C.1 Rotational motion
Figure C.2: Sketch of the coupling (blue) between the motor drive (rotation axis at ~M ) with rotation angle ϕ2 and the rotation of the platform (rotation axis at coordinate (0,0) and rotation angle ϕ1) in a right handed coordinate system (left). The connection rod of length s2 transmits the motion between the coupling points C1 and C2 at the distances s1 and rM from the respective rotation axes.
An image of the motion table is overlain by the two rotation axes and the considered distances (right).
The motion table for rotational motion used for experiments in section 3.2.1 is driven by an electric motor. By means of its supply voltage, the motion period is freely adjustable. The coupling between the motor and the platform is outlined in figure C.2 within the right-handed coordinate system (¯x, ¯y). The angle ϕ1 of the platform is a function of the rotation angle ϕ2 of the motor. The lengths of the con-rods s1, s2 and rM can be set by screws in elongated mounting hole and are fixed during an experiment. The length of s3 changes continuously by the position of the coupling point C2 related to the angle ϕ2. The coordinates of the coupling point C1 follow immediately from that.
The coordinates of the coupling point C2 are given by the formula
C~2 = ~M + rM · cos ϕ2 sin ϕ2
!
. (7.4)
With that, the length s3 can be deduced from the relation
s23 = C2¯2x+ C2¯2y = (Mx¯+ rM · cos ϕ2)2+ (My¯+ rM · sin ϕ2)2, (7.5) and the angle ϕ01 from
tan ϕ01 = C2¯y
C2¯x = My¯+ rM · sin ϕ2
Mx¯+ rM · cos ϕ2. (7.6)
With equation (7.5) and the law of cosines adapted for the triangle with the sides s1, s2 and
s3 the angle ϕ001 is given in the formula
cos ϕ001 = s21− s22+ s23
2 · s1s3 = s21− s22+ (Mx¯+ rM · cos ϕ2)2+ (My¯+ rM · sin ϕ2)2 2 · s1
q
(M¯x+ rM · cos ϕ2)2+ (My¯+ rM · sin ϕ2)2
. (7.7)
Finally, the rotation angle ϕ1 of the motion platform in dependence of ϕ2can be calculated by using equations (7.6) and (7.7)
ϕ1 = ϕ01+ ϕ001
= arctanMy¯+ rM · sin ϕ2
Mx¯+ rM · cos ϕ2 + arccoss21− s22+ (M¯x+ rM · cos ϕ2)2+ (My¯+ rM · sin ϕ2)2 2 · s1
q
(M¯x+ rM · cos ϕ2)2+ (My¯+ rM · sin ϕ2)2 .
(7.8) It has to be noted that the angle ϕ1 in equation (7.8) is not the rotation angle α of the platform which is used in the transformation matrices. The angle α is defined in the left-handed coordinate system of the PET scanner which is here indicated with the subscript.
Following relations hold true: α = 0 for yPET = ¯x = 0 and α = π2 for xPET = ¯y = 0. That means |α| = ϕ1− π2 .
C.2 Motion table with stepping motor for precise 1D motion patterns A motion table has been developed to emulate motion patterns as given by (2.13). This in-house made motion table is shown in figure C.3. A stepping motor drives a dedicated target holder made up of PMMA along a linear axis. The protruding target support of 6 mm thick-ness allows for well-defined attenuation correction. It is the only part of the whole system that is placed in the FOV when the motion table has been adjusted to perform a target
mo-Figure C.3: Motion table for 1D motion patterns driven by a stepping motor along a linear axis.
Components are labelled in the left figure. The right image shows the motion table with displacement sensor operating at the treatment site including the double-head PET scanner. The marked (left-handed) coordinate systems allow for better understanding of the motion table orientation in between the two detector heads. Motion proceeds parallel to the y-axis.
