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Correspondence Noninvasive Young’s Modulus Evaluation of Tissues Surrounding Pulsatile Vessels Using Ultrasound Doppler Measurement Simone Balocco, Olivier Basset, Guy Courbebaisse, Enrico Boni, Piero Tortoli, and Christian Cachard Abstract—This paper presents an indirect approach to estimating the mechanical properties of tissues surrounding the arterial vessels using ultrasound (US) Doppler measurements combined with an inverse problem-solving method. The geometry of the structure and the dynamic behavior of the inner fluid are first evaluated using a novel dualbeam US system. A numerical phantom associated with a parametric finite element simulator that calculates the hydrodynamic pressure and the displacement on the walls’ boundaries is then built. The simulation results are iteratively compared to the US measurement results to deduce the value of the unknown parameters, i.e., the Young’s modulus and the pressure resulting from the downstream load. The feasibility of the proposed approach was experimentally tested in vitro using a phantom composed of a latex tube surrounded by a cryogel tissue-mimicking material.

I. Introduction nflammation of arterial walls in humans, often responsible for the development of atherosclerotic plaques, is characterized by a change in their elasticity [1]. Periodically monitoring the vessels’ mechanical properties could help physicians classify a patient’s disease [2], but performing this task noninvasively is difficult. During routine patient examination, physicians typically measure the systolic-diastolic arterial diameter and the intima-media thickness (IMT) [3], and empirically derive the Young’s modulus from these parameters, as proposed by Selzer [4]. The local pressure can be assessed through a pressure catheter inserted into the artery. However, the invasive nature of this technique could influence local flow conditions and vessel geometry, making the technique unsuitable for routine examination. The use of pressure waveforms assessed at other sites in the arterial tree is questionable because the pressure waveform changes with the location in the arterial tree. To solve this problem, the pressure waveform can be derived from the measured diameter wave-

I

Manuscript received March 26, 2006; accepted January 22, 2007. This work was supported by the Italian Ministry of Education, University and research (MIUR - PRIN 2005) and the 2006 BQR of INSA Lyon. S. Balocco, O. Basset, and C. Cachard are with CREATIS, Universit´ e de Lyon, Universit´e Lyon 1, CNRS UMR 5220, INSERM U630, Villeurbanne, France (e-mail: [email protected]). G. Courbebaisse is with CREATIS, INSA Lyon, CNRS UMR 5220, INSERM U630, Villeurbanne, France. S. Balocco, E. Boni, and P. Tortoli are with the Microelectronic Systems Design Laboratory, Universit` a di Firenze, Italy. Digital Object Identifier 10.1109/TUFFC.2007.379

form by using an empirical exponential relationship between them [5]. Brands [6] and Zhang [7] have suggested evaluating the Young’s modulus by indirectly measuring the pulse wave velocity. This noninvasive method can assess the segmental arterial stiffness, but the precision of the approach is affected by path length measurement errors and by waveform distortion with pulse propagation [8]. Ultrasound (US) elastographic techniques have recently been proposed for superficial arteries [9] or by means of an intravascular approach [10]. The latter method is less affected by patient or tissue movements than the classic elastographic approach but presents the same disadvantages as the pressure catheter. Doyley [11] presented an inverse elastography reconstruction technique based on a modified Newton Raphson iterative scheme and a finite element method that is used for computing the spatial distribution of the Young’s modulus from within soft tissues. Since this indirect method cannot express the Young’s modulus in a closed form, the initial condition employed to solve the iterative process becomes relevant. Another noninvasive US method has recently been introduced [12], proposing a multilayer model based on an iterative inverse problem solution. Although the method can only reconstruct the relative Young’s modulus of each layer and assumes that the clot cross-section is circular, the convergence of the minimization procedure is fast, and preliminary results look promising. This study presents an indirect noninvasive approach to quantifying the mechanical properties of a vessel’s surrounding tissue, consisting of the arterial membrane, the possible plaque, and the muscle tissues around the artery, using US Doppler measurements and an inverse problemsolving method. The Young’s modulus is proposed to represent the composite stiffness of the region as a ratio between stress and strains. To reach this goal, an artery model, capable of reproducing the complex fluid-structure interaction between the blood and the tissue, is first introduced. The model is based on the following measured physiological data: the fluid velocity profile, the vessel wall displacement, and the vessel geometry, which are simultaneously acquired with a novel dual-beam US system [13], [14]. A parametric finite element simulator then iteratively compares the simulated wall displacement to measured data and deduces the values of the two unknown parameters, the Young’s modulus of the surrounding tissue and the diastolic pressure value resulting from the downstream load. The next section introduces the physical model and the minimization method. Section III describes an in vitro validation experiment conducted on a vessel phantom. The results are discussed in Section IV.

