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Alignment procedure of silicon pixel detectors for
ion-beam therapy applications
C.-A. Reidel, Ch. Finck, C. Schuy, M. Rovituso, U. Weber
To cite this version:
Alignment procedure of silicon pixel detectors for
ion-beam therapy applications
C.-A. Reidela,b,∗, Ch. Finckb, C. Schuya, M. Rovitusoc, U. Webera
aGSI Helmholtzzentrum f¨ur Schwerionenforschung GmbH, Planckstrasse 1, 64291
Darmstadt, Germany
bUniversit´e de Strasbourg, CNRS, IPHC UMR 7871, F-67000 STRASBOURG, France cTrento Institute for Fundamental Physics and Applications, TIFPA, Povo, Italy
Abstract
In ion-beam therapy, the elastic and inelastic interactions of the primary beam interacting with different mediums are of strong interest. Cross sec-tion and scattering measurements are important in order to provide accurate basic data for the treatment planning system. Using several planes of a high spatial resolution vertex detector based on monolithic CMOS pixel sensor technology is a common concept for measuring precise trajectories of charged particles before and after interactions. To reach high spatial resolution, the alignment of the sensors is mandatory. In this work, an alignment proce-dure based on a global χ2 cost function, that simultaneously optimizes the alignment parameters for all events is presented and shows accurate results which allow the detectors to reach high spatial resolution. The procedure was benchmarked with simulated and experimental data for the Mimosa28 pixel detector and compared to a state of the art alignment algorithm. The results of the alignment were in agreement with the requirements needed for precise measurements in particle therapy. The spatial resolution reached after the alignment was better than 10 µm.
Keywords:
Alignment procedure, Mimosa28, CMOS pixel detector, Protons, Ion-beam therapy
∗Corresponding author.
1. Introduction
1
Charged particles in the energy range of typically 50 to 200 MeV for
pro-2
tons and 100 to 400 MeV/u for Carbon ions are used for the treatment of
3
deep-seated tumors due to their favorable depth-dose profile (Bragg curve).
4
Moreover, state of the art treatment techniques such as pencil beam-scanning
5
are able to deliver a prescribed dose to a complex target volume while sparing
6
healthy tissues [1]. In order to deliver the prescribed dose inside the tumor
7
and to avoid unnecessary dose outside the target volume, the interactions
8
of the primary particles with the penetrated tissue need to be taken into
9
account by the treatment planning system. Especially for ions heavier than
10
protons, i.e. Helium or Carbon ions, fragmentation of the projectile creates
11
secondary fragments with different ranges and angles with respect to the
pri-12
mary beam. These fragments contribute to the dose inside and outside of the
13
tumor [2, 3]. The characterization of fragmentation processes, for instance
14
the double differential fragment production cross-section, and scattering of
15
the primary beam due to elastic Coulomb interactions are still a field of
re-16
search. The combination of typical nuclear physics detectors, as scintillators
17
of different kinds for Time-Of-Flight (TOF) or energy loss (∆E)
measure-18
ments, combined with high resolution pixel sensors is a common concept for
19
large nuclear physics campaigns [4] and for smaller dedicated setups used to
20
improve basic data in particle therapy [5, 6, 7].
21
The Mimosa28 (Minimum Ionizing MOS Active pixel sensor) detector based
22
on CMOS technology is a high resolution pixel sensor [8]. This detector is
23
composed of 928 rows × 960 columns with a pixel size of 20.7 µm and has
24
a readout time of 186.5 µs (∼ 5 kHz frame rate). The binary output of the
25
chip is delivered after discrimination of the signal. The threshold voltages for
26
the discriminators are adjustable via a JTAG controller. The total thickness
27
of the sensor is 50 µm with an epitaxial layer of 14 µm. The performance
28
of the Mimosa28 sensor was evaluated with a 120 GeV π− beam where the
29
single point resolution of the detector was found to be better than 4 µm [8].
30
Although the Mimosa28 sensor has a high intrinsic resolution, the
resolu-31
tion of the reconstructed particle trajectories depends on the given setup. In
32
this work, the track resolution is defined as the difference between the
re-33
constructed track position and the measured particle position given by each
34
sensor. The latter can be better than 10 µm by using this type of pixel sensor
35
[5]. However, a precise track reconstruction of the particles is only ensured
36
when, in a previous step, an alignment procedure was performed. Since the
sensors are normally used as a stack of several detectors, the alignment is
38
needed to determine their translational and rotational shifts (referred to as
39
alignment parameters) relative to each other. It can be performed by a
so-40
called alignment run with a particle beam where no target is placed in front
41
or in-between the sensors and without magnetic field. The alignment
pa-42
rameters can then be determined using the positions of the primary particles
43
passing through the stack of sensors.
44
Several alignment methods are commonly used, usually based on iterative
45
procedures that sequentially process the given amount of tracks [9, 10]. These
46
algorithms fit the particle tracks by a linear regression and iterate
transla-47
tional and rotational shifts of the sensors. A clear disadvantage of this
iter-48
ative concept is the time consumption. Therefore, e.g. Blobel introduced a
49
so-called matrix method for the alignment, where the alignment parameters
50
are optimized in one step by matrix operations [9, 11]. This procedure is
51
able to optimize a large number of parameters in a simultaneous linear least
52
square fit for a certain number of tracks. This algorithm uses the
simulta-53
neous optimization approach for the local and global parameters, where the
54
local parameters define the track parameters (e.g. slope and intersection for
55
a straight line) and the global parameters define the alignment parameters
56
(correction of detector positioning common for all tracks).
