Det medisinske fakultet


Angle Dependency of Systolic Strain Measurements using Strain Rate Imaging.

ByAndreas Heimdal, dr. Ing.


Last updated: Monday, 24-Aug-1998 16:54:44 MET DST

Abstract

To find the angle dependency of Strain Rate Imaging we assumed that the principal strain directions in a myocardial muscle segment during systole were the circumferential, the meridional and the transmural direction. Using the equation for incompressibility of the tissue, and assuming a linear relationship between two of the strain components we calculated the strain in the direction of the ultrasound beam. We found that the strain in the meridional direction is measured with the correct sign up to between 27 and 35 degrees, depending on the strain component model, where zero degrees is in the meridional direction, and 90 degrees is in the transmural direction.

Introduction

    As all other Doppler based imaging methods, Strain Rate Imaging is also angle dependent. The method measures only the strain rate in the direction of the beam. If the principal strain components are in other directions, the measured strain rate, and the calculated strain, will depend on the beam angle.
    In this paper we give a theoretical description of the angle dependence of systolic strain measurements.

Theory

Definitions

Strain is in this work defined as the relative change of length:
epsilon = S = (L-L0)/L0
(1)
Strain rate is defined as the rate of deformation:
SR = dS/dt
(2)
The strain rate in a spatial direction x can be shown to equal the spatial gradient of the velocity component in that direction:
SRx = dvx/dx
(3)
     We ignore shear strain and shear strain rates in this work. Therefore, only one coordinate is needed to define each strain component, and to simplify the notation we write Sx = Sxx where x is any coordinate. Locally for each muscle segment, we define the coordinates: where u, v and w will be approximatley perpendicular, as shown in Figure 1.

Figure 1. Definition of the coordinates r, u, v, and w, and the insonation angle "a". Notice that the angle "a" is negative in this illustration, which is the usual case when imaging from the apex.

The origo (u,v,w)=(0,0,0) does not need to be defined in relation to the macroscopic ventricle geometry, and can be anywhere in the imaged muscle segment.
    Furthermore we define "a" as the angle between the v-axis and the r-axis, so that zero degrees corresponds to measuring along the muscle in the meridional direction. We assume that the angle is in the v-w-plane (long axis or apical views), so the problem becomes two-dimensional. Notice that the angle is negative in Figure 1. 

Relation between strain and strain rate

    The strain after a time T is calculated from the measured strain rate as
S = exp(integral(SR dt)) - 1
(#)
where the integral is taken from t = 0 to t = T. If one assumes that the strain rate is constant during that time, the formula can be simplified to:
S = exp(SR T) - 1
(#)
and the reverse relation becomes:
SR = (ln(S +1))/T
(#)
 

Angle dependence

    Without loosing generality, we can assume that the point (v,w)=(0,0) is not moving. If the strain rate is spatially homogeneous in the muscle segment, the muscle point (v,w) will then move with the velocity components:
vv = v SRv
vw = w SRw
(#)
  These velocity components are shown in Figure 2.
Figure 2.   Illustration of the velocity components vv , vw andvr, the distance dr and the angle "a" in a small muscle segment. All the parameters are drawn positive, but notice that the angle "a" is usually negative when imaging from the apex, and that vv , and consequently vr,  normally are negative during systole.

We want to find the strain rate along the ultrasound beam 
SRr = dvr/dr
(#)
Notice that the velocity vr is defined positive towards the transducer, i.e., in negative r-direction. This is the usual definition in Doppler imaging. Since the center of the muscle is not moving, this can be simplified to 
SRr = vr/dr
(#)
where dr now is the distance over which the strain rate is measured. Since the beam has the angle "a" to the v-axis, the velocity along the ultrasound beam becomes 
vr = v SRv cos(a) + w SRw sin(a)
(#)
The coordinates can also be written 
v = dr cos(a)
w = dr sin(a)
(#)
so the measure strain rate becomes: 
SRr = SRv cos2(a) + SRw sin2(a)
(#)
Notice that when imaging from the apex, the angle "a" will be close to zero and negative for most of the ventricle.
    The same formulas will apply if one is imaging in the u-w-plane (short axis view) by interchanging u and v, and redefining "a" as the angle between the u-axis and the r-axis. In that case "a" will have values from -180 to +180 degrees.

