TITLE Statistical analysis of labelling patterns of mammary - PowerPoint PPT Presentation

TITLE Statistical analysis of labelling patterns of mammary carcinoma cell nuclei on histological sections AUTHORS Torsten Mattfeldt 1 Stefanie Eckel 2 Frank Fleischer 3 Volker Schmidt 2 DEPARTMENTS 1 Department of Pathology, Ulm University 2 Institute of Stochastics, Ulm University 3 Medical Data Services/Biostatistics, Boehringer-Ingelheim Pharma GmbH & Co. KG Gemeinsame Arbeitstagung der AGs Bayes-Methodik, R¨ aumliche Statistik und ¨ Okologie und Umwelt 27.–29. September 2007, Schloss Reisensburg

OVERVIEW OF THE LECTURE • Explorative point pattern analysis • Point process modelling • Distance-dependent Simpson indices • Monte Carlo rank tests

Ductal invasive mammary carcinoma Immunohistochemistry for MIB-1

Ductal invasive mammary carcinoma Immunohistochemistry for MIB: Detection of nuclei

Case 3, Image 1: Point pattern of unlabelled nuclei

Case 3, Image 1: Point pattern of labelled nuclei

Case 3, Image 1: Point pattern of unlabelled and labelled nuclei

Ductal invasive mammary carcinoma Immunohistochemistry for MIB

Ductal invasive mammary carcinoma Immunohistochemistry for MIB: Detection of nuclei

MATERIAL AND METHODS CASES • Breast cancer: 20 routine cases • Operative specimens • Domains with invasive ductal adenocarcinomas MICROSCOPY • Paraffin sections • Light microscopy • Immunohistochemistry Ki 67: nuclear protein associated with proliferation MIB-1: monoclonal antibody versus Ki 67

IMAGE EVALUATION Sampling Two rectangular fields per case Size: 1240 × 1000 pixels = 440 µ m × 354 µ m Explorative point pattern analysis Interactive detection of centres of tumour cell nuclei on sections Labelled and unlabelled nuclei → marked point pattern Estimation of g ( r ) for r = 1–200 pixels using kernel methods Epanechnikov kernel � ˆ Bandwidth: h = 0 . 1 / λ Software Library geostoch under Java (Mayer et al., 2004; http://www.geostoch.de) Package spatstat under R 2.2.0 under Linux (Baddeley & Turner, 2005)

EXPLORATIVE ANALYSIS OF PLANAR POINT PATTERNS • Stationary planar point process X = { X n } with intensity λ • Second order K -function, reduced second moment function K ( r ) K ( r ) = E (number of other points with distance ≤ r | ( x, y ) ∈ X ) λ K Poi ( r ) = πr 2 • Pair correlation function g ( r ) g ( r ) = ̺ (2) ( r ) 1 dK ( r ) = λ 2 2 πr dr g Poi ( r ) = 1

ESTIMATION OF THE PAIR CORRELATION FUNCTION • Estimation of the product density � 1 k h ( r − || X i − X j || ) � ̺ (2) ( r ) = 2 πr | W X i ∩ W X j | X i ,X j ∈ W i � = j 4 h (1 − x 2 k h ( x ) = 3 h 2 )1 ( − h,h ) ( x ) • Estimation of the squared intensity λ 2 = X ( W )( X ( W ) − 1) � | W | 2 • Estimation of g ( r ) � ̺ (2) ( r ) � g ( r ) = � λ 2

Case 3, Image 1: g-function of unlabelled nuclei

Case 3, Image 1: g-function of labelled nuclei

Case 3, Image 1: g-functions of  unlabelled and --- labelled nuclei

Case 3: Mean g-functions of  unlabelled and --- labelled nuclei

Estimated g-functions of  unlabelled and --- labelled nuclei Mean values of all 20 cases

Local comparisons of g -functions Mean values for labelled and unlabelled nuclei Unlabelled Labelled Level of D ¯ ¯ g ( r ) g ( r ) significance r � � 5 0.00000 0.29258 0.29258 p < 0 . 001 10 0.00127 0.70074 0.69947 p < 0 . 05 15 0.24768 1.33178 1.08410 p < 0 . 001 20 1.13593 1.94351 0.80758 p < 0 . 001 25 1.46384 2.33754 0.87370 p < 0 . 001 30 1.35353 2.29150 0.93797 p < 0 . 001 35 1.22940 2.00732 0.77792 p < 0 . 001 40 1.17549 1.77813 0.60265 p < 0 . 001 45 1.16746 1.61141 0.44395 N.S. 50 1.15192 1.50813 0.35621 N.S. 55 1.13997 1.42883 0.28886 N.S. 60 1.13603 1.35491 0.21888 N.S. 65 1.12911 1.30302 0.17391 N.S. 70 1.11672 1.28056 0.16385 N.S. 75 1.10675 1.27387 0.16712 N.S. 80 1.09886 1.28003 0.18117 N.S. 85 1.10246 1.26078 0.15832 N.S. 90 1.10575 1.22666 0.12091 N.S. 95 1.08833 1.17965 0.09132 N.S. 100 1.10102 1.13606 0.03504 N.S. 150 1.05846 1.13965 0.08120 N.S. 200 1.04896 1.07587 0.02691 N.S.

