The impact of these volumes on the calculation of ${k_{\vec B,{Q_{\text{msr}}}}}$ was investigated by comparing simulations of response curves in which the dose scored in dead volumes is neglected with simulations in which it is included. It is shown that, depending on the chamber type, small dead volumes with an extent of at most 1 mm might change the response curves (e.g. Figure 10) by several percent. The influence of the dead volume is larger for chambers with a smaller collecting volume. Finally, the experimental results obtained by Agnew et al (2017) could be reproduced by a proper choice of the size of the dead volume.

A similar approach has been followed by Spindeldreier et al (2017) who determined the response curves of five custom-built and one commercial (PTW 30013) Farmer-type ionization chambers with increasing radius of the collecting volume, both experimentally and by Monte Carlo simulations using the EGSnrc toolkit. By varying the dead volume of the investigated chambers in Monte Carlo simulations the experimentally obtained response curves could be reproduced. These dead volumes were then used in Monte Carlo simulations to calculate magnetic field correction factors for orientations of the ionization chamber which could not be realized experimentally.

A more detailed treatment of the dead volume in the Monte Carlo calculation of correction factors was reported by Pojtinger et al (2019) who applied the idea formulated by Ross (2009). In the first step, the electric field inside a Farmer-type chamber PTW 30013 was calculated by FEM simulations. From these calculations the shape and size of the dead volume was deduced. It was found that the dead volume has a doughnut shape (see figure 11) which is different from the cylindrically shaped dead volumes assumed by Malkov and Rogers (2017) and Spindeldreier et al (2017). In the second step, ${k_{\vec B,{Q_{\text{msr}}}}}$ was calculated by means of Monte Carlo simulations, in which, the collecting volume of the ionization chamber was defined as the geometrical volume excluding the dead volume obtained from the FEM simulations. The calculated ${k_{\vec B,{Q_{\text{msr}}}}}$ were then compared to experimentally obtained factors. When the dead volume was not considered, a discrepancy of more than 1% was found between measured and Monte Carlo calculated correction factors. After accounting for the FEM calculated dead volume the measured and simulated ${k_{\vec B,{Q_{\text{msr}}}}}$ agreed on the 0.1% level which is well within the uncertainties of the experiments and simulations.

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These studies demonstrate that the dead volume must be considered in Monte Carlo simulations of detector response in the presence of magnetic fields. Based on the described literature above the following approaches in modelling the dead volume can be distinguished:

• Approach 1: in which the shape of the dead volume (e.g. cylindrical) is fixed. Dimensions are estimated based on the agreement between simulations and experiments.
• Approach 2: in which shape and dimension of dead volume is based on the electrical field line distribution from FEM simulations by assuming that electrical field lines that do not hit the collecting electrode are part of the dead volume.

The disadvantage of the first approach is that the Monte Carlo result is not completely independent from the measurements. In the last approach, it is assumed that the charge produced in the cavity has no kinetic energy and that the trajectory of the charge to the collecting electrode is not affected by the magnetic field. An approach in which these effects are also accounted for would require Monte Carlo simulations to a much lower energy range of the electrons than currently done in simulations of detector response, for which the cut-off energy is typically 10 keV, and in which the electric field is included. Given the good agreement between measurements and simulations it is anticipated that the impact of these improvements is small.

6.1.1. Monte Carlo simulation of charged particle transport in magnetic fields

The two major aspects in the simulation of charged particle transport are the simulation of multiple soft scattering events and crossing of the boundaries between two bodies. Generally, modern Monte Carlo codes use a mixed scheme for particle transport in which condensed history (CH) steps alternate with hard discrete interactions (or hard events, HE). In a CH step multiple soft scattering events are modelled as a single soft event (SE) (figure 12).

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In this approach first the step length, s, between two hard events is sampled from the path length probability distribution. Then the position of the SE is simulated by sampling a value for a from a uniform distribution with a range of (0,1) (figure 12). In the next step, the polar scattering angle, θ, of the soft event is sampled. The energy transferred from the particle to the medium before the SE is calculated using the CSDA approach and scored at the position of the SE. The energy transferred after the SE is calculated similarly and scored at the end of the CH step. Together with the energy scoring the velocity of the particle is reduced at these points, which results in three points of the trajectory where the velocity is known.

Most Monte Carlo codes that have implemented charged particle transport in static magnetic and electric fields base this implementation on the theoretical framework formulated in the work of Bielajew (1988). The Lorentz force acting on the charged particle depends on the velocity and position (if the magnetic field is not uniform) of the particle. Exact calculation of the impact of the Lorentz force on particle position and direction requires an integration of the Lorentz force along the particle trajectory (which reduces to an integration of the particle velocity for the case of a uniform magnetic field).

