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Inferring baseline opticalproperties of the human head
Alex [email protected]
MGH/MIT/HMS AAM NMR CBI
Inferring baseline optical properties of the human head – p.1
Aims
� How well can we measure baseline optical tissueproperties? Given. . .
� 3d anatomical MRI data
� optically-uniform segmented tissue types
� time-resolved measurements
� single optical
�
� Motivations:
� functional imaging requires accurate baselineproperties
� more
�
’s absolute [Hb] and [HbO].
� sets an upper bound on capability withoutMRI data.
Inferring baseline optical properties of the human head – p.2
Outline1. Bayesian method overview
2. simple layer system
3. likelihood
4. results in layer
5. optode calibration & location
6. preliminary head
7. issues & conclusion
Inferring baseline optical properties of the human head – p.3
Method overviewWhat do measurements � tell you about parameters � ?Inference � probability distribution functions (PDFs)
y
x
p(x|y)
= modeljoint p(y,x)
y measured
posterior likelihood prior
Constant prior look for peaks in likelihood.
No artificial regularization
Peak widths give all errorbars & correlations
Currently: testing with numerically-generated noisymeasurements
Inferring baseline optical properties of the human head – p.4
Method overviewWhat do measurements � tell you about parameters � ?Inference � probability distribution functions (PDFs)
y
x
p(x|y)
= modeljoint p(y,x)
y measured
� ��� ��� � � � �� � ��
� �� � � �� � ��� �
posterior � likelihood prior
Constant prior look for peaks in likelihood.
No artificial regularization
Peak widths give all errorbars & correlations
Currently: testing with numerically-generated noisymeasurements
Inferring baseline optical properties of the human head – p.4
Method overviewWhat do measurements � tell you about parameters � ?Inference � probability distribution functions (PDFs)
y
x
p(x|y)
= modeljoint p(y,x)
y measured
� ��� ��� � � � �� � ��
� �� � � �� � ��� �
posterior � likelihood prior
Constant prior look for peaks in likelihood.
� No artificial regularization
� Peak widths give all errorbars & correlations
Currently: testing with numerically-generated noisymeasurements
Inferring baseline optical properties of the human head – p.4
Method overviewWhat do measurements � tell you about parameters � ?Inference � probability distribution functions (PDFs)
y
x
p(x|y)
= modeljoint p(y,x)
y measured
� ��� ��� � � � �� � ��
� �� � � �� � ��� �
posterior � likelihood prior
Constant prior look for peaks in likelihood.
� No artificial regularization
� Peak widths give all errorbars & correlations
Currently: testing with numerically-generated noisymeasurements �
Inferring baseline optical properties of the human head – p.4
Simple 2-layer system
10 20 30 40 50 60 700
100
200
300
400
m = 1...M
ym
10 20 30 40 50 60 70
100
102
m = 1...M
ym
fm
(x)y
m
source detectors
layer 1: 8 mm layer 2: 18 mm
measurement vector y
56 mm
det3 det441
det1
det2
~20 ‘timegates’ from 0.2 − 2.0 ns
S-D separations of 7, 14, 21, 28 mm
Parameter vector � � ��� ���� �� � �
��� �� �
�
�� �� �
�
��� �
Inferring baseline optical properties of the human head – p.5
Likelihood
�
� �� � forward model (signal expectation)
� �
noise
noise model
y
x
f(x)σ(x)
� �� � � � �
noise
�� � � ��� � �
uncorr. gaussian �
�
�� �� ���� � � � ��
� � � � ��� �� �
� � � ��� �
� is some (growing) function of
�
, giving detectionstatistics. Inferring baseline optical properties of the human head – p.6
Look at sensitivity
20 40 6010
−2
100
102
104
m
sign
al e
xpec
tatio
n f m
mua1 = 0.010 per mmmua1 = 0.003 per mmmua1 = 0.030 per mm
20 40 6010
−2
100
102
104
m
sign
al e
xpec
tatio
n f m
mua2 = 0.010 per mmmua2 = 0.003 per mmmua2 = 0.030 per mm
20 40 6010
−2
100
102
104
m
sign
al e
xpec
tatio
n f m
musp1 = 2.000 per mmmusp1 = 0.667 per mmmusp1 = 6.000 per mm
20 40 6010
−2
100
102
104
m
sign
al e
xpec
tatio
n f m
musp2 = 1.000 per mmmusp2 = 0.333 per mmmusp2 = 3.000 per mm
Inferring baseline optical properties of the human head – p.7
Sensitivity compared to noise
0 20 40 60 80−20
0
20
40
60
80
m
β m
