Classification of bone microstructures using PCA and 2 ......2 The composition of bone tissue is...

i

Classification of Bone Microstructures Using PCA and

2-Point Statistics

A Thesis

Submitted to the Faculty

Of

Drexel University by

Alka Basnet

in partial fulfillment of the

requirements for the degree

of

Masters of Science

September 2012

iii

ABSTRACT

A database framework that could automatically classify and organize characterized microstructure

datasets based on a rigorous reduced variable representation of the microstructure rather than relying

on ad-hoc selected metrics, can be a powerful tool in the field of materials science and engineering.

Classification using lower order descriptors has been shown to improve the quantification of

microstructures. Principal component analysis over a classification structure is suggested as an

effective method for reduced order representation of a microstructure.

In this paper, 2 point correlation functions and Principal Component Analysis (PCA) is used as an

effective tool for classification of 3-D microstructures. [19].The central hypothesis of this paper is

that on the first 3 PC weights, bone microstructure would depict some classification based on the

wedge location within the cross. Although there was no distinct classification based on wedge

location, we see that microstructures clustered together in the low dimensional PCA representations

actually look similar to each other i.e. visually, compared to the microstructures that are far apart

from each other.

iv

ACKNOWLEDGEMENT

I would like to thank my advisor Prof. Surya Kalidindi, who supported and instructed my

performance throughout my research work with much patience and guidance. I also would like to

thank David Turner, my mentor for his immense support and guidance in all the steps as well as

Anne Hanna for all her help.

I would also like to thank Dr. Haviva Goldman and Naomi Hampson for providing the data used in

my thesis. In the mean time I would like to acknowledge all the help from my lab mates.

At last but not the least, I would like to thank my family for their immense support.

v

TABLE OF CONTENTS

Abstract ii

Acknowledgement iv

List of Figures vi

Chapter 1: Introdcution 1

1.1 Background: 1

1.2 Bone Composition and Remodeling 2

1.3 Sample Details 5

1.4 Classification Using Data Science Approaches 7

Chapter 2: Extracting cubical RVE’s from Micro Ct Imgaes 8

Chapter 3: Two Point Statistics 10

Chapter 4: Principal Component Analysis 12

Chapter 5 Results 16

Chapter 6: Conclusion and Future Studies 23

Appendix A: 25

Appendix B: 36

Appendix C: 36

Appendix D: 40

References 49

vi

LIST OF FIGURES

Figure 1 Compact Bone 2

Figure 2 Bone remodeling 4

Figure 3 pQCT: Tibia length sectioned at 38% and 66 6

Figure 4 Radial sections of bone 6

Figure 5 Binarized posterior wedge with 3 radially extracted boxes 9

Figure 6 Phase microsructure and Autocorrelation 11

Figure 7 Schematic representation of principal component analysis 12

Figure 8 Explained Variance for First 10 PCA Weights 14

Figure 9 Classification of Microstructure in PCA space 16

Figure 10 Categorization of Tibia Wedges based on Classification Showed in Fig.9 17

Figure 11 Tibia Length 17

Figure 12 Tibia Crosssection divided into 6 wedges 17

Figure 13 Pore Structure of Wedge 1_38% 18






vii




Figure 22 Pore Structure of Wedge 6_66%..........…..……………..…………………………… 20

Figure 23 Pore Structure of Wedge 6_66%...........….……………………………...…………... 20

Figure 24 Pore Structure of Wedge 6_66%.............……….………………………………........ 20

Figure 25 Pore Structure of Wedge 1_66% …………………………………………………...... 21

Figure 26 Pore Structure of Wedge 2_66%………………………..……………………………. 21

Figure 27 Pore Structure of Wedge 1_66%………………………………………...………….... 21

Figure 28 Pore Structure of Wedge 3_66%……….……………………………………….......... 21

Figure 29 Pore Structure of Wedge 4_66%…………………………………………………...... 22

Figure 30 Pore Structure of Wedge 5_66%………………..……………………………………. 22



1

CHAPTER 1: INTRODUCTION

BACKGROUND:

As new and rapid materials characterization tools became widely available, large datasets

are been generated at a faster pace which results into challenges. Some of which, are

rapid acquisition of large datasets, predictive capabilities in materials systems using the

information from dataset, and analysis of the increasing volumes of data. One of the

solutions to these challenges can be combination of classification methodology and

principal component analysis for effective reduced-order representation of 3D

microstructures which is demonstrated in the following paper. Instead of using several

higher-order measures for the mathematical representation of 3D microstructures, a

principal component analysis (PCA) technique is introduced for enabling reduced-order

representation. In this case, classification technique based on lower-order descriptors is

shown to increase the efficiency of representation. [20]. The dataset used in the analysis

in this thesis is from 10 individuals (average age 36.7 +/-8.4 years) from 2 specific

location 38% and 66% of total tibia length.

