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Approximately half of patients experiencing congestive heart failure present with a normal left ventricular ejection fraction. Perturbations in material properties affecting ventricular pressure/volume relationships likely play an important role in the “stiff heart syndrome” yet noninvasive tools permitting the accurate assessment of myocardial elasticity are extremely limited. We developed an MRI-based technique to examine regional left ventricular stress/strain relationships by incorporating displacement-encoding with stimulated-echoes (DENSE) and phase-contrast (PC) velocity mapping and compared regional elastic moduli (EM) and viscous delay time constants (VDTCs) (N = 10) with immediate postmortem direct strain gauge measurements (N = 8) and global chamber compliance (literature) in normal dogs. EMs by MRI were significantly greater in papillary muscle columns when compared with lateral wall and septal locations by MRI (7.59 ± 1.65 versus 3.40 ± 0.87 versus 2.55 ± 0.93 kPa, P < 0.0001) and were in agreement with direct strain gauge measurements (3.78 ± 0.93 and 2.96 ± 0.88 kPa for the lateral wall and the septum, P = ns for both versus MRI). MRI-determined VDTCs were similar in the three regions (VDTC = −1.15 ± 12.37 versus 3.04 ± 7.25 versus 4.17 ± 5.76 ms, P = ns) and did not differ from lateral and septal wall strain gauge assessment (VDTC = 3.09 ± 0.40 and 4.57 ± 1.86 ms, P = ns for both versus MRI). Viscoelastic measurements obtained in six normal volunteers demonstrated the feasibility of this technique in humans. Noninvasive, regional assessment of myocardial stiffness using DENSE and PC velocity mapping techniques is accurate in a canine model and feasible in humans.
Between 35 and 55% of patients presenting with symptoms of congestive heart failure have a normal left ventricular (LV) ejection fraction (1,2), suggesting that limitations in the dynamics of chamber filling may be responsible for their symptoms. While heart failure with a normal ejection fraction (HFNEF) or “stiff heart syndrome” was first described 2 decades ago (3–5), its mechanistic underpinnings remain controversial and are variously ascribed to: (a) delayed or incomplete LV relaxation, (b) reduced LV compliance (6), and/or (c) abnormal ventricular–vascular coupling (7,8). Such limitations in pathophysiologic understanding are largely based upon a paucity of reliable hemodynamic data in this group of patients.
While easily accessible techniques such as Doppler echocardiography or radionuclide ventriculography can document rates and degree of LV relaxation and filling, these measurements fail to adequately assess chamber compliance, an indicator of material ventricular properties underlying the stiff heart syndrome. Measurement of chamber compliance is rarely performed outside of research laboratories due to its dependence upon invasive interrogation with expensive, complex equipment requiring management by expert personnel. In fact, only two reports, both recent, have attempted to implement this technology in the setting of HFNEF, resulting in somewhat disparate conclusions (6,7). If the mechanisms of HFNEF and other forms of “diastolic dysfunction” are to be adequately understood, then a noninvasive, high-throughput technique musto be developed that will provide a comprehensive analysis of chamber stiffness.
In this report we describe an MRI-based technique for measuring regional viscoelastic diastolic properties of LV myocardium. Traditionally, LV chamber compliance has been measured through assessment of the relationship between absolute pressure and volumes, which correspond, respectively, to the mechanical terms of “stress” and “strain.” While MRI cannot currently measure absolute values, we propose to measure myocardial wall compliance based on the relationship between transient gradients of ventricular pressure and regional myocardial strain, both of which are attainable with existing MRI methods.
Diastolic filling of the ventricles can be regarded as an interactive process between elastic myocardial lengthening and dynamic ventricular flow. Transient pressure gradients associated with the acceleration and deceleration of blood lead to heterogeneous stress (stress gradients) along the myocardial wall and resultant heterogeneous strain (strain gradients). The relative timing and amplitude of the stress/strain gradients are dependent upon the viscoelastic constants of the myocardium. Using displacement encoding with stimulated echo (DENSE), a validated MRI-based, phase-shift displacement technique (9 –15), in combination with phase-contrast (PC) velocity mapping, we assessed regional stress/strain gradients in the LV wall and papillary muscle and compared the estimates of the viscoelastic parameters with (1) direct strain gauge and (2) canine global chamber compliance as reported in the literature. We further demonstrated the feasibility of this method in humans.
