Rheological Basis of Magnetic Resonance Elastography



Fig. 7.1
Schematic view of MRE system. The MRE system is fabricated by retrofitting an MRI system with a vibration exciter for exciting transverse waves and an MSG generator



1.

Transverse waves are created in the living body by an external periodic force acting on surface of the living body.

 

2.

By applying a sinusoidally oscillating magnetic field with a gradient called the motion-sensitizing gradient (MSG) on the living body, the waveform of transverse waves rippling in the living body is measured by the phase shift of the magnetic resonance (MR) signal, where transverse waves excited in the living body and the MSG should oscillate in the same phase.

 

3.

From the waveform derived from the phase shift, the stiffness of the living body is obtained position-dependently.

 


From the viewpoint of its hard ware, the MRE system is simply an extended version of the MRI system; it is fabricated by retrofitting an MRI system with a vibration exciter for exciting transverse waves and an MSG generator. However, the physical quantities measured by MRI and MRE differ in quality. By MRI, the material distribution (more precisely, the proton density distribution) in the living body is measured; thus, MRI is grouped with computed tomography (CT) and X-ray photography. From the viewpoint of medical practice, MRI is regarded as a visual diagnostic tool. On the other hand, by MRE, the stiffness distribution in the living body is measured. MRE is regarded as a palpation method. Palpation is an effective method of detecting tumors. However, it is restricted to body areas accessible to the physician’s hand. Obtaining quantitative diagnostic outcomes by palpation is difficult. To develop strengths and eliminate weaknesses of palpation, various types of elastography for disease diagnosis have been developed [3]. MRE has the advantage over other types of elastography in that the stiffness in all areas of the living body can be measured quantitatively in principle; namely, a complete viscoelastic parameters map of the living body can be obtained.

The original procedure of MRE for measuring the stiffness is as follows (See Fig.7.2):

A314557_1_En_7_Fig2_HTML.gif


Fig. 7.2
Excitation of a transverse sinusoidal wave in the living body. Assume that the displacement of a transverse wave excited in the living body at position x and time t is expressed by the sinusoidal form with amplitude A: $$ u\left(x,t\right)=A \cos \left(\frac{2\pi }{\varLambda }x-\omega t+{\delta}_0\right) $$. From the wavelength Λ and the angular frequency ω of the sinusoidal wave, the stiffness (modulus of the rigidity) μ of the living body is expressed by $$ \mu =\rho {\left(\varLambda \frac{\omega }{2\pi}\right)}^2 $$ on the basis of a linear elastic body picture, where ρ is the density of the living body




  • (P1) Generate a transverse wave with an angular frequency ω in the living body using a vibration exciter and a magnetic field with MSG whose phase is matched to that of the vibration exciter.


  • (P2) Obtain the phase shift θ of the MR signal induced by the transverse wave position-dependently.


  • (P3) Derive the wave shape u(x, y, z) at the position (x, y, z) using the proportional relation between the wave shape and the phase shift $$ u\propto \theta $$.


  • (P4) Obtain the wavelength Λ from the wave shape u(x, y, z) by assuming that the transverse wave is a sinusoidal wave such that $$ u\left(x,y,z\right)=A \cos \left(\frac{2\pi }{\varLambda }x+{\delta}_0\right) $$, where the wave propagation direction is chosen to be the x-direction.


  • (P5) Obtain the stiffness (modulus of rigidity) μ from the relationship among the modulus of rigidityμ, the density ρ, the wavelength Λ, and angular frequency ω, i.e., $$ \mu =\rho {\left(\varLambda \frac{\omega }{2\pi}\right)}^2 $$, which is derived on the basis of the assumption that the living body is a linear elastic body.

Since the living body is a rheologically complex system, the elastic body picture of the living body is too simple. Therefore, the above procedure for obtaining the stiffness distribution in the living body has the following problems.

1.

The modulus of rigidity is derived from the wavelength. However, the wavelength of a transverse wave is so long that position-dependent measurement is quite difficult.

 

2.

In the above simple procedure, it is assumed that all the points of the living body oscillate in the same phase. However, the viscosity shifts the phase position-dependently.

 

3.

A transverse wave cannot reach deep areas of the living body since the viscosity damps the amplitude of the transverse wave rapidly.

 

4.

Since the living body is soft, it is difficult for the vibration exciter to excite accurately a sinusoidal vibration on the living body surface.

