Basic concepts in myocardial strain and strain rate

Misty fjords, Alaska

This section:

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

• Relations between tissue velocities and strain rate
• Apex to base velocity gradient
• Velocity gradient
• Is there an apex to base gradient in strain / strain rate as well?
• Strain and strain rate by myocardial level (basal, midwall, apical) in the HUNT study
• Differences between walls
  
• Strain, strain rate and velocity; differences between walls in the HUNT study
• Tethering
• Translational effects
  
• What are the differences between strain rate and strain?
• Contractility Basically, strain is the total systolic shortening, equivalent to the isotonic shortening in experimental models, and thus very afterload dependent. Peak systolic Strain rate, on the other hand has been shown to be more closely related to contractility, but the physiological limits of this correlation is discussed.
  
• Stroke volume It has been shown that strain relates best to the stroke volume, and thus is both afterload and volume dependent.
  
• Timing Strain rate, being the temporal derivative of strain shows more detectable shifts, especially in colour M-mode, making it more useful for timing purposes.
• Normal left ventricular dimensions Left ventricular dimensions and geometry is closely related to the geometry of left ventricular strain. Thus, normal values and the relation to body size, age and gender are included here. Normal values provided from the HUNT study. The data from the HUNT study suggests that relative wall thickness are both body size and age dependent, that while wall thickness increases with age, LV length decreases, invalidating previously findings of M-mode based LV mass increase with age. Finally, the ratio of LV length and external diameter is BSA independent and (nearly) gender independent, but decreases with age, being a measure of age related remodeling. Tables of normal findings from the HUNT study are added.
• Relative wall thickness
• Left ventricular length and external diameter:
• Ratio between LV length and external diameter
• Wall lengths per wall
• Left ventricular volumes derived from linear measures and an ellipsoid LV model.
  
• LV volumes and age. Due to the simultaneous increase in wall thickness and decrease in length, there is no increase in LV volume by age, when the age dependent increase in BP is considered.
  
• Geometry of myocardial strain
• Myocardial directions - normal strains
• Longitudinal strain
• Linear strain
• Speckle tracking strain tracks not only wall shortening, but also wall thicening. And as the wall thickens, both the midwall and the endocardial (even more) alyers moves inwards. This inward motion will in itself make the lines shorter, which is an additional effect to the true wall shortening. Thus Speckle tracking strains overestimates true wall shortening.
  
• Segmental strain
• Inter vendor differences in speckle tracking
• There is no gold standard for Global longitudinal strain. As there is no universal algorithm for global strain, of course the concept of global strain as a universal measure of ventricular function has no exact meaning. It is only a theoretical concept.
  This means there is no reference standard, they are method dependent. This means that strain values cannot be validated, and different methods cannot be compared in terms of validity, and finally, normal values do only have validity within the method used.
  There is no universal definition, and thus no "ground truth" at all for GLS. Methods cannot be validated, and GLS has to be considered only within the method used.
  
• Is there layer specific longitudinal strain, and can we measure it?
  
  • Normal transmural gradient of longitudinal strain The gradient in longitudinal strain seen by speckle tracking methods, can simply be explained by the speckle tracking, the inward motion is most pronounced at the endocardial surface, less at the midwall, and least at the outer surface. This apparent shortening is not true shortening, but a systematic error due to tracking of wall thickening, and can explain the gradient of longitudinal strain.
    
  • Even though analysis software will produce layer values at request, layer strain separation depends on line density, line width, direction in relation to the wall (in order to avoid pericardial echoes), and focussing. Number of lines is again dependent on frame rate. It is difficult to achieve a sufficient line density as well as narrow enough lines.
• Transmural strain which some calls "radial strain". The term "radial, however, is ambiguous, as in general ultrasound terminology this means "in the direction of the ultrasound beam".There is no such thing as "radial function". Radial strain means wall thickening, but there are no myocardial fibres going in the radial direction. Wall thickening is a function of wall shortening, and circumferential shortening. 
• Circumferential strain Circumferential strain do NOT reflect circumferential fibre contraction. There would have been circumferential shortening even without circumferential fibres. Circumferential strain is mainly the circumferential shortening due to inward movement of midwall or endocardial circumference as the wall thickens. As different vendors use different definition of circumferential strain (midwall or endocardial), there is no standard circumferential strain.
• Circumferential strain is an ambiguous term because
• There is a gradient of both transmural and circumferential strains towards the endocardium, but this is trivial, and solely an effect of geometry.
• But circumferential strain is simply a measure of diameter reduction, the global circumferential strain is the same as the negative value of fractional shortening.
  
• Circumferential strain by speckle tracking
• Linear strains in three dimensions from the HUNT study
• Inter relations of all three major strain components As the myocardium is nearly incompressible, the three strains must be interrrelated. Longitudinal shortening will give transmural thickening, and transmural thickening (inwards) will give inward motion of endocardial and midwall circumferences. In addition, there is a small outer circumferential shortening, so all thickening occurs inwards. This is the cause of the inwards gradient of transmural and circumferential strain.
• Incompressibility and the myocardial strain tensor there is no gold standard for strain, and different sets of assumptions as well as specific methods will give different values. This also means that both the inter relations of strains, as well as the relations to the myocardial volumes and incompressibility calculations will vary, and at the present level of technology, strains cannot be used to decide if the myocardium is incompressible.
• Area strain Area strain is neither the sum, nor the product of circumferential strain. A slightly simplified modelling will give the formula A = L * C + L + C . Thus, area strain is a function of longitudinal and circumferential strain, not a "new" parameter.As different vendors use different definition of area strain (midwall or endocardial), there is no standard circumferential strain.
• Incompressibility and stroke volume
• Is the myocardium compressible?
  