57
For the Mimosa28 sensor the alignment algorithm that was originally
imple-58
mented in the reconstruction software Qapivi [12] (referred to as TrackAlign)
59
uses a sequential fit of the global parameters for a given amount of tracks.
60
This procedure iterates until the difference of the alignment parameters
be-61
tween two consecutive iterations is smaller than a given requirement. This
62
procedure requires iteration steps and leads in some cases to non-convergence.
63
In addition, a pre-alignment is necessary in order to find the tracks.
64
To improve the accuracy and efficiency of the alignment, a new procedure
65
was implemented in the Qapivi software (referred to as ClusAlign). In this
66
work the general concept of Blobel (Solution II, [9]) was followed. However,
67
the procedure proposed in this work is explicitly adapted to the situation of
68
a tracker system based on a set of parallel Mimosa28 sensors. The whole
69
optimization of the alignment parameters uses only the measured positions
70
of the particles in the sensors, which means that the alignment can be
per-71
formed without any initial track reconstruction. The concept simplifies the
72
workflow and can be implemented easier than the Millepede algorithm
73
[11] where a linearization around initial local and global alignment
parame-74
ters is mandatory. The mathematical formulas are optimized for a minimum
amount of algebraic operations. Since the primary particles undergo multiple
76
scattering in the sensors and in the air gaps between them, the alignment
77
procedure described in this work includes an event rejection loop based on
78
simple scattering calculations which improves the quality of the results.
79
The alignment procedure and formulas are described in detail (section 2).
80
The experimental setup and analysis package are presented in section 3. In
81
section 4, the efficiency and accuracy of the procedure is described by
applica-82
tion to measured and simulated data. In addition, a comparison between this
83
procedure with the originally implemented alignment procedure in Qapivi
84
and a state of the art alignment algorithm Millepede II [13] is presented
85
in section 4 with experimental data.
86
2. Alignment procedure
87
2.1. Alignment algorithm
88
In this work, the linear regression parameters of single track are not explicitly
89
computed; they are rather implicitly dependent of the alignment parameters
90
which are simultaneously optimized within a global χ2 function. This makes
91
the method very efficient and fast, even for a high number of analyzed events.
92
In this work, the measured positions of a single particle traversing the sensors
93
are referred to as event whereas the fit of these positions is referred to as track.
94
The size of the matrix-equation to be solved (inverted) is independent of the
95
number of events. The χ2 function, depending on the alignment parameters,
96
and the resulting matrix are derived and stepwise described in this section.
97
2.1.1. Concept
98
In the ideal case, the sensors would be placed without misalignment and the
99
track reconstruction could be performed without correction (Figure 1(a)). In
100
a more realistic case, the sensors are placed with initial mechanical
misalign-101
ments which need to be corrected by software before the track reconstruction.
102
Figure 1(b) illustrates a reconstructed track without any alignment
correc-103
tions of the sensors. After applying the alignment parameters to the sensors,
104
tracks can be reconstructed with the required resolution (Figure 1(c)).
105
The free alignment parameters defined in this work are two translations dx
106
and dy and a rotation dφ around the z-axis (parallel to the beam axis) for each
107
sensor. For a precise mechanical setup, the sensors can be positioned with
108
an accuracy of typically < 0.5 mm lateral to the beam axis and < 30 mrad
109
around the x-axis and y-axis. Thus, the alignment for these rotations can be
z 1 2 3 4 5 6 (a) z 1 2 3 4 5 6 (b) z 1 2 3 4 5 6 (c)
Figure 1: Reconstructed track with 6 sensors positioned along the beam axis in the ideal case (a). Reconstructed track before (b) and after (c) the alignment procedure with 6 sensors positioned along the beam axis. The points represent the interaction of the particle with the sensors and the line represents the reconstructed track.
neglected since the geometrical correction factor is smaller than 1 − cos(0.03)
111
and is not relevant for 20 × 20 mm2 sensors. It should also be noted that
112
the influence of the longitudinal parameters is very small compared to the
113
transversal ones. For a RMS of 10 mrad for the linear track slopes, the
114
sensitivity of the longitudinal alignment parameters is a factor 100 (1/RMS)
115
smaller compared to the transversal parameters.
116
Assuming a set of certain parameters dxi, dyi and dφi, the corrected
coordi-117
nates (Xij,Yij) relative to the original coordinates (xij,yij) of sensor i for a
118
track j are calculated as follows:
119 Xij Yij = cos(dφi) sin(dφi) − sin(dφi) cos(dφi) xij yij +dxi dyi . (1)
Since the rotation dφi around the z-axis is small (dφi 1), equation (1) can
120 be simplified to: 121 Xij Yij = 1 dφi −dφi 1 xij yij +dxi dyi . (2) 2.1.2. Degrees of freedom 122
It is important to fix the whole tracker in the global coordinate system.
