1/18

Measuring T1 and T2 in the Presence of B0 Inhomogeneities

Vladimir A. Kobzar1

joint work with Carlos Fernandez-Granda12 and Jakob Asslander34

1NYU Center for Data Science

2Courant Institute of Mathematical Sciences

3Center for Biomedical Imaging, Dept. of Radiology, NYU School of Medicine

4Center for Advanced Imaging Innovation and Research, NYU School of Medicine

May 16, 2019

2/18

Declaration of Financial Interests or RelationshipsSpeaker Name: Vladimir KobzarI have no financial interests or relationships to disclose with regard to the subjectmatter of this presentation.

3/18

Banding artifacts in bSSFP

Con

tras

ts

Standard shim(∆B0 ≈ 0) Corrupted shim

Du

alG

rad

ien

tE

cho

−1000

0

300

γ 2∆B0

(Hz)

B0 map without ”wrapping”

4/18

Banding artifacts in bSSFP

Con

tras

ts

Standard shim(∆B0 ≈ 0) Corrupted shim

Du

alG

rad

ien

tE

cho

−1

0

1

(γ2

∆B0TR

mo

d2π

)/π

B0 map with ”wrapping”

5/18

bSSFP Magnetization Trajectory

0

0.05

0.1

0.15

|M⊥|

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−1

0

1

∆ωzTR/π

ϕ/π

Transverse magnetization is smooth and nonzero everywhere except near the ”stoppingbands”.

6/18

Summary

Since balanced hybrid-state free precession (bHSFP) sequences have the same issuewith banding artifacts, we address it as follows:

I Incorporate B0 inhomogeneities bias into a measure of T1,2-encoding quality ofthe signal (Cramer-Rao Bound or CRB);

I Vary flip angles and phase offsets of the driving field (sweeping); and

I Optimize the CRB over the set of possible flip angles.

7/18

Cramer-Rao Bound (CRB)

A simplified illustrative model of the signal: S = r(Ti ,B0) + N where

I ri is magnetization during the ith read-out period;

I r ∈ RNpulse where Npulse is the number of pulses;

I N ∼ N(0, σI ).

If B0 is known,

I the CRB bounds the mean squared error (MSE) of any unbiased estimator t(S) ofTi :

MSE (t(S)) ≥ σ2

‖∂Tir‖2

I Thus, the MSE of an efficient estimator isI proportional to the noise, andI inversely proportional to the sensitivity of the magnerization to Ti .

8/18

Cramer-Rao Bound (CRB)R

Npulse ∂Tir

θ

∂B0r

If B0 is not known,

I the CRB provides that

MSE (t(S)) ≥ σ2

(1− cos2 θ)‖∂Tir‖2

where cos θ is the correlation between the derivatives ∂Tir and ∂B0r .

I Thus, if the magnetization trajectory r can be designed to reduce the correlation,this CRB will be close to the CRB where B0 is known.

9/18

Optimal bHSFP sequenceI Asslander et al. (2018): r(α, φnom,B0,Ti ) controlled by α, φnom.I Thus, we can control ∂Ti

r and ∂B0r , and therefore the CRB.

(a) Optimized for ∆B0 = 0

x

zφnom = .2π

x

z

φnom = .2π

x

z

φnom = π

x

z

φnom = π

x

z

(b) Optimized for unknown ∆B0

0

0.2

0.4

0.6

0.8

1

|r|

0

0.2

0.4

0.6

0.8

1

|r|

0

0.2

0.4

0 3.8

t(s)

·/π

ϑ

34.42 38.25

ϑ α

0

0.2

0.4

179.77 182.4

·/π

t (s)

10/18

Optimal bHSFP sequence duration

I The optimal duration TC = 3.8s of a pulse sequence is comparable to that for thecase where B0 variations are known.

I In each period of length TC , a different phase offset φnom is introduced, withφnom uniformly distributed between −π and π.

I See, generally, Benkert (2015) and Scherbakova (2018) (phase-cycling in bSSFP).

11/18

rCRB as a function of B0 and B1

∆B0 need not be determined separately (B1 is assumed to be determined separately)

γ∆B0TR/π

Optimized for ∆B0 = 0rCRB for unknown ∆B0

Optimized for ∆B0 = 0rCRB for known ∆B0

Without φnom sweeping

Optimized for ∆B0 = 0CRB for unknown ∆B0

Optimized for unknown ∆B0

CRB for unknown ∆B0

102

103

104

rCRB

(T1)

With φnom sweeping

0.8 1 1.2−2

−1

0

1

2

B1/Bnom1

γ∆B0TR/π

0.8 1 1.2

B1/Bnom1

0.8 1 1.2

B1/Bnom1

0.8 1 1.2

B1/Bnom1

102

103

104

rCRB

(T2)

12/18

In vivo experiments

I Flip angles αnom are determined by numerical optimization discussed above;

I Acquire the data in 20 cycles, each 3.8 s long;

I Phase offset of the driving field φnom varied over (”sweeps”) [−π, π] from cycle tocycle; and

I Other parameters are the same as in Asslander et al. (2018), id.

13/18

MRF with standard shim

No

swee

pin

g

Du

alG

rad

ien

tE

cho

−1 0 1

(γ2∆B0TR mod 2π)/πS

wee

pin

g

.1 .5 3

T1 (s)

.05 .3 .5

T2 (s)

−1 0 1

(γ2∆B0TR mod 2π)/π

14/18

MRF with corrupted shim (sweeping)

No

swee

pin

g

Du

alG

rad

ien

tE

cho

−1 0 1

(γ2∆B0TR mod 2π)/πS

wee

pin

g

.1 .5 3

T1 (s)

.05 .3 .5

T2 (s)

−1 0 1

(γ2∆B0TR mod 2π)/π

T1 artifacts arise due to the inversion pulse fading at extreme off-resonance frequencies

15/18

MRF with corrupted shim (sweeping)

No

swee

pin

g

Du

alG

rad

ien

tE

cho

−1000 0 300γ2∆B0 (Hz)

Sw

eep

ing

.1 .5 3

T1 (s)

.05 .3 .5

T2 (s)

−1 0 1

(γ2∆B0TR mod 2π)/π

T1 artifacts arise due to the inversion pulse fading at extreme off-resonance frequencies

16/18

MRF with corrupted shim (B0 corrected externally)

B0

corr

ecte

dex

tern

ally

Du

alG

rad

ien

tE

cho

−1 0 1

(γ2∆B0TR mod 2π)/πS

wee

pin

g

.1 .5 3

T1 (s)

.05 .3 .5

T2 (s)

−1 0 1

(γ2∆B0TR mod 2π)/π

T1 artifacts arise due to the inversion pulse fading at extreme off-resonance frequencies

17/18

Future work

I More robust inversion or omit inversion

I 3D sequence for efficient sampling

18/18

Questions: vkobzar@cims.nyu.eduAcknowledgements

I V.A.K. is supported by the Moore-Sloan Data Science Environment at NYU

I C.F. is supported by NSF award DMS-1616340.

I J.A. is supported by the research grants NIH/NIBIB R21 EB020096 andNIH/NIAMS R01 AR070297, and this work was performed under the rubric of theCenter for Advanced Imaging Innovation and Research (CAI2R, www.cai2r.net), aNIBIB Biomedical Technology Resource Center (NIH P41 EB017183).