Arterial Plaque Characterization Using IVUS Radio-Frequency Time Echoes

The reported yearly worldwide death tolls and especially in the US about cardio-vascular disease is the leading factor that motivated scientists to invest time and money in order to find innovative ways that enable accurate and early detection of such diseases (American Heart Association [AHA], 2006; AHA, 2008). Since CVD occurs within the human body, imaging modalities were invented to present a picture of the artery as close as possible to its real status. These imaging methods accompanied with continuous developments have helped tremendously in the improvement of proactive health care and early interventions before aggravations. Intravascular Ultrasound (IVUS) is one of the cost effective modalities that have been extensively used for diagnostics purposes (Gaster et al., 2003; Mueller et al., 2003). Although IVUS has the advantage of differentiating between all the artery crosssection components in terms of geometry (Nissen & Yock, 2001), the composition and mechanical properties of each component is still a subject of discussion and research. In fact, an objective classification of plaques based on both mechanical and acoustic properties is not reached yet. For instance, echogenicity of luminal tissue in most cases is the same for a plaque composed of high lipid depositions (Stary, 1992; Stary et al., 1994; Stary et al., 1995). Thus ultrasound wave propagation and transmission should be studied fundamentally in the human tissues and specifically through the artery cross-sections. The only way these ultrasonic waves could be studied meticulously is by solving the propagation governing equations, i.e., through the wave equation. Finding the analytical solution of these waves contributes definitely in understanding the propagation fashion and behavior inside the medium. Finding analytical solution of the wave equation is highly dependent on the medium. In the case of artery cross-section in a human body, there exist numerous geometric irregularities. These irregular shaped components of the artery make the wave equation unsolvable analytically. Consequently numerical methods are employed to find discrete solutions of the ultrasound waves. The inability of discriminating most plaque types from the grayscale images has fostered researchers to think about the content and information that the IVUS backscattered radiofrequency (RF) ultrasound waves could offer. Studies have shown that RF signals possess valuable information in terms of plaque composition (Normal, fatty, fibro-fatty, fibrous, fibrous with calcification). It has been noticed that there is a difference between these RF echoes coming from various plaque types (Urbani et al., 1993).


