Transcript SimSET: a Simulation System for Emission Tomography

Simulation of emission
tomography
Robert L. Harrison
University of Washington Medical Center
Seattle, Washington, USA
Supported in part by PHS grants CA42593 and CA126593
What is emission
tomography?
Nuclear Medicine
Radiology
Radiotherapy
(www.imaginis.com)
(Stieber et al)
(Wikipedia)
The difference between
transmission and emission
X-ray CT
X-ray computed
tomography
Radiology
PET
Positron emission
tomography
Nuclear Medicine
Unclear
(Wikipedia)
Different information
X-ray CT
PET
X-ray computed
tomography
Positron emission
tomography
Anatomy/
Form
Metabolism/
Function
PET/CT
Complementary
Information
(Wikipedia)
Different information
What’s the diagnosis?
Dead…
Emission tomography:
what should we simulate?
Digital phantom
Patient
(Segars)
Emission tomography:
what should we simulate?
PET scanner
Half scanners with and without collimation
(Suetens)
SPECT scanner
Single photon emission computed tomography
(George et al)
Emission tomography:
what should we simulate?
Signal processing / output event position
y
a
blue photon
position =
(x1,y1,z1)
x
d
pink photon
position =
(x2,y2,z2)
An example: SimSET
A Simulation System for Emission Tomography
• Goals
- Flexible
- Extensible
- Portable
- Easy-to-use
- FAST
SimSET overview
Object = patient
Geometric
description
Attenuation
Activity
Voxelized
description
Object: voxelized
• Voxelized objects:
– Easier to define complex objects.
• Patient scans are voxelized.
– Faster tracking in complex objects.
• Obvious which the next voxel is.
• However, the time tracking through the object
does increase as the voxelization grows finer.
Object: processes
• Generate decays.
• Decay products.
• Tracking particles/photons.
–
–
–
–
–
–
Compton scatter.
Coherent scatter.
Photoabsorption.
Pair production.
Fluorescence.
Brehmstrahlung.
Object: generate next decay
• Which voxel? Two options:
- Make a list of all the voxels with the
sum of activity in them to that point:
voxel1 activity1;
voxel2 activity1 + activity2; ….
voxelN TotalActivity.
- Pick a random number, u, between 0
and TotalActivity.
- Next decay generated in the last voxel
with the summed activity < u.
- Generate all the decays in voxel1;
- Generate all the decays in voxel2;
- …
- SimSET uses this method: it
is faster.
- The decays are not generated in
the correct order timewise.
Object: generate next decay
• Choose a random location in the voxel.
How?
- 3 random numbers, one each for x, y, z.
Object: generate next decay
• When?
– The mean number of decays in a voxel is the product of the
scan time and the activity in the voxel.
– The distribution of the actual number of decays is Poisson.
• What distribution do we use to determine the elapsed
time to the next decay?
- The exponential distribution (1 random number).
- Keep generating decays until the sum of the elapsed times
is ≥ the scan time.
Object: generate next decay
• What now?
• Depends on the isotope: some combination of
– alpha (Helium nuclei)
– beta (electrons or positrons)
– gamma (photons)
• SimSET only produces one particle per decay:
– positron (PET); or
– photon (SPECT) < 1000 keV, all photons one energy.
– No 124I, a positron emitter that also emits photons:
MeV of pho ton
% of d ecays
1.37
3
1.51
4
1.69
14
2.09
2
2.26
1.5
Object: annihilate positron
(PET only)
• SimSET does not track
positrons.
– Too many interactions; too
computationally intensive.
– Uses a probabilistic range
model instead.
– Positron/electron
annihilation at end of range.
– Two (almost) anti-parallel
511 keV photons produced.
– Photon polarization not
modeled.
Object: pick photon direction
• Generate a random 3D unit
vector. How?
• 2 random numbers:
– One picks an azimuthal
angle between 0 and .
– The other picks the cosine
of the inclination angle
between -1 and 1.
– This results in a uniform
distribution over the unit
sphere.
Object: track photons
• How far will a
photon travel in a
uniform medium?
  is the material’s
attenuation coefficient.
