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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