Introduction
The reported yearly worldwide death tolls and especially in the US about cardio-vascular disease is the leading factor that motivated scientists to invest time and money in order to find innovative ways that enable accurate and early detection of such diseases (American Heart Association [AHA], 2006;AHA, 2008). Since CVD occurs within the human body, imaging modalities were invented to present a picture of the artery as close as possible to its real status. These imaging methods accompanied with continuous developments have helped tremendously in the improvement of proactive health care and early interventions before aggravations. Intravascular Ultrasound (IVUS) is one of the cost effective modalities that have been extensively used for diagnostics purposes (Gaster et al., 2003;Mueller et al., 2003). Although IVUS has the advantage of differentiating between all the artery crosssection components in terms of geometry (Nissen & Yock, 2001), the composition and mechanical properties of each component is still a subject of discussion and research. In fact, an objective classification of plaques based on both mechanical and acoustic properties is not reached yet. For instance, echogenicity of luminal tissue in most cases is the same for a plaque composed of high lipid depositions (Stary, 1992;Stary et al., 1994;Stary et al., 1995). Thus ultrasound wave propagation and transmission should be studied fundamentally in the human tissues and specifically through the artery cross-sections. The only way these ultrasonic waves could be studied meticulously is by solving the propagation governing equations, i.e., through the wave equation. Finding the analytical solution of these waves contributes definitely in understanding the propagation fashion and behavior inside the medium. Finding analytical solution of the wave equation is highly dependent on the medium. In the case of artery cross-section in a human body, there exist numerous geometric irregularities. These irregular shaped components of the artery make the wave equation unsolvable analytically. Consequently numerical methods are employed to find discrete solutions of the ultrasound waves. The inability of discriminating most plaque types from the grayscale images has fostered researchers to think about the content and information that the IVUS backscattered radiofrequency (RF) ultrasound waves could offer. Studies have shown that RF signals possess valuable information in terms of plaque composition (Normal, fatty, fibro-fatty, fibrous, fibrous with calcification). It has been noticed that there is a difference between these RF echoes coming from various plaque types (Urbani et al., 1993).
Integrated backscatter IVUS (IB IVUS) was introduced and calculated based on these RF signals. Despite the accurate differentiation between all plaque types (calcification, mixed lesion, fibrous tissue, lipid core and thrombus) using IB IVUS, the angle dependency between ROI's and catheter axis makes this classification unstable and sensitive (Urbani et al., 1993;Picano et al., 1983;Sarnelli et al.,1986;Landini et al., 1986;Barzilai et al., 1987;De Kroon et al., 1991;Picano et al., 1994). Moreover the high resolution dictated for plaque detection necessitates high frequency ultrasound signals which affect the penetration depth of these transmitted ultrasonic pulses. In addition to the use of IB as a determinant parameter by which plaques were classified, new research directions has been inspired from the elastic property that could characterize each plaque type. This gave rise to what is called intravascular elastography (also known as IVUS elastography) (Cespedes et al., 1991). In fact, investigators have taken advantage of the possibility of recording the RF echoes to use them for displacements or strain determination (Ryan & Foster, 1997;De Korte et al., 1998;Schaar et al., 2003;Saijo et al., 2004;Shapo et al., 1996). Despite the striking difference of the strain values for various plaques, most of the studies were performed in vitro where the temperature is different from in vivo case. Additionally excised arteries were used after freezing and thawing which influences the values of elastic modulus. Elastography has also been criticized for its inability of discriminating between normal artery and Fibrous caps (De Korte et al., 2000). The use of the RF echoes content has also been used further. A method called virtual histology (also called VH IVUS) based on these IVUS-RF-ultrasonic waves has been developed. This tissue characterization technique is based on the frequency spectrum analysis (Koenig & Klauss, 2007;Nair et al., 2002). Several limitations accompanied the development of VH IVUS. In fact, virtual histology gives an axial resolution which is too low to detect fibrous cap thickness. In addition the detection between soft plaque material and thrombus is not possible. Despite this simplicity of image construction from IVUS and the development of some signal processing procedures to overcome the lack of plaque characterization, there are significant challenges which limited the accuracy and clarity of the images produced via IVUS. These challenges are always present in the imaging process. For example, the omnipresence of noise related to the acquisition of the ultrasound waves (due to electronic devices) plays an important role on hiding useful information during the detection process. Moreover uncertainties due to sound speed variation, eccentricity of the transducer as well as scattering (related to small particles such as cells, and irregular surfaces) are important factors which contribute to the limitations of the developed IVUS techniques for the identification of vulnerable plaques. This certainly influences the issues of resolution and inability to adequately discriminate between fibrous and lipid-rich plaques. The nature of this imaging modality (irregularities in the geometry of the tissues and movement of the catheter tips) and the sensitivity of the recorded RF echoes have motivated other research groups to work on modeling the ultrasound wave propagation in biological tissues and specifically IVUS. Since the IVUS imaging method is based on ultrasound wave propagation, the only way to sketch a model for this propagation will be dictated from the constitutive laws that govern these waves. To know the complete behavior of the waves towards each tissue component, a solution should be found in time and space for these signals.

