Bayesian estimation of templates in computational anatomy: Difference between revisions

Revision as of 00:13, 11 November 2018

Revision as of 00:07, 11 January 2019

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{{Main|Computational anatomy}}

[[Statistical shape analysis]] and [[Computational anatomy#Statistical shape theory in computational anatomy|statistical shape theory]] in [[computational anatomy]] (CA) is performed relative to templates, therefore it is a local theory of statistics on shape.[[Computational anatomy#Template Estimation from Populations|Template estimation]] in [[computational anatomy]] from populations of observations is a fundamental operation ubiquitous to the discipline. Several methods for template estimation based on [[Bayesian probability|Bayesian]] probability and statistics in the [[Computational anatomy#The random orbit model of computational anatomy|random orbit model of CA]] have emerged for submanifolds<ref>{{Cite journal|title = A Bayesian Generative Model for Surface Template Estimation|url = https://dx.doi.org/10.1155/2010/974957|journal = Journal of Biomedical Imaging|date = 2010-01-01|issn = 1687-4188|pmc = 2946602|pmid = 20885934|pages = 16:1–16:14|volume = 2010|doi = 10.1155/2010/974957|first = Jun|last = Ma|first2 = Michael I.|last2 = Miller|first3 = Laurent|last3 = Younes}}</ref><ref>{{Cite journal|title = Atlas Generation for Subcortical and Ventricular Structures with its Applications in Shape Analysis|journal = IEEE Transactions on Image Processing|date = 2010-06-01|issn = 1057-7149|pmc = 2909363|pmid = 20129863|pages = 1539–1547|volume = 19|issue = 6|doi = 10.1109/TIP.2010.2042099|first = Anqi|last = Qiu|first2 = Timothy|last2 = Brown|first3 = Bruce|last3 = Fischl|first4 = Jun|last4 = Ma|first5 = Michael I.|last5 = Miller}}</ref> and dense image volumes.<ref>{{Cite journal|title = Bayesian Template Estimation in Computational Anatomy|journal = NeuroImage|date = 2008-08-01|issn = 1053-8119|pmc = 2602958|pmid = 18514544|pages = 252–261|volume = 42|issue = 1|doi = 10.1016/j.neuroimage.2008.03.056|first = Jun|last = Ma|first2 = Michael I.|last2 = Miller|first3 = Alain|last3 = Trouvé|first4 = Laurent|last4 = Younes}}</ref>

== The deformable template model of shapes and forms via diffeomorphic group actions ==

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== The Bayes model of computational anatomy ==

The central statistical model of [[computational anatomy]] in the context of [[medical imaging]] is the source-channel model of [[Shannon theory]];<ref>{{Cite journal|title = Statistical methods in computational anatomy|url = http://smm.sagepub.com/content/6/3/267|journal = Statistical Methods in Medical Research|date = 1997-06-01|issn = 0962-2802|pmid = 9339500|pages = 267–299|volume = 6|issue = 3|doi = 10.1177/096228029700600305|language = en|first = Michael|last = Miller|first2 = Ayananshu|last2 = Banerjee|first3 = Gary|last3 = Christensen|first4 = Sarang|last4 = Joshi|first5 = Navin|last5 = Khaneja|first6 = Ulf|last6 = Grenander|first7 = Larissa|last7 = Matejic}}</ref><ref>{{Cite book|title = Pattern Theory: From Representation to Inference|author = U. Grenander and M. I. Miller |url = https://www.amazon.com/Pattern-Theory-Representation-Inference-European/dp/0199297061|publisher = Oxford University Press|date = 2007-02-08|isbn = 9780199297061|language = English}}</ref><ref>{{Cite book|title = Bayesian Multiple Atlas Deformable Templates|author = M. I. Miller and S. Mori and X. Tang and D. Tward and Y. Zhang | series = Brain Mapping: An Encyclopedic Reference|url = https://books.google.com/books?id=ysucBAAAQBAJ|publisher = Academic Press|date = 2015-02-14|isbn = 9780123973160|language = en}}</ref> the source is the deformable template of images <math> I \in \mathcal {I} </math>, the channel outputs are the imaging sensors with observables <math> I^D \in {\mathcal I}^{\mathcal D} </math> . The variation in the anatomical configurations are modelled separately from the Medical imaging modalities [[Computed axial tomography|Computed Axial Tomography]] machine, [[Magnetic resonance imaging|MRI]] machine, [[Positron emission tomography|PET]] machine, and others. The [[Bayes' theorem|Bayes theory]] models the prior on the source of images <math> \pi_{\mathcal{I}} (\cdot) </math> on <math>I \in \mathcal{I} </math>, and the conditional density on the observable imagery <math> p(\cdot |I) \ \text{on} \ I^D \in {\mathcal I}^{\mathcal D} </math>, conditioned on <math> I \in \mathcal{I} </math>. For images with [[Computational anatomy#Groups and group actions|diffeomorphism group action]] <math>I \doteq \phi \cdot I_\mathrm{temp}, \phi \in Diff_V</math>, then the prior on the group <math>\pi_{Diff_V} (\cdot)</math> induces the prior on images <math>\pi_{\mathcal{I}} (\cdot)</math>, written as densities the log-posterior takes the form

