Basic concepts.

Motion and deformation

Asbjørn Støylen, Professor, Dr. med.

Department of Circulation and Medical Imaging,
Faculty of Medicine,
NTNU Norwegian University of Science and Technology

Contact address: asbjorn.stoylen@ntnu.no

 Motion. Floating iceberg in hurricane, Antarctic sound. Deformation. Calving glacier in Marguerite Bay, Antarctic peninsula.

Website updated: September 2015.     This section updated: September 2015.

This section:

Deals with the basic concepts of motion (displacement and velocity) and deformation (strain and strain rate), and how these concepts are inter related. Previous sections from the maths section about mathematical relations as well as the extended discussion on Lagrangian vs Eulerian strain rate has been included. Further, how strain behaves in three dimensional objects, and how the different strain components are inter related in an incompressible object. It is important to realise that even if speckle tracking is in principle angle independent, strain measurement by speckle tracking is not.

References for all sections

Motion vs deformation

When considering the different modalities of echocardiography, the distinction between motion and deformation imaging is important. Displacement and velocity are motion. A stiff object may move, but not deform. A moving object does not undergo deformation so long as every part of the object moves with the same velocity. An object that deforms may not move in total relative in space, but different parts has to move in relation to each other for the object to deform. The object may then be said to have pure translational velocity, but the shape remains unchanged. Over time, the object will change position – this is displacement. Velocity is a measure of displacement per time unit.

,Strain and strain rate are deformation measures. If different parts of the object have different velocities, the object has to change shape. This is illustrated below.

Motion imaging.  Parametric (colour) image. A train staring, running and stopping.  The engine starts first, the connection between carriages has to stretch, before the next carriage is brought into motion. When all carriages are in motion, the train runs evenly. In stopping, the engine stops fist, then the connection between carriages has to be compressed before the next carriage stops, until all carriages are motionless. In this parametric image all carriages that are in motion are coloured red. However, both at standstill (the whole train is white) and running evenly (whole train red), there is no deformation, only motion. The engine keeps the connectors between the coaches stretched to a fixed length by pulling at constant speed, so the engine and coaches remain in the same position relative to each other.  The deformation occurs when any two carriages are moving with different velocities. This is shown below.

Deformation imaging.  Parametric (colour) image. This is the same figure as above, but in his image, the two carriages between which deformation occurs are shown in either cyan (stretching) or orange (compression), while the other carriages where no deformation occurs are shown in green. When the train is immovable, there is no deformation, the whole train is green. AS the engine starts, there is stretching between that and the first carriage (cyan). Once the first engine is at the same velocity as the engine, no further stretching (deformation) of that connection occurs, while the stretching has moved backwards in the train to the next connection. The stretching can be seen as a wave of deformation (cyan) moving backwards in the train. (Another example of this is given in the section on diastolic strain rate). Once all carriages move with the same velocity, no further deformation occurs, and the whole train has even motion, and is coloured green again. When all parts of the object have the same motion, there is no deformation.  In stopping the opposite occurs, there is compression between engine and first carriage, then between first and second carriage, and so forth.  Again the compression can be seen as an orange wave moving backwards through the train.  When the train is at standstill, no further deformation occurs. When different parts of the object have different motion, there is overall deformation of the object.  Deformation is thus differential motion.

Comparing the two images above, one thing is evident. As the carriages have acquired a motion, even if this is the same as the neighboring carriage, they are all visualized in the same colour. In this case, the passive moving carriages is tethered to the carriage in front. The deformation image below is able to separate those carriages that move with different velocities, where there is stretching or compression of the connection between them, and those that are passively moving along. Thus there is an additional spatial resolution in deformation imaging compared to that of motion.

In the myocardium, as there is a stationary apex and motion of the base, there is differential motion, i.e. deformation as discussed here.

The images below shows how the impression from motion may be erroneous, while deformation gives the correct answer:

 Short axis view. The inferior wall shows systolic outward motion, thus apparent pathological motion; dyskinesia. Reconstructed M-mode through the inferior wall from the same loop. The inferior wall shows systolic thickening, thus normal deformation, normokinesia.

This example is more discussed in detail here.