Object: track photons
• How far will a
photon travel in a
changing medium?
• Sample a
dimensionless distance,
free paths, p, from the
exponential distribution
with  = 1.
• Weight the distance
traveled by the true ’s.
Travel until
dii = p
Object: photon interaction
• If the photon leaves the object before traveling
the sampled number of free paths, we pass it to
the collimator module.
• Otherwise randomly choose an interaction from:
– Photoabsorption.
– Compton scatter.
– Coherent scatter.
• If the photon scatters, continue tracking.
• SimSET does not model pair production or
secondary photons.
Object: choosing interaction
type
• If the probability of:
– Photoabsorption is p < 1;
– Compton scatter is c < 1 - p;
– Coherent scatter is 1 - p - c.
• Sample u randomly from (0,1).
• If
u < p then photoabsorb;
p < u < p + c then Compton scatter;
u ≥ p + c then coherent scatter.
Object: simulating
interactions
• Photoabsorption:
– Discard photon.
• Compton scatter:
–
–
–
–
Klein-Nishini density function to determine the scatter angle.
Acceptance-rejection method.
Klein-Nishina is a free electron approximation.
Photon loses energy as function of scatter angle.
• Coherent scatter:
– Table lookup with linear interpolation to determine scatter angle.
– No energy lost.
– Generally very small angles (< 5 degrees).
Collimators
• Tracking is the same as through object.
– Fluorescence (ignored in SimSET) is an
issue for Thallium SPECT and deadtime.
• Efficiency is a problem. Of decays in
the FOV,
– PET: only 1/20 - 1/200 detected.
– SPECT: only 1/10000 - 1/1000000 are
detected.
Collimators
• SPECT collimators
– Hunk of lead with
hexagonal holes.
– Collimator and
detector circle patient.
– SimSET models
geometric collimator.
• PET collimators
– Cylindrical annuli of
lead or tungsten to
reduce randoms and
scatter.
– Trend towards no
collimation in FOV.
Collimators
• SimSET models only the collimators shown
on previous slide.
• Other collimation possibilities (mainly
SPECT):
–
–
–
–
Pinhole.
Rotating slat.
Slit.
Electronic.
• When (if) the photon escapes the collimator,
SimSET passes it to the detector module.
Detectors
• Tracking remains the
same, but our interests
change.
– We are now interested in
where/how much energy
is deposited.
Detectors/electronics
• When a photon interaction
deposits energy in the
detector crystal, the energy
is converted into a shower of
scintillation photons.
• The photomultiplier tubes
convert (some of) these
photons into a electrical
signals.
• The electronics convert the
signals into a detected
position and energy.
• Multiple interactions usually
lead to incorrect positioning.
Detectors/electronics
• SimSET ignores the
scintillation photons.
– These could be tracked.
• Detected position is
computed using the energyweighted centroid of the
interactions in crystal.
• Detected energy is the sum
of the energies deposited in
crystal. It can be ‘blurred’
with a Gaussian.
• For PET, time-of-flight offset
is computed - it can be
blurred as well.
Binning
•
•
•
•
Line-of-response or crystal pair.
Detected energy.
True, scatter, or random (PET only) state.
Time-of-flight position (PET only).
(Schmitz)
Take away
• For emission tomography, the patient is injected
with (or ingests, etc.) a radio-labeled tracer.
• Emission tomography is used to explore
metabolism.
• One type of simulation tracks individual photons
through the ‘patient’, collimators and detectors.
• Designing such a simulation requires knowledge of
photon interactions with matter.
• Some details may be skipped to improve efficiency,
but this will bias the results and should be done
with care.
References
J.T. Bushberg, The essential physics of medical imaging, Lippincott Williams & Wilkins, 2002.
K.P. George et al, Brain Imaging in Neurocommunicative Disorders, in Medical speech-language pathology: a
practitioner's guide, ed. A.F. Johnson, Thieme, 1998.
D.E. Heron et al, FDG-PET and PET/CT in Radiation Therapy Simulation and Management of Patients Who Have
Primary and Recurrent Breast Cancer, PET Clin, 1:39–49, 2006.