• Long axis shortening, and LV volumes in the HUNT study
• MAPSE contribution to the stroke volume: MAPSE contributes about 75% to the total SV (although this is not normative as the geometrical model has limitations. However, while MAPSE decreases with age, there is no compensatory increase in short axis function (which actually also declines), to explain maintained EF. EF is maintained with increasing age by the simultaneous decrease in LVEDV and SV, keeping the ratio constant. Also, MAPSE remains a constant percentage of the decreasing SV. The notion that circumferential function is the main contributor to SV, is bases on the lack of uncerstanding that circumferential strain (except external CS) is partly due to wall thickening, which again is mainly due to longitudinal shortening.
  
• Deformation and stroke volume
• Strains and fibre direction It is evident from the discussion of strain components, that they to a very little degree are related to fibre directions.
• Shortening is not equivalent with contraction, contraction can happen without shortening (creating only tension  - isometric contraction), but shortening can also happen without contraction
• Myocardial shear strains
  
• The eggshell model
• The eggshell mechanism
• The eggshell model and atrial filling.

Relations between tissue velocity and strain rate

Apex to base velocity gradient

As the apex is stationary, while the base moves toward the apex in systole, away from the apex in diastole, the ventricle has to show differential motion, between zero at the apex and  maximum at the base. Longitudinal strain will be negative (shortening) during systole and positive (lengthening) during diastole (if calculated from end systole).

M-mode lines from an M-mode along the septum of a normal individual. These lines show regional motion. It is evident that there is most motion in the base, least in the apex. Thus, the lines converge in systole, diverge in diastole, showing differential motion, a motion gradient that is equal to the deformation (strain).  This difference in displacement from base to apex is also evident in the displacement image shown above.

AS motion decreases from apex to base, velocities has to as well. Thus, there is a velocity gradient from apex to base, which equals deformation rate.

Spatial distribution of systolic velocities as extracted by autocorrelation. This kind of plot is caled a V-plot (247).  It may be usefiul to show some of the aspects of strain rate imaging. The plot shows the walls with septal base to the left, apex in the middle and lateral wall base to the right. As it can be seen again the velocities are decreasing from base to apex in both walls. There is some noise resulting in variation from point to point, but the over all effect is a more or less linear decrease. The slope of the decrease equels the velocity gradient. (Image courtesy of E Sagberg). However, this shows only one point in time, and all values are simultaneous.

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.

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.

Diagram showing that for the whole ventricle, v(x) is apical velocity = 0, and v(x+x)  = S', then SR = -S'/WL

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  .

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.

Strain rate and strain assessed by offset between velocity curves

Strain rate and strain can be visually assessed by the offset between the curves, when the velocity curves are obtained from points with a known (and equal) distance.

Segmental strain rate from velocities: Velocity curves from four different points of the septum. The image shows the decreasing velocities from base to apex. The distances between the curves show the strain rate of each space between the measurement points (segments).	Segmental strain from displacement. Displacement curves from the same four different points of the septum, obtained by integration of the velocity curves. The image shows decreasing displacement from base to apex. The distances between the curves show the strain of each space between the measurement points (segments).

If the curves are taken from the segment borders, this is a representation of the segmental strain rate and strain. Thus, it is evident that the strain rate and strain can be visualised (qualitatively) by the spacing of the velocity and displacement curves, even without doing the derivation.

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

Is there an apex to base gradient in strain and strain rate as well?

It has been maintained that as the curvature is larger (smaller radius both in cross sectional and longitudinal planes) in the apex, the wall stress (i.e. load) is lower, and hence shortening higher, in accordance with the law of Laplace. However, this reasoning do not take the varying wall thickness into account. As the wall is thickest at the base, and thinnest at the apex (62), the wall thickness decreases as the radius decreases, and no conjectures about the wall stress can be made.

Good quality V-plot of venlocities  from the septal base to the left through the apex in the middle to the lateral base to the lateral base to the right,  shows velocities as near straight lines, and thus, a constant velocity gradient. This should mean that there is no strain rate gradient from base to apex.	A nearly straight line. Blue eyed shags (cormorants) at Cabo de Hornos (Cape Horn), Chile.

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.

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.  

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).

Looking back to the animal experiments with ultrasonomicrometry, Urheim (8) found 12% in the base and 16% near apex under baseline conditions. Korinek et al (428) found 15.8% in the mid posterior segments vs 13.1% in the apical segment under baseline with chrystals, and 11.9 vs 13.7 with 2D strain. However, ultrasonic chrustal measurements are also subject to geomatreic distortion, especially when placing the chrystals in the sub endocardium, they will follow the inward movement. In straight segments, (basally), this will not result in shortening, but in curved segments (apically) this will lead to an apparent segmental shortening, which will come in addition to the true longitudinal shortening of the segment. This is illustrated below:

Simplified image illustrating the effect of inward motion (due to wall thickening) on a pair of sub endocardial crystals close to the base and close to the apex, respectively. The longitudinal shortening, and thus true segmental shortening is omitted. Blue: end diastole, Red: end systole. In the base, the crystals are aligned with the wall, and the inward motion simply displaces the crystal pair, with no reduction of the distance between them. In the apex, the crystals are placed on a curved surface. The inward motion is thus also a reduction in the curvature radius, and thus the crystals converge (dotted black lines). This reduces the distance between them, resulting measurement of an apparent shortening of the segment, which the is added to the true segmental shortening.