Infarcted segments may be totally akinetic, but still being pulled along by active segments, showing motion without deformation. The passive motion due to active contraction of other parts is called tethering, the passive parts being tethered to the active. In the example with trains above, when the train moves the coaches are tethered to the engine, and moves by tethering when all of the train has the same speed.
In the heart tethering is seen where passive segments are pulled along by active segments as shown below.

 ﻿﻿ Tethering: The basal and midwall segments are infarcted, and are being pulled along by the active apical segment. The whole inferior wall seems stiff. Motion curves from basal, midwall and apical segments. All three segments move with about the same motion, meaning that there is no deformationof the two basal segments (no differential motion). Strain curves from the same segments, showing no systolic deformation in the basal segment, some slight shortening of the midwall segment, and shortening of the apical segment, which, by contraction is the main source of the movement of the two tethered segments.

But this means that tethering can be seen by the lack of offset between the displacement curves (middle image), or by the zero strain values (left image) in the base. This is described in detail here.

Strain and strain rate

Strain, in daily language means, “stretching”. Basically, the strain is the deformation itself, not the force that cause stretching, this is "stress". The relation between the stress and the strain is the compliance.

 Strain. Thingvellir, Iceland is situated on the rift between the Eurasian and American continental plates, which are sliding apart. Thus the area is expanding (positive strain), which can be seen by the ground cracking up.

In scientific usage, the definition is extended to mean “deformation”. The concept of strain is complex, but linear strain can be defined by the Lagrangian formula:
which describes deformation relative to baseline length.

Where  is strain, L0 = baseline length and L is the instantaneous length at the time of measurement as shown below. Thus strain is deformation of an object, relative to its original length. By this definition, strain is a dimensionless ratio, and is often expressed in percent. From the formula, it is evident that positive strain is lengthening or stretching, in accordance with the everyday usage of the term, negative strain is shortening or compression, in relation to the original length. By using this definition, however, when an object is stretched from Lo, strain will remain positive during compression as long as the object remains longer than Lo, and vice versa after compression, strain will be positive during stretching so long as the object remains shorter than Lo.  (This is treated in more detail here).

The strain rate is the rate by which the deformation occurs, i.e. deformation or strain per time unit. This is equivalent to the change in strain per time unit.
The unit of strain rate is /s, or s-1. The strain rate is negative during shortening, positive during elongation. Thus, two objects can have the same amount of strain, but different strain rates as shown below:

 Strain An object undergoing strain. In this case there is a 25% elongation from the original length (L0). The Lagrangian strain is then: Thus, according to the Lagrangian formula there is positive strain of 25% or 0.25. Strain rate. Both objects show 25% positive strain, and both corresponds to the object to the left, but with different strain rates, the upper has twice the strain rate of the lower.  If the period is one second in the upper object, the strain rate is 25% or 0,25 per second, giving a strain rate of 0.25 s-1. The lower object has twice that period, i.e. half the strain rate, which then is 0.25 / 2 seconds = 0.125 s-1 . In these cases, the strain rate is constant.

The main point is that there may be motion without deformation, and deformation without much motion, the deformation is due to the differential motion within an object:

This means that there can be motion without deformation, but no deformation without motion - differential motion:
• In an object where no parts are moving, there is no deformation.
• When the object is moving, but all parts are moving with the same velocity, there is no deformation no differential motion.
• When an object is not moving over all, but different parts of an object are moving - differential motion, there is deformation.
• When an object is moving overall, and different parts of an object is moving differently - differential motion, there is deformation.

 Strain rate. In these four cases there are different instances of deformation and motion. Object A does not move, none of the end points has motion (V1 = V2 = 0), and thus, there is no motion and no deformation (SR = 0). Object B has motion, but the two points 1 and 2 move with the same velocity. Thus there is motion, but no differential motion, and thus no deformation. (SR = 0). No elongation of the object can be seen. In object C point 1 has no velocity (V1 = 0), but point 2 has velocity, thus the two velocities are different, the object has differential velocities and motion, and thus there is deformation (SR does not equal 0), elongation is very visually evident. There is little motion, although one might argue that the midpoint does move a little. Object D shows velocities at both end points, thus there is definitely motion, and in addition V1 and V2 are different. Thus there are differential velocities and differential motion, and there is deformation, visually, elongation in addition to the motion is evident. Looking at displacement instead of velocity, the change in length of the objects is the difference in displacement of the two end points: L = L0 + (D2 - D1), L = D2 - D1,  and  = (D2 - D1)/L0

Thus, the strain rate is equal to the differential velocities of the object, strain is equal to the differential displacement, in the examples above the difference between points 1 and 2.

