Bayesian estimation of optical properties of the human head via 3D structural MRI p.1

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1 Bayesian estimation of optical properties of the human head via 3D structural MRI June 23, 2003 at ECBO 2003 Alex Barnett Courant Institute, New York University Collaborators (NMR Center, Mass. Gen. Hosp., Boston) David Boas Joe Culver Anna Custo (MIT) Gregory Sorensen Anders Dale Funding: CIMS, NIH, CIMIT Bayesian estimation of optical properties of the human head via 3D structural MRI p.

2 The big picture Diffuse Optical Tomography (DOT): I(t) fibers t Sources (near IR laser pulses) params brain? x signals y I(t) 0 ns Detectors (photodiodes) t Bayesian estimation of optical properties of the human head via 3D structural MRI p.2

3 The big picture Diffuse Optical Tomography (DOT): I(t) fibers t Sources (near IR laser pulses) params brain? x signals y I(t) 0 ns Detectors (photodiodes) t vector x = spatial absorption & scattering info vector y = components of measured signals Inverse problem: given y find x. Bayesian estimation of optical properties of the human head via 3D structural MRI p.2

4 The big picture Diffuse Optical Tomography (DOT): I(t) fibers t Sources (near IR laser pulses) params brain? x signals y I(t) 0 ns Detectors (photodiodes) t vector x = spatial absorption & scattering info vector y = components of measured signals Inverse problem: given y find x. Many wavelengths [HbO 2 ], [HbR], activation... Bayesian estimation of optical properties of the human head via 3D structural MRI p.2

5 Baseline optical measurement Assuming head tissues optically homogeneous: How well could we measure their baseline properties? absolute cortical absorption µ a cerebral oximetry, neonatal, stroke, trauma... required for quantitative brain activation studies Bayesian estimation of optical properties of the human head via 3D structural MRI p.3

6 Baseline optical measurement Assuming head tissues optically homogeneous: How well could we measure their baseline properties? absolute cortical absorption µ a cerebral oximetry, neonatal, stroke, trauma... required for quantitative brain activation studies small # unknowns (N = 6): x {µ a, µ s} scalp, skull, brain time-resolved DOT 0 mm 0 mm 0 mm D D2 D3 D4 S S2 small system eg 2 Src, 4 Det 0 mm 0 mm Bayesian estimation of optical properties of the human head via 3D structural MRI p.3

7 Baseline optical measurement Assuming head tissues optically homogeneous: How well could we measure their baseline properties? absolute cortical absorption µ a cerebral oximetry, neonatal, stroke, trauma... required for quantitative brain activation studies small # unknowns (N = 6): x {µ a, µ s} scalp, skull, brain time-resolved DOT 0 mm 0 mm 0 mm D D2 D3 D4 S S2 small system eg 2 Src, 4 Det 0 mm 0 mm Numerical study: simulated noisy signals y Bayesian estimation of optical properties of the human head via 3D structural MRI p.3

8 MRI segmented geometry segment Bayesian estimation of optical properties of the human head via 3D structural MRI p.4

9 MRI segmented geometry segment tissue µ a (mm ) µ s (mm ) shape scalp mm layer skull mm layer CSF folded 3 mm sheet brain cm folds (sulci) Much uncertainty. Diffusion, we use µ s,eff 0.4 mm. Bayesian estimation of optical properties of the human head via 3D structural MRI p.4

10 Diffusion forward model c φ t = ( ) 3µ s(r) φ Finite difference, lattice size h Forward Euler, timestep t ps accuracy O(h 2 ), typ few % error h = 2 mm: cells, 0s CPU µ a (r)φ + q(r, t) Signals: time-dep fluence at detectors = vector f(x) h µ s µ a φ Bayesian estimation of optical properties of the human head via 3D structural MRI p.5

11 Diffusion forward model c φ t = ( ) 3µ s(r) φ Finite difference, lattice size h Forward Euler, timestep t ps accuracy O(h 2 ), typ few % error h = 2 mm: cells, 0s CPU µ a (r)φ + q(r, t) Signals: time-dep fluence at detectors = vector f(x) h µ s µ a φ S S2 D D2 D3 D4 D D2 D3 D4 detect f m m Bayesian estimation of optical properties of the human head via 3D structural MRI p.5

12 Bayesian inverse problem f m x n sing. vals. 0 : ill-posed (many x equally valid) Incomplete info on x probability density function Bayesian estimation of optical properties of the human head via 3D structural MRI p.6

13 Bayesian inverse problem f m x n sing. vals. 0 : ill-posed (many x equally valid) Incomplete info on x probability density function y model: p(x,y) x Bayesian estimation of optical properties of the human head via 3D structural MRI p.6