2

The composition of bone tissue is more complex than most engineering composites. [4]

The bone is composed of mineralized collagen fibrils which are composed of fibrous

protein collagen. At the macro scale bone is made of two basic structures, cortical bone

and trabecular bone. Cortical bone is densely packed with around 5-10% porosity and

forms the outer shell of all bones. The shafts of long bones are made primarily of cortical

bone with the midshaft being almost entirely cortical. My research is focused on human

cortical bone, specifically found at the midshaft of the tibia. 80% of the skeleton is made

of cortical bone and it plays a vital role in the majority of the skeleton’s supportive and

protective function.

Figure 1: Compact Bone

1.1 Bone Composition and Remodeling:

Bone composition and organization at the tissue level can play a very important role in

mechanical strength of the bone at the whole bone level. At the finest level of bones

hierarchical structure, bone has an organic matrix which is deposited as unmineralized

3

osteoid and contains primarily Type I collagen, proteoglycans and water. The collagen

provides flexibility as well as tensile strength to the bone. The organization, orientation,

and cross-linking of the collagen fibrils have been shown to be related to the mechanical

properties of the bone [11]. The mineral adds compressive strength and rigidity to the

bone and the degree of mineralization can have a huge impact on the mechanical

properties. Heavily mineralized bone has a higher breaking stress compared to less

mineralized bone, however overly mineralized bone can become brittle. [12] Bone

mineralization density, collagen fiber organization and porosity are discussed below,

relative to their relationship to bone remodeling and bone strength. [3]

There are two ways to measure mineralization - volumetric mineralization and specific

mineralization. Volumetric mineralization, otherwise known as bone mineral density is a

measure of the amount of mineral per unit volume of bone and is a function of both

mineralization and porosity. It can be measured with computed tomography which is a non-

invasive technique. [13, 14] Specific mineralization is the amount of mineral in a bone. It

can be measured a number of ways (ex vivo), ash fraction analysis being [15] the one used

in the present study. As bone ages, the amount of mineral contained in the bone matrix

increases and this causes the bone to be brittle. Tibia is also known as the long bone.

Proximally and distally of the midshaft will mostly will be cortical but have some

cancellous bone.

As bone grows it needs to be reshaped and thus starts remodeling. Bone is deposited on

one surface and reabsorbed on another. It gives bones its adult shape. The bone formed

during growth and development is very different in organization and properties from the

bone of an adult.

4

Figure 2: Bone Growth and Remodeling

Bone is reabsorbed and deposited. The remodeling rates are different at different

locations around the cortex. Bone gets more mineralized over time. New bone can be

right next to older bone. More osteons lead to greater porosity. As an individual grows

the bone does not just increase in length, the shape also changes, with areas that had been

very wide such as the bone ends becoming narrow and part of the shaft. During this

5

process bone must be removed through osteoclastic activity in some areas while in other

areas bone is laid down via osteoblastic activity. This process is called “growth

remodeling” [6, 7, 8] or modeling. During this process, bone is deposited on one cortex

and resorbed on the opposite cortex which leads to shift in the position of bone laid down

at an earlier time compared to newer bone.

1.3 SAMPLE DETAILS:

Our research sample acquired from the Musculoskeletal Transplant Foundation (Edison,

NJ, USA) represents a subset of an existing collection of donor tibiae with a total of 10

average age 37 +/- 8 years old and 17 male donors average age 33 +/- 10 years old. The

information on donor body weight and height are available, and only donors with no

known skeletal pathology were included in the collection. The tibiae were freshly

harvested, wrapped in wet gauze and stored frozen at -40 degrees Celsius in plastic bags.

Each of the tibiae in the sample were thawed, sectioned at 33% and 68% of total tibial

length, resulting in 3mm thick diaphyseal cross-sections at each location (see Figure 4).