Although, the particle velocity is not known at each point of the particle trajectory, and consequently the Lorentz force acting on it, Bielajew (1988) has shown that under certain conditions and applying a first order integration of the Lorentz force along the particle trajectory, the simulation of the CH event can be performed by superimposing equations of motion of a charged particle under the influence of the Lorentz force in vacuum to the 'field-free' (i.e. without magnetic and electric field) equations of motion of charged particles in media based on those from statistical treatments such as by Moliere (1947) to describe the multiple soft scattering events.

The conditions described by Bielajew (1988) are represented as a set of constraints that restrict the relative change of the following quantities in a single CH step: particle energy and velocity, magnetic and electric field strength. All these constraints affect the maximum distance that a particle is allowed to travel in the CH event.

Although boundary crossing is treated differently in most Monte Carlo codes (Kawrakow et al 2013, Salvat et al 2014), they have one aspect in common. In each step made by the particle close to the boundary, the algorithm checks whether the particle crosses a boundary by calculating the intersection of the intended step with the nearest interface(s). In this check the intended step is assumed to be a straight path. Therefore, this method is not applicable for application in the presence of magnetic fields in which steps are curved paths. This might lead to missing boundaries between bodies and wrong registration of particle position and body and erroneous scoring of dose to the body.

6.1.1.1. EGSnrc

Simulation of charged particles in magnetic fields was implemented in EGS4. In the later releases of EGSnrc special actions were required to enable these simulations (Kawrakow et al 2013). The method in EGS4 and earlier EGSnrc releases is based on the same first-order integration method as Bielajew proposed, but differ in the implementation of the constraints (Kawrakow et al 2013). This is because EGS has only implemented a uniform magnetic field. Therefore, the change in particle direction is only proportional to the particle gyroradius of equation (2). This allows for the replacement of the set of four constraints to limit the step length, s, by a single constraint on the relative change in particle gyroradius along the CH step.

Malkov and Rogers (2016) proposed a further improvement in the implementation of the magnetic field in EGSnrc. Using the available velocity vectors at three points in a CH step (figure 12) they proposed a third order integration method, in which the velocity vector is integrated over the full CH step. Since the third order integration increases computation time, an adaptive algorithm is proposed that uses first order integration for small steps (low value of s) and third order integration for larger steps. The authors also describe how the constraint on the relative change in particle gyroradius along the CH step can be applied for the third-order integration method and show that this constraint becomes less restrictive.

In addition, Malkov and Rogers (2016) developed a boundary crossing algorithm compatible with simulations in magnetic fields. The algorithm transports the particle along the calculated step length by smaller steps in an iterative process. In each loop of the iteration the shortest distance from particle position to the nearest boundary is calculated. Then the particle is transported a small step with a step length equal to this shortest distance using an analytical solution for the particle trajectory in a magnetic field. This process is repeated until the shortest distance is smaller than a threshold or until the full step length has been reached. In the first situation, the particle is transported in a straight line to the other side of the body interface. The current release of EGSnrc has implemented the Malkov and Rogers (2016) third order integration method and boundary crossing algorithm.

6.1.1.2. PENELOPE

The PENELOPE (Salvat et al 2014) package contains routines for simulation of charged particle transport in both magnetic and electric fields. Implementation in the user codes requires only little software coding which is described in the routines. Similar to EGS4, PENELOPE uses first-order integration of the Lorentz force to simulate the direction change of charged particles in magnetic fields. However, in a single CH event this integration is performed twice; once before the SE and once after the SE (figure 12). In PENELOPE, in each step s is limited by constraints on the relative change in electric field, magnetic field, kinetic energy and velocity. In a homogeneous magnetic field this set reduces to the constraint on relative change in velocity or direction. Although this constraint has the same physical meaning as the one used in EGSnrc, due to the different implementation of the magnetic field in the code and due to differences in radiation transport, its value required for accurate simulations might be different.

6.1.1.3. GEANT4

Geant4 (Agostinelli et al 2003) uses a generic approach to calculate the curved trajectory of a charged particle in the presence of a field (Geant4 Collaboration 2017a, 2017b). This approach is applied to all fields implement in Geant4 (i.e. (in)homogenous electric and magnetic fields). This is done for each CH step by either using a numerical integration of the equation of motion (Runge-Kutta) or an approximate analytical calculation as for the case of a uniform magnetic field. In both approaches, the curved trajectory is approximated by a sequence of linear segments. These segments are used to determine a boundary crossing. A set of geometrical parameters can be used to constrain the deviation of the segments from the 'real' curved trajectory. This set of parameters is the following (see also Simiele and Dewerd (2018)):

• MissDistance: the maximum distance between the linear segment and the curved trajectory;
• δ_intersection: the maximum distance between the intersection of the segment with a boundary and the 'real' trajectory with the boundary. Because the linear segments are always on the 'inside' of the curved trajectory, this can introduce a bias in boundary crossing; and
• δ_{one step}: the accuracy for the endpoint of segments, which do not intersect a volume boundary.