Illustration of this convergence on measured segment shortening. In this image, similar systolic segment shortening is shown. The convergence in the apex due to the convergence is shown in red. Without this convergence (using parallels - green), would result in true segmental shortening, which is less in the apex, more similar to the base.

Differences between walls

Although Höglund did not find any difference in systolic mitral annular displacement between different walls (30), other authors have found such differences, with lateral displacement higher than the septal (167). In the large HUNT study, the same differences were found in systolic annular velocities (165), with differences between septum and lateral wall was of the order of 10%, but not in deformation parameters (153), where the same difference was on the order of 4% in strain rate and only 1% (relative) in strain.

Normal annular velocities, strain rate and strain per wall in the HUNT study. (From  153 and 165)

	Anteroseptal	Anterior	(Antero-)lateral	Inferolateral	Inferior	(Infero-)septal
PwTDI S' (cm/s)		8.3 (1.9)	8.8 (1.8)		8.6 (1.4)	8.0 (1.2)
cTDI S' (cm/s)		6.5 (1.4)	7.0 (1.8)		6.9 (1.4)	6.3 (1.2)
SR (s-1)	-0.99 (0.27)	-1.02 (0.28)	-1.05 (0.28)	-1.07 (0.27)	-1.03 (0.26)	-1.01 (0.25)
Strain (%)	-16.0 (4.1)	-16.8 (4.3)	-16.6 (4.1)	-16.5 (4.1)	-17.0 (4.0)	-16.8 (4.0)

Results from the HUNT study (153, 165) with normal values based on 1266 healthy individuals. Values are mean values (SD in parentheses).  Velocities are taken from the four points on the mitral annulus in four chamber and two chamber views, while deformation parameters are measured in 16 segments, and averaged per wall.  The differences between walls are seen to be smaller in deformation parameters than in motion parameters, although still significant due to the large numbers.

This is illustrated below.

M-modes from the septal an lateral mitral ring, showing that systolic displacement is higher septally.	Pw tissue Doppler from the septal and lateral mitral ring, showing the lateral peak systolic velocities to be highest.

As this is not the case for strain and strain rate, this is illustrated below:

Colour Doppler from the four chamber view, traces from the septum (yellow) and lateral wall (cyan). In this image, the peak velocity and displacement shows bigger differences than peak strain rate and strain

Tethering

Velocity, displacement, strain rate and strain from three different points, apex, midwall and base, in the septum of a normal person. These curves all represent the same data set. It is evident that motion (velocity and deformation) increases from apex to base, showing a gradient, while deformation (strain rate and strain) is more constant, in fact a direct measure of the motion gradient.  Diastolic deformation is far more complex, and is discussed below.

1. Infarcted segments may be totally akinetic, but still being pulled along by active segments, showing motion without deformation. In this case, no offset between displacement curves, means no strain. This is usually evident in the inferior wall. A perfect example of a totally passive, tethered segment moving close to normally, can be seen below, and in more detail here. It may also be pertinent to the basal part of the right ventricle. In both cases, the annular motion may be near to normal due to hyperkinesia in the neighboring segment, as this segment is offloaded as explained here.  

   Tethering: The basal and midwall segments are infarcted, and are being pulled along by the active apical segment. The whole inferior wall seems stiff.
   The stiffness is evident in velocity and displacement curves. All of the wall has motion, which must be due to the apical segment, but as all curves lie on top of each other, the whole wall moves as a stiff object, i.e. there is no deformation below the apical point, and thus akinesia.	Strain rate and strain curves, however, show that the findings are more differentiated, showing akinesia basally (yellow), hypokinesia in the middle (cyan) and hyperkinesia in the apex (red).

1. Thus; in this case, the passive segment is tethered, showing motion and masking the pathology to some degree. Deformation imaging will show this.
2. If there is pathological contraction at some time in the heart cycle (e.g. post systolic shortening), the shortening of a pathological segment may impart motion to a whole wall.
   

   Velocity images showing motion towards the apex in red,  away from apex in blue.  Left, systolic 3D reconstructed image, showing normal motion in the septum and inferior wall, and paradoxical motion in the inferolateral, lateral and anterior wall. Right, o top are bull's eye from systole, showing the same, as well as early diastole showing inverse motion during the e-phase, i. e motion of the whole wall towards the apex in diastole. Apparently, the whole anterolateral half of the ventricle is ischemic .

   Strain rate images from the same recording, left systole, right early diastole, showing that the ischemia is due to a smaller ischemic area in the inferolateral, lateral and anterior apex, where there is stretching during systole (blue).  This stretching, results in the midwall and basal segments moving away from the apex, despite contracting normally. In early diastole there is recoil in the ischemic area (yellow), resulting in anterior diastolic motion in the whole of the wall.  In this case, the ischemia is obviously limited to a part of the apex, the rest of the motion abnormalities being due to tethering.

Translational effects:

In this video the rocking motion of the left ventricle is evident, the whole heart rocks toward the left in systole. (However, this is NOT due to conduction delay).	However, looking at deformation (wall thickening - transmural strain) in this cross sectional recording, the wall thickening can be seen to be normal and symmetric in both onset and extent.

