In-vivo dosimetry for conformal arc therapy using several MOSFET in stereotactic radiosurgery computed by an inverse model

In-vivo dosimetry is still a challenge in stereotactic radiosurgery since most of treatments are delivered using rotational technique with small fields. A realistic and practical solution for these treatments delivered in conformal radiotherapy is proposed to control the absorbed dose at isocentre, using multiple surface MOSFET measurements over an arc. On the one hand, a forward method was developed to optimize the location of the detectors at the patient surface, taking into account arc length, prescribed isocentre dose, collimator and field size. On the other hand, an inverse method was used to compute the dose at isocentre for conformal arc therapy in stereotactic radiosurgery, using MOSFET measurements. Finally, the reconstructed dose at isocentre was compared to real measurement, obtained for several detectors positioned at a phantom surface. Results show that the inverse method gives good results with five MOSFET equispaced positioned within the arc beam course: deviation between prescribed and computed average total dose at isocentre was below 2% both for 30×30 mm2 and 18×18 mm2 field size La dosimétrie in-vivo est toujours difficilement applicable pour la radiochirurgie stéréotaxique car la plupart des traitements sont délivrés en technique rotationnelle avec des petits champs. Une solution réaliste et pratique pour ces traitements délivrés par radiothérapie conformationnelle est proposée afin de contrôler la dose absorbée à l’isocentre, en utilisant plusieurs mesures de MOSFET positionnés à la surface d’un arc. Dans un premier temps une méthode directe a été développée afin d’optimiser le positionnement des détecteurs à la surface du patient, en intégrant la longueur de l’arc, la dose prescrite à l’isocentre, la taille du champ. En second lieu, une méthode inverse a été utilisée pour calculer la dose à l’isocentre par technique d’arcthérapie conformationnelle en radiochirurgie stéréotaxique en utilisant les mesures de MOSFET. Finalement, la dose reconstruite à l’isocentre a été comparée à la mesure réelle, obtenue à partir DOI: 10.1051/ 00007 (2016) , 124 12400007 EPJ Web of Conferences epjconf/2016


INTRODUCTION
In vivo dosimetry for intracranial radiosurgical treatments, usually delivered in a single instalment over a small field remains a current problem.Small radiation beams are used increasingly in modern therapy and the dosimetric measurements required for deploying such techniques are a topic of great interest in the literature [1,2,3,4].However, the methodology for in vivo dosimetry by point detectors placed on the surface is not described for stereotactic radiosurgical treatments.
Despite the difficulties with the use of the dose at a point quantity in non equilibrium radiation beams, the ideal dosimeter for small fields should have the following features: biophysical tissue equivalence, independence in respect of energy and dose rate, minimal disturbance to the radiation field, very high spatial resolution, and small active volume, in order to minimise the effects of partial volume and to avoid errors due to measurement [5].Moreover, in clinical conditions the detector must be convenient to use, well adapted to small field conditions and to the irregularities of the head's skin surface.Therefore diodes and films are not employed.Using thermoluminescent diodes (TLDs) could offer an alternative, but several studies have reported their fragility and the need for lengthy calibration and tedious reading [6].The sensitive surface of a MOSFET is so small that it can be considered as a point measure.Therefore, MOSFETs could be suitable for in-vivo assessment of radiosurgery microbeams [7,6,8].In the literature their use has been described extensively for such in vivo procedures as: in vivo dosimetry for external radiotherapy [9,10,11], dose verification in intensity modulated radiotherapy (IMRT) [12,13,14], measurements in head and neck Tomotherapy [15], skin dose measurement [16], entry dosimetry [17], implantable detectors for in-situ testing during radiation therapy treatment [18], dose verification of permanent low-dose-rate implants [19] and as a dosimeter for imaging in radiological procedures [20].Additional studies have reported excellent linearity, dosimetric accuracy in the build-up region and directional independence [9,21].Besides, MOSFET detectors do not exhibit dependence on dose power [9,22,23].
In this paper, we suggest the possibility of using an inverse model for reconstructing the dose delivered at the isocentre by way of several surface measurements made with MOSFET detectors in rotational technique of a conforming arc.First, we describe an experimental method, using a spherical phantom, allowing the characterisation of the MOSFET response in off-axis conditions in order to determine the detection window of each detector in terms of angle for fields from 18×18 mm² to 42×42 mm².Different positioning solutions (i.e. at the entry or the exit measurements) compatible with clinical conditions have been examined and evaluated.
Then we describe a forward model allowing optimisation of the positioning of detectors depending on the treatments parameters: length of the arc, dose prescribed at the isocentre, and fieldsize.
Lastly, an inverse model has been developed and experimentally evaluated on a phantom for classical stereotactic treatments which, allow dealing with issues such as loss of lateral electronic equilibrium 2 Methods