c 2007 IEEE 0885–3010/$25.00 

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II. Modeling Approach In pulsatile flow conditions, arterial wall displacement ∆H(t) can be expressed at any point of the tube boundary as a function of the fluid’s hydrodynamic pressure p, the Young’s modulus E, and the thickness H of the surrounding tissue according to Hooke’s law [15]. The hydrodynamic pressure can be related to the flow velocity profile in the vessel through the Navier-Stokes equation, provided that the initial pressure offset resulting from the downstream load P0 , the viscosity η, and the mass density ρ of the fluid are known. Accordingly, the wall displacement at time t is described by ∆H(t) = Φ{H, E, Ψ(v(r, t), P0 , ρ, η)},

(1)

where Φ is a function describing Hooke’s law, Ψ is a function reporting the hydrodynamic pressure map p[(x, y, z); t] inside the tube from the Navier-Stokes equation [16], when v(r, t) is the instantaneous fluid velocity in the inflow section at a distance r from the vessel axis. As reported in the sections below, the tissue thickness, the velocity profile, and the wall displacement ∆Hexp are experimentally assessed using US measurements. The Young’s modulus and the pressure remain the only unknown variables of the problem. A finite element simulation tool, based on a combined fluid dynamics and mechanical model, calculates the theoretical wall displacement waveform ∆H(t) for a geometrical shape and inflow conditions equal to the experimental conditions. Once the simulation is initialized with arbitrary values of E and P0 , an iterative minimization method is implemented to find the pair of parameters minimizing the distance between ∆Hexp (t) and ∆H(t). This approach is described in more detail in the following sections. A. Ultrasound Measurements The geometry of the investigated structure is reconstructed through B-mode US imaging. H is the distance between the artery boundary and the US transducer. Since the wall displacement ∆Hexp (t) due to pulsation is typically negligible compared to the distance H between the transducer and the artery, the value of H can be considered as constant in (1) [17]. As detailed in Section III, the fluid velocity profile v(r, t) and the amplitude of the time variation of tissue thickness ∆H(t) in a section of the tube are assessed through an US dual-beam multigate Doppler system [14]. In this type of system, one US beam is perpendicularly oriented to the vessel to optimally detect the wall movement. The second beam is oriented at a suitable Doppler angle to simultaneously measure the fluid velocities. B. Geometrical Model The geometry considered in our model is shown in Fig. 1. The inner part of the cylinder corresponds to the

Fig. 1. Geometrical model of the vessel; longitudinal (left) and transverse (right) views. The physical dimensions H and D are measured from ultrasound B-mode images.

blood vessel and the outer part represents a homogeneous surrounding tissue. Let D be the diameter of the artery, H the surrounding tissue thickness, L the vessel length,  and r = y 2 + z 2 the radial coordinate of a point in the vessel. The finite element simulation tool (FEMLAB [18]) provides a meshed representation of the three-dimensional (3-D) structure and computes the dynamic displacement of each node of the mesh. The resolution of the mesh must be compatible with the resolution of the wall displacement measurement. C. Fluid Dynamics Model and Mechanical Analysis Inside the vessel, a fluid dynamics model is applied to calculate the fluid kinetics. To consider the effects of turbulence and possibly complex geometry tube shapes, a generic model based on the Navier-Stokes nonlinear fluiddynamics equation was implemented. In each point of the 3-D space, this model describes the movement of incompressible fluids using ρ

