Computational anatomy: an emerging discipline

This paper studies mathematical methods in the emerging new discipline of <italic>Computational Anatomy</italic>. Herein we formalize the Brown/Washington University model of anatomy following the global pattern theory introduced in [1, 2], in which anatomies are represented as deformable templates, collections of 0, 1, 2, 3-dimensional manifolds. Typical structure is carried by the template with the variabilities accommodated via the application of random transformations to the background manifolds. The anatomical model is a quadruple <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis normal upper Omega comma upper H comma upper I comma upper P right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo>,</mml:mo> <mml:mi>H</mml:mi> <mml:mo>,</mml:mo> <mml:mi>I</mml:mi> <mml:mo>,</mml:mo> <mml:mi>P</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\left ( \Omega , H, I, P \right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the background space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Omega ModifyingAbove equals With dot upper U Subscript alpha Baseline upper M Subscript alpha"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mover> <mml:mo>=</mml:mo> <mml:mo>˙</mml:mo> </mml:mover> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>α</mml:mi> </mml:msub> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>M</mml:mi> <mml:mi>α</mml:mi> </mml:msub> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\Omega \dot = {U_\alpha }{M_\alpha }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of 0, 1, 2, 3-dimensional manifolds, the set of diffeomorphic transformations on the background space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H colon normal upper Omega left-right-arrow normal upper Omega"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>H</mml:mi> </mml:mrow> <mml:mo>:</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo stretchy="false">↔</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">{H} : \Omega \leftrightarrow \Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the space of idealized medical imagery <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I"> <mml:semantics> <mml:mi>I</mml:mi> <mml:annotation encoding="application/x-tex">I</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper P"> <mml:semantics> <mml:mi>P</mml:mi> <mml:annotation encoding="application/x-tex">P</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the family of probability measures on <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The group of diffeomorphic transformations <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper H"> <mml:semantics> <mml:mi>H</mml:mi> <mml:annotation encoding="application/x-tex">H</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is chosen to be rich enough so that a large family of shapes may be generated with the topologies of the template maintained. For <italic>normal anatomy</italic> one deformable template is studied, with <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis normal upper Omega comma upper H comma upper I right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi mathvariant="normal">Ω</mml:mi> <mml:mo>,</mml:mo> <mml:mi>H</mml:mi> <mml:mo>,</mml:mo> <mml:mi>I</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\left ( \Omega , H, I \right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> corresponding to a <italic>homogeneous space</italic> [3], in that it can be completely generated from one of its elements, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I equals upper H upper I Subscript t e m p Baseline comma upper I Subscript t e m p Baseline element-of upper I"> <mml:semantics> <mml:mrow> <mml:mi>I</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>H</mml:mi> <mml:msub> <mml:mi>I</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mi>e</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>I</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mi>e</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo>∈</mml:mo> <mml:mi>I</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">I = {HI_{temp}}, {I_{temp}} \in I</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. For <italic>disease</italic>, a family of templates <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper U Subscript alpha Baseline upper I Subscript t e m p Superscript alpha"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>α</mml:mi> </mml:msub> </mml:mrow> <mml:msubsup> <mml:mi>I</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mi>e</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> <mml:mi>α</mml:mi> </mml:msubsup> </mml:mrow> <mml:annotation encoding="application/x-tex">{U_\alpha }I_{temp}^\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are introduced of perhaps varying dimensional transformation classes. The complete anatomy is a collection of homogeneous spaces <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper I Subscript t o t a l Baseline equals upper U Subscript alpha Baseline left-parenthesis upper I Superscript alpha Baseline comma upper H Superscript alpha Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>I</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>t</mml:mi> <mml:mi>o</mml:mi> <mml:mi>t</mml:mi> <mml:mi>a</mml:mi> <mml:mi>l</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msub> <mml:mi>U</mml:mi> <mml:mi>α</mml:mi> </mml:msub> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>I</mml:mi> <mml:mi>α</mml:mi> </mml:msup> </mml:mrow> <mml:mo>,</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>α</mml:mi> </mml:msup> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">{I_{total}} = {U_\alpha }\left ( {I^\alpha }, {H^\alpha } \right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.