Statistical Aspects of Imaging Cancer with PETyandell/stat/50-year/OSullivan_Finbarr.pdf · Statistical Aspects of Imaging Cancer with PET Finbarr O’Sullivan Department of Statistics

Statistical Aspects of Imaging Cancer with PET

Finbarr O’Sullivan

Department of StatisticsUniversity College Cork

Ireland

50’th CelebrationMadison, WIJune, 2010.

Collaborators: Janet Eary, Ken Krohn, David Mankoff, Mark Muzi, Alex Spence (UW)Jian Huang, Niall Fitzgerald, Eric Wolsztynski (UCC)

Supported by: MI-2007 (SFI), P01-CA-42045 & RO1-CA-65537 (NIH)

Positron Emission Tomography (PET) BASICS

Data ~ Poisson(S+ARλ)λ is the target isotope emission distribution (where the tracer ends up)

R (Radon Transform); A (Attenuation); S (Scatter)

Dose Limited Resolution -> Statistical Aspects are Important (Vardi et al,…Nychka, Wahba…Leahy..)

Imaging Model

CLINICAL PET IMAGINGScanner (PET/CT)

ThyroidBrain

Heart

Bladder

Metabolic State of Cancer?

Normal Glucose (FDG) Pattern Source: Radiological Society of North America

PET Scans used in Cancer Medicine

• Diagnosis/Staging

• Treatment Response

• Recurrence Assessment

Increasing Emphasis on Clinical Validation: PET measurements Patient Outcomes [Survival, Disease Progression,Morbibity ]

18 year PET-FDG study at UW ~ 900 Sarcoma patients (scans and outcome data)

Human Sarcoma

• Class of malignant tumors affecting soft conjonctive tissue, cartilage and bone

• Can arise anywhere in the body, frequently hidden deep in the limbs

• Represents ~1% of adult cancers, more prevalent with children (~15-20%), ~10% of all cancers overall

• 5-year mean survival rate: ~90% (stage 1), ~75% (stage 2), ~54% (Stage 3) [statistics for the USA]

• Soft tissue sarcomas usually appear as a lump or mass, rarely cause pain, swelling, or other symptoms. Often misdiagnosed. Sometimes thought to be sports injuries.

• “Late detection” is not unusual → potentially advanced stage of development

PET-FDGSarcomaStudies

Soft TissueHigh Grade

Bone TissueHigh Grade

Soft TissueHigh Grade

Soft TissueHigh Grade

Heterogeneity Measurement Evaluate Conformity to a Pattern in the Spatial Distribution of the Metabolically Active Elements.

Homogeneous Heterogeneous

CV Spatially Insensitive

Spatially Coherent Spatially Incoherent

0.0 0.2 0.4 0.6 0.8 1.0

010

020

030

040

0

Histogram

CV is 0.71 for Both!

Ellipsoidal Model for Homogeneous Tumor

x1

0.1

0.2

0.3

0.4

0.5

0.60.7

0.8

0.9

1

1

2

Heterogeneity

| , (( ) ' ( ))

(monotone); ( , )

1-

( ) x g g x x

g

H R

θ µ µ

θ µ

λ −− Σ −

= Σ

=

≈

H=0.06g

O’Sullivan, Roy, Eary et al (2003,2005,2009)

HeterogeneityMeasurements

H=0.02 H=0.06

H=0.26 H=0.34

Heterogeneity

and

Patient Outcome

PredictorVariable (X)

Scale %Change in Risk(unitchange in X)

95%C.I.

P-value

AGE(years)

16.8 34 (-12,101) 0.150

SUV(max)(ml/gm)

6.14 -38 (-60,-29) 0.037

Heterogeneity 7.4% 87 (35,160) 0.0002

Roose, Chapman and Maini, SIAM Review, 2008.Cristini, Gatenby, Sutherland, Casciari, Rasey, Krohn 1986...2010

Necrosis

NecroticCenter

Tumor Synthesis (Growth Pattern)

Transverse Sagittal

Coronal

1 2 3 1 2 3

1 2 3

1 2 3

, , ) ( , , )( , , )

Uptake Model

, , )( )

Co-ordinate TransformationsPrincipal Axes : ( Flexible Cyclinder : ( , , )

( ) ( , , ) (h hh

x x x z z zz z z h r

rx x x h r gσ

θ

λ λ θ α τθ

→→

= ≈ +

Sarcoma

Bladder

PhaseInformation

Chemotherapy Response

MODEL:

PRE POST

GLM-Test:ˆ

ˆRESPONSE

β

βσ

=

Pre

) PRE quasi-Poisson( )POST quasi-Poisson(eβ

µµ

::

Co-

Reg

istra

tion

MarginMargin

Post

CorrelationAdjusted!

