Silicon planar structures as detectors for microbeam radiation therapy

Abstract

Calculations of responses for experimentally studied silicon planar structures (epitaxial single-strip silicon diodes) have been carried out for various profiles of microbeams. The spatial distribution of responses on the microbeam radiation therapy (MRT) beams has been calculated taken into account both electrical field distribution inside of the detector and recombination in the diffusion region. Contributions to the responses from the space charge region (SCR) and the diffusion region are compared. It has been shown that the spatial resolution and the detector efficiency in edge on mode of the detector relative to X-ray microbeam can be higher than for the face on irradiation geometry. The response functions of diodes under MRT irradiation for a simple physical model have been obtained analytically. That allows calculating responses using the convolution procedure for different beam profiles.

Full text

Silicon planar structures as detectors
for microbeam radiation therapy
Igor E. Anokhin, Member, IEEE, M. Lerch, Member, IEEE, M. Petasecca, Member, IEEE,
O. Zinets, A. Rosenfeld, Senior Member, IEEE
Abstract– Calculations of responses for experimentally studied
silicon planar structures (epitaxial single-strip silicon diodes)
have been carried out for various profiles of microbeams. The
spatial distribution of responses on the microbeam radiation
therapy (MRT) beams has been calculated taken into account
both electrical field distribution inside of the detector and
recombination in the diffusion region. Contributions to the
responses from the space charge region (SCR) and the diffusion
region are compared. It has been shown that the spatial
resolution and the detector efficiency in edge on mode of the
detector relative to X-ray microbeam can be higher than for the
face on irradiation geometry. The response functions of diodes
under MRT irradiation for a simple physical model have been
obtained analytically. That allows calculating responses using the
convolution procedure for different beam profiles.
I. INTRODUCTION
owadays microbeam radiation therapy is widely used for
cancer treatments [1]. It allows minimizing the side effects
of irradiation using highly spatially fractioned X-ray beams of
micrometers sizes. The extremely high dose rate (~20 kGy/s),
laterally fractionated radiation field and steep dose gradients
utilized in this therapy require the real-time readout and a high
spatial resolution.
In the present paper responses of silicon planar structures
under different geometry of irradiation (front and lateral) have
been studied. The spatial resolution and the efficiency of
detectors are determined by the charge collection in the space
charge region and in the diffusion region. Spatial resolution
depends on geometry and sizes of the sensitive volume. The
results of the calculations of the diode responses for the edge
on irradiation geometry for a simple physical model have been
presented.
II. X-RAY FOR MICROBEAM RADIATION THERAPY AND
PROPERTIES OF THE SI SINGLE STRIP DETECTOR
The specifications for dosimetry come from the
characteristics of the beam itself where intense blades of x-ray
light have a pitch from 50 µm to 400 µm, are 25 µm to
500 µm wide and intensity of the beam can reach the dose rate
of 25kGy/s. A dosimetry system for such peak/valley/peak
distribution of the intensive microbeam array requires
Manuscript received November 09, 2013.
Igor E. Anokhin is with the Institute for Nuclear Research, Kiev,
UKRAINE (e-mail: anokhin@kinr.kiev.ua).
O. Zinets is with the Institute for Nuclear Research, Kiev, UKRAINE.
M. Lerch, M. Petasecca and A. Rosenfeld are with the Centre for Medical
Radiation Physics, University of Wollongong, NSW 2522, AUSTRALIA.
excellent spatial resolution, a dynamic range of 104-105, tissue
equivalency, real time response and be water proof for Quality
Assurance (QA) in a water phantom.
A. Beam profile
Using of high resolution epitaxial silicon single strip
detector (SSD) for MRT beam analyzing was demonstrated in
paper [2]. The lateral dose profile of the MRT radiation field
which incorporates X-ray microbeams was measured. All
microbeam peaks and valley regions between two microbeams
are clearly resolved (Fig.1).
