Strain by speckle tracking:
Strain and strain rate can be measured by speckle
tracking, which is pattern recognition and tracking of
the speckle pattern through the cardiac cycle. The
fundamentals
of speckle tracking is given in the basic
ultrasound section.
Humpback whale
diving. Each humpback has an unique (speckle)
pattern on the underside of the tail (and flukes).
Thus each individual can be identified by its
speckle pattern. Photographs at different times
and places can thus track the wandering of each
individual all over the area it wanders, without
recourse to anything else than the pattern.
 Speckle tracking! This is thus a method with low
frame rate, giving mainly the extent of wandering
over a long time period (the sampling interval).
To measure swimming velocity, a Doppler sonar
would have been useful.


Kernel displacement Motion
of one kernel can thus be measured throughout
one heart cycle. Displacement curve obtained by
tracking through a whole heart cycle shown to
the right, derived velocity curve shown
below.Velocity can be derived from the motion
curve or calculated by the motion divided by the
frame interval. 
From two different kernels,
the relative displacement per
distance and hence,
strain can be derived.
Likewise the relative velocity per distance
(velocity gradient / strain rate) can be
calculated or derived from the strain curve. 
Strain by speckle tracking has been validated by
ultrasonomicrometry in the longitudinal direction (
124)
as well as for rotation (
125).
One way of using this approach, is to place defined
ROI in the myocardium at the segmental borders and measure
segmental
strain and strain rate directly by changes in segment
length.


With kernels
at all segmental borders, segmental motion and
deformation can be tracked, as shown to the right. 
Segmental
strain by speckle tracking, applying the
principle shown to the left. The length
variations of the segments between the kernels
kan be followed through the heart cycle. 
From strain, the strain rate can be derived by temporal
derivation.
In speckle tracking, the relation to
Lagrangian and
Eulerian strain may vary:
If speckle tracking is used to track the relative positions
of two kernels, the strain will be derived form the relative
displacement, divided by the distance between the kernels.
If the denominator is the initial (end diastolic) distance,
this gives the Lagrangian strain, if it is the instantaneous
distance, it will be the Eulerian strain. Temporal
derivation of Lagragian strain will result in Lagrangian
strain rate, derivation of Eulerian strain the Eulerian
strain rate.
To comply with custom (which is Lagrangian strain, but
Eulerian strain rate), a conversion formula has to be
applied.
In principle, pure speckle tracking is direction
independent, and can track crosswise. However, lateral
resolution is important in delineating the speckles in the
lateral direction. If the lateral resolution is low,
the interpolation will result in a "smeared" picture, with
speckles that are nor so easily tracked in the lateral
direction. In addition the lateral resolution decreases in
depth with sector probes.


LOngitudinal
speckle tracking in apical 4 chamber view. The
resulting tracking of the kernels shown in motion.
As can be seen, with a drop out
apicolateral, this ROI tracks less than perfect,
giving too low strain both in LA and MA segments. 
Speckle
tracking can be applied crosswise. In this
parasternal long axis view, the myocardial motion
is tracked both in axial and transverse
(longitudinal) direction. It is evident that the
tracking is far poorer in the inferior wall, due
to the poor lateral resolution at greater depth. 
Also, despite the more or less angle independent speckle
tracking, strain by speckle tracking is not entirely angle
independent, due to the geometric effect.
Insonation
angle and strain
Both tracking of tissue and Doppler are affected by the
angle between the ultrasound beams and the true direction of
longitudinal shortening.
Angle distortion in strain.
As we have seen measuring or tracking along an ultrasound
line at an (insonation) angle to the direction of the true
direction of the motion, results in an over estimation by
the cosine of the isonation angle, as discussed in
the
basic ultrasound section:
However, in strain, being relative deformation, this angle
problem is eliminated (in principle):
As both L1 and L2 are over estimated with the
same factor, the ratio between the estimated lengths
is the same, and the strain is not affected.
However, this argument is only valid in one dimensional
strain. In three dimensional objects, there are simultaneous
strain in
more
than one direction, and in
incompressible
objects, some strains will be positive, other negative
as volume is constant, and thus will detract from each other
as more than one strain component is measured at an angle.
Geometric distortion
The main point of strain measurement is that strain is
basically measured along one line. If this is not parallel
to the direction of one of the
normal
strains, the alignment error may add to or detract
from the main strain measurement as shown below: In an
incompressible object,
there is simultaneous strain in the transverse direction, in
order to keep the volume constant, and the two strain
components are opposite and will detract from each other (
1,
2,
7).
This
results in further reduction in the measured strain and
strain rate (
7).
It is evident that the insonation angle becomes important
again, as the relative contributions of each strain
component depends on the insonation angle:


