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Red blood cells as the main source of scattering in blood
The RBCs represent 98% of the cellular content in blood, so ultrasonic scattering due to other cells (i.e., leukocytes and platelets) is neglected. A red blood cell can be considered as a membrane containing hemoglobin. The effect of the RBC membrane on ultrasonic backscattering is neglected because of its extremely small thickness about 10 nm (in comparison with the characteristic size of a red blood cell around few micrometers). Hemoglobin and plasma are acoustically described as fluid and non-viscous media with mass density and compressibility (or in an equivalent way with impedance Z = p = and speed of sound c) whose values are given in Table 1.1. The acoustic parameters , and Z have the subindices e for erythrocyte and p for plasma. Fluctuations of density and compressibility between RBC and plasma are defined by = e p e and = e p p.
The Born approximation
The features in Table 1.1 show that impedance contrast between plasma and RBC is low, about 13%. When a heterogeneous medium has scatterers with low impedance contrast ( Z < 0:15), one can consider that the incident wave propagates without being perturbed by the heterogeneities. This means that the scattered field is negligible when compared to the incident field: this is the Born approximation [Shung and Thieme, 1992]. Therefore, multiple scattering is neglected. This hypothesis is commonly accepted in literature for blood QUS characterization [Insana et al., 1990, Fontaine et al., 1999, Fontaine et al., 2002], and will be adopted in the framework of our study.
Ultrasonic scattering model for a single RBC
Let us consider an isolated RBC insonified by a plane wave of unit pressure amplitude pi (r) = eikni :r, where k is the wavenumber and ni is the unit vector in the direction of propagation of the incident field, as depicted in Fig. (1.4). The interaction between the incident wave and the particle results in a deviation of the incident wave that redistributes the acoustical energy in space. By considering the far-field regime (i.e., the observation distance is large compared to the size of scattering volume V ) and the Born approximation, the field of pressure scattered in the observation point r=rn can be expressed as [Shung and Thieme, 1992] ps(r) = eikr r (k(n ni )).
Ultrasonic scattering models for an ensemble of RBCs
For an ensemble of RBCs, the relevant quantity to be measured is the differential backscattering cross-section per unit volume also known as the backscatter coefficient, BSC. There are several theoretical approaches to model the backscattering from an ensemble of RBCs: the continuum model, the particle model and a combination of them. In the continuum model [Angelsen, 1980,Atkinson and Berry, 1974], the medium containing the RBCs is considered to be a continuum with spatial inhomogeneities in density and compressibility. Thus, the scattering is due to the fluctuations given by these inhomogeneities around mean values m and m. In addition, the BSC is described in function of the mean value of the wave number km. In the particle model [Lucas and Twersky, 1987, Twersky, 1987], the source terms are considered to be the RBCs, modeled as identical discrete scatterers much smaller than the wavelength with density e and compressibility e in a homogeneous surrounding background (i.e. the plasma) with density p and compressibility p. The backscattering from the ensemble of particles is the sum of the field backscattered by each particle, and interference effects caused by correlations among particle positions are modeled by a structure factor, as detailed in the following paragraph 1.5.1. The third approach is called the hybrid model that combines both continuum and discrete models [Mo and Cobbold, 1992]. In this approach, several RBCs in an elemental volume (voxel), which is smaller than wavelength, are treated as a scattering unit. Then the contribution of each voxel is obtained by means of the particle approach. The BSC is modeled taking into account the mean values of scatterers in each voxel and the variance of this number in the voxel assembly. In the remaining of this work, all the theoretical developments will be based on the particle model.
Simplified models for disaggregated RBCs
The most simple model is obtained by assuming that RBCs are randomly and independently distributed, it means that their positions are uncorrelated. In this case, the BSC is proportional to the number density of scatterers n as: BSC(k) = nb(k): (1.16).
