Citation Context

...s adopted to describe the mechanical behaviour of the tissue in quasi-static elastography for the computational simplicity, and the relationship of the stress σ and the strain obeys the Hooke’s law =-=[25]-=-: σ = S (1) where S is the strain-stress matrix. Let u(x, y) and v(x, y) be the displacement along the x- and y-axis of a point, the infinitesimal strain tensor of the point is: = ⎡ ⎢⎢⎢⎢⎢⎢⎣ ∂u ∂x ∂...

Citation Context

...d through the Kalman filter in the minimum mean square error (MMSE) sense. Introduction A tumour or a suspicious pathological growth is normally much stiffer than the background of normal soft tissue =-=[1]-=-. So when a mechanical compression or vibration is applied, the tumour deforms less than the surrounding tissue, i.e. the strain in the tumour is less than the surrounding tissue. Hence a strain image...

Citation Context

... various nature. It is thus necessary to develop a strategy, the stochastic finite element method (SFEM), which has been used widely for structural dynamic analysis in probability analysis frameworks =-=[28,29]-=-. In SFEM, structural material properties are described by random fields, possibly with known prior statistics. In our implementation, a Delaunay triangulated finite element mesh is constructed at the...

Citation Context

...ochastic finite elementmethod In the past decades, the deterministic finite element method has been able to provide an effective and convenient platform for bio-mechanical studies in the past decades =-=[26,27]-=-. However, it does not have the capability to process the uncertainties of material properties, and kinematic observations, i.e. measurements of the displacement. Especially the ultrasound imaging-der...

Citation Context

...ntation. For example, as one of incompressible materials, the Poisson’s ratio of tissue can be set close to 0.5. The state space representation of Equation (4) becomes ẋ(t) = Acx(t) + BcW (t) + n(t) =-=(8)-=- where n(t) is the process noise and it is white noise (E[n(t)] = 0; E[n(t)n(s)′] = Qn(t)δts, where Qn is the process noise covariance). The system matrices Ac, Bc and input forces W (t) are: Ac = [ 0...

Citation Context

... Bc = [ 0 0 0 −M−1 ] W (t) = [ 0 R(t) ] An associated measurement equation, which describes the observations y(t) provided by the ultrasonic RF data, can be expressed in the form: y(t) = Hx(t) + e(t) =-=(9)-=- where e(t) is the measurement noise and it is white noise (E[e(t)]= 0;E[e(t)e(s)′]= Re(t)δts, and Re is the measurement noise covariance), independent of n(t). H is the measurement matrix which shoul...

Citation Context

...rement vector y(t). In order to run the dynamic equation (Equation (4)) in computer, it should be transformed into a discrete state-space equation, typically seen in control and estimation literature =-=[30,31]-=-. We discretize Equation (8) and (9) over a constant time interval T. Since the interval T is always a known constant, we can replace kT with k in following equations x(k + 1) = Ax(k) + Bw(k) + n(k), ...

Citation Context

...asurement noise. In order to evaluate the performance quantitatively, we define the following two factors, the lateral displacement relative error (DRE) and contrast-to-noise ratio (CNR): DRE = ē d̄ =-=(14)-=- CNR = √ 2(s̄b − s̄t)2 σ 2b + σ 2t (15) where ē is the average nodal displacement error, d̄ is the average nodal displacement value, s̄t and s̄b are the spatial strain average of the target and backg...

Citation Context

...romRF signals using different approaches [18]. Numerous techniques have been proposed to solve the reconstruction problem of strain images for quasi-static ultrasound elastography in the last decades =-=[19]-=-, but the estimation of strain images in ultrasound elastography still remains as a challenging researching topic at present because of its ill-posed nature [20]. The general procedure of quasi-static...

Citation Context

...rement vector y(t). In order to run the dynamic equation (Equation (4)) in computer, it should be transformed into a discrete state-space equation, typically seen in control and estimation literature =-=[30,31]-=-. We discretize Equation (8) and (9) over a constant time interval T. Since the interval T is always a known constant, we can replace kT with k in following equations x(k + 1) = Ax(k) + Bw(k) + n(k), ...