Each cross section was cut into 6 wedges and labeled 1 through 6 starting with the

anterior wedge and ending with the medial anterior wedge. Cuts were based of the line

from the anterior tip through the centroid as shown in fig 4. Three 200 x 200 x 200 cubic

matrix with planes parallel to the subchondral plate was extracted from each wedge -

periosteal, midosteal and endosteal.

6

Figure 3: pQCT: Tibia length sectioned at 38% and 66%

Figure 4: Tibia cross section divided into 6 radial sections

7

1.5 CLASSIFICATION USING DATA SCIENCE APPROACHES:

Often in Materials Science, we deal with a large set of data especially while defining

a microstructure. In the study of human tibia properties, it is crucial to have a

classification of datasets we use to perform the analysis, based on their

microstructure. An efficient classification based on the microstructure saves time and

reduces redundancy in the acquired datasets. Developing a framework for accurate

analysis of this complex microstructure dataset has vital clinical applications as it can

provide insights into indications of high bone fragility. Given the several hundred

thousands of distinct engineered and natural materials of interest, the critical need for

new computationally efficient approaches for archival, retrieval, and real-time

exploration of the microstructure datasets by the broader scientific community is

self-evident. Efforts in these activities are hindered by the lack of a rigorous

mathematical definition of the internal structure or microstructure of a material.

The n-point correlation functions have been shown to be capable of recovering the

original micrograph to within a linear translation and/or an inversion, and as such

will serve as a primary descriptor of the statistics underlying the stochastic nature of

the microstructure [1]. Decomposition of the n-point statistics via principal

component analysis (PCA) offers a highly efficient data science approach for

classifying and cataloguing the microstructure datasets.

8

CHAPTER 2: EXTRACTING CUBICAL ROI’S FROM MICRO CT IMGAES

The samples studied in this research were prepared by a member of my research group,

Naomi Hampson. Naomi obtained the subset of donor tibiae that were obtained from the

Musculoskeletal Transplant Foundation, and were used in related studies at Mount Sinai

School of Medicine. For this project, nine donor (5 M, 4 F, 23-46 years.) tibia cross

sections were used, each 2.5 mm thick and extracted from a distance of 38% and 66% of

the total length of the bone from its distal end. The samples were pre-cut into six radial

sections and all the wedges were used for the study. These (number of wedges) were

imaged by micro computed tomography using a Sky-Scan 1172 at 100 kV and a 5 micron

voxel size. To facilitate image processing, a four-fold reduced versions of the micro-CT

scans were used so that every wedge had a corresponding set of approximately 110

images of 992 x 992 pixels each and a voxel size of 20 microns. The images from each

wedge were reconstructed to create a set of binarized images in the form of a pliable 3D

matrix model.

Using a matlab program shown in Appendix A, developed as a part of my thesis, cubic

regions of interest were radially extracted from each wedge. Three cubes, each of size

200 by 200 by 200 pixels were extracted from each rotated and binarized wedge

(Appendix A has the matlab code that shows how the images are rotated) which gave a

total of 18 cubes from a single cross section. Figure 5 depicts a binarized rotated posterior

wedge with 3 regions of interest.

9

Figure 5: Binarized posterior wedge with 3 radially extracted boxes

The data sets obtained from processing images for all 10 individuals were fairly large and

required a lot of time to be processed and saved. Thus, to save time and obtain objective

reduced-order representations of the microstructure, we extracted principal components

of the 2-point statistics from each microstructure. Such a database can dramatically

increase the speed and efficacy with which we can build datasets that can be shared by

the broader scientific community, while minimizing duplication of effort.

Endosteal

Box 1

Periosteal

(external)

Outline

Periosteal

Box 3

Mid-cortex Box 2

10

CHAPTER 3: TWO POINT STATISTICS

Due to recent advances in computational algorithms, it has been possible to obtain

higher-order statistical descriptions of microstructure with advanced computational

resources. More specifically, a microstructure is quantified through a selected list of

statistical measures. [20] In this thesis, this is explored in the framework of n-point

statistics. In this approach, the statistics of the microstructure are described by a set of

distributions that systematically provide more information about the microstructure with

increasing order of statistics [16, 17]. For example, 1-point statistics capture the

probability associated with finding a specific local state of interest at any point selected

randomly in the microstructure. At the next hierarchical level, 2-point statistics describe

the probability density associated with finding the specific local states at two ordered

points, specified by a vector, thrown randomly into the microstructure [1]

Among the various techniques to update the 2-point statistics, use of Fast Fourier

Transforms (FFTs) for rapid computation of two point statistics shows tremendous

promise [1]. We employ FFTs to calculate the 2-point statistics with an assumption that

the structure is periodic. We calculated the 2-point correlation by using an FFT

convolution. [1, 2]

The discretized two point statistics for a two phase material system is expressed as

1' '

0

1 snn n n

t s s t

s

f m ms

(Equation1)

where the superscripts n and n’ denote the local states of interest and the subscript t

denotes the vectors that can be thrown into the microstructure. The enumerator "s + t "

11

denotes the grid point reached in the microstructure by adding the vector t to the grid

point s. The details on how to calculate 2 point statistics is described in [1, 2].

