Bayesian estimation of acoustic aberrations in highintensity focused ultrasound treatment

Bamdad Hosseini 1, Charles Mougenot 2, Samuel Pichardo 3 , ElodieConstanciel 4, James M. Drake 4, John M. Stockie 1

1Simon Fraser University

2Philips Healthcare

3Thunder Bay Regional Research Institute

4Hospital for Sick Children

June 28, 2016

www.sfu.ca/~bhossein

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What is HIFU?

High intensity focused ultrasound (HIFU).

A focused beam of acoustic waves converging in a small volume.

The generated heat ablates diseased tissue.

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What are the challenges?

Clinical success in treatment of prostate cancer, liver tumours, uterinefibroids, etc.

Treatment of brain tumours remains a challenge.

Strong aberrations due to skull bone → defocused beam.

Estimate aberrations → compensate phase shift → refocus.

Phas

e sh

ift

(deg

)

−20

0

20

Att

enuat

ion (

%)

90

100

110

−40

−20

0

20

60

80

100

120

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An approximate forward model

The pressure field due to a single piezoelectric element:

p(x) = p0 exp

(i

[ωt+

ω

c0|x|

])× µ µ = ζ exp(iφt).

p0 signal amplitude.

ω frequency.

c0 speed of sound.

µ aberration due to tissue

ζ is attenuation

φ phase shift

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Energy based measurements

Assemble in to a linearforward map:

p = FZa

F free field matrix(Green’s function).

Z input phase andamplitude (experimentaldesign).

a aberration due totissue.

Measure the amplitudeof pressure

d = diag(p)p∗.

Phas

e des

ign (

deg

)

0

10

20

30

Am

pli

tude

des

ign (

%)

0

50

100

Dat

a

0

2

4

0

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30

0

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0

2

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0

2

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0

2

4

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The inverse problem

Estimate the vector of aberrations a given the data d and matrices Fand Z.

Use Bayes’ rule1:π(a|d) ∝ π(d|a)π0(a).

π(d|a) likelihood.

π0(a) prior.

π(a|d) posterior.

1J. Kaipio and E. Somersalo. Statistical and Computational Inverse Problems.Springer Science and Business Media, New York, 2005.

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The likelihood

π(a|d) ∝ π(d|a)π0(a).

Additive noise model

G(a) := diag(FZa)(FZa)∗ d = G(a) + εεε, εεε ∼ N (0,ΣΣΣ).

Likelihood:

π(d|a) ∝ exp

(−1

2

∥∥∥ΣΣΣ−1/2(d− G(a))∥∥∥22

)

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Choosing the prior π0

π(a|d) ∝ π(d|a)π0(a).

Ph

ase

shif

t (d

eg)

−20

0

20

Att

enu

atio

n (

%)

90

100

110

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1

x

y

γ =

1

0.2

0.4

−0.4

−0.2

0.2

0.4

−0.6

−0.4

−0.2

γ =

0.1

−0.5

0

0.5

1

1.5

−1.5

−1

−0.5

0

0

0.5

1

1.5

−1.5

−1

−0.5

γ =

0.0

1

−4

−2

0

2

−4

−2

0

2

4

−4

−2

0

2

4

−4

−2

0

2

4

Measured aberrations hint at an underlying continuous field.

Construct π0(a) by pointwise evaluation of a Gaussian random field:

N (0, (I − γ∆)−2)

(I − γ∆)−2 biharmonic operator with Neumann boundary condition.

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Sampling from π0

Pick σ > 0.

Discretize (I −∆)−2.

Sampleu ∼ N (0, (I − γ∆)−2), α1 ∼ N(0, σ)

v ∼ N (0, (I − γ∆)−2), α2 ∼ N(0, σ)

Set a = diag(α21u) exp(iα2

2v).

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Sampling from the posterior π(a|d)

Forward problem is nonlinear → posterior π(a|d) is not Gaussian.

