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!)