Apparent asynchrony: Looking at mitral valve velocities, the lateral wall (cyan) seems to have a delayed contraction compared to the septum (yellow), both looking at onset and peak velocities, indicating either asynchronous activation or initial akinesia of the septum	Looking at multiple sites in the lateral wall, it seems that the delay in early ejection phase corresponds to positive velocity in the base (yellow), zero velocity a little more apical (cyan), and increasingly negative velocities toward the apex, i.e. possible apical initial dyskinesia (which might be ischemia).	The curved M-mode from the base of the septum through the apex to the base of the lateral wall shows the same effect, normal timing of the velocities in the septum, inverted velocities in the apical two thirds of the lateral wall.

Comparing tissue velocity with strain rate in the base and apex, however, , we see that the apparent delayed motion in the lateral wall has no corresponding delay in deformation, wheteher looking at onset of, or peak negative strain rate.  All four parts shortens synchronously and normally. Thus, it illustrates that the rocking motion velocities are added to the velocities, the subtraction algorithm of the velocity gradient subtracts these velocities again, showing the true timing of regional deformation.

What are the differences between strain rate and strain?

Contractility

Stroke volume

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

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

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

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.

Left ventricular length and external diameter:

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

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.
   

Wall lengths per wall

Left ventricular volumes

Applying the linear measures to an elliptical model of the left ventrcle, allowed the estimation of LV volumes (471).

Ellipsoid model of the left ventricle. All basic measures are linear, and the ellipsoid model assumes symmetrical wall thickness, declining to half in the apex, mitral annular diameter constant; equal to ventricular end systolic diameter, as LV diameter decreased by 12.8% is systole while the fibrous mitral annulus may be assumed to be more constant.

The ellipsoid model has some limitations. Being symmetric, it do not conform totally to the shape of the LV, which is assymmetric, as in other model studies.

An indication of this was that while all linear measurements were near normally distributed, there was a greater skewness in the calculated volunes:

Comparing skewnesses of the distributions of the linear measures (which is small), with the calculated volumes (which is significantly (greater), seems to indicate a systematic error in the volume data from the model.

LV volume and age

A: Mean BP showing an age related increase, above 60 about half is in the hypertensive level >140/90. B: LV volume in the different BP groups (results were not different if 140/90 was used). There is significant higher volumes in the >130/80 group, but in neither group was there any significant increase with increasing age.

Geometry of myocardial strain

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

Myocardial directions - normal strains

Left, the xyz coordinate systems relating to the deformation of a cube. Right the LTC coordinates of the LV myocardium, which in principle is equivalent as shown here. In  both cases there is ONE deformation of a three dimensional object, which deforms in three dimensions, and the three normal strains are the coordinates of this deformation. As wen would not talk about the xyz deformations of a cube as three independent functions, it is not appropriate to talk about three independent myocardial functions. Below are the strain tensors for the xyz and ltc deformations. In both cases, there is one tensor with three normal components.

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.

• Longitudinal systolic strain     =   relative longitudinal systolic shortening.
• Transmural systolic strain       =   relative wall thickening.
• Circumferential strain             =   relative shortening of one of the circumferences (external, midwall or endocardial).
  

Linear strains in three dimensions. Longitudinal shortening. Longitudinal strain can be measured by systolic and diastolic left ventricle (LV) lengths (A) or by Annular motion (B) divided by wall lengths (A). Transmural strain to be a truly segmental measure (C), the quantitative equivalent of wall motion score. The circumferential strains can be seen to be related to outer circumferential shortening as well as wall thickening, and endocardial circumference can be seen to move most, external most.  As circumferences can be calculated from diameters, circumferential strains can be calculated from fractional shortening. Midwall and external circumferential strains were calculated from endocardial diameters and wall thicknesses.

Longitudinal strain

Longitudinal shortening of the left ventricle. Lagrangian strain is the relative shortening normalised for the end diastolic length.	LV shortening can be measured by M-mode as the MAPSE, the relative shortening is the normalised MAPSE = MAPSE / L0.

An example:

Systolic strain is normalised MAPSE. The normalised MAPSE for this ventricle with an end diastolic length of 9.8 cm and an MAPSE of 17 mm is 15 / 92  = 17.3. This corresponds to a longitudinal strain of -17.3% in this example.


In the HUNT study, MAPSE as mean of 4 walls was 1.58 cm (417,  456). Mid ventricular end diastolic length was 9.24 (), and longitudinal strain by this method was -17.1%.
However, this is not ambiguous. In the HUNT study, we measured the distance from the apex to the mitral points, in lieu of wall length (WL). This ensured better reproducibility, the mitral points being more defined that the mid LV point.


Over all ventricular strain should be as illustrated above. Basically, Global LV strain (GLS) strain is LV shortening normalised for LV length, GLS = MAPSE / LVL (normalized displacement), and for Lagrangian strain, this the denominator is end - diastolic length (L₀).

Linear strain

Strain by WL is numerically smaller than by LV length, as the denominator is bigger (WL>L).	But following the curvature of the wall, would result on an even longer WL, a higher denominator and a numerically even smaller strain.

In the HUNT study, Mean diastolic WL was 9.47 cm, and mean strain by MAPSE/WL (calculated per subject an wall an then averaged, was -16/3%, as WL is longer than LV length, as evident from the figure above. We did not do the curved wall exercise at that time, as the data quality was less, and automated edge detection was not so good. It might be done now, in later databases with newer generation scanners, if deemed worth while.


