Det medisinske fakultet


Basic concepts in myocardial strain and strain rate

by 

Asbjørn Støylen, Professor, Dr. med.

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

The page is part of the website on Strain rate imaging

Contact address:
asbjorn.stoylen@ntnu.no


This section updated:      January 2017



This section:

This section extends the basic concept of strain into the specific geometry of the left ventricle. It is important to understand that the effects seen by strain rate imaging has geometrical explanations. This means that over all geometry governs the changes and relations between strain components. This is true both of transmural and circumferential as well as area strain. Also, the strain gradient across the wall seen both in transmural and circumferential strain is mostly due to geometry, not differential fibre action. Back to website index
References



Relations between tissue velocity and strain rate

Strain rate is calculated at the velocity difference per length unit /velocity gradient) between two points in the myocardium:







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.


This is shown in more detail here. Peak velocity (left, A) is earlier than peak strain rate (Middle, B), but from the figurte to the right, it is shown that B is the point of maximum ditansce between the curves.


Thus the distances between the two curves is an indication of the strain rate:


Left: velocity curves. Middle: strain rate curves from the two segments between the velocity curves. Right, the areas between the velocity curves corresponding to, and shaded with the corresponding strain rate curves. Peak strain rate is not simultaneous in the two segments, peak velocity is more simultaneous due to the tethering effects. This is described in more detail here.

But this means that the global strain rate (of a wall or the whole ventricle), equals the normalised, inverse value of the annular velocity:


If the two points are at the apex and the mitral ring, the apical velocity , apex being stationary, and  is annular velocity.  then equals wall length (WL),
thus and peak  .

Thus, peak strain rate is peak annular velocity normalised for wall length.


Comparison between velocity and strain rate. Left, strain rate of most of the length of the septum, right spectral Doppler of the mitral annulus of the same wall. The two curves can be seen to be very similar, although the strain rate  curve is inverted as explained above. Also, the values and units are different, as strain rate is divided by the ventricular wall length. The summed strainrate curve has peak strain rate very close to the time of peak velocity, but tihis is due to the averaging effect, as peak strain rates differ between segments.


Exactly the same is the case for basal displacement vs strain, of course as shown in the basic concepts section.:

The difference in displacement varies during the heart cycle, and the distances are shaded red, always being negative (d1<d2). The resulting strain curve is shown to the left, strain rate being negative during the whole heart cycle, isshown in red. Mark that as opposed to peak strain rate and peak velocities, peak displacement and peak strain are simultaneous, being near end ejection.


Thus, basal velocities are equivalent to wall strain rate, and basal displacement, are equivalent to wall strain:

Septal strain and strain rate (right) from (nearly) the whole septum,  and basal septal velocity and displacement (left). As the apex is (nearly) stationary, the basal velocity and displacement is a motion inscribing the whole of the shortening of the wall, the deformation curves from of the whole wall is very near the inverted motion curves from the base as described elsewhere. The negative deformation curves is from the original Lagrangian definition where shortening is baseline length + resulting length, becoming negative when there is shortening.  Motion measures are absolute, deformation measures are relative. Peak shortening can be measured as either peaks systolic annular displacement (MAPSE) and peak systolic strain, and shortening rate as peak systolic basal velocity, the S' or peak systolic strain rate, SR. All four measures are in clinical use with ultrasound.


The strain rate being the difference between the decreasing velocities from base to apex, means that 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.


What are the differences between strain rate and strain?

Contractility

Basically peak systolic strain rate is peak rate or velocity of shortening. This occurs after ejection start. Thus, both peak rate of shortening, and maximal shortening are afterload dependent, as shown below.


Left: Twitches in isolated papillary muscle from (208). Top, twitches with increasing afterload, showing the isometric phases before tension equals load, and whan tension equals load, further contraction is shortening under constant tension (isotonic). Below are the corresponding length diagrams of the same twitches. From the diagram it is evident that:

- Peak rate of force development occurs during the isometric phase, i.e. before onset of shortening, except in the completely unloaded twitch
- Peak rate of shortening occurs at start of isotonic shortening, i.e. later than peak rate of shortening
- With increasing afterload, onset of shortening is delayed, peak rate of shortening as well as total shortening is reduced
Right: strain rate (top) and strain curves from a healthy subject. The similarity of the strain curve to the shortening curve to the left. The differences are due to the interaction of the ventricle with valves, blood and atria.

- Initial shortening occurs before mitral valve closure  (350, 351). This means that the initial contraction is near unloaded, and thus show an initial shortening
- With MVC, the ventricle enters an isovolumic (i.e.) isometric phase. Peak RFD occurs in this phase, and corresponds to peak dP/dt.
- With AVO, the ventricle enters the ejection phase, corresponding to the isometric phase, (although it is not completely isometric, as seen from the pressure curve). As seen from the strain rate curve, however, there is a delay after AVO, before peak rate of shortening (peak strain rate), which may be an inertial effect as the blood pool being ejected is accelerated first.

Peak rate of force development is the peak dP/dt, closely related to contractility (241) and afterload dependent (208, 209, 409), although preload dependent (395, 409, 410). However, this occurs during  during IVC (241), when ther eis isometric contraction, and hence, no hsortening, i.e. no strain or strain rate.

Peak rate of shortening occurs later, in the twitch model at the transition from isometric (isovolumic) to isotonic work, and is a function of the time from peak RFD to initial shortening, in the intact ventricle a little later, probably due to inertia. Total shortening, on the other hand, is also a function of the time where tension is equals the total load. This means, it is an end systolic measure, an expression of the total systolic work (at least the ejection part). Thus, it will be load dependent to a great degree. Peak strain rate, is peak systolic measure, the peak rate of deformation during ejection. It is simultaneous with peak ejection rate, thus early in ejection, closer to the time of peak dP/dt, (which is during IVC), the peak rate of force development. Thus, it is less afterload dependent, although shortening velocity is still load dependent as shown already by Sonnenblick (209). The relation of strain rate to contractility was shown experimentally by Greenberg (80). Greenberg found a 94% correlation of SR with LV elastance Emax, 82% with preload recruitable stroke work PRSW and 78% with dP/dt, in a study comparing baseline to low and high dose esmolol, baseline and and low and high dose dobutamine. However, HR increased as well, and inotropic stimulation increases.

Clinically,
Thorstensen found that early (peak) systolic measures were more responsive to changes in contractility (223) than end systolic measures.

In an elaborate study using both esmolol and Dobutamine, but controlling for heart rate by atrial pacing, Weidemann (79) did show that while strain strain rate was a closer correlate of contractility, as in the study by Greenberg, Strain was a correlate of stroke volume. Thus, strain is both volume and afterload sensitive. Peak strain rate is still preload sensitive (via the Frank-Starling mechanism), and afterload sensitive, but to a lesser degree. The same was found in animals exeriments by Ferferieva (408).


Stroke volume

The close relation between strain and stroke volume seems evident, when looking at the volume and strain curves below.



This has recently been supported by a work showing changes in strain during chemotherapy may be due to volume changes rather than contractility changes (396).

Timing

Longitudinal strain is negative during systole, as the ventricle shortens. Peak strain is in end systole, after this, the ventricle lengthens again. But the strain remains negative until the ventricle reaches baseline length. thus the values of the strain are less sensitive to event timing. Strain rate on the other hand, is negative when the ventricle shortens, shifting to positive when the ventricle lengthens, irrespective of the relation to baseline length. Thus events with changes in lengthening or shortening rate are much more evident by the strain rate crossing over between positive and negative. This is most evident in colour M-mode, which also can differentiate timing of events at different depths.


Looking at the strain rate and strain curves from one singe heart cycle to the left, it is evident that while strain (bottom) remains negative throughout the heart cycle, strain rate (top) shifts between positive and negative. It can be seen that the shifts from positive to negative (zero crossings), in strain rate, corresponds to the shifts from increase to decrease, or vice versa in strain (i.e. the peaks and troughs in the curve). The peaks of the strain rate curve on the other hand, corresponds to the change in the rate of increase in the strain curve (of course), seen as the shifts from concave to convex (or vice versa). The correspondences are not perfect, as the strain rate is Eulerian, while the strain is recalculated to Lagrangian, as is the common convention. To the left are colour M-modes. Strain rate (top) can identify the events by the positive-negative shifts (blue-orange), while the peaks are not discernible. But the colour M-mode discerns the differences between event shifts in different depths. Strain colour M-mode is not very useful in timing events.



