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Proc IEEE Int Symp Biomed Imaging. Author manuscript; available in PMC 2010 November 9.

Published in final edited form as:

Proc IEEE Int Symp Biomed Imaging. 2009 August 7; 2009: 366–369.

doi: 10.1109/ISBI.2009.5193060PMCID: PMC2975998

NIHMSID: NIHMS224202

A. A. Joshi,^{1} A. J. Chaudhari,^{2} Changqing Li,^{2} D. W. Shattuck,^{1} J. Dutta,^{3} R. M. Leahy,^{3} and A. W. Toga^{1}

See other articles in PMC that cite the published article.

Estimation of internal mouse anatomy is required for quantitative bioluminescence or fluorescence tomography. However, only surface range data can be recovered from all-optical systems. These data are at times sparse or incomplete. We present a method for fitting an elastically deformable mouse atlas to surface topographic range data acquired by an optical system. In this method, we first match the postures of a deformable atlas and the range data of the mouse being imaged. This is achieved by aligning manually identified landmarks. We then minimize the asymmetric *L*^{2} pseudo-distance between the surface of the deformable atlas and the surface topography range data. Once this registration is accomplished, the internal anatomy of the atlas is transformed to the coordinate system of the range data using elastic energy minimization. We evaluated our method by using it to register a digital mouse atlas to a surface model produced from a manually labeled CT mouse data set. Dice coefficents indicated excellent agreement in the brain and heart, with fair agreement in the kidneys and bladder. We also present example results produced using our method to align the digital mouse atlas to surface range data.

Anatomical atlases with appropriate co-registration schemes can be useful tools for small animal studies involving modalities that are incapable of imaging anatomy. Specifically, in optical fluorescence tomography (OFT) and bioluminescence tomography (BLT) studies in small animals, where estimation of the internal organ optical properties via all-optical techniques is non-trivial [1], deformable anatomical atlases may be used with published optical properties [2]. The atlas must first be aligned with the optical images of the individual mouse being studied.

In optical studies, anatomical images such as those from computed tomography (CT) or magnetic resonance imaging (MRI) are often not acquired due to the inavailability of the scanners in the same facility, concerns such as radiation dose in case of CT, viability and associated cost [3]. The surface geometry of the animal can be estimated using optical systems [4, 5]. These systems typically produce a height map of the animal consisting of discrete points (called range data), contours, or silhouettes that can then be used to generate a 3D representation of the animal volume. For a finite element method (FEM) based solution to the diffusion equation of light propagation, a volumetric tessellation of the animal needs to be generated for the surface map [6]. This process is non-trivial since these range data are often incomplete or under-sampled [7]. Minimization of commonly used symmetric distance metrics such as *L*^{2} or Hausdorff distance [8] is not suitable when the measured range data are incomplete. Asymmetric *L*^{2} pseudo-distance minimization between the two data offers an attractive alternative since the local minima will occur when incomplete surface range data match with part of the complete data set.

In this paper, we present a volumetric registration scheme that warps the Digimouse atlas [9] to an optically-imaged mouse based only on the atlas data and the measured surface topography of the imaged animal. We achieve this registration in three stages:

- 1) The Digimouse is repositioned and its posture is corrected to match the position and posture of the mouse in the acquired data set.
- 2) The posture-corrected atlas is then warped to the available surface topographic data using the asymmetric
*L*^{2}pseudo-distance. - 3) The internal anatomy data in the warped atlas is transformed corresponding its deformed surface. The volumetric tessellation is also transformed using the computed deformation field.

We performed a validation of our method based on micro-CT data acquired from the optically imaged animal.

We used the Digimouse atlas [9] as our anatomical template. The Digimouse was generated using co-registered CT and cryosection images of a 28g normal male nude mouse. Seventeen anatomical structures were labeled in the Digimouse. A corresponding volumetric tetrahedral mesh was provided by the Digimouse atlas. It was generated using the constrained-Delaunay method which conforms to organ boundaries. This mesh contains *N* = 58, 244 vertices and *T* = 306, 773 tetrahedral faces.