:<math>

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:<math>

\hat \theta \doteq \arg \max_{\theta \in \Theta} \log p(\theta \mid I^D). </math>

[[File:Xiaoying Tang ADNI template.png|thumb|Shown are shape templates of amygdala, hippocampus, and ventricle generated from 754 ADNI samples` Top panel denotes the localized surface area group differences between normal aging and Alzheimer disease (positive represents atrophy in Alzheimer whereas negative suggests expansion). Bottom panel denotes the group differences in the annualized rates of change in the localized surface areas (positive represents faster atrophy rates (or slower expansion rates) in Alzheimer whereas negative suggests faster expansion rates (or slower atrophy rates) in Alzheimer); taken from Tang et al.<ref name="Tang 599–611">{{Cite journal|last=Tang|first=Xiaoying|last2=Holland|first2=Dominic|last3=Dale|first3=Anders M.|last4=Younes|first4=Laurent|last5=Miller|first5=Michael I.|date=2015-01-01|title=Baseline Shape Diffeomorphometry Patterns of Subcortical and Ventricular Structures in Predicting Conversion of Mild Cognitive Impairment to Alzheimer’sAlzheimer's Disease|journal=Journal of Alzheimer's Disease|volume=44|issue=2|pages=599–611|doi=10.3233/JAD-141605|issn=1387-2877|pmc=4474004|pmid=25318546}}</ref><ref name="Tang 2093–2117">{{Cite journal|last=Tang|first=Xiaoying|last2=Holland|first2=Dominic|last3=Dale|first3=Anders M.|last4=Younes|first4=Laurent|last5=Miller|first5=Michael I.|last6=for the Alzheimer's Disease Neuroimaging Initiative|date=2015-06-01|title=The diffeomorphometry of regional shape change rates and its relevance to cognitive deterioration in mild cognitive impairment and Alzheimer's disease|url=http://onlinelibrary.wiley.com/doi/10.1002/hbm.22758/abstract|journal=Human Brain Mapping|language=en|volume=36|issue=6|pages=2093–2117|doi=10.1002/hbm.22758|issn=1097-0193|pmc=4474005|pmid=25644981}}</ref><ref name="Tang 645–660">{{Cite journal|last=Tang|first=Xiaoying|last2=Holland|first2=Dominic|last3=Dale|first3=Anders M.|last4=Miller|first4=Michael I.|last5=Alzheimer's Disease Neuroimaging Initiative|date=2015-01-01|title=APOE Affects the Volume and Shape of the Amygdala and the Hippocampus in Mild Cognitive Impairment and Alzheimer's Disease: Age Matters|journal=Journal of Alzheimer's Disease|volume=47|issue=3|pages=645–660|doi=10.3233/JAD-150262|issn=1875-8908|pmid=26401700|pmc=5479937}}</ref>

]]This requires computation of the conditional probabilities <math>p(\theta\mid I^D) = \frac{p(I^D,\theta)}{p(I^D)}</math>. The multiple atlas orbit model randomizes over the denumerable set of atlases <math>\{ I_a, a \in \mathcal{A} \}</math>. The model on images in the orbit take the form of a multi-modal mixture distribution