Normal left ventricular dimensions

Dimensions of the ventricle is closely related to the functional measures. While the motion indices of displacement and velocity are dimension unrelated, strain and strain rate are relative deformation measures, and thus related to dimensions. Thus changes in dimensions will relate to changes in strain and strain rate. The HUNT study, being ta large study of normals has published normal values, related to age and gender (386):

Conventional left ventricular cross sectional measures from M-mode in the HUNT study by age and gender, raw and indexed for BSA. SD in parentheses. From (386).
Age (years) N IVSd
(mm)
IVSd/BSA
(mm/m
2)
LVIDd
(mm)
LVIDD/BSA
(mm/m
2
FS (%) LVPWd
(mm)
LVPWd/BSA
(mm/m
2)
RWT RWT/BSA
Women
<40
207
7.5 (1.2) 4.2 (0.6) 49.3 (4.2) 27.5 (2.6) 36.6 (6.1) 7.7 (1.4) 4.3 (0.6) 0.31 (0.05) 0.17 (0.03)
40–60 336
8.1 (1.3) 4.5 (0.7) 48.8 (4.5) 27.3 (2.8) 36.5 (6.9) 8.3 (1.3) 4.6 (0.7) 0.33 (0.05 0.19 (0.03)
> 60 118
8.9 (1.4) 5.1 (0.8) 47.8 (4.8) 27.4 (3.1) 36.0 (9.1) 8.7 (1.4) 5.1 (0.8) 0.37 (0.07) 0.22 (0.04)
All 661
8.1 (1.4) 4.5 (0.8) 48.8 (4.5) 27.4 (2.8) 36.4 (7.1) 8.2 (1.4) 4.6 (0.8) 0.34 (0.06) 0.19 (0.04)
Men
<40 128
8.8 (1.2) 4.3 (0.6) 53.5 (4.9) 26.1 (2.6) 35.5 (6.9) 9.2 (1.3) 4.5 (0.7) 0.34 (0.06) 0.17 (0.03)
40–60 327
9.5 (1.4) 4.6 (0.7) 53.0 (5.5) 26.0 (3.0) 35.8 (7.4) 9.7 (1.4) 4.7 (0.7) 0.37 (0.07) 0.18 (0.03)
> 60 150
10.1 (1.6) 5.1 (0.9) 52.1 (6.4) 26.3 (2.9) 36.0 (8.0) 10.0 (1.3) 5.1 (0.7) 0.39 (0.07) 0.20 (0.04)
All 605
9.5* (1.5) 4.6† (0.8) 52.9* (5.6) 26.0† (2.9) 35.8 (7.5) 9.6* (1.4) 4.7† (0.7) 0.37 (0.07) 0.18 (0.04)
Total 1266
8.7‡ (1.6) 4.6 (0.8) 50.8‡ (5.4) 26.7 (2.9) 36.1 (7.3) 8.9 (1.6) 4.7 (0.7) 0.35 (0.07) 0.18 (0.04)
*p<0.001 compared to women. †p<0.01 compared to women. ‡Overall p<0.001 (ANOVA) for differences between age groups.

Wall thicknesses and LVIDD correlated with BSA (R from 0.41 - 0.48), Thus, all values were consistently higher in men due to this. FS, of course, did not correlate with BSA, and was thus gender independent.  Wall thicknesses increased with age (R=0.33), while LVIDD and FS remained constant between age groups, in accordance with other studies (387, 388, 389, 390).

Normal range is generally considered the interval between the 2.5 and 97.5 percentiles, ie. more or less mean ± 2SD.



Wall thicknesses and chamber diameters. RWT = (IVSd + LVPWd)/LVIDd, but there was no difference if we used LVPWd x 2 / LVIDd. FS = (LVIDd - LVIDs)/LVIDd. Left ventricular external diameter; LVEDd = IVSd + LVIDd + LVPWd.
Left ventricular length. Wall lengths were measured in a straight line (WL) in all six walls from the apex to the mitral ring. This wil underestimate true wall lengths (dotted, cirved lines), but will be more reproducible, as the curvature may be somewhat arbitrary. LVL was calculated as mean of all four walls, thus overestmating true LVL (yellow line) slightly, but again the arbitrary placement in the middle of the ostium will result in lower reproducibility, while taking the mean of six measurements will increase it.

Relative wall thickness

Relative wall thickness is generally considered to be a body size independent measure, as both wall thicknesses and LVIDD are body size dependent, the RWT, supposedly, is normalised for heart size, and hence, for body size. Interenstingly, in the HUNT study this was not the case, although correlation with BSA was very modest (R=0.18). This probably do not warrant normalising RWT for BSA. More pronounced was correlation with age (R=0.34). The age dependency is a logical consequence of the unchanged LVIDd and increasing wall thickness, and has been shown also previously (391).




Relation of RWT and BSA. This shows that RWT is not perfectly aligned with body size.
RWT and age. This shows a more marked dependence of RWT and age, so age related normal values is probably warranted.

Current guidelines recommend a cut off value of 0.42 between normal and concentric geometry (146) without taking age into consideration. In HUNT, however, the normal upper limit is also closer to 0.52 over all.

The age relation is not taken into account either, as upper normal limit is increases with age, from 0.41 to 0.54 in women and 0.44 - 0.54 in men, so age related values is warranted, unless one will consider that all > 60 years have concentric geometry.


Left ventricular length and external diameter:

Left ventricular length and external diameter is also important in an evaluation of the total strain images. We measured these in the HUNT study as well:

Left ventricular length and external diameter by age and gender from the HUNT study, raw and indexed for BSA. From (386).
Age (years) N LVEDD (cm)
LVEDD/BSA (mm/m2)
LVL (cm)
LVL/BSA (cm/m2)
LVL/LVEDD
Women
<40
207
6.45 (0.48)
35.9 (2.7)
9.4 (1.6)
5.23 (1.00)
1.46 (0.26)
40–60 336
6.52 (0.52)
36.5 (3.2)
9.1 (1,7)
5.08 (0.95)
1.40 (0.27)
> 60 118
6.52 (0.52)
37.7 (3.5) 8.9 (1.3) 5.08 (0.79) 1.36 (0.23)
All 661
6.51 (0.51)
36.5 (3.2)
9.1 (1.6)
5.13 (0.93)
1.41 (0.27)
Men
<40 128
7.16 (0.53)
35.0 (2.9)
10.3 (1.7)
5.02 (0.88)
1.44 (0.25)
40–60 327
7.22 (0.58)
35.0 (3.2)
10.0 (1.8)
4.84 (0.89)
1.39 (0.26)
> 60 150
7.22 (0.68)
36.5 (3.1)
9.5 (1.8)
4.80 (0.97)
4.80 (0.97)
All 605
7.21 (0.59)
35.3 (3.1)
9.9 (1.4)
4.86 (0.91)
1.38 (0.27)
Total 1266
6.84 (0.65)
36.0 (3.2)
9.5 (1.8)
5.00 (0.93)
1.40 (0.27)

Left ventricular external diameter, is simply the sum of the wall thickensses and LVIDd, so it is logical that this increased both with BSA (R=0.60) and modestly with age (R=0.11, the unchanged LVIDd being part of it, dilutes the effect of wall thickness) (386).

Left ventricular length, on the other hand, increased with BSA (R=0.29), but decreased with age (R = -0.12).

Fundamental findings are summarised below:



Fundamental findings in the HUNT study: With increasing BSA, both wall thickness, internal diameter (and hence, external diameter) and relative wall thickness increase, showing that neither measure is independent of body size (or heart size). The length / external diameter, however, remains body size independent, being a true size independent measure. Differences are exaggerated for illustration purposes.
With increasing age, both wall thickness (and hence, external diameter) increase, while internal diameter is age independent. Left ventricular length decreases, and hence length / external diameter decreases, and i a measure of age dependent LV remodeling. This has implication for LV mass calculation. Dimension changes are exggerated for illustration puposes.


Ratio between LV length and external diameter

The ratio L/D did not correlate with BSA, was near gender independent (although the difference was significant due to the high numbers), but declined somewhat more steeply with age (R = -0.17).

This has some important corollaries:
  1. LV shape in healthy adults, is in itself a physiological measure
  2. Normalising cross sectional measures to LV length, corrects better for heart size than normalising for BSA
  3. The ratio L/D is a measure of age dependent remodeling in healthy adults
  4. LV mass calculations based on cross sectional (M.mode measures), will over estimate LV mass increasingly with age, and the assumption of age dependent mass increase with age may not be valid.
The L/D ratio may be a new measure of LV hypertrophy.