  ∂v − ∇η ∇v + (∇v)T + ρ(∇v)v + ∇p = 0, ∂t

(2)

where v is the velocity of an infinitesimal mass element and T indicates the transposed operation. By solving the discrete Navier-Stokes equations, the pressure variations p[(x, y, z); t] and the velocities v[(x, y, z); t] of the fluid in any node of the geometrical mesh for each time step of the pulsating cycle can be calculated. These equations are used to model the flow conditions for any geometry. In the specific case of a Newtonian flow and laminar inflow conditions in a cylindrical tube, the nonlinear term of the Navier-Stokes equation is equal to zero, and the flow along the tube is a Hagen-Poiseuille laminar flow. In this case, (2) can be conveniently described using cylindrical polar coordinates x, r, ϑ, where ϑ = (z/y):   1 ∂ ∂v 1 ∂2v ρ dp r + 2 2 = . (3) r ∂r ∂r r ∂ϑ η dx Let us now assume that the flow has a radial symmetry, so that v is a function of r only; then (∂ 2 v/∂ϑ2 ) = 0 and (3) becomes:   1 ∂ ∂v ρ dp r = . (4) r ∂r ∂r η dx

balocco et al.: young’s modulus evaluation of tissues surrounding pulsatile vessels

Finally, for a Newtonian flow in a cylindrical tube, the general Navier-Stokes equation can be simplified to  dp 1ρ 2 v=− R − r2 , 4η dx

with λ0 =

(5)

which represents the parabolic velocity profile typical of the Hagen-Poiseuille flow [16]. The pulsating time-dependent velocity profile acquired through experimental Doppler measurements, v(r, t), is defined as an inflow condition, whereas the hydraulic downstream pressure P0 is assigned to the outlet boundary. Following the energy conservation principle, the hydrodynamic pressure p[(x, y, z); t] calculated at each point of the surface wall of the tube is equal to the normal mechanical pressure σ(t) applied to the surrounding tissue boundaries. The arterial wall kinetics is governed by Hooke’s law for an elastic isotropic material: σ(t) E= . ∆H(t)/H

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D. Minimization Method The wall displacement obtained from the finite element simulation is a function of two unknown constants, the hydraulic downstream pressure P0 and the Young’s modulus E. For each set of P0 and E values, the resulting simulated displacement ∆Hopt is compared to the experimental measurements ∆Hexp . An iterative gradient descent method [19] is used to determine the values of the variables E and P0 , which minimizes the distance between the experimental measurement and the simulated values of wall displacement. The function to minimize is the following:  d(E, P0 ) = max |max (∆Hexp ) − max (∆Hopt (E, P0 ))| ,  |min (∆Hexp ) − min (∆Hopt (E, P0 ))| , (7) where δHexp (t) is the vector of the experimental US measurements of the wall displacement, ∆Hopt (t) is the vector of the finite element displacement, and max(∆H) returns the maximal value of ∆H during a cycle of the pulsating wave. At each iteration, the variables’ values are updated as follows:  k k  k+1 K+1   k k  k ∇d E , P0  E , P0 = E , P0 − λ  , ∇d E k , P0k  (8) where ∇ is the gradient of the function, k is the iteration step, and λk is a value setting the algorithm convergence velocity. The variable λk is adapted at each step in order to fasten the convergence and is chosen as follows: k+1     λ = 1.1 · λk ⇔ d E k+1 , P0k+1  < d E k , P0k  , λk+1 = λk /2 ⇔ d E k+1 , P0k+1 ≥ d E k , P0k (9)

(10)

