Elsevier

Medical Image Analysis

Volume 11, Issue 6, December 2007, Pages 648-662
Medical Image Analysis

Efficient computational fluid dynamics mesh generation by image registration

https://doi.org/10.1016/j.media.2007.06.011Get rights and content

Abstract

Most implementations of computational fluid dynamics (CFD) solutions require a discretisation or meshing of the solution domain. The production from a medical image of a computationally efficient mesh representing the structures of interest can be time consuming and labour-intensive, and remains a major bottleneck in the clinical application of CFD. This paper presents a method for deriving a patient-specific mesh from a medical image. The method uses volumetric registration of a pseudo-image, produced from an idealised template mesh, with the medical image. The registration algorithm used is robust and computationally efficient. The accuracy of the new algorithm is measured in terms of the distance between a registered surface and a known surface, for image data derived from casts of the lumen of two different vessels. The true surface is identified by laser profiling. The average distance between the surface points measured by the laser profiler and the surface of the mapped mesh is better than 0.2 mm. For the images analysed, the new algorithm is shown to be 2–3 times more accurate than a standard published algorithm based on maximising normalised mutual information. Computation times are ∼18 times faster for the new algorithm than the standard algorithm. Examples of the use of the algorithm on two clinical examples are also given. The registration methodology lends itself immediately to the construction of dynamic mesh models in which vessel wall motion is obtained directly using registration.

Introduction

It is increasingly recognized that there is strong correlation between haemodynamic parameters such as wall shear stress and the state of health of the vasculature. Direct visualisation of flow in 3D is difficult in the clinical environment, particularly on the requisite length scales. Computational fluid dynamics (CFD) permits the computation of detailed flow fields given appropriate boundary conditions. Most implementations of CFD solutions require a discretisation or meshing of the fluid domain. The production from a medical image of an appropriate mesh for CFD remains a major bottleneck in the clinical application of CFD. The quality of the CFD solution depends on the mesh. One critical property is the density of the mesh and how this density varies throughout the mesh. Another is how geometrically regular the mesh elements are. In general, in the context of a patient-specific simulation, it is necessary to start from a segmented image, and the segmentation process is often non-trivial and sometimes labour-intensive. Sophisticated mesh generation tools are available, often supported within commercial CFD suites, but the whole procedure needs to be streamlined and automated as far as is possible if patient-specific CFD is to become viable in clinical practice. One of the advantages of the method proposed in this paper is that segmentation can be achieved automatically as part of the image registration process.

This paper describes the application of a novel image registration algorithm, based on the concepts of optical flow but with the addition of a new method for dealing with amplitude differences between images, which facilitates the construction of computational meshes of patient-specific vasculature from a template mesh. A primary advantage of the image registration algorithm described in this paper is that of speed. Section 2 briefly describes the image registration algorithm, how a reference image is constructed from the template mesh and how image registration is used to produce a patient-specific mesh from the template. Section 3 explains the methodology used to validate the algorithm, the comparison of the surfaces generated from images of phantoms of the vasculature with measurements made by laser profiling. Section 4 presents and discusses the results, including comparison with a widely accepted registration algorithm based on an information theoretic cost function (Normalised Mutual Information, NMI). Section 5 demonstrates use on real data. Section 6 draws conclusions about the usefulness of this approach to segmentation and meshing. More details of the registration algorithm are given in Appendix.

This paper does not discuss in detail how the template mesh might be produced, but the important fact is that this needs be done only once. There is an inherent assumption in the current approach that the fundamental characteristics of the template mesh (such as an increase in mesh density in areas of high velocity or pressure gradients and the quality of the individual mesh elements) are preserved under the mapping operations derived from the image registration. Theoretical considerations based on the smoothness of the mappings and actual assessment of mesh quality suggest that this assumption is generally justified.

Steinman and Taylor, 2005, Steinman, 2002 present reviews of patient-specific simulations of the vasculature. There are two broad approaches to the creation of a patient-specific model. The first is to segment the image, identify the vessel and then mesh. Examples of this approach include those presented by Antiga et al., 2003, Cebral et al., 2002, Cebral et al., 2005. In each case a degree of manual intervention is required.

An alternative approach, closer to the methods proposed in this paper, is to deform a model already based on a mesh to fit the vessel. Baghdadi et al. (2005) adopt this approach to deform a template surface mesh produced from a baseline study to monitor changes of the vasculature in follow-up studies. After a manual alignment step to improve robustness, virtual forces computed from image gradients are used to register the template surface to the vessel surface. A smoothing operation is carried out, and a volume mesh is produced by standard methods. Wolters et al. (2005) segment the abdominal aorta by inflating a tubular template model aligned with the vessel centreline to fit the previously segmented surface (represented as a triangular surface mesh), preserving mesh quality by smoothing.

The methodology described in this paper is based on image registration. Zitova and Flusser (2003) give a comprehensive review of the principles of image registration. Maintz and Viergever, 1998, Hill et al., 2001 concentrate on rigid or affine registration of medical images. Lester and Arridge, 1999, Crum et al., 2004 focus on non-linear registration. Although most studies have focused on the brain, there is increasing interest in non-linear registration for other organs, often in the context of motion correction (McLeish et al., 2002, Rohlfing et al., 2004, Slomka et al., 2004). The principal practical issue is how the mapping which achieves registration of the images can be derived. The general approach is to define a cost function which will take some extreme value when the images are in registration. For example, if corresponding points on two images are defined by landmarks or other extracted features e.g. Maintz et al. (1996) and in the specific context of vasculature Aylward et al. (2003), then the cost function would be based on how well these points match after registration.