Derivation and integration

• As an object moves, it has velocity (V). The velocity is displacement (D) per time unit, but as velocity seldom is constant this means in general: , the temporal derivative of displacement.
• The rate of deformation is equal to the differences in velocity per lenght unit: , where d is the distance between two points of velocity measurement, i.e. the spatial derivative of velocity.
• The deformation, (strain) is equal to the differences in displacement (D) per length unit: , i.e. the spatial derivative of displacement.
• And finally, as has been mentioned, strain rate is the rate of strain: , i.e. the temporal derivative of strain.
All of the derivations can of course be reversed by integration, so velocity, displacement, strain rate and strain are all interrelated:

Relation between strain rate, strain, velocity and displacement. From one dataset (e.g. a velocity field), all three other parameter sets can be derived.

The derivation and integration processes are illustrated graphically with curves below, although the meaning of the shape and relations between curves is discussed more in detail here.

The spatial derivation process can be seen applied to velocity curves (top) and displacement curves (bottom), in the horisontal direction from left to right. In both cases the derivation is between two points, and the spatial derivative is the instantaneous difference in velocity or displacement between the two points, divided by the instantaneous distance between them. Thus, the strain rate derived from the two velocity curves is the deformatin rate of the area between them, demarcated by the red ROI top left. (The offset between the two curves is marked in red, but in this case do not mean that the derivative is the area between the curves, each value is the instantaneous distance. ) This is equal to the velocity gradient or displacement gradient, and results in strain rate and strain, respectively. The curves are shown to the left.

The temporal integration is shown in the vertical direction, applied to velocity (left) and strain rate (right). The integration is applied to one curve at a time, but both curves are shown to the left. The integrstion amounts to the sum of (all velocity or strain rate values times the sampling interval). This process results in the area under the curve. The value can be expresses as a new curve. The value of this curve increases when the original velocity or strain rate curve is positive, decreases when they are negative.

Velocity gradient

As we have seen the strain is the differential displacement of the object, while strain rate is the differential velocities of the object.

Any object having differential velocities, have a velocity gradient. For instance, as the outer contour of the LV wall moves least and the endocardial surface most, there has to be a velocity gradient across the wall.

The concept of velocity gradient was introduced by Fleming et al (14). The velocity gradient is defined as the slope of the linear regression of the myocardial velocities along the M-mode line across the myocardial wall. If velocities are linearly distributed through the wall, this is equal to the difference in endocardial and epicardial velocities divided by the instantaneous wall thickness (W).

The definition was extended by Uematsu et al (15) to include the transmural velocity gradient across the parts of the wall where the scan line is not perpendicular to the wall, by the cosine correction of the velocities. The velocity gradient measured in this way, was transmural (radial). As transmural strain rate is the rate of change in wall thickness, the strain rate is:

in other words, the velocity gradient is an estimator of the transmural strain rate, strain per time unit approximates velocity per length unit. The reason this is an approximation only, is that while strain rate is defined in relation to the initial (diastolic) wall thickness, the velocity gradient is a function of the instantaneous wall thickness as shown in the formula.

The velocity gradient can also be applied to the longitudinal velocities.

In the heart, the apex is stationary, while the base of the ventricles move towards the apex (ventricular shortening) in systole. If the velocities are distributed evenly, it means that there will be a velocity gradient along the wall. The spatial derivation of velocities can be approximated:

i.e. change in velocity over a finite distance. If (and only if) velocities are evenly distributed, i.e. the velocity gradient is constant along the wall, this resolves into:

i.e. velocity difference per length unit.