:<math>p(I^D, \theta) = \textstyle \sum_{a \in \mathcal{A}} p(I^D,\theta\mid I_a) \pi_{\mathcal A}(a) \ .</math>

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== Surface templates for computational neuroanatomy and subcortical structures ==

The study of sub-cortical neroanatomy has been the focus of many studies. Since the original publications by Csernansky and colleagues of hippocampal change in Schizophrenia,<ref>{{Cite journal|last=Csernansky|first=John G.|last2=Joshi|first2=Sarang|last3=Wang|first3=Lei|last4=Haller|first4=John W.|last5=Gado|first5=Mokhtar|last6=Miller|first6=J. Philip|last7=Grenander|first7=Ulf|last8=Miller|first8=Michael I.|date=1998-09-15|title=Hippocampal morphometry in schizophrenia by high dimensional brain mapping|journal=Proceedings of the National Academy of Sciences of the United States of America|volume=95|issue=19|pages=11406–11411|issn=0027-8424|pmc=21655|pmid=9736749|doi=10.1073/pnas.95.19.11406}}</ref><ref>{{Cite journal|last=Csernansky|first=John G.|last2=Wang|first2=Lei|last3=Jones|first3=Donald|last4=Rastogi-Cruz|first4=Devna|last5=Posener|first5=Joel A.|last6=Heydebrand|first6=Gitry|last7=Miller|first7=J. Philip|last8=Miller|first8=Michael I.|date=2002-12-01|title=Hippocampal deformities in schizophrenia characterized by high dimensional brain mapping|journal=The American Journal of Psychiatry|volume=159|issue=12|pages=2000–2006|doi=10.1176/appi.ajp.159.12.2000|issn=0002-953X|pmid=12450948}}</ref><ref>{{Cite journal|last=Wang|first=L.|last2=Joshi|first2=S. C.|last3=Miller|first3=M. I.|last4=Csernansky|first4=J. G.|date=2001-09-01|title=Statistical analysis of hippocampal asymmetry in schizophrenia|journal=NeuroImage|volume=14|issue=3|pages=531–545|doi=10.1006/nimg.2001.0830|issn=1053-8119|pmid=11506528}}</ref><ref>{{Cite journal|last=Csernansky|first=John G.|last2=Schindler|first2=Mathew K.|last3=Splinter|first3=N. Reagan|last4=Wang|first4=Lei|last5=Gado|first5=Mohktar|last6=Selemon|first6=Lynn D.|last7=Rastogi-Cruz|first7=Devna|last8=Posener|first8=Joel A.|last9=Thompson|first9=Paul A.|date=2004-05-01|title=Abnormalities of thalamic volume and shape in schizophrenia|journal=The American Journal of Psychiatry|volume=161|issue=5|pages=896–902|doi=10.1176/appi.ajp.161.5.896|issn=0002-953X|pmid=15121656}}</ref> Alzheimer's disease,<ref>{{Cite journal|last=Csernansky|first=J. G.|last2=Wang|first2=L.|last3=Swank|first3=J.|last4=Miller|first4=J. P.|last5=Gado|first5=M.|last6=McKeel|first6=D.|last7=Miller|first7=M. I.|last8=Morris|first8=J. C.|date=2005-04-15|title=Preclinical detection of Alzheimer's disease: hippocampal shape and volume predict dementia onset in the elderly|journal=NeuroImage|volume=25|issue=3|pages=783–792|doi=10.1016/j.neuroimage.2004.12.036|issn=1053-8119|pmid=15808979}}</ref><ref>{{Cite journal|last=Wang|first=Lei|last2=Miller|first2=J. Philp|last3=Gado|first3=Mokhtar H.|last4=McKeel|first4=Daniel W.|last5=Rothermich|first5=Marcus|last6=Miller|first6=Michael I.|last7=Morris|first7=John C.|last8=Csernansky|first8=John G.