We model the atlas mouse body as an elastic volume (Ω) and therefore, deformations to it will be governed by the elastic equilibrium equation, i.e. at equilibrium, the elastic energy *L*(*u*) corresponding to deformation *u* equals the external forces *f* applied on the body [10]:

$$L\left(u\right)=f=-\mathrm{div}\left[(I+\nabla u)\widehat{S}\right]\phantom{\rule{1em}{0ex}}\widehat{S}:\mathrm{\Omega}\to {\propto}^{3},$$

(1)

where *Ŝ* denotes the second Piola-Kirchoff stress tensor defined by *Ŝ* = λTr(*Ĝ*)*I* + 2*μĜ* with $\widehat{G}=\frac{1}{2}({{\nabla}_{u}}^{T}+{\nabla}_{u}+{{\nabla}_{u}}^{T}{\nabla}_{u})$ representing the Green-St. Venant strain tensor. The coefficients λ and *μ* are Lamé's elastic constants. Linearization of (1) using Frèchet derivatives leads to

$$L\left(u\right)=-\mathrm{div}\left(S\right)=f,$$

(2)

where, *S* = λTr(*G*) + 2*μG* is the linearized stress tensor and $G=\frac{1}{2}\phantom{\rule{thinmathspace}{0ex}}({\nabla}_{u}+{{\nabla}_{u}}^{T})$ is the linearized strain tensor [10]. The elasticity operator *L* is discretized using a finite element method. In brief, the equilibrium equation in (2) is formulated by using a variational principle in an energy minimization, which leads to a quadratic form *U ^{T} KU*, where

$${E}_{\mathit{elastic}}\left(U\right)={U}^{T}\mathit{KU}+\alpha \underset{i\in \partial \mathrm{\Omega}}{\Sigma}{\parallel {U}_{i}-{V}_{i}\parallel}^{2},$$

(3)

where α > 0 is a mismatch penalty parameter.

This mathematical formulation allows the whole mouse atlas volume deform elastically when displacements are applied on only the surface nodes of its mesh. The external forces in this formulation are non-zero at the mouse surface Ω and will be used to guide the volumetric deformations in section 2.3 and section 2.4.

There are a wide variety of postures in which mice are imaged at various imaging facilities since a standard has not been established. Typically, the positions of limbs, the position of the head and the orientation of the animal vary greatly. As an initial step, the limbs and head of the Digimouse need to be repositioned to match those of the mouse being imaged. We used a landmark-based method with the elastic mouse model from section 2.2 for posture correction. We selected five landmarks, denoted here by *p _{i}*

$${E}_{s}\left(U\right)={\parallel {\mathrm{\Delta}}_{d}U\parallel}^{2}+\beta \underset{i=1}{\overset{5}{\Sigma}}{({U}_{i}-{W}_{i})}^{2},$$

(4)

where Δ_{d} denotes the discretized Laplacian operator matrix for the atlas surface Ω [11] and *β* > 0 is a mismatch penalty parameter. Let the vector ${U}_{s}^{p}:\{s\in \partial \Omega \}$ be the minimizer of the energy in (4) and Ω* ^{d}* = Ω+

$${E}_{\mathit{elastic}}\left(U\right)={U}^{T}\mathit{KU}+\alpha \underset{s\in \partial \mathrm{\Omega}}{\Sigma}{({U}_{s}-{U}_{s}^{p})}^{2}$$

(5)

leads to a displacement field *U ^{p}* at the volumetric points, that when applied to the atlas Ω, leads to a warping of the internal organs consistent with the warped surface. The posture-corrected atlas is now given by Ω

In order to be able to register the surface topography data recovered from the optical setup to the atlas surface, the matching problem is formulated as an asymmetric *L*^{2} pseudo-distance minimization, where the distance is computed from the incomplete surface to the complete surface. We define the asymmetric *L*^{2} pseudo-distance metric *d* between an incomplete acquired surface *P* and the posture-corrected atlas surface Ω* _{p}* by:

$$d(\partial {\mathrm{\Omega}}_{p},\partial P)=\underset{p\in \partial P}{\Sigma}\left(\underset{e\in \partial \mathrm{\Omega}}{\mathrm{inf}}{\parallel p-e\parallel}^{2}\right).$$

(6)

Our objective, then, is to deform the posture matched atlas surface Ω* _{p}* from section 2.3 such that the distance metric in (6) is minimized. Additionally, we want the displacement field for this operation

$${C}_{s}\left(U\right)=\underset{p\in P}{\Sigma}{\left(\underset{e\in \partial \mathrm{\Omega}}{\mathrm{inf}}d(p,a+{U}_{e})\right)}^{2}+{\parallel {\mathrm{\Delta}}_{d}U\parallel}^{2}$$

(7)

where Δ_{d} denotes the discrete Laplacian [11]. The minimization of the pseudo-distance is performed by a searching strategy over the point-set and results in a displacement vector *U ^{s}*. The displacement field

Similar to section 2.3, the surface warp is then extrapolated to the entire mouse volume using the elastic mouse model. Then, the energy minimization in (3):

$${E}_{\mathit{elastic}}\left(U\right)={U}^{T}\mathit{KU}+\alpha \underset{s\in \partial {\mathrm{\Omega}}_{p}}{\Sigma}{({U}_{p}-{U}_{p}^{s})}^{2}$$

(8)

leads to a displacement field *U ^{m}* at the volumetric points that, when applied to the posture-corrected atlas Ω

The warping method was implemented in MATLAB^{®} . The conjugate gradient minimization procedure was used for the cost function minimizations in sections 2.3 and 2.4. We empirically chose Young's modulus = 1 and Poisson ratio = 0.3 for computing the elasticity matrix *K* [12], *α* = 3, *β* = 1.