163
Nonlinear propagation mathematical models have been introduced. The most widely used model for modeling finite amplitude sound beam propagation is the so-called Khokhlov, Zabolotskaya and Kuznetsov (KZK) equation (Kuznestov, 1971). Numerous methods to solve this non-linear model have been proposed (Lee & Hamilton, 1995;Tavakkoli et al., 1998). One of the difficulties for these numerical models was the computational aspect. Huge memory and supercomputing machines have to be allocated for the implementation of such nonlinear models. Even using advanced equipments in terms of performance, the computation process of these models can take several hours and even days. This is far from simulating a real time propagation of the ultrasonic waves. Given the complexity in solving these nonlinear models, other groups of researchers have resorted to the adoption of simple linear wave equation models (Kendall & Weimin, 2001). In fact ultrasound that is propagating in biological tissues generates small fluctuations. The propagation of these fluctuations is governed by what is called the wave equation. The closed form solution of the wave equation here above is unavailable. Thus two main numerical methods were proposed; which are the finite differences and the finite element methods (Guenther & John, 1996;Kendall & Weimin, 2001). The complexity of the models presented problems in terms of real time simulation and computational burden. A reduced order model called Transmission Line matrix (TLM) method is developed in this chapter to simulate the ultrasound wave propagation. The foundation of the computational method (Transmission Line Matrix Method) that is used to model IVUS in a simple regular medium (rectangular shape) is first developed. A new TLM model in polar coordinates (circular shape to model the artery cross-section) is then outlined. The TLM model will subsequently be modified to model IVUS. The system identification methodology used to construct a parametric model that characterizes a plaque for specific mechanical and acoustic properties is demonstrated in the last part of the chapter.

TLM model
Transmission Line Matrix concept was based on transmission lines (Fig. 1).

Fig. 1. A simple transmission line circuit between points x and x+Δx
The idea of using electrical circuits for TLM came from the analogy that was established between the current or voltage propagating in this line from one point to another and the electromagnetic field (EMF) governing equations (Christos, 1995). Besides this analogy, TLM was based on the Huygens principle where each point of a wave front is regarded as a secondary wave source point and the surface tangent to the secondary wave fronts is used to determine the future position of wave front (Fig. 2). TLM models this physics principle by discretizing the medium into a mesh grid and replacing the wave amplitudes by voltages and currents traveling from one node to another (Fig. 3).

Rectangular TLM
The 2-D TLM method has been widely developed and studied in the literature because of its importance in terms of treating wave propagation in the two dimensional space (Christos, 1995;De Cogan et al., 2006). As mentioned previously the use of TLM method for wave propagation purposes was first inspired from the analogy that was found between the electric circuit variables and the EMF problem. A two dimensional element of a medium of dimensions u and v , is represented by a node intersected by two transmission lines in the x and y directions (Fig. 4). If C is the total capacitance of this element and x L , y L are the total inductances for the two lines in x and y directions respectively, then the voltage/current differential equations are www.intechopen.com where x I and y I are the current traveling in the lines in x and y directions respectively and z V is the total voltage in the element. Recall that the governing equations for EMF and Sound wave problems in the 2-D configuration are the following. ( ( Where E and H are the Electric and magnetic fields, µ and ε are the permeability and permittivity of the space. P is the pressure, x U and y U are the pressure velocity components in x and y directions, ρ is the density of the medium and σ is the compressibility of the medium. The system of equations in (1) Here w is an arbitrary distance inserted to retain the correct dimensionality when dividing x I and y I by v and u respectively (Al- Mukhtar & Sitch, 1981). Comparing equations (2), (3) and (4), the analogy between EMF, sound propagation and electric circuits is established in Table 1. From this analogy the electric and magnetic fields of the EMF problem could be solved through the TLM method by considering them as the transmission line voltage and currents respectively. The same thing applies for the sound wave problem ( In a regular mesh, each node is characterized by equally spaced nodes related to each other by four lines (in the x and y directions) as illustrated in Fig There are applications, such as arteries, when the medium geometry is complex (curved or circular shapes), where the use of the regular TLM method is not appropriate to solve numerically the sound wave equation. This restriction on the shape and size of the mesh affects the capability of the conventional regular TLM method. Moreover the employment of the conventional TLM where the shape is irregular necessitates the use of finer meshes to represent these irregularities accurately. This represents a burden in terms of memory use and run time for numerical computations. Hence a new TLM mesh and model will be developed for the whole artery cross-section in the next section.