Strain rate is calculated at the velocity difference per length unit /velocity gradient) between two points in the myocardium:
Longitudinally the velocity gradient is the velocity difference between two material points, divided by the instantaneous difference between them:

Longitudinal velocity gradient, where v1 and v2 are two different velocities measured at points 1 and 2, and L the length of the segment between those points.

The distance L changes with time, if v1 and v2 are different. The unit of the velocity gradient is cm/s/cm, which is equal to s-1. The gradient was found as the slope of the linear regression of the tissue velocities. (The linear regression assumes that the velocity distribution is homogeneous.) under the assumption that the velocity gradient is constant over the length L (spatially homogeneous).

Thus, we have seen that
strain rate equals the velocity gradient:

or

If the presumption of constant velocity gradient is not taken into account, although the derivation of the velocity gradient by tissue Doppler is slightly different as discussed below.

But as the the velocities of the two points is the displacement per time , the difference in displacement is the difference in length, and the difference in velocities (velocity gradient) is the difference in length per time.

By this, the longitudinal velocity gradient is:

But as:

then:

and:

In other words, the velocity gradient equals the rate of Eulerian strain and the Eulerian strain equals the time integral of the velocity gradient.

Relation to tissue Doppler:

 The strain rate can be described by the instantaneous velocity gradient, in this case between two material points, but divided by the instantaneous distance between them. In this description, it is the relation to the instantaneous length, that is the clue to the Eulerian reference. Strain rate is calculated as the velocity gradient between two spatial points. As there is deformation, new material points will move into the two spatial points at each point in time. Thus, the strain that results from integrating the velocity gradient, is the Eulerian strain. In this view, the relation to the spatial, rather than material reference is very evident.

The two methods shown above are not identical, But it can be shown that they are equivalent.
In the material velocity gradient the L changes in the spatial it is constant. Thus, they are not equal, except at the point in time when x equals L, then v(x) = v2 and v(x+x) = v1. However, Usually, however, L will differ from x, for most frames and objects, and the velocities will hence differ too. Under the assumption that the strain is equally distributed over the length of the object (spatially constant), SR will still be equal to the velocity gradient, i e the values of the two ratios will be the same. Strain being spatially constant means that the velocity increases linearly along the length as shown in the diagram:

For any L that is different from x, v2 – v1 will be greater or smaller than v(x) – v(x + x) by the same ratio. In the figure, this is evident, as the slope of the curve is the same wherever it is measured. As v1 and v2 are the velocities of the end points of L, the ratios SR and VG will be the same, and thus the expressions are equivalent: SR = VG and the strain rate by tissue Doppler (SR) equals Eulerian strain rate.

But this, of course means that strain rate and strain can be visually assessed by the offset between the curves, as discussed in the display section

The velocity difference varies during the heart cycle, and the distances are shaded red when the differences are negative (v1<v2), and blue when they are positive (v1>v2). The resulting strain rate curve is shown to the left, with negative strain rate shown in red, positive shown in blue. Mark also that the peak strain rate and peak velocities are not simultaneous in this segment.

Then, the longitudinal velocity gradient, velocity per length unit is a measure of longitudinal strain rate, but this is only valid if velocities are evenly distributed, i.e. if the velocity gradient is constant. If not, this is an approximation, which becomes more precise the shorter the L. However, Longer L improves signal to noise ratio as described here, especially if strain rate is derived by linear regression of velocities along the whole of the L. However, the longitudinal velocity gradient seems to be fairly constant:
 Systolic velocity plot through space, from the septal base to the left through the apex in the middle to the lateral base to the lateral base to the right. The velocities seem to be distributed along  fairly straight lines, i.e. there seems to be a fairly constant velocity gradient (in space, but not in time). A nearly straight line. Blue eyed shags (cormorants) at Cabo de Hornos (Cape Horn), Chile.

Thus, while velocities decrease, strain rate is more or less constant ´from base to apex as described below.

 Decreasing velocities from base to apex. Constant systolic strain rate from base to apex, i.e. velocity fgradient is constant (linear) as discussed below.

This is discussed further here.

(In fact: this might be so in the longitudinal direction. However, transmurally, there is a gradient of strain, and thus strain rate across the wall, as shown in the section on myocardial strain. This means firstly, that the transmural velocity gradient is not constant, and secondly that the transmural endo- to epicardial velocity gradient only gives the average strain rate across the wall).