A micro-CT scan of an anesthetized normal adult mouse (nu/nu, weight = 26 g) was acquired using the MicroCAT II scanner (Siemens Preclinical Solutions, Knoxville, TN). The experiment was performed under a protocol approved by the University of California - Davis Animal Care and Use Committees. We extracted the mouse surface from the CT image using BrainSuite's [13] mask tool and surface generation tool. We manually erased the nose cone from the binary mask. This mouse surface served as a target topography surface *P* for evaluation. The brain, the heart, the two kidneys and the bladder were segmented manually by an experienced observer using BrainSuite [13].

Our surface profiling scheme used a conical mirror with a horizontal stage that held the animal in an axial orientation within it [14]. A normal anesthetized adult mouse (nu/nu, weight = 24 g) was used for this experiment. The dorsal portion of the animal was scanned by the three line-laser setup. Using geometrical optics formulae, and by translating the laser source horizontally, the complete animal was scanned. The complete procedure lasted 25 min. No data were acquired for the ventral surface.

We used the mouse surface extracted from the CT scan as our target surface. We matched the posture of the Digimouse atlas to that of the target animal using the procedure described in section 2.3. This produced a repositioned and posture-matched atlas. The warping field generated for the tetrahedral mesh was used to resample the labeled CT of the Digimouse using nearest neighbor interpolation. An overlay of the warped labels and the mouse CT is shown in figure 2(b). The asymmetric *L*^{2} pseudo-distance based minimization described in section 2.4 lead to an improved surface and volume warping shown in figure 2(c). The whole method took approximately 20 – 30 min of runtime on a Pentium IV 3.6 GHz machine with 4GB RAM.

Results from the performance evaluation study. In each case, three orthogonal sections from the CT scan of the target are overlaid with the corresponding sections of the Digimouse (a) after only rigid registration, (b) after posture-correction and, (c) **...**

We compared the accuracy of our method by computing Dice coefficients [15] of set overlap between warped atlas labels and manually determined labels for the brain, the heart, the two kidneys and the bladder. Our results are tabulated in Table 1.

Range data were acquired using the optical technique described in section 2.7. Figure 3(a) shows the point cloud representing the top surface of the animal. Figure 3(b) shows an overlay of the point cloud on the posture-corrected Digimouse surface (shown in green). This warping is further improved by the asymmetric *L*^{2} distance minimization procedure to obtain the results in figure 3(c). The internal organs were then elastically warped using the surface as a guide to produce figure 3(d).

We have presented a deformable mouse atlas based registration scheme for estimating internal anatomy of an animal when only surface topography information is available. We evaluated our proposed scheme against results from an anatomical imaging modality. For the heart and the brain, high Dice coefficients between the anatomical image and the warped atlas imply that the head and the chest region of the warped atlas had excellent alignment even though only surface information was used. The bladder and kidneys show significant shape variability across subjects but overlap metric greater than 50% were achieved using the proposed method, even in these regions. The proposed scheme could allow detailed investigation of anatomical variability on optical source reconstruction and facilitate comparisons with methods that derive anatomical information from all-optical techniques. Additionally, the organ map derived from the proposed method can serve as an anatomical prior for diffuse optical tomography.

Exact alignment of mouse surfaces is not enforced in our method since we wanted the deformation of the atlas surface to be smooth. Thus, we have a Sobolev regularizing term in the matching energy equation (4). This term tends to avoid fold-overs in the deformed surface and leads to topologically correct deformations of the mouse atlas. Quasi-rigid and bending priors can be enforced in our method. Such priors allow rigid movement of the bones and the skeleton while still relaxing the rigidity constraints for other organs. In cases where internal anatomical information is available, our proposed method can provide a good initialization for atlas-based registration. Additionally, the proposed method can provide a good initialization for segmentation in cases where an intensity-based anatomical image such as CT or MR is available.

This work was supported by the National Cancer Institute under grants R01CA121783, R44CA138243 and National Institutes of Health through the NIH Roadmap for Medical Research, grant U54 RR021813.

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