Fig. 1. Acquisition of 49 microbeams by a scan of approximately 10
seconds [2]
The schematic view of the formation of microbeams is
shown in Fig.2
Peak
width o f microplane s: 25
width of microplanes: 25 -
-100
100 μ
μm
m
C
C-
-to
to-
-C = centre
C = centre-
-to
to-
-centre distance (50
centre distance (50 -
-400
400 μ
μm)
m)
Peak
C-to-C distance
ValleyValley
Fig. 2. Schematic view of the main elements of the ID17 beamline; the
primary slit collimator (PSC) is followed by the Slit Horizontal Gap (SHG)
and then by the Multi Slit Collimator (MSC) which creates the microbeams
B. Planar silicon diode structure
MRT requires a dosimetry system with:
• Excellent spatial resolution: at least 10 µm
• Large dynamic range: 10,000
• Tissue equivalent: QA of MRT treatment plan
• On-line: MRT beam set up and monitoring
• Fast: peak-to-valley dose ratio (PVDR) measurements
for all microbeams in real-time
• Water proof (QA procedure compatible).
N

The investigated single strip detector is a planar p-type
silicon detector based on a 50 m thick, 100 ·cm resistivity,
epitaxial silicon layer grown on a lower resistivity
(0.001 ×cm) silicon substrate. Dimensions of the central
strip of the SSD are 20 m×900 m and it is surrounded by a
guard ring situated 20 m from the edge of the strip. The
dimensions of the chip are 1 mm×1.4 mm. Schematic view of
the silicon planar diode is presented in Fig. 3. More details are
available in [2].
n+ n+n+
p+
Al
p+
n+ n+n+
p+
Al
p+
100 Ω·cm p-type Si
50 µm
0.001 Ω·cm p-type Si
Strip Junction Guardringsubstrate
Fig. 3. Model end side view of a single strip silicon planar structure
To resolve narrow beams the scanning of the detector
across microbeams was used. The scheme of the experiment is
shown in Fig. 4 [3]. The Single Strip Detector (SSD) is
positioned within the phantom in edge-on mode, on the
goniometer. The linear stage controlled motor allows one to
precisely move the detector across the microbeams at constant
speed; inset is a zoom of the detector.
Fig. 4. The scheme of the experiment
C. Efficiency and spatial resolution of detector at different
geometry of irradiation
The schemes of geometry of irradiation are shown in the
Fig. 5.
face on mode edge on mode
Fig. 5. Schematic view of the geometry of irradiation
One can see that irradiated volumes of the diode in the case
of face on (left panel) and edge on (right panel) irradiation are
~Lbeam×wbeam×Ldiff and ~Lbeam×wbeam×(Hstrip+2Ldiff)
correspondently. So, one can see that the edge on case is
preferable.
Also, in the case of the edge on geometry of irradiation one
has potentially better spatial resolution (of order (Hstrip+2Ldiff)
and (wscr+Ldiff) in the cases of face on and edge on irradiation
geometry respectively, where wscr and Ldiff are width of the
space charge region (SCR) and diffusion length respectively).
Spatial resolution of the detector is determined by the
response function U(x, x0), which is the detector response on
the -function beam (beam intensity 0
~( )–xx
δ
), where x0 is a
center of mass of the beam intensity in a rigid frame
associated with the detector. Usually this response function
can be approximated by the Gaussian function or rectangular
(stepwise) ones
()( )
()
2
0
2
0
00
,
x
x
U
UxxUxx e
σ
πσ
−
−
=−= , (1.a)
() ()()
0
000
, /2 /2
U
Ux x x x x x
σσ
σ
+− +Θ=Θ −. (1.b)
Values of the σ and U0 depend on sizes and properties of
sensitive regions of sensors and peculiarities of the charge
collection in detectors.
Presented response function is generally used in a sensors
response [4] however response function for considered
epitaxial diode will be derived further using diffusion and drift
equations.
D. Calculations of responses
Measured response of a detector R(x0) for a beam with the
intensity distribution function F(x) can be written as follows
00 0
() ( )() ()( )
R
xUxxFxdxUxFxxdx
∞∞
−∞ −∞
′′′
=− = −
∫∫. (2)
Analysis of the response with known functions F(x) and
U(x0 - x) allows the obtaining required position resolution of
detectors. Fig. 6.a shows typical response on a beam with a
rectangular profile.