In addition, several other parameters can be used in Geant4 to steer the accuracy of a full CH step by limiting the step length. A CH step will only be accepted if the constraints on the following parameters are fulfilled:

• _max: maximum accepted relative error on particle position and momentum at the end of a CH step;
• _min: minimum tolerated relative error on particle position and momentum at the end of a CH step;
• maxStep: the maximum accepted CH step length for a particle in a magnetic field; and
• minStep: the minimum tolerated CH step length for a particle in a magnetic field.

6.1.2. Benchmarks for Monte Carlo simulation in the presence of magnetic fields

6.1.3.  ${k_{\vec B,{Q_{\text{msr}}}}}$ data based on Monte Carlo simulations.

The general applied method to calculate ${k_{\vec B,{Q_{\text{msr}}}}}$ factors is described in O'Brien et al (2016) and is similar to the calculation of ${k_{Q,{Q_0}}}$ factors as applied, for example, by Wulff et al (2008). According to this method, which is graphically explained in figure 13, four simulations using a detailed geometry model of the ionization chamber are performed: two simulations to calculate the dose to the air cavity of the ionization chamber, D_cavity, with and without a magnetic field; and two simulations to calculate the dose-to-water in the absence of the chamber at the reference point of the chamber with and without a magnetic field. The method of O'Brien et al (2016) is based on the assumptions that:

• (a)  
  the measured charge, M, is proportional to the dose to cavity; and
• (b)  
  and this factor of proportionality is independent on magnetic field strength and chamber orientation.

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Several papers describe Monte Carlo based methods to determine detector response changes as a function of magnetic field strength (Meijsing et al 2009, Reynolds et al 2013, O'Brien et al 2016, Shukla et al 2017, Spindeldreier et al 2017, Malkov and Rogers 2018, Pojtinger et al 2018). The earlier papers (Meijsing et al 2009, Reynolds et al 2013) focused on general parameters affecting ionization chamber response in magnetic fields such as their orientation. In these papers the simulations to calculate the dose-to-water in the absence of the chamber at the reference point of the chamber with and without the magnetic field are ignored. These results are not in line with the definition of the formalisms (section 4.3) and therefore the results of these papers are classified in table 4 as a ratio of readings. Several of the later published papers (O'Brien et al 2016, Spindeldreier et al 2017, Malkov and Rogers 2018, Pojtinger et al 2018) have described the calculation of ${k_{\vec B,{Q_{\text{msr}}}}}$ in line with the formalisms and with other further refinements in methodology due to improved understanding such as: inclusion of the dead volume (see section 5.1) and avoidance of air gap effects (see section 5.2). In most studies the simulations have been done for several cylindrical ionization chambers. An overview of papers on Monte Carlo based ${k_{\vec B,{Q_{\text{msr}}}}}$ determination is given in table 4 together with the relevant characteristics of the applied methodology.

Table 4. Overview of papers on ${k_{\vec B,{Q_{\text{msr}}}}}$ simulations with chamber types investigated and the relevant characteristics of the applied methodology. The definition for the orientation of the chamber with respect to the magnetic field vector is in line with section 4.1 and figure 7.