Citation Context

...se. In addition, errors will be introduced when the tissue incompressibility assumption is invalid. The recovery of strain and elasticity of tissue from the sparse displacement is one inverse problem =-=[2,23,24]-=-, therefore, it is necessary to develop a robust framework with a more meaningful bio-mechanical constraint to recover a full displacement field from the ultrasonic measurements. The rest of this pape...

Citation Context

...se. In addition, errors will be introduced when the tissue incompressibility assumption is invalid. The recovery of strain and elasticity of tissue from the sparse displacement is one inverse problem =-=[2,23,24]-=-, therefore, it is necessary to develop a robust framework with a more meaningful bio-mechanical constraint to recover a full displacement field from the ultrasonic measurements. The rest of this pape...

Citation Context

...he performance quantitatively, we define the following two factors, the lateral displacement relative error (DRE) and contrast-to-noise ratio (CNR): DRE = ē d̄ (14) CNR = √ 2(s̄b − s̄t)2 σ 2b + σ 2t =-=(15)-=- where ē is the average nodal displacement error, d̄ is the average nodal displacement value, s̄t and s̄b are the spatial strain average of the target and background, σ 2t and σ 2b are the spatial st...

Citation Context

... able to remotely generate mechanical vibration sources radiating low-frequency, shear waves inside tissues, and then reconstruct the viscoelastic properties of tissue from the propagated shear waves =-=[17]-=-. Among different forms of ultrasound elastography, quasi-static ultrasound elastography is one popular technique because of its simplicity: two sets of radio-frequency (RF) echo frames are first coll...

Citation Context

... ⎡ ⎢⎢⎢⎢⎢⎢⎣ ∂u ∂x ∂v ∂y ∂u ∂y + ∂v ∂x ⎤ ⎥⎥⎥⎥⎥⎥⎦ (2) Under the plane strain condition [25], strain-stress matrix S can be derived as: S = E (1 + ν)(1 − 2ν) ⎡ ⎢⎢⎢⎣ 1 − ν ν 0 ν 1 − ν 0 0 0 1 − 2ν2 ⎤ ⎥⎥⎥⎦ =-=(3)-=- S is a material-dependent matrix, E stands for the Young’s modulus, and ν stands for the Poisson’s ratio. Here, the Young’s modulus E and the Poisson’s ratio ν are two materialspecific parameters. Th...

Citation Context

.... We discretize Equation (8) and (9) over a constant time interval T. Since the interval T is always a known constant, we can replace kT with k in following equations x(k + 1) = Ax(k) + Bw(k) + n(k), =-=(10)-=- where A = eAcT , (11) B = A−1c (eAcT − I)Bc (12) The associated measurement equation can be: y(k) = Dx(k) + e(k) (13) where y(k) is the measurement vector contained the displacement extracted from ul...

Citation Context

...deformation caused by external compressions, and then the displacement field, strain field and even the distribution of tissular elasticity are reconstructed fromRF signals using different approaches =-=[18]-=-. Numerous techniques have been proposed to solve the reconstruction problem of strain images for quasi-static ultrasound elastography in the last decades [19], but the estimation of strain images in ...

Citation Context

...a state-space representation of the continuous-time linear stochastic system. First the kinematic vector x(k) and the material parameter vector θ are defined as: x(t) = [ U(t) U̇(t) ] (6) θ = [ E ν ] =-=(7)-=- where the kinematic vector x(t) is consisted of displacement U(t) and velocity U̇(t) information, and the material parameter vector θ is consisted of Young’s modulus E and Lu et al. BioMedical Engine...

Citation Context

... we can replace kT with k in following equations x(k + 1) = Ax(k) + Bw(k) + n(k), (10) where A = eAcT , (11) B = A−1c (eAcT − I)Bc (12) The associated measurement equation can be: y(k) = Dx(k) + e(k) =-=(13)-=- where y(k) is the measurement vector contained the displacement extracted from ultrasound RF data. In most quasi-static elastography, only theaxial component of displacement vector is extracted. Ther...