Solution of the integral equation (2) for the unknown
function F(x), when the function U(x) and the response R(x0)
are known, allows finding the beam intensity distribution. A
simple way to find an approximate solution of equation (2) is
the following one. If there are the measured response R(xi) in
N+1 points xi one can obtain a system of N linear equations
for N unknown variables F(xj)
1
() ( )( )( )
N
iijjjj
j
R
xUxxFxxx
+
=− −
∑. (3)
Fig. 6.b shows result of calculations the beam intensity F(x)
for typical response data and the Gaussian approximation of
U(x).
s
Gaussian
=
0.25
s
Gaussian
=
0.1
0 1 2 3 4 5
2
4
6
8
10
12
Reconstracted
Beam Pro file
Respons e of the S S D
R
H
x
i
L
Beam P rofi le
F
H
x
L
Gaussian Response
Function U
H
x
0
-
x
L
0.6 0.8 1.0 1. 2 1.4
2
4
6
8
10
1
2
Fig. 6.a Response of the detector on
a rectangular beam for Gaussian
response function with different 
per unit intensity of beam (U0=1),
where x is a detector position in
dimensionless units
Fig. 6.b Reconstructed beam inten-
sity distribution function F(x).
Function R(xi) was used from the
results of simulations for the Fig.6.a

Comparisons of the Fig. 6.a and Fig. 6.b shows that the
distribution of the beam intensity F(x) (assuming it is a step
wise function) can be restored sufficiently well from
experimentally measured data using procedure of the solving
integral equation (2) described above. Similar procedure can
be used for calculation of function U(x) when the beam
intensity distribution and measured response are known. For
obtaining more precise results it is necessary to calculate
response function from the real physical model of the detector
structure.
III. PHYSICAL MODEL AND MAIN EQUATIONS
The diode response R(x0) is determined by absorption x-ray
in the diode and charge collection in the space charge region
(SCR) and the diffusion region. The diode response is
proportional to the diode current.
Usually the condition of slow scanning (l/v>>τrelaxatio n,
where v is the scanning velocity, l/v is the characteristic time
of scanning, τrelaxation is the time of the achievement of steady
state in diode under irradiation) is fulfilled and one can
consider the stationary response at different value of x0.
Under the condition 1/α>>hdiode ( is the absorption
coefficient of x-ray in silicon) in the first approximation one
can consider the one-dimensional model.
A. Space charge region
The contribution to the diode response of the generation in
the space charge region is equal to the current
0
0
(, )
scr
w
SCR
J
gxx dx=∫, (4)
where g(x,x0) is the generation rate of electron-hole pairs
under irradiation, x0 is the beam scanning position.
The response function U(x, x0) for the Dirac -function
()
0
()xxgx
δ
=− is given by the (1.b) where σ=wscr. The
contribution to the resolution is equal to the width of SCR.
For the Gauss distribution function
2
0
2
(x x )
2
0
1
(x, x , ) exp
2
π
−
−Ω
ΓΩ=
Ω the response is as follows
0
0
0
(2
)1
22
,scr
Gauss
Rwx x
Erf Erfx⎛⎞
−
⎡⎤⎡⎤
+
⎜⎟
⎢⎥⎢⎥
ΩΩ
⎣⎦⎣⎦
⎝
=
⎠
Ω
In this case the contribution to the resolution is determined
by the higher of two values  and wscr.
The normalized rectangular form of beam can be written in
the next form:
000
1
(, ) ( /2 ) ( /2 )
beam beam
beam
g
xx x w x x w x
w
=Θ+ −Θ+ −,
where ( )x
Θ is Heaviside theta-function. The responses of
the SCR for different beam profile are shown in Fig. 7.
All calculations below were performed in dimensionless
units: current is normalized on the beam generation function
and the unity length is the diode width without substrate
(50
diode
wm
μ
=). Also for the better graphically present of
beam pulses we use /5
beam
wΩ= ratio.