Wall lengths per wall

Different walls has different lengths. In the HUNT study, the wall lengths differed:
Diastolic lengths of different walls (Mean and SD) measured in a straight line from apex to mitral ring, by age and gender. Only over all values are published in (386):
Age (Years)
Septum
Lateral
Mean of two;
septal and lateral
Anterior
Inferior
Mean of four;
Septal, lateral,
anterior, Inferior
Anteroseptal
Inferolateral
Mean of all six
Women
<40
9.0 (1.6)
9.4 (1.6)
9.2 (1.6)
9.4 (1.7)
9.3 (1.6)
9.2 (1.6)
9.2 (1.9)
9.8 (1.6)
9.4 (1.6)
40-60
8.8 (1.6)
9.1 (1.7)
9.0 (1.6)
9.2 (1.7)
9.1 (1.7)
9.0 (1.6)
8.9 (1.9)
9.6 (2.1)
9.1 (1.7)
>60
8.5 (1.3)
9.0 (1.4)
8.7 (1.3)
8.9 (1.4)
8.8 (1.3)
8.8 (1.3)
8.5 (1.6)
9.4 (1.7)
8.9 (1.3)
All
8.8 (1.5)
9.2 (1.6)
9.0 (1.6)
9.2 (1.6)
9.1 (1.6)
9.1 (1.6)
8.9 (1.9)
9.6 (2.0)
9.1 (1.6)
Men
<40
9.9 (1.7)
10.3 (1.8)
10.1 (1.7)
10.2 (1.8)
10.3 (1.8)
10.2 (1.7)
10.1 (1.8)
10.8 (1.9)
10.3 (1.7)
40-60
9.7 (1.7)
10.2 (1.8)
9.9 (1.7)
10.0 (1.8)
10.1 (1.8)
10.0 (1.7)
9.5 (1.9)
10.6 (2.2)
10.0 (1.8)
>60
9.1 (1.8)
9.7 (1.9)
9.4 (1.9)
9.4 (2.1)
9.5 (2.1)
9.4 (1.9)
9.1 (1.9)
10.2 (2.1)
9.5 (1.9)
All
9.6 (1.8)
10.1 (1.8)
9.8 (1.8)
9.9 (1.9)
10.0 (1.9)
9.9 (1.8)
9.5 (1.9)
10.5 (2.1)
9.9 (1.8)
Total
9.2 (1.7)
9.6 (1.8)
9.4 (1.7)
9.5 (1.8)
9.5 (1.8)
9.5 (1.7)
9.2 (1.9)
10.1 (2.1)
9.5 (1.8)
All lengths in cm.

The lateral and inferolateral walls were significantly longer than all other walls (including each other). The septum and anteroseptal walls were significantly shorter than all other walls. Means of two, four and six walls were all significantly different from each other, but the differences were negligible, considering that the limit for measurement accuracy is 1 mm.

Geometry of myocardial strain

Still preaching my personal litany: Strain is geometry. (Cormorant, Galway, Ireland). Normal strains, longitudinal, transmural, circumferential.


Myocardial directions - normal strains

Asa describe in "basic concepts"section, the strain tensor has three normal strains (11) in the x, y and z directions in a Cartesian coordinate system. Also, in an incompressible object, meaning that deformation doesn't affect volume, the three strains have to balance by the incompressibility equation: .

Strain in the heart also has three main components, but the directions are customary related to the most common coordinate system used in the heart: Longitudinal, circumferential and transmural. (The term "radial" is often used to describe transmural direction, but as this in ultrasound terms also means in the direction of the ultrasound beam in the ultrasound specific coordinate system, "radial" strain is ambiguous and should be avoided. Transmural strain is unambiguous).



Strain in three dimensions. In the heart, the usual directions are longitudinal, transmural and circumferential as shown to the left. In systole, there is longitudinal shortening, transmural thickening and circumferential shortening. (This is an orthogonal coordinate system, but the directions of the axes are tangential to the myocardium, and thus changes from point to point.) This long axis video shows how the apex is stationary, while the base moves toward the apex in systole, away from the apex in diastole. This means the ventricle shows strain between apex and base. Longitudinal strain will be negative (shortening) during systole and positive (lengthening) during diastole (if calculated from end systole).  This short axis video shows both transmural and circumferential strain. Systolic transmural strain equals wall thickening. Systolic circumferential strain is the systolic shortening of any of the countours; outer, midwall or endocardial, The change in outer contour is least, while the endocardial contour shortens most, thus, there is a gradient of circumferential strain across the wall. This is explained below.



Thus, systolic deformation in the heart occurs in all three dimensions simultaneously. 


It is evident that Lagrangian strain is well suited to describe systolic deformation. Diastolic thinning or elongation, however, is not so well described by Lagrangian strain as Lo is defined in end diastole.

Thus:

The concepts transmural displacement and transmural velocity are in reality meaningless in a physiological sense. The displacement and velocity in the transmural direction is dependent on where across the wall it is measured, i.e. the transmural depth of the ROI placement. Different data sets from tissue Doppler in the transmural direction is thus not comparable, and the measurements have little clinical value. Some applications like 2D strain will give the segmental average value for transmural velocity and displacement. They may have a clinical meaning, in that they may separate normal from reduced function, but the use of clinical measurements that are physiologically unsound, is doubtful.


In addition, as the myocardium is (basically) incompressible, the volume do not change (much) during systole, so the strains are inter related.


Incompressibility in the heart:

If  an object is incompressible, the volume (not mass!) remains constant during deformation. This is the true definition of incompressibility. Thus, compression in one dimension has to be balanced by expansion in others, i.e. strain in the three dimensions in a coordinate system cancel out as described in the basic concepts section. This means that cartesian strains balance:

In the heart, this is equivalent with:


If , the myocardium is expanding, and if , the myocardium is shrinking during systole.

Although there may be some diminishing of volume due to collapse of small intramural vessels, and crypts, but this is limited, and still, there has to be substantial inter relation between the strain components as illustrated below.

In two larger studies of strain by speckle tracking in three dimensions (393, 394) the strain product values were 0.87 and 0.91, respectively. Due to inherent limitations in speckle tracking, however, there may be systematic over estimations of longitudinal strain, or under estimation of transmural strain, or both, so this is not resolved.

Longitudinal strain

Longitudinal strain, being relative longitudinal systolic shortening, is, from the Lagrangian definition of strain, it follows that longitudinal strain is: . It follows from the formula, that as there is systolic shortening, systolic longitudinal strain is negative (systolic length smaller than diastolic).





Longitudinal shortening of the left ventricle. The absolute shortening is the MAPSE, while the relative shortening is the normalised MAPSE = MAPSE / L0.

However, this is not ambiguous. Over all ventricular strain should be as illustrated above. However, wall strain would be measured along the walls, and the mean wall strain would be a measure of global LV strain. As shown above, wall strain  length can be measured in different ways, which has consequences for strain measuremnt.

Left ventricular length, is LVL, ,easured from the apex to the mid-annulus. Wall length is either measured in a straight line from apex to the mitral ring, or in a curved along the wall. The mean of the straight line leasures, will over estimate LV length, but under estimate the true wall length. The curved line will further over estimate the LVL, thus being further from the LV strain, and will also be less robust, as the curvature may be somewhat arbitrary.
Strain being (L - L0) / L0 will thus relate to the length used.  Both the strain length, L0 and the shortening (L - L0) will be different when measured along the LV length (Black/grey), a skewed line (red) and even longer along a line following the wall curvature (blue).  This will be especially evident depending on how shortening is used.

It's important to realise that different applications may measure strain in different ways as indicated in the above right figure, and as shown elsewhere. 2D strain measures along the curved line following the wall, the M-mode method as well as Tissue Doppler will measure along the ultrasound beam, being a straight line, while segmental strain will measure along a straight line in each segment, thus being somewhat in between, as shown by this figure. Also, there is a slight difference between longitudinal stain measured in the midwall compared to endocardial measurement, due to the inward shift being more pronounced in the endocardium as discussed below, as well as due to the fact that the midwall line is slightly longer than the endocardial, thus giving a larger denominator in the strain expression.

Thus, global longitudinal strain will vary with processing software (vendor).

Now, the EACVI?EAE task force has recommended that for speckle tracking, the denominator should be the line following the myocardial wall, whether it is is the endocardial or midwall, and also that the level should always be reported by the software (287).

This is discussed further in the pitfalls section.



Transmural strain


Transmural strain is simply relative wall thickening. There is no such thing as "transmural myocardial function", as there are no transmural fibres. Wall thickening is solely due to incompressibility; as the wall shortens, in the longitudinal, and eventually also in the circumferential direction, it must thicken in the transverse direction to conserve volume.


Wall thickening. Systolic wall thickening equals systolic transmural strain: WT = (WS - WD)/WD =
Wall thickening, illustrated from the loop shown to the left. The outer (red) and endocardial (yellow) contours and wall thicknesses are shown in the diastolic image to the left, and transferred to the systolic image on the right, shown as dotted lines of the same colour. The systolic contours are shown as solid lines. The systolic wall thickness is then (more or less) the dotted plus the solid blue lines, and the wall thickening the solid blue lines.

Thus, transmural strain is a purely segmental measure. Global transmural strain either has to be measured in all 16 segments, (three parasternal short axis planes), or inferred from assumptions of symmetry.

What is very evident here, is that transmural strain in fact is a quantitative equivalent to the semi quantitative wall motion score.