Markov Chain Monte Carlo.

Requires sampling from the prior π0 and evaluating likelihood π(d|a).

π(d|a) ∝ exp

(−1

2

∥∥∥ΣΣΣ−1/2(d− G(a))∥∥∥22

).

Differentiable in real arguments, not differentiable in the complexarguments.

MALA + Random walk block sampler2 (at every step):

(1) Fix v and α2 and sample u and α1 using Metropolis adjusted Langevinalgorithm (uses gradient information).

(2) Fix u and α1 and sample v and α2 using preconditionedCrank-Nicolson random walk algorithm.

2S. L. Cotter et al. “MCMC methods for functions: modifying old algorithms tomake them faster”. In: Statistical Science 28.3 (2013), pp. 424–446.

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Synthetic experiment

Philips Sonalleve device with 256 piezoelectric elements.512 unknowns.Discretize the continuous field on a 8× 8 mesh (64 dimensionalparameter space).16 sonication tests.19× 19 voxel MRI window for each sonication test.SNR = 5.3× 105 burn-in + 5× 105 steps of MCMC.

Phas

e des

ign (

deg

)

0

10

20

30

Am

pli

tude

des

ign (

%)

0

50

100

Dat

a

0

2

4

0

10

20

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100

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0

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Synthetic experiment

Phase Target (deg)

−60

−40

−20

0

Attenuation Target (%)

70

80

90

100

Phase PM (deg)

−60

−40

−20

0

Attenuation PM (%)

60

80

100

Phase error (deg)

5

10

15

20

Attenuation error (%)

10

20

30

40

Phase std (deg)

2

4

6

8

10

Attenuation std (%)

6

8

10

12

14

3 4 5 6 7 8

x 105

8.63

8.632

8.634

8.636

8.638

iteration

Φ Φ := −12

∥∥ΣΣΣ−1/2(d− G(a))∥∥22

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Synthetic experiment

Phase Target (deg)

−60

−40

−20

0

Attenuation Target (%)

70

80

90

100

Phase PM (deg)

−60

−40

−20

0

Attenuation PM (%)

60

80

100

Phase error (deg)

5

10

15

20

Attenuation error (%)

10

20

30

40

Phase std (deg)

2

4

6

8

10

Attenuation std (%)

6

8

10

12

14

0 5 10 15 20−0.5

−0.45

−0.4

−0.35

−0.3

−0.25

α1

PM = −0.33542std = 0.023736

0 5 10 15 200.55

0.6

0.65

0.7

0.75

α2

Density

PM = 0.62485std = 0.022769

PM

pre

dic

tio

n

Dat

a

2

4

6

0

2

4

6

0.5

1

1.5

0

0.5

1

1.5

0.2

0.4

0.6

0.8

1

1.2

1.4

0

0.5

1

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Synthetic experiment

Phase Target (deg)

−60

−40

−20

0

Attenuation Target (%)

70

80

90

100

Phase PM (deg)

−60

−40

−20

0

Attenuation PM (%)

60

80

100

Phase error (deg)

5

10

15

20

Attenuation error (%)

10

20

30

40

Phase std (deg)

2

4

6

8

10

Attenuation std (%)

6

8

10

12

14

Ph

ase

(deg

)

−40

−20

0

Att

enu

atio

n (

%)

80

90

100

−40

−20

0

60

80

100

−40

−20

0

70

80

90

100

−60

−40

−20

0

70

80

90

100

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Physical experiment

The same Philips Sonalleve device.

Artificial aberrator.

Discretize the Gaussian field on a 8× 8 grid.

32 sonication tests.

7× 7 voxel MRI window for each test.

3× 105 burn-in + 5× 105 steps of MCMC.