123
For this, several alignment parameters need to be fixed. Two translational
124
alignment parameters in x and two in y will constrain the translation and
125
rotation in x and y and around the x and y-axis respectively. Since the
z-126
shift is neglected, only one rotational parameter around the z-axis needs to
be fixed. In this work, the first and the last planes are defined as references
128
such as dx1 = 0, dy1 = 0, dφ1 = 0, dxn = 0 and dyn = 0 where n is the
129
number of planes. The
130
2.1.3. Linear regression analysis for a single track
131
A reconstructed track is defined as the linear fit—also called linear regression—
132
of a single particle passing through the sensors. The corresponding S is the
133
square sum of the differences between the measured positions in the sensors
134
and the fitted track positions defined as:
135 S(a, b) = n X i=1 wi2[(a + bzi) − pi]2, (3)
where pi is the coordinate (either xi or yi) of the measured particle for the
136
sensor i at position zi. wi is the weight given by the expected uncertainty
137
of the measurement of an event. In order to define the weight, the spatial
138
distribution σi (induced by multiple Coulomb scattering) at sensor i is
calcu-139
lated as a function of the angular scattering distributions σα from the sensors
140
and the air gaps, as given by eq. (30) of [14]. The applied weight, such as
141
wi = 1/σi, is the same for all events and the same in x and y since the
142
Mimosa28 is composed of square pixels. The regression coefficients a and b
143
of the fit need to be defined. In order to have the best possible fit, the least
144
squares method is used to estimate a and b which minimizes S leading to χ2S
145
[15].
146
The optimized regression coefficients a and b become the least squares
esti-147
mators ˆa and ˆb and are calculated as:
148 ˆa ˆ b = A−1B p1 .. . pn , (4)
where A is a 2×2 matrix and B is a 2×n matrix:
By replacing the regression coefficients by the least squares estimators, χ2 S is 150 written as follows: 151 χ2S = n X i=1 wi 1 zi ˆa ˆ b − pi 2 . (6)
Combining equations (4) and (6), χ2
S becomes: 152 χ2S = − →w (KL − I n) − →p2, (7)
where In is the n×n identity matrix and L is the product L = A−1B. The
153
symbol ” ” defines the element-wise product. The n-vectors −→w and −→p and
154
the K matrix of dimension n×2 are defined as:
155 − →w = w1 .. . wn , − →p = p1 .. . pn , K = 1 z1 .. . ... 1 zn . (8)
In order to simplify the readability, the n×n matrix Q is introduced as
156
Q = −→w (KL − In). Q is a constant matrix for a given setup with certain
157
positions zi and weights wi and equation (7) can be written in the form:
158 χ2S = Q−→p 2 . (9)
2.1.4. Definition of the χ2 for the global alignment parameters optimization
159
The alignment algorithm is based on the minimization of a global χ2 which
160
is the combination of the linear regression analysis for all tracks. In this
161
work, Xi and Yi are the corrected coordinates explicitly depending on the
162
searched alignment parameters dxi, dyi and dφi (see equation (2)) common
163
for all tracks. For a single track, χ2SX and χ2SY respectively in x and y define
164
the optimal linear fit as follows:
165 χ2 SX = n P i=1 w2 i[(ˆaX + ˆbXzi) − Xi]2 χ2SY = n P i=1 wi2[(ˆaY + ˆbYzi) − Yi]2 , (10)
where ˆaX, ˆbX, ˆaY and ˆbY are the least squares estimators of the regression line
166
of the corrected positions Xi and Yi and differ for each independent track.
The global χ2 function of the alignment parameters is then the sum over all
168
k tracks (j being the track index and i being the sensor index):
169 χ2(dxi, dyi,dφi) = k X j=1 n X i=1 w2i[(ˆaXj+ ˆbXjzi) − Xij] 2 + n X i=1 wi2[(ˆaYj + ˆbYjzi) − Yij] 2 . (11)
This χ2 defines the goodness of the linear fit of all tracks and should yield
170
a minimum when the searched alignment parameters are optimized. The
171
least squares estimators ˆaX, ˆbX, ˆaY and ˆbY implicitly depend on the searched
172
alignment parameters (equation (4)).
173
2.1.5. Alignment parameters calculation
174
The alignment parameters can be explicitly calculated by transforming the
175
χ2 in equation (11) in a single matrix equation to find a solution with a
176
matrix inversion. From equation (2), the corrected coordinates Xi and Yi
177
can be calculated for a single event as:
178 X1 .. . Xn = x1+ dφ1y1+ dx1 .. . xn+ dφnyn+ dxn , Y1 .. . Yn = y1− dφ1x1+ dy1 .. . yn− dφnxn+ dyn . (12)
By defining the vectors −→X ,−→Y , −→x and −→y , equation (12) can be rewritten to:
179 − → X = −→x + PX−→m, − → Y = −→y + PY−→m, (13) where PX and PY are n×3n matrices which can be derived from equation (12).
180
−
→m is the searched 3n-vector containing the common alignment parameters
181
being valid for all events:
−
→m = (dx
1, · · · , dxn, dy1, · · · , dyn, dφ1, · · · , dφn)T.
(14) By combining equations (4) and (13), the minima χ2
SX and χ 2 SY become: 183 χ2 SX = Q−→x + QPX−→m 2 , χ2 SY = Q−→y + QPY−→m 2 . (15)
Introducing the n×3n matrices CX and CY and the n-vectors −d→X and−d→Y as:
184
CX = QPX, CY = QPY,
−→
dX = Q−→x , −d→Y = Q−→y , (16) the χ2in equation (11) can now be rewritten as a quadratic matrix equation:
185 χ2(−→m) = k X j=1 h (CjX−→m +−d→Xj )2+ (CjY−→m +−d→Yj )2i, (17) where the index j was introduced again in order to treat all events. The
186
gradient of this χ2 with respect to −→m is then a term linear in −→m:
187 − → ∇χ2(−→m) = C m−→m + − → dm , (18)
where the 3n×3n matrix Cm and the 3n-vector − →
dm are built by summing
188 all events: 189 Cm = k X j=1 (CjXTCjX) + (CjXTCjX)T+ (CjYTCjY) + (CjYTCjY)T , − → dm = k P j=1 2(−d→Xj T CjX +−d→Yj T CjY)T . (19)
As mentioned in section 2.1.1, five of the alignment parameters have to be
190
fixed in order to get a defined geometrical situation, therefore the alignment
parameters of the first plane and the transversal parameters of the last plane
192
are set to 0. This choice is best suited for the coarse mechanical alignment
193
since the first and the last sensor can be easier accessed than intermediate
194
ones.