Yuan & Shung [Yuan and Shung, 1988] conducted experiments on porcine RBCs suspended in a saline suspension in order to measure the BSC as a function of the hematocrit. Note that RBCs suspended in saline solution do not exhibit aggregation. Their results showed that this model is valid for disaggregated RBCs with hematocrit values less than 8%. A non-linear relationship between the backscatter amplitude and hematocrit was observed as shown in Fig. (1.8). It suggest that the effects of spatial correlation between RBCs can no longer be neglected for hematocrit greater than 8%.
Backscattering cross-section by a single prolate-shaped aggregate of RBCs
Before modeling the ultrasound backscattering by an ensemble of anisotropic aggregates, this work focuses on the modeling of the backscattering by a single anisotropic aggregate of RBCs. The backscattering cross-section by a single spherical aggregate of RBCs was previously developed during the PhD work of Romain de Monchy at LMA [de Monchy et al., 2016b]. As part of the present work, this model was extended for the case of aggregates with prolate ellipsoidal shape, as presented in this section. This theoretical extension was assessed using 3D computer simulations of ultrasound backscatter from a single aggregate of RBCs.
Theory: coherent and incoherent ultrasound backscatter from a single prolateshaped aggregate
In the following, the incident wavelength is assumed to be larger than the RBC size, such that the RBC shape can be approximated by a sphere of radius a with an equivalent volume Vc = (4=3)a3. The RBCs are described in terms of their mass density c and compressibility c, and their surrounding medium, the plasma, is characterized by its mass density o and compressibility o. The parameters = (c0)=0 and = (c 0)=c correspond to the fractional variations in compressibility and mass density, respectively. It is assumed that an aggregate contains N identical RBCs randomly distributed within it. The shape of an aggregate is approximated by a prolate ellipsoid having a semi-minor axis bag and a semi-major axis agbag (with ag defined as the axial ratio). Using Born and far-field approximations, the backscattering amplitude from a single RBC aggregate ag can be expressed as (see Eq. (4) in Ref. [de Monchy et al., 2016b]) ag(k) = k2( ) 4 VcF0(k;a) XN j=1 e2ik n0rj .
Table of contents :
List of symbols
List of acronyms
INTRODUCTION
I Theoretical part
1 State of the art: scattering modeling for ultrasound blood characterization
1.1 Phenomenon of RBC aggregation
1.2 Quantitative ultrasound techniques based on backscatter coefficient measurements
1.3 Assumptions for modeling ultrasound backscattering by blood
1.3.1 Red blood cells as the main source of scattering in blood
1.3.2 The Born approximation
1.4 Ultrasonic scattering model for a single RBC
1.5 Ultrasonic scattering models for an ensemble of RBCs
1.5.1 Particle model
1.5.2 Simplified models for disaggregated RBCs
1.5.3 Simplified models for aggregated RBCs
1.5.3.1 Structure factor size estimator (SFSE)
1.5.3.2 Effective medium theory combined with the structure factor model (EMTSFM) 14
1.6 Motivation: limitations of current models
2 Ultrasonic scattering modeling of red blood cells aggregates taking into account anisotropy
2.1 Backscattering cross-section by a single prolate-shaped aggregate of RBCs
2.1.1 Theory: coherent and incoherent ultrasound backscatter from a single prolate-shaped aggregate
2.1.1.1 The coherent component
2.1.1.2 The incoherent component
2.1.2 3D simulation method
2.1.3 Results and discussion: comparison between theoretical and simulated ag
2.2 Backscattering by an ensemble of prolate-shaped aggregates
2.2.1 Effective Medium Theory combined with the Local Monodisperse Approximation (EMTLMA) for perfectly aligned prolate ellipsoids