 

Further improvements of the instruments for the MRE system and of the analysis algorithm of the MR signal could resolve some of the above programs. Actually, various efforts to improve the instruments [4] and algorithm [5] have been made and some MRE devices are actually used in clinical practice [1, 6]. The aim of these efforts is to develop excellent medical diagnostic devices. This aim is, of course, the starting point of the development of MRE, but it might limit our options of improving the MRE device. To avoid patient discomfort, measurement time should be as short as possible. To utilize existing MRI devices effectively, a small-scale modification of MRI devices is acceptable. Besides its clinical usage, the MRE device is useful for the nondestructive measurement of the viscoelastic properties of soft materials in general [7, 8]. MRE can therefore become a powerful tool for determining the viscoelastic structure of nonuniform gels and for observing the characteristic slow dynamics in gels. MRE is expected to contribute to the progress of gel science. To expand the fields in which MRE is utilized beyond the field of medicine, it is necessary to consider what kind of physical quantity is actually measured and what kind of principle relates the measured quantity to the viscoelastic properties, from the viewpoint of basic science. The approach from basic science would also promote the development of MRE as a medical diagnostic device. Here, we try to introduce the concept of rheology in MRE and to develop a purely theoretical approach to MRE.

This chapter is organized as follows. First, the quantities measured by MRE are explained. In MRE, the measurable quantity is the phase shift of the MR signal. The relationship between the phase shift and the motion of the concerned point in the living body is determined independently of the details of the MRE device. The relationship among the motion, MSG, and the phase shift is explained. The shape of standing waves in the living body is visualized from the phase shift data. The visualization mechanism is explained.

To connect the shape of a wave excited in the living body to the viscoelastic structure, the modeling of the living body based on Newton’s equation of motion is required. Next, the modeling of the living body is discussed. The living body is regarded as a viscoelastic body and its viscoelastic structure is described in terms of the Voigt model. Of course, a more complex model can be adopted. In complex models, however, many viscoelastic parameters must be introduced. The multitude of such parameters disturbs the unique determination of viscoelastic properties. Therefore, we choose the Voigt model, which is the simplest of all viscoelastic models. Here, we must note that all viscoelastic parameters are “defined” through the modeling based on the Voigt model. The Voigt picture of the living body is described by Lagrangian formalism for general versatility and simplicity in the mathematical treatment of such formalism.

By the above modeling, the motion at each point in the living body can be described in terms of viscoelastic parameters. In principle, the values of the viscoelastic parameters can be derived from measured data of the motion. All the quantities expressing the mechanical properties of materials such as rigidity and viscosity are determined through this “inverse” problem. Unfortunately, in MRE, the motion cannot be measured, but the shape of an excited steady-state wave can. The solution of the inverse problem based on the sinusoidal-wave assumption cannot be adopted because of the long wavelength [Problem (I)] and the viscosity of the living body [Problem (II)]. A novel method of solving the inverse problem and of reconstructing the map of the modulus of rigidity in the living body from the measured data is required. The reconstruction formula necessary to comply with this requirement is explained.

As a way of resolving Problem (III), the method of exciting a transverse wave using a longitudinal wave has been invented. A theoretical basis of transverse wave excitation by a longitudinal wave is also developed. For hard body organs and tissues such as bone and tendons, the viscoelastic properties are anisotropic and often uniaxial. Finally, a method of measuring the rigidity of a uniaxial system is introduced.



7.2 Visualization of Wave Shape by MSG



7.2.1 Original Idea for MRE


The original idea of Muthupillai et al. [2] is briefly reviewed.

Let us denote the displacement of an infinitesimal element of the living body at a position $$ \mathbf{x}=\left({x}_1,{x}_2,{x}_3\right) $$ at a time t by


$$ \mathbf{u}\left(\mathbf{x},t\right)=\left({u}_1\left(\mathbf{x},t\right),{u}_2\left(\mathbf{x},t\right),{u}_3\left(\mathbf{x},t\right)\right). $$

(7.1)
Let us choose the direction of the static magnetic field B 0 that induces the Larmor precession of protons as the x 3-direction: $$ {\mathbf{B}}_0=\left(0,0,{B}_0\right) $$. The magnetic field superimposed on the static magnetic field B 0 for inducing MSG is given by


$$ {\mathbf{B}}_{\mathrm{MSG}}\left(\mathbf{x},t\right)=\left(0,0,\mathbf{G}(t)\cdot \mathbf{x}\right). $$

(7.2)
Superimposing the magnetic field B MSG on the magnetic field B 0, we obtain the phase shift of MR signal at x,


$$ \theta \left(\mathbf{x}\right)=\varGamma {\displaystyle \underset{0}{\overset{\tau }{\int }}\mathbf{G}(t)\cdot \mathbf{u}\left(\mathbf{x},t\right)dt}, $$

(7.3)
where τ is the “measurement time” and Γ is a constant proportional to proton density (See Fig. 7.3).