195
This step leads to write the final gradient of the χ2 as follows:
196 − → ∇χ2(−→mf) = Cf m − →mf +−→df m , (20)
where the (3n−5)×(3n−5) matrix Cf
m and the (3n−5)-vector − →
df
m are
ob-197
tained by cancelling the 5 degrees of freedom (i.e. dx1, dy1, dφ1, dxn and
198
dyn) and the corresponding columns, rows and elements in Cmand − →
dm. The
199
vector −→mf contains the searched alignment parameters.
200
In order to obtain a χ2 minimum, its gradient with respect to −→mf is set to
201
0. By inverting the matrix Cmf, the searched alignment parameters are then:
202 − →mf opt = −C f m −1−→ dfm. (21)
It should be noted that without the mentioned cancellation of the 5 degrees
203
of freedom, the matrix Cf
m cannot be inverted. The matrix Cmf −1
is the
so-204
called covariance matrix. The alignment parameters in −→mfopt define now an
205
optimal correction for all events with a minimum weighted (by the
uncer-206
tainties) mean (quadratic) deviation between the corrected measured points
207
Xij and Yij and their corresponding individual regression line. A package of
208
script examples of the alignment algorithm is presented in the supplementary
209
material (http://dx.doi.org/10.17632/9g5r3ypcb6.1).
210
2.2. Refinement of the algorithm
211
The described algorithm is able to determine the alignment parameters in
212
one step after the accumulation of a given number of events. The tracking is
213
not explicitly done which allows the alignment procedure to be independent
214
of the initial misalignments. The basic principle exploited by the algorithm is
215
that the particles are traversing almost straight through the sensors.
There-216
fore, strongly scattered events degrade the quality of the results. In order to
217
avoid a relevant influence on the alignment parameters due to the scattered
218
particles, two further calculation steps are performed with the same
align-219
ment algorithm by rejecting the strongly scattered events (e.g. due to nuclear
220
interaction) applying the alignment parameters from the previous solution.
221
Therefore, for each particle, the measured angular deflection from one sensor
i to the next one is compared to the theoretical width σi
α of the angular
223
distribution from the scattering at sensor i. σα is calculated by the Highland
224
approximation [16, 17] which considers the multiple Coulomb scattering as
225
a Gaussian process and provides an estimate for its standard deviation. In
226
this work, the threshold for rejecting the strongly scattered events was set
227
to 20σiα (pre-filtering) and 4σiα for the second and third calculation step
re-228
spectively. In each step, the calculated alignment parameters become more
229
accurate. Typically 10–40% of the events are eliminated by this filter
proce-230
dure. These additional steps are especially necessary for the accuracy of the
231
rotation parameter which is more sensitive to the scattering. Table 1
quan-232
tifies the accumulated angular distribution [14] from a set of six Mimosa28
233
sensors for protons, 4He and 12C ions for 150 and 300 MeV/u.
234
Table 1: Accumulated angular distribution due to multiple Coulomb scattering for a set of 6 Mimosa28 sensors for protons,4He and12C ions for 150 and 300 MeV/u.
Ion Protons Helium Carbon
Energy (MeV/u) 150 300 150 300 150 300
σα (mrad) 2.1 1.1 1.1 0.6 1.0 0.5
3. Experimental setup & Analysis package
235
3.1. Experimental setup
236
Measurements were performed in the experimental room at the Proton
Ther-237
apy Center in Trento [18] and at the Heidelberg Ion Therapy Center (HIT).
238
The tracker consisted of 6 sensors placed with an angle of 0◦ with respect to
239
the beam axis. The whole setup was optimized for high mechanical stability
240
and reproducibility. For this, the tracker was positioned on an optical bench
241
and each sensor was placed inside an individually adapted holder. Reference
242
marks on the holders were used to align the tracker with the room laser. In
243
addition, cables attached to the sensors were strain-relieved. The alignment
244
runs were performed without target (and without magnetic field) and at low
245
intensity (< 5 kHz) to avoid pile-up in the sensors. The beam particles used
246
were protons and Helium ions (4He) in the therapeutic energy range of 80 to
247
220 MeV/u. The energies and the Full Width Half Maximum (FWHM) of
248
the beam at the isocenter for both experiments are listed in Table 2. Two
249
different setup geometries were tested: one with a distance of 35 cm between
the first and the last sensor (see Figure 2(a)) and one with a distance of only
251
8.8 cm (see Figure 2(b)). For both experiments, the detector threshold was
252
set to 6 times the noise level [8].
253
Table 2: Energy and FWHM of the proton beams (Proton Therapy Center in Trento) and Helium ion beams (HIT).