E.G.A. Aird and J. Conway, CT simulation for radiotherapy treatment planning, British J Radiology, 75:937-949, 2002.
R. McGarry and A.T. Turrisi, Lung Cancer, in Handbook of Radiation Oncology: Basic Principles and Clinical
Protocols, ed. B.G. Haffty and L.D. Wilson, Jones & Bartlett Publishers, 2008.
R. Schmitz et al, The Physics of PET/CT Scanners, in PET and PET/CT: a clinical guide, ed. E. Lin and A. Alavi,
Thieme, 2005.
W.P. Segars and B.M.W. Tsui, Study of the efficacy of respiratory gating in myocardial SPECT using the new 4-D
NCAT phantom, IEEE Transactions on Nuclear Science, 49(3):675-679, 2002.
V.W. Stieber et al, Central Nervous System Tumors, in Technical Basis of Radiation Therapy: Practical Clinical
Applications, ed. S.H. Levitt et al, Springer, 2008.
P. Suetens, Fundamentals of medical imaging, Cambridge University Press, 2002.
depts.washington.edu/simset/html/simset_main.html
www.wofford.org/ecs/ScientificProgramming/MonteCarlo/index.htm
www.impactscan.org/slides/impactcourse/introduction_to_ct_in_radiotherapy
What is SimSET used for?
• Optimizing patient studies.
• Assessing and improving quantitation.
• Prototyping tomographs.
Variance reduction /
importance sampling
Variance reduction goal
• Increase the precision of the simulation
output achieved for a given effort:
– Precision of the output is partly dependent
on the number of detections.
– Effort is the amount of CPU time we need
for the simulation.
Variance reduction concepts
• Increasing the efficiency of photon
tracking.
• Bias.
• Data correlations.
• Importance sampling.
• Measuring efficiency.
Variance reduction concepts:
photon tracking efficiency.
• Decrease the amount of time we spend
per photon
OR
• Increase the likelihood that each photon
will be detected.
Variance reduction concepts:
photon tracking efficiency.
• In general, decreasing the time spent
tracking a photon is considered code
optimization (not variance reduction).
• Most variance reduction methods
increase the likelihood that photons will
be detected.
– In emission tomography only 1/20th (3D
PET) to 1/100000th (SPECT) of decays are
detected.
Variance reduction
concepts: bias
• Variance reduction methods can be
unbiased or biased.
– Unbiased methods are safer.
– Biased methods can greatly increase
apparent efficiency.
Variance reduction concepts: bias
Variance reduction concepts: bias
Variance reduction concepts: bias
Variance reduction concepts:
data correlations
• In experimental data, different events
are uncorrelated.
• Many variance reduction methods add
correlations between events.
• In choosing variance reduction methods
to use, be clear about how much
correlation is acceptable.
Variance reduction concepts:
importance sampling
Variance reduction concepts:
importance sampling
• We can create more decays in cone A than
cone B,
• but this would bias our output data.
• To avoid bias we give each decay a weight
that tells us how many ‘real world’ decays it
represents.
• For our output data we sum weights rather
than incrementing counts.
• In all variance reduction techniques, the
weight of a decay/photon is adjusted to
eliminate bias.
Variance reduction concepts:
measuring efficiency
• If we oversample cone A by a factor of
2, and undersample cone B by a factor
of 10, we will collect a lot of events with
weight 0.5.
Variance reduction concepts:
measuring efficiency
• But occasionally an
event from cone B
will scatter and be
detected with weight
10.
Variance reduction concepts:
measuring efficiency
• A list of N non-uniformly weighted
events is not as valuable as a list of
N uniformly weighted events (e.g.
counts).
• How valuable are non-uniformly
weighted events?
Variance reduction concepts:
measuring efficiency
• We value the data by its signal-to-noise.
• We define a ‘quality factor’, 0<Q≤1
which gives the relative value of a list of
events as compared to a list of
uniformly weighted events.
Variance reduction concepts:
measuring efficiency
• The cost of producing data is the CPU
time, T, required to generate it.