A314557_1_En_7_Fig3_HTML.gif


Fig. 7.3
Phase shift of the MR signal induced by the magnetic field with MSG and excited transverse wave. The magnetic field with MSG B MSG superimposed on the static magnetic field B 0 that induces the Larmor precession of protons and the transverse wave excited in the living body change the MR signal from S 0 to $$ S={S}_0 \exp \left(i\theta \right) $$. From the phase shift θ, the shape of the transverse wave is obtained

Let us assume that we generate a transverse sinusoidal wave with an angular frequency ω, a wave vector k, and a phase constant δ 0 in the living body by a vibration exciter, whose displacement is expressed by


$$ \mathbf{u}\left(\mathbf{x},t\right)=\mathbf{A} \cos \left(\mathbf{k}\cdot \mathbf{x}-\omega t+{\delta}_0\right). $$

(7.4)
We oscillate the gradient vector G(t) at the angular frequency ω harmonically and synchronize it with the vibration motion of the vibration exciter as


$$ \mathbf{G}(t)={\mathbf{G}}_0 \cos \omega t. $$

(7.5)
Choosing the measurement time of an integral multiple of the period of the harmonic oscillation, $$ \tau =2\pi N/\omega $$ (N is a positive integer), we obtain the phase shift as


$$ \theta \left(\mathbf{x}\right)={\mathbf{G}}_0\cdot \mathbf{A}\frac{\pi N\varGamma }{\omega } \cos \left(\mathbf{k}\cdot \mathbf{x}+{\delta}_0\right). $$

(7.6)
The above expression shows that the position dependence of the phase shift indicates a snapshot of the sinusoidal wave, i.e., the wave shape. Then, the wave motion is visualized by means of MSG.

In terms of the wave vector k obtained from the wave shape, the wavelength is expressed as $$ \varLambda =2\pi /\left|\mathbf{k}\right| $$. Assuming a linear isotropic elasticity in the living body, the displacement satisfies the wave equation


$$ \varDelta \mathbf{u}-\frac{1}{c^2}\frac{\partial^2\mathbf{u}}{\partial {t}^2}=0, $$

(7.7)
where c is the propagation velocity of the wave and is given by


$$ c=\sqrt{\frac{\mu }{\rho }}. $$

(7.8)
In the above, ρ is the density of the living body and is estimated as $$ \rho \approx 1 $$ g/cm3. Substituting Eq. (7.4) into Eq. (7.7), we obtain the dispersion relation


$$ \omega =2\pi c/\varLambda $$

(7.9)
and obtain the expression of the rigidity


$$ \mu =\rho {\left(\varLambda \frac{\omega }{2\pi}\right)}^2=\rho {\left(\frac{\omega }{\left|\mathbf{k}\right|}\right)}^2. $$

(7.10)
This expression allows us evaluate rigidity.


7.2.2 Utilization of Visualization Ability of MSG


The abovementioned method of deriving rigidity is very simple, but the validity of the result strongly depends on the wave shape of Eq. (7.4). However, the sinusoidal-wave assumption is not always valid. The nonuniformity of rigidity and/or viscosity makes the amplitude A, the phase constant δ 0 , and the wave vector k depend on the position x. Hence, the displacement is generally described as


$$ \mathbf{u}\left(\mathbf{x},t\right)=\mathbf{A}\left(\mathbf{x}\right) \cos \left(\mathbf{k}\left(\mathbf{x}\right)\cdot \mathbf{x}-\omega t+{\delta}_0\left(\mathbf{x}\right)\right). $$

(7.11)
Substituting Eqs. (7.5) and (7.11) into Eq. (7.3), we obtain the phase shift


$$ \theta \left(\mathbf{x}\right)={\mathbf{G}}_0\cdot \mathbf{A}\left(\mathbf{x}\right)\frac{\pi N\varGamma }{\omega } \cos \left(\mathbf{k}\cdot \mathbf{x}+{\delta}_0\left(\mathbf{x}\right)\right). $$