mua1 = 0.003 per mmmua1 = 0.030 per mm
0 20 40 60 80−15
−10
−5
0
5
10
15
20
m
β m
mua2 = 0.003 per mmmua2 = 0.030 per mm
0 20 40 60 80−100
0
100
200
300
400
500
m
β m
musp1 = 0.667 per mmmusp1 = 6.000 per mm
0 20 40 60 80−15
−10
−5
0
5
10
15
m
β m
musp2 = 0.333 per mmmusp2 = 3.000 per mm
� -normalized changes : � � �� � �
Inferring baseline optical properties of the human head – p.8
Maximizing Likelihood
0.0080.01
0.0120.014
0.8
1
1.2
1.40
0.2
0.4
0.6
0.8
1
mua2 (per mm)
slice through relative likelihood peak
musp2 (per mm)
p lik(x
)
0.009 0.0095 0.01 0.0105 0.011 0.0115 0.012 0.01250.9
0.95
1
1.05
1.1
1.15
1.2
1.25
mua2 (per mm)
mus
p2 (p
er m
m)
objective funct ( −log plik
(x)) difference from min
0.10.31231030100
x = true parametersexpt
minimization
Minimize ‘objective function’ NLL � � � � � �� � � �
� gaussian noise � ‘weighted least squares’
� peak very narrow in �
layer 1 I show only �
layer 2
� 1-2 minutes per optimizationInferring baseline optical properties of the human head – p.9
Results : photon numbertyp tissue properties �
expt
����
� � � � �� � � � � ��
mm � �
0 0.005 0.01 0.015 0.020
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
mua2 (per mm)
mus
p2 (p
er m
m)
# photons in det 4 = 0.72.0
6.7
20
67
Ellipses define likelihood peak width:
200
� more photons narrower peak
� true �expt rarely outside peak — good!
Inferring baseline optical properties of the human head – p.10
Results : varying �
0 0.005 0.01 0.015 0.020
0.2
0.4
0.6
0.8
1
1.2
mua2 (per mm)
mus
p2 (
per
mm
)less photons survive to distant detectors
so greater uncertainty in parameters
� other 3 parameters held constant
� photon # : 67 photons at det4
� realistic inference of errorbarsInferring baseline optical properties of the human head – p.11
Results : varying
��
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
xexpt
: musp2 (per mm)
p(x
|y)
: m
usp
2 (
pe
r m
m)
0.007 0.008 0.009 0.01 0.011 0.012 0.013 0.014 0.015
0.5
1
1.5
2
2.5
mua2 (per mm)
mu
sp
2 (p
er m
m)
Generally good agreement. Reliability problems. . .
� noise model mismatch ? / optimization gettingstuck
Inferring baseline optical properties of the human head – p.12
Integrating out free parametersWidth in �
layer 1 is much less than in �
layer 2.We only care about �
layer 2 (e.g. cortex in head).Once peak found, use gaussian approx: analytic integral over
�
layer 1:
xlayer 1
x layer 2
marginalize (integrate)multidimensional gaussian
λλ−1/2
2
1−1/2
�� � � �� �T ��� �� � � � ��
�� � � ��
This illustrates the general Bayesian recipe for free parameters :integrate over them.
Inferring baseline optical properties of the human head – p.13
Optode calibration & placementOptode calibration :
� � � � �
free scaleparameters
� As for layer 1, they will be narrow-width
� integrate out with gaussians (fast)
Placement : choose best source/detector locations
� use peak volume
�� � � � ��as objective func.
� fix � �
expt, and optimize over locations.
For gaussian noise model � � � diag
��� � � �� �
.with jacobean
� ����� � � �
�� ��� � .
Inferring baseline optical properties of the human head – p.14
Noise model detailsUsed uncorrelated gaussian model:
py f = gaussian approx to Poisson
5% model errorσf
fpy = signal of 1 photon / timegate
� gaussian approx to poisson, clipped at both ends
� Collect more photons model error dominates
� Other more robust noise models (power law tails,etc) possible, easy to implement in Bayesianformalism.
Inferring baseline optical properties of the human head – p.15
Forward model detailsTime-resolved detector signals
�
given params � .Written finite-difference time-domain (FDTD) code:
time
∆
∆
x
t
space (3d)
boundary conditions
initial condition (source)
EVOLUTION METHOD
� arbitrary 3d tissue geometries
� 0.5s per source, small system 6cm � 6cm � 3cm � 2ns
� Diffusion Approx, validated against Monte Carlo
� Robin BCs, surface normals only
��� ��
.
� evolution: ‘forward-Euler’
� ��
, small � � slows it down.Inferring baseline optical properties of the human head – p.16
Forward model issuesThere are
� �� �
methods (‘implicit’, e.g. ADI) :
� faster (less timesteps), but nonsmooth fluence bad!
