Fig. 7. 3D IVUS strain palpography of a patient in vivo. A Angiogram of the right coronary artery. B 2D IVUS strain palpogram superimposed on the IVUS echogram, which was taken at the location indicated by arrow in A. The echogram shows an eccentric plaque and the high strain regions with adjacent low strains at the shoulders of this plaque suggest that it is vulnerable. C A map of multiple 2D IVUS strain palpograms stacked after each other. The dotted line corresponds to the palpogram shown in B. This map provides an overview of the deformability of the inner layer of the arterial wall.

Fig. 7. 3D IVUS strain palpography of a patient in vivo. A Angiogram of the right coronary artery. B 2D IVUS strain palpogram superimposed on the IVUS echogram, which was taken at the location indicated by arrow in A. The echogram shows an eccentric plaque and the high strain regions with adjacent low strains at the shoulders of this plaque suggest that it is vulnerable. C A map of multiple 2D IVUS strain palpograms stacked after each other. The dotted line corresponds to the palpogram shown in B. This map provides an overview of the deformability of the inner layer of the arterial wall.

The main conclusion was that 3D IVUS palpography detects vulnerable plaque-specific strain patterns in human coronary arteries that correlated both with clinical presentation and levels of C-reactive protein.

Ultrasound Modulus Elastography

IVUS strain elastography has proven to be a clinically available tool that is able to detect the presence of vulnerable plaques in vitro with high sensitivity and specificity [21]. In vivo animal experiments and in vitro human experiments demonstrated that discrimination between fibrous and fatty plaques is possible [56, 57].

However, a strain elastogram (and also a palpogram) cannot be interpreted directly as a morphology and material composition image of a plaque, since there is no one-to-one relation between the local radial strain value in a strain elastogram and the local plaque component type (calcified, fibrous, fatty or tissue weakened by macrophage inflammation). The underlying reason for this is that the stresses that induce local strain depend upon the structural build-up of the artery, the stiffness (i.e., Young's modulus) and geometry of its plaque components; furthermore, the radial component of the strain depends upon the catheter position used during imaging [60, 61]. Figure 6 exemplifies this. Histology shows a TCFA that consist of a soft homogeneous lipid pool covered by a stiff fibrous cap. Because the lipid pool is soft and homogenous, one would expect high radial strain throughout the same region in the IVUS strain elastogram. However, due to the stiffness of the cap and its circumfer-entially distributed geometry, stress concentrations occur at the shoulders of the lipid pool [41] resulting in local high strain. Furthermore, the presence of the cap hinders deformation behind it, which results in the low-strain region at the center of the lipid pool.

To overcome this limitation, one could image the Young's modulus distribution of a plaque. In general, modulus elastography (also called modulogra-phy) is the name for methods that compute a Young's modulus image (also called a modulogram) from a strain (or displacement) image. The Young's modulus E [kPa] is a material parameter, which can be loosely interpreted as the ratio between the normal stress S [kPa] (tensile or compressive) enforced upon a small block of tissue and its resulting strain (elongation or compression) [62]. The Young's modulus of soft tissue (e.g., lipid pool) is low and of stiff tissue (e.g., media or fibrous cap) high. There are two main reasons for performing modulography: (i) a modulogram can be interpreted as a material composition image because there is large difference between the Young's moduli of various tissue components, including plaque components [63], and (ii) a modulogram shows the modulus of tissue, which is a material property and, therefore, the appearance of the modulogram is independent of the geometry of tissue components, in contrast to strain.

In this section, the technique behind the methods and the results with IVUS modulus elastography are discussed.

The general approach to perform ultrasound modulus elastography is to firstly use ultrasound displacement/strain elastography to measure one or more components of the displacement vector and/or strain components of the deformed tissue. Next, a deformation model for computing the deformation, strain and/or stress of tissue is defined. This model consists of (i) a set of mathematical (partial differential) equations that describe the equilibrium of tissue, (ii) the relation between displacement and strain of tissue and, finally (iii) the constitutive equation, which defines the relation between stress and strain of tissue [62]. Many researchers approximate the behavior of biological tissue by a linear, isotropic, nearly incompressible (Poisson's ratio >0.49) elastic material. In those cases, the constitutive relation contains only one material param eter, namely the Young's modulus. Finally, the deformation model and the measured displacement/strain components are used to compute the modulo-gram by a 'direct reconstruction approach' or by an 'iterative reconstruction approach'.

Direct. In the direct approach the measured displacement/strain data are plugged in the deformation equations, which are mathematically manipulated so that the moduli can be considered and expressed as the unknowns. Next, the moduli are computed using a discretization [64] or numerical integration of the manipulated deformation equations [65, 66].

Iterative. In the iterative approach, the deformation model is treated as a finite element (computer) model (FEM). The FEM fills the space of the tissue with a mesh that consists of small discrete (finite) elements (e.g. triangles, bricks) and each element is given a constitutive relation, i.e. Young's modulus. Next, an initial modulus value for each element is defined. Finally, the modulus value of each individual mesh element or groups of mesh elements in the FEM are iteratively changed such that the computed FEM deformation output eventually closely matches the measured deformation (displacement/strain data). This matching is fully automatically performed by a minimization algorithm [67].