Transmural strain can be measured by speckle tracking as shown below. It has to be measured from short axis images, as the decreasing lateral resolution with depth precludes transmural measurements from apical images. In fact, that option was removed after we pointed it out.





Short axis cine loop.
Speckle tracking in the same cine loop
Resulting peak strain values and strain curves from the tracking.


The basics of this method is given in detail in basic strain ultrasound section, and the limitations of the speckle tracking method is discussed in the pitfalls section.

Speckle tracking, however,  is not necessary for transmural strain. Wall thickening can be measured by simple caliper measurements of wall thickness in systole and diastole. It is still segmental, but can be generalised from fewer measurements under assumptions of symmetry, as has been done from M-mode. The transmural strain can be measured in M-mode from systolic and diastolic wall thickness, which will give wall thickening in only two segments, but may be taken as representative as the mean wall thickening in this plane where there is no segmental dysfunction. However, in this case, generalizing from M-mode measurements, the sepal and inferiolateral wall should be averaged, as septal thickening is less than inferiolateral wall thickening (392):




Transmural strain by M-mode. The M-mode measurement is more accurate than 2D measurements, but are only feasible in the septum and inferolateral (posterior) wall. Thus, the transmural strain can only be extrapolated in symmetric ventricles. Strain by  tissue Doppler is also only feasible in the two walls perpendicular to the ultrasound beam as indicated by the arrows. If using M-mode, however, the average of septal and inferolateral wall should be used, as septal thickening is less than inferiolateral wall thickening.

However, it is very evident that longitudinal and transmural strain are not independent. Longitudinal shortening can easily be demonstrated in apical echo images as shown above, as well as measured as shown below. Transmural thickening is equivalent to wall thickening, but from the images below, it is evident that the wall has to thicken as it shortens in order to conserve volume (NOT MASS!!!).


As the ventricle shortens, the wall has to thicken in order to maintain the wall volume, as the myocardium is more or less incompressible. And as the outer contour of the ventricle either do not change (13, 59, 60), or changes very little (158), this means that the wall thickening has to be inwards.



Ventricular strain. Diastolic and systolic images of the heart. Systolic shortening of the left ventricle relative to diastolic length, is the systolic strain of the ventricle.  From the Lagrangian definition of linear strain; , it follows that systolic longitudinal strain is:


However, it is also evident from this image, that as the wall shortens, it also thickens, to conserve the volume. Heart muscle is generally assumed to be incompressible.
Schematic diagram of the left ventricle, showing the relation between shortening and wall thickening (exaggerated for illustration purposes), with a model of unchanging outer contour in an incompressible myocardium.



Thus, one source of the wall thickening is simply that the volume has to be conserved when the walls shorten. Circumferential fibre shortening would also cause the wall to thicken, because moving the outer contour inwards would cause the wall to thicken, as there is less room for the myocardium within a smaller outer circumference, thus circumferential shortening will also contribute to wall thickening. So even inward motion of the outer circumference would cause the wall to thicken, even without longitudinal shortening:


Wall thickening as a function of longitudinal shortening. Calculated from a symmetric half ellipsoid model with a diastolic mid wall thickness of 0.9 mm (decreasing towards apex), an outer diastolic diameter of 60 mm, a diastolic length of 95 mm. Wall thickening is calculated from longitudinal shortening and conservation of wall volume, given different degrees of outer contour change (outer circumferential strain or shortening). Longitudinal strain given in negative values; i.e. wall thickening increases as THE VALUE of longitudinal strain increases. As seen here, if there is no outer diameter reduction, the wall thickening is solely a function of wall shortening.


Thus:

There is no such thing as transmural function. Transmural strain is thus in itself not a function measure. This is hardly surprising, as there are no transmurally directed fibres. Wall thickening reflects the thickening of the individual muscle fibers inn all directions as they shorten.

Depending on how much or little change there is in outer contour, the transmural strain will mainly be a function of longitudinal shortening. Wall thickness and cavity diameter are also geometric determinants of wall thickening. However, in any given ventricle with a given cavity diameter and end diastolic wall thickness, the transmural (radial) strain is a function of longitudinal strain, not an independent measure.

In conclusion:

If there is no outer circumferential shortening, meaning that there is no change in outer diameter (and circumference), transmural strain will be solely a function of longitudinal shortening, and both globally and regionally those two parameters give the same information. Then, the only source of wall thickening will be the wall shortening. As the outer contour changes little during systole, this means that as the ventricle shortens, the wall has to thicken inwards. This has been shown in semi quantitative assessment of regional function (7), which supports this.

Circumferential strain



Circumferential strain means shortening of a circumference in the ventricle.


As transmural strain, circumferential strain must be measured in short axis planes.
External circumference is shown in red, midwall circumference in blue, and endocardial circumference in orange. The circumferences from the diastolic left frame are shown as dotted lines of the same colour in the systolic frame to the right, to compare with the systolic contours in unbroken lines. The inward motion is evident, and there is a gradient from outer to inner contour.

However, Circumferential strain is an ambiguous term.

The circumferential strain has no meaning except as a shortening of a defined circumference. And this is dependent on which circumference, as circumferential shortening increases from the epicardium  to the endocardium. Thus, there is a gradient of circumferential strain from the outer to the inner contour, (due to geometry NOT to layer specific function).

Thus, in order to talk about circumferential strain, first, the question has to be answered: Which circumference? (external, midwall or endocardial) as illustrated below.
Different software today use different definitions, some measuring endocardial, others midwall circumferential shortening. Thus, there is no standard circumferential strain, it is is method dependent.

When considering circumferential shortening, three points is important:
  1. Circumferential shortening is to a large degree due to the inward shift of the circumferences as the wall thickens. thus:
  2. Even without any change in outer contour, the endocardium will shift inwards as the wall thickens, and there will be both endocardial and midwall circumferential shortening
  3. There would have been circumferential shortening even if there had been no circumferential fibres, as the wall thickening due to shortening would give this inward shift.
This means that there is fairly little relation between circumferential fibre action and circumferential strain, except for the outer contour. The main function of the circumferential vectors seems to be balancing of the intracavitary pressure, but this is isometric, and do not necessarily cause shortening.

The circumferential strain in a normal ventricle is the shortening of a circumference due to the inward shift caused by the wall thickening. Even if there had been no circumferential fibres, there would have been wall thickening and thus circumferential strain as shown in the figures below.


Illustration of the geometric mechanism for circumferential shortening. Outer contour is shown in black, midwall contour in blue and endcardium in orange. Even without presupposing circumferential fibre shortening as shown by the invariant outer contour, there is inward shift of midwall and endocardial contours as the wall thickens. This is geometry, not fibre shortening. Also as the endocardial circumference is pushed further inwardss, being on the inside of the whole wall, while the midwall contour is only pushed inwards by half the wall thickening, there is a gradient of circumferential shortening, and thus the two are different.
Schematic diagram of the left ventricle, showing the relation between shortening and wall thickening (exaggerated for illustration purposes), with a model of unchanging outer contour in an incompressible myocardium. Wall thickening  again relates to circumferential shortening. Also, relative diameter shortening (fractional shortening) equals relative circumferential shortening.


The circumferential fibre shortening contribute to circumferential strain, depending on how much reduction there is in outer diameter. This will increase not only the shortening of the midwall and endocardial surfaces, but also the gradient of shortening from outer to inner surface. If the outer circumference shortens, there is less room for the myocardium which has no alternative than expanding inwards.

But again: If transmural strain is mainly a function of longitudinal shortening, and circumferential shortening mainly a function of transmural thickening, this means the three are inter related:

This means: Circumferential strain is partly a function of wall thickening (and outer circ shortening)
Wall thickening is a function of longitudinal shortening (and outer circ shortening).



Calculated from a symmetric half ellipsoid model with a diastolic mid wall thickness of 0.9 mm (decreasing towards apex), an outer diastolic diameter of 60 mm, a diastolic length of 95 mm. Wall thickening is calculated from longitudinal shortening and conservation of wall volume, given different degrees of outer contour change (outer circumferential strain or shortening). Longitudinal and circumferential strains are given in negative values; i.e. wall thickening increases as THE VALUE of longitudinal strain increases.
Midwall and endocardial strain as functions of wall thickening, for 0%, 5% and 10% outer diameter reduction.
As wall thickening also is a function of longitudinal strain, midwall and endocardial strain as functions of longitudinal strain, for 0%, 5% and 10% outer diameter reduction.

Finally, as the circumference is simply a function of the diameter (C = * D), circumferential strain can be computed directly from the diameter fractional shortening (i.e. midwall or endocardial, respectively):

C = (C - C0)/C0 = ( * D - * D0) / * D0 = (D - D0) / D0 = ÷ FS

Thus, circumferential strain equals fractional shortening!
(I.e. either endocardial or midwall)




Circumferential strain again is available by speckle tracking in short axis images:






Short axis cine loop.
Speckle tracking in the same cine loop
Resulting peak circumferential strain strain values and strain curves from the tracking. There is abnormal swtrain curves in the inferior segments due to imperfect lateral tracking in the remote region (reduced lateral resolution with depth) as discussed int the pitfalls section.