Detectors and dosimetry system
The mobile MOSFET dose verification system (TN-RD-70W, Best Medical Canada, Ottawa, Canada) has been used for dose measurements.The MOSFET dosimeters used in this study are p-channel and of the dual-MOSFET type (TN-502RD type, Best Medical Canada, Ottawa, Canada).From one to five MOSFET detectors can be simultaneously connected to the reader.The MOSFETs studied have the following dimensions: 1.3 mm (thickness) × 2.5 mm (width) × 8 mm (length).In all measurements described here, detectors were oriented with dome upwards for entrance measurement and dome downward for exit measurements.The MOSFETS were irradiated using the standard sensitivity mode (1 mV.cGy -1 ) so as to increase their lifetime.Fuller details on the physical detection principle can be found here [23].Dose measurements were made on a Novalis accelerator (Varian) equipped with a micro multi-leaf (micro-MLC) m3 collimator (BrainLab®) delivering a beam of 6 MV photons.The treatment planning system (TPS) used for simulating treatments was Brainscan version 5.3.1 (BrainLab ® ) with a pencil beam dose calculation.
The MOSFET was embedded in a groove made in hemispherical cap of the RW3 material (ρ = 1.04 g.cm -3 and diameter = 13 mm) to provide electronic equilibrium.Indeed the inherent thickness of the MOSFET in a water thickness equivalent to the electronic equilibrium was estimated at 0.8 mm [24,9].Lastly, from the geometrical dimensions of the cap and of the MOSFET, the total thickness equivalent water for establishing the electronic equilibrium above the sensitive surface of the receiver in measurement was estimated at 7.1 mm.
The detectors were calibrated [25] at the surface of phantom XWU-IMRT (ρ = 1.18 g.cm -3 ; 20×20×20 cm 3 ) with the dome oriented towards the radiation source using the accelerator output at zmax.The sensitivity coefficients SCzeq,A×A in mV.cGy -1 were obtained at the zeq equivalent depth and with a given field size A×A.As the MOSFET detectors are of the dual type, temperature dependence is minimal [26,27].The maximal discrepancy in sensitivity measured was 1.8%.Batch uniformity was found to be similar to the range provided by the manufacturer (2%).

MOSFET angular response
The aim of this section is to characterise the response of the MOSFET for given angular positions of the detector in respect of beam axis, for different beam entry or exit fields.Arc therapy consisting in dynamic isocentric radiation over an arc may be considered as a degree-by-degree succession of static beams converging on a point.Thus the angle of detection sensitivity was determined on a spherical Lucy © SRS (Standard Imaging, Middleton, WI) phantom centred at the SSDref = 93cm of the beam and at 7 cm of depth.A MOSFET detector equipped with an RW3 cap and centred on the axis was irradiated by a beam of 0° angle.With the MOSFET fixed at beam entry (0°), then the accelerator gantry rotated in steps by angles α of 1, 2, 3, 4, 5, 7, 10, 15, 20 and 25°.This operation was repeated for the detector placed at the beam exit for the same α values of 5, 10, 15, 20, 25, 30 and 40°.For each angle position α the MOSFET was irradiated with 100 monitor units (MU), keeping the angle of the arm fixed at 0° for fields of 18×18 mm², 24×24 mm², 30×30 mm², 36×36 mm² and 42×42 mm².The angular response (Sα)A×A normalised at 0° for a given angle α was obtained at the entry and exit of the phantom applying equation ( 1): (S ) × = ቀ with MOSFETα being the MOSFET reading obtained for the angle position and MOSFETα = 0° being the MOSFET reading obtained when the axis of the beam and the axis of the detector are aligned, with the angle of the arm fixed at 0°. Considering the symmetrical geometry of the detectors along the axis of the beam, the angular gradient was investigated experimentally.Angular response profiles (Sα)A×A were interpolated for each size of field in 1° steps from -25 to +25° in respect of the beam entry, and from -40 to +40° in respect of the beam exit.A program written in Matlab ® enables to obtain the interpolated profiles of angular response (Sα,i)A×A(cf.figure 1).For a given angle position of the detector, the value of the dose at the point, was calculated by the treatment planning system, TPS, at the intersection of the axis of the MOSFET and of 7.1 mm of depth in the spherical phantom, with the Point Kernel algorithm of TPSIsogray (DosiSoft ® ) and extracted in 1° steps.The correlation coefficients between the dose profiles thus obtained and the response as a function of the angle (Sα,i)A×A were determined for each field size.