As shown above, wall length can be measured in different ways, which has consequences for strain measurement. The lowest denominator as illustrated above, is the mid chamber line (L₀), giving the highest numerical GLS value. Normalising for wall lengths instead, will give a higher denominator, and thus a lower numerical GLS value. One approximation to wall lengths is to use the straight line from apex to the mitral points. This will give a high reproducibility by using clear anatomical landmarks, and the straigh line is reproducible. In addition, there will be little angle difference between the M-mode line and the wall line, in effect eliminating the systematic angle error. This is the linear strain method (417, 444) described here, resulting in a mean strain of 16.3 (2.4)%.


This is also illustrated by the examples below. .

For any given MAPSE, the global strain will be determined by the choice of denominator. In this case, mean MAPSE is 1.7 cm. End diastolic length will be the denominator in the strain equation. Using the mid ventricular line (blue), gives the smallest denominator and thus the highest global strain value of 17.3% in this example. Using wall length, will result in a higher denominator, resulting in lower GLS value, the straight line approximation (green) gives an intermediate denominator and a GLS value in this example of 16.3%, while the curved lines (red) following the walls gives the highest denominator, and thus the lowest GLS value, in this example 14%.

Another example:

Illustration on how the choice of reference length will affect the strain value. The curved lines, representing the longest wall measurements, will give the lowest GLS value, the straight lines will be in between, while the mean ventricular length will be the shortest, and thus give the highest strain value.

Choosing instead the curved wall line, will give a truer length, at the cost of higher variability, but the length will be systematically higher, resulting a higher denominator and lower value of GLS as shown above. In this case, there is a lagrangian strain using only the end diastolic length, as opposed to speckle tracking as shown below.

Speckle tracking strain

Speckle tracking strain in general not only have curved ROIs, but also tracks crosswise motion of the speckles, which is due to wall thickening. As the wall thickens, this means that speckles move inwards in the cavity. Let us consider a hypothetical example where there is wall thickening without wall shortening. (This would mean volume expansion, but illustrates still the effect of wall thickening.):

Hypothetical wall thickening without wall shortening. As the wall thickens, both the midwall (red unbroken) and the endoicardial (blue unbroken)line moves inwards. This is true for both the curved line (depending on the curvature) and the straight lines (depending on the cosine of the angle change), the inward motion shortens the lines. 

Simultaneous wall shortening and thickening. As the wall shortens, it has to thicken, due to conservation of the myocardial volume. As illustrated above, the mid and endocardial lines shorten not only due to wall shortening, there is a shortening due to the inward movement as well, which again is caused by wall thickening. And as wall thickening is due to wall shortening, this means that the shortening is speckle tracking strain is over estimating the true wall shortening.

Segmental strain

However, there are additional assumptions that will differ between vendors of speckle tracking programs. Using mean strain over the ROI will result in a value close to the mid ROI line. Some vendors, however, trace the endocardial line, which will result in higher absolute values. The thickness of the ROI is often assumed to be constant, while the wall is thinner in the apex. As the apex is the most curved part, a ROI in the apex that is thicker than the wall, will result in a higher absolute GLS. The curvature in the apex may also vary, even in the same software, as shown here.

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.

Finally smoothing, software algorithms, such as choice of kernel sizes, selection and weighting of acoustic markers, stability of speckles, and drift compensation during heart cycle are all assumptions that are guarded as industrial secrets.

Thus, it is not surprising that there are inter vendor differences, even in speckle tracking derived strain.

Inter vendor differences in speckle tracking

• However, this is for the speckle tracking applications only, as it is not feasible (or indeed possible) for tissue Doppler, nor for normalised MAPSE. And some of the comments are not valid for segmental strain by speckle tracking either. (The recommendations explicitly are for 2D speckle tracking only).
• Secondly, some ambiguity remains, as applications where the ROI has the same width in the base and apex, despite the differences in thickness of the myocardium. In this case, at least one side of the ROI will not follow the wall, and the mid line of the ROI will not follow the midline of the wall (or, if the ROI are suited to the midline, the endocardial side of the ROI will not follow the endocardium). If so, the results may vary as shown in the figure below:

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

Normal transmural gradient of longitudinal strain.

Hypothetical wall thickening without wall shortening. As the wall thickens, both the midwall (red unbroken) and the endoicardial (blue unbroken)line moves inwards. This is true for both the curved line (depending on the curvature) and the straight lines (depending on the cosine of the angle change), the inward motion shortens the lines. 

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.

Transmural strain

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.

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.

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

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.

Circumferential strain

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.

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, as long as the pericardium would counteract the ecpansion by the pressure generated by the longitudinal shortening.
   

Circumferential strain is an ambiguous term.

Normal transmural and circumferential strain gradients

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 due to shortening, 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.

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.

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

Circumferential strain is a function of diameter reduction.

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

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

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.

Circumferential strain by speckle tracking

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.

• either segmental, measuring strain in separate segments,
  
• or regional, the average circumferential strain in a short axis plane is representative for that plane only,
• and that the FS of that plane as measured by M-mode, is equal to the average circumferential strain in the same plane.

Linear strains in three dimensions from the HUNT study

Inter relations of all three major strain components

Deformation in three dimensions, diastole left, systole right, with end diastolic contours shown as dotted lines. Longitudinal shortening as well as outer circumferential shortening will contribute to the total volume reduction (yellow). Both will result in wall thickening, which goes inwards. Wall thickening displaces the midwall (dark red) and endocardial (dark blue) circumferences inwards. As the outer layer (light red) thickens, it pushes the inner layer (light blue) inwards, causing it to thicken, and in addition the layer has it's own intrinsic thickening due to the longitudinal shortening. Thus, the endocardial circumference is pushed even more inwards.

There is ONE strain tensor, with three normal strains, which are the coordinates of ONE deformation in three dimension.