(7.12)
This expression shows that the phase shift indicates a snapshot of the wave even if the wave is not sinusoidal. MSG can visualize the wave shape generally. Note, however, that the relationship between the wave shape expressed by Eq. (7.12) and the rigidity μ(x) is not clear since the velocity expression Eq. (7.8) and the dispersion relation Eq. (7.9) no longer hold. To derive the precise relationship between wave shape and rigidity, the modeling of the rheological structure of the living body is necessary. The transverse wave expressed by Eq. (7.11) is rewritten as


$$ \mathbf{u}\left(\mathbf{x},t\right)=\boldsymbol{\varphi} \left(\mathbf{x}\right) \cos \omega t+\boldsymbol{\psi} \left(\mathbf{x}\right) \sin \omega t, $$

(7.13)
where


$$ \begin{array}{l}\boldsymbol{\varphi} \left(\mathbf{x}\right)=\mathbf{A}\left(\mathbf{x}\right) \cos \left(\mathbf{k}\left(\mathbf{x}\right)\cdot \mathbf{x}+{\delta}_0\left(\mathbf{x}\right)\right)\\ {}\boldsymbol{\psi} \left(\mathbf{x}\right)=\mathbf{A}\left(\mathbf{x}\right) \sin \left(\mathbf{k}\left(\mathbf{x}\right)\cdot \mathbf{x}+{\delta}_0\left(\mathbf{x}\right)\right),\end{array} $$

(7.14)
Using the two types of MSG, the phases of whose oscillation differ by only π/2, i.e.,


$$ \left\{\begin{array}{l}\mathbf{G}(t)={\mathbf{G}}_0 \cos \omega t\\ {}{\mathbf{G}}^{\mathbf{\prime}}(t)={\mathbf{G}}_0 \sin \omega t,\end{array}\right. $$

(7.15)
we can obtain two types of phase shift:


$$ \begin{array}{l}{\theta}_0\left(\mathbf{x}\right)=\frac{\pi N\varGamma }{\omega }{\mathbf{G}}_0\cdot \boldsymbol{\varphi} \left(\mathbf{x}\right)\\ {}{\theta}_{\pi /2}\left(\mathbf{x}\right)=\frac{\pi N\varGamma }{\omega }{\mathbf{G}}_0\cdot \boldsymbol{\psi} \left(\mathbf{x}\right).\end{array} $$

(7.16)
The above expression shows that we can obtain the rigidity μ(x) position-dependently by determining the relationship between the two amplitudes of the vibration, i.e., φ(x) and ψ(x), and the rheological structure of the living body. Hence, the visualization ability of MSG can be effectively utilized by modeling the rheological structure of the living body.


7.3 Modeling of Living Body



7.3.1 Voigt Model Picture


The amplitude of the excited transverse wave is about 1–100 μm and the wavelength is about a few centimeters. Since the displacement and the gradient of the displacement are very small, the nonlinear effect on the rheology of the living body is negligible. Therefore, the rheological properties of the living body are described in terms of a linear viscoelastic structure. Let us describe the viscoelastic structure of the living body on the basis of the Voigt model. Introducing the strain tensor defined as


$$ {u}_{ij}\left(\mathbf{x}\right)=\frac{1}{2}\left(\frac{\partial {u}_i}{\partial {x}_j}+\frac{\partial {u}_j}{\partial {x}_i}\right) $$

(7.17)
and expressing the density by $$ \rho \left(\approx 1\kern0.5em \mathrm{g}/{\mathrm{cm}}^3\right) $$ and the elasticity tensor at x by λ ijkl (x), we express the elastic property of the living body by the Lagrangian [9, 10]


$$ L={\displaystyle {\int}_{\varOmega }{d}^3\mathbf{x}\left[\frac{1}{2}\rho {\displaystyle \sum_{j=1}^3{\left({\dot{u}}_j\left(\mathbf{x}\right)\right)}^2-\frac{1}{2}}{\displaystyle \sum_{i,j,k,l=1}^3{\lambda}_{ijkl}\left(\mathbf{x}\right){u}_{ij}\left(\mathbf{x}\right){u}_{kl}\left(\mathbf{x}\right)}\right]}, $$