Applicable forward solution for optimal detectors placement
First, one calculates the reading of a MOSFET positioned on the skin of a patient treated with an arc therapy whose length, dose at the isocentre and field size are known.This model requires prior knowledge of the angular response of the MOSFET.In a second step, four spatial arrangements of detectors are investigated which allow choosing the most appropriate one for a given irradiation geometry and dose prescribed at the isocentre.

Forward method to calculate the reading of the MOSFET
After the planning stage of a conformational arc therapy the following parameters are known: dose prescribed at the isocentre, arc length, angle at the beginning and end of arc to be delivered, field size.From these entry parameters the dose delivered for each α angle degree Dc,deg for an length of arc Aarc in degrees is given by: Where Diso is the dose prescribed at the isocentre in cGy; Mc,deg is the calculated reading of the MOSFET by degrees in mV; and SCzeq,A×A the response expressed in mV.cGy -1 obtained at the depth equivalent zeq and for field size A x A.
In arc therapy the reading of the MOSFET comprises the sweep of the beam over an angular window whose size [-αmax à +αmax] is a function of field size and distance from the source.The dose calculated Dc, in Gy, delivered at a MOSFET positioned at the entry or the exit of the beam is given by the relation: Where Sα,i is the matrix of interpolated angular response normalised at 0° obtained by experiment from surface measurements made on the spherical phantom at the SSDref for a field size A x A, FSSD is the correction factor of the angular response with the distance to the source, Fatt is the correction factor for the response diminution of the MOSFET when it is placed at the beam exit, and Mc is the calculated reading of the MOSFET in mV after irradiation.
With FSSD = 1 if the SSD is identical to SSDref With (Fatt)α = 1 if the detector is positioned at the entry of the beam Where MOSFETA and MOSFETC are the MOSFET reading obtainedat the positions A and C on the phantom surface.Figure 3 represents the principle of obtaining the calculated reading of a MOSFET, Mc, after the irradiation of an arc whose angle at the beginning and ends, and the field size, are known.

Optimization of the positioning of MOSFET detectors
It is necessary to optimize the positioning of MOSFET detectors for a given irradiation geometry to obtain a balance between maximizing in-vivo measurement efficiency and minimizing MOSFETS detectors setup time.With the mobileMOSFET system, it is possible to connect into the reader housing for measuring an arc.For clinical conditions, four spatial arrangements of detectors were used as constraints in resolving the forward problem in the algorithm.Figure 4   By incorporating the angular response, (Sα,i)A×A, the forward method allows automatic calculation of positioning detectors on the surface for each constraint for a given length of arc and field size.As the detector is not ideal an evaluation of the method was made in near-clinical conditions.The equation for the forward solution allows the theoretical dose received by each MOSFET placed in a given angular position to be reconstructed.
A large set composed of 252 simulations was used to validate the method.The following irradiation parameters were used: -Length of arc in 10° steps from 90° to 170° -Dose at the isocentre from 1 Gy to 7 Gy in 1 Gy steps -Field sizes 18×18 mm², 30×30 mm² and 42×42 mm².
For each forward solution obtained through the simulations the constraint is deemed clinically viable if the theoretical dose to the detector Dc is greater than a dose of 8cGy.This criterion corresponds to an acceptable measurement variation of ~2%.Forward solutions allow the minimal dose at the isocentre to be defined, thus allowing each constraint to be examined as efficiently as possible for given irradiation parameters.