An object
undergoing longitudinal shortening (negative
strain). If the object is incompressible,
there will be simultaneous transverse
expansion (positive strain) as well. This
means that the direction of measurement (along
an ultrasound beam or an ROI) matters, the
more skewed the beam, the more transverse
strain is incorporatedinto the measurement,
detracting from the absolute value of the
longitudinal strain. The in between line will
measure both to the same degree, i.e. zero
strain, while the beams in the more transverse
direction will measure increasing positive
strain, detracting from the absolute value of
the longitudinal strain. 
The effect of skewed
insonation on the measurement of longitudinal
strain. The skewed line defines a rectangular
section. The longitudinal strain of this section
is the result of the displacement of the two
ends. In the skewed line, however, the total
displacement is the sum of the longitudinal and
transverse displacement vectors, and thus a
larger shortening than the longitudinal strain
along the skewed line. The greater the angle the
greater the error.

Looking at the figure to the left, the longitudinal strain
is (L
_{1}L
_{2})/L
_{1}. But the
shortening
L is also equal to the total
displacement D
_{L} of the ends of the rectangle, D
_{L}
= D
_{L1} + D
_{L2}. The baseline length
measured along the skewed beam, is L
_{1M} = L
_{1}
/ cos (
). The total measured displacement,
however, is the sum of the longitudinal and transverse
vectors, and thus D
_{LM} = D
_{L} / cos (
) + D
_{T} / sin (
).
Thus, the transverse component will add to the total
shortening, but not to the baseline length, and thus, will
detract from the magnitiude of the longitudinal shortening.
This
simplified argument is for illustration purposes only:
 It presupposes a symmetrical object as in the
discussion on imcompressibility above.
 In a three dimensional object, this two dimensional
reasoning is only valid if the imaging plane is the
zplane, or else there will be a z component
detracting further from the longitudinal measured
strain.
 The myocardial
strain geometry is more complex, as both
systolic longitudinal and circumferential strains are
negative, and the wall thickening is far greater in
magnitude than the other two.
Angle dependency of speckle tracking. It all
depends on the application's ability to track along
the wall, and across the ultrasound beams.
This is illustrated below.




Apical long axis view from
a healthy person. The base of the inferolateral
wall is positioned at an angle to the
insonation. 
An Mmode line along the
ultrasound beam shows positive strain, by the
divergence of the Mmode lines (speckles). The
wall thickening is predominant. Interpreted as
longitudinal strain, this vould mean systolic
elongation . dyskinesia.

Adjusting the direction of
the Mmode, in this image the longitudinal and
transverse strain balance, as seen by the
parallel lines (speckles). This results in near
zero strain. Interpreted as longitudinal strain,
this would mean akinesia.

Further adjustment of the
Mmode line, will result in best alignment of
the line with the wall, and here there is
negative strain seen by the convergence of the
lines (speckles). This shows true longitudinal
shortening, although if there is a residual
component of transverse strain is uncertain.

Angle in speckle tracking
In speckle tracking, in principle, if tracking is perfect,
there should be no angle effect, as the tracking occurs in
the direction of the longitudinal shortening. However, as
explained elsewhere, the lateral resolution is far poorer
than the axial, and thus there is an angle dependency as
well. And in practice, the effect on speckle tracking
depends on many factors, such as the orientation of the ROI,
the ability of the algorithm to take ROI direction into
account when tracking (and interpretation) , and the lateral
resolution in the image. This is illustrated below, where 2D
strain is applied to the same loop as above.


The basal inferolateral
segment seems to follow fairly well, but still
there is some thickening that interferes with
the analysis of the shortening.