Energy (MeV/u) FWHM (mm) Protons 90.8 7.8 125.3 6.2 148.5 5.4 164.4 4.9 188.8 4.1 219.8 3.5 Helium 80.64 11.8 130.25 7.6 220.51 4.9
3.2. Software package for the analysis of Mimosa28 pixel sensors
254
The energy deposited by an ionizing particle creates charges which are
col-255
lected by a certain number of pixels (cluster) in the sensor [5]. The analysis
256
software Qapivi [12], based on the ROOT [19] and Geant4 [20] libraries,
257
reconstructs the clusters from the fired pixels whose analog signal is above a
258
given threshold. In this work, the signal to noise ratio threshold is set at 6
259
times the noise. By calculating the cluster position defined as the center of
260
mass of the fired pixels, a straight line matching the clusters in the different
261
planes is determined (tracking). Finally, the vertices of the tracks when a
262
target is placed in front or in-between the tracker are computed. In this work,
263
only the first reconstruction step (clustering) is necessary for the alignment
264
procedure. The clusters are reconstructed in an iterative way based on the
265
first neighbor search with an efficiency better than 99% [5].
266
The x-y position of the particle is defined as the cluster position. The new
267
alignment procedure (ClusAlign) was implemented in C++ in the Qapivi
268
software and the mathematical calculations are performed with the ROOT
269
libraries.
Vacuum Beamline
Figure 2: Experimental setup dedicated for the alignment runs with a spread geometry tested with protons at the Proton Therapy Center in Trento a) and a compact geometry tested with Helium ions at HIT b).
The Monte-Carlo (MC) simulations were performed with Qapivi. To prove
271
the accuracy of the algorithm, several tests have been done with the
simu-272
lation package implemented in Qapivi based on Geant4 10.04. The Binary
273
Cascade light ion model (BIC) [21] physics list was used for the simulations
274
presented in this work. The ion optics of the beam (FWHM and beam
di-275
vergence) were adapted to reproduce comparable beam characteristics to the
276
ones used in therapy (see Table 2). Each sensor is simulated by a volume
277
composed of different material layers and with a sensitive volume (epitaxial
278
layer of Silicon) with an area defined as the active area of the sensor. The
279
deposited energy of the particle in the sensor determined by the simulation
280
is converted in the corresponding number of fired pixels [22]. A so-called
281
digitizer was implemented to compute a realistic cluster shape as a function
282
of the deposited energy in the sensor [5]. Noisy pixels were randomly created
283
as a function of the signal to noise ratio threshold.
284 285
In order to have comparable conditions to the experimental ones, different
ion beam configurations listed in Table 3 were simulated for two different
287
setup geometries. The values for the divergence of the beam were defined as
288
the angular distribution in front of the vacuum window. The sensor positions
289
along the beam axis are listed in Table 4 for a compact geometry (referred
290
to as Geo1 ) and for a spread geometry (referred to as Geo2 ). For both
291
geometries, the start position of the MC simulations was set to 50 cm before
292
the first sensor.
293
Table 3: List of the different ions, energies and ion-optical beam parameters (FWHM and divergence) at the vacuum window used for the MC simulation.
Ion Protons Helium Carbon
Energy (MeV/u) 150 300 150 300 150 300
FWHM (mm) 7.1 7.1 7.1 5.9 5.9 4.7
Divergence (mrad) 5 3 3 2.5 2.5 2
Table 4: Position of the sensors along the beam axis for a compact geometry (Geo1) and for a spread geometry (Geo2).
Position (cm)
Sensor Geo1 Geo2
1 −1.25 −12.50 2 −0.75 −7.50 3 −0.25 −2.50 4 0.25 2.50 5 0.75 7.50 6 1.25 12.50
In order to test the alignment algorithm, some arbitrary set-values for the
294
misalignments dx, dy and dφ were implemented for each plane. As explained
295
in section 2.1.1, the translations and the rotation of the first sensor (primary
296
reference plane) and the translations of the last sensor (secondary reference
297
plane) were fixed to 0. Since the rotation is the most sensitive alignment
298
parameter, a first study only on the translations dx and dy (dφ set to 0) was
299
performed for a scenario with small misalignment values slightly bigger than
300
the detector resolution (referred to as Mis1 ) and bigger misalignment values
301
in the order of the mechanical positioning uncertainty during an experiment
302
(referred to as Mis2 ) (∼1 mm) (Table 5).
Table 5: Two misalignment scenarios for a set of 6 sensors. Mis1 describes a scenario with small misalignment values, whereas Mis2 gives larger misalignment values. Rotational misalignments are set to zero in both cases.
Translational misalignments (µm) Mis1 Mis2 Sensor dx dy dx dy 1 0.00 0.00 0.00 0.00 2 −5.00 5.00 −753.00 928.00 3 3.00 −4.00 850.00 −789.00 4 −15.00 −8.00 −978.00 842.00 5 6.00 12.00 925.00 −891.00 6 0.00 0.00 0.00 0.00
The accuracy of the alignment algorithm as a function of the number of
304
events (from 2500 to 650000 events) was determined with the misalignment
305
scenarios listed in Table 6 (referred to as Mis3 ), where the rotational
mis-306
alignment is now set to non-zero values.
307
Additionally, since the rotation matrix in equation (2) was approximated
308
for small values of dφ, the alignment algorithm results for larger rotational
309
misalignments (up to 10◦) were also studied.
310
Table 6: Translational and rotational misalignment scenario (Mis3) for a set of 6 sensors.