Inverse Solution
The inverse method consists first in estimating the delivered dose by degree, from the MOSFET measurement at the surface after an arc irradiation when the arc size and the field size are known.This computed dose by degree is directly linked to the dose at the isocentre.

Inverse method
For each detector the dose per degree Dc,deg, is sought which will estimate the value of the dose measured Dm depending on the position of the detectors.To this end we shall minimise the residue Rk, representing the difference between the measured dose Dm and the total dose at iteration k : Where Rk is the residue between the dose measured Dm and the dose calculated Dc for each MOSFET.
The value of the dose by degree Dc,deg,k, is modified at each iteration k by the algorithm by calculating the solution Mc,deg,k which allows for adjustment of the dose calculated on the angular window for the dose profile obtained by the forward method.The modification is based on the relation: Where β is the iterative step.
So the residue Rk+1 at the k+1 th iteration may be expressed: In this study the solution (Mc,step)k is obtained at the kth iteration when the value of the stop criterion C is reached.When this happens the stop criterion C is relative, in conformity with the recommendations of 'Numerical Recipes' defined b: Through this calculation the dose at the isocentre detected by each MOSFET on its angular sector Diso,MOS, can be reconstructed.The global dose DT delivered at the isocentre on the arc is obtained by the relation: Where n=5 MOSFETs used in measuring the dose on an arc.
Using the equations 6 to 11 a programme was developed under MATLAB © to resolve the inverse problem from given angular positions of MOSFETs on the surface and parameters of irradiation

Numerical evaluation of the inverse solution
The accuracy of the inverse solution depends on the accuracy of the forward model.First, the forward problem is resolved in order to calculate the dose delivered to the MOSFET detectors for a given spatial configuration and irradiation parameters.Then the 5 solutions are entered into the inverse algorithm in order to calculate in iterative fashion the dose at the isocentre detected by each MOSFET.Convergence of the model is achieved if the value of the stop criterion, C, is reached.In this investigation, after numerous convergence tests for the algorithm, the value of C was fixed at 10 -5 .The forward solutions obtained as in section 3.2 (Optimisation of the positioning of MOSFET detectors) were entered into the inverse method in order to determine the corresponding inverse solutions.The gap between the forward dose and the inverse (ΔDinv-for) n for a given MOSFET n: Where n is the number of the MOSFETs (1 to 5), Dc,inv is the dose calculated by the inverse method, and Dc,for is the dose calculated by the forward method.
The calculation time for the inverse method is less than one second for each configuration of the MOSFET.The calculation uncertainty for each detector VDiso,MOSD, was determined in order to estalish the overall uncertainty of the method total, σT(for all the MOSFETs) by:

Experimental evaluation of the inverse solution
The experimental evaluation was conducted on the spherical phantom Lucy © in limit conditions of lateral electronic equilibrium at 6 MV (lateral electronic equilibrium, L.E.E) for a field size 30×30 mm².With the aim of testing the method in the least favourable conditions, the absence of lateral electronic equilibrium (non-L.E.E.) was also investigated.A field size of 18×18 mm² was chosen, which corresponds to the smallest field size for SCzeq,A×A.Thus, the following irradiation conditions were fixed as entry parameters for the forward problem in order to establish positioning for the detectors at the surface of the phantom relating to each constraint for a prescribed dose at the isocentre of 7 Gy.
Then the MOSFETS, fitted with their RW3 caps, were positioned on the axis of the arc with the aid of the accelerator reticle for each constraint, as shown in Figure 5.The constraints were investigated from 1 to 4 for 30×30 mm² field size, and from 4 to 1 for the 18×18 mm² field size.MOSFETs (calibrated by the optimised calibration method previously described) were readout 30 s after each irradiation in order to obtain a stable readout without fading effect [25].For all constraint geometries the time between two consecutive measurements was 3.5 minutes and never more than 5 minutes.The detectors' response was then entered into the inverse algorithm in order to obtain the corresponding dose Diso,MOS.Then the total dose at the isocentre DT was derived from the 5 solutions given by each MOSFET.Using the interpolated sensitivity angle Sα,i chart, the width of the angular detection window WWconstraint of the detector positioned for a given constraint was defined for the angle -α à + α when the response of the detector for the angle α was more than 1% of the maximal detection sensitivity, i.e. when the detector is placed on the axis of the beam.For each constraint C, the percentage of arc length efficiently explored (%) was calculated in relation to the width of the angular detection window: Possible drift of the angular windows of the detectors used in the program on entry and/or on entry and exit was corrected.For the different field sizes the variation between the Diso,MOS obtained for each detector was compared with the total dose prescribed.Then the average total dose DT of the 5 detectors was calculated and compared with the dose prescribed.The variations in dose were compared in respect of the type of measurement considered for -measurement at the beginning and end of the arc -measurement at the entry of the arc -measurement at the exit of the arc The PinPoint ionisation chamber (PTW, volume 0.015 cm3) was placed in a vertical position at the centre of the phantom in order to measure the attenuation linked to the presence of detectors on the surface for each constraint, and to verify the fluence of the beam.Figure 6 shows the experimental setup.