Definition of motion parameters User-defined motion limits (to avoid crash with laser sensor)
Emergency stop Reference movement (left or right limit switch) Free movement for defined start position y0
Online monitor Start after certain
spill or time Counted spills and nextpoint signals
Immediate motion start or with certain spill
Figure C.4: Screenshot of the graphical user interface of CosMoS in an early version 1.1 with some highlighted functionalities.
tion in the left-right direction in BEV at the fixed horizontal beamline at GSI. This motion along the y-axis in the left-handed coordinate system of the PET scanner would correspond to a cranio-caudal motion within a patient treated in conventional supine position. An at-tached laser displacement sensor (Model OD100-35P840, SICK Vertriebs-GmbH, Düsseldorf, Germany) records the actual elongation of the target holder and generates a surrogate signal which controls the motion mitigated beam delivery by means of gating or tracking.
The developed LabVIEW control software CosMoS (which stands for COSinusartige MO-torSteuerung) for the stepping motor records the motion and other signals like the nextpoint signals, the trigger signals for a predefined gating window and the beam status from the accelerator. The latter one allows an optional triggering of the motion start at exhalation (Φ0 = π/2) or inhalation (Φ0 = 0) with the begin of the first spill. Arbitrary values can be chosen for the start position y0, the peak-to-peak amplitude a and the period time τ , both for n equal to 1 or 2. A screenshot of the graphical user interface of CosMoS version 1.1 is shown in figure C.4. The motion table is equipped with several safety measures. Besides an emergency stop, inner limit switches can be set by the user to avoid an accidental crash with the mounted displacement laser sensor or other equipment nearby. These switches can be turned off for reference movement when the platform is driven to one of the limit switches at
the end of the linear axis. An online motion monitor gives a visual feedback to the user who operates the motion table from the control room, i.e. outside of the experiment bunker.
The functionality of the motion table has been considerably increased in the recent past.
In the current version 2.0, an analytically described variation of amplitude, frequency and baseline has been incorporated. The related investigations are beyond the scope of this thesis.
This thesis does only refer to experiments with regular motion patterns described by (2.13).
C.3 Motion table enabling relative target movement
Figure C.5: Motion table for relative motion between two targets. Components are labelled in the right image which shows the setup for one of the presented experiments in section 5.1.2. A bridge device spans the moving platform and carries a static target (here made of 100 mm×100 mm×30 mm PMMA).
Another one (labelled in orange and consisting of 80 mm × 50 mm × 47 mm PMMA) is positioned on the platform in front of the static target. A laser displacement sensor measures continuously the distance to the platform which is driven by an electric motor. A draft (not to scale) on the left shows the coupling between the motor drive with rotation centre at M and the motion platform by means of the connection rod of length s.
For performing experiments with targets moving relative to each other (where one is at rest), a motion table from GSI, also dedicated for 1D motion patterns, has been used. As shown in figure C.5, it consists of a moving platform which is slightly tilted against the treatment couch by about 2.2◦ to move exactly parallel to the incident beam. This angle is neglected within all other experiments. An additional bridging device allows to position phantoms at arbitrary fixed positions in the field of view of the camera directly next to moving targets. The motion of the platform is provided by a coupling to an electric motor.
A sketch of that coupling is also shown in figure C.5 with a right-handed coordinate system (¯x, ¯y). The location of the coupling point C of the connection rod with length s is described by the radial distance r from the centre of the rotation axis M and the continuously changing rotation angle ϕ. The distance of the moving platform xp from the origin of the coordinate system (¯x, ¯y) is described as
xp = x1+ x2
= r cos ϕ + q
s2− (y1+ y2)2
= r cos ϕ +qs2− (r sin ϕ + y2)2,
(7.9)
where y2 and s are fixed quantities of about 25 mm and 155 mm and r can be adjusted on a limited scale. For emulating a peak-to-peak amplitude of 20 mm during experiments a radial distance r of 10 mm was chosen. The exact motion calculation according to (7.9) results in a peak-to-peak amplitude of the moving platform of 20.27 mm. Considering the manufacturing accuracy of the motion table and the accuracy for length determination of s and y2 a peak-to-peak amplitude of 20 mm has been assumed for data evaluation when interpreting the voltage signal of the laser displacement sensor.