|date=2006-03-01|title=Abnormalities of hippocampal surface structure in very mild dementia of the Alzheimer type|journal=NeuroImage|volume=30|issue=1|pages=52–60|doi=10.1016/j.neuroimage.2005.09.017|issn=1053-8119|pmc=2853193|pmid=16243546}}</ref><ref>{{Cite journal|last=Wang|first=Lei|last2=Swank|first2=Jeffrey S.|last3=Glick|first3=Irena E.|last4=Gado|first4=Mokhtar H.|last5=Miller|first5=Michael I.|last6=Morris|first6=John C.|last7=Csernansky|first7=John G.|date=2003-10-01|title=Changes in hippocampal volume and shape across time distinguish dementia of the Alzheimer type from healthy aging|journal=NeuroImage|volume=20|issue=2|pages=667–682|doi=10.1016/S1053-8119(03)00361-6|issn=1053-8119|pmid=14568443}}</ref> and Depression,<ref>{{Cite journal|last=Posener|first=Joel A.|last2=Wang|first2=Lei|last3=Price|first3=Joseph L.|last4=Gado|first4=Mokhtar H.|last5=Province|first5=Michael A.|last6=Miller|first6=Michael I.|last7=Babb|first7=Casey M.|last8=Csernansky|first8=John G.|date=2003-01-01|title=High-dimensional mapping of the hippocampus in depression|journal=The American Journal of Psychiatry|volume=160|issue=1|pages=83–89|doi=10.1176/appi.ajp.160.1.83|issn=0002-953X|pmid=12505805}}</ref><ref>{{Cite journal|last=Munn|first=Melissa A.|last2=Alexopoulos|first2=Jim|last3=Nishino|first3=Tomoyuki|last4=Babb|first4=Casey M.|last5=Flake|first5=Lisa A.|last6=Singer|first6=Tisha|last7=Ratnanather|first7=J. Tilak|last8=Huang|first8=Hongyan|last9=Todd|first9=Richard D.|date=2007-09-01|title=Amygdala Volume Analysis in Female Twins with Major Depression|journal=Biological Psychiatry|volume=62|issue=5|pages=415–422|doi=10.1016/j.biopsych.2006.11.031|issn=0006-3223|pmc=2904677|pmid=17511971}}</ref> many neuroanatomical shape statistical studies have now been completed using templates built from all of the subcortical structures for depression,<ref>{{Cite web|url=https://www.researchgate.net/publication/271515359_Amygdala_and_Hippocampal_in_ADHD_Volumetric_and_Morphometric_Analysis_and_Relation_to_Mood_Symptoms271515359|title=Amygdala and Hippocampal in ADHD: Volumetric and Morphometric Analysis and Relation to Mood Symptoms.|website=ResearchGate|access-date=2016-03-22}}</ref> Alzheimer's,<ref name="Tang 599–611"/><ref name="Tang 2093–2117"/><ref>{{Cite journal|last=Qiu|first=Anqi|last2=Fennema-Notestine|first2=Christine|last3=Dale|first3=Anders M.|last4=Miller|first4=Michael I.|date=2009-04-15|title=Regional shape abnormalities in mild cognitive impairment and Alzheimer's disease|journal=NeuroImage|volume=45|issue=3|pages=656–661|issn=1053-8119|pmc=2847795|pmid=19280688|doi=10.1016/j.neuroimage.2009.01.013}}</ref><ref>{{Cite journal|last=Qiu|first=Anqi|last2=Younes|first2=Laurent|last3=Miller|first3=Michael I.|last4=Csernansky|first4=John G.|date=2008-03-01|title=Parallel Transport in Diffeomorphisms Distinguishes the Time-Dependent Pattern of Hippocampal Surface Deformation due to Healthy Aging and the Dementia of the Alzheimer’sAlzheimer's Type|journal=NeuroImage|volume=40|issue=1|pages=68–76|doi=10.1016/j.neuroimage.2007.11.041|issn=1053-8119|pmc=3517912|pmid=18249009}}</ref><ref>{{Cite journal|last=Miller|first=Michael I.