Paper	Chamber type	Chamber geometry	Dead volume	Beam model	B (T)	Orientation chamber	Monte Carlo code	Fano test	Experimental benchmark	Result
Meijsing et al (2009)	NE2571	In-air with build-up cap	No	Spectrum 6 MV Elekta SL25 (parallel beam)	0–2.0	a or a b and d or e	Geant4.9.1.p01	No	Yes	$\frac{{M_Q^{\vec B}}}{{{M_Q}}}$
Reynolds et al (2013)	NE2571 PR06 C	In-water	No	Spectrum 6 MV Elekta SL25 (parallel beam) and Varian 600 C 6 MV	0–2.0	e	PENELOPE	No	Yes (up to 1.2 T)	$\frac{{M_Q^{\vec B}}}{{{M_Q}}}$
O'Brien et al (2016)	NE2571, b PTW30013, PTW30012, PTW30011, PTW30010, Exradin A19	In-water	No	Point-source, Elekta Unity™ spectrum	1.5	a, b, d and e	Geant4.9.6p04	Yes	Yes	${k_{\vec B,{Q_{\text{msr}}}}}$
Spindeldreier et al (2017)	PTW30013, 5 custom built chambers	In-water	Yes	PSF 6 MV Siemens Primus	0–3.0	d and e	EGSnrc c	No	Yes	${k_{\vec B,{Q_{\text{msr}}}}}$
Malkov and Rogers (2018)	All chamber of (Muir and Rogers 2010)Muir et al	In-water	Yes	Spectra of 60Co beam, Varian TrueBeam (6 MV, FFF), Elekta Unity™, Varian Clinical (6 and 10 MV).	0.35, 1.5	b and e	EGSnrc d	Yes	No	${k_{\vec B,{Q_{\text{msr}}}}}$
Pojtinger et al (2018)	NE2571, PTW30013, PTW30016, PTW30015, Roos, NACP-02, Adv. Markus, Markus	In-water	No	Head 6 MV Elekta FFF linac (0.35 T), Spectrum Elekta Unity™ (1.5 T)	0.35, 1.5	and e	EGSnrc d	Yes	Yes	${k_{\vec B,{Q_{\text{msr}}}}}$
Pojtinger et al (2019)	PTW30013	In-water	Yes	Head 6 MV Elekta FFF linac (0.35 T), Spectrum Elekta Unity™ (1.5 T)	1.5	e	EGSnrc d	Yes e	Yes	${k_{\vec B,{Q_{\text{msr}}}}}$
Shukla et al (2017)	Exradin A1SLMR	In-water	No	Spectrum 6 MV linac	0–1.2 T	a, b, c and d	EGSnrc d	No	Yes	$\frac{{M_Q^{\vec B}}}{{{M_Q}}}$

^a The applied direction cannot be derived from the paper. ^b Non-waterproof chambers were modelled with a 1 mm thick layer of PMMA surrounding them, to represent a waterproof sleeve. The sleeves were assumed to be in full contact with the chambers with no air gaps in between. ^c With customised magnetic field macro. ^d With new EEMF routines from Malkov and Rogers (2016). ^e From Pojtinger et al (2018).

6.1.4.  Intra-type variability of ${k_{\vec B,{Q_{\text{msr}}}}}$

6.2.1. Based on primary standards

6.2.2. Cross-calibration against reference detector

6.2.3. Monitor dependent method

A third method for ${k_{\vec B,{Q_{\text{msr}}}}}$ determination of ionization chambers, here coined as the monitor dependent method, is described in several papers (Smit et al 2013, van Asselen et al 2018, Pojtinger et al 2019). In contrast to the two previous described methods, for this method D_w is not measured separately in the presence and in the absence of the magnetic field. Instead, by means of simulations, it determines a ratio of D_w at the point of measurement between magnetic field present and magnetic field absent, under the condition of a constant photon fluence, φ. Next the reading of the chamber is measured in the presence and in the absence of the magnetic field. The beam photon fluence in the presence and in the absence of the magnetic field is thereby monitored, $M_{\text{monitor}}^{\vec B}$ and ${M_{\text{monitor}}}$ respectively, to correct for possible photon fluence variation with and without a magnetic field, ${\varphi ^{\vec B}}$ and $\varphi$, during the ionization chamber measurements. Therefore, an underlying assumption is that the ratio $\frac{\varphi }{{{M_{\text{monitor}}}}}$ should be independent from the magnetic field.

The measurement equation for determination of ${k_{\vec B,{Q_{\text{msr}}}}}$ is written as the product of two factors, ${c_{\vec B}}$ and ${k_{\vec B,M,{Q_{\text{msr}}}}}$ as shown in the following equation:

$$\begin{equation}{k_{\vec B,{Q_{\text{msr}}}}} = \frac{{N_{D,{\text{w}},{Q_{\text{msr}}}}^{\vec B}}}{{{N_{D,{\text{w}},{Q_{\text{msr}}}}}}} = {c_{\vec B}} {k_{\vec B,M,{Q_{\text{msr}}}}}\end{equation} \tag{ 13 }$$

where ${c_{\vec B}}$ is given by

$$\begin{equation}{c_{\vec B}} = \frac{{\frac{{D_{{\text{w}},{Q_{\text{msr}}}}^{\vec B}}}{{{\varphi ^{\vec B}}}}}}{{\frac{{{D_{{\text{w}},{Q_{\text{msr}}}}}}}{\varphi }}}.\end{equation} \tag{ 14 }$$

$D_{w,{Q_{\text{msr}}}}^{\vec B}$ the absorbed dose-to-water for beam quality Q_msr with magnetic field present at the same reference point as the absorbed dose-to-water without the magnetic field present, ${D_{\text{w},{Q_{\text{msr}}}}}$. This factor depends on the magnetic field strength. It can be determined by scoring the dose per history with Monte Carlo simulations in a homogenous water phantom. Therefore, details of the ionization chamber geometry and materials are not required, which simplifies the simulations.

