Stereology: Theory and Applications

✍ Scribed by Luis Manuel Cruz-Orive

• Publisher: Springer
• Year: 2024
• Tongue: English
• Leaves: 498
• Series: Interdisciplinary Applied Mathematics, 59
• Edition: 1
• Category: Library

✦ Synopsis

This book presents a comprehensive set of methods for quantifying geometric quantities such as the volume of a tumor, the total surface area of the alveoli in a lung, the length of plant roots, or of blood vessels, the number of neurons in a brain compartment, the connectivity number of trabecular bone, the mean size of grains in a rock, etc.. The methods, illustrated by twenty solved case studies, are based on properly sampled slices, sections, or projections of the material, observable under light, laser, or electron microscopy, or under non-invasive radiological devices such as ecography, computed tomography, or magnetic resonance imaging. Thus, the input usually consists of flat images, and the output consists of relevant quantities defined in three dimensions. Stereology is the discipline of providing sampling designs which warrant unbiased estimation of the corresponding quantities, that is, estimation with zero mean deviation from the target. Sampling is usually systematic (i.e., with regularly spaced probes), and sparse (as opposed to reconstructions) and it is thereby efficient and easy to implement.

Stereology is essentially geometric sampling, grounded on integral geometry. The necessary elements of both disciplines are detailed in textbook style, and may be used for postgraduate courses, or to serve the interest of scientists in general. Hitherto no other book on stereology has appeared which encompasses the theory, methodology, and applications of stereology in an interconnected and comprehensive way. The currently available error variance prediction formulae under systematic sampling, and their (non-obvious) derivation, are all gathered, for the first time, in the last chapter. The exposition is augmented by 127 line drawings for the theory, and 27 color pictures of real materials for the case studies.