Angular response of MOSFETs
In Figure 7, the angular responses of the MOSFETs sited at the entry of the beam at the phantom surface and interpolated with MATLAB © and the angular response calculated with the Point Kernel at the depth equivalent of the cap have been normalised and presented as a function of the field size.The range of the angular response increases with the field size.The MOSFET response is nil when the angle between the beam and the axis of the detector is, for each increasing field size, 12°, 15°, 17°, 23° and 25° to the surface.For the same field sizes the correlation coefficient (R²) is respectively: 0.967, 0.952, 0.997, 0.961 and 0.999.
In Figure 8 the angular responses measured by MOSFETs positioned at the exit of the beam at the phantom surface and interpolated with MATLAB © and the angular response profiles calculated with the Point Kernel at the depth equivalent of the cap have been normalised and shown in relation with Xα and field size.For α =40° the response of the MOSFET obtained at 42×42 mm² corresponds to 3.14% of the response of the MOSFET obtained at α = 0°.The width of the angular profiles increases with field size.The response of the MOSFET is nil when the angle between the beam and the detector for increasing field sizes is 22°, 30°, 36° and 40° to the surface (SSDα=0° = 93 cm).For these field sizes, the correlation coefficient is respectively: 0.984, 0.996, 0.999, 0.987 and 0.997.

Forward solution: Optimization of the positioning of MOSFET detectors
In order to determine the best position for the detectors (i.e. the constraint configuration) the minimal detection level was investigated.Figure 9 shows the value of the minimal dose required at the isocentre to give a theoretical dose of more than 8 cGy at all the detectors at a given field size and length of arc for each constraint.For all constraints, field sizes and amplitudes of arc investigated the minimal dose required at the isocentre to satisfy the criterion of acceptability is between 1 and 4 Gy.It is obvious from Figure 9 that Constraint 2 is the best configuration.Constraint 2 allows investigating all the combinations of field size and length of arc for the lowest dose at the isocentre.For field size 36×36 mm² and the same parameters (result not shown on figure 9), the minimal dose required at the isocentre to give a theoretical dose at the detectors of more than 8 cGy is similar to the results for 42×42 mm².

Numerical accuracy of the inverse solution
For all combinations of field size, length of arc and constraints studied, the dose discrepancy due to the use of the inverse method is less than 0.5% at each position of the MOSFET, for a stop criterion value fixed at 10-5.The convergence time is less than 1 second for each constraint, and the overall uncertainty of the method is less than 1%.

Experimental inverse solution
Figure 10 shows the percentage of arc length efficiently explored calculated for each constraint at field 30×30 mm² and 18×18 mm².For these two field sizes, better results are obtained with Constraint 2 (100% and 71.4%).The percentage of arc length efficiently explored diminishes with field size for each constraint.For these two field sizes, the poorest arc length efficiently explored was obtained with Constraint 1 (85.8% and 42.9%).
The results of calculations by the inverse method of the dose at the isocentre detected by each MOSFET Diso,MOS and of the total dose at the isocentre DT for field sizes 30×30 mm² and 18×18 mm² are reported in Tables 1 and 2. For field size 30×30 mm² the differences between the total dose at the isocentre DT through separate measurements for the constraints 1, 2, 3 and 4 are respectively: -2.5%, -1.3%, -9.4% and -3.7%.
The attenuation due to the presence of detectors measured by the Pin Point for each constraint is shown in Figure 11.The attenuation value varies between 0.45% and 0.69% at 30×30 mm² and between -1.07% and -2.57% at 18×18 mm².

