Angle Dependency of Systolic Strain
Measurements using Strain Rate Imaging.
ByAndreas Heimdal, dr. Ing.
Last updated: Monday, 24Aug1998 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:
= S = (LL_{0})/L_{0}

(1)

Strain rate is defined as the rate of deformation:
The strain rate in a spatial direction x can be shown to equal the
spatial
gradient of the velocity component in that direction:
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 S
_{x} = S
_{xx}
where x is any coordinate. Locally for each muscle segment, we define
the
coordinates:
 r  along the ultrasound beam, positive towards the transducer
 u  circumferential, clockwise seen from the apex
 v  meridional (longitudinal), from apex to base
 w  transmural, from endo to epicard
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
vaxis and the raxis, so that zero degrees corresponds to measuring
along
the muscle in the meridional direction. We assume that the angle is in
the vwplane (long axis or apical views), so the problem becomes
twodimensional.
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:
and the reverse relation becomes:
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:
v_{v} = v SR_{v}
v_{w} = w SR_{w}

(#)

These velocity components are shown in Figure 2.
Figure 2. Illustration of the velocity
components
v_{v} , v_{w} andv_{r},
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 v_{v} , and
consequently
v_{r}, normally are negative during systole.
We want to find the strain rate along the ultrasound beam
Notice that the velocity v
_{r} is defined positive towards the
transducer, i.e., in negative rdirection. This is the usual definition
in Doppler imaging. Since the center of the muscle is not moving, this
can be simplified to
where dr now is the distance over which the strain rate is measured.
Since
the beam has the angle "a" to the vaxis, the velocity along the
ultrasound
beam becomes
v_{r} = v SR_{v} cos(a) + w SR_{w}
sin(a)

(#)

The coordinates can also be written
v = dr cos(a)
w = dr sin(a)

(#)

so the measure strain rate becomes:
SR_{r} = SR_{v} cos^{2}(a) + SR_{w}
sin^{2}(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 uwplane (short axis view) by interchanging u and v, and
redefining
"a" as the angle between the uaxis and the raxis. 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:
(S_{u} + 1)(S_{v} + 1)(S_{w} +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.
S
_{u} 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. S
_{v} has also
been
measured directly using MR tagging, but the fractional shortening of
the
major axis can also be used as an indication. S
_{w} 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:
S_{x} = (2E_{xx} + 1)^{1/2}  1

(#)

Several studies have showed that there is a difference between S
_{w}
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 
S_{u} 
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 
S_{v} 
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 Mmode echo 
Long axis fractional shortening 
S_{w} 
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 S
_{u} = 0.19, S
_{v}
= 0.17 , S
_{w} = 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 S
_{u} and S
_{v} for the
systole:
Using the incompressibility equation we then get the relation
S_{w} = S_{v}(kS_{v}+1+k)/((S_{v}+1)(kS_{v}+1))

(#)

Figure 3 shows the relation for k = 0.5, 1.0, 1.5, 2.0 and 2.5.
Figure 3. Assumed relation between S_{w}
and S_{v} for the systole. The parameter k is the linearity
constant
in the assumed linear relationship between S_{u} and S_{v}.
Methods and Results
The measured strain (S
_{r}) was calculated for angles of
insonation
between 0 and 90 degrees, using the following method:
 A value for S_{v} is assumed, and the following is
repeated for
each angle "a".
 S_{u} and S_{w} are found using the
relations S_{u}
= k S_{v} and S_{w} = S_{v}(kS_{v}+1+k)/((S_{v}+1)(kS_{v}+1)).
 The corresponding strain rates are found using SR = (ln(S +1))/T
 The strain rate in the radial direction is found using SR_{r}
=
SR_{v} cos^{2}(a) + SR_{w} sin^{2}(a)
 Finally, the measured strain is found as S_{r} = exp(SR_{r}
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 S
_{v}, 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 S
_{w}.
Figure 4. The angle dependence for k
=
0.5. Each curve represents different values for the meridional systolic
shortening S_{v}.
Figure 5. The angle dependence for k
=
1.0. Each curve represents different values for the meridional systolic
shortening S_{v}.
Figure 6. The angle dependence for k
=
1.5. Each curve represents different values for the meridional systolic
shortening S_{v}.
Figure 7. The angle dependence for k
=
2.0. Each curve represents different values for the meridional systolic
shortening S_{v}.
Figure 8. The angle dependence for k
=
2.5. Each curve represents different values for the meridional systolic
shortening S_{v}.
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:
 Firstly, the strain rate has been assumed to be constant during
the time
T. This was done to get a simple formula to calculate strain rate from
strain. The strain rate is needed in the calculations of the angle
dependence.
 To be able to get simple expressions we have also assumed that
the strain
rate is spatially homogeneous. In the muscle, the strain rate is
probably
changing during time and space, and this will change the shape of the
curves.
 The formula for incompressibility assumes that u, v and w are the
normal
strain directions. This is not necesseraly true, especially since the
ventricle
is known to have about 15 degrees rotation from the base to the apex
around
the major axis during the systole. When we insert values from the
literature,
we see that the formula is not fulfilled. Therefore the relations
between
S_{w} and S_{v} are probably somewhat different than
what
we have used, and thus the curves will be different.
These uncertainties make it difficult to exactly
predict
the angle dependence, but still the curves give an impression of the
behaviour.