14 Bayesian inverse problem f m x n sing. vals. 0 : ill-posed (many x equally valid) Incomplete info on x probability density function y noise σ(f) f(x) x Bayesian estimation of optical properties of the human head via 3D structural MRI p.6

15 Bayesian inverse problem f m x n sing. vals. 0 : ill-posed (many x equally valid) Incomplete info on x probability density function y noise σ(f) f(x) prior p( x) x Bayesian estimation of optical properties of the human head via 3D structural MRI p.6

16 Bayesian inverse problem f m x n sing. vals. 0 : ill-posed (many x equally valid) Incomplete info on x probability density function y measured y noise σ(f) f(x) prior p( x) x Bayesian estimation of optical properties of the human head via 3D structural MRI p.6

17 Bayesian inverse problem f m x n sing. vals. 0 : ill-posed (many x equally valid) Incomplete info on x probability density function y measured y noise σ(f) f(x) prior p( x) posterior p( x y) x Bayesian estimation of optical properties of the human head via 3D structural MRI p.6

18 Bayesian inverse problem f m x n sing. vals. 0 : ill-posed (many x equally valid) Incomplete info on x probability density function y measured y prior p( x) noise σ(f) posterior p( x y) f(x) x Bayesian inference p(x y) p(x, y) = p(y x) p(x) posterior likelihood prior Bayesian estimation of optical properties of the human head via 3D structural MRI p.6

19 Bayesian inverse problem f m x n sing. vals. 0 : ill-posed (many x equally valid) Incomplete info on x probability density function y measured y prior p( x) noise σ(f) posterior p( x y) f(x) x Bayesian inference p(x y) p(x, y) = p(y x) p(x) posterior likelihood prior assumptions about noise width of likelihood Embraces ill-posedness, statistically rigorous. Need to explore N-dim posterior: many f(x) evals required (> 0 2 ). Bayesian estimation of optical properties of the human head via 3D structural MRI p.6

20 Why full PDF? Common Bayesian method for DOT inversion: find single best-fit x = x MAP. (MAP = maximum a-posteriori) Bayesian estimation of optical properties of the human head via 3D structural MRI p.7

21 Why full PDF? Common Bayesian method for DOT inversion: find single best-fit x = x MAP. (MAP = maximum a-posteriori) However x MAP can be moved by reparametrization! Bayesian estimation of optical properties of the human head via 3D structural MRI p.7

22 Why full PDF? Common Bayesian method for DOT inversion: find single best-fit x = x MAP. (MAP = maximum a-posteriori) However x MAP can be moved by reparametrization!.5 p(µ s ) p(κ) µ s (/mm) Example: broad PDF on µ s Normal, σ = 30% of mean. 6% shift in MAP between p(µ s) and p(µ s ) κ (mm) Bayesian estimation of optical properties of the human head via 3D structural MRI p.7

23 Why full PDF? Common Bayesian method for DOT inversion: find single best-fit x = x MAP. (MAP = maximum a-posteriori) However x MAP can be moved by reparametrization!.5 p(µ s ) p(κ) µ s (/mm) Example: broad PDF on µ s Normal, σ = 30% of mean. 6% shift in MAP between p(µ s) and p(µ s ) κ (mm) CPUs advance faster than DOT instrumentation best to make maximal use of data Statistical answers multimodal imaging. Bayesian estimation of optical properties of the human head via 3D structural MRI p.7

24 Realistic new noise model Each signal component f m (x) independent noise. Photons Poissonian: gaussian approx σ(f) = f /2 E.g. 0 6 photons = 0.% relative error But: we do not trust forward model to 0.%! Bayesian estimation of optical properties of the human head via 3D structural MRI p.8

25 Realistic new noise model Each signal component f m (x) independent noise. Photons Poissonian: gaussian approx σ(f) = f /2 E.g. 0 6 photons = 0.% relative error But: we do not trust forward model to 0.%! σ( f ) noise standard deviation vs signal (photon units) log log εf forward model limited f /2 detection limited (shot noise) /ε 2 f ε = relative forward model error e.g. 5% (errors: physics, numerical... ) Bayesian estimation of optical properties of the human head via 3D structural MRI p.8

26 Result: marginal posterior PDF Applied Optics, special issue on biomedical optics, June µ s a) true b) c) Gaussian PDF approx via Levenburg-Marquardt, ellipse encloses 65% prob µ s µ a µ a SCALP SKULL BRAIN µ a Bayesian estimation of optical properties of the human head via 3D structural MRI p.9

27 Result: marginal posterior PDF Applied Optics, special issue on biomedical optics, June µ s a) true b) c) Gaussian PDF approx via Levenburg-Marquardt, ellipse encloses 65% prob µ s µ a µ a SCALP SKULL BRAIN µ a Sampling exact PDF Markov chain Monte Carlo 40 f(x) evals per sample (CPU intensive) Bayesian estimation of optical properties of the human head via 3D structural MRI p.9