Fig. 6. (a) An example of two-phase microstructure. (b) Autocorrelation for the

black phase.

12

CHAPTER 4: PRINCIPAL COMPONENT ANALYSIS

Principal Component Analysis (PCA) is a way of sorting high dimensional data based on

the directions exhibiting the highest variances. This is most easily understood as a

projection of a high dimensional data set into a new orthogonal (perpendicular)

coordinate frame where the axes are defined by the directions of highest variance. These

directions are known as the principal components of the dataset. Each principal

component is a linear transformation of the entire data set. The principal components are

identified such that the first principal component contains the highest variance. The

second principal component contains the second highest variance and is orthogonal

(uncorrelated) to the first principal component, so on and so forth, as depicted in Fig. 10.

The importance of each principal direction is given by its associated eigenvalue. PCA has

been proven to be the optimum linear technique at reduced order representation. Basic

background on the PCA technique can be found in [18].

Figure 7: Left – Schematic representation of principal component analysis of

multidimensional data. Right – Projection of the data onto the 1 and 2 principal

directions

13

PCA can be performed either by using Covariance matrix or by the method of eigen

vector decomposition. Using the latter, the data set is defined by the 2-point statistics on

the tibia bone samples from 10 individuals. From a total of 10 individuals, 93 distinct

microstructure datasets were extracted (excluding the 3 data sets that were not usable -

1226_66_Part 4, 6318_38_Part 2 and 6318_ 38_Part 5) and their 2-point statistics were

computed.. The steps involved in the method are as follows:

If we consider an ensemble of L vectors, the PCA decomposition of l the member is

given by

1

1

Ll l

i i

i

f a f (Equation 2)

where f represents the mean vector. In Eq. (2), i represent the orthogonal basis set and

l

ia represent the corresponding weights of the lth

member. Mathematically, the

decomposition can be envisioned as a few basic steps

1. Mean center the data.

l lf f

2. Calculate the covariance matrix of the data.

1( )l l T

LC

L

3. Perform eigenvalue decomposition.

i i iC b

4. Project mean centered data into the eigenspace to find the weights.

( )l T l

i ia

14

Our approach differs in that the PCA decomposition is used to provide a reduced

representation of the n-point correlations of microstructure.[11] The advantage of our

approach is that the statistics are smooth analytic functions with a natural origin. PCA

decomposition results in an uncorrelated orthogonal basis that spans the same space as

the original n-point correlations of microstructures.

The following figure projects the weights for first 10 Principal components.

Figure 8: Explained Variance for first 10 PCA weights

It is evident that first three Principal components account for 98.5 % of the variance in

the observed variables. This basically means that the first principal accounts for the

maximum amount of variance in the data set which we assume to be the volume fraction

of the bone. The second component extracted will account for a maximal amount of

variance in the data set that was not accounted for by the first component and will be

15

uncorrelated to the first component. Thus for my preceding analyses, I only took into

account the first PCA as the rest of them contain trivial information.

16

CHAPTER 5: RESULTS

Figure 9: Classification of Bone Microstructure on First 3 PCA Weights

Data from all the individuals were plotted in first three PCA space, using a convex

representation. It was seen that Wedge 1_38%, Wedge 3_38%, Wedge 5_38%,Wedge

6_38% clustered together that is labeled as Group 1, Wedge 6 separated out as Group 2

and the remaining sample from Individuals were clustered together that is labeled as

Group 3in Figure 11 as well as Figure 12. The size of the hull varies within the samples

and represents the variance in the data. In group 3, the large blue hull i.e. Wedge 1_66%

Group 3 Group 2

Group 1

17

has much larger variance in the data set compared to the small green hull i.e. Wedge

3_66% of the same group 3.