However, again, speckle tracking is not necessary for measuring circumferential strain. As circumferential strain equals the negative value of fractional shortening, it can be generalised from fewer measurements from assumptions of symmetry, as has been done from M-mode. If the cross section of the LV is assumed circular, the CS equals - FS. Angulation of the M-mode line wil not matter, as this is the relative shoirtening, which will remain constant.


Illustration of the circumferential shortening from diameter measurement, and how these can be derived from M-mode.






Area strain


Strain area. The Thingvellir Rift Valley in Iceland is the rift between the North American and the Eurasian continental plates. The plates are diverging, so the rift is expanding and the area undergoes positive strain.


Hypothetically, with the advent of 3D echocardiography, it would also be possible to measure simultaneously in all direction, enabling the measurement of composite measures. One candidate for such composite measures is  area strain. However, as discussed elsewhere, there are serious shortcomings in 3D speckle tracking, due to low frame rate and line density.

Both area strain as well as transmural and circumferential strain can in principle be assessed by 2D acquisitions, if they are processed into a 3/4D reconstruction.
This, however, requires tracking in both longitudinal and transverse directions, ans thus has to be done with either speckle tracking alone , or combined tissue Doppler and speckle tracking, as shown below. It also includes some assumptions about the angle between the planes and simultaneity of events in the loops that are acquired sequentially, but processed into a simultaneous image.




3D strain rate mapping. Reconstructed 3/4D image with longitudinal tracking from tissue Doppler. (This is described in detail below). Yellow represents shortening, blue elongation and green no strain. In this case only longitudinal strain is tracked and displayed, as can be seen from the diameter circumference of the grid, it doesnt change during the heart cycle.
Apical four chamber view with B-mode and tissue Doppler data. Longitudinal shortening is tracked by tissue Doppler. In this image both sides of the LV wall are marked and tracked,  thus the wall thickening is tracked as well, by speckle tracking. In this analysis both longitudinal and transmural strains are available, but for circumferential strain 3/4D reconstruction is necessary, and requires three planes.
3/4D reconstruction from three sequential planes to a thick walled model analysed as shown in the image in the middle. In this case, the endocardial and midwall circumferences are given in the grid, and circumferential and area strains can be calculated. (The colours in this image, however, are tissue Doppler derived strain rate, i. e. longitudinal strain rate).

Giving the present sorry state of 3D speckle tracking, this may still be an option, especially as B-mode has improved substantially with new computing techniques, giving both higher line density and frame rate.


However, as area strain is not part of the original Lagangian definition, the concept needs a definition, one reasonable candidate is simply the systolic relative reduction in area, giving an analogous definition to the one concerning one dimensional strain:









Area strain. As the one dimensional strain is relative change in length, the area strain should have the same definition: relative change in area.

However, just as circumferential strain, the area strain is dependent on which level of the wall it is measured. Epicardially, there is very little circumferential shortening at all, and the area strain would be equal to the longitudinal strain, as the area will shorten by length only.



Area strain. As the ventricle contract, the end diastolic area of the selected region (red) would be reduced in both the longitudinal and circumferential direction. Assuming a cylindrical shape of the segment, the area will be equivalent to a flat geometry. In the apex, the shape would be more triangular, which means the area is only half that. Both the cylinder and triangle will underestimate the true area, as the surface is curved, but the underestimation will be similar in end systole and end diastole, so the area strain approximation will be closer to the real area strain.
Area strain is a function of longitudinal strain.


Simple geometry will then show that the area strain is a function of longitudinal circumferential strain, and that the relation is: A = L * C + L + C 

One dimensional strain is defined as = (L - L0)/L0 The equivalent for the change in area is thus A = (A - A0)/A0
Then, in an approximately cylindrical segment: A0 = C0 * L0 and A = L * C
L = (L - L0)/L0 and C = (C - C0)/C0
L - L0 = L * L0 and C - C0 = C * C0
L = L * L0 + L0 = L0 ( L + 1) and C = C * C0 + C0 = C0 (C + 1)

Thus:
A = L0 ( L + 1) * C0 (C + 1)
And:
A = (L0 ( L + 1) * C0 (C + 1)) - (C0 * L0 ) / C0 * L0 = ( L + 1) * (C + 1) - 1 = A = L * C + L + C


Thus the area strain is:





As area strain is a function of circumferential and longitudinal strain, and in the "eggshell model" circumferential strain again is solely a function of longitudinal strain, area strain itself can be seen as solely a function of longitudinal strain. But even if there is dependency on both variables, this is stillnot added information, just a composite.

Thus, for global function, area strain does not seem to add new information. Also, for area strain, the 3D speckle tracking technique may render it inferior to single measures from 2D or tissue Doppler.

Where there is regionally reduced function, however, the situation may be different. The circumferential shortening may be reduced in a sector, and the area strain would then be a compound of reduced longitudinal and circumferential shortening. However, it could still be computed to  certain degree, as endocardial circumferential shortening can be computed from the fractional shortening through the hypokinetic area. The limitations in area strain, however, will still persist.

However, in a recent study (279) of myocardial infarcts, 3D strain did not show incremental diagnostic value to the other modalities. 3D longitudinal strain was inferior to 2D longitudinal strain, and 3D Circumferential, longitudinal and area strain did not add information, as opposed to infarct area by tissue Doppler (243).



The eggshell model

In order to see which consequences the incompressibility of myocardium has for cardiac mechanics, it is important to look at the eggshell model of left ventricular function.

The concept that the heart functions as a double pump, with the atrioventricular plane as a piston, rather than pumping by squeezing, is indeed a concept dating back to Leonardo da Vinci (57).In 1951 Rushmere was able to show by means of implanted iron filings in dog hearts inserted in the wall of the ventricles, that the pumping action of the right ventricle was predominantly in the long axis direction, while the left ventricle apparently pumped by an inward squeezing action (58). The inward motion of the markers, however, is dependent on how deep into the myocardium (close to the endocardium) the markers are placed. The concept of inward squeezing motion has been confirmed by innumerable ventriculographies (59), blinding the viewers to what happens the outer contour of the heart during systole.

Already in 1932, Hamilton and Rompf (59) argued from experimental studies that the heart worked mainly by the movement of the atrioventricular plane toward apex in systole, away from apex in diastole, while the apex remained stationary and the outer contour of the heart relatively constant. The heart will the work by the principle of a reciprocating pump, alternately expanding the atria and the ventricles, without moving the surrounding tissue.  Their hypothesis was confirmed by Hoffman and Ritmann in CT studies in dogs in 1985 (60), showing a stationary apex, constant outer contour and motion of the AV-plane. They also stressed that this mode of action minimised the energy expenditure as the ventricular volume rediction in systole moves blood into the heart, rather than moving the surrounding tissue during systole. If the heart should be pumping by inward squeezing, reducing the outer contour of the heart this would be unfavourable energetics, as this means moving the surrounding tissue (lungs and mediastinum) inward by each heartbeat, without regaining this energy in diastole. Mitral ring movement was first demonstrated by echocardiography from the apical position by  Zacky in 1967 (61). Working before the time of MR and second harmonic 2D echo, Stig Lundbäck, in a series of elegant human studies  using both  gated myocardial scintigraphy, echocardiography and coronary angiography (Demonstrating the outer heart contour by tangential cine angiograms of the LAD), documented the invariant outer contour and the AV-plane mode of working (13).
It has been established that the longitudinal shortening of the left ventricle, and thus the longitudinal measures is closest related to the stroke volume and EF, i.e. to the total left ventricular volume change (30 - 35, 56, 59, 60, 64 - 67, 116).

This is seems to be the case when looking at modern imaging such as MR or high quality echocardiography as f.i. above.

The radial motion of the septum in diastole is determined by the differences in filling pressure of the left and right ventricles. In systole, If the filling pressures are reasonably similar, as in the normal situation, the septum has little radial displacement in diastole.  In systole, the pressure induces a circular cross section, as the most energetically feasible shape. Thus, during systole, the left ventricle operates without much change in the outer contour.