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Physical experiment

Phase Target (deg)

−60

−40

−20

0

Attenuation Target (%)

70

80

90

100

Phase PM (deg)

−100

−50

0

Attenuation PM (%)

99.992

99.994

99.996

99.998

100

Phase error (deg)

10

20

30

40

Attenuation error (%)

10

20

30

Phase std (deg)

2

4

6

8

10

12

Attenuation std (%)

0.999

0.9995

1

0 200 400 600 800−2

−1

0

1

2x 10

−3

α1 std = 0.0005178

PM = −1.2685e−06

std = 0.0005178

0 2 4 6 80.7

0.8

0.9

1

1.1

α2

Density

PM = 0.89489

std = 0.047994

PM

pre

dic

tio

n

20

40

60

80

Dat

a

0

20

40

60

5

10

15

0

5

10

5

10

15

20

0

10

20

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Physical experiment

Phase Target (deg)

−60

−40

−20

0

Attenuation Target (%)

70

80

90

100

Phase PM (deg)

−100

−50

0

Attenuation PM (%)

99.992

99.994

99.996

99.998

100

Phase error (deg)

10

20

30

40

Attenuation error (%)

10

20

30

Phase std (deg)

2

4

6

8

10

12

Attenuation std (%)

0.999

0.9995

1

Phas

e (d

eg)

−80

−60

−40

−20

Att

enuat

ion (

%)

99.99

99.995

100

−80

−60

−40

−20

99.99

99.995

100

−80

−60

−40

−20

99.99

100

−80

−60

−40

−20

99.9995

100

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Selling points

For the practitioner:

The “state of the art”3 uses 256 sonication tests as compared to 32tests used here.

Patient spends less time in the MRI machine.

Save energy.

flexible design of experiments (choice of matrix Z).

For the mathematician:

Well-posed inverse problem.

Estimate uncertainty.

Compute in 2-4 hours on a laptop.

Experimental design.

3E. Herbert et al. “Energy-based adaptive focusing of waves: application tononinvasive aberration correction of ultrasonic wavefields”. In: IEEE Transactions onUltrasonics, Ferroelectrics and Frequency Control 56.11 (2009), pp. 2388–2399.

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Challenges and future directions

Challenges:

Large discrepancy between the model and physical process

Calibration of F.

Better data.

Regularizing the likelihood.

Better sampling algorithm.

Future direction:

Phase retrieval techniques4.

Matrix completion to estimate the free-field F.

Design of experiments.

4E. J. Candes et al. “Phase retrieval via matrix completion”. In: SIAM Review 57.2(2015), pp. 225–251.

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Acknowledgement

Fields institute and the organizers of Fields-Mprime IndustrialProblem Solving Workshop, August 2014.

NSERC

Brain Canada Multi-Investigator Research Initiative

Focused Ultrasound Foundation

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References

Candes, E. J. et al. “Phase retrieval via matrix completion”. In: SIAM Review57.2 (2015), pp. 225–251.

Cotter, S. L. et al. “MCMC methods for functions: modifying old algorithms tomake them faster”. In: Statistical Science 28.3 (2013), pp. 424–446.

Herbert, E. et al. “Energy-based adaptive focusing of waves: application tononinvasive aberration correction of ultrasonic wavefields”. In: IEEETransactions on Ultrasonics, Ferroelectrics and Frequency Control 56.11(2009), pp. 2388–2399.

Hosseini, B. et al. “A Bayesian approach for energy-based estimation of acousticaberrations in high intensity focused ultrasound treatment”. In: arXiv preprintarXiv:1602.08080 (2016).

Kaipio, J. and E. Somersalo. Statistical and Computational Inverse Problems.Springer Science and Business Media, New York, 2005.

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MCMC performance

3 4 5 6 7 8

x 105

8.63

8.632

8.634

8.636

8.638

iteration

Φ

0 5 10

x 104

0

0.5

1

lag

Φ a

uto

corr

elat

ion

3 4 5 6 7 8

x 105

10.3765

10.377

10.3775

iteration

Φ

0 5 10

x 104

0

0.5

1

lag

Φ a

uto

corr

elat

ion

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