And the strain curve from
the basal inferolateral segment (yellow curve)
shows apparent elongation during systole, as the
algorithm interprets the thickening as
shortening. 
As seen there is positive strain during the whole systole in
the posterobasal segment as well, just as in the simple
Mmode tracking above. In this case, we see that the
speckles are very wide in that area, indicating a very low
ability to track in the lateral direction.
Angle independence of speckle tracking
presupposes good lateral resolution. It is influenced
by depth, line density and near shadows.
Angle in tissue Doppler
In tissue Doppler the angle distortion of both velocity and
length measurement must be taken into account. The velocity
measurement is affected inversely by the insonation angle:
Insonation angle deviation. The angle alpha
reduces the velocity difference by the cosine to
alpha, and increases the length measurement by the
same, thus the velocity gradient is reduced by the
square of the cosine to alpha. However, the geometric
distortion that detracts transverse thickening from
longitudinal shortening, is also reduced (by the
square of the cosine to the angle beta, but this is
equal to the square of the sinus to the angle alpha,
so the angle effect is in fact less.
Velocity (and thus velocity difference is reduced by the
cosine of the insonation angle. The length measurement,
however, is increased by the cosine of the same angle, and
thus:
Longitudinal velocity and motion are
v_{l }and_{ }x_{l},
respectively. Transverse velocity and motion are v_{t
}and_{ }x_{t}.
Along the ultrasound beam, the longitudinal
and transverse velocity components are:
and
the total velocity is the sum of
the two components:
The total distance is the
diagonal in the rectangle:
The formula is valid
for velocity differences, as can be
shown by setting v_{2} = 0.
Then:
It must be emphasised that this argument is
simplified, to illustrate the principles, and not an
algorithm for actual angle correction. The same
reservations apply here as to basic geometry:
This simplified argument:
 It presupposes a symmetrical object as in the
discussion on imcompressibility above.
 In a three dimensional object, this two
dimensional reasoning is only valid if the
imaging plane is the zplane, or else there will
be a z component detracting further from the
longitudinal measured strain.
 The myocardial
strain geometry is more complex, as both
systolic longitudinal and circumferential
strains are negative, and the wall thickening is
far greater in magnitude than the other two.
For a more complete mathematical analysis, see (1).
The angle problem is also specially analysed in:
Andreas
Heimdal. Angle dependency of systolic strain
measurements using strain rate imaging.
In addition there are shear strain tensors and
finally there are angle related artefacts that are
not subject to mathematical analysis, which may be
even more important as discussed here.
However, as the two methods are different, also
relatihg to the fundamental principles, they will
react differently to the angle problem.
And
it is not always the tissue Doppler method that
has the greatest angle problem, as seen here:
A tissue Doppler recording from the same patient as
above shows about the same skewness as above,
however, there is no positive systolic strain in the
posterobasal segment.


Surprisingly,
the tissue Doppler in this case
doesn't show the same effect. In
this tissue Doppler based strain image
(systolic tracking only), the
inferolateral myocardium, can be seen to
be red (negative strain) through the
whole systole, thus no inverted strain
measurement. .

And this is
confirmed by the segmental strain
curves. This is due to differences in
the ROI size, placement and strain
length.

In this case, the speckle tracking is more angle
dependent than tissue Doppler, although this may
vary, as shown in
the
pitfalls section.
Tissue Doppler is still angle dependent, but
segmental
strain
by tissue Doppler has only the basic
limitation common to all Doppler measurement. And
using it in
combination
with transverse speckle tracking, eliminates
this angle dependency also, to the same degree as in
speckle tracking (meaning that
speckle
tracking is not altogether angle independent).
However, drop outs and reverberations will affect the
tracking, and in the lateral direction, low lateral
resolution will "
smear"
the speckles in the lateral direction, making tracing less
perfect, as can be seen in the parasternal long axis image
above. It also means that the lateral tracking will be
poorer with increasing depth (as the lines diverge as well
as becoming wider), as also discussed
in
the measurements section.
Other applications, using the whole 2D field, may use the
tracking to acquire a velocity field by dividing by the FR
^{1},
thus starting with the velocity field as in tissue Doppler,
which means deriving strain rate, and then integrating to
strain. Again, however, it will be possible to divide by the
initial (end diastolic) length, which gives Lagrangian
strain rate and strain, or by the instantaneous length (as
material points are tracked), giving Eulerian strain and
strain rate.