Discussion
We propose an inverse method, allowing the determination of the dose at the isocentre delivered through a conformational arc therapy, and based on point measurements at the surface of the patient made via 5 MOSFET detectors in stereotactic radiosurgical conditions.The angular response of the MOSFET was determined experimentally at the entry and exit of the beam on the surface of a homogeneous spherical medium for field sizes between 18×18 mm² and 42×42 mm².
The angular dependence showed very small variations in the response of the detector with arm angulation : the variation was estimated to 1.62% for beam angles 0° to 90° and 0.78% for beam angles 150° to 180°.
Previous studies reported that variation in the response of the MOSFET was negligible for beam angles 0° to 90°.The anisotropy of the MOSFET, as for other in-vivo dosimeters may be explained by the attenuation caused by the geometry of the detector and has been estimated at between 1% and 3 % from 0° to 180° for the most recent models TN-502-RD [17,28,29].Therefore, when the angle of the beam is 0 ° to 25 ° on entry and between 0 ° to 30 ° at exit, the angular dependence may be considered as negligible.As was expected the results showed that the width of the angular detection window of MOSFETs increases with field size.Besides, as a result of the divergence of the beam the angular profiles obtained at the exit are wider than on entry.
The angular profiles measured were compared with the profiles of dose calculated at the depth equivalent of a MOSFET equipped with an RW3 cap.For all field sizes studied a good correlation was observed between the measured angular profiles interpolated at entry and exit and the dose profiles calculated by TPS: the risk value α remained low, i.e. p <0.05.These results validated the method of experimental determination of angular responses.A forward method was developed to calculate the dose delivered to MOSFET detectors placed on the phantom surface for given irradiation parameters.Then an investigation was undertaken to determine optimal positioning of the detectors for four constraints easily reproducible in clinical practice.The maximal arc course explored here was fixed at 170°, as it is commonly done in stereotactic radiosurgery.The irradiation field size and the length of the arc have an effect on the value of the dose given by degrees and hence the acceptable response of the detectors.In general, when the field size diminishes and the arc length increases the dose diminishes for each degree, which means that the dose at the isocentre has to be increased to attain the acceptable criterion (i.e. a measurement at one of the detectors of at least 8 cGy).
For all constraints, field sizes and amplitudes considered the minimal measurable dose at the isocentre (between 1 and 4 Gy) remains lower than doses delivered through each arc in most sessions of single dose stereotactic radiosurgery.At this stage, this means that a measurable reading from detectors in experimental irradiation conditions can be expected.When the detectors are positioned at the entry of the beam, for configurations 1 and 2, the results showed that the minimal dose at the isocentre and the arc course to be investigated are less critical than for configurations 3 and 4, which presume a positioning of two detectors at the exit of the arc.Positioning the detectors according to Constraint 2 enables investigation of dose levels at the isocentre comparable with those delivered in a conventional radiotherapy session, for field sizes encountered in stereotactic radiosurgery, and also offers the advantage of measuring the beginning and end of the arc.However, for Constraint 2 the detectors are not positioned at the angle of neither the beginning nor the end of the arc.Thus measurement of the first or the last angular section of the arc takes place at the limits of the angular window of the MOSFETs where angular response diminishes.This could be a limitation since at the beginning of irradiation the flow of the accelerator is slightly increased to compensate the rise in the rate of the number of UM/min, which occurs at the start of irradiation.Hence, a possible fluctuation in this parameter, which directly conditions the dose delivered per degree, is more difficult to detect in the case of Constraint 2. Constraint 1 can alleviate this deficiency but on the other hand increasing the dose at the isocentre as half the angular window of the detectors placed at the beginning and end of the arc is not irradiated.
For Constraints 3 and 4, which rely on two measurements at the beam exit, the minimal dose required at the isocentre to give a measurable signal for all the detectors of at least 8 cGy, is higher.Because of attenuation, these constraints are most sensitive over the length of the arc course and the width of the detection window.For 18×18 mm² and 30×30 mm², the dose at the isocentre is between 2 and 4 Gy for the course of the arc.
An inverse method was developed to determine the dose at the isocentre by means of 5 detectors on the surface.The method was evaluated numerically and then experimentally.