Group 1 Group 2 Group 3

Wedge 1_38% Wedge_6_66% Wedge 2_38%

Wedge 3_38% Wedge 4_38%



Wedge 3_66%

Wedge 4_66%

Wedge 5_66%

Figure 10: Categorization of Tibia Wedges based on Classification Showed in Fig.9

Figure.11 Tibia length Figure.12 CS divided into 6 wedges

The initial hypothesis was that bone microstructure would depict some fine classification

based on their placement within the cross section i.e. wedge location. Plotting the first 3

PC weights (as it covered 98.5% of total variance in the data set) of all the

microstructures we did not see any classification based on the wedge location. However,

we see that microstructures clustered together in the low dimensional PCA

18

representations actually look similar to each other i.e. visually, compared to the

microstructures that are far apart from each other. The pore structures of the

microstructures shown in Group 1, Group 2 and Group 3 vary from each other with

respect to the pore size. Pore space from each category (Group1, 2 and 3) are shown

below. Figures 14 – 29 were plotted using isosurface, as shown in Appendix D, such that

we can visualize the pore space of each microstructure.

Group 1

Figure 13: Wedge 1_38% Figure 14: Wedge 1_38%


19



Group 2

20



Group 3

21



22



23

CHAPTER 6: CONCLUSION AND FUTURE STUDIES

Summing the results obtained from Principal Component Analysis and two point

statistics on a set of binarized micro-CT scan images we see that microstructures

clustered together in the low dimensional PCA representations actually look similar to

each other i.e. visually, compared to the microstructures that are far apart from each

other. If we compare the pore structure of all three categories or group, we see that the

pore size is the basis of classification. Group 3 has few but large pore size compared to

Group 1 which has more but smaller pore size. Group 2 does not have a distinct has some

large and some small pore size. There is a possibility that a better classification of

microstructure would have been possible with a larger data size. I used 10 individuals

with 9 data sets for 38% and 7 for 66% location of the tibia length which is fairly small

compared to large amount of data being gathered routinely.

The continuation to this work should be using a larger data set to see if a better

classification could be achieved. A better classification would result in better predictive

capabilities and analysis in materials science and engineering system. It would also be

very interesting work to see if the relation between reduced order representation of

microstructure using Two point statistics and PCA, vs. mechanical properties like

effective modulus and strain energy distribution on different locations in the tibia length

to see if there is any distinct pattern of bone remodeling. Obtaining such results will aid

in better understanding of fracture risk and aging, improving our understanding of

structure / property relationships in bone and better understanding of relationships ahead

24

of improved clinical imaging techniques. I think a larger data set can depict a better

classification of microstructure.

25

APPENDIX A:

A1: EXTRACTING ROI FROM THE WEDGE_I

% Function that picks out parameters for boxes of specified side lengths

% for an image that is already binarized and lined up correctly.

% Returns box parameters for user with the other images.

function [xMin xMax yMin yMax] = lineUp(saveFile, name, numBoxes, sides, buffer)

%[xMin xMax yMin yMax] = lineUp(saveFile, name, numBoxes, sides, buffer)

%saveFile = str, directory where to save files

%name = str, name of files to be saved as

%numBoxes = int, number of boxes to take up for each section

%sides = int, number of pixels of each box side

%buffer = int, number of pixels to cushion on extreme sides of first and

%last boxes

% yMin, yMax = int, int, locations of y limits for string

% xMin = [numBoxes x 1], location of x minimun limit for boxes

% xMax = [numBoxes x 1], location of x maximum limit for boxes

% Get images

[fileName,pathName] = uigetfile('*.bmp','Select a good file to process');

imag = imread([pathName fileName]);

imag = ~imag; %inverting image

buffSides = sides;

26

%%

% Get outline

outlines = bwboundaries(imag);

index= 1;

for ii=1:size(outlines,1)

if size(outlines{ii},1) > size(outlines{index},1)% change it to more than equal to

index= ii;

end

end

imgOutline = [outlines{index}(:,2) outlines{index}(:,1)];

clear outlines;

centroid = round(mean(imgOutline));

yCentroid = centroid(2);

xCentroid = centroid(1);

%%

%Find locIn, locOut that will delineate the strip

indexIn = find(round(yCentroid-buffSides/2)<=imgOutline(:,2)...

& round(yCentroid+buffSides/2)>=imgOutline(:,2)...

& xCentroid<imgOutline(:,1));

indexOut = find(round(yCentroid-buffSides/2)<=imgOutline(:,2) ...