Comparison between velocity and strain rate. left: Velocities from two points; v(x) and v(x+x), separated by the distance x. The strain rate curve is then the instantaneous difference between the two curves, divided by the distance:

As the ratio is the apical minus the basal velocity, the strain rate curves are inverted, compared to the velocities.

Strain rate by linear regression of velocities

Looking at the V-plot again, it is evident that there is a substantial variability in velocity estiamates from pixel to pixel. This, of course, will result in a greater variability of strain rate measurements, as the variability will be the sum of the relative variabilities of the single measures, while the strain rate is the difference, resulting in a substantially worse signal-to-noise ratio.

V-plot, showing the real variability from pixel to pixel of velocities.

 Velocity measures with some amount of noise. Unsmoothed strain rate curves from the same loop. The increase in noise is evident.

Instead of measuring just the velocities at the ends of the offset distance; or respectively, the velocity gradient / strain rate can be calculated as the slope of the regression line of all velocities along the offset distance as described originally (14). With perfect data, the values will be identical, both formulas defining the slope. With imperfect data, this method will tend to make the method less sensitive to errors in velocity measurements, as the value is an average of more measurements.

Strain rate calculated over an offset (strain length) of 12 mm (L). "True" velocities at the end points are v1 = 0 and v2 = 1.2 cm s-1 giving a strain rate of -1.0 s-1 (blue squares), the strain rate is actually the
slope of the line between the points, being equal to (v2 - v1)/L (blue line). Due to random variability of the measurements, the measured values deviate from the slope. Here velocities are sampled for each 0.5 mm along the
strain length (red points), and are seen to be dispersed around the true strain rate line. The regression line through the points (red line) is fairly close to the true strain rate line, and results in a strain rate measurement
of -1.14 s-1. This makes the measurement far less vulnerable to measurement variability than simply measuring the two velocities at the end of the strain length (points in the green open squares), and compute
SR = (v2 - v1)/L shown by the green line, yielding a strain rate of -1.63 s-1.

Strain in three dimensions

Strain in real objects occur in three dimensions. This complicates the matter, as the strain tensor then has more components, the number of components increase by the square of the number of dimensions. This is shown in full for 1- and 2-dimensional strain below:

One - dimensional Lagrangian strain. The length is the only strain component, and thus L is measured along the only coordinate axis, thus L = x

Strain in two dimensions. Above are the two normal strains along the x and y axes, where each strain component can be seen as Lagrangian strain along one main axis.  Below are the two shear strain components, movement of the borders relative to each other. Here there are two strain components, characterised by the tangent to the shear angle alpha.

Thus in two dimensions, the strain tensor has four components, two normal:

And two shear components:

From the figure, it is also evident that:

The whole strain tensor can be written as a matrix:

There may be simultanous shear strain along both axes as well:

In this case there is shear strain in both directions, but still:

Three dimensional objects  can deform along all three axes. Thus, there may be more than one component of strain.
In a three dimensional object, there is the possibility of deformation in three directions. Normal strain is the deformation components along the main axes of a coordinate system. To complicate matters further, there are also shear deformations, which means displacement of the surface borders relative to each other. In fact, 3-dimensional strain is a tensor with three normal and six shear components (11). This is further explained in the mathemathics section. As all strain components are interrelated, one component may be representative of all of the regional function (7), but the 3-dimensional nature of the strain tensor is important to understand the specific problems of insonation angle in strain rate imaging compared to velocity imaging.

The basic direction in three dimensions are given by the coordinate system given. In a Cartesian coordinate system, the directions are x, y, z, somewhat randomly chosen. In relation to the ultrasound system the coordinates of the ultrasound system are often used: Axial (depth - i.e. along the ultrasound beams also often called radial), lateral (In-plane angle or distance - i.e. across the beam; also called azimuth) and elevation (out of plane distance or angle), while in relation to the ventricle, the coordinates are longitudinal, circumferential and transmural (also confusingly called "radial") as discussed in the next section..

