Multi-Scale Geometric Flow forSegmenting Vasculature in MRI:

Theory and Validation

Maxime Descoteaux, Louis Collins, and Kaleem Siddiqi

Centre for Intelligent Machines

Montreal Neurological Institute

McGill University, Montréal, Canada

Plan of the talk

• Problem and motivation• Previous work• Theory

– Extracting local shape information– Vesselness measure– Flux maximazing flow– Construction of the vector field– Multi-scale geometric flow

Plan of the talk

• Experiment– Qualitative results and examples– Vessel extraction details– MRA and MRI image acquisition on the same

subject– Quantitative validation

• Discussions• Conclusions• Selected References

Blood Vessel Segmentation

Input: 3D data sets (MRA, PD, GadoliniumMRI,…)

Output: Binary volume containing vascular tree

Useful?• Automatic (no manual slice by slice manipulation)• Image-guided neurosurgery• Pre-surgical planning• Clinical analysis

What’s been done in the past

• Bullit – Aylward group• Frangi et. al.• Krissian et. al.• Lorigo – Westin et. al.• Vasilevskiy – Siddiqi

Work onangiographicdata sets ONLY(MRA and CTA)

Problem

• These methods cannot be used on standard MRIvolumes such as PD

• Why?– Initialization based on thresholding original volume– No explicit tubular structure models– Not multi-scale– Based on the assumption that the gradient of the

original image is strong ONLY at vessel boundaries• FAILS for PD weighted volumes

MRA vs PD

Phase-Contrast(PC)

Time-Of-Flight(TOF)

Proton Density(PD)

Easier problem on MRA than PD

Consequence: Cannot base the segmentationonly on image intensity contrast

Needed: tubular shape model to constrain thesegmentation

Harder: Many dark->bright structures in PD otherthan vessels (white and gray matter)

Easier: Clear bright->dark contrast change inTOF and PC data sets

Goal of our work

Propose a segmentation algorithm for standard MRIvolumes that overcomes limitations of existingmethods applied to MRA

If successful,• Vascular model available for surgical planning and

clinical analysis• Eliminates the need for an additional scan• Save time and burden on patient

Extracting local shape information• We want to know the local behaviour of a 2D/3D image I• It is common to consider the Taylor expansion of an image

I in the neighborhood Δx of a pixel (2D) or a voxel (3D)x0

where

and

Gradient operator

• Very very well-known and used everywhere• Its magnitude gives an edge detector• Its direction gives a normal to an implicitly

defined isosurface

Hessian operator

• Usefulness is less known!• 2nd order derivatives encode the shape information

i.e. How the normal of an isosurface changes

Eigen analysis of the Hessian

Finding e-values and e-vectors of the Hessian matrixis related to finding the direction with extremevalues of 2nd derivativesi.e. directions where there is extreme change in the

normal to the isosurface

e-vector corresponding to the smallest e-valueis the direction of the blood vessel locally

Eigen analysis of the Hessian

Eigen value classification

Key observations at locations centered withintubular structures:

1) smallest e-value (λ1) is close to zero

(low curvature along vessel)

2) other two e-values (λ2, λ3) are are high andvery close

(high curvature of the circular cross-section)

Eigen value classification

Frangi’s e-value ratios

• Three quantities are defined to differentiate bloodvessels from others:

Frangi’s local structurediscrimination

max for blob-like max for vessel-like max for *-likezero for vessel-like zero for sheet-like zero for noise

Multi-scale vesselness measure

max for blob-like max for vessel-like max for *-likezero for vessel-like zero for sheet-like zero for noise

Multi-scale vesselness measure

• Designed to maximum along centerlines oftubular structures and close to zero outsidevessel-like regions

• Natural introduction of scale, σ. We computeentries of the Hessian matrix using derivatives ofLindeberg’s γ-parametrized normalized Gaussiankernels.

After vesselness computation

We have:

• The local radius of the blood vessel whichcorresponds to the scale at which the maximumvesselness measure was computed

• The local orientation of the blood vessel givenby the e-vector corresponding to the smalleste-value

Some Examples

Vesselness map on PD

Flux maximizing flowIdea:The inward flux of a vector field V through a closed curve (orsurface in 3D) S, given by an equation of the form

provides a measure of how well the normal vectors of thesurface are aligned with the vector field. A(t) is the surfacearea of the evolving surface.

Flux maximizing flow (initial state)

Flux maximizing flow (final state)

Flux maximizing flow• It is used to direct the evolution of a curve so that the

normals of the curve are aligned with a given vector field.

Theorem (Vasilevskiy, Siddiqi PAMI 2002):if a closed surface S in 3D flows according to

then the inward flux through the boundary of the fixedvector field V increases as fast as possible. Here N is thenormal vector field of S.

Choosing the vector field V

If we choose V to be1) Static (does not depend on evolving surface)2) Maximum along the surface of vessel boundaries

and zero outside vessel regions

Then we can use the flux maximizing flow to evolveseed points placed inside vessels.