• We divide the data’s value by its cost to
get a computational figure-of-merit:
Variance reduction techniques
•
•
•
•
•
•
•
•
Stratification.
Forced detection.
Photon splitting.
Russian roulette.
Fictitious interaction tracking / delta scattering.
Convolution forced detection.
Forced non-absorption.
Forced first interaction in detectors.
Variance reduction techniques:
stratification
• In stratification we sample the starting
location/direction of decays/photons
based on the probability of detection.
• A decay/photon is weighted to account
for any over- or under-sampling.
• Ideally locations/directions are over/under-sampled in proportion to their
probability of detection (productivity).
Variance reduction techniques:
stratification
Variance reduction techniques:
stratification
Variance reduction techniques:
stratification
Variance reduction techniques:
forced detection
• Force a copy of a photon from its
current position/direction to the detector.
Variance reduction techniques:
forced detection
• At photon creation and after each scatter:
–
–
–
–
–
Create a copy of the photon.
Force an interaction in the field-of-view.
Force the interaction to be a scatter.
Force the scatter to be in a detectable direction.
Force the photon through the attenuating material to the
detector.
• Continue tracking original photon.
Variance reduction techniques:
forced detection
– Create a copy of the photon.
– Force an interaction in the field-of-view (FOV).
Variance reduction techniques:
forced detection
– Force the interaction to be a scatter.
– Force the scatter to be in a detectable direction.
Variance reduction techniques:
forced detection
– Force the photon through the attenuating material to the detector.
– (This step is also done for true photons before any other tracking is
done.)
Variance reduction techniques:
forced detection
• Continue tracking original photon normally:
– If it exits the object, discard it.
– If it is absorbed, discard it.
– If it scatters, repeat forced detection steps.
Variance reduction techniques:
stratification and forced detection
• Stratification and
forced detection
are
complimentary
techniques.
– The weight
differences
introduced by
stratification tend
to be reduced by
forced detection.
Variance reduction techniques:
photon splitting
1
1
5

1
1
1
Variance reduction techniques:
Russian roulette
0.2

1
20%
or
(nothing)
80%
Variance reduction techniques:
splitting and roulette
Splitting doesn’ t
make sense
Start tracking
a photon
Roulette doesn’t
make sense
CPU time
Splitting may be
dangerous
Finish tracking
a photon
Variance reduction techniques: fictitious
interaction tracking (delta scattering)
• Tracking through a
voxelized phantom
takes time.
Variance reduction techniques: fictitious
interaction tracking (delta scattering)
• Fictitious interaction tracking pretends
that the everything has the same
attenuation coefficient as bone.
• A new interaction possibility is added
for each tissue, the fictitious interaction.
• The distance to travel can then be
computed directly.
• If a fictitious interaction is selected, the
photon continues in the same direction,
unchanged.
Variance reduction techniques:
convolution forced detection
• Convolution forced detection is mainly
used for SPECT.
• Tracking is similar to regular forced
detection until the forced scatter
direction is chosen.
Variance reduction techniques:
convolution forced detection
• The direction is chosen perpendicular to the current
collimator position.
• The photon’s weight is distributed over the detector
by convolving with a depth dependent point-spread
function.
Variance reduction techniques:
forced non-absorption
• At interactions in the object and collimator, do
not allow the photon to be absorbed.
Variance reduction techniques:
forced first interaction
• In the detector, force at least one interaction to occur.
Variance reduction techniques:
bias and correlation
• With the exception of convolution forced detection, all
the techniques discussed are unbiased.
• Forced detection adds minimal correlations to the
output data.
• Photon splitting can add noticeable correlations to the
output data if done too late in the photon tracking.
• Convolution forced detection adds noticeable
correlations between neighboring bins in the output
data.
Variance reduction: closing
thoughts
• Variance reduction methods can improve the
efficiency of emission tomography simulations.
• They require substantially more effort to implement
than normal Monte Carlo.
• Efficiency gains using variance reduction are very
problem dependent.
– As little as 1.5 - 2 for some 3D PET simulations.
– 1000+ for some SPECT simulations.
• Events with extremely high weights can be a
problem.