 Strain in three dimensions.  In this case there is deformation along the X axis,, the strain  is: All three-dimensional objects can be deformed in three dimensions (along all three axes).  The y and z strains will be exactly the same and can be imagined by rotating the x images. The three main strain components (principal strains) are: Shear strain. The linear X-strain is shown to the left for comparison: . The two shear strains along the x axis are shown as well:

The three main deformation components are :
.

Thus, linear strain is deformation along the main axes.
Shear strain along one axis measured relatively to an orthogonal axis. There are three shear deformations, but each can be measured relative to two different orthogonal axes, thus giving six shear strains.
In three dimensions there are three normal and six shear strain components:

However, as can be seen by the figures above, the deformation may be measured relative to either y or z axes (resulting in different strain values if the object's dimensions are different in the two directions), while the absolute deformation is identical. Thus there are only three shear deformations, is the same in and , is the same in and and is the same in and but six shear strains, as the strains are relative.

Incompressibility.

 The cylinder shows strain (compression along its long axis) , which can be described as Lagrangian strain from L0 to L. However, the figure also shows simultaneous thickening or expansion in the two transverse directions. This also illustrates the principle of incompressibility. An incompressible object must maintain an unchanged volume, thus compression along one axis has to be balanced by extension along at least one other. In this case both diameters increase simultaneously. Incompressibility in the XYZ coordinate system. Usually this comprises simultaneous strain in all three directions: The cube increases its length along the x axis (positive strain), while the x and y lengths decreases (negative strain), in such a way as to keep the volume constant.

If  the object is incompressible, the volume (not mass!) remains constant during deformation as shown in the illustrations above.This is the true definition of incompressibility. Thus, compression in one dimension has to be balanced by expansion in others as shown in the figures above, i.e. strain in the three dimensions in a coordinate system cancel out. This means that strain in three dimensions are interrelated, so strain in one direction is representative of regional deformation in more than one direction, as has been shown for heart muscle where wall thickening and wall shortening gives the same information about regional function (7).

Simultaneous strain in three dimensions.  In this case the object expands in the X direction, and shrinks in the y and z directions, so in this case, x is positive, y and z are negative.

Then, the volume of the cube is:
V = x × y × z
After deformation, the volume is:

As the volume remains constant during deformation, this means that:

hence:

and:

and thus:

It is just as evident that if the object expands, (i.e. volume increase, )       and then
and if it compresses (i.e. volume decrease)                                                  and then

In a symmetric, isotropic object like the cylinder or rectangle above, the transverse strains are equal, and both opposite to the longitudinal.
Thus, as:
and ,
then

which resolves to a standard second degree equation:

and thus:

In cardiac mechanics, the incompressibility equation will then translate into:

However, the inter relations are different. Systole is longitudinal shortening (negative), circumferential shortening (negative) and transverse thickening (positive) as discussed in the section on the geometry of myocardial strain. The n ormal strains are from different studies longitudinal ca - 15 to - 20%, midwall circumferential strain from -20 to -25% and transmural strain from 50 to 60%.

Lagrangian and Eulerian strain

There are two different ways of describing strain and strain rate: Lagrangian and Eulerian (named after the two mathematicians Joseph-Louis Lagrange and Leonhard Euler, respectively.

Lagrangian strain is the strain defined above;   the change in length divided by the original length, while Eulerian strain is the strain divided by the instantaneous length; .

Some prefer to use the term "Natural strain" instead of "Eulerian", however, I fail to see how one reference system is more "natural" than another. Using both mathematicians' names, the nomenclature will at least be more symmetrical.

 Lagrangian strain (top) and Eulerian strain (below). Visually, it is evident that both objects undergo the same strain at the same strain rate. Thus, the physical reality is the same, but the two figures show the two different ways of describing the deformation, as the Lagrangian strain shows an increasing deformation relative to the constant baseline length, while Eulerian strain describe the deformation (in this case constant, as the strain rate is constant, but this is not a condition), relative to the continually changing length. Lagrangian strain (top) and Eulerian strain (bottom). Only four point in time is shown, to illustrate how this means that by Lagrangian strain at any time is the sum of all length increments up to that time, divided by the baseline length, while Eulerian strain at any time is calculated as the sum of all ratios of length increments and the instantaneous length up to the actual time.