Mis3 Sensor dx (µm) dy (µm) dφ (◦) 1 0.00 0.00 0.00 2 438.00 −225.00 0.823 3 −349.00 574.00 −0.150 4 −913.00 −682.00 −1.730 5 166.00 751.00 −0.340 6 0.00 0.00 1.228
All data sets were processed with an Intel Core i3-2100 CPU @ 3.10 GHz (2
311
physical cores 4 threads).
312
4. Results
The accuracy which is expected from the alignment procedure should be
bet-314
ter than the required spatial resolution of the reconstructed tracks.
There-315
fore, the deviations of the calculated alignment parameters from the real
316
misalignments should be small enough to reach the required track
resolu-317
tion. In the following section, the performance of the alignment procedure is
318
benchmarked against this requirement. It should be noted that the deviations
319
(indicated by the letter δ) are defined as the absolute value of the difference
320
between the calculated alignment parameters and the ones set in the MC
321
simulations (referred to as set-misalignments) such as δ = |dcalc− dset|. In
322
addition, the standard deviations of the alignment parameters were computed
323
for different sets of data obtained from the experiments.
324
In this section, the residuals are presented for experimental data and are
325
used as a control of the alignment procedure success. In addition, the track
326
resolution is defined by the width of the residuals. As presented in Table 1,
327
the angular distribution of the multiple Coulomb scattering for a set of 6
328
sensors is small for the types and energies of ions used in ion-therapy.
How-329
ever, its influence on the track resolution depends on the setup geometry
330
(distance of the air gaps between the sensors). For ion energies presented in
331
Table 1, the track resolution for a set of six Mimosa28 sensors placed along
332
the beam axis with air gaps < 3 cm is better than 10 µm. For bigger air
333
gaps, the track resolution is less good. Setup geometries with small air gaps
334
between the sensors (compact geometry) and bigger ones (spread geometry)
335
are presented since the goal of this work is to have an alignment procedure
336
aligning the sensors with good accuracy independent of the setup geometry.
337
It is important to note that for the setup with bigger air gaps between the
338
sensors, the track resolution is > 10 µm and could be improved by using a
339
different track model than a linear fit. However, the tracking does not belong
340
to the scope of this work. Therefore, the track fit uses a simple linear fit for
341
all cases which is suited to this work.
342
4.1. Alignment procedure performance for MC simulation
343
The alignment procedure was performed for all beam configurations (Table 3),
344
for both setup geometries (Table 4) and for all scenarios Mis1, Mis2 (Table 5)
345
and Mis3 (Table 6). The results in Table 7 show the mean deviations δx, δy
346
and δφ between the calculated alignment parameters and the set-misalignments
347
for different scenarios (averaged over the 6 given beam configurations). These
348
results fulfill the requirement mentioned above and show the good accuracy
349
and robustness of the algorithm with average deviations such as δx, δy < 2 µm
and δφ < 0.05◦. The uncertainties of the alignment parameters can be
de-351
rived from the elements of the covariance matrix Cmf−1defined in section 2.1.5.
352
These values were compared to the ones averaged over the 6 beam
config-353
urations and have shown smaller uncertainties. Since the results from the
354
covariance matrix are less conservative, the results from the averaged
param-355
eters from the different configurations are presented in Table 7. It can be seen
356
that the uncertainty of the rotational parameter increases proportionally to
357
the z-direction. This systematic can be induced by different types of so-called
358
weak mode [9, 13] (e.g. torsion of the tracker). The rotational parameter has
359
a weak correlation with the other translational parameters which can induce
360
small errors on the rotational alignment parameter results. Since the fixed
361
rotation is the one from the first plane, the error on the rotational parameter
362
increases proportionally to the z-direction. To verify the influence of fixing
363
the rotation of another plane, the tracking was performed for the case when
364
only the rotation of the first plane was fixed versus the case when the first
365
and the last rotation were fixed. Even so the deviations δ on the rotational
366
alignment parameters were smaller in the second case, the resulting χ2 of the
367
tracks were the same for both cases. In this work, only the first rotational
368
parameter was fixed in order to correct the initial torsion of the setup. The
369
CPU time of the alignment procedure was ∼ 50 s in single-threaded mode
370
for 50000 events.
371
To determine how fast the alignment parameters converge to the required
372
values, the alignment parameters were calculated for a certain amount of
373
events used by the algorithm for protons, 4He and 12C ions at 150 MeV/u
374
for the setup geometry Geo2. A comparison between the two setup
geome-375
tries for 150 MeV/u4He was also performed. The average deviations between
376
the calculated alignment parameters and the set-misalignments were
calcu-377
lated. As shown in Figure 3 (top), the average deviation of the translation
378
is smaller than 2 µm in all cases. However, the rotational parameter, being
379
more sensitive to the scattering, converges slower (Figure 3 (bottom)) and at
380
least 50000 events are needed for an agreement between the calculated
align-381
ment parameters and the real misalignment values. As expected, the heavier
382
the particle is, the faster the algorithm converges because of the stronger
383
scattering for lighter particles. The results of the alignment parameters for
384
the setup geometry Geo1 were very similar. The alignment parameter results
385
are independent of the ion optics of the beam (FWHM and divergence). The
386
results remained the same even if the divergence values of Table 3 were
in-387
creased to a larger value of 20 mrad.
Table 7: Average deviations between the calculated alignment parameters and the set-misalignments. The deviations were averaged over 6 beam configurations listed in Table 3. The standard deviations of the δ-values are given in parentheses and indicates their fluc-tuations.