The factor ${k_{\vec B,M,{Q_{\text{msr}}}}}$ is the ratio of ionization chambers readings and is defined as

$$\begin{equation}{k_{\vec B,M,{Q_{\text{msr}}}}} = \frac{{\frac{{{M_{{Q_{\text{msr}}}}}}}{{{M_{\text{monitor}}}}}}}{{\frac{{M_{{Q_{\text{msr}}}}^{\vec B}}}{{M_{\text{monitor}}^{\vec B}}}}}\end{equation} \tag{ 15 }$$

where ${M_{{Q_{\text{msr}}}}}$, the corrected reading in the absence of the magnetic field, and $M_{{Q_{\text{msr}}}}^{\vec B}$, the corrected reading obtained in the presence of a magnetic field are a measure for the fluence in both situations. ${k_{\vec B,M,{Q_{\text{msr}}}}}$ depends on the magnetic field strength and the ionization chamber orientation relative to the magnetic field and photon beam (sections 4.1 and 5.1).

The ${k_{\vec B,M,{Q_{\text{msr}}}}}$ factor can be determined for any ionization chamber during ramp-up and ramp-down procedures. For an MRI-linac, this can be done for a single magnetic field strength for any orientation of the ionization chamber (Smit et al 2013, van Asselen et al 2018). For a radiation facility (linac or ⁶⁰Co-source) combined with an electromagnet, this can be done for varying magnetic field strengths and, due to the limited space between the poles, for the perpendicular orientations ((d) and (e), figure 7) (Pojtinger et al 2018). The correction for possible variation of photon fluence output in both conditions (i.e. with and without magnetic field) can be based on an independent monitor in the beam in a low magnetic field area (Smit et al 2013) or on the linac internal monitor (van Asselen et al 2018, Pojtinger et al 2019).

6.2.4.  Evaluation of measurement-based methods for determination of $k_{{Q_{\text{msr}}},{Q_0}}^{{f_{\text{msr}}},{f_{\textrm{ref}}}}$ and ${k_{\vec B,{Q_{\text{msr}}}}}$ factors.

6.3.1.  Data for ${k_{\vec B,{Q_{\text{msr}}}}}$

Since 2009, many papers have been published which investigate quantitatively the change of response of ionization chambers in magnetic fields or determine the magnetic field correction factors, ${k_{\vec B,{Q_{\text{msr}}}}}$, needed for absorbed dose measurements (see section 4.3)—either experimentally or by means of Monte Carlo simulations. From today's perspective, most of the early publications on this topic have some flaws caused by the lack of knowledge about peculiarities of dosimetry in magnetic fields which have been realized at a later stage. Such flaws are, for instance:

• the use of waterproof sleeves or solid phantoms without considering the air gap effect (see section 5.2);
• the use of Monte Carlo algorithms or simulation parameters which have not been benchmarked thoroughly for their use in magnetic fields (see section 6.1.2); and
• the negligence of the influence of the dead volume on the results of Monte Carlo calculations (see section 5.3).

Furthermore, the orientation of the chamber, magnetic field and beam direction is not always specified in sufficient detail and sometimes it is not clearly stated if the correction factors ${k_{\vec B,{Q_{\text{msr}}}}}$ or ${k_{\vec B,M,{Q_{\text{msr}}}}}$ are determined (see sections 4.3 and 5.1), i.e. whether the change of the absorbed dose due to the magnetic field ${c_{\vec B}}$ is considered or not (see equation (13)). Such flaws, together with a large variety of measurement setups and conditions (phantom, SSD, field size, beam quality etc.), make it difficult to compare the results from earlier and recent publications and to judge their applicability for reference dosimetry in MRI-linacs. Moreover, in most cases a proper uncertainty evaluation of the determined correction factor is lacking.