1. Longitudinal shortening will lead to wall thickening. This is true even if there is only longitudinal fibres.
   
2. Circumferential shortening will lead to a modest reduction in outer diameter, and circumferential shortening.
   
1. Both leads to inwards expansion (thickening) of the wall, which occurs inwards.
   
2. This thickening leads to inwards displacement of the midwall and endocardial circumferences, and thus circumferential shortening.
1. AS there is less room for expansion further in, this mus result in more thickening in the inner layers.
   

Relations between longitudinal shortening (MAPSE), wall thickening, transmural and circumferential. A modest decrease in outer diameter is postulated, measured at the mid ventricular level. It is less credible that the mitral ring, being fibrous will contract. This is shown in 3D (bottom), cross sectional, above, and the corresponding M-modes to the right. Endiocardial surfaces: dark blue, midwall surfaces, dark red, outer surfaces, black. Light yellow: volume reduction in systole, light red: outer myocardial layer, light blue: inner myocardial layer. The wall thickening is less than the reduction in diameter, which also has a component of outer diameter shortening.

Incompressibility and the myocardial strain tensor

Deformation of the myocardium. There is simultanous shortening and wall thickening (which also results in midwall circumferential shortening), showing the inter relationship of the strains.

Area strain

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).

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.

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

Incompressibility and stroke volume

Looking at walls, it seems that the cavity volume reduction (i.e.) the stroke volume, depends on both shortening and wall thickening as well as outer diameter reduction. But the wall thickening is simply a function of outer diameter reduction, and shortening. If the myocardium is incompressible, the myocardial volume has to be the same in both systole and diastole. Thus, as the total (outer) volume is the sum of myocardial and cavity volume, the systolic reduction of cavity volume has to equal the reduction in total volume, and thus the stroke volume is only determined by the longitudinal and outer diameter shortening. If the myocardium is partly compressible, the myocardial compression will detract from the stroke volume.

The total volume in diastole is the sum of the blood inside, and the myocardium. If the myocardium is incompressible, and the outer contour absolutely constant, 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.


When the left ventricle deforms in systole, the total volume is reduced by the outer longitudinal shortening, and the OUTER diameter shortening. If the myocardium is nearly incompressible, it means that the myocardial volume inside the end systolic volume is the same as in diastole, and the full outer volume reduction will be nearly equal to the stroke volume. Thus, the stroke volume is given by the outer diameter and the systolic longitudinal ventricular shortening (56).

If there is systolic compression of the myocardium, the outer volume reduction will be higher than the SV, by the compression factor.

Long axis shortening, and LV volumes in the HUNT study

Entering the linear measurments into an ellipsoid model of the LV, gave a possibility to study age dependency of LV volumes, as well myocardial compressibility (471).

Ellipsoid model of the left ventricle. All basic measures are linear, and the ellipsoid model assumes symmetrical wall thickness, declining to half in the apex, mitral annular diameter constant; equal to ventricular end systolic diameter, as LV diameter decreased by 12.8% is systole while the fibrous mitral annulus may be assumed to be more constant. As shown above the MAPSE contribution to the SV would be external mitral annulus x MAPSE, while the transverse diameter shortening would contribute the rest.

The ellipsoid model has some limitations. Being symmetric, it do not conform totally to the shape of the LV, which is assymmetric, as in other model studies.

An indication of this was that while all linear measurements were near normally distributed, there was a greater skewness in the calculated volunes:

Comparing skewnesses of the distributions of the linear measures (which is small), with the calculated volumes (which is significantly (greater), seems to indicate a systematic error in the volume data from the model.

Despite this, it was interesting findings:

Age	MAPSE (cm)	MAPSE vol(ml)	LVEDV(ml)	SV(ml)	EF(%)	MAPSE% of SV	Endocardial FS(%)	Outer FS (%)
Women
<40	1.73(0.20)	56.5(9.9)	111.6(21.6)	76.3(16.4)	68(6)	75.4(11.9)	36.6(6.1)	14.1(3.3)
40-60	1.58(0.23)	53.3(11.7)	106.9(21.7)	72.7(17.0)	68(6)	74.9(13.5)	36.5(6.9)	13.2(4.2)
>60	1.33(0.26)	45.2(10.1)	97.9(19.7)	65.4(16.9)	66(9)	72.0(21.9)	36.0(9.1)	12.1(4.2)
Total	1.58(0.26)	52.9(11.5)	106.8(21.8)	72.6(17.3)	68(6)	74.6(14.9)	36.4(7.1)	13.3(4.0)
Men
<40	1.72(0.22)	70.1(14.9)	144.8(30.5)	96.1(22.9)	66(8)	74.9(14.2)	35.5(6.9)	12.6(3.7)
40-60	1.58(0.22)	65.1(14.2)	138.1(31.1)	92.2(23.8)	67(8)	72.8(14.8)	35.8(7.4)	12.2(3.8)
>60	1.45(0.21)	60.3(14.7)	126.3(33.7)	84.1(25.7)	66(8)	74.9(19.0)	36.0(8.0)	11.8(4.4)
Total	1.58(0.24)	64.9(14.7)	136.6(32.2)	91.0(24.4)	67(8)	73.8(15.8)	35.8(7.5)	12.2(3.9)
All	1.58(0.24)	61.5(13.0)	121.1(31.1)	81.4(22.9)	67(8)	75.2(12.8)	36.1(7.3)	12.8(4.0)

Basically, the findings of the functional measures were:

In this study, SV calculated from EDV-ESV was 81.4ml, while Mitral area x MAPSE was 61.5 ml, = 74.2% of total SV. Circumferential shortening due to OUTER circ. (diameter) shortening, was 12.8%, and must make up the rest, 25.8% of SV.