& round(yCentroid+buffSides/2)>=imgOutline(:,2)...

27

& xCentroid>imgOutline(:,1));

%Keep the most extreme boundaries on the outline for the boxes

locIn = min(imgOutline(indexIn,1));

locOut = max(imgOutline(indexOut,1));

figure; imshow(imag); hold on;

plot([locOut locIn], [yCentroid yCentroid], '+c');

% Ask if user would like to manually designate edges

% ask = input('Manually designate edge of rightmost box (y or n): ', 's');

% if ask == 'y'

title(gca,'Click location of right side of rightmost box');

LocIn = ginput(1); % picks the coordinates of the first click

plot(LocIn(1), LocIn(2),'*m');

pause(0.5);

% end

% To draw lines across the image

S = size(imag);

plot ([1 S(1)],[LocIn(2)-100 LocIn(2)-100], '-m');

plot ([1 S(1)],[LocIn(2)+100 LocIn(2)+100], '-r');

% Ask if user would like to manually designate edges

% ask = input('Manually designate edge of leftmost box (y or n): ', 's');

28

% if ask == 'y'

title(gca,'Click location of left side of leftmost box');

LocOut = ginput(1);

plot(LocOut(1), LocIn(2), '*m');

pause(0.5);

% end

% Take out boxes

y1=round(LocIn(2)-sides/2);

y2=round(LocIn(2)+sides/2);

yMin = min(y1,y2);

yMax = max(y1,y2);

%Find the centers of the boxes

center(1) = LocIn(1) - round(buffSides/2);

center(numBoxes) = LocOut(1) + round(buffSides/2);

diff = abs(center(1)-center(numBoxes));

for m=2:numBoxes-1

center(m) = center(1)-diff/2/(numBoxes-2)*(m-1);

end

close all;

figure(1)

29

imshow(imag)

hold on;

for j=1:numBoxes

%find verteces of boxes

x1=round(center(j) - sides/2);

x2=round(center(j) + sides/2);

figure(1)

hold on;

%plot location of boxes on image

plot([x1 x2], [y1 y1], 'r-', 'LineWidth', 2);




plot(center(j), LocIn(2), '*c');

%plot actual box images

xMin(j) = min(x1,x2);

xMax(j) = max(x1,x2);

figure(2)

hold on;

30

subplot(1,numBoxes, j);

imshow(imag(yMin:yMax,xMin(j):xMax(j)));

end

%save image with boxes

saveas(1, [saveFile, name, '_3DImage.bmp']);

end

31

A2 : EXTRACTING ROI FROM THE WEDGE

% LineUp_Multiple.m

% Script.m file that processes a set of image files (already binarized,

% rotated, and cropped). It allows the user to pick one sample file to

% create parameters in order to pick out a specified number of boxes out of

% each image to create a cube of image data that can be used for FEM. Also

% saves a text file specifying the information for the parameters used.

% Boxes_NAME.mat will contain cell array of size 1xn of all necessary image

% data for n boxes of size s for all m images:

% { [s x s x m] [s x s x m] [s x s x m] ... }

%%

%Get user inputs

numBoxes = input('Number of Boxes to pull out: ');

sides = input('Number of pixels for each side of each box: ');

buffer = input('Buffer on extreme sides of first and last box (zero): ');

%Get all image files (bmp)

filesLocation = uigetdir(pwd, 'Pick folder with image files: ');

cd(filesLocation);

32

files = ls('*bmp');

parentFolder = uigetdir(pwd, 'Select directory to place new folder with saved data');

%Specify Sample and Wedge Name

%This should be changed to either show what the name of the imags chosen is

%or get the name automatically from the file name or folder name

% SampleName = input('Name of Sample and Wedge Number (e.g. 1226-38Part1) No

Underscores: ', 's');

%Specify location of saved data

name = input('Name of images (to be saved as): ', 's');

%name='results';

saveFile = [parentFolder,'\', name,'\'];

mkdir(parentFolder,name);

%Line up one piece to get parameters

cd(..);

[xMin xMax yMin yMax] = lineUp(saveFile, name, numBoxes, sides, buffer);

cd(filesLocation);

box = cell([1, numBoxes]);

fprintf('\n Number of images to be loaded: %d \n', size(files,1));

33

%%

% Process each image file

tic

for jj=1:size(files,1)