The first measurement of magnetic field correction factors in a (prototype) Elekta Unity^™ MRI-linac has been reported by Smit et al (2013) for a set of NE2571 ionization chambers in a polystyrene slab phantom with a perpendicular orientation, using (partially) the monitor dependent method (section 6.2.3). Although it is mentioned in the publication that the doses applied to all NE2571 ionization chambers with and without magnetic field were equal, the ${c_{\vec B}}$ factor is not used in the determination of ${k_{\vec B,{Q_{\text{msr}}}}}$. The correction factors determined here are therefore ${k_{\vec B,M,Q}}$. Due to the use of the solid phantom, an influence of the air gap effect on the results (see section 5.2) cannot be excluded. For the four ionization chambers investigated here, values for ${k_{\vec B,M,Q}}$ between 0.951 and 0.955 (average: 0.953) were obtained, which suggests a relatively small intra-type variation for chambers of type NE2571 and a neglectable influence of the air-gap effect in this experiment. Furthermore, because the four chambers were positioned at three different depths in the phantom, it can also be concluded that the magnetic field correction factors are largely independent on depth. The correction factors obtained by Smit et al (2013) agree with the results obtained by Meijsing et al (2009) and Reynolds et al (2013); a preliminary relative uncertainty of 0.5% is estimated for the correction factor.

The first correction factors for Farmer-type ionization chambers (PTW 30013 and IBA FC65-G chamber) based on measurements using a water calorimeter as a primary standard for absorbed dose to water in an Elekta Unity^™ MRI-linac have been reported by de Prez et al (2016b). No measurements in the Elekta Unity^™ MRI-linac without magnetic field have been done, therefore the calibration coefficient in a ⁶⁰Co beam quality has been used which results in the product $k_{{Q_{\text{msr}}},{Q_0}}^{{f_{\text{msr}}},{f_{\textrm{ref}}}}\, \cdot \,{k_{\vec B,{Q_{\text{msr}}}}}$. From these values, ${k_{\vec B,{Q_{\text{msr}}}}}$ was determined using a generic ${k_{Q,{Q_0}}}$ value taken from IAEA TRS-398 (Andreo et al 2000) for $k_{{Q_{\text{msr}}},{Q_0}}^{{f_{\text{msr}}},{f_{\textrm{ref}}}}$. In a later publication de Prez et al (2019b) presents updated data for ${k_{\vec B,{Q_{\text{msr}}}}}$ which have now been determined based on calorimeter measurements in an Elekta Unity^™ MR linac with and without a magnetic field. These correction factors agree within the uncertainty with the previously reported values; the deviation between both sets of correction factors for the two types of chambers investigated in both studies is 0.2% and 0.4%, respectively.

Wolthaus et al (2016) and van Asselen et al (2018) report results for ${k_{\vec B,{Q_{\text{msr}}}}}$ using the monitor dependent method (section 6.2.3). The obtained results agree well with the calorimeter-based correction factors obtained by de Prez et al (2016b, 2019b).

Monte Carlo calculated and experimental data are also presented in a study described by Shipley et al (2019). This study calibrated a Famer-type ionisation chamber (PTW 30013) in a conventional 8 MV Elekta Synergy beam, in an electromagnet and at different magnetic field strengths, using alanine as a transfer standard. Alanine was traceably calibrated against the NPL's primary standard of absorbed dose to water, a graphite calorimeter, and corrected for the effect of the magnetic field based on Billas et al (2020) (see section 6.2.2). In addition, Billas and Duane (2018) determined ${k_{\vec B,{Q_{\text{msr}}}}}$ data with the same method as described in section 6.2.2.

Tables 5 and 6 present a selection of published ${k_{\vec B,{Q_{\text{msr}}}}}$ data for orthogonal MRI-linacs. Some publications publish $k_{{Q_{\text{msr}}},{Q_0}}^{{f_{\text{msr}}},{f_{\textrm{ref}}}}\, \cdot \,{k_{\vec B,{Q_{\text{msr}}}}}$ or $k_Q^{mag}$ factors (e.g. Iakovenko et al (2020)). Recalculation to ${k_{\vec B,{Q_{\text{msr}}}}}$ would require the selection of a $k_{{Q_{\text{msr}}},{Q_0}}^{{f_{\text{msr}}},{f_{\textrm{ref}}}}$ value and makes the resulting ${k_{\vec B,{Q_{\text{msr}}}}}$ value dependent on the selected data set (de Prez et al 2016b). Therefore, these publications are not included in tables 5 and 6.

Table 5.  ${k_{\vec B,{Q_{\text{msr}}}}}$ data for orthogonal MRI-linacs with B = 0.35 T fulfilling the selection criteria. Orientations are defined according to section 4.1 and figure 7. The bold-faced values in the column Parallel orientations are values for orientation b, all other values are for orientation a. References with (MC) indicate that the ${k_{\vec B,{Q_{\text{msr}}}}}$ has been determined with Monte Carlo simulations.

