optimization of biopsy protocols for detection of prostate cancer ariela sofer, george mason...

Optimization of Biopsy Protocols for Detection of Prostate Cancer

Ariela Sofer , George Mason University

Jianchao Zeng, Georgetown University

Brett Opell, Georgetown University

With acknowledgement to

John J. Bauer, Xiaohu Yao, Wei Zhang, Isabel A. Sesterhenn

Judd W. Moul, John Lynch, Seong K. Mun

GU, Walter Reed Army MC, AFIP, CPDR USUHS

Prostate Cancer & Biopsy

• Prostate cancer is the second leading cause of cancer-related death among American men.

• In US alone, about 220,900 new cases are expected to be diagnosed in 2003, and 28,900 men are expected to die of the disease.

• Unfortunately, no imaging modality can effectively differentiate cancerous tissue from normal prostate tissue

• Gold standard for prostate cancer detection: transrectal ultrasound guided needle biopsy

• Problem: current biopsy protocols are not adequate in terms of detection rate

Systematic Prostate Biopsy

• A biopsy protocol designates the number of needles to be used and their location on the prostate.

• Currently adopted protocol is the “sextant” – misses 20% or more of cancers

• Recently some alternative protocols have been shown empirically to have better detection rates

• Our goal: determine an optimal needle biopsy protocol

The Approach • Use real prostate specimens obtained from prostatectomies

to reconstruct 3-D prostate models (currently 301 prostate specimens)

• Superimpose a 3-D grid over each model and calculate cancer presence within the grid

• Develop a 3-D distribution map of tumor location • Use the map to determine the biopsy protocols that

maximize the probability of detection. Protocols should be identifiable by the physician to within the resolution of ultrasound

• Develop a 3-D simulation platform for comparing the optimal protocol to existing protocols. System allows for automatic simulation by computer and interactive virtual biopsy by urologists

3-D Surface Modeling

a b

c d

Prostate Division for Biopsy Protocols48 Zones

Z

Y

XA

Base

Apex

Mid

Left

Anterior

Posterior

Right

Prostate as Seen by Physician in Biopsy

Distribution Map of Cancer By Zone – 301 Patients

Base

Mid Apex

No. of Patients

Optimal Protocol for a Prescribed Number of Needles

• Our goal: to determine the protocol that maximizes the probability of detecting cancer in a biopsy with a prescribed number of needles.

• Some restrictions: Physicians want left-right symmetry in probes The anterior regions are harder and more

uncomfortable to probe. Biopsies restricted to the posterior, or to the rear 3/4’s (posterior + rear half of anterior) would be desirable

What is the Probability of Detecting Cancer in a Zone?

• A probe in a cancerous zone will not necessarily yield a positive(cancerous) diagnosis

• Given that a zone is cancerous,what is the probability that a probe will detect cancer?

• Difficulties:– Prostates have varying volumes– Variability in physician’s placement

of needle– Cancer is not distributed randomly in

zone– Needle does not draw a random selection of cells

0.5-1.2 cm

Needle core:1.6 cm long0.16cm diameter0.016 cc

Typical zone0.14 -1.7 cc

Estimating the Probability of Detection in a Zone

• Partition each zone into a “sufficiently large” number of subzones, with each subzone small enough in volume to be contained in a needle core.

• Identify the presence of cancer for each patient in each sub-zone

• Assume that the longitudinal position (z) of the needle insertion point the depth (y) the firing angle of the needle (only one degree of freedom)

are independent Gaussian variables.

• Combine the above model with the prostate volume and needle core volume information to estimate the probability that a needle probe in a zone will be positive

yx

z

Initial Approximation• Divided each zone into 53 =125 sub-zones for a total grid

of 6000 sub-zones.

• The table below summarizes

• Proportion of patients who had cancer in each zone (blue)

• Estimated probability that a needle in zone will detect cancer (black);

15% 7%

19% 8%

19% 8%

13% 7%

29%13%

28%11%

28%12%

32%15%

 43%22%

38%17%

38%14%

43%21%

38%25%

36%19%

35%15%

34%21%

27% 13%

33%16%

32%16%

23%12%

47%24%

44%21%

44%20%

46%23%

61%34%

61%29%

62%27%

60%34%

58%38%

64%37%

61%32%

58%36%

15% 8%

29%14%

30%13%

18%10%

41%21%

51%26%

50%221%

42%20%

 51%29%

63%33%

65%33%

55%31%

41%29%

57%34%

57%31%

44%28%

Base Mid Apex

Optimal Protocol for a Prescribed Number of Needles

• Given that a patient has cancer, determine the protocol that maximizes the probability of detection in a biopsy of k needles.