( r max , g max ) + + ( r min , g min ) ∆ = g max − g min M = (g max − g min )/(r min − r max )

Group comparisons of explorative summary characteristics Estimate Unlabelled nuclei Labelled nuclei Level of ¯ ¯ significance x SD x SD N (nucl/field) 741 217 89 47 p < 0 . 001 λ (points/pixel 2 ) 0.0005957 0.000175 0.00007177 0.0000379 p < 0 . 001 r 0 (pixel) 14.75 1.18 18.25 1.81 p < 0.001 r max (pixel) 25.70 3.55 27.13 5.66 N.S. g max (pixel) 1.582 0.295 2.559 0.655 p < 0.001 r min (pixel) 40.46 6.53 53.18 12.69 p < 0.001 g min (pixel) 1.113 0.162 1.244 0.377 N.S. 0.035 0.017 0.051 0.030 N.S. M ∆ g 0.469 0.191 1.314 0.699 p < 0 . 001

Estimated K-functions of  unlabelled and --- labelled nuclei Mean values of all 20 cases versus Poisson process

POINT PROCESS MODELLING Model Gibbs processes Stationary Strauss hard core process Methods Package spatstat under R 2.2.0 under Linux (Baddeley & Turner, 2005) Explorative statistics ( λ , r 0 , g ( r )) Parametric model fitting

FITTING OF THE STATIONARY STRAUSS HARD CORE MODEL PROPERTIES Parameters λ, r 0 , R, γ No point pairs within minimal interpoint distance r 0 r < h c Interval of interaction distances h c ≤ r < R if ( γ < 1): Repulsion if ( γ = 1): Classical hard core point process if ( γ > 1): Clustering ’Some radius beyond which influence is inconceivable’ r ≥ R IRREGULAR PARAMETERS Hard core distance r 0 Estimator: minimum interpoint distance Interaction radius R Method: profile maximum pseudolikelihood REGULAR PARAMETER Interaction parameter γ

Group comparisons of model parameters Unlabelled nuclei Labelled nuclei Level of ¯ ¯ significance x SD x SD Intensity N (nucl/field) 741 217 89 47 p < 0.001 λ (points/pixel 2 ) 0.0005957 0.000175 0.00007177 0.0000379 p < 0.001 Strauss hard core model r 0 (pixel) 14.75 1.18 18.25 1.81 p < 0.001 R (pixel) 39.12 18.94 44.52 14.28 N.S. 0.874 0.334 3.164 2.010 p < 0.001 γ

Distance-independent characteristics of diversity Simpson index D m λ 2 � i D = 1 − λ 2 i =1 • Definition Probability to select a point pair at random belonging to different components • Measure of diversity • Distance-independent → Generalisation to distance-dependent Simpson indices

Distance-dependent characteristics of diversity n λ 2 i K ii ( r ) � α ( r ) = 1 − λ 2 K ( r ) i =1 • Probability to select a point pair at random belonging to different components conditional to the event that it has distance less than r • Random labelling → α ( r ) = D • α ( r ) < D : point pattern has smaller diversity for distances below r than in the case of random labelling • α ( r ) > D : point pattern has larger diversity for distances below r than in the case of random labelling

Distance-dependent characteristics of diversity n λ 2 i g ii ( r ) � β ( r ) = 1 − λ 2 g ( r ) i =1 • Probability to select a point pair at random belonging to different components conditional to the event that it has distance r • Random labelling → β ( r ) = D • β ( r ) < D : point pattern has smaller diversity at distance r than in the case of random labelling • β ( r ) > D : point pattern has larger diversity at distance r than in the case of random labelling

Distance-dependent characteristics α (r) Simpson index D, 95% CI of α (r), α (r) for pattern1 of case 1 of mammary cancer

Distance-dependent characteristics α (r) Simpson index D, 95% CI of α (r), α (r) for 20 cases of mammary cancer

Distance-dependent characteristics β (r) Simpson index D, 95% CI of β (r), β (r) for 20 cases of mammary cancer

Testing individual point patterns on random labelling • Random labelling K 11 ( r ) = K 22 ( r ) • Test statistic with equal weights � s T (1) = | � K 11 ( r l ) − � K 22 ( r l ) | l =1 • Weighted test statistic s � T (2) = w ( r l )( � K 11 ( r l ) − � K 22 ( r l )) 2 l =1 w ( r l ) = ( � V ar ( K 11 ( r l ) − K 22 ( r l ))) − 1

Monte Carlo rank test on random labelling • Simulate 9999 realisations based on independent labelling Given the realisation of locations and the observed values of the marks, assign the marks completely randomly to the locations • Estimate the functions K 11 and K 22 at distances r 1 , ..., r s • Compute values of the test statistic T 1 , ..., T 9999 for the 9999 simulations and T for the single real pattern • Sort the resulting 10000 values in ascending order • Determine the rank of T in this sequence • Reject the hypothesis of random labelling, if the rank of T ∈ [9501 , 10000]

Simulation of random labelling Case 3, image 1 Case 3, image 1 Real pattern Simulated pattern #1 of 9999 Coordinates identical Labelling fraction identical Labels random

Case 3, image 1 K-functions of unlabelled and labelled nuclei Estimated from the real pattern Case 3, image 1 K-functions of unlabelled and labelled nuclei Estimated from simulation #1 of 9999

Case 3, image 1 g-functions of unlabelled and labelled nuclei Estimated from the real pattern Case 3, image 1 g-functions of unlabelled and labelled nuclei Estimated from simulation #1 of 9999