The Theoretical Basis for Quantitative Ultrasound from Pulse-Echo Data

BRP Support from NIH/NCI RO1CA111289

Timothy J. Hall

Medical Physics Department

University of Wisconsin-Madison

Context for Quantitative Ultrasound

Clinicians use images to describe lesion morphology This lesion is described as “Hypo-echoic” and “Shadowing”

Typical Clinical Breast Ultrasound

Motivation: Quantitative Ultrasound

Mass

Orientation -parallel -not parallel

Shape -round -oval -irregular

Margins -circumscribed -not circumscribed -indistinct -angular -microlobulated -spiculated

Vascularity Special Cases

Calcifications

Posterior acoustic features -none -enhancement -shadowing -combined pattern

Lymph node Intramammary

Boundary -abrupt interface -echogenic halo

ACR BI-RADS®

Ultrasound Lexicon

Clustered microcysts

Foreign body

Mass in skin

Surrounding Tissue -duct changes -Cooper’s ligament changes -edema -architectural distortion -skin thickening -skin retraction/irregularity

Complicated cyst

Lymph node Axillary

-micro-calcification -macro-calcifications -micro-calcification in a mass

-not present (or NA) -present -present adjacent to lesion -diffusely increased surrounding

Echo Pattern -anechoic -hyperechoic -complex -hypoechoic -isoechoic

American College of Radiology Breast Imaging and Reporting Data System

Courtesy of Dr. Elizabeth Burnside, Radiology Dept. UW-Madison

Motivation: Quantitative Ultrasound

American College of Radiology Breast Imaging and Reporting Data System

Mass

Orientation -parallel -not parallel

Shape -round -oval -irregular

Margins -circumscribed -not circumscribed -indistinct -angular -microlobulated -spiculated

Vascularity Special Cases

Calcifications

Posterior acoustic features -none -enhancement -shadowing -combined pattern

Lymph node Intramammary

Boundary -abrupt interface -echogenic halo

ACR BI-RADS®

Ultrasound Lexicon

Clustered microcysts

Foreign body

Mass in skin

Surrounding Tissue -duct changes -Cooper’s ligament changes -edema -architectural distortion -skin thickening -skin retraction/irregularity

Complicated cyst

Lymph node Axillary

-micro-calcification -macro-calcifications -micro-calcification in a mass

-not present (or NA) -present -present adjacent to lesion -diffusely increased surrounding

Echo Pattern -anechoic -hyperechoic -complex -hypoechoic -isoechoic

Motivation: Quantitative Ultrasound

American College of Radiology Breast Imaging and Reporting Data System

80% prevalent 40% malignant Low level echoes

throughout (relative to fat)

12% prevalent 16% malignant

Same echogenicity as

fat

•Anechoic •Hyperechoic •Complex •Hypoechoic •Isoechoic

Hong AS, Rosen EL, Soo MS, Baker JA. BI-RADS for sonography: positive and negative predictive values of sonographic features. AJR Am J Roentgenol 2005; 184:1260-1265.

Figures: Courtesy of Dr. Elizabeth Burnside, Radiology Dept. UW-Madison

• Anechoic • Hyperechoic • Complex • Hypoechoic • Isoechoic

Echo Pattern

Motivation: Quantitative Ultrasound

American College of Radiology Breast Imaging and Reporting Data System

• To be predictive of benign and malignant disease • Help to communicate, facilitate research, and

provide better patients care Many descriptors are subjective and qualitative

Inter-observer disagreement for some descriptors has been shown in the literature1

1Elizabeth Lazarus et al., BI-RADS Lexicon for US and Mammography: Interobserver Variability and Positive Predictive Value, Radiology, 239-2, pp385-391,2006

Goal of QUS

Reduce/remove subjectivity in clinical description of lesions This lesion is described as “Hypo-echoic” and “Shadowing”

Typical Clinical Breast Ultrasound

QUS Theory

Imaging System Properties

Soft Tissue Properties

RF Echo Signal

Ultrasonic (B-mode)

Image

Generalized Image Formation Model

QUS Theory

• We’ve known for many decades about the limiting conditions in acoustic wave propagation – Compare the size of the scattering source (d) with the

acoustic wavelength (λ) • λ << d “specular reflection” (Snell’s law) • λ >> d “Rayleigh scattering” (proportional to f4 d6)

• Physics is more interesting between these limiting

conditions

• Use models for acoustic interactions with tissue to extract physically descriptive parameters – Quantitative ultrasound (QUS)

QUS Theory

• First some definitions – Scattering is an elastic interaction resulting in a change in the

amplitude, frequency, phase or direction of the acoustic wave as a consequence of interacting with a spatial or temporal non-uniformity in the propagation medium

– Absorption is an inelastic process by which some portion of the acoustic wave energy is irreversibly converted into internal energy of the propagating medium structure.