Neither endocardial FS nor EF declined with age, as has been shown so many times before. (387, 391, 392, 458, 473, 474) (158, 423, 449, 475). Thus there is no compensatory increase in transverse (short axis/circumferential) function to maintain EF, it's simply the simultaneous decline in LVEDV and SV, that maintains their ratio. And this is not due to decrease in short axis internal diameter, but to a decrease in length (386).

MAPSE declined with age as described before (417, 456). But just as for EF, the simultaneous decline in MAPSE and SV, maintains their ration, so the percentage of the SV due to MAPSE doesn't decline, and thus there is no need for any increase in transverse function. In fact, there is a decrease in midwall and outer FS, so there is a decline in transverse function with age (456).

In addition, MAPSE was independently related to DBP (Age Beta = - 47%, DBP Beta -0.13, both p<0.001), but not to SBP. This means, that the (hypertrophic?) effect of hypertension, which was increasingly present with increasing age as discussed here, had an impact on MAPSE, but far less than age per see, and in this limited range, afterload (SBP) was not a factor.

MAPSE is nearly body size independent (417, 456), while GLS is inversely related to body size (417) as explained here. The percentage of MAPSE contribution to SV, showed no correlation to BSA, whatsoever.

MAPSE contribution to the stroke volume:

Thus the MAPSE contributes somwhere between 60 (420) and 85% (158) of the total SV, with our findings in between (471). However, due to the limitations of the geometrical model, our findings were not meant to be normative, but was more a study of the relations to each other, to BP and age.

The notion that circumferential function is the main contributor to SV, is bases on the lack of understanding that circumferential strain (except external CS) is partly due to wall thickening, which again is mainly due to longitudinal shortening. Circumferential component is due to OUTER circ. (diameter) shortening, was 12.8%. Midwall and endocardial circ. shortening is not circ. fibre shortening, but increasingly a function of wall thickening, which again is a function of wall shortening:

Figure showing that while outer volume decrease is due to outer circumferential and longitudinal shortening, the apparent cavity decrease due to midwall or endocardial decrease, is mainly a function of wall thickening, which again is wall shortening.

Deformation and stroke volume:

In modern MR studies (420, 430) the longitudinal shortening contributed about 60% to stroke volume, outer contour reduction the rest, even when the AV-plane motion was depressed (431). A newer ultrasound studyy found the longitudinal shortening to contribute as much as 83% of the stroke volume (158). The correlation between MAPSE 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 , although it may be more in real situations.The HUNT study found about 75% long axis contribution, but being a stidy of a geometrical model, that may be an over estimation, although the volumes were skewed towards the lower range (471).

Thus, 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, 420, 430, 431, 471).

And, if the myocardium is incompressible, the wall shortening and thickening, and thus the internal diameter reduction have to be interrelated (7), and both would be valid measures of stroke volume. This is little compared to wall thickening, showing that the main inner contour diameter reduction is due to longitudinal shortening, as discussed above. Thus, the eggshell model is fairly accurate, and the long axis function describes most of the pumping action of the heart.



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, contributing between 60 and 80% of the total stroke volume.

Interestingly, in the HUNT study, The percentage was similar across all age groups, despite MAPSE declining with age. This is explained by the fact that LVEDV decreased with age, by reduction of LV length , while LVIDD remained constant(386, 471). As SV decreased by the same magnitude (471), both the ratio of SV/LVEDD (=EF) and the ratio of MAPSE / SV remained constant. The hypothesis that EF was maintained by a compensatory increase in short axis function is erroneous, as endocardial FS remained constant (386, 456), while external FS actually decreased (456, 471), showing a reduction in short axis function with age.

It was a reasonable hypothesis that strain (and strain rate) was shortening normalised for heart size, and thus was size independent measures of LV shortening. However, it has always been known that global longitudinal strain was gender dependent (153). However, as body size, and presumably heart size is gender dependent as well, gender differences is just an effect of body size, while linear regression showed that only body size was an independent variable in the HUNT study (417). What was more interesting was that non-normalised MAPSE was not gender dependent.

Comparing global strain, normalised global strain (MAPSE / LV length) and global longitudinal strain, a weak correlation of MAPSE with BSA was noted, while normalised MAPSE and GLS was stronger, but negatively correlated with BSA (417). This was also the main reason for the sex difference in MAPSE.


Findings are summarised in the following picture:

As shown by the boxplot, no significant gender differences in MAPSE, but in normalised MAPSE and GLS (women highest). All three measures are age dependent. Lower panels shows weak positive correlation between MAPSE and BSA, stronger negative correlations between BSA and normalised MAPSE and GLS. Gender differences only due to differences in BSA, no independent contribution.

And as the ratio between LV diameter and length are nearly constant across body sizes (386). With increasing body size the heart size increases both in length and diameter, while the ratio between them stays constant (386). With increasing diameter, the same MAPSE will result in increased stroke volume, as MAPSE is responsible for about 75% of SV, thus annular motion (S' and MAPSE) do not have to increase as much with body size to increase SV. Thus LV length increases more with SV than annular motion, inducing a negative relation between global strain / strain rate and body size (417), as illustrated below.