2.2.2 Methods
2.2.2.1 Computer simulations based on the Structure Factor Model
2.2.2.2 QUS parameter estimation
2.2.3 Results
2.2.3.1 Influence of structural parameters on the frequency-dependent BSC when using the anisotropic EMTLMA
2.2.3.2 Forward problem: comparison between simulated and theoretical BSCs
2.2.3.2.1 Influence of the polydispersity in terms of aggregate sizes
2.2.3.2.2 Influence of the alignment: randomly oriented versus perfectly aligned prolate ellipsoids
2.2.3.3 Inverse problem: estimation of QUS parameters
2.2.4 Discussion
2.2.4.1 Limitation to estimate QUS parameters for randomly oriented prolate ellipsoids
2.2.4.2 On the behavior of the cost function
2.2.4.3 On the use of the anisotropic EMTLMA in vivo
2.3 Conclusion
3 Evaluation of the anisotropic EMTLMA in estimation of structural parameters of partially aligned aggregates
3.1 Material and Methods
3.1.1 Experiments on porcine blood sheared in microfluidic shearing system
3.1.2 Computer simulations based on the Structure Factor Model
3.1.3 Theoretical EMTLMA modeling for partially aligned prolate ellipsoids
3.1.4 QUS parameter estimation
3.2 Results
3.2.1 Actual size and orientation distributions of RBC aggregates
3.2.2 Forward problem: comparison between simulated and theoretical BSCs
3.2.3 Inverse problem: estimation of QUS parameters
3.3 Discussion
3.3.1 Limitations of numerical simulations
3.3.2 Qualitative comparison between the simulated and measured BSCs
3.3.3 On the assumption of perfectly aligned aggregates in the estimation of QUS parameters
3.3.4 On the assumption of gamma (truncated) size distribution
3.3.5 On the cost function behavior
3.4 Conclusion
II Experimental part
4 State of the art: Measuring the BSC in soft tissues using an ultrasound imaging system
4.1 Ultrasound imaging system and transducers
4.1.1 Ultrasonic transducers
4.1.2 Ultrasound beamforming
4.2 Measurement techniques of the BSC in soft tissues
4.2.1 The substitution method
4.2.2 The reference phantom technique
4.2.3 Selection of the ROI to compute the BSC
4.3 Measurement techniques of local attenuation in soft tissues
4.4 Measure of BSC and other spectral-based quantitative ultrasound (QUS) parameters using medical ultrasound imaging systems
4.5 Motivation: Evaluation of the performance of a CMUT probe and a piezoelectric probe in
measuring tissue anisotropy
5 Performance comparison between piezoelectric and CMUT probes in measuring the backscattering coefficient
5.1 Material and Methods
5.1.1 Hydrophone system, ultrasound scanner and transducers
5.1.2 Transmit beam pattern measurements
5.1.2.1 Estimation of the beam steering direction
5.1.2.2 Sources of error and uncertainty in the beam direction estimation
5.1.3 Measurements of spectrum-based parameters on tissue-mimicking phantoms
5.1.3.1 Tissue-mimicking phantoms
5.1.3.2 Acoustical parameters of the tissue-mimicking phantoms
5.1.3.3 Acquisition of the backscattered raw radio frequency signals
5.1.3.4 Measurement of the backscatter coefficient and integrated backscatter coefficient
5.1.3.5 Analysis of spectrum-based parameter measurements
5.2 Results
5.2.1 Estimation of the beam steering direction
5.2.1.1 Beam pattern measurements
5.2.2 Measurements of the spectrum-based parameters
5.2.2.1 Isotropic medium
5.2.2.2 Anisotropic medium
5.3 Discussion
5.3.1 Limitation in hydrophone measurements
5.3.2 Estimation of the beam steering direction
5.3.3 Limitation in spectrum-based parameter measurements
5.3.3.1 Attenuation measurements
5.3.3.2 Measured and theoretical BSCs
5.3.4 Spectrum-based parameters measurements
5.3.4.1 Power spectral densities
5.3.4.2 BSC and iBSC
5.3.4.3 On the use of piezoelectric probe for anisotropic backscatter measurements .
5.4 Conclusion
General conclusion and perspectives
References