To obtain a quick convergence and to be sure to converge to the global minimum, the initial values of E and P0 need to be initialized as close as possible to the solution [11]. The chosen initial values of E and P0 are discussed in Section III. In vivo experimental reference values were found in the medical literature [20]. Since the function ∆Hopt (t) comes from the finite element simulation, the gradient could not be analytically calculated but is obtained as a differential coefficient: ∂d d(E + ∆E, P0 ) − d(E, P0 ) = , ∂E (E + ∆E, P0 ) − (E, P0 ) ∂d d(E, P0 + ∆P0 ) − d(E, P0 ) , = ∂P0 (E, P0 + ∆P0 ) − (E, P0 )

(6)

The external boundaries are assumed to be constrained in the normal and tangential directions; the vessel boundaries are free to move only in the direction normal to the vessel axis.

1 . ∇d (E 0 , P00 )

(11) (12)

where ∆P0 =

1 1 and ∆E = E. 100 100

(13)

III. Experimental Measurements and Young’s Modulus Estimation A. Experimental Setup In order to reproduce the dynamic behavior of a human artery and to simulate the tissue resistance to the vessel dilatation [21], a 100-mm-long cylindrical-holed phantom made of cryogel tissue-mimicking material, with 6-mm and 36-mm inner and outer diameters, respectively, was integrated into a hydraulic circuit. Since the cryogel is not waterproof, a 0.3-mm thin latex tube, which is assumed to have very little influence on the elastic behavior of the phantom, was inserted inside the cylinder [22]. A gear pump forced a pulsating flow inside the tube. A downstream load was applied by narrowing the end of the hydraulic circuit, to simulate the vascular system’s resistance. Accordingly, a dilatation of the inner diameter up to 7.23 mm was induced. The liquid used in the simulation was a 1% solution of Orgasol (ELF Atochem, Paris, France) backscattering particles in water. The 2-D phantom geometry is visible in the B-mode scan of the experimental structure shown in Fig. 2. The US measurements were taken by using a novel dualbeam multigate Doppler system [14] providing the simultaneous, real-time assessment of both blood velocity profile and wall displacements in vessels. The system consists of a modified US machine (Megas, Esaote SpA, Florence, Italy) and a PC add-on board based on a high-velocity digital signal processor. The Megas software was customized so that two independent M-lines in the B-mode display could be selected simultaneously. These two lines were set at optimal angles, θ, for detection of wall (θ = 90◦ ) and blood (θ < 90◦ ) movements, respectively.

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Fig. 3. Experimental velocity profile and second-order polynomial interpolation, in the mean square error sense, obtained with the ultrasound beam oriented at 72 degrees. Fig. 2. Ultrasound B-mode scan of the experimental vessel embedded in tissue-mimicking cryogel. Before the application of the downstream load, the diameter of the tube was 6 mm. It reached 7.23 mm after the downstream narrowing.

For the beam oriented at a right angle to the vessel, the displacements of the walls were estimated using the following automatic procedure. The approximate positions of the anterior and posterior walls were first roughly identified by finding the depths (gates) at which the A-mode signal presented the maximum gradients. The modified 2D-autocorrelation algorithm [23] was applied to the signals that originated in correspondence to these gates, to provide the instantaneous wall velocities. This algorithm was shown to be mathematically identical to the crosscorrelation algorithm, the difference being that the former is applied on complex RF data, whereas the latter is applied on complex demodulated RF data. By integrating the wall velocities, the positions of either the anterior and posterior walls or the distension waveform were estimated in real time. The wall gate position was continuously updated on the basis of the estimated wall displacement, according to the classic tracking procedure [24], [25]. Accuracy on the order of micrometers has been demonstrated to be possible for this procedure [14]. The echo-signals corresponding to the second (Doppler) beam were sampled with 14-bit resolution over 128 adjacent depths. For the Doppler signal corresponding to each depth, the power spectral density was computed, and the mean velocity was estimated from the spectrum’s first moment [13]. B. Experimental Results Seventy velocity profiles v(r, t) were acquired during each pulsation cycle along the Doppler line (θ = 72◦ ). Fig. 3 shows a typical instantaneous profile detected during the diastolic phase. Each velocity profile was approximated by a parabola in the mean square error sense (Fig. 3). The