|last2=Younes|first2=Laurent|last3=Ratnanather|first3=J. Tilak|last4=Brown|first4=Timothy|last5=Reigel|first5=Tommy|last6=Trinh|first6=Huong|last7=Tang|first7=Xiaoying|last8=Barker|first8=Peter|last9=Mori|first9=Susumu|date=2012-10-01|title=Amygdala Atrophy in MCI/Alzheimer’sAlzheimer's Disease in the BIOCARD cohort based on Diffeomorphic Morphometry|journal=Medical imageImage computingComputing and computerComputer-assisted interventionIntervention : MICCAI ... International Conference on Medical Image Computing and Computer-Assisted Intervention|volume=2012|pages=155–166|pmc=4063307|pmid=24955432}}</ref><ref>{{Cite journal|last=Miller|first=Michael I.|last2=Ratnanather|first2=J. Tilak|last3=Tward|first3=Daniel J.|last4=Brown|first4=Timothy|last5=Lee|first5=David S.|last6=Ketcha|first6=Michael|last7=Mori|first7=Kanami|last8=Wang|first8=Mei-Cheng|author8-link= Mei-Cheng Wang |last9=Mori|first9=Susumu|date=2015-01-01|title=Network neurodegeneration in Alzheimer’sAlzheimer's disease via MRI based shape diffeomorphometry and high-field atlasing|url=http://journal.frontiersin.org/article/10.3389/fbioe.2015.00054/full|journal=Computational Physiology and Medicine|pages=54|doi=10.3389/fbioe.2015.00054|pmc=4515983|pmid=26284236|volume=3}}</ref> Bipolar disorder, ADHD,<ref>{{Cite journal|last=Qiu|first=Anqi|last2=Crocetti|first2=Deana|last3=Adler|first3=Marcy|last4=Mahone|first4=E. Mark|last5=Denckla|first5=Martha B.|last6=Miller|first6=Michael I.|last7=Mostofsky|first7=Stewart H.|date=2009-01-01|title=Basal Ganglia Volume and Shape in Children With Attention Deficit Hyperactivity Disorder|journal=The American Journal of Psychiatry|volume=166|issue=1|pages=74–82|doi=10.1176/appi.ajp.2008.08030426|issn=0002-953X|pmc=2890266|pmid=19015232}}</ref> autism,<ref>{{Cite web|url=http://www.jaacap.com/article/S0890-8567(10)00248-0/abstract|title=Basal Ganglia Shapes Predict Social, Communication, and Motor Dysfunctions in Boys With Autism Spectrum Disorder - Journal of the American Academy of Child & Adolescent Psychiatry|website=www.jaacap.com|access-date=2016-03-22}}</ref> and Huntington's Disease.<ref>{{Cite journal|last=Younes|first=Laurent|last2=Ratnanather|first2=J. Tilak|last3=Brown|first3=Timothy|last4=Aylward|first4=Elizabeth|last5=Nopoulos|first5=Peg|last6=Johnson|first6=Hans|last7=Magnotta|first7=Vincent A.|last8=Paulsen|first8=Jane S.|last9=Margolis|first9=Russell L.|date=2014-03-01|title=Regionally selective atrophy of subcortical structures in prodromal HD as revealed by statistical shape analysis|journal=Human Brain Mapping|volume=35|issue=3|pages=792–809|doi=10.1002/hbm.22214|issn=1097-0193|pmc=3715588|pmid=23281100}}</ref><ref>{{Cite webjournal|url=http://f1000research.com/posters/1096125|title=f1000research.com/posters/1096125|website=f1000research.com|doi=10.7490/f1000research.1096125.1|access-date=2016-03-22|doi-broken-date=2019-01-11}}</ref> Templates were generated using Bayesian template estimation data back to Ma, Younes and Miller.<ref>{{Cite journal|last=Ma|first=Jun|last2=Miller|first2=Michael I.|last3=Younes|first3=Laurent|date=2010-01-01|title=A Bayesian Generative Model for Surface Template Estimation|journal=International Journal of Biomedical Imaging|volume=2010|doi=10.1155/2010/974957|issn=1687-4188|pmc=2946602|pmid=20885934|pages=1–14}}</ref>