JDelta
JStepWise
JGauss
-
1.0
-
0.5 0.5 1.0 1.5 2.0
0.2
0.4
0.6
0.8
1.0
JDelta
JStepWise
JGauss
0.5 1.0 1.5 2.0
0.2
0.4
0.6
0.8
1.0
Fig. 7 The response of the SCR for different beam profile: -function,
stepwise and Gauss distribution for next parameters:
a) wbeam = wscr = wdiode = 1; b) wdiode = 1, wscr = 0.06 wdiode, wbeam = 0.6 wdiode
B. Diffusion region
Contribution of the generation in the diffusion region to the
diode current is
0
(, )
s
cr
diff n
xw
dnxx
JeD dx =
Δ
=, (5)
where 0
(, )nxxΔ is the excess carriers density in the
diffusion region, e is the electron charge.
In the case of a low resistivity base of a diode one can use
the diffusion equation for the excess carrier density in the
following form
()
0
2
20–+ ,
n
n
Dgx
dn
dx x
τ
Δ=. (6)
The following boundary condition may be used
0
00
(, ) 0
(, ) 0 (, )/ 0
scr
scr base scr base
xw
xw w xw w
nxx
nxx or nxx x
=
=+ =+
⎧Δ=
⎪
⎨Δ=∂Δ∂=
⎪
⎩
, (7)
where wbase is the width of the base of the detector.
The response function U(x0) for our model can be obtained
analytically by solving (6) with (7) in dependence on
parameters wscr, Ldiff and wbase.
IV. RESULTS AND DISCUSSIONS
The total response of diode is
0
000
() () (,)
total SCR
x
x
Jx JxJxx
=
=+ .
Results of tabulation of U(x0) and responses on the
Gaussian and the Rectangular beam profiles for different
values of wscr and Ldiff are shown in Fig. 8.
StepWise
Gauss
Delta
-
0.5 0.0 0.5 1.0 1.5
0.2
0.4
0.6
0.8
1.0
a)
StepWise
Gauss
Delta
-
0.5 0.0 0.5 1.0 1.5
0.2
0.4
0.6
0.8
1.0
b)
StepWise
Gauss
Delta
-
0.5 0.0 0.5 1.0 1.5
0.2
0.4
0.6
0.8
1.0
c)
a
b
c
-
0.5 0.0 0.5 1.0 1.5
0.2
0.4
0.6
0.8
1.0
-function
a
b
c
-
0.5 0.0 0.5 1.0 1.5
0.2
0.4
0.6
0.8
1.0
Stepwise beam profile
a
b
c
-
0.5 0.0 0.5 1.0 1.5
0.2
0.4
0.6
0.8
1.0
Gauss beam profile
Fig. 8 Results of tabulation of U(x0) and responses on the Gaussian and the
Rectangular beam profiles for different values of wSCR and Ldiff:
a) wdiode = 1, wbeam = 0.6, wscr = 0.06 (~100  cm), Ldiff = 1.2 (~1 µs)
b) wdiode = 1, wbeam = 0.6, wscr = 0.18 (~1 k cm), Ldiff = 0.4 (~0.1 µs)
c) wdiode = 1, wbeam = 0.6, w
s
c
r
= 0.50 (~10 k cm), Ldi
ff
= 0.12 (~0.01 µs)

The contributions of the SCR and the diffusion region (JSCR
and Jdiff) into the total current (Jtotal) for different generation
functions are shown in Fig. 9.a and Fig. 9.b.
StepWise
Gauss
Delta
-
0.5 0.0 0.5 1.0 1.5
0.2
0.4
0.6
0.8
1.0
Total response
Stepwise beam profile
Gauss beam profile
Jtotal
Jbase
Jscr
-
0.5 0.0 0.5 1.0 1.5
0.2
0.4
0.6
0.8
1.0
-function
Jtotal
Jbase
Jscr
-
0.5 0.0 0.5 1.0 1.5
0.2
0.4
0.6
0.8
1.0
Stepwise beam profile
Jtotal
Jbase
Jscr
-
0.5 0.0 0.5 1.0 1.5
0.2
0.4
0.6
0.8
1.0
Gauss beam profile
Fig. 9.a Total current (Jtotal) and comparison of contributions of the JSCR and
Jdiff into the Jtotal for different generation function for the following
parameters:
wdiode = 1 (50 µm), wbeam = 0.6 (30 µm),
w
s
c
r
= 0.06 (3 µm, ~100  cm), Ldi
ff
=1.2 (60 µm, ~1 µs)
StepWise
Gauss
Delta
-
0.5 0.0 0.5 1.0 1.5
0.2
0.4
0.