Fig. 4. Maximum fluid velocity waveform v(0, t) on the axis of the tube measured over a pulsating cycle.

peak velocity corresponding to the vessel axis, v(0, t), is plotted in Fig. 4. The flow was given by the superposition of a constant (0.35 m/s) fluid velocity and an additive pulsating flow, producing a maximum velocity of 0.7 m/s. The measured wall displacement ∆Hexp (t) is shown in Fig. 5. C. Finite Element Simulation Results For a given set of input parameters as listed in (1), the finite element solver determines the corresponding values of the wall displacement. In the simulation, the fluid was considered as a water-like substance with dynamic viscosity η equal to 1.002 10−2 Pl and density of 103 kg m−3 . The Poisson coefficient was υ = 0.49; E 0 = 30 103 Pa and P00 = 1, 000 Pa were set as initial values. These values were selected in accordance with those given in the literature [20], [21].

balocco et al.: young’s modulus evaluation of tissues surrounding pulsatile vessels

Fig. 5. Radial arterial wall displacement measured over a pulsating cycle obtained with the transverse ultrasound beam.

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Fig. 7. Finite element simulation and experimental displacements over a pulsating cycle.

TABLE I Measured and Estimated Young’s modulus (E) and Hydraulic Downstream Pressure (P0 ) Values.

Estimated value Reference value Relative error

Fig. 6. Wall displacement direction (arrows) and fluid velocity field (grayscale) plots produced by the finite element simulation at the beginning of the pulsating cycle. The dilated geometry is superimposed.

Fig. 6 illustrates the outcome of the finite element analysis at the beginning of the pulsating cycle. The grayscale represents the axial velocity of the fluid, whereas the arrows symbolize the wall displacement direction. The geometry of the tube at the time of maximum dilatation is represented through dashed lines. D. Convergence of the Minimization Algorithm The Young’s modulus estimation was obtained by iterating the gradient’s descent technique. The algorithm stopped when the distance d between the measured and the simulated wall displacement was lower than an arbitrary threshold value dt+1 . dt+1 < 0.01 [max (∆Hexp ) − min (∆Hexp )] . (14)

E

P0

32 590 Pa 34 307 Pa 5.3%

1 278 Pa 1 407 Pa 9.4%

Fig. 7 reports the simulated and the measured displacements, showing a good match between the amplitude of the two curves. The difference between the two waveforms that can be noted in the interval 0.4–0.6 s may have different origins. The delay between the simulated and experimental signals can be explained either by the viscoelastic nature of the material and the related relaxation time or by a small dissymmetry between the inflation and deflation of the pump used in the experiment, or by the rebound in the fluid velocity, which can be observed in Fig. 4, leading to a backflow response. E. Validation The convergence of the minimization method leads to a Young’s modulus estimate of 32 590 Pa and to a downstream load pressure P0 of 1 278 Pa. The estimated Young’s modulus was compared to the reference value measured with a mechanical traction machine on a sample of the cryogel mimicking the surrounding tissue. The strain-stress diagram gives an approximately straight line whose slope represents the elastic coefficient (6). The linear regression curve (Fig. 8) gives a Young’s modulus of 34 307 Pa. As reported in Table I, the estimated and measured values exhibit relatively small differences (5.3%). Once the Young’s modulus value is known, the pressure P0 can be derived from the tube diameter measurements. At a constant flow of 0.35 m/s, by substituting the mea-

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Fig. 8. Diagram of the displacement as a function of the applied strain, measured on the cryogel phantom sample with a traction machine. The slope of the linear regression curve represents the Young’s modulus.

sured Young’s modulus E in Hooke’s law equation, the pressure P0 can be calculated as: P0 =

|D1 − D2 | E = 1407 Pa, 2H

(15)

where H = 15 mm, D1 is the original diameter of the tube (6 mm), and D2 the dilated diameter (7.23 mm) after filling the tube with the liquid.

