Robust reconstruc,on of flow in MRI C. Bertoglio, H. Carrillo, J. - - PowerPoint PPT Presentation

Robust reconstruc,on of flow in MRI

• C. Bertoglio, H. Carrillo, J. Garay,
• D. Nolte, A. Osses, S.Uribe

Bernoulli Institute – University of Groningen www.math.rug.nl/~bertoglio

Inverse hemodynamics

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2D Velocity-encoded MRI 3D Geometry 3D flow This talk!

Magne,c resonace imaging (MRI) MRI scanner

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Modeling of the MRI signal Current through coils generate magnetic fields

B0

B1(t)

Static field Radiofrequency field Magnetic resonance

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∂ϕ ∂t + (u · r)ϕ = γBz(x, t)

Velocity-encoded MRI

• Magne,za,on vector:

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Phase

ϕ(t)

m = [mx, my, mz]

my

mx

• Bloch equa,on in Eulerian form:

with

Sta,c gradient Velocity Velocity-encoding gradient

Bz(x, t) = B0 + G(t) · x

Standard velocity es,ma,on in MRI

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• Measure phases for gradients G1 6= G2
• Model of phase:
• Phase-contrast velocity:

Reference phase Flow velocity Both unknown!

User-selected velocity

ˆ ϕG1, ˆ ϕG2

Encoding gradient

= ˆ ϕG1 − ˆ ϕG2 π venc1,2

ϕG = ϕ0 + uM1(G)γ, M1(G) = Z TE G(t)t dt

ˆ u = ˆ ϕG1 − ˆ ϕG2 M1(G1)γ − M1(G2)γ

venc1,2 := π M1(G1)γ − M1(G2)γ

The problems of phase-contrast MRI Example with volunteer data (ascending aorta)

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1st Big problem: noise in grows with venc

< u venc = 150

Average of 3 scan repe,,ons

venc = 150

venc = 100

V NR ∝ SNR venc

The problems of phase-contrast MRI 2nd Big problem: aliasing when Example with volunteer data (ascending aorta)

venc = 150

venc = 100

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venc1,2 < u

venc = 75

Phase aliasing

• If then

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venc1,2 > u

ˆ ϕG1 − ˆ ϕG2 < π

k ∈ Z

ˆ uk = ˆ ϕG1 − ˆ ϕG2+2πk π venc1,2 ˆ uk=0 is the correct solu,on!

venc1,2 < u

• If

ˆ ϕG1 − ˆ ϕG2 > π

We need more informa8on for finding k

Standard dual-venc reconstruc,on Goal: Improve velocity/noise with addi,onal measurement Dual-venc

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(0, G1) (0, G2) (0, G1, G2)

venc = 150

venc = 100

High venc Low venc S,ll needed!

venc > u

Schnell et al 2017

The key: angle es,ma,on

• Usual computa,on of angle from vector components

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Phase

mx

ϕ

my

mx = m cos ϕ my = m sin ϕ

tan−1 my mx = ϕ

• Least-squares computa,on of angle

J(ϕ) = (mx − m cos ϕ)2 + (my − m sin ϕ)2 dJ dϕ = 0

tan−1 my mx = ϕ

⇔

Velocity es,ma,on from least squares

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JG(u) = ⇣ ˆ mG

x − ˆ

mG cos(ϕG(u)) ⌘2 + ⇣ ˆ mG

y − ˆ

mG sin(ϕG(u)) ⌘2

• Least squares for velocity es,ma,on:

−2 −1 1 2 0.5 1 1.5 2 2.5 3 3.5 4 Velocity Functionals utrue venc=0.9 venc=0.6

2venc

ϕG(u) = ˆ ϕ0 + πu/venc

Velocity es,ma,on from least squares ˆ u = argmin JG(u)

Phase-contrast velocity

ˆ uk = ˆ ϕG − ˆ ϕ0 π venc + k · 2 venc , k ∈ Z

Period of

JG(u)

• Least squares for velocity es,ma,on

−2 −1 1 2 0.5 1 1.5 2 2.5 3 3.5 4 Velocity Functionals utrue venc=0.9 venc=0.6

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Mul,ple encoding gradients

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• Now suppose we have 3 measurements with:

−3 −2 −1 1 2 1 2 3 4 5 6 7 8 Velocity Functionals utrue venc=0.6 venc=0.45 Sum

G = 0, G1 6= G2

JΣ(u) = JG1(u) + JG2(u)

• Sum func8onal from 3 measurements:

Global minima of smallest magnitude! Dual-venc es8mate Larger periodicity

Take-home message Aliasing-free es,ma,on when

venc = 100

Dual-venc

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venc < u

(0, G1) (0, G2) (0, G1, G2)

venc = 75

Automa8cally chosen by the MRI scanner! Carrillo, Bertoglio++ (to appear)

Thanks for your a`en,on!

www.math.rug.nl/~bertoglio