Chamber type	Reference	Perpendicular orientations
		Orientation d		Orientation e		Parallel orientations
		${k_{\vec B,{Q_{\text{msr}}}}}$	$u\left( {{k_{\vec B,{Q_{\text{msr}}}}}} \right)$	${k_{\vec B,{Q_{\text{msr}}}}}$	$u\left( {{k_{\vec B,{Q_{\text{msr}}}}}} \right)$	${k_{\vec B,{Q_{\text{msr}}}}}$	$u\left( {{k_{\vec B,{Q_{\text{msr}}}}}} \right)$
PTW 30013	Spindeldreier et al (2017) (MC)	0.9675	0.31%	0.9830	0.31%	0.9970/0.9965	0.30%
	Pojtinger et al (2019)	0.9649	0.15% a	0.9770	0.15% a
	Malkov and Rogers (2018) (MC)					0.9957	0.10%
	Shipley et al (2019) (MC)	0.9750	0.10%	0.9770	0.10%
	Shipley et al (2019)	0.9780	0.20%	0.9730	0.20%

^a The uncertainty on the calculated ${c_{\vec B}}$ was estimated to be similar to the uncertainty on the Monte Carlo calculated value for ${k_{\vec B,M,{Q_{\text{msr}}}}}$(0.15 %), and is added in quadrature to the reported uncertainties on both the experimental and simulated ${k_{\vec B,M,{Q_{\text{msr}}}}}$.

Table 6.  ${k_{\vec B,{Q_{\text{msr}}}}}$ data for orthogonal MRI-linacs with B = 1.5 T fulfilling the selection criteria. Orientations are defined according to section 4.1 and figure 7. The bold-faced values in the column Parallel orientations are values for orientation b, all other values are for orientation a. References with (MC) indicate that the ${k_{\vec B,{Q_{\text{msr}}}}}$ has been determined with Monte Carlo simulations.

Chamber type	Reference	Perpendicular orientations
		Orientation d		Orientation e		Parallel orientations
		${k_{\vec B,{Q_{\text{msr}}}}}$	$u\left( {{k_{\vec B,{Q_{\text{msr}}}}}} \right)$	${k_{\vec B,{Q_{\text{msr}}}}}$	$u\left( {{k_{\vec B,{Q_{\text{msr}}}}}} \right)$	${k_{\vec B,{Q_{\text{msr}}}}}$	$u\left( {{k_{\vec B,{Q_{\text{msr}}}}}} \right)$
IBA FC65-G	Wolthaus et al (2016)	0.9512 a	1.00%
	van Asselen et al (2018)	0.9520	0.21%			0.9970	0.30%
	Malkov and Rogers (2018) (MC)					0.9917	0.10%
	de Prez et al (2019b)	0.9560	0.34%			0.9950	0.34%
	Billas and Duane (2018)	0.9570	1.00%			1.003/1.002	1.00%
PTW 30013	Spindeldreier et al (2017) (MC)	0.9540	0.30%	0.9590	0.30%	0.9920/0.9930	0.30%
	Pojtinger et al (2019) (MC)	0.9597	0.21% b			0.9963	0.16%
	Pojtinger et al (2019)	0.9607	0.16% b	0.9575	0.05% b
	Malkov and Rogers (2018) (MC)					0.9881	0.10%
	van Asselen et al (2018)	0.9630	0.21%			0.9920	0.20%
	de Prez et al (2019b)	0.9630	0.34%			0.9850	0.34%
	Billas and Duane (2018)	0.9660	1.00%			0.9980/0.9980	1.00%
	Shipley et al (2019) (MC)	0.9700	0.10%	0.9580	0.10%
	Shipley et al (2019)	0.9600	0.60%	0.9630	0.60%

^a Recalculated from the data in the abstract using equation (13). ^b The uncertainty on the calculated ${c_{\vec B}}$ was estimated to be similar to the uncertainty on the Monte Carlo calculated value for ${k_{\vec B,M,{Q_{\text{msr}}}}}$(0.15 %), and is added in quadrature to the reported uncertainties on both the experimental and simulated ${k_{\vec B,M,{Q_{\text{msr}}}}}$.

The criteria applied for the selection of chamber types (and ${k_{\vec B,{Q_{\text{msr}}}}}$ values) were:

• reference type ionization chamber (criteria according to McEwen et al (2014));
• ${k_{\vec B,{Q_{\text{msr}}}}}$ correction factors determined by measurement and Monte Carlo simulation or by two different Monte Carlo codes;
• dead volume considered in Monte Carlo simulations;
• no air gap effect in the experimental determination of ${k_{\vec B,{Q_{\text{msr}}}}}$; and
• estimation of uncertainty for ${k_{\vec B,{Q_{\text{msr}}}}}$.