Cylindrical TLM
A two dimensional element of a medium having differing dimensions u and v , is represented by a node intersected by two transmission lines in x and y directions (Fig. 6).  Table 1, the following relationships are obtained Time synchronism is conserved in the TLM (Christos, 1995). In fact, the transmitted pulse from one node must reach its surrounding nodes at the same time regardless the lines lengths linking these nodes in all direction ( x and y directions in the 2-D case). This means that the velocity of propagation of the voltage is dependent on the lines lengths. Since the velocity is function of the total inductance and capacitance of the line ( 1 V LC = ), the inductance term is fixed and capacitance is calculated for each direction by taking into account the relationship between inductance, capacitance and velocity.
Given the inductances in both directions, the capacitances per unit length in these lines are obtained.
Due to the irregular mesh grid, the total capacitances found in x and y directions, x C and y C are sometimes less than the total capacitance of the element C . A residual capacitance s C is defined and modeled by an open-circuit stub at each node of the mesh (Fig. 7).

Fig. 7. Stub represented by a new line at each node
The artery cross-section is presented by a disk. The coordinates system is characterized by the angle θ and the radial position r (polar coordinates). In the TLM method, the discretization process of the medium will lead to an irregular gridding (Fig. 8). In general for an arbitrary node that has a radial position r the element is characterized by the following lines lengths Obviously the nodes in this case are not equally spaced. Therefore the TLM model corresponds exactly to the same formulation. Consequently a stub line must be added to compensate for the residual capacitance caused by the length difference of the radial and angular lines of each element in the mesh. The element is characterized by radial and angular impedances and a stub admittance r Z , Z θ and 5 Z .  For both Cartesian and cylindrical geometry, each TLM element is thus characterized by a circuit structure of five transmission lines (Fig. 10). The equivalent Thevenin circuit is composed of five Thevenin sources in series with five equivalent impedances corresponding to the five lines (Fig. 11). Fig. 11. Thevenin equivalent circuit for the 5 lines intersecting an arbitrary node of the irregular mesh The scattering matrix equation in this case is

TLM IVUS model
Ultrasound waves have the same properties as sound waves. They obey the same wave propagation law. In this section a physics-based numerical model is developed using TLM method to mimic the ultrasound wave propagation inside the arterial wall. Both rectangular and cylindrical TLM models will be employed to model IVUS. The first TLM model is used by considering a small portion of the artery cross-section that has a rectangular shape (Fig.  14). However the second TLM model (cylindrical) takes into account the whole artery crosssection geometry. These codes were developed in Matlab. In order to construct the TLM models many basic parameters should be provided. The size of the medium, the gridding rate, the boundary termination, the wave source location and the traveling process are the most important data to be known in the modeling process. The medium is automatically generated from the developed TLM codes once the dimensions are specified. Given the inner and outer radii of the artery cross-section, an automated mesh is generated. Since TLM model is based on the nodes and lines of the discretized medium, the number of nodes in both directions as well as the lines lengths are function of the indicated gridding rates. In the cylindrical TLM model, the choice of the gridding rate is essential for capturing the propagation of the ultrasound wave. The wave should reach the outer edge of the artery cross-section. In fact for an arbitrary mesh the propagation trend of the ultrasound wave is more pronounced in the angular direction of the circular medium. The choice of the mesh grid must be generated in a way that guarantees a conic wave propagation of the TLM method. This means that the propagation should occur in a conic way in the radial direction. The propagation as developed previously in the TLM model depends on the scattering matrix which is function of the lines impedances. These impedances are function of the gridding rates in both radial and angular directions. Thus a specific mesh grid should be developed to ensure this numerical stability. The propagation of the sound wave is expected to be like the one illustrated by Fig. 12. Investigations have shown that the angular deviation of the propagation occurs when the radial position is characterized by 0.633 , after this radial position, the ratio between the gridding rates has this condition 0.8 dr Rd < θ . The mesh grid is designed such that 0.8 dr Rd ≥ θ . By imposing this numerical condition on the mesh grid, the propagation occurs in a conic fashion as illustrated in Fig. 12.