Geo1–Mis1 Geo1–Mis2
Sensor δx (µm) δy (µm) δφ (◦) δx (µm) δy (µm) δφ (◦)
1 0.00 0.00 0.000 0.00 0.00 0.000 2 0.01 (0.01) 0.02 (0.03) 0.005 (0.003) 0.21 (0.24) 0.19 (0.20) 0.008 (0.005) 3 0.03 (0.01) 0.03 (0.03) 0.010 (0.007) 0.38 (0.45) 0.41 (0.49) 0.017 (0.010) 4 0.02 (0.01) 0.02 (0.02) 0.015 (0.010) 0.60 (0.70) 0.71 (0.84) 0.025 (0.014) 5 0.02 (0.02) 0.02 (0.01) 0.020 (0.013) 0.83 (0.96) 0.87 (1.03) 0.034 (0.019) 6 0.00 0.00 0.025 (0.016) 0.00 0.00 0.042 (0.023) Geo2–Mis1 Geo2–Mis2
Sensor δx (µm) δy (µm) δφ (◦) δx (µm) δy (µm) δφ (◦)
1 0.00 0.00 0.000 0.00 0.00 0.000 2 0.08 (0.09) 0.07 (0.03) 0.006 (0.008) 0.12 (0.09) 0.12 (0.06) 0.005 (0.004) 3 0.13 (0.11) 0.10 (0.06) 0.011 (0.012) 0.18 (0.15) 0.14 (0.08) 0.010 (0.007) 4 0.14 (0.13) 0.10 (0.07) 0.017 (0.015) 0.23 (0.19) 0.31 (0.17) 0.014 (0.011) 5 0.13 (0.09) 0.05 (0.06) 0.022 (0.018) 0.28 (0.28) 0.29 (0.25) 0.019 (0.014) 6 0.00 0.00 0.028 (0.015) 0.00 0.00 0.024 (0.016) 389
The performance of the alignment algorithm as a function of the rotational
390
misalignment of the sensors were computed for 150 MeV protons and 300 MeV/u
391
12C with the setup geometry (Geo2). The translations dx and dy were set as
392
Mis3 (Table 6) and the rotational parameter was varied from 0.5◦ up to 10◦
393
compared to the first sensor. Figure 4 shows (for 150 MeV protons) that the
394
deviations of the translations significantly increase when δφ becomes larger
395
than 3◦. The same test was done for 300 MeV/u 12C and has shown very
396
similar results. The large deviations of the calculated alignment parameters
397
for δφ > 3◦ can be explained by the inaccurate approximation of cos(φ) ≈ 1.
398
However in practice, the alignment accuracy of ± 3◦ for a stable mechanical
399
setup is easily feasible.
400
4.2. Experimental results
401
The alignment procedure was tested on experimental data obtained with the
402
setups described in section 3.1. To show the effect of the alignment
Number of events 0 20 40 60 80 100 3 10 × m) µ ( x δ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 protons He 4 C 12 Number of events 0 20 40 60 80 100 3 10 × ) ° ( φδ 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 protons He 4 C 12
Figure 3: Average deviations δy (top) and δφ (bottom) between the calculated alignment parameters and the set-misalignments in the MC simulations, respectively for the transla-tion in x (top) and the rotatransla-tion around the z-axis (bottom) as a functransla-tion of the number of events for 150 MeV/u protons, 4He and 12C with the scenario Geo2–Mis3. The δ-values
are averaged over the sensors and the error bars represent the range (min–max) of the δ-values.
cedure, the tracking was computed before and after the alignment and the
404
residuals in x and y for 220.51 MeV/u4He before and after the alignment
pro-405
cedure for all sensors are shown in Figure 5. The residual distribution shows
406
the success of the alignment procedure with a track resolution of 5.1 µm in
407
x and y for the presented experimental data.
) ° Rotational misalignment ( 0 2 4 6 8 10 m) µ x ( δ 0 2 4 6 8 10 12 14 Sensor 4 Sensor 2 Sensor 5
Figure 4: Deviation δx as a function of the rotational set-misalignment for different sensors for 150 MeV protons for the setup geometry Geo2.
To test the robustness of the algorithm, the alignment parameter results
409
were studied for a certain setup geometry for both experiments with
pro-410
tons and Helium ions for different energies. Table 8 shows the average value
411
of the alignment parameter results calculated with the procedure described
412
in this paper (ClusAlign), the original algorithm implemented in Qapivi
413
(TrackAlign) and the Millepede II algorithm for the protons data set. The
414
Millepede II algorithm was used as a stand-alone program where the
in-415
put parameters were given after fitting the same selection of data used for
416
ClusAlign.
417
The alignment parameter results of the procedure ClusAlign and Millepede II
418
are in agreement. The original algorithm implemented in Qapivi shows
dif-419
ferences of the average value with ClusAlign and Millepede II up to 25 µm
420
for the translations and 0.4◦ for the rotation. As explained in Section 4.1,
421
the deviation of the rotational parameter increases proportionally to the
z-422
direction. The verification of the effect of fixing the last rotation in addition
423
to the first one was also performed for the experimental data. As for the
424
simulated data, the small error on the rotational parameter has no influence
425
on the χ2 of the tracks.