Ventricular filling (Fig. 1) during sinus rhythm is biphasic and characterized by (a) isovolumic relaxation leading to acceleration and deceleration of blood into the ventricles following mitral valve opening in early diastole and (b) late diastolic transport of a second pulse of blood during atrial systole. The acceleration and deceleration phases of ventricular inflow are due to modulation of LV pressure gradients from the base to the apex and vice versa, which are more prominent in the early diastole. These pressure gradients exert heterogeneous forces on the elastic ventricular walls: in the early phase elevated pressure near the base squeezes the basal ventricular wall more than the apical portion, resulting in increased stretch or less strain; in the later period of early diastole the pressure gradients are reversed as are the influences on the walls. This process can generally be regarded as a pulse-wave phenomenon in which the elastic walls conform to the pressure waves in the ventricles. Mechanically, the base-to-apex gradients of wall strain conform to the ventricular pressure gradients as portrayed in Fig. 1.
In the simplest model the myocardial wall can be regarded as an incompressible, Hookian elastic material of an average elastic modulus (EM) E and viscosity μ across its thickness. The relationship between pressure and strain gradients in an equatorial wall segment is
where τ, the viscous delay time constant (VDTC), is: μ/E. The derivation of this relationship is detailed in the Appendix.
The strain gradient within the wall is, by definition, a second-order spatial derivative of the displacement fields, as measured with the DENSE technique (described under “DENSE Imaging” below (11)). The mean pressure gradient on both surfaces of the wall and inertial forces due to tissue acceleration were derived from PC velocity maps described under “Phase-Contrast Velocity Mapping” below (16–21). The sum of the two is effectively the stress gradient as detailed in the Appendix. Viscoelastic constants were then determined through fitting of the stress–strain gradient relationship.
Given the anisotropic elasticity of individual muscle fibers, we also measured the viscoelastic constants of an LV papillary column, in which the fibers and perimysial collagen are aligned in parallel along its length. This simpler structure resulted in a mathematically modified relationship between stress and strain gradients (see “Treatment of the Papillary Column” in the Appendix).
The animal protocol was approved by the National Institutes of Health Animal Care and Use Committee and conformed to all relevant institutional and federal guidelines. Two groups of animals were involved in this study: a normal group undergoing MRI (N = 10) and a second normal group assessed by strain gauge (N = 8). Closed-chested beagles (10 –12 kg), which were initially given an intramuscular (i.m.) dose of acepromazine (0.1 mg/kg) and an i.v. dose of thiopental (2.27 mL/kg), underwent anesthesia with i.v.-administered sodium pentothal (13 mg/kg). Then cephalic vein, jugular vein, and femoral artery catheters were placed for blood gas sampling, hemodynamic monitoring, and drug administration. Body temperature was monitored and maintained with an air heater. The animals were intubated and ventilated at 25–30 breath/min in order to perform acquisitions during the static end-expiratory plateau, without holding the breath of the animal for prolonged periods of time. Anesthesia was maintained by isoflurane (1 to 1.5%).
In the MRI normal group an MR-compatible pacing catheter (Millar, Houston, TX, USA) was inserted under fluoroscopic guidance into the right atrium via the femoral vein and placed against the sinoatrial node in order to maintain a constant heart rate. The pacing rate was chosen at 10 to 15% above the intrinsic rate and phase-locked to the ventilation rhythm to maintain a fixed number of beats within one respiratory cycle. Cardiac pacing and ventilation were electronically synchronized. The details of this preparation and pacing scheme have been previously presented (22).
For the strain gauge group, anesthetized dogs, including one from the MRI group, underwent diastolic arrest through infusion of heparin and potassium chloride. The heart was immediately excised and immersed in physiologic saline. The right ventricle was opened with a longitudinal cut along the posterior wall to allow access to the interventricular septum. The strain gauge was of the type developed by Parsons et al. (23) for use in hydrated matrix gels. It consisted of a 2 French needle as the oscillating element and an infrared probe for readout (Fig. 2). The needle was inserted 5 mm into the myocardium, and the other end was tapped to induce a dampened small amplitude oscillation, which was recorded with the infrared probe. The period and the decay rate of the oscillation were converted to the EM and the VDTC of the muscle against calibration curves obtained in gelatin blocks. This measurement was repeated at three sites on the epicardial side of the lateral wall and on the right side of the inter-ventricular septum. The 5-mm depth of insertion was over 2/3 of the wall thickness, so the results represented the average over most of the thickness. These measurements were performed at room temperature (20°C) and within 30 min of euthanasia, in all cases, to avoid rigor mortis.