Relation between stroke volume and mitral annular plane systolic excursion (MAPSE) in ventricles of different size. It has been shown that while heart size increases with body size, the ratio between length (LVEDL) and external diameter (LVEDD) does not. As the stroke volume is mainly determined by the systolic shortening (MAPSE), a larger ventricle has a larger radius, and thus, a larger stroke volume (increasing proportional to the square of the radius) even without any differences in MAPSE, as shown by the very low correlations between MAPSE and BSA. Thus, in the ventricle to the right, for the same MAPSE, the SV is far higher. As length increases proportional to the diameter, GLS being MAPSE/ LVEDL, GLS actually decreases with increasing heart size. This is a systematic error that occurs due to the one-dimensional normalization. As there is a strong correlation between S′ and MAPSE, this is also the case for S′ vs GLSR, even if those measures are more closely related to contractility than stroke volume

Is the myocardium compressible?

In the HUNT study, using the strain product on linear measures, the strain product, being equal to the volume ratio: was 1.009 (1.0136 - 0.99851) using straight line wall measures (longitudinal strain -16.3%), and 0.9957 (1.003 – 0.98896) using mid ventricular line (longitudinal strain -17.1%). However, speckle tracking tends to measure higher GLS, because of the shortening due to inward tracking of the wall thickening, and wall thickening varies too much between studies to give any meaning of the strain product at all.

In the model study, however, myocardial volumes could be measured directly. Here, we found a myocardial volume reduction in systole of 3.28 ml, or 2.5% of myocardial volume, 4.8% of SV.
This corresponds to a Vs/Vd of 0.975 (SD 0.112), 95%CI ((0.969-0.981) . 

But as the model has limited accuracy, this is not normative either. Our main finding was that this compressibility, however, was not related to age, BP or BSA.

Strains and fibre direction

Schematic representation of fibre directions after Greenbaum (62) and and Streeter (424) showing the spiral course of the sub epicardial fibres (red) in a counter clockwise direction as seen from the apex, the circular course of the midwall fibres (green) and the clockwise course of the spiral fibres in the sub endocardium (blue), forming three sheets 256, 257.	Schematic representatin of fibre course after ....... where the sub epicardial fibres (red)  dive into the midwall, continuing into the midwall circular sheet (green), and then this again continuing into the sub endocardial spiral fibres (blue), considering the sheets contiguous and not as three separate.

Schematic representation of the effects of compression on the helix. Green: the original helix. Comopressing the helix in the longitudinal direction makes the spiral's components to be more horizontal, i.e. a smaller angle with the horizontal plane. Compressing the helix in the transverse direction (red) makes the spiral to run in a more vertical direction, i.e. increasing the angle with the horizontal plane. In this case, the blue spiral is compressed longitudinally by 50% (blue), and then transversally by 50%(red, which restores the original angle.

Shifting fibre angles in systole. The sub epicardial fibres do not move much inwards, thus there is a mainly longitudinal compression of the helix, which will reduce the angle with the horizontal plane. The sub endocardial fibres, on the other hand,, are shifted inwards, which will tend to oppose the effect, being a tranverse compression of the helix.

Diagram of fibre angles according to location in the wall in diastole and systole after Streeter (424) . The sub epicardial fibres are considered to run downwards towards the apex, and thus the angle in the paper is designed as negative, while the sub endocardial fibres are considered to run upwards towards the base, and thus a positive angle. The diagram both shows how the fibres run mostly longitudinally in the sub endocardium and sub epicardium, and mostly horizontally in the mid myocardium. The shift from diastole and systole is consistent with the predictions from the model to the left. The angle decreases in absolute values in the sub epicardium, consisent with longitudinal compression. The sub endocardial fibres on the other hand, actually increases the angle, indicating that the transverse compression dominates.

The helical arrangement of fibres running between endo- and epicardial surfaces would meant that even if there had been no fibre deformation, the wall would thicken.

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.

Illustration of contraction. In an isometric contraction, there is no sarcomere shortening, the tension is converted into configuration shape within the sarcomere, while the isotonic contraction, the tension is converted into shortening of the sarcomere itself.

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. .

The eggshell model

Four chamber view showing the outer contour of the heart (yellow) to be fairly stable. Not much inward squeezing is evident, and there is no need for energy expenditure in moving the surrounding tissue in and out. Apex remains stationary, and the main movement is the AV-plane motion. This serves as the main mechanism for pumping. However, due to myocardial incompressibility, the wall thickens as it shortens, and thus reduces chamber volume by inward motion of the endocardium. At the same time, the AV plane motion in systole expands the atria, thus contributing to the atrial filling in systole.

If the diastolic pressures are similar on both sides of the septum, there is little diastolic motion of the septum. The systolic position of the septum is determined by the circular cross section of the ventricle. Thus, in normal ventricles, there is little motion of the vseptum, and the left side of the heart follows the eggshell principle as well.

The eggshell mechanism

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.

In a total invariant outer contour, the systolic apical AV-plane motion would shrink the ventricles, and expand the atria equally, and systolic ejected volume from the ventricles andvenous inflow to the ventricles would be the same.	But that means that during early cdiastole, there would be no net volume changes, as the volume shift from atria to ventricles would be due to AV-plane shift alone. the ventricles simply taking over part of the atrial volume.

The eggshell model and atrial filling.

From colour flow, it is very visible that there is intra atrial flow closely related to the apical AV-plane motion during ejection. This flow starts immediately at ejection, and propagates all the way down to the pulmonary vein, showing how PVs flow is a contiguopus event through the atrium, originating in the suction from the AV-plane motion and atrial expansion.