Much research has focused on applying these two approaches on non-vascular tissue geometries such as a cross section of a homogeneous rectangular medium with a circular or rectangular inclusion, or a breast, brain, heart. To date, only a few groups have investigated modulography for vascular geometries. Most of them used an adjusted iterative reconstruction method [68-71] and [72] some others an adjusted direct reconstruction method [73, 74]. All groups encountered difficulties in computing a modulus elastogram (related to uniqueness and continuity), which may be caused by noisy measurements, a limited number of measured displacement/strain components, type of boundary data [75], using an inadequate deformation model for the tissue, non-uniqueness of the inverse problem [76], converging to non-optimal local minima by the minimization algorithm.

Baldewsing et al. [28] focused on performing modulography of atherosclerotic vascular geometries by using a newly developed iterative reconstruction approach with geometric constraints. To this end, they used the arterial radial strain, as measured with IVUS strain elastography, and an a priori parametric plaque geometry model. Their motivation for using a priori information is threefold. Firstly, they want to compute a modulogram of an atherosclerotic plaque that is diagnostically useful and easy to interpret in clinical settings. Secondly, the computation should suffer at least as possible from

Fig. 8. Parametric finite element model for a vulnerable plaque. A Each circle is parameterized by its center (X, Y) and a radius R. The dynamic control points P, Q, R, S, T, and U are used to define the three plaque component regions. B Finite element mesh regions corresponding to geometry in A. Each finite element in a region has the same material property values as other elements in that region. L = Lipid, c = cap.

Fig. 8. Parametric finite element model for a vulnerable plaque. A Each circle is parameterized by its center (X, Y) and a radius R. The dynamic control points P, Q, R, S, T, and U are used to define the three plaque component regions. B Finite element mesh regions corresponding to geometry in A. Each finite element in a region has the same material property values as other elements in that region. L = Lipid, c = cap.

converging problems. Finally, they want to be able to investigate and quantify reconstruction difficulties (uniqueness and continuity) and limitations for computing a modulogram of plaques in a structured manner.

Their approach is specially suited for TCFAs [1, 77]; To this end, the deformation output calculated with a parametric finite element model (PFEM) representation of a TCFA (fig. 8) is matched to the plaque's radial strain, as measured with IVUS strain elastography. The PFEM uses only six morphology and three material composition parameters, but is still able to model a variety of these TCFAs. The computed modulogram of the TCFA shows both the morphology and Young's modulus values of three main plaque components, namely lipid, cap and media, and should therefore be easy to interpret.

In the next three subsections, the main parts of their iterative solution approach are discussed, namely the PFEM for a TCFA, the used deformation model, and used minimization algorithm.

PFEM Geometry for a Plaque. An idealized TCFA [77] is used as a model for a plaque and is an extension of the PFEM model used by Loree et al. [40]; The PFEM geometry consists of a media area containing a lipid pool, which is covered by a fibrous cap. The borders of the lipid, cap and media areas are defined using circles (fig. 8) . Lipid is defined by region QTQ, cap by region PQTSP, and media by the remaining area. Each circle is parameterized by its center with cartesian coordinates (X, Y) and radius R, resulting in a total of six morphology parameters.

Material Deformation Model. Baldewsing et al. [78] used coronary arteries (n = 5) to demonstrate that radial strain elastograms measured in vitro using IVUS strain elastography could be simulated with a (finite-element) computer model. Their material deformation model treated the arterial tissue as a linear elastic, isotropic, plane strain, nearly incompressible material with a Poisson's ratio of 0.4999 [79] .The computer-model geometry and material properties were determined from histology (collagen, smooth muscle cells and macrophages). The agreement between a simulated and measured elastogram was performed upon features of high-strain regions. Statistical tests showed that there was no significant difference between simulated and corresponding measured elastograms in location, surface area and mean strain value of a high strain region (n = 8).

The same material deformation model is used for the PFEM. Lipid, cap and media region are assumed to have a constant Young's modulus value EL, EC, and EM, respectively. This results in a total of only three material composition parameters.

The PFEM radial strain deformation is computed using the finite element package SEPRAN (Sepra Analysis, Technical University Delft, The Netherlands), with the catheter center as origin. This radial strain field is called a PFEM elastogram. The whole process from defining the PFEM morphology and material composition parameters up to the calculation of the PFEM strain elastogram is fully automatic.

Minimization Algorithm. The modulus elastogram of a plaque is determined by a minimization algorithm. This algorithm tries to find values for the six morphology and three modulus parameters of the PFEM such that the corresponding PFEM elastogram 'looks similar' to the measured IVUS strain elastogram. The similarity between elastograms is quantified as the root-mean-squared (RMS) error, between PFEM strain elastogram and measured IVUS strain elastogram. The sequential-quadratic-programming minimization algorithm fully automatically searches a local minimum of the RMS by iteratively updating the nine PFEM parameters. Each update gives a lower RMS error. The algorithm stops when the either the RMS itself or each of a few consecutive RMS values is below a threshold value. When the resulting final PFEM elastogram has qualitatively enough strain pattern features in common with the measured elastogram, the corresponding modulus elastogram is considered as a good approximation of the real plaque.

IVUS Modulus Elastography of Vulnerable Plaques: Results

Baldewsing et al. [28, 80] have shown the feasibility and robustness of their approach by successfully applying their modulography approach to radial strain elastograms of vulnerable plaques that were (a) simulated, (b) mea-

Echogram

Echogram

Young's modulus image at iteration 25

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