Fig. 12. Conic propagation of the sent wave through the circular medium
As the TLM is a numerical method, the medium should be of a finite size. This is known as medium termination. This is presented by a wall that is characterized by the same acoustic impedance of the terminated medium. For example if the external medium that comes after the artery is air, then the wall that models the termination of air is characterized by the acoustic impedance of the air. For all the developed simulations the surrounding media impedances will be taken in such a way the boundary reflection effect is attenuated so that these reflected components do not influence the wave propagation inside the artery cross-section.
The design of the wave source has been widely studied for the IVUS. In the TLM model, flexibility is given in the code to design any kind of source waves. Since Matlab is the platform where the code was developed, SIMULINK toolbox is used for the source wave specifications. This toolbox offers a variety of signals some of them are illustrated in Fig. 13. In addition to the wide range of source waves, this model gives the possibility of specifying any location in the mesh to be the source point where the ultrasonic wave starts to propagate. For instance the source could be in one node of the edges of the medium, inside the medium or in different nodes at the same time.

Fig. 13. Examples of different source waves designed by Simulink
The propagation process is based in the Huygens principle where for a given node in the mesh, the reflected amplitudes going out at time will serve as the incident waves to the surrounding nodes at time + . For the rectangular and cylindrical TLM models, it is given by equations (12) and (13).     The IVUS TLM models can represent both healthy and abnormal artery cross-sections. In the healthy case the TLM model is described by a medium that has constant acoustic impedance ( = , is the medium density and is the acoustic speed in the medium). However the abnormal artery is modeled by an inclusion that is inserted in the healthy medium. This inclusion is characterized ban acoustic impedance that is different from the one of the healthy medium. Illustrated in Fig. 14 is the case where artery contains a plaque (Yellow colored region).

Fig. 14. Plaque in the artery
The TLM model will consider the plaque as a rectangular object that is inserted inside the artery portion presented by the rectangular medium. The plaque (inclusion) that is formed inside the artery is characterized by different acoustic properties. In fact the ultrasound speed and density are not the same as of the normal tissue (i.e. different acoustic impedances). This will produce a boundary between the healthy portion of the artery and the abnormality edge. This boundary characterizes a change in impedances. A reflection/transmission phenomenon takes place due to this change along the four edges of the plaque. Considering The transmission and reflection that occur on both sides of the boundary are function of these coefficients. Depending on the acoustic characteristics of the plaque the propagation of the wave will show different behavior. As an example to illustrate the effect of the plaque, three different acoustic impedances are taken. A swept sine wave is designed to be the source signal at the node (1 st horizontal, 45 th vertical). The medium is composed of 751 by 76 nodes and the plaque is located between the 20 th and 30 th node in the horizontal direction and the 25 th and 65 th node in the vertical direction (Fig. 15).  Also using the cylindrical TLM model, an inclusion was inserted in the circular medium between the 70 th and 130 th angular nodes and 20 th and 30 th radial nodes. A sine wave is injected from the inner radius (node (1 st radial direction, 100 th angular direction)) and the wave propagation is recorded. It can be shown from Fig. 17, the effect of the acoustic properties of the inclusions that form inside the artery. These models will be employed in the next section to analyze these recorded signals and build a lumped parameter model of the different tissues types based on their size and acoustic properties.

Plaque characterization
This section departs from the traditional approaches appearing in the literature by considering the inclusion within a medium to be a dynamic system. A system identification approach will be adopted to characterize the dynamics of the plaque (Soderstrom & Stoica, 2001). Thus, the TLM models developed in the previous sections are used as a mean of acquiring the time signals transmitted and reflected in a region of interest within the medium (plaque or inclusion location). These signals will be considered as inputs and outputs to the dynamic system where digital signal processing techniques are employed to identify a parametric model for these plaques.