(7.18)
where Ω is the space region occupied by the living body and $$ {\dot{u}}_j\left(\mathbf{x}\right) $$ is the j-th component of the velocity (the time derivative of the j-th component of the displacement $$ {\dot{u}}_j=\partial {u}_j/\partial t $$). Note the elasticity tensor has the following symmetry property:


$$ {\lambda}_{ijkl}={\lambda}_{jikl}={\lambda}_{ijlk}={\lambda}_{klij}. $$

(7.19)
In terms of the viscosity tensor η ijkl (x), the viscous property of the living body is expressed by the dissipation function given by [11]


$$ D={\displaystyle \underset{\varOmega }{\int }{d}^3\mathbf{x}\left[\frac{1}{2}{\displaystyle \sum_{i,j,k,l=1}^3{\eta}_{ijkl}\left(\mathbf{x}\right){\dot{u}}_{ij}\left(\mathbf{x}\right){\dot{u}}_{kl}\left(\mathbf{x}\right)}\right]}, $$

(7.20)
where $$ {\dot{u}}_{ij} $$ is the time derivative of the strain tensor:


$$ {\dot{u}}_{ij}\left(\mathbf{x}\right)=\frac{1}{2}\left(\frac{\partial {\dot{u}}_i}{\partial {x}_j}+\frac{\partial {\dot{u}}_j}{\partial {x}_i}\right). $$

(7.21)
Note that the viscosity tensor has the following symmetry property:


$$ {\eta}_{ijkl}={\eta}_{jikl}={\eta}_{ijlk}={\eta}_{klij}. $$

(7.22)
The rheological structure of the living body is expressed by the elasticity tensor “field” λ ijkl (x) and the viscosity tensor “field” η ijkl (x). From the viewpoint of rheology, palpation is performed to obtain the map of the viscoelastic parameters: λ ijkl (x) and η ijkl (x).


7.3.2 Equation of Motion


The equation of motion of the living body is obtained from the Lagrange equation [12]


$$ \frac{d}{dt}\frac{\delta L}{\delta {\dot{u}}_j\left(\mathbf{x},t\right)}-\frac{\delta L}{\delta {u}_j\left(\mathbf{x},t\right)}+\frac{\delta D}{\delta {\dot{u}}_j\left(\mathbf{x},t\right)}=0, $$

(7.23)
where δL/δu j and $$ \delta L/\delta {\dot{u}}_j $$ stand for the functional derivatives. We can easily calculate all the terms on the left-hand side of the above equation and obtain


$$ \frac{d}{dt}\frac{\delta L}{\delta {\dot{u}}_j\left(\mathbf{x},t\right)}=\frac{d}{dt}\kern0.8em \rho {\dot{u}}_j\left(\mathbf{x},t\right)=\rho {\ddot{u}}_j\left(\mathbf{x},t\right), $$

(7.24)
where $$ {\ddot{u}}_j=\partial {\dot{u}}_j/\partial t $$ and


$$ \begin{array}{c}\frac{\delta L}{\delta {u}_j\left(\mathbf{x},t\right)}\kern-1em =-{\displaystyle \underset{\varOmega }{\int }{d}^3{\mathbf{x}}^{\boldsymbol{\prime}}\frac{1}{2}{\displaystyle \sum_{\kern0.3em i,k,l,m}\frac{\delta }{\delta {u}_j\left(\mathbf{x},t\right)}\left({\lambda}_{iklm}{u}_{ik}{u}_{lm}\right)}}\\ {}=-{\displaystyle \underset{\varOmega }{\int }{d}^3{\mathbf{x}}^{\boldsymbol{\prime}}\frac{1}{2}{\displaystyle \sum_{i,k,l,m}{\displaystyle \sum_{\alpha, \beta}\frac{\delta {u}_{\alpha \beta}\left({\mathbf{x}}^{\mathbf{\prime}},t\right)}{\delta {u}_j\left(\mathbf{x},t\right)}\frac{\partial }{\partial {u}_{\alpha \beta }}\left({\lambda}_{iklm}{u}_{ik}{u}_{lm}\right)}}}\\ {}=-{\displaystyle \underset{\varOmega }{\int }{d}^3{\mathbf{x}}^{\boldsymbol{\prime}}{\displaystyle \sum_{l,m}{\displaystyle \sum_{\alpha, \beta}\frac{\delta {u}_{\alpha \beta}\left({\mathbf{x}}^{\mathbf{\prime}},t\right)}{\delta {u}_i\left(\mathbf{x},t\right)}{\lambda}_{\alpha \beta lm}\left({\mathbf{x}}^{\mathbf{\prime}}\right){u}_{lm}\left({\mathbf{x}}^{\mathbf{\prime}},t\right)}}}\end{array} $$