• Seeds will evolve and cling to vessel boundaries

Construction of vector field

• Natural choice: combination of vesselnessmeasure and gradient vector field

• Problem: vesselness measure is maximum oncenterline of vessels.

Extending vesselness measure tovessel boundaries

We have local orientation and radius at every voxel

Distribute the vesselness to vessel boundaries => ϕ

New vector field

1) The magnitude of ϕ is maximum at vesselboundaries and the ellipsoidal extensionperforms a type of local integration

2) The normalized gradient vector field of theoriginal image captures the direction informationwhich is expected to be high at boundaries ofvessels as well as orthogonal to them

The multi-scale geometric flow

• The new vector field

• Expending the flux maximizing flow expression

Properties of the flow

1) First term acts like a doublet term

2) Second term behave like a regularization term

3) No leaking because vector field is zero outsidevessel regions

Implementation details

• Level set methods– Adaptive to topological change– Efficient, if original volume not too BIG!

• Use at most 3 MINC volumes at all time– MINC voxel loop is of great help!

• Obtain convergence in 5000 iterations in mostcases. Depends on initialization

Some qualitative results

1) MRA (Anders Lassen)

2) Cropped portion of gadolinium enhanced MRI(Ingerid’s pre-operation scans)

3) Full PD weighted data set (ICBM subject 100)

4) Movie of segmentation on PD weighted volume

MIP of MRA

Initial seeds (t =0)

t = 55

t = 100

t = 200

t = 2000

MIPs of gadolinium MRI

Thresholding

Our segmentations

MIP of vesselness measure of thePD weighted MRI

Blood vessel extractions

Geometric flow in action

Validation Experiment

• Image Acquisition of angiographic and non-angiographic on the same patient => ME!

• MRA• 3D axial Time-Of-Flight angiogram

(0.43mm x 0.43mm x 1.2mm)

• 3D axial Phase-Contrast angiogram(0.47mm x 0.47mm x 1.5mm)

• PD/T2-weighted dual turbo spin-echo (sagittal)(1 mm3)

Image acquired

Phase-Contrast(PC)

Time-Of-Flight(TOF)

Proton Density(PD)

Vessel extraction

• Registering and resampling all volumes in thesame space (0.5mm3 resolution)

volumes are HUGE!

• Crop common 259 x 217 x 170 region

• Run multi-scale geometric segmentation algorithm

Quantitative Validation

Alignment error: how well does the test dataexplains the ground truth data

• Take Euclidean DT of test data. Every voxelcontains the closest distance to vessel structure

• For every vessel label in ground truth, add up thetest data distances and compute the average

Results: average alignment error is 1mm!!!(PD is the test data and angiograms ground truth)

Kappa measure

Kappa coefficient: degree to which the agreementexceeds chance levels: 2a / (2a + b + c)

a = number of voxels agreeing in both volumei.e within 2 voxels each other (red labels)

b = number of vessel voxels only in ground truth(green labels)

c = number of vessel voxels only in test data(blue labels)

Results: kappa ~= 75%

Ratio

a / (a + b)measures the degree to which the ground truth datais accounted by the test data.

Results:1) PD data accounts ~90% of the vasculature

segmented from either angiograms

2) ~30% of the vasculature obtained from PDdata are recovered from either angiograms

Table of results

Expected results???

• Resolution is higher in angiographic volumes• Should pick up smaller vessels

• MRA is designed to image blood vessels• Are the angiographic acquisition optimal?

• Visual examination of the PD data in fact showsstronger evidence for vessel of small size

Phase-Contrast(PC)

Time-Of-Flight(TOF)

Proton Density(PD)

Conclusions

• Our results suggest that the algorithm can beused to improve upon the results obtained fromangiographic data

• Promising alternative when MRA is not available.The PD-weighted MRI data are usuallly availablewhen planning brain tumor surgery

What’s next

• Would be nice to go back in the scan andcompare GADO vs MRA vs PD

• Use this for other medical imaging problems– Ingerid’s vessel driven correction of brain shift– Other tubular structure detection… bone?– Suggestions?

Selected references1) A. Vasilevskiy, K. Siddiqi. Flux maximizing geometric flows.

IEEE Transactions PAMI, vol. 24, 2002.

2) A. Frangi, W. Niessen, K.L. Vincken, M.A. Viergever. Multi-scale vessel enhancement filtering. Proc. MICCAI'98,pp.130-137, 1998.

3) K. Krissian, G. Malandain, N. Ayache. Model-baseddetection of tubular structures in 3D images. ComputerVision and Image Understanding, 80:2, pp. 130-171, Nov. 2000

THANK YOU!

ϕ-distribution• We distribute vesselness measure at (x,y,z) to

every (x_e, y_e, z_e) on ellipsoid surface byscaling it with the projection on the cross-sectional plane.

• We define the superposition of the extensionscarried out independently at all voxels to be thescalar function ϕ.