IV. Discussion and Conclusion This paper has presented a US-based method to measure the elastic modulus of tissues surrounding arterial vessels. The Young’s modulus is considered here to be representative of the composite stiffness of the region, although it is not characteristic of the local stiffness of each layer. Hence, the information obtained could be an indicator of the general state of the arterial disease and not a local characterization of the plaque or of the different tissue layers. The method is based on the comparison between data measured with a dual-beam multigate system and data simulated with a finite element method. Such data include the velocity profile detected along the axis of a first US beam, and the wall displacements measured by the second US beam. The latter are obtained through a modified (2D) autocorrelation algorithm. This is a widely used, computationally low-cost method [26], assuming that the motion is in the direction of the US beam axis. If there are variations in the Young’s modulus along the wall, such as with atherosclerosis, the displacement may not be normal to the vessel axis. In this case, a fine adjustment of the interrogating beam direction, e.g., aiming at maximizing the energy reflected from the walls, could be necessary. The simulation describes a model that incorporates a fluid-dynamics and mechanical elastic analysis by means of

a parametric finite element approach. A gradient descent method was implemented to determine the set of values leading to the best match between the simulated and the measured vessel wall displacement waveforms. The feasibility of the method has been illustrated through a phantom consisting of a cylindrical tube, in which the Navier-Stokes equation was reduced to the Poiseuille Law. The experiment carried out has provided encouraging results. The Young’s modulus assessment presents an estimation error of 5.3%. The pressure load value calculated from the estimated Young’s modulus gives an estimation error of 9.4%. More complex structures, corresponding to realistic anatomical shapes of carotid arteries, possibly including a plaque or a stenosis, could be added to the model as well. The fluid-dynamics model described in Section II makes it possible to calculate the velocity map in each point of the vessel using the information from the inflow profile and the acquired geometry. On the other hand, the dual-beam multigate system has been shown to be capable of working in vivo [27] and of acquiring any complex fluid velocity profile [28]. In order to obtain the stream map inside the vessel and the normal stress amplitude for any flow condition, the geometry of the vessel and the boundary conditions need to be accurately accessed using imaging methods. Since the velocity profile is acquired along one direction, the velocity inside the vessel should be mapped. In conditions in which the velocity profile cannot be considered axisymmetric, such as during turbulent flow, a 3-D meshed structure should be reconstructed from multiple transversal scans of successive slices along the vessel. Although the experiment was conducted on a simplified, single phantom, and the measurement and computation uncertainties could not be evaluated, these preliminary results suggest that the new method is a possible alternative to elastographic and pulse wave velocity techniques to evaluate the stiffness of the composite region surrounding arterial vessels.

References [1] R. S. Reneman, J. M. Meinders, and A. P. Hoeks, “Non-invasive ultrasound in arterial wall dynamics in humans: What have we learned and what remains to be solved,” Eur. Heart J., vol. 26, no. 10, pp. 960–966, 2005. [2] K. V. Ramnarine, T. Hartshorne, Y. Sensier, M. Naylor, J. Walker, A. R. Naylor, R. B. Panerai, and D. H. Evans, “Tissue Doppler imaging of carotid plaque wall motion: A pilot study,” Cardiovasc. Ultrasound, vol. 1, no. 1, p. 17, 2003. [3] Y. Aggoun and M. Beghetti, “Noninvasive assessment of arterial function in children: Clinical applications,” Pediatr. Cardiol., vol. 13, pp. 12–18, 2002. [4] R. Selzer, W. Mack, P. Lee, H. Kwong-Fu, and H. Hodis, “Improved common carotid elasticity and intima-media thickness measurements from computer analysis of sequential ultrasound frames,” Atherosclerosis, vol. 154, no. 1, pp. 185–193, 2001. [5] J. M. Meinders and A. P. Hoeks, “Simultaneous assessment of diameter and pressure waveforms in the carotid artery,” Ultrasound Med. Biol., vol. 30, no. 2, pp. 147–154, 2004. [6] P. J. Brands, J. M. Willigers, L. A. Ledoux, R. S. Reneman, and A. P. Hoeks, “A noninvasive method to estimate pulse wave velocity in arteries locally by means of ultrasound,” Ultrasound Med. Biol., vol. 24, no. 9, pp. 1325–1335, 1998.