Shown in the accompanying Figure is an example of subcortical structure templates generated from T1-weighted [[Magnetic resonance imaging|magnetic resonance imagery]] by Tang et al.<ref name="Tang 599–611"/><ref name="Tang 2093–2117"/><ref name="Tang 645–660"/> for the study of Alzheimer's disease in the ADNI population of subjects.

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== Surface estimation in cardiac computational anatomy ==

[[File:Siamak atlas.tif|alt=Showing population heart atlases with superimposed hypertrophy.|thumb|Showing population atlases identifying regional differences in radial thickness at end-systolic cardiac phase between patients with hypertrophic cardiomyopathy (left) and hypertensive heart disease (right). Gray mesh shows the common surface template to the population, with the color map representing basilar septal and anterior epicardial wall with larger radial thickness in patients with hypertrophic cardiomyopathy vs. hypertensive heart disease.<ref name="Semantic Scholar">{{Cite web|url=https://www.semanticscholar.org/paper/Shape-analysis-of-hypertrophic-and-hypertensive-Ardekani-Jain/4073fc2af6ef7fb978d5203be3e84c999e1a0dbd|title=Semantic Scholar|website=www.semanticscholar.org|language=en-US|access-date=2016-04-05}}</ref>]]

Numerous studies have now been done on cardiac hypertrophy and the role of the structural integraties in the functional mechanics of the heart. Siamak Ardekani has been working on populations of Cardiac anatomies reconstructing atlas coordinate systems from populations.<ref>{{Cite journal|last=Ardekani|first=Siamak|last2=Weiss|first2=Robert G.|last3=Lardo|first3=Albert C.|last4=George|first4=Richard T.|last5=Lima|first5=Joao A. C.|last6=Wu|first6=Katherine C.|last7=Miller|first7=Michael I.|last8=Winslow|first8=Raimond L.|last9=Younes|first9=Laurent|date=2009-06-01|title=Computational method for identifying and quantifying shape features of human left ventricular remodeling|journal=Annals of Biomedical Engineering|volume=37|issue=6|pages=1043–1054|doi=10.1007/s10439-009-9677-2|issn=1573-9686|pmc=2819012|pmid=19322659}}</ref><ref>{{Cite journal|last=Steinert-Threlkeld|first=Shane|last2=Ardekani|first2=Siamak|last3=Mejino|first3=Jose L. V.|last4=Detwiler|first4=Landon Todd|last5=Brinkley|first5=James F.|last6=Halle|first6=Michael|last7=Kikinis|first7=Ron|last8=Winslow|first8=Raimond L.|last9=Miller|first9=Michael I.|date=2012-06-01|title=Ontological labels for automated ___location of anatomical shape differences|journal=Journal of Biomedical Informatics|volume=45|issue=3|pages=522–527|doi=10.1016/j.jbi.2012.02.013|issn=1532-0480|pmc=3371096|pmid=22490168}}</ref><ref>{{Cite journalbook|last=Ardekani|first=Siamak|last2=Gunter|first2=Geoffrey|last3=Jain|first3=Saurabh|last4=Weiss|first4=Robert G.|last5=Miller|first5=Michael I.|last6=Younes|first6=Laurent|date=2014-01-01|title=Estimating dense cardiac 3D motion using sparse 2D tagged MRI cross-sections|journal=Conference proceedingsProceedings: ... Annual International Conference of the IEEE Engineering in Medicine and Biology Society. IEEE Engineering in Medicine and Biology Society. Annual Conference|volume=2014|pages=5101–5104|doi=10.1109/EMBC.2014.6944772|issn=1557-170X|pmc=4474039|pmid=25571140|isbn=978-1-4244-7929-0}}</ref> The figure on the right shows the computational cardiac anatomy method being used to identify regional differences in radial thickness at end-systolic cardiac phase between patients with hypertrophic cardiomyopathy (left) and hypertensive heart disease (right). Color map that is placed on a common surface template (gray mesh) represents region ( basilar septal and the anterior epicardial wall) that has on average significantly larger radial thickness in patients with hypertrophic cardiomyopathy vs. hypertensive heart disease (reference below).<ref name="Semantic Scholar"/>

== MAP Estimation of volume templates from populations and the EM algorithm ==