✦ Table of Contents

Preface
Contents
Chapter 1 Basic Results of Integral Geometry
1.1 Introduction
1.2 Points and Unbounded Probes
1.3 Unbounded Probes Hitting a Target Set
1.4 Crofton Formulae for Unbounded Probes of a Fixed Orientation
1.5 Crofton Formulae for Motion-Invariant Unbounded Probes
1.6 Surface Area From Vertical Sections
1.7 Formulae of Local Stereology
1.8 Surface Area and Volume With the Invariator
1.9 Blaschke–Petkantschin Formulae for Powers of Area and Volume
1.10 The Euler–Poincaré Characteristic of a Planar Domain
1.11 Crofton Formula for the Integral of Mean Curvature
1.12 The Euler–Poincaré Characteristic in 3D
1.13 Bounded Probes: The Kinematic Density
1.14 Crofton Formulae for Bounded Probes: Purpose and Preliminaries
1.15 Crofton Formulae for Bounded Probes of a Fixed Orientation
1.16 Crofton Formulae for Bounded Invariant Probes
1.17 Surface Area From Vertical Sections and Cycloids
1.18 The Euler–Poincaré Characteristic From Bounded Probes
1.19 Hitting Measures and Projection Formulae
1.20 Hitting Measures for Bounded Probes. Kinematic Formulae
1.21 Test Systems
Chapter 2 Basic Ideas of Geometric Sampling
2.1 Background and Purposes
2.2 A Single Uniform Random Test Point
2.3 Weighted Sampling
2.4 FUR and IUR Test Lines in 2D
2.5 FUR and IUR Test Planes, and Lines, in 3D
2.6 Mean Values and Ratios for a Test Plane
2.7 Mean Values and Ratios for a Test Line in 3D
2.8 Test Planes and Lines Conditional on Hitting a Set
2.9 Sampling With a Vertical Plane
2.10 Sampling With a FUR Slab
2.11 FUR, and IUR, Bounded Test Probes
2.12 Mean Values and Ratios for a Bounded Test Probe
2.13 Bounded Test Probes Conditional on Hitting a Set
2.14 Sampling With a Cycloid Test Curve in a Vertical Plane
2.15 Particle Number
2.16 Particle Number and Size From Slab and Plane Probes
2.17 Connectivity
2.18 Sampling With a Local Probe
2.19 Point-Sampled Intercepts
2.20 The Invariator
2.21 Blaschke–Petkantschin Formulae for Higher Moments of Particle Volume
2.22 Isotropic Projections
2.23 Vertical Projections
2.24 Test Systems: Preliminary Comments
2.25 FUR Test Systems
2.26 IUR Test Systems
2.27 Test Systems of Cylinders
2.28 Vertical Designs
2.29 Spatial Grids
2.30 Ratios Based on Test Systems
2.31 Classical Ratio Designs for Mean Particle Size
2.32 Local Stereology for Particle Size
2.33 Pappus–Guldin Identities for Volume
Chapter 3 Model and Second-Order Stereology
3.1 Random Processes of Geometric Objects: Basic Concepts
3.2 Stationarity and Isotropy
3.3 Motion-Invariant Process Hit By a Fixed Bounded Probe
3.4 Motion-Invariant Process Hit By a Fixed r-Plane Probe
3.5 Intersection of Two Motion-Invariant Processes
3.6 Second-Order Functions
3.7 Local Stereology for Second-Order Measures
3.8 Second-Order Measures for a Planar Domain
3.9 Stationary Point Processes. The Poisson Point Process
3.10 Motion-Invariant Line Processes in the Plane
3.11 The Motion-Invariant Poisson Line Process in the Plane
3.12 Germ–Grain and Boolean Models
3.13 Particle Size Weighting and Size Distributions
3.14 Band and Membrane Models
Chapter 4 Sampling and Estimation for Stereology
4.1 Estimation in Design Stereology: Basic Ideas
4.2 Estimation of Global Quantities
4.3 Discrete Sampling
4.4 Uniform Sampling
4.5 Planning and Optimizing a Stereological Design
4.6 Planar Area and Boundary Length With a Square Grid
4.7 Volume By Fluid Displacement
4.8 Volume From Cavalieri Sections
4.9 Volume From Cavalieri Slabs
4.10 Number and Mean Size of Planar Particles From Systematic Quadrats
4.11 Number By the Fractionator Method
4.12 Curve Length in Space From ICav Sections
4.13 Surface Area, Volume, and Mean Barrier Thickness By a Ratio Design
4.14 Neuron Number With the Optical Disector
4.15 Connectivity of Trabecular Bone
4.16 Surface Area From Vertical Cavalieri Sections
4.17 Cortical Surface Area and Mean Thickness From Digitized Vertical Sections of a Human Brain
4.18 Curve Length From Vertical Projections
4.19 Stereology of Articular Cartilage From Local Vertical Sections: I. Global Size Properties
4.20 Stereology of Cartilage From Local Vertical Sections: II. Lacunae Number and Mean Size
4.21 Volume-Weighted Mean Nuclear Volume From Point-Sampled Intercepts
4.22 Mean Neuron Volume With the Optical Nucleator
4.23 Volume and Surface Area of an Isolated Object With the Invariator
4.24 Mean Neuron Volume and Surface Area With the Invariator
4.25 Volume-Weighted Mean Grain Volume and Surface Area With the Invariator
4.26 Estimation of the K-Function for Volume, Surface Area, and Number
Chapter 5 Variance Predictors for Systematic Sampling
5.1 Introduction
5.2 Cavalieri SamplingWith Section Areas Measured Exactly
5.3 Cavalieri Section Areas Affected By Local Errors
5.4 Cavalieri Slabs Affected By Local Errors
5.5 The Splitting Design for Cavalieri Slabs
5.6 Precision of the Estimation of Particle Number in the Plane With Systematic Quadrats
5.7 Precision of the Estimation of Planar Curve Length With a Square Grid
5.8 IUR Test Systems to Estimate Planar Area, or Volume: Preliminary Comments
5.9 Isotropic Cavalieri Lines in the Plane
5.10 Isotropic Cavalieri Planes
5.11 Isotropic Square Grid of Test Points in the Plane
5.12 Isotropic Fakir Probe to Estimate Volume
5.13 Isotropic Cubic Grid of Test Points
5.14 Isotropic Cavalieri Stripes in the Plane
5.15 Isotropic Cavalieri Slabs
5.16 Isotropic Systematic Quadrats in the Plane
5.17 Isotropic Grid of Straight Line Segments in the Plane
5.18 Apparent Paradoxes in Geometric Sampling
Appendix
A.1 Prerequisites of Probability and Statistics
A.2 Approximate Mean and Variance of Non-Linear Functions of Random Variables
A.3 The Fourier Transform of rω
List of Notation
References
Author Index
Subject Index