Given an invariant outer contour, the whole of the stroke volume is described by the longitudinal shortening, as wall thickening is simply a function of wall shortening. The total volume in diastole is the sum of the blood inside, and the muscle wall. When the left ventricle shortens in systole, the total volume is reduced by the volume of the cylinder  shown in grey: . But the myocardium, comprising a part of this volume is incompressible, thus maintaining a constant volume.  Thus, the whole volume reduction  is the reduction in blood volume, in other words the stroke volume:  Thus, the stroke volume is given by the outer diameter and the systolic longitudinal ventricular shortening (56). But as the myocardium is incompressible, the wall shortening and thickening, and thus the internal diameter reduction have to be interrelated (7), and thus both would be valid measures of stroke volume. In a newer study, the correlation between MAE and stroke volume in healthy adults was seen to be about 90%, corresponding to an explained 82% of the stroke volume compared to the reference (Simpson). Thus, an outer contour systolic reduction of about 3% should be present to explain the rest of the stroke volume (158), and may be more in real situations. This is little compared to wall thickening, showing that the main inner contour diameter reduction is due to longitudinal shortening and incompressibility, as discussed above. Thus, the eggshell model is fairly accurate, and the long axis function describes most of the pumping action of the heart.


M-mode as well as short axis cross sections, may sometimes show greater inward motion of the outer contour, due to the out of plane motion of the base of the heart.





As can be seen, the base of the heart moves through the M-mode line during the heart cycle.
This means that measurements in fact are taken from different part of the ventricle in end diastolie and end systole. It seems to indicate that systolic measurements are done in a part of the ventricle withsmaller diameter, thus over estimating  inward motion of the outer contour.



The eggshell model and atrial filling.

In the eggshell model, the atrioventricular plane has to be the piston of a reciprocating pump as discussed ), expanding the atria while the ventricle shortens and shortening the atria while the ventricle expands. This is energetically feasiblel, as the work used to decrease the volume, in additon to ejection, also moves the blood from the veins into the atria. If the heart had worked by squeezing changing outer contour to a high degree, the work would have been used to shift the rest of the thoracic contents especially lungs inwards in each systole, work that would have been wasted. Thus, most of the filling volume to the ventricles, is a function of the AV-plane pumping, as also discussed it the section of strain in the atria.

The eggshell mechanism

But how is this possible, even if energetically favorable, the pericardium is not stiff, and the surrounding lung tissue is highly compliant. The muscle forces would tend to reduce both inner and outer contour, as the circumferential fibres contract. If the pericardium had been stiff, this would generate a pressure drop, and the vacuum would hold the myocardium against the pericardium. But as the pericardium is pliable, this would not work. And Smiseth et al has shown that pericardial pressure actually increases during systole, if measured by proper techniques (63). Allso, the apex beat is a clinical empirical fact, meaning that the apex moves towards the chest wall in systole, thus not creating a suction at the apical location:


The answer may lie in the recoil forces. The pericardium is soft, but non-compliant. During ejection, the ventricle impels a momentum to the blood volume being ejected, generating a momentum of similar magnitude, but opposite direction according top Newton's third law (mv = - mv where m is mass and v is velocity). The recoil, pressing the heart toward the chest wall as can be felt by the apex beat and demonstrated by apexcardiography and has been demonstrated by echocardiography as well (33). And the pericardium, although pliant, is not elastic, and pressing the heart into the pericardial sac will give a constraint and pressure increase as previously shown (63).

Recoil forces.  The momentum away from the apex is ejection of the stroke volume. The displacement of the ejected volume is equal to the stroke velocity integral (measured by Doppler flow in the left ventricular outflow), which is about 15 to 20 cm. The motion of the opposite momentum is displacement of the annular plane, which  is between 1 and 1,5 cm (30) at the same time, and the mass being displaced also equals the (mass of the) stroke volume. The mass is the same. The mean velocity, and thus, the momentum, being mv, being generated by ejection is at least ten times the momentum pushing in the other direction, thus generating the forces pushing the heart into the pericardium, which is non compliant. This can be felt as the apex beat, shown here in an apexcardiogram taken with a pressure transducer, demonstrating that the beat is a systolic event. (Image modified from Hurst: The Heart).
The apex beat can also be demonstrated by M-mode echocardiography and tissue Doppler.




A recent study demonstrates the importance of the pericardium in accordance with the above arguments in an elegant way (122). Following the velocity and strain rate by TEE during an operation, they show that when the apex was dislodged from the pericardium, the basal velocities changed direction, so the base and apex moved toward each other in systole, without any change in strain, i.e. the myocardium still shortening at the same rate. The motion of all basal regions toward the apex was reestablished after the heart was repositioned within the pericardium.




The volume (and mass) being ejected, is equal to the volume being moved towards the apex as shown here. 




However, the septum is not contained in the pericardial sac. But the motion of the septum is small compared to the wall thickening, and some of the motion may be apparent as shown above. Thus, the pumping action of the left ventricle can be described by the long axis changes, and is a measure of  the systolic pumping function. Even so, much of the ventricular work is not taken into account by this, namely the work that is used for increasing the pressure from low filling pressure to high ejection (aortic) pressure. However, this is true whether measures of cavity size such as stroke volume, ejection fraction, shortening fraction. or measures of longitudinal shortening such as mitral annulus displacement, systolic annulus velocity, longitudinal strain or longitudinal strain rate is used.


















As stated above, transmural strain is a measure of deformation, not of function, due to the incompressibility.




Simultaneous strain in three dimensions. Relation of long axis shortening and wall thickening.  As the heart muscle is generally considered incompressible, longitudinal shortening must give transmural thickening.. Thus as the ventricle shortens, the wall has to thicken correspondingly in order to preserve wall volume, the thickening shown in blue. In this case, the outer contour of the left ventricle is assumed fairly constant, as described below.

 
However, transmural strain will be very much influenced by processing, especially ROI size (276), as discussed here.

There will be a gradient of transmural strain from the epi- to the endocardium. As the wall thickens, the endocardial layers expand in a space with a smaller circumference, and thus they have to thicken more for the same volume increase. But this is due to geometry, not to any gradient in layer function, as discussed below.


This so-called "eggshell" model of the left ventricle has been supported by other studies (13, 59, 60, 116) as discussed below.

If there is a component of circumferential fibre shortening, this must mean that there will be a decrease in outer diameter, which then also contributes to wall thickening.




Speckle tracking in short axis image. The thickness follows the wall thickening,
and the mid line in the ROI shows midwall circumferential shortening.



 


This means that the measure of circumferential strain is




Midwall (blue) and endocardial (orange) circumferential strain is equalt o the negative value of fractional shortening, and thus, the mean circumferential shortening of the short axis plane can be measured from M-mode.


Myocardium is incompressible, and the incompressibility equation then works out to:

Thus, all normal strains are interrelated.


In the "eggshell" model of the left ventricle, if there is no outer diameter change, circumferential strain is solely a function of wall thickening, which again is solely a function of longitudinal shortening. Then, all information about LV deformation is given by the longitudinal strain, the other principal strains are simply derivatives. The function of the circumferential fibres are mainly to balance pressure, thus circumferential function is mainly given by the peak pressure (and diameter - wall stress), and without sortening, this is tension without deformation, i.e. isometric muscle work.

Looking at the ventricular volume curve shown below left, it is evident how much the volume curve reflects a longitudinal strain curve, showing the close relation between longitudinal deformation and pumping volume.


Left ventricular volume curve from MUGA scan (gated blood pool imaging  by 99Tc labelled albumin. The total volume is proportional to tne number of counts, thus making MUGA a true volumetric method, but averaged from several hundred beats.) It is evident that there is volume reduction corresponding to ejection, then there is early and late filling. Thus this might seem to correspond to contraction - relaxation. The temporal resolution of MUGA is low, and the isovolumic phases are poorly defined.
(Longitudinal) strain (shortening) curve from left ventricle. Note the close correspondence to the volume curve on the left, but due to higher temporal resolution, the isovolumic phases are visible.  It is evident that the longitudinal shortening describes most of the volume changes. Again the shortening might seem to be contraction, and the (early) elongation relaxation.

Thus, the pumping action of the heart, i.e. the ejection volume can be described mainly by the long axis function.



If myocardium is incompressible the three strains anyway must balance by:


  If myocardium is partly compressible;
if the object expands, (i.e. volume increase, )                                       and then 
if it compresses (i.e. volume decrease)                                                  and then 

Thus, neither transmural nor circumferential strain are independent measures of ventricular function. However, the relations will change not only with longitudinal strain, but also with ventricular size and wall thickness, still dependent on the geometry of the ventricle.

As strain measurements are software dependent, inter vendor consistency is low, although best for global longitudinal strain (277, 278), as might be expected as the sources of differences are smaller.

Are there apex to base differences in strain and strain rate?

As apex is stationary and the base of the ventricle moves, there has to be a gradient in velocity and motion from base to apex. As strain rate actually is that velocity gradient, the presence of a gradient in strain rate depends on whether the velocity gradient is constant or not.

Looking at the V-plot, the curve seems fairly straight, i.e. the velocity gradient seems fairly constant along the wall:




Good quality V-plot shows velocities as near straight lines, and thus, a constant velocity gradient. This seems to exclude that there is a strain rate gradient from base to apex.
A nearly straight line. Blue eyed shags (cormorants) at Cabo de Hornos (Cape Horn), Chile.
 