Due to technical limitations related to sequence gating as detailed in the Discussion below, data acquired in this pilot study only covered the early filling phase of diastole. Magnetic resonance imaging was performed on a 1.5-T clinical scanner (Sonata, Siemens, Erlangen, Germany). Displacement imaging of the myocardial wall incorporated the DENSE technique (9–14), a phase-shift method that utilizes the stimulated-echo acquisition mode to extend the phase coherence of spins over the diastolic period (Fig. 3). The three-dimensional spatial coordinates of the proton nuclear spins are encoded in their phase values and the phase coherence is preserved over hundreds of milliseconds with the stimulated-echo mode. Phase shifts during this time period are directly proportional to the displacement of the protons and provide a measurement of such. In its implementation (Fig. 3), DENSE consists of a position encoding and an image acquisition segment separated by the mixing time. Field gradients in the encoding segment impose spatial phase ramps across the heart. Subsequent movement of the protons in the tissue over the mixing period results in phase shifts such that, in the acquired image, the phase change of a pixel, relative to its initial phase, indicates its displacement along the gradient direction. The three-dimensional (3D) displacement vector is obtained from three such measurements. In DENSE images, the blood pool appears dark and wall segmentation is relatively accurate.
In dogs, the displacement imaging parameters were as follows: single long axis slice bisecting the septum and the lateral wall, temporal resolution = 10 ms, voxel size = 1.48 × 1.98 × 8.0 mm3, image matrix dimensions = 128 × 96, in-plane displacement encoding resolution = 1.86 mm/π, through-plane displacement encoding resolution = 3.18 mm/π. A segmented k-space, true-FISP readout scheme was used (TR = 3.1 ms), in which a train of 32 k-space lines were acquired in each shot (24). The total scan time was 15 min.
Ventricular velocity imaging employed a standard cine phase-contrast (cine-PC) sequence (16 –21). Only the in-plane velocity components in the long axis slice were acquired, which were sufficient for pressure gradient estimation (21). The scan parameters were as follows: single long axis slice bisecting the septum and the lateral wall, temporal resolution = 10 ms, voxel size = 1.48 × 1.98 × 8.0 mm3, image matrix = 128 × 96, velocity encoding strength (Venc) = 100 cm/s/π, total scan time = 8 min.
Six normal volunteers (5 female, 1 male, 29–58 years old) were scanned. For adaptation to humans, both the DENSE and the PC velocity sequences were modified so as to be gated by the respiratory signal from an air bellow strapped to the waist of the volunteer and data were acquired during end-expiration. The DENSE sequence parameters were as follows: single long axis slice bisecting the septum and the lateral wall, temporal resolution = 20 ms, voxel size = 1.48 × 2.96 × 8 mm3, image matrix = 256 × 96, in-plane displacement encoding resolution = 3.7 mm/π, through-plane displacement encoding resolution = 3.18 mm/π, scan time = 14 min. The PC velocity sequence parameters were as follows: single long axis slice bisecting the septum and the lateral wall, temporal resolution = 20 ms, voxel size = 1.48 × 2.96 × 8 mm3, image matrix = 256 × 96, Venc = 150 cm/s/π, scan time = 5 min.
For myocardial wall strain mapping, the displacement vector fields were obtained from the DENSE phase images then differential movement of every four adjacent pixels were used to define the strain tensor of the tissue element enclosed by these pixels. The strain tensor was then projected onto the transmural and longitudinal directions to give the corresponding normal strains in these directions (25). The user manually outlined the equatorial segments in the septum, the lateral wall, and the papillary muscle for the viscoelastic measures.
For ventricular pressure mapping, the Navier–Stokes equation relating local flow acceleration and relative pressure distribution was used (16 –21). Thompson and McVeigh showed that LV pressure gradients within a long-axis slice were dominated by flow acceleration in the slice, and computations from in-plane velocity components gave accurate estimates of these gradients (21). This was the basis for pressure gradient calculations in this study.
Once strain and pressure maps were obtained, they were synchronized to the R-wave of the ECG signal, and the curves of strain and stress gradients were fit to Eq.  to give estimates of EM and VDTC.
All results are expressed as means ± SD. Because the MRI and strain gauge groups had only one shared animal, the comparison between the two was performed with unpaired, two-tailed t tests. Comparisions between ventricular locations within the MRI and strain gauge groups were based on paired, two tailed t tests. Probability was accepted at P < 0.05.
A complete canine DENSE data set contained myocardial wall displacement vectors in multiple cardiac phases from which the strain fields were calculated. The normal strain components used in the viscoelastic measures included both transmural (radial) εrr and longitudinal εzz components. An example of corresponding displacement and strain data at three diastolic time points is displayed