(7.25)



$$ \begin{array}{c}\frac{\delta D}{\delta {\dot{u}}_i\left(\mathbf{x},t\right)}\kern-1em ={\displaystyle \underset{\varOmega }{\int }{d}^3{\mathbf{x}}^{\mathbf{\prime}}\frac{1}{2}{\displaystyle \sum_{i,k,l,m}\frac{\delta }{\delta {\dot{u}}_i\left(\mathbf{x},t\right)}\left({\eta}_{iklm}{\dot{u}}_{ik}{\dot{u}}_{lm}\right)}}\\ {}={\displaystyle \underset{\varOmega }{\int }{d}^3{\mathbf{x}}^{\mathbf{\prime}}\frac{1}{2}{\displaystyle \sum_{i,k,l,m}{\displaystyle \sum_{\alpha, \beta}\frac{\delta {\dot{u}}_{\alpha \beta}\left({\mathbf{x}}^{\mathbf{\prime}},t\right)}{\delta {\dot{u}}_i\left(\mathbf{x},t\right)}\frac{\partial }{\partial {\dot{u}}_{\alpha \beta }}\left({\eta}_{iklm}{\dot{u}}_{ik}{\dot{u}}_{lm}\right)}}}\\ {}={\displaystyle \underset{\varOmega }{\int }{d}^3{\mathbf{x}}^{\mathbf{\prime}}{\displaystyle \sum_{l,m}{\displaystyle \sum_{\alpha, \beta}\frac{\delta {\dot{u}}_{\alpha \beta}\left({\mathbf{x}}^{\mathbf{\prime}},t\right)}{\delta {\dot{u}}_i\left(\mathbf{x},t\right)}{\eta}_{\alpha \beta lm}\left({\mathbf{x}}^{\mathbf{\prime}}\right){\dot{u}}_{lm}\left({\mathbf{x}}^{\mathbf{\prime}},t\right)}}}\end{array} $$

(7.26)
where


$$ \frac{\partial }{\partial {u}_{\alpha \beta }}{\displaystyle \sum_{i,k,l,m}\frac{1}{2}{\lambda}_{iklm}{u}_{ik}{u}_{lm}}={\displaystyle \sum_{l,m}{\lambda}_{\alpha \beta lm}\left({\mathbf{x}}^{\mathbf{\prime}}\right){u}_{lm}\left({\mathbf{x}}^{\mathbf{\prime}},t\right)}\equiv {\sigma}_{\alpha \beta}\left({\mathbf{x}}^{\mathbf{\prime}},t\right) $$

(7.27)
is the stress tensor and


$$ \frac{\partial }{\partial {\dot{u}}_{\alpha \beta }}{\displaystyle \sum_{i,k,l,m}\frac{1}{2}{\eta}_{iklm}{\dot{u}}_{ik}{\dot{u}}_{lm}}={\displaystyle \sum_{l,m}{\eta}_{\alpha \beta lm}\left({\mathbf{x}}^{\mathbf{\prime}}\right){\dot{u}}_{lm}\left({\mathbf{x}}^{\mathbf{\prime}},t\right)}\equiv {\sigma}_{\alpha \beta}^{\prime}\left({\mathbf{x}}^{\mathbf{\prime}},t\right) $$

(7.28)
is the dissipative stress tensor (the share stress tensor). From


$$ \frac{\delta {u}_{\alpha \beta}\left({\mathbf{x}}^{\mathbf{\prime}},t\right)}{\delta {u}_i\left(\mathbf{x},t\right)}=\frac{\delta }{\delta {u}_i\left(\mathbf{x},t\right)}\frac{1}{2}\left(\frac{\partial {u}_{\alpha }}{\partial {x}_{\beta}^{\prime }}+\frac{\partial {u}_{\beta }}{\partial {x}_{\alpha}^{\prime }}\right)=\frac{1}{2}\left({\delta}_{i\alpha}\frac{\partial }{\partial {x}_{\beta}^{\prime }}\kern0.3em {\delta}^3\left({\mathbf{x}}^{\mathbf{\prime}}-\mathbf{x}\right)+{\delta}_{i\beta}\frac{\partial }{\partial {x}_{\alpha}^{\prime }}{\delta}^3\left({\mathbf{x}}^{\mathbf{\prime}}-\mathbf{x}\right)\right) $$

(7.29)
and
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Mar 22, 2018 | Posted by in BIOCHEMISTRY | Comments Off on Rheological Basis of Magnetic Resonance Elastography

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