 ...Indicating that there is no gradient in strain rate, and thus not in strain.


Motion (velocity and displacement - left) and deformation (strain rate and strain - right) traces from the base, midwall and apex of the septum in the same heart cycle. It is evident that there is highest motion in the base (yellow traces), and least near the apex (red trace), and this is seen both in velocity (top - actually both in systolic and diastolic velocity) and systolic displacement (bottom). The distance between the curves are a direct visualization of strain rate and strain, showing fairly equal width of the intervals. Strain rate (top and strain (bottom) curves are shown to the left, showing no difference in systolic strain rate or strain between the three levels.

Some of the earliest strain rate studies found no base - to apex gradient (10, 19, 341), although later studies seem to find differences with lowest values in the apex (124). However, the angle error is also greatest in the apex (206). In the comparative study between methods in HUNT (153), using tissue Doppler velocity gradient,  there was lower values in the apex, but only  only when the ROI did not track the myocardial motion through the heart cycle. Tracking the ROI eliminated this gradient:


Velocity gradient (stationary ROI)
Dynamic velocity gradient (tracked ROI)

Peak Strain rate End systolic Strain Peak Strain rate End systolic Strain
Apical -1.46 (0.85)
-14.6 (9.0)
-1.31 (0.73)
-17.2 (9.1)
Midwall
-1.29 (0.56)
-18.2 (7.4)
-1.40 (0.58)
-16.9 (7.1)
Basal
-1.71 (0.94)
-19.6 (9.3)
-1.59 (0.74)
-17.1 (8.6)
Mean
-1.45 (0.79)
-17.7 (8.5)
-1.43 (0.67)
-16.7 (8.1)
Comparison between standard tissue Doppler velocity gradient and tracked ROI. Standard deviations in parentheses.

Thus, it seems fairly reasonable to conclude that this finding is artificial.

With 2D strain, some authors have found a reverse gradient of systolic strain as well, highest in the apex, lower in the base (207). However, in that application, measurements are curvature dependent, the apparent curvature being highest in the apex and lowest in the base, and the discrepancy between ROI width and myocardial thickness being greatest:



Curvature dependency of strain in 2D strain by speckle tracking.
The two images are processed from the same loop, to the right, care was taken to straighten out the ROI before processing, while the left was using the default ROI. In both analyses the application accepted all segments. It can be seen that the apical strain values are far higher in the right than in the left image (27 and 21% vs 19 and 17%).  However, the curvature of the ROI even affects the global strain, as also discussed above in the basic section.


The large HUNT study (153) found no such gradient either way with the combined speckle tracking -TDI method:

Basal
Mid ventricular
Apical
Strain rate (s-1)
-0.99 (0.27)
-1.05 (0.26)
-1.04 (0.26)
Strain (%)
-16.2 (4.3)
-17.3 (3.6)
-16.4 (4.3)
Results from the HUNT study (153) with normal values based on 1266 healthy individuals. Values are mean values (SD in parentheses).  Differences between walls are small, and may be due to tracking or angular problems.  No systematic gradient from apex to base was found.


In the subset of 50 analysed for comparison of the methods, taking care to avoid both foreshortened images and excessive curvature, there were no level differences in 2D strain either:

Segment length by TDI and ST
2D strain (AFI)

Peak Strain rate
End systolic Strain
Peak Strain rate End systolic Strain
Apical -1.12 (0.27)
-18.0 (3.6)
-1.12 (0.37)
-18.7 (6.6)
Midwall
-1.08 (0.22)
-17.2 (3.2)
-0.99 (0.23)
-18.3 (4.7)
Basal
-1.03 (0.24)
-17.2 (3.5)
-1.12 (0.36)
-18.0 (6.2)
Mean
-1.08 (0.25
-17.4 (3.4)
-1.07 (0.33)
-18.4 (5.9)
Comparison between methods. Standard deviations in parentheses.
In addition, in the  comparative study
, there was no gradient using the 2D strain application, in this case care was taken to align ROI shapes as much as possible.

MR studies have also found various results.
Bogaert and Rademakers (171) in a study of healthy subjects (N=87) found lowest longitudinal strain in the midwall segments, higher in both base and apex, but no systematic gradient from base to apex. Moore et al (384) in a study of healthy volunteers (N= 31) found a systematic gradient, but with the lowest strain in the apex, highest in the base. Venkatesh et al in a healthy subset from the MESA study (N= 129) (385) examined only transmural and circumferential strains in cross sectional planes, and found decreasing transmural strains from base to apex in all layers. As segmental shortening and thickening are very closely related through incompressibility, this should amount to a decreasing strain from base to apex too.

Circumferential strains, on the other hand, seemed to be less systematic, and the apex to base gradient varied between both layers and walls. This, however, is counterintuitive, as wall thickening causes inwards displacement of the circumference, wall thickening is equivalent to shortening, as the findings should show the same gradient.

MR measurements have processing issues as well. Using short axis planes, the planes will show an increasing deviation from the 90° angle with the wall, towards apex, causing an over estimation of wall thickness in the apical planes. Using magnetic tagging, this is usually done in a grid with 90° angles, at least in the transcverse/longitudinal direction, while the radial might vary, although usually at 90° with the horisontal plane. This might cause angle deviations as shown below.


Diagram illustrating MR planes and magnetig tagging grids and relation to myocardial directions. Horizontal planes and grud lines (red) are usually cross sectional, causing increasing angulation with the transverse direction of the wall (green) towards the apex. Longitudinal grid lines deviate increasingly from the longitudinal direction of the wall toward the apex as well (orange).

MR tagging may include algortihms for calculation of the local coordinates, but this again will introduce new uncertainties in the angle calculations, causing both over- and under corrections depending on the calculation. Shear strain may affect the motion of tags, and attempts to calculate shear strains and separate them from the normal strains, will again increase the complexity of calculations and possible uncertainties.

Thus, the presence of a base to apex gradient in deformation parameters has so far not been established.



Is there layer specific strain, and can we measure it?


The advance of speckle tracking have enabled analysis of deformation in all directions, although with severe limitations inherent in ultrasound itself as well as due to the specific applications for analysis Speckle tracking also gives the possibility of measuring smaller regions of the myocardium. This may be subject to severe restrictions, however. Also, measurements are related to geometry, which do not necessarily relate to differences in fibre function.

Normal transmural and circumferential strain gradients

There is a normal gradient of strain from outer to inner contour. This has been confirmed emprically (255).  This, however, has got nothing to do with differences in fibre function,  but is simply due to geometric factors,and is already discussed inder the paragraph on circumferential strain.



transmural gradient of strain. The thickening of the outer layer displaces the inner layer inwards. This alobe will cause the inner layer to thicken, due to being pushed into a region where the circumference is smaller, and thus thickening has to compensate in order to preserve layer volume. The thickening of the inner layer comes in addition to this, and thus the inner layer has to thicken more than the outer layer. Thus, there is a gradient of ttransmural strain across the wall, increasing towards the endocardium. But this also is the case for circumferential strain. The mid circumference of the outer layer moves inwards (and hence, shortens) according to the thickening of the outer layer. The midwall line of the inner layer moves inwards (and hence, shortens) both due to the inward shift of the inner alyer, and due to the increased thickening of the inner layer. Thus there is a gradient of circumferential strain increasing towards the endocardium as well.

The gradients of transmural and circumferential strains are thus a function of geometry alone in the normal ventricle, simply as the myocardium nearest the inner wall is pushed more inwards, and thus have to both thicken and shorten more due to reduction in available space.

Normal transmural gradient of longitudinal strain.

A transmural gradient of longitudinal strain has likewise been published (371). This again, is solely a function of the geometry of myocardial thickening, and the way strain is measured, and the fact that the endocardial parts have to thicken more, due to the decreasing space analoguous with the circumferential strain. If there had been an additional increase in endocardial longitudinal strain, this would have resulted in systolic torsion of the mitral ring, which, being part of the larger fibrous AV-plane, is inconceivable.



Longitudinal layer strain. The mid layer lines in diastole (unbroken lines), move inwards in systole (dotted lines), both using straight lines and curved lines, and the endocardial moves most, due to the inner layer thickening most as there is less room. Colurs are for differentation only, and have nothing to do with the colours seen in parametric images.
Proposed geometry if there should have been more absolute shortening in the inner layer, this would mean a torsion of the mitral plane in systole as illustrated here by the rise of the inner part, but this is inconceivable, the mitral plane is part of the larger fibrous annular plane.
Wall layers in relation to beams. Even though the beam width is exaggerated for illustration purposes, it show that one beam may cross different layers at different depths, and that at some depths the beam may overlap more than one layer, which will cause "smearing" of speckles, which may be allocated to the wrong layer, While focusing may increase lateral resolution in some level, this is only if line density is adequate, if not the interpolation described elsewhere will take over. .