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[19] P. Leray, “Analyse num´erique probl`emes pratiques,” Optimisation, vol. 11, 2003. (in French) [20] Y. C Fung, Mechanical Properties of Living Tissues. 2nd ed. New York: Springer, 1993. [21] J. Fromageau, E. Brusseau, D. Vray, G. Gimenez, and P. Delachartre, “Characterization of PVA cryogel for intravascular ultrasound elasticity imaging,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 50, no. 10, pp. 1318–1324, 2003. [22] K. V. Ramnarine, B. Kanber, and R. B. Panerai, “Assessing the performance of vessel wall tracking algorithms: The importance of the test phantom,” in J. Phys.: Conf. Series (Adv. Metrol. Ultrasound Med.), vol. 1, 2004, pp. 199–204. [23] T. Loupas, J. T. Powers, and R. W. Gill, “An axial velocity estimator for ultrasound blood flow imaging, based on a full evaluation of the Doppler equation by means of a two-dimensional autocorrelation approach,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 42, pp. 672–688, 1995. [24] P. J. Brands, A. P. Hoeks, L. A. Ledoux, and R. S. Reneman, “A radio frequency domain complex cross-correlation model to estimate blood flow velocity and tissue motion by means of ultrasound,” Ultrasound Med. Biol., vol. 23, no. 6, pp. 911–920, 1997. [25] H. Kanai, M. Sato, Y. Koiwa, and N. Chubachi, “Transcutaneous measurement and spectrum analysis of heart wall vibrations,” IEEE Trans. Ultrason., Ferroelect., Freq. Contr., vol. 43, no. 5, pp. 791–810, 1996. [26] S. I. Rabben, S. Bjærum, V. Sørhus, and H. Torp, “Ultrasoundbased vessel wall tracking: An auto-correlation technique with RF center frequency estimation,” Ultrasound Med. Biol., vol. 28, pp. 507–517, 2002. [27] P. Tortoli, T. Morganti, G. Bambi, C. Palombo, and K. V. Ramnarine, “Noninvasive simultaneous assessment of wall shear rate and wall distension in carotid arteries,” Ultrasound Med. Biol., vol. 32, no. 11, pp. 1661–1670, 2006. [28] G. Bambi, F. Guidi, S. Ricci, P. Tortoli, M. R. Cirelli, and L. Pedrini, “Ultrasound blood flow imaging in carotid arteries before and after endarterectomy,” Acoust. Imag., vol. 27, pp. 471– 476, 2004.

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HITCHIN–KOBAYASHI CORRESPONDENCE FOR ...
SL(2,C)-equivariant holomorphic bundles and coherent sheaves over X × P1 —where the action of SU(2) has been extended to its complexification, SL(2,C)).

Correspondence-1949-1975-New-Heidegger-Research.pdf ...
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Hitchin–Kobayashi Correspondence, Quivers, and ... - Springer Link
prove a Hitchin–Kobayashi correspondence, relating the existence of ... This correspondence provides a unifying framework to study a number of problems.

Committees of Correspondence QUestions.pdf
Questions. 1. What was the purpose of the Committees of Correspondence? ___. 2. Do you think the Committees were part of the resistance movement? Explain. ___ ...