From a numerical point of view, the choice to limit the positions of the detectors according to 4 constraints allows a reduction in the search space and in calculation time.The value of the stop criterion can be defined precisely while keeping to the rapid convergence conventions of the algorithm (< 1 s).Because of this, it was not deemed necessary to use inverse optimisation algorithms to reduce calculation time and avoid falling into local minima within the margin for errors.As a result the dose uncertainty in respect of each detector is low (< 0.5 %).Using the inverse method developed in this investigation, the total uncertainty in the inverse calculation of the dose at the isocentre was estimated from ideal values yielded by the forward problem at less than 1%.
From the point of view of the experiment the inverse method was evaluated for a dose at the isocentre of 7 Gy, typical of a dose delivered in stereotactic arc radiosurgery.In accordance with the investigation of the forward problem, this dose allowed satisfaction of the acceptability criterion for all constraints.Two field sizes, 30×30 mm² and 18×18 mm², were investigated on an arc to evaluate the method both with and without lateral electronic equilibrium.
The study showed that the difference between the theoretical dose and the real dose measured by the detector (Diso, MOS)i is greater when the measurements are taken at the beginning and at the end of the arc course and at the exit of the arc.So as the average total dose is reconstructed by combining the responses of all detectors the gap between the average dose delivered at the isocentre (DT) and the theoretical dose is smaller for Constraint 2. When the detectors are positioned according to Constraints 1 and 3 the doses measured is smaller because only half of the angular detection window is irradiated and furthermore the response of the detectors is diminished by the thickness of the phantom.Consequently, the difference in relation to the dose absorbed increases when the value of the dose measured by the detector diminishes.This effect is larger as the field size diminishes and as the profiles of the dose calculated at each detector flattens out.This experimental result confirms the hypothesis of the theoretical study on the forward problem.
The results showed that the gaps between the prescribed dose (7 Gy) and the inverse solutions obtained from each MOSFET's measurements are explained in all likelihood by the more difficult positioning of the detector at the arc exit.Notably this was the case for the 30×30 mm² investigation conducted for Constraints 3 and 4 with MOSFET #5 placed at 110°.The same was done for the investigation of Constraints 3 and 4 for an 18×18 mm² field.This explains the larger difference in exit dose in the reconstruction of DT from the measurement of MOSFET #5.Similarly, a dose difference of more than 10 %, linked to poor positioning of MOSFET #4 (angle = 70°) with 18×18 mm² field size was obtained for Constraint 3. In fact the positioning of the detector was modified in this last case (angle = 55°, difference of dose -1.1 %).Combining the response of all the detectors in calculating the total dose DT delivered at the isocentre, the inverse method gives satisfactory results for each constraint.The dose difference was between -2.4 % and 0.75 % for field size 30×30 mm² and from -2.5%, to -3.7% for field size 18×18 mm² (for constraint 4 two of the 5 measurements were erroneous).
For each constraint, the attenuation due to the presence of detectors on the surface was measured at 18×18 mm² and 30×30 mm² with a PinPoint chamber in a vertical position in order to minimise disturbance to dose measurements in small field sizes.In fact, when the PinPoint is in a vertical position it is possible partly to compensate the partial volume effect due to the presence of the chamber [30].If the attenuation is negligible for all constraints at 30×30 mm², it becomes quantifiable at 18×18 mm² (between -1.07 % and -2.57%).At 18×18 mm², the attenuation is maximal using Constraint 2, where all the detectors are positioned at the beam entry.The attenuation thus becomes comparable with that attained with fixed beams [25].For comparison, the attenuation of OSLs equipped with electronic water-equivalent caps was estimated at 4.6 %, and for the diodes the attenuation varies between 5.4 % and 12 % [31].The response anisotropy of the PinPoint chamber used in this investigation was deemed lower than ± 0.5 % for rotations of the beam round the axis of the chamber, and lower than ± 1 % when the axis of the chamber is inclined ± 20 ° in radial incidence and ± 15 ° in axial incidence (PTW, 2011).
Each Constraint described offers advantages and limitations depending on the irradiation parameters investigated such as field size, arc length, the dose at the isocentre and the angle position at the beginning and end of the arc.To our knowledge, it does not exist in the literature any method analogous to the reconstruction of the dose at the isocentre in arc therapy using only detectors positioned on the surface.It is nonetheless possible to note the intra-cavity use of MOSFET detectors in arc therapy techniques involved in treatments to the rectal wall [32].Reconstruction of the dose at the isocentre is in fact mainly described by way of transmission measurements using EPID detectors, but nonetheless relatively rarely.An accuracy of ± 4 % was reported for the study by Piermattej et al conducted with patients, but this involved field sizes between 30×30 mm 2 and 50×50 mm² using arc therapy and doses at the isocentre of 17 Gy in instalments over three days (51 Gy) [33].