Persistency of excitation of the input signal
Any input signal used in the context of system identification must meet certain condition. In particular, the input signal should be persistently exciting (Ljung, 1999). The frequency domain condition enforcing a persistency of excitation is: Φω is said to be persistently exciting if Thus the spectrum may be zero on a set of number of points (almost all definition).
One classical frequency-rich signal is the swept sine signal which is also known as the "chirp signal". The chirp signal is a single sine wave with a frequency that is changing continuously as a function of time. The general mathematical presentation of the swept sine wave is . The target time, denoted as arg te t t , is the time for which the upper bound of the frequency range is achieved. During the system identification process of the plaque and healthy tissue characterization, a swept sine will be designed as the source signal (the choice of the swept sign came from the fact that this wave was the only signal that was consistent in terms of constancy of the model structure). From literature, the frequency range used in the signal processing for the soft tissue is in the range of 57 10 ,10 Hz   . The swept sine is therefore designed such that the frequency covers this range. This signal will be combined with a zero-element vector. As illustrated in Fig. 18, a swept sine is designed to be the source signal of the simulation part for the TLM models. The sampling time of this signal is equal to 10 -9 sec.

System identification method
This process is composed of two stages. In the first stage, the identification technique that is used for modeling purposes is the so-called Orthogonal Least Squares (OLS) method (Korenberg et al., 1988;Chen et al., 1989). This method is a discrete time domain based approach where the time signals are used for model determination. In fact, the transient time domain information generated from the TLM artery models and their delayed components will be used as possible input regressors. This method will determine the most significant regressors from a broad range of possible candidates to identify the best parametric model. The general model structure can be linear or nonlinear. However for this work, it is anticipated that only a linear model will be needed. A discrete time model generally resembles a polynomial structure. Considering a general case, a discrete system model with unknown coefficients (model parameters to be estimate) can be transformed into the linear-in-the-parameters representation by means of the expansion Given this formulation, a linear least squares problem emerges. The proposed OLS method systematically searches the entire regressor space (monomials) as precised in equation (16) to find the best error reduction set, transforms these optimal regressors into orthogonal components, and then perform final regressor identification based on the new orthogonal system. These estimates will be mapped back to the original parameter estimates with their relative regressors given in equations ,(15) and . (16). This is called Parameter Estimation (PE). Structure Selection (SS) algorithm is then applied where it is anticipated that a linear model will emerge. This procedure takes the estimated parameters and statistically prioritizes these regressors into the most significant regressor, second most significant regressor and so on until accuracy of the model output is realized. The second stage is characterized by the estimation of the model coefficients. A second system identification technique called recursive least squares (RLS) is used (Ljung, 1999). This method departs from a system of linear equations in the matrix form where 1 M×

Θ∈ℜ
is the unknown parameters of the model to be determined, is the information matrix which is function of the input and output vectors. The utility of this method is to calculate the inverse of the information matrix. Based on an algorithm that is composed of three equations, the inverse matrix is simply calculated by means of additions and multiplications. Finally once the inverse of the information matrix is found, the model coefficient vector Θ can be calculated from equation (17).