The layer structure is well established (62, 256, 257). Due to different fibre direction (62, 257), they may have different longitudinal tension also in the natural situation. As fibre directions vary across the wall, the longitudinal tension has to be unequally distributed; specifically it will probably be lowest in the middle layer, where the fibre direction is mostly circular.So, again from anatomy, it is evident that layer strain do not measure layer function.

Finally, measurement of layer strains depend on an adequate beam width to separate the layers, This is not the case all over the field, as the lines broadens with depth, and have different widths depending on the focussing. this is discussed in more detail elsewhere. This might mean wrongly allocating deformation to different layers, as well as picking up stationary echoes from the pericardium on the outside. (The beam problem may change with newer generations where increased processing enables both higher MLA factor and focusing along the whole beam. Beam broadening with increasing depth, however, remains a fact of geometry).

Thus, studies of longitudinal layer strain from apical full sectors older than about 2016 may be dubious, and if focus and line density is not reported, actually valueless.

Myocardial shear strains

As explained in the basics section, there may, at least theoretically be shear strains in the myocardium as well. In the myocardium the principal deformations should be as for the principal strains, longitudinal, circumferential and transmural. (this is evident, force being a vector can only have three spatial components). But as measured relatively, there will be six different shear strains. If shear strains will be available for measurements, some may have more practical implications than others. Measuring shear strains means that one will be able to measure differential strain across a cross section of the image. This is related to measurement of layer strains as discussed above.

With some degree of layer independence, and differential tension both across as well as along the wall, there may be differential layer strain. The difference in longitudinal strain across the wall is will then be longitudinal shear deformation, and measured relatively to wall thickness, it will be longitudinal/transmural shear strain.

The shear strain has been demonstrated experimentally by applying differential stress to isolated tissue (i. e;. passive strain), showing that the tissue strains most easily in the direction l  the myocardial layers (258). Differential tension restricted to regions in the myocardial wall is what is expected from non transmural ischemia. Thus, shear strain might be demonstrable in these situations, and has been demonstrated experimentally (259)


Approximation to the normal tension distribution of the tension, with least longitudinal tension in the middle layer. With a deformable mitral ring and independent layers, the deformation would be unequal as well (orange, high longitudinal deformation, yellow less longitudinal deformation), causing the mitral ring to buckle in the middle (A). As discussed above, this is undocumented as well as improbable, the more probable model being homogeneous deformation across the wall, as a resultant of the different forces. Hypothetical model of shear strains with non transmural loss of force. In both cases, the weakened layer in the affecte dsegment(s) will shorten less (yellow), but this must be compensated by more shortening of the non affected segment in the same layer (red), as the mitral ring doesn't torque. This must mean that there has to be inverse shear strains in hte affected vs non affected segments in the same wall.

If there are non transmural infarcts, this might in principle cause shear strains especially in the longitudinal-transmural direction. However, as discussed in the section on regional function, this must happen within the framework of the AV-plane. This means, that the different segments must interact, without deforming the mitral ring, and will result in differential shear strains between the different segments of the same wall. .

Hypothetically, measuring sub endocardial longitudinal strain selectively, if possible, might increase sensitivity for non transmural infarcts / ischemia, as the endocardial layer will be the most affected. However, this remains to be proven. Also it may hypothetically be a method for differentiating transmural and non transmural akinesia, in the acute situation demonstrating transmural ischemia. Transmural ischemia in the acute situation may be an indication of coronary occlusion as opposed to non transmural ischemia.



Strain and fibre direction


It has been a popular misconception that strain in the different directions have to do with the actions of different muscle fibers, i.e. circumferential and transmural (radial) strain reflects the action of circular fibers, while longitudinal shortening reflects the function of the longitudinal fibers. It seems to be something almost "everybody knew". While the latter is partially true, the first is not.  There would have been circumferential shortening even if there had been no circumferential fibres. Mean circumferential strain must be taken to mean midwall circumferential shortening. As shown above, the midwall circumferential shortening is almost entirely the function of diameter shortening, which again is a function of wall thickening. This is due to the finding that the LV outer contour is nearly invariant from diastole to systole (13, 59, 60) as shown in the example above, the diameter reduction being a function of wall thickening inside a virtual "eggshell". The reduction in outer contour contributes only to a small part of the circumferential strain.

The fibre directions are diverse, and varies throughout the thickness of the heart, the middle layer being more circular, while the endo- and epicardial layers being more longitudinal, although helically ordered (62, 257). In dealing with the principal strains, the wall is treated as isotropic, which it is not. Thus, there may be differential strain as well as shear strain.


Thus the three principal strains are totally interrelated and does not convey separate information about different fibre function. The information is about the myocardial volume deformation in ejection phase.

The longitudinal fibers are responsible for the longitudinal shortening, and any process that mainly affect longitudinal shortening (f.i. sub endocardial ischemia), will result in reduced longitudinal shortening. It is also true that the ejection work (stroke volume and ejection fraction) is closely correlated with longitudinal strain as discussed in long axis function. In fact, the longitudinal shortening can explain most (but not absolutely all (158)) of the stroke volume. This is mainly the work of the longitudinal fibers (or the longitudinal component of the spiral fibers) both in the endo- and epicardium and represents mainly isotonic work. This is what we measure by longitudinal displacement, velocity and longitudinal deformation measures.


Thus, circumferential shortening is related to wall thickening, which is due to the thickening of the individual muscle fibres.
In addition, as the inner circumference decreases, the longitudinal fibers gets less room, especially in the endocardial parts, and thus the longitudinal fibers have to shift inwards during systole.  This also contributes to the wall thickening as illustrated below. Wall thickening is thus greater than the sum of the individual fibre thickenings.

Transmural strain is not only due to wall thickening, but also of inward displacement of the inner layers. Simplified and exaggerated diagram showing the relation between fiber thickening and wall thickening. As the fibers shorten, they thicken. Thus, the sub epicardial  longitudinal fibers will thicken, displacing the circular fibers in the mid wall inwards. In addition, as the fibre become thicker, they will need more room, thus necessitating some rearrangement of the fibres, making the layer thickening even more than the individual fibres. They will also displace the circular fibres inwards, thus making the shorten and also thicken as they contract. Finally the sub endocardial longitudinal fibers will be displaced inward. The sub endocardial fibers will also, thicken. But the circumference has been decreased at the same time due to the thickening of the outer fibers,  and thus there has to be an extra inward shift of longitudinal fibers for them to have room. Assuming a systolic reduction in outer diameter will only enhance this effect. By this, it's evident that wall thickening is not equivalent to the sum of fibre thickening alone. The circumferential strain is thus mainly the shift of the midwall line inwards due to wall thickening.


The circumferential fibers,  mainly contributes to the pressure increase, i.e. isometric work, which takes place mainly during the  isovolumic contraction phase. Isometric contraction cannot be measured by deformation along the fibers, and thus by no imaging modality at all. As they contract, however, there will also be a slight inward shift, due to the displacement of the fibres, which also results in a shortening and thickening of the fibres. In addition, the circumferential fibers may be responsible for whatever there is of outer contour diameter reduction . If so, they contribute to the ejection work, and in addition slightly to wall thickening, as the wall has to thicken even more in order to retain wall volume with a reduced outer diameter. If there is loss of longitudinal contractile function, either regionally (typical ischemia) or globally as in cardiomyopathia with sub endocardial affection (e.g. Fabry), there may be a shift toward circumferential pumping, with an increase in the variations of outer circumference. Then there will be true radial compensation for loss of longitudinal function. But in hypertrophic states, there is usually loss of longitudinal function and circumferential function both, but due to the increased wall thickness the fractional shortening may be increased. This has been called "radial compensation", but as explained belowthis is a total  misunderstanding of geometry.

It is also extremely important that if longitudinal and "radial function" are compared, care should be taken that the measurements are comparable. To compare for instance fractional shortening of the LV diameter with longitudinal strain (wall shortening), is comparing two different measures, and may lead to completely erroneous conclusions as shown below, where fractional shortening increases but wall thickening decreases.



In terms of energetics, the ejection work may be described as the kinetic energy in the blood being ejected is 1/2 m v2, which is less than 20% of the potential energy (P*V). Thus, almost 80% of the work is pressure buildup, and this is done by tension increase, before onset of shortening (deformation).


Thus, deformation analysis, whether it is factional shortening, EF, longitudinal shortening, or deformation, all measure myocardial deformation in one way or other, and thus only a fraction of the work done by the heart. The greatest  great part of the ventricular work - the isometric work, cannot be described by deformation analysis (or any imaging modality) at all as all functional analysis by cardiac imaging is about deformation.



The full description of LV work need to incorporate a measure of load, either by invasive measures, or by externally measured pressure (eventually pressure traces) in combination with mathematical models.


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Editor: Asbjørn Støylen Contact address: asbjorn.stoylen@ntnu.no