%Read each image

file = files(jj,:);

imag = imread(file);

imag = ~imag;

%Take out boxes

for ii = 1:numBoxes

imagBox = imag(yMin:yMax,xMin(ii):xMax(ii));

box{ii}(:,:,jj) = imagBox;

end

end

% Record information

fid = fopen([saveFile, name, '_3DBoxParameters.txt'],'a');

fprintf(fid, '3D Box Parameters for: %s\r\n', name);

fprintf(fid, 'Total number of images: %d\r\n', size(files,1));

fprintf(fid, 'Dimensions of boxes: %d pixels (%d pixel buffer)\r\n\r\n', sides, buffer);

34

fprintf(fid, 'y lower bound: %d\r\n', yMin);

fprintf(fid, 'y upper bound: %d\r\n', yMax);

for j=1:numBoxes

fprintf(fid, 'x lower and upper bounds for Box %d: (%d, %d)\r\n', ...

j, xMin(j), xMax(j));

end

fprintf(fid, '\r\nBoxes go from endsteal outward radially to periosteal\r\n');

fclose(fid);

% Save boxes as mat file

save([saveFile, 'Boxes_', name, '.mat'], 'box');

%%

% Extract the names of the file

Samples = uigetdir(pwd, 'Pick folder to name the image: ');

for ii=length(Samples):-1:1 ;

if(Samples(ii) == '\') break;

end;

end;

SampleName = Samples(ii+1:end);

%%

% Create folders to save the images

cd (saveFile);

mkdir('periosteal');

35

mkdir('midcortex');

mkdir('endosteal');

cd...

% To generate the images for each ROI

for j = 0 : size(files,1)-1 % number of images from each ROI

data =(box{1}(:,:,j+1) );

data1=imrotate(data,-90);

imwrite(data1,['endosteal/', SampleName,'_ENDO_', num2str(j,'%04i'),'.bmp']); %

change part number for each wedge

end

for k = 0 : size(files,1)-1 % number of images from each ROI

data =(box{2}(:,:,k+1) );


imwrite(data1,['midcortex/', SampleName,'_MID_', num2str(k,'%04i'),'.bmp']); % change

part number for each wedge

end

for l = 0 : size(files,1)-1 % number of images from each ROI

data =(box{3}(:,:,l+1) );


imwrite(data1,['periosteal/', SampleName,'_PERI_', num2str(l,'%04i'),'.bmp']); % change

part number for each wedge

end

36

APPENDIX B:

CALCULATING TWO POINT STATISTICS:

% This m file creates the elements and defines the nodes for the two point

% stats of ROI.

function [nodes elements] = MakeMesh(M) % nodes and elements are output and M is

the input from twopoint stats

%meshgrid -

[X Y Z] = meshgrid(0:size(M,1),0:size(M,2),0:size(M,3));

A = size(X)

% creates the mesh

nodes = [X(:) Y(:) Z(:)];

[nodes sort_map] = sortrows(nodes, [-1 -3 -2]);

% Sort map tells us where a row in the sorted matrix comes from in the

% unsorted one

% Inverse sort map tells us where a row in the unsorted matrix wound up in

% the sorted one.

inv_sort_map(sort_map) = 1:length(sort_map);

37

%elements = zeros(sum(M(:)), 8);

num_elements = 0;

for ii=1:size(M, 1)

for jj=1:size(M, 2)

for kk=1:size(M, 3)

if(M(ii, jj, kk) == 1)

num_elements = num_elements + 1;

els = [ii, jj, kk; ii+1, jj, kk; ii, jj+1, kk; ii+1 jj+1 kk; ...

ii, jj, kk+1; ii+1, jj, kk+1; ii, jj+1, kk+1; ii+1 jj+1 kk+1;];

% keyboard

% This is the point when the 3d subscripts for each node

% are being converted to linear indices. The error is being

% caused by the passing of the incorrect size to the

% sub2ind function. This error will occur when the matrix

% that we are tryin to convert has unequal dimension

els_linear = sub2ind(size(X), els(:, 1), els(:, 2), els(:, 3));

elements(num_elements, :) = inv_sort_map(els_linear);

end

end

38

APPENDIX C

CALCULATING PCA WEIGHTS

close all

D = dir('/mnt/nfs01/users/alka/matlab/stats');

% Get rid of this LATER!!!! When files are done copying!!!!