Many approaches to registration use a cost function based on voxel intensities, several of which have been proposed. The simplest, the sum of the square of the differences of voxel intensity (SSD) assumes that when the images are registered the value of this cost function (ignoring noise) is zero. Only a few types of images can be reliably registered using unmodified SSD. If the SSD cannot be reduced to zero, i.e. there are significant differences in voxel intensity between the two images at corresponding anatomical points, then minimising the SSD may not produce the best registration. Cost functions based on information theory (Viola and Wells, 1995, Collignon et al., 1995) appear to provide a solution to the problem of differing intensities, and have been used successfully to register images from different modalities, which SSD cannot do. Pluim et al. (2003) reviewed the use of mutual information based cost functions in medical image registration. Pluim et al. (2004) investigated the performance of a variety of information theoretic related cost functions, of which normalised mutual information (NMI) is one.

The registration mapping is a function which defines the position of corresponding voxels in the images being registered. Generally, mappings are defined by a combination of parameters and basis functions, where the parameters are the coefficients of the basis functions in the mapping. The basis functions can be global (Friston et al., 1995, Barber et al., 1995) or local (Barber, 1999, Vemuri et al., 1998). For non-linear mappings the resolution of the mapping can in principle go down to the level of the voxel dimensions (Stefanescu et al., 2004). The registration mapping is computed by minimising (or maximising) the cost function with respect to the parameters. The speed of registration is largely dependent on the efficiency of this process. For general cost functions the minimum is invariably found using a function minimisation (maximisation) algorithm and these approaches can be computationally costly (Crum et al., 2004), largely because the cost function has to be evaluated many times. Improvements in speed may be obtained through parallel implementations (Stefanescu et al., 2004). In some circumstances degenerate, singular or non-unique mappings are produced; one approach which has shown promise in dealing with this problem is to model the deformation as a fluid flow problem driven by image derived forces. The physical model ensures that the mapping behaves properly. D’Agostino et al. (2003) derived a formulation of this approach using mutual information.

The computational cost is important; long registration times may make use impractical in the clinical environment. A computationally efficient algorithm based on ‘optical flow’ (Horn and Schunck, 1981) has been proposed (Barber, 1992, Barber et al., 1995, Barber, 1999, Hayton et al., 1999, Dougherty et al., 1999). This method can be formulated in terms of a SSD cost function. A significant limitation of the original optical flow concept, that intensity values of points in the image did not change as the images moved, was relaxed by Gennert and Negahdaripour (1987). Extending the cost function to cope with differences in intensity between the images being registered can help to overcome the fact that the SSD cost function can be sensitive to such differences. An example of the use of an earlier version of the algorithm described in the current paper for correcting image motion is given by Froh et al., 2006, Martel et al., 2007.

Section snippets

Methodology

There are four principal steps in the generation of a patient-specific mesh using the template approach proposed in this paper. These are

  • Building the reference mesh.

  • Building the reference image.

  • Registering the patient image to the reference image.

  • Mapping the reference mesh to the patient image.

Template models were constructed from geometric primitives, and meshes were produced using ANSYS. Four examples are described and three illustrated (Figs. 1a, 7a and 9a).

A binary image v is formed from

Experimental data

Dental material casts (3M ESPE Garant II Dental Impression Media, 3M-ESPE, St Paul, MN, USA) of the lumen of two vessels, one from a bovine carotid artery bifurcation and one from a human aorta at the junction with the superior mesenteric artery, were prepared. The casts were supported by taut suspension in a supporting frame. The models were imaged using a 3D MR sequence. A ‘gold standard’ characterization of their geometry was provided by a laser surface-profiling system, which generated a

D and λ

Fig. 6 shows values of dmean for D = 2 and 4 and for a variety of λ for both sets of model data. Some selected results are tabulated in Table 1 (SSD). The results given in Fig. 6 and Table 1 suggest that, for the model data at least, the exact choice of D and λ is not critical. It is worth recalling that the resolution of the MR images is ∼1 mm and the voxel dimension is 0.488 mm. The values of dmean are of the order of 30–40% of the voxel dimensions and 20% of the resolution of the MR image. This

Clinical study

A complete analysis of blood flow should include the temporal variation of flow within the cardiac cycle. The vessel will change shape as flow and pressure change in the vessel and a full analysis would need to model both flow and the mechanical behaviour of the vessel walls. Such an analysis, though technically possible, will require patient-specific knowledge of the mechanical properties of the vessel wall tissue and this may not be available. If the wall motion can be measured from gated

Discussion and conclusions

A registration algorithm based on the concepts of optical flow has been developed which allows the registration of idealised geometric meshes of anatomy to patient images, so generating patient-specific meshes. In the present work the emphasis has been on the construction of meshes for flow computation in blood vessels, and the accuracy of this application has been validated using physical models of blood vessels. Elsewhere the use of this approach for constructing finite-element models of the

Acknowledgements

The authors wish to thank Dr. P.V. Lawford for making the casts of the bovine aorta and human carotid, C. Sinclair and S. Olley for scanning the casts in the laser profiler and R. Gillott for performing the MR scans on these models.

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