Dynamic PET Studies: Scans after Tracer Input

Quantitative Data Analysis:Separate Delivery and Retention

0( )( () ) ( )T B P

tPC t V Ct s sC dR t s= −∆ + −∆− ⋅∫

Residue

AIF

•Parametric (compartmental)

•Non-Parametric (non-compartmental)O’Sullivan et al. JASA (2009)

•Directly Sampled

•Image Extracted (Statistically Guided)O’Sullivan et al. IEEE-TMI (2010)

Data

Quantitative Analysis of Dynamic PET Data

BLOOD

FDG

TISSUE

FDG FDG-6-P

0.0

0.2

0.4

0.6

0.8

1.0

0 30 60 90 120Time (minutes)

Cp FLT Cp metArterial Input (AIF)

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0 30 60 90 120

Time (minutes)

Tumor BrainTissue Data

Residue(Impulse Response)

0( )( () ) ( )T B P

tPC t V Ct s sC dRt s= −∆ + −∆− ⋅∫

Flux Flow

VD Extraction

Functionals

Nonparametric Residue Analysis

0

0( ) (( )

1

extraction

)

( ) (Survival Function)

is the residence density for tracer label is flow, is blood volume an

(

d

)

( )t

P Pt

T B

B

C t V t s ds

V

C C

t d

hK

R

hR

t s

γ

γ

τ τ

= −∆ + −∆

= −

− ⋅

∫

∫

Meier and Zierler (1954), Bassingthwaighte (1971), Ostergaard et al. (1996)

Estimation based a cross-validated regularization procedure involving Positivity/Monotonicity and Smoothness Constraints.

1 2

1 1 2 2

1 2

1 1 2 2 3

( ) ( ) ( ) .... ( )

( ) ....

( ) ( ) ( ) .... ( )

p

B p p

C p

M p

h B B B

h e e e

h h h h

λ τλ τ λ τ

τ φ τ φ τ φ τ

τ α α α

τ π τ π τ π τ

−− −

≈ + + +

≈ + + +

≈ + + +

B - splines

C om partm ental

M ixtures

Numerical Approximations for Residence

Mendelsohn and Rice (1984); Cunningham and Jones(1993) , O’Sullivan et al (2009)

Most Widely Used Compartmental Model for PET

BLOOD

FDG

K1

k2

TISSUE

FDG FDG-6-PO4 k4

k3

1 2

1 1 2 2

1 2

1 1 2 2 3

( ) ( ) ( ) .... ( )

( ) ....

( ) ( ) ( ) .... ( )

p

B p p

C p

M p

h B B B

h e e e

h h h h

λ τλ τ λ τ

τ φ τ φ τ φ τ

τ α α α

τ π τ π τ π τ

−− −

≈ + + +

≈ + + +

≈ + + +

B - splines

C om partm ental

M ixturesMay be reasonable in-vitro, but for in-vivo PET ROI data???

Implies a Residence Density of the form:

PET FDG Data from Normal Brain ROIs

Nonparametric and Compartmental Analysis[ A formal statistical test rejects the compartmental model, p-value=0.046 ]

Cerebellum ROI

Nonparametric Residue Analysisvs Parametric Compartment Model

120 TACs:10 Brain Regions and 12 Subjects

(analysis uses a reference distribution constructed by simulation –c.f. Cox, Wahba, Yandell, Wang, Li, Raz etc)

Adaptation for Parametric Mapping

Tissue Concentration Model (voxel x and time t)

Mixture Analysis of Residence Density

Residue Function

Residue Analysis of Segment Time Course Patterns

A

Residence

Residence

Time-Course

Time-Course

B

RMSData

ResidualPCA

RMSResidual

Temporal

Spatial

Diagnostic Assessment:

Voxel-Level Residuals

ml/gm

l/g

ml/g

/min

K VD VP

ml/1

0g/m

in

min

Flux MTT Ext0.0

0.0 0.0 0.0

0.00.0

0.5

0.52.0

0.60.3

0.5

Thym

idin

eVe

rapa

mil

mL

/g/m

in

0.0

Wat

er0.025

0.0

0.20

Uptake

Uptake

Uptake

mL

/g

0.0

2.21

mL

/g/m

in

0.0

0.37

2.60

K

K

K

mL

/g/m

inm

L/g

/min

mL

/g

0.0

0.13

VP

VD

Flux

Variance of Residues

(Greenwood’s Formula)

Approximation:

Flow AIFMean

-> Variation in Functionals by the Delta-Method

RegionalVoxel-LevelResidues andFlow Distribution

StandardizedVoxel-LevelResidues(Measured)

Some Analysis

Poisson with mean

Asymptotic Variance of MLEs

Summary

• PET in Cancer Imaging Diagnosis/StagingResponse AssessmentTreatment Planning

• Spatial and Temporal Aspects of PET Data Important

• Detailed Measurement and Modeling of the Disease Process is key to adaptive treatment

Statistics (Wisconsin style) has much to offer.(Please keep it going for another 50… at least!)