Conclusion
We developed an inverse model for reconstructing the dose delivered at the isocentre from five surface measurements made with MOSFET detectors rotated in a conformational arc.The model relies on the experimental characterisation and reconstruction of the response outside the axis of the detectors.The investigation conducted with the model allows optimising the positioning of the detectors for a given geometry of radiation in a realistic fashion.Our results showed that the inverse method is satisfactory for beam fields of up to 18×18 mm² in conditions of classic stereotactical treatment.The next stage of the investigation will be the taking into account of the movement of micro-MLCs which conform to the geometry of the tumour during the movement of the beam through the arc.These realistic conditions will allow the combining of treatments delivered in dynamic arc therapy with small beams.

Figure 1 .
Figure 1.Diagram of experimental assembly used to determine the angular response of MOSFET detectors on entry, at the left and on exit (in the middle), for a given size of field at a fixed fluence: the positions A and C correspond to the position of the MOSFET detector at 0° at the entry and at the exit of the beam.: the position B corresponds to the MOSFET detector at the angle α to the beam axis.

Figure 1 .
Figure 1.b.Profile of the measured angular response (Sα)A×A then interpolated (Sα,i)A×A at the entry of a beam of the MOSFET, enabling the response of a detector positioned on the surface and irradiated through an arc (for a field size A x A) to be predicted.

Figure 2 Figure 2 .
Figure 2. Diagrammatic representation of the irradiation geometry induced at the level of the detector when it is placed either on the axis or on the edge of the beam on the surface of the homogeneous spherical phantom Lucy©.The dotted line represents the coordinates of the points of dose calculation by the Point Kernel algorithm.One sees the diminution of the signal at the edge of the beam, which represents the sensitivity angle in Figure 1b.
represents the spatial configurations of the MOSFETs for each constraint: -Constraint 1: Positioning 5 MOSFETs covering the whole arc at the beam entry.-Constraint 2: Positioning 5 MOSFETs covering the arc at the beam entry but avoiding the edges.-Constraint 3: Positioning 3 MOSFETs covering the whole arc at the entry of the beam and positioning 2 MOSFETs at the beam exit (in respect of the angle at the beginning and the angle at the end of the arc).-Constraint 4: Positioning 3 MOSFETs covering the whole arc at the beam entry but avoiding the edges and positioning 2 MOSFETs at the beam exit (in respect of the angle at the beginning and the angle at the end of the arc).

Figure 4 .Figure 3 .
Figure 4. Diagram representing, from left to right, the situation of the MOSFETs on the surface for constraints 1, 2, 3, and 4 for measuring the dose in arc therapy conditions.

DOI: 10 Figure 5 .
Figure 5. Angular positioning of each detector for field sizes of the irradiation beam of 30×30 mm² and 18×18 mm² and the constraints applied to the forward problem.

Figure 6 .
Figure 6.Illustration of the positioning of the 5 MOSFET detectors in accordance with Constraint 1 on the surface of the spherical phantom Lucy©

Figure 7 - 8 .
Figure 7-8.Profiles of interpolated angular response (Sα,i)A×A obtained by the MOSFET placed at the entry (top) and exit (bottom) of the beam and profiles of the dose calculated with the Point Kernel algorithm of TPS as a function of the angle α in degrees at a depth of 7.1 for field sizes 18×18 mm², 24×24 mm², 30×30 mm², 36×36 mm² and 42×42 mm².

Table 1 .
Results of inverse calculations for each restraint, with a field size 30×30 mm² for a prescribed dose at the isocentre of 7 Gy.L.A.R.D.2015

Table 2 .
Results of inverse calculations at each constraint with a field size 18×18 mm² for a prescribed dose at the isocentre of 7 Gy.