Boundary Conditions
� do matter.
� ‘Stiffness’ tricky forFDTD stability 0 5 10
x 10−10
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
t (sec)
sign
al
4: MC7: MC4: FD dir7: FD dir4: FD rob7: FD rob
Avoid large system (head) by matching to :fluence components � � ��� obey Helmholtz eqn with fixed
� � � � �� � � . So, ‘radiative’ BC is just Robin BC.
Inferring baseline optical properties of the human head – p.17
Nonlinear optimization issuesForward model with discontinuities (jumps) = bad :
ridges in
� ��� �
fake local minima
Had to be removed!
Derivative info vastly improves speed/robustness:� Adjoint (‘reverse’) differentiation: get
���
�
wrt all� � withlittle more effort than
�
(e.g. Hielscher, Klose, Hanson1999)
Inferring baseline optical properties of the human head – p.18
Future directions
� MRI-segmented heads: 5 tissue-types, effect ofCSF.
Forward model improvements: adjointdifferentiation, ADI timestepping.
Use peak width (Hessian) to optimize optodelocations
Test integrating out free params: optodecalibration, start time. . .
Noise models: Poisson for low photon counts,detector saturation.
More sources, experimental phantom verification,heads. . .
Inferring baseline optical properties of the human head – p.19
Future directions
� MRI-segmented heads: 5 tissue-types, effect ofCSF.
� Forward model improvements: adjointdifferentiation, ADI timestepping.
Use peak width (Hessian) to optimize optodelocations
Test integrating out free params: optodecalibration, start time. . .
Noise models: Poisson for low photon counts,detector saturation.
More sources, experimental phantom verification,heads. . .
Inferring baseline optical properties of the human head – p.19
Future directions
� MRI-segmented heads: 5 tissue-types, effect ofCSF.
� Forward model improvements: adjointdifferentiation, ADI timestepping.
� Use peak width (Hessian) to optimize optodelocations
Test integrating out free params: optodecalibration, start time. . .
Noise models: Poisson for low photon counts,detector saturation.
More sources, experimental phantom verification,heads. . .
Inferring baseline optical properties of the human head – p.19
Future directions
� MRI-segmented heads: 5 tissue-types, effect ofCSF.
� Forward model improvements: adjointdifferentiation, ADI timestepping.
� Use peak width (Hessian) to optimize optodelocations
� Test integrating out free params: optodecalibration, start time. . .
Noise models: Poisson for low photon counts,detector saturation.
More sources, experimental phantom verification,heads. . .
Inferring baseline optical properties of the human head – p.19
Future directions
� MRI-segmented heads: 5 tissue-types, effect ofCSF.
� Forward model improvements: adjointdifferentiation, ADI timestepping.
� Use peak width (Hessian) to optimize optodelocations
� Test integrating out free params: optodecalibration, start time. . .
� Noise models: Poisson for low photon counts,detector saturation.
More sources, experimental phantom verification,heads. . .
Inferring baseline optical properties of the human head – p.19
Future directions
� MRI-segmented heads: 5 tissue-types, effect ofCSF.
� Forward model improvements: adjointdifferentiation, ADI timestepping.
� Use peak width (Hessian) to optimize optodelocations
� Test integrating out free params: optodecalibration, start time. . .
� Noise models: Poisson for low photon counts,detector saturation.
� More sources, experimental phantom verification,heads. . .
Inferring baseline optical properties of the human head – p.19
Conclusions
� Time-resolved data with few detected photonscan infer optical parameters in layer 8mm belowsurface, to
� �
Full posterior (errorbars & correlations) can behandled
Developed & validated rapid 3d diffusionforward model
Bayesian optode calibration and optimal locationrecipes
Inferring baseline optical properties of the human head – p.20
Conclusions
� Time-resolved data with few detected photonscan infer optical parameters in layer 8mm belowsurface, to
� �
� Full posterior (errorbars & correlations) can behandled
Developed & validated rapid 3d diffusionforward model
Bayesian optode calibration and optimal locationrecipes
Inferring baseline optical properties of the human head – p.20
Conclusions
� Time-resolved data with few detected photonscan infer optical parameters in layer 8mm belowsurface, to
� �
� Full posterior (errorbars & correlations) can behandled
� Developed & validated rapid 3d diffusionforward model
Bayesian optode calibration and optimal locationrecipes
Inferring baseline optical properties of the human head – p.20
Conclusions
� Time-resolved data with few detected photonscan infer optical parameters in layer 8mm belowsurface, to
� �
� Full posterior (errorbars & correlations) can behandled
� Developed & validated rapid 3d diffusionforward model
� Bayesian optode calibration and optimal locationrecipes
Inferring baseline optical properties of the human head – p.20