Then, as described above left, Lagrangian strain is the cumulated deformation, divided by the initial length, Eulerian strain is the cumulated ratios between the instantaneous deformation and the instantaneous length:
Lagrangian strain is:                                   while Eulerian strain is:

The point is that the two formulas will result in slightly different values. The positive Lagrangian strain of 25% in the example above, will be equivalent to 22% Eulerian strain (and not 20%, as one might believe).

As described above left, Lagrangian strain is the cumulated deformation, divided by the initial length, , or at any given time: but it still resolves to the total deformation divided by the initial length. As:

then the instantaneous change in Lagrangian strain is:

Eulerian strain is the cumulated ratios between the instantaneous deformation and the instantaneous length:     ,   i.e. at any time t, the Eulerian strain is ,

The instantaneous change in Eulerian strain is:

However, in continuous moving material points through spatial points, i.e. continuous deformation, the Eulerian strain is exact only when the increments and time intervals are small, i.e.:

The instantaneous increase in length is:

Summing all increments from t = 0 to t gives (L(t+dt)-L(t)) + ( L(t+dt+dt) - L(t+dt))+.......    , and as L(t) = L0 at t=0 and L(t) = L at t, gives: dL = L - L0 and thus:

As:

then:

and:

and finally:

and

The customary use is Lagrangian strain, but eulerian strain rate. This has historical reasons; Lagrangian strain was first used by Mirsky and Parmley in describing myocardial strain (12). Strain rate was first measured by tissue Doppler velocity gradient (4, 14) which is equal to the Eulerian strain rate (as can be seen by the formula, the denominator is L, not L0). Then, integrating strain rate to strain, gives Eulerian strain, the value has to be converted in order to derive Lagrangian strain.

Practical consequences of the difference between Eulerian and Lagrangian strain

The point is that the two formulas will result in slightly different values:

 Lagrangian versus Eulerian strain. Lagrangian strain will give slightly higher values, i.e. negative values are lower in absolute values, while positive values are higher. Lagrangian and Eulerian strain curves. As myocardial strain in general is negative, the Eulerian strain curve lies below the Lagrangian.

In general, peak strain may be up to 4% higher (absolute values but a relative difference of up to about 20%) by Eulerian strain than by Lagrangian strain, but the two references can be easily converted:
and

Lagrangian and Eulerian strain rate

The relation between Eulerian and Lagrangian strain rate is:
and

.

Lagrange vs Eulerian strain rate. As Eulerian peak systolic strain is lower (more negative) than Lagrangian, the peak Eulerian systolic strain rate has to be lower (more negative) too, in order to reach lower strain values. In diastole, however, the peak strain rate has to be higher (positive), in order to return from deeper negative values.

In this case, the difference in peak systolic and diastolic strain rates are smaller, about 12% relative.

Conversion is simple and can be applied by the scanner and analysis software instantly.

This means that any scanner or analysis software need to report which reference is used (287).

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 (L1-L2)/L1. But the shortening L is also equal to the total displacement DL of the ends of the rectangle, DL = DL1 + DL2.  The baseline length measured along the skewed beam, is L1M = L1 / cos (). The total measured displacement, however, is the sum of the longitudinal and transverse vectors, and thus DLM = DL / cos () + DT / 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:
1. It presupposes a symmetrical object as in the discussion on imcompressibility above.
2. In a three dimensional object, this two dimensional reasoning is only valid if the imaging plane is the z-plane, or else there will be a z component detracting further from the longitudinal measured strain.
3. 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 M-mode line along the ultrasound beam shows positive strain, by the divergence of the M-mode lines (speckles). The wall thickening is predominant. Interpreted as longitudinal strain, this vould mean systolic elongation . dyskinesia. Adjusting the direction of the M-mode, 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 M-mode 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 M-mode 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 vl and xl, respectively. Transverse velocity and motion are vt and xt.  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 v2 = 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:
1. It presupposes a symmetrical object as in the discussion on imcompressibility above.
2. In a three dimensional object, this two dimensional reasoning is only valid if the imaging plane is the z-plane, or else there will be a z component detracting further from the longitudinal measured strain.
3. 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.

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