6
0.8
1.
0
Total response
Stepwise beam profile
Gauss beam profile
Jtotal
Jbase
Jscr
-
0.5 0.
0
0.
5
1.
0
1.
5
0.
2
0.4
0.6
0.8
1.0
Response on -function
Jtotal
Jbase
Jscr
-
0.5 0.0 0.5 1.0 1.5
0.2
0.4
0.6
0.8
1.0
Response on Stepwise
Jtotal
Jbase
Jscr
-
0.5 0.0 0.5 1.0 1.5
0.2
0.4
0.6
0.8
1.0
Response on Gauss
Fig. 9.b Total current (Jtotal) and comparison of contributions of the JSCR and
Jdiff into the Jtotal for different generation function for the following
parameters:
wdiode = 1 (50 µm), wbeam = 0.6 (30 µm),
wscr = 0.057 (2.85 µm, =100  cm), Ldiff = 0.1224 (6.12 µm, =0.01 µs)
Comparing results presented in Fig. 9.a and Fig. 9.b one can
see that in the case when Ldiff >> wbeam >> wscr the diffusion
current makes the major contribution to the total response.
Also in this case the response curve is broader than the beam
profile. Alternatively, when the diffusion length is small
Ldiff << wbeam the spatial resolution is good and edge on
detector response is almost corresponding to the beam profile
and contributions of the diffusion and SCR currents become
comparable.
The sequence of beams can be resolved when distance
between peaks is larger than width of the diode response
function. Thought the measured response reflects the real
shape of beams at a high resolution, solving integral equation
(2) we can restore shape of real beams at arbitrary known
response function.
In the case of linear model we can obtain response on any
beam profile using the convolution of the response function
and the generation function without solving the diffusion
equation. The comparisons of the calculated Jtotal and response
obtained by convolution of the response function with
different beam profiles Rtotal are shown on Fig. 10.
J
total
J
convolution
-
0.5 0.0 0.5 1. 0 1.5
0.2
0.4
0.6
0.8
1.
0
a) Gauss beam profile
J
total
J
convolution
-
0.5 0.0 0.5 1.0 1. 5
0.2
0.4
0.6
0.8
1.
0
b) Stepwise beam profile
Fig. 10 The comparisons of the calculated Jtotal and response obtained by
convolution of the response function with different beam profiles Rtotal
(wdiode = 1, wbeam = 0.6, wscr = 0.06, Ldiff = 1.2)
Comparing graphics presented in Fig. 10 one can see that
responses calculated directly and obtaining by convolution
coincide.
The response of experimentally measured sequences of
microbeams [3] (Fig. 11.a) with edge on SSD may be
described analytically using the stepwise response function
(2.a). The analytically calculated response on the sequence of
rectangular microbeams is shown in Fig. 11.b.
10 20 30 40
0.1
0.2
0.3
0.4
0.5
0.6
a) b)
Fig. 11 a) The experimentally measured five central microbeams
b) Analytically calculated response for rectangular beams
(wbeam = 62 µm, wscr + Ldiff = 62 µm, wdiode = 50 µm)
Comparison of the experimental results with the analytical
ones shows that real beam have rectangular profile with beam
width approximately equal to the width of the response
function (~wscr + Ldiff).
V. CONCLUSIONS
The possibility of calculations the response functions of
diodes is shown when the properties of beams and
characteristics of diodes are known. The real intensity profiles
of beams can be calculated using the measured beam intensity
profile and known response function of the diode.
Calculations of responses for experimentally studied planar
structure (epitaxial one-strip silicon diode) in case of edge on
irradiation geometry by narrow X-ray beams have been
carried out. The contributions to responses from the space
charge region and the diffusion region have been compared.
The experimentally observed response can be well described
in a simple physical model.
The spatial distribution of the intensity of the MRT beam
has been calculated taken into account both charge collection
from the space charge region and the diffusion region of
diodes. It has been shown that spatial resolution for the edge
on irradiation geometry of the detector can be higher than in
the face on irradiation geometry. Also, the efficiency of the
charge collection in the edge on irradiation geometry can be
better than for the face on irradiation.

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