• Our decision variables

• Let pij = probability that a needle in zone j detects cancer in patient i.

• Let qij = 1 - pij

• Here i = 1,…,m and j = 1,…,n where m = number of prostates models in study (m = 301)

n = number of zones in 3-D map (n = 48)

Maximizing the Probability of Detection

• Then

• Thus the probability of diagnosing cancer in patient i is

• The k-biopsy protocol that maximizes the probability of detection solves:

Maximizing the Probability of Detection (Cont’d)

• Equivalently

• Denote

• Then the protocol solves the nonlinear integer program

NLIP via Generalized Bender’s Decomposition

Assume that f is convex and g is concave for each fixed x. Then problems are equivalent

Generalized Bender’s Decomposition (cont’d)

Termed the “primal problem. Assume for simplicity it has a solution for all x.

Gives an upper bound on f*

Relax these constraints:Rather than all and the best y, just choose previous ’s and the related y’sResulting problem termedthe “master problem”

Get a lower bound on f*If f and g are linear in x, the master problem is an IP – “easy”

The Algorithm

• Given an initial feasible point , and an upper bound UB on the objective

• Iteration t = 0, 1, …:– Solve “primal” for xt. Set

– Solve the IP (relaxed “master problem”):

– Set

– If (UB - LB) < terminate.

Our NLIP: Generalized Bender’s Decomposition

Problems are equivalent

The NLIP: Generalized Bender’s Decomposition

Termed the “primal problem”

Solution:

Gives an upper bound on f*

Relax these constraints:Rather than all and the best y, just choose previous ’s and the related y’sResulting problem termedthe “master problem”

Get a lower bound on f*

The Algorithm

• Given an initial feasible point , and an upper bound UB on the objective

• Iteration t = 0, 1, …:– Set

– Solve the IP (relaxed “master problem”):

– Set

– If (UB - LB) < terminate.

Results

• Estimated detection rate for sextant method: 67.3%• Estimated detection rated for optimized biopsies:

• Note that in 6 of the patients cancer is restricted to anterior

Number of biopsies

Posterior only Posterior + Rear Anterior

6 78.8% 79.3%

8 81.6% 82.9%

10 84.2% 85.5%

Updated Biopsy Protocols

Base Mid Apex

x x

     x

x

x x

 x x

x x

     x

x 

x x

x x

x x 

x x

6-Needle Biopsy. Estimated Probability of Detection: 79.3%

8-Needle Biopsy. Estimated Probability of Detection 82.9%

10-Needle Biopsy. Estimated Probability of Detection 85.5%

 x x

x x

Further Work: Patient Specific Groups

• We are studying the detection rate of our optimized protocols on patient-specific groups classified by age, race, PSA level and prostate size.

• Preliminary analysis of detection rates of the optimal schemes for a crude division to small- and large- size prostates:

General Protocols Specific Protocols

No. of Cores Prostates < 30 cc

Prostates > 30 cc

Prostates < 30 cc

Prostates > 30 cc

Sextant 75.8% 47.8%

6 optimal 86.2% 66.7% 86.2% 68.1%

8 optimal 88.7% 72.0% 88.9% 73.1%

10 optimal 89.7% 75.1% 91.1% 79.8%

Future Research

• Enhance the current distribution model by Using finer division for probability estimates Increasing the number of prostate models

• Perform more comprehensive study for biopsy protocols by race, age, prostate size, grade of cancer & re-biopsy

• Test protocols via interactive virtual biopsy by urologist using biopsy simulation system

• Apply the 3-D prostate tumor distribution map protocol to real-time in vivo image-guided biopsy procedures

• Apply distribution map for dose escalation in brachytherapy

A New Generation of Image-Guided Prostate Biopsy

Biopsy guidance:A patient-specific 3-D prostate model is reconstructed from the prostate outlines and matched to the 3-D tumor distribution map with optimal biopsy protocols. The tumor distribution information, and the optimal biopsy protocols are highlighted or color-coded as grids on top of the ultrasound images during biopsy.

US Beams

Prostate

US Images

US Machine

SuperimposedDisplay

US Probe

Tracker

US Beams

US Machine

Computer

US ImagesProstate

US Probe

Tracker