– Acoustic attenuation is the sum of scattering and absorptive losses

QUS Theory

• To estimate an absolute parameter that describes acoustic scattering we compare our data to a model – “Inverse Problem” approach – A model for what?

• The differential scattering cross section

- Defined as the acoustic power scattered per solid angle per unit incident intensity

- Units are (length2 steradian-1)

- Generally normalized to the volume contributing to scattering

- (units become length-1 steradian-1)

QUS Theory

• The differential scattering cross section per unit volume

- (Normalize DSC to the volume contributing to scattering with units of length-1 steradian-1)

- An intrinsic material property

• Special Case – Scattering in the 180o direction

(like pulse-echo ultrasound) • Backscatter coefficient • The acoustic power scattered per solid angle per unit

incident intensity per unit volume in the 180o direction

• Now we have a quantity to model: the BSC

QUS Theory

• The Backscatter Coefficient – The acoustic power scattered per solid angle per

unit incident intensity per unit volume in the 180o direction

• What do we need to know to model it? – Incident intensity – Details of the scattering interaction

QUS Theory

• Incident intensity – Closed-form solutions for simple radiating

surfaces (single-element transducers) – Freely available programs for simulation complex

surfaces (array transducers)

• Typical assumption is incident plane waves

QUS Theory

• Model for backscatter – Continuum model

(continuously varying impedance distribution)

– Discrete scatterer model

QUS Theory

Details of the scattering interaction

• Numerous approaches – Random inhomogeneities – Inhomogeneous wave equation

= Spatial variation from local average compressibility

= Spatial variation from local average mass density

QUS Theory

Details of the scattering interaction – Discrete scatterers

• Numerous approaches – Geometry of the scattering source – Boundary conditions

• Closed-form solutions – Infinite and finite cylinders – Spheres – Prolate spheroids

QUS Theory

• Cylinders (infinite, finite, bent, etc.) – Lax and Feshbach, JASA 20(2): 108-124, 1948. – Faran, JASA 23(4): 405-418, 1951. – Su, et al., JASA 68(2): 687-691, 1980. – Flax, et al., JASA 68(6): 1832-1835, 1980. – Stanton, JASA 94(6): 3454-3462, 1993. – Ye, et al. JASA 102(4): 1964-1976, 1997.

• Assume incident plane wave (normal or oblique) • Compute scattered pressure as a function of angle

Closed-form solutions for scattered pressure

The publication list is examples only and is not exhaustive

QUS Theory

• Single Sphere – Faran, JASA 23(4): 405-418, 1951. – Hickling, JASA 34(10): 1582-1592, 1962.

• Spherical shell and oblate spheroid

– Stanton, JASA 88(3): 1619-1633, 1990. – Stanton, JASA 94(6): 3454-3462, 1993.

• Assume incident plane wave (normal or oblique) • Compute scattered pressure as a function of angle

Closed-form solutions for scattered pressure

The publication list is examples only and is not exhaustive

QUS Theory

• Why so much interest in closed-form solutions for scattering functions? – Scattering theory based on first principles – Comparison between experiment and theory

• Many of the publications on scattering from

cylinders included experimental results

• Early work on scattering from spheres contained comparisons with experiments

QUS Theory

• Discrete models for backscatter – Burke, et al., Ultrasonic Imaging 6: 342—247, 1984.

Ability to accurately measure scattering from single spheres

QUS Theory

Narrowband measurements of scattering from a “cloud” of spherical glass beads in agar Davros, et al. JASA 80(1): 229-237, 1986

4MHz 6MHz 7MHz

QUS Theory

• Comparisons between Faran’s theory for backscatter from spheres Hall, et al., UMB 22(8): 987-997, 1996.

3 transducers required to cover this bandwidth

Scattering from a “cloud” of glass spheres

QUS Theory

Summary • There is a need for quantitative estimates of acoustic

scattering – A common clinical task requires subjective description of

relative echogenicity • Scattering theory has been described for simple

structures • Measurements of phantoms agree very well with

predictions from scattering theory • This is a major step toward the goal of quantifying

scattering in unknown media

Thank you This work was funded, in part, by NIH BRP grant R01CA111289