In addition, only chamber types with more than one data point are included in tables 5 and 6. For the data shown in tables 5 and 6, the standard deviation of the mean values were calculated per orientation (d, e or parallel) and per chamber type. The maximum standard deviation of the mean was 0.21%. The reported data and their uncertainties is consistent with these mean values on the level of k= 2 for 18/20 data points.

Pojtinger et al (2019) calculated and measured ${k_{\vec B,M,Q}}$ factors using a linac combined with an electromagnet. Their Monte Carlo model uses a dead volume determined by FEM simulations and is therefore fully independent from measurements. Calculated ${c_{\vec B}}$ values for the same beam have been published elsewhere (Delfs et al 2018). ${k_{\vec B,{Q_{\text{msr}}}}}$ values in tables 5 and 6 were calculated from the data of both studies using equation (13). For B = 0.35 T only experimental ${k_{\vec B,M,Q}}$ data is reported by Pojtinger et al (2019).

O'Brien et al (2016) calculated magnetic field correction factors for different types of Farmer chambers based on Monte Carlo simulations using Geant4. The observed discrepancies of >1% between their values for the PTW 30013 chamber with orientation d (0.976) and other more recent values given in tables 5 and 6 might be explained by not considering the dead volume. Monte Carlo calculations of magnetic field correction factors which take into account the dead volume (see section 5.3 and table 4) have been done by Spindeldreier et al (2017), Malkov and Rogers (2017, 2018) and Pojtinger et al (2019).

6.3.2.  Data for $k_{{Q_{\text{msr}}},{Q_0}}^{{f_{\text{msr}}},{f_{\textrm{ref}}}}$

7.1.1. Formalism and reference conditions

7.1.2. Chamber characteristics and correction factors

Even though the literature on dosimetry for MRI-linacs contains less references for inline systems than for orthogonal systems, it mainly covers similar topics. The main differences between the two type of facilities observed from the literature is discussed in the remainder of this section.

Because of the different configuration of the magnetic field and radiation field (section 2.3), radiation field characteristics are considerably different for inline MRI-linacs (Liney et al 2016, Begg et al 2019). This is mainly driven by the focussing effect as a result of the parallel direction of the beam with the magnetic field. This becomes mainly apparent in the build-up region of the depth dose profile where a dose increase is observed. From the perspective of reference dosimetry, it can be expected that beam quality specification based on %dd(10)_x is considerably affected by this. However, as shown by Begg et al (2019), TPR_20,10 remains largely unaffected by the magnetic field.

Several papers have investigated response changes of ionization chambers as a function of magnetic field for inline configurations (Reynolds et al 2013, 2015, 2017, Shukla et al 2017, Malkov and Rogers 2018). An overview is provided in table 8. Since conventional electromagnets do not allow for an inline beam configuration to a high field strength, because of the metal pole caps, very little experimental studies on detector response changes have been carried out and are limited to magnetic field strengths below 0.2 T. The Monte Carlo studies of table 8 show that the increase in response change is not higher than 2.5% for the investigated configuration.

For inline MRI-linacs the number of chamber orientations reduces to two, i.e. perpendicular or parallel to the beam. Although only the former configuration is compatible with existing CoPs, the data in table 8 shows that the differences between the two configurations are small (<1.5%). Note that for inline facilities negative magnetic field strengths are symmetrically equivalent to positive magnetic field strengths, which is also supported by the overview in table 8. The study of Spindeldreier et al (2017) investigated the impact of dead volumes for an inline configuration of radiation field and magnetic field. Their data suggests that the impact of dead volumes on Monte Carlo detector response simulations for inline facilities is negligible which is in line with the observations of Malkov and Rogers (2018).

The only set of ${k_{\vec B,{Q_{\text{msr}}}}}$ data that can be compared is that for a PTW 30013 chamber from the Monte Carlo studies of Malkov and Rogers (2018) and Spindeldreier et al (2017). This data shows differences in the order of 0.4% for magnetic field strengths. No comparison with measured data ${k_{\vec B,{Q_{\text{msr}}}}}$ is possible currently.

With future progress to clinical inline MRI-linacs it can be expected that the literature for these facilities will further develop. For now, there is no indication that reference dosimetry for inline MRI-linacs requires techniques or knowledge on chamber characteristics, which are not already available from orthogonal facilities. With further development in this field and comparisons between new data sets from experiments and simulations, the level of consistency of methods and data for MRI-linacs will become apparent.