D = D(3:end-16);

D(10) = [];

D(51) = [];

NUM_SAMPLES = size(D, 1);

A = [];

%NUM_SAMPLES = 20;

kk = 1;

for ii=1:NUM_SAMPLES

fprintf(1, 'Loading Sample %d of %d\n', ii, NUM_SAMPLES);

in_file = D(ii).name;

G = load(sprintf('/mnt/nfs01/users/alka/matlab/stats/%s', in_file), 'G'); G = G.G;

if(ii == 1)

39

fprintf(1, ' Allocating A Matrix ... ');

A = zeros(size(G,2)*size(G,3)*size(G,4), NUM_SAMPLES);

fprintf(1, 'Done\n');

end

%

% T = squeeze(G(1, :, :, :)); A(:, kk) = T(:);

% T = squeeze(G(2, :, :, :)); A(:, kk+1) = T(:);

% T = squeeze(G(3, :, :, :)); A(:, kk+2) = T(:);

T = mean(G,1) ; A(:,ii) = T(:);

% kk = kk + 3;

end

fprintf(1, 'Mean Centering Data ... ');

A = A - repmat(mean(A, 2), [1 size(A, 2)]);


[~, S, V] = pca(A, size(A, 2));

[~, S, V] = pca(A, 10);

L = S * V';

E = S * S';

fprintf(1, 'Saving Results ... ');

save -v7.3 results_averaged.mat L S V E


40

APPENDIX D

PLOTTING CONVEX HULL FOR EACH DATA SET

clc

%Reading data from Excel

X_1 = xlsread('MeanPCA','wedge1','F4:H12');

X_2 = xlsread('MeanPCA','wedge1','N4:P10');

Y_1 = xlsread('MeanPCA','wedge2','N4:P10');

Y_2 = xlsread('MeanPCA','wedge2','F4:H12');

Z_1 = xlsread('MeanPCA','wedge3','F4:H12');

Z_2 = xlsread('MeanPCA','wedge3','N4:P10');

A_1 = xlsread('MeanPCA','wedge4','F4:H12');

A_2 = xlsread('MeanPCA','wedge4','N4:P10');

B_1 = xlsread('MeanPCA','wedge5','F4:H12');

B_2 = xlsread('MeanPCA','wedge5','N4:P10');

C_1 = xlsread('MeanPCA','wedge6','F4:H12');

C_2 = xlsread('MeanPCA','wedge6','N4:P10');

% calculating Convex Hull indices

K1 = convhulln(X_1);

K2 = convhulln(X_2);

K3 = convhulln(Y_1);

K4 = convhulln(Y_2);

K5 = convhulln(Z_1);

K6 = convhulln(Z_2);

41

K7 = convhulln(A_1);

K8 = convhulln(A_2);

K9 = convhulln(B_1);

K10 = convhulln(B_2);

K11 = convhulln(C_1);

K12 = convhulln(C_2);

colormap('default')

trisurf(K1,X_1(:,1),X_1(:,2),X_1(:,3), ones(9,1))

hold on

trisurf(K2,X_2(:,1),X_2(:,2),X_2(:,3), 2*ones(7,1))

hold on

trisurf(K3,Y_1(:,1),Y_1(:,2),Y_1(:,3), 3*ones(9,1))

hold on

trisurf(K4,Y_2(:,1),Y_2(:,2),Y_2(:,3), 4*ones(7,1))

hold on

trisurf(K5,Z_1(:,1),Z_1(:,2),Z_1(:,3),5*ones(9,1))

hold on

trisurf(K6,Z_2(:,1),Z_2(:,2),Z_2(:,3),6*ones(7,1))

hold on

trisurf(K7,A_1(:,1),A_1(:,2),A_1(:,3),7*ones(9,1))

hold on

trisurf(K8,A_2(:,1),A_2(:,2),A_2(:,3),8*ones(7,1))

hold on

trisurf(K9,B_1(:,1),B_1(:,2),B_1(:,3),9*ones(9,1))

hold on

trisurf(K10,B_2(:,1),B_2(:,2),B_2(:,3),10*ones(7,1))

42

hold on

trisurf(K11,C_1(:,1),C_1(:,2),C_1(:,3),11*ones(9,1))

hold on

trisurf(K12,C_2(:,1),C_2(:,2),C_2(:,3),12*ones(7,1))

xlabel('PCA 1');

ylabel('PCA 2');

zlabel('PCA 3');

title('Classification of Bone Microstructure')

hold off

43

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