Incompressible material

    Since the cardiac muscle tissue can be considered incompressible, the following relation holds: 
(Su + 1)(Sv + 1)(Sw +1) = 1
(#)
under the assumption that u, v and w are the principal strain directions, and that there are no shear strains. There are unlimited solutions that satisfy this equation, but each parameter has only a small range that is clinically relevant.
    Normal strain values can be found in the literature. Su has been measured directly using MR tagging, but since the circumference of the ventricle has an approximately linear relationship to the radius, the systolic midwall fractional shortening of the minor axis in the left ventricle can also be used. Sv has also been measured directly using MR tagging, but the fractional shortening of the major axis can also be used as an indication. Sw is systolic wall thickening, and has been measured in numerous studies. The values and references are given in Table I. In some publications, the strain was given as the Greens strain tensor E. The strains in Table I were then converted from the diagonal components of E using: 
Sx = (2Exx + 1)1/2 - 1
(#)
Several studies have showed that there is a difference between Sw in the different walls, thus the large standard deviation (48, 49).
 Table I    Normal values for systolic strain in the human myocardium

Mean value Standard deviation Reference Method Derived from
Su -0.18 0.03 Young AA, et al.,  1994. (50)
MR tagging Local Lagrangian strains
-0.21 0.04 Shinzu G, et al. 1991. (51)
Cineangiogram Midwall short axis fractional shortening
Sv -0.16 0.03 Young AA, et al.,  1994. (50)
MR tagging Local Lagrangian strains
-0.24 0.05 de Simone G et al., 1997. (52)
2D and M-mode echo Long axis fractional shortening
Sw 0.53 0.24 Pandian et al.,  1983. (53)
Short axis 2D echo Wall segment thickening
    These normal values do not exactly satisfy the incompressibility equation, but by introducing a small offset in each value the relation holds. One solution is for instance Su = -0.19, Sv = -0.17 ,  Sw = 0.48. These are all values at a distance of 0.2 standard deviations from the normal mean value.
    A reasonable proposal might be that there is a linear relationship between Su and Sv for the systole: 
Su = k Sv
(#)
Using the incompressibility equation we then get the relation 
Sw = -Sv(kSv+1+k)/((Sv+1)(kSv+1))
(#)
Figure 3 shows the relation for k = 0.5, 1.0, 1.5, 2.0 and 2.5.
Figure 3.    Assumed relation between Sw and Sv for the systole. The parameter k is the linearity constant in the assumed linear relationship between Su and Sv.

Methods and Results

The measured strain (Sr) was calculated for angles of insonation between 0 and 90 degrees, using the following method:
  1. A value for Sv is assumed, and the following is repeated for each angle "a".
  2. Su  and Sw are found using the relations Su = k Sv  and Sw = -Sv(kSv+1+k)/((Sv+1)(kSv+1)).
  3. The corresponding strain rates are found using SR = (ln(S +1))/T
  4. The strain rate in the radial direction is found using SRr = SRv cos2(a) + SRw sin2(a)
  5. Finally, the measured strain is found as Sr = exp(SRr T) - 1
The time T was set to 0.1 sec. The results are shown in Figures 4 to 8 for different values for k. For each curve in the figures, the strain given at zero degrees is Sv, i.e., the strain in the meridional direction (wich is the one we assume to be imaging when we image from the apical view). The strain at 90 degrees is Sw.
 
Figure 4.    The angle dependence for k = 0.5. Each curve represents different values for the meridional systolic shortening Sv.
 
Figure 5.    The angle dependence for k = 1.0. Each curve represents different values for the meridional systolic shortening Sv.
Figure 6.    The angle dependence for k = 1.5. Each curve represents different values for the meridional systolic shortening Sv.
Figure 7.    The angle dependence for k = 2.0. Each curve represents different values for the meridional systolic shortening Sv.
Figure 8.    The angle dependence for k = 2.5. Each curve represents different values for the meridional systolic shortening Sv.

Notice that the angle where the strain changes sign is reduced from 40 degrees for k = 0.5 to 27 degrees for k = 2.5.
 

Discussion and conclusions

    Several assumptions have been made in the calculations, and these have to be taken into account when discussing the results:   These uncertainties make it difficult to exactly predict the angle dependence, but still the curves give an impression of the behaviour.




Editor: Asbjørn Støylen, Contact address: asbjorn.stoylen@medisin.ntnu.no, Updated: May 2004.