426
The residuals in x and y for all sensors computed after the alignment
proce-427
dure with the different procedures (ClusAlign, TrackAlign and Millepede II)
428
for a compact geometry for 220.51 MeV/u 4He are shown in Figure 6. The
m) µ Residual x ( 600 − −400 −200 0 200 400 600 Counts 0 50 100 150 200 250 300 350 m) µ Residual x ( 50 − −40 −30 −20 −10 0 10 20 30 40 50 Counts 0 500 1000 1500 2000 2500 3000 3500 4000 m µ 0.0 ± : 5.1 σ 0.0 % ± Tail: 8.8 m) µ Residual y ( 600 − −400 −200 0 200 400 600 Counts 0 50 100 150 200 250 300 m) µ Residual y ( 50 − −40 −30 −20 −10 0 10 20 30 40 50 Counts 0 500 1000 1500 2000 2500 3000 3500 4000 m µ 0.0 ± : 5.1 σ 0.0 % ± Tail: 8.5
Figure 5: Histograms of the residuals in x (top) and y (bottom) before (left) and af-ter (right) the alignment procedure for 220.51 MeV/u 4He (from the experimental data recorded at HIT) for all sensors and all tracks. The red dotted line show the Gaussian fit of the residuals.
residuals in x and y show the improvement of the performance of ClusAlign
430
compared to TrackAlign. In addition, the residuals computed after ClusAlign
431
and Millepede II are comparable leading to a track resolution of 5.1 µm.
432
5. Conclusion
433
An accurate and robust alignment procedure for high spatial resolution
ver-434
tex detectors was developed and evaluated for a tracker composed of a stack
435
of Mimosa28 sensors with arbitrary distances. The performance of the
pro-436
cedure was tested on simulated and experimental data for different ions and
437
different setup geometries in the energy range of interest for hadrontherapy.
438
For an accurate alignment, it is important to have a stable mechanical setup
439
with a rotation angle < 3◦ compared to the primary reference plane and
440
enough statistics (more than 50000 events) in order to have an error on the
441
alignment parameters small enough to reconstruct tracks with the required
442
resolution. In this work, all alignment parameter results were estimated
Table 8: Average value of the alignment parameter results over 6 energies in the range of 80–220 MeV protons (from the experimental data recorded at the Proton Center Ther-apy in Trento) calculated with ClusAlign, TrackAlign and Millepede II. The standard deviation of the average value of the alignment parameters is given in parenthesis.
Sensor dx (µm) dy (µm) dφ (◦) ClusAlign 1 0.0 0.0 0.00 2 −125.5 (1.5) 159.9 (2.7) −0.25 (0.03) 3 −5.4 (1.5) 157.7 (2.9) 0.14 (0.03) 4 122.6 (0.3) 291.2 (0.7) −0.44 (0.08) 5 165.1 (0.7) 270.3 (0.9) 0.17 (0.08) 6 0.0 0.0 0.17 (0.09) TrackAlign 1 0.0 0.0 0.00 2 −129.7 (1.4) 123.5 (6.5) 0.03 (0.17) 3 −9.0 (1.6) 122.9 (5.5) 0.04 (0.06) 4 114.5 (2.3) 263.9 (3.9) −0.16 (0.27) 5 157.4 (3.0) 239.5 (3.9) −0.04 (0.03) 6 0.0 0.0 −0.08 (0.02) Millepede II 1 0.0 0.0 0.00 2 −126.0 (2.9) 155.5 (5.5) −0.18 (0.13) 3 −5.8 (3.0) 153.1 (5.7) 0.21 (0.14) 4 123.0 (1.3) 287.4 (2.9) −0.28 (0.28) 5 165.4 (1.8) 266.5 (2.9) 0.33 (0.28) 6 0.0 0.0 0.36 (0.34)
with uncertainties < 3 µm for the translations and < 0.1◦ for the rotation.
444
The alignment procedure described in this work is able to locally align
sev-445
eral planes with high accuracy so that the spatial track resolution can be
446
achieved. The new procedure was compared to the one originally
imple-447
mented in Qapivi and has shown significant improvements of the alignment
448
parameter results. An additional comparison was performed with a state
449
of the art alignment algorithm Millepede II validating the results of the
450
alignment procedure implemented in this work which has the advantage of
451
aligning the sensors without preliminary tracking. The track residuals
per-452
formed after the alignment procedure were used to control the alignment
m) µ Residual x ( 150 − −100 −50 0 50 100 150 Counts 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 ClusAlign TrackAlign MILLEPEDE II m) µ Residual y ( 150 − −100 −50 0 50 100 150 Counts 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 ClusAlign TrackAlign MILLEPEDE II
Figure 6: Histograms of the residuals in x (left) and y (right) after the alignment procedure with ClusAlign (black circle), TrackAlign (blue star) and Millepede II (red triangle) for 220.51 MeV/u 4He (from the experimental data recorded at HIT) for all sensors and all
tracks.
success. In addition, the width of the residuals showed that the track
reso-454
lution was better than 10 µm for a compact setup geometry. This procedure
455
can then improve the particles track resolution needed for experiments like
456
range monitoring [12] or studies of nuclear fragmentation processes [5].
457
Acknowledgements
458
We would like to thank the Proton Therapy Center in Trento and TIFPA
459
for the opportunity to conduct our measurements and we thank Dr. C. La
460
Tessa for the help and organization for the beam time. We also thank the
461
Heidelberg Ion Therapy Center (HIT) for completing these measurements
462
with Helium beam. We want to thank the mechanical work shop at GSI
463
for their help for building an improved mechanical setup. We would like
464
to acknowledge Prof. J. Baudot and M. Goffe for their support for the
465
commissioning of the Mimosa28 detectors.
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467
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