28 Result: marginal posterior PDF Applied Optics, special issue on biomedical optics, June µ s a) true b) c) Gaussian PDF approx via Levenburg-Marquardt, ellipse encloses 65% prob µ s µ a µ a SCALP SKULL BRAIN µ a Sampling exact PDF Markov chain Monte Carlo 40 f(x) evals per sample (CPU intensive) Pancake-like PDF: major-to-minor axis ratio 0 2 Bayesian estimation of optical properties of the human head via 3D structural MRI p.9

29 Results: confidence intervals 2 68% confidence interval (.0 σ) µ mua mua2 mua3 musp musp2 musp3 true MCMC sam s MCMC conf brain gauss conf posterior marginal distributions (precon units) n µ s,brain: Gaussian approx bad, need MCMC sampling Bayesian estimation of optical properties of the human head via 3D structural MRI p.0

30 Tracking brain parameters measured tracks true optical params, little crosstalk µ a,brain (mm ) 0.03 a) µ s,brain (mm ).3 b) (0) (mm ) µ a,brain (0) (mm ) µ a,brain µ a,brain (mm ) c) µ s,brain (mm ) d) (0) (mm ) µ s,brain (0) (mm ) µ s,brain Bayesian estimation of optical properties of the human head via 3D structural MRI p.

31 Tracking brain parameters measured tracks true optical params, little crosstalk µ a,brain (mm ) 0.03 a) µ s,brain (mm ).3 b) (0) (mm ) µ a,brain (0) (mm ) µ a,brain µ a,brain (mm ) c) µ s,brain (mm ) d) (0) (mm ) µ s,brain (0) (mm ) µ s,brain det photons: errorbars 3% µ a,brain, 0% µ s,brain for ε = 3%, flat prior Bayesian estimation of optical properties of the human head via 3D structural MRI p.

32 How many photons needed? ε a) 30 b) % Error in N p % Error in µ a,brain µ s,brain Baseline brain errorbars vs... N p = total det photons ɛ = fwd model accuracy ε Can optimize design of DOT apparatus N p Bayesian estimation of optical properties of the human head via 3D structural MRI p.2

33 Partial geometric info multilayer slab diffusion forward model: /µ s S D D2 D3 Crank-Nicholson z novel logarithmic timesteps <% error Fast! typ. 0.2 sec CPU Bayesian estimation of optical properties of the human head via 3D structural MRI p.3

34 Partial geometric info multilayer slab diffusion forward model: /µ s S D D2 D3 Crank-Nicholson z novel logarithmic timesteps <% error Fast! typ. 0.2 sec CPU Preliminary results... slab fits MRI head signals well (if no CSF) Bayesian estimation of optical properties of the human head via 3D structural MRI p.3

35 Partial geometric info multilayer slab diffusion forward model: /µ s S D D2 D3 Crank-Nicholson z novel logarithmic timesteps <% error Fast! typ. 0.2 sec CPU Preliminary results... slab fits MRI head signals well (if no CSF) can also fit for unknown scalp+skull thickness but µ a,brain is 3 more accurate if MRI geom used Bayesian estimation of optical properties of the human head via 3D structural MRI p.3

36 Partial geometric info multilayer slab diffusion forward model: /µ s S D D2 D3 Crank-Nicholson z novel logarithmic timesteps <% error Fast! typ. 0.2 sec CPU Preliminary results... slab fits MRI head signals well (if no CSF) can also fit for unknown scalp+skull thickness but µ a,brain is 3 more accurate if MRI geom used Optode amplitude & time-offset calibrations work (Bayes marginalize over the nuisance params) Bayesian estimation of optical properties of the human head via 3D structural MRI p.3

37 Conclusions Optical tissue model from structural MRI can infer baseline tissue parameters errors < 0% µ a,brain, µ s,brain, no crosstalk noise model: shot noise + fwd model error Bayesian estimation of optical properties of the human head via 3D structural MRI p.4

38 Conclusions Optical tissue model from structural MRI can infer baseline tissue parameters errors < 0% µ a,brain, µ s,brain, no crosstalk noise model: shot noise + fwd model error Forward models in complex geometry need to be fast will call many times known errors include in noise model Bayesian estimation of optical properties of the human head via 3D structural MRI p.4

39 Conclusions Optical tissue model from structural MRI can infer baseline tissue parameters errors < 0% µ a,brain, µ s,brain, no crosstalk noise model: shot noise + fwd model error Forward models in complex geometry need to be fast will call many times known errors include in noise model Bayes: understand full PDF on unknowns predict all errorbars, correlations CPU intensive but optimal use of data handle calibration (nuisance) params Bayesian estimation of optical properties of the human head via 3D structural MRI p.4