Tissue characterization using regular and cylindrical irregular TLM models
A set of inputs were designed. This set was composed of a swept sine, a Schroeder wave, a pulse and a band limited white noise. Swept sine and Schroeder wave signals were the more appropriate candidates since they are characterized by their frequency-rich content and their persistent excitation property. During the model structure identification all the considered inputs except the swept sine gave different model structure of the plaque while its properties were changed. That is why, the swept sine signal was considered to be the best input that captures the dynamics of the studied plaques. The OLS algorithm is applied to the input/output sets of data and a discrete parametric model is established for the plaque. The following simple first order linear model is generated for the different plaques where () y k is the model output of the plaque and () uk is the input to the plaque. In order to investigate the effect of plaque acoustic properties variation on the model, a first order continuous model is derived from the input/output data sets d is the DC gain and τ the time constant. The DC gain and the time constant are determined using different approximations of the first derivative of the output, () y t  . These approximations calculations were performed to test the robustness of the model coefficients when recovering continuous model from discrete model. Multiple first time derivate approximations were performed to recover the continuous model calculations (Centered first order derivative approximation, Forward second order derivative approximation, Backward second order derivative approximation and Centered fourth order derivative approximation). The model coefficients were calculated and compared using these approximations and the error variation was negligible. The characterization of the plaque will be linked to the model coefficients and mainly the DC gain. The DC gain is expected to decrease if the impedance is increasing. This is explained by the fact that if the acoustic impedance is going up then the density of the plaque is increasing, meaning that this portion of tissue is getting harder and denser mechanically. A denser material has a repulsive effect and the transmitted signal into it is minor. This means that the denser the material is the more resistive effect it shows. Therefore if the impedance goes up then the DC gain is expected to decrease. In the TLM model, the plaque acoustic impedance is increased gradually and the input/output data sets to this plaque are recorded. The model coefficients (DC gain and pole location) are calculated. Table 2 illustrates the results of the first order model of the plaque that is obtained from the rectangular TLM model. This model is characterized by a medium that is composed of 90 by 300 nodes in the horizontal and vertical directions respectively and a plaque that is located between the 10 th and 30 th horizontal nodes and 25 th and 65 th vertical nodes. The designed swept sine signal is composed of 600 data points with a sampling time that is equal to 10 -9 sec and a frequency range of 47 10 ,10 Hz     . This input was injected at the 1 st horizontal and 45 th vertical node. The 45 th vertical direction coincides with the middle line crossing the plaque. The input (10 th radial position) and output (31 st radial position) signals to the plaque were recorded.  Table 2 confirms that the DC gain is indeed decreasing when the acoustic impedance of the plaque in increased. Moreover, the time constant (indicator of the time response of the system) is decreasing likewise the DC gain when the acoustic impedance is increased. This is explained physically by the fact that the increase of the hardness of the tissue (the plaque) will affect the speed of the system response. The coefficients percentage variations are summarized in Table 3. Using the cylindrical irregular TLM model, the same plaque size and location as of the regular TLM model is considered. The plaque is located between the 10 th and 30 th radial nodes and 25 th and 65 th angular nodes. The same designed swept sine signal is injected at the 1 st radial and 45 th angular node. The 45 th angular direction coincides with the middle line crossing the plaque. The input (10 th radial position) and output (31 st radial position) signals to the plaque are recorded. To study the plaque type effect on the model coefficients, the variation of the DC gain and pole location is studied as function of the plaque density variation. Therefore the speed is set constant and the density of the plaque is varying with respect the healthy tissue density. The same approximations are used to determine the coefficients of the continuous model. As in the case of the regular TLM model section, it is expected that both the DC gain and the time constant to decrease when the acoustic impedance of the plaque is increased and this is what is presented in Table 4. It can be seen from both models that a plaque could be characterized by a first order system. In fact the coefficients variation of this model is inversely proportional to the acoustic impedance variation. The coefficients variations are summarized in Table 5. Considering the model structure found during the tissue characterization, the plaque can be viewed as a first order low pass filter. This first order filter illustrated in Fig. 19 is characterized by a DC gain and a time constant that decrease when the resistance 1 R is increasing.
The resistance 1 R and 2 R can be linked to the acoustic impedance of the medium. From the transfer function above, if 1 R and 2 R increases the DC gain and the time constant decrease.

Conclusions
In this paper, ultrasound wave propagation through biological tissues and specifically through arterial wall was studied. This modeling work was twofold. First it departed from a simple regularly-shaped medium (rectangular geometry) where a transmission line matrix (TLM) model was employed to capture the ultrasound wave propagation. In a second step an irregular TLM model was constructed based on a circular-shaped medium (disk) to simulate IVUS. Both of these models were based on the discretization of the medium where the Huygens principle was applied in the propagation process from one node to another. Moreover the propagating ultrasound waves were recorded in a digitized format in any location at any time specified during the numerical simulation.
Advanced system identification techniques were, then, introduced and applied to characterize tissues. This approach used specific simulated waves in terms of locations to serve as the input/output data sets for the dynamic identification of these regions of interest. This characterization was based upon the construction of parametric models in the form of transfer function where its coefficients are directly related to each plaque type. A first order structure was, thus, found for all plaque types with different DC gain and time constant values depending on the properties of these inclusions. Finally, a quantitative study was performed to link the variation of these two parameters with respect to acoustic properties. It has been shown consequently that these plaque characteristics were identified quantitatively based on the ultrasound waves using the system identification approach.