Citation Context

...olution in the axial direction is much higher than that in the lateral direction in ultrasound imaging, that is why the estimation of the axial component of the motion has received the most attention =-=[1,21]-=-. Recently, quite a few approaches have been proposed to recover the lateral and shear strain fields (i.e. reconstruct the strain tensor), Poisson’s ratio and Young’s modulus [21], but their performan...

Citation Context

...se. In addition, errors will be introduced when the tissue incompressibility assumption is invalid. The recovery of strain and elasticity of tissue from the sparse displacement is one inverse problem =-=[2,23,24]-=-, therefore, it is necessary to develop a robust framework with a more meaningful bio-mechanical constraint to recover a full displacement field from the ultrasonic measurements. The rest of this pape...

Citation Context

... various nature. It is thus necessary to develop a strategy, the stochastic finite element method (SFEM), which has been used widely for structural dynamic analysis in probability analysis frameworks =-=[28,29]-=-. In SFEM, structural material properties are described by random fields, possibly with known prior statistics. In our implementation, a Delaunay triangulated finite element mesh is constructed at the...

Citation Context

...g that the material parameters E (Young’s modulus) and ν (Poisson’s ratio) are temporally constant but varying spatially, we can have the following dynamic governing equation [25]: MÜ + CU̇ + KU = R =-=(4)-=- where U is the displacement vector, and R is the load vector. Mass matrix M here is a known function of the material density and is temporally and spatially constant. Stiffness matrix K is a function...

Citation Context

... = b be known from the imaging data at selected sampling nodes of the boundary, then the additional constraining equation μUb = μb is enforced on the system dynamics through MÜ+ CU̇+ KU+ μUb = R+ μb =-=(5)-=- where weighting coefficient μ depends on the confidence of each displacement, with large μ values (1 × 104 in this study) indicating highly trustworthy data points and small μ values for others. In t...

Citation Context

...on (8) and (9) over a constant time interval T. Since the interval T is always a known constant, we can replace kT with k in following equations x(k + 1) = Ax(k) + Bw(k) + n(k), (10) where A = eAcT , =-=(11)-=- B = A−1c (eAcT − I)Bc (12) The associated measurement equation can be: y(k) = Dx(k) + e(k) (13) where y(k) is the measurement vector contained the displacement extracted from ultrasound RF data. In m...

Citation Context

...ant time interval T. Since the interval T is always a known constant, we can replace kT with k in following equations x(k + 1) = Ax(k) + Bw(k) + n(k), (10) where A = eAcT , (11) B = A−1c (eAcT − I)Bc =-=(12)-=- The associated measurement equation can be: y(k) = Dx(k) + e(k) (13) where y(k) is the measurement vector contained the displacement extracted from ultrasound RF data. In most quasi-static elastograp...

Citation Context

...sound elastography in the last decades [19], but the estimation of strain images in ultrasound elastography still remains as a challenging researching topic at present because of its ill-posed nature =-=[20]-=-. The general procedure of quasi-static elastography is first to recover internal tissue displacements from two ultrasound radio-frequency (RF) frames before and after tissue deformation due to a quas...

Citation Context

...ochastic finite elementmethod In the past decades, the deterministic finite element method has been able to provide an effective and convenient platform for bio-mechanical studies in the past decades =-=[26,27]-=-. However, it does not have the capability to process the uncertainties of material properties, and kinematic observations, i.e. measurements of the displacement. Especially the ultrasound imaging-der...

Citation Context

...ransformed into a state-space representation of the continuous-time linear stochastic system. First the kinematic vector x(k) and the material parameter vector θ are defined as: x(t) = [ U(t) U̇(t) ] =-=(6)-=- θ = [ E ν ] (7) where the kinematic vector x(t) is consisted of displacement U(t) and velocity U̇(t) information, and the material parameter vector θ is consisted of Young’s modulus E and Lu et al. B...

Citation Context

...ne Elasticity QA Phantom (Model 049, CIRS Inc. USA) using a PC-based Ultrasonix RP ultrasound machine (UltrasonixMedical Corporation, Burnaby, BC, Canada). One phase-shift method with prior estimates =-=[32]-=- is used to estimate the axial displacements from the RF signals, and the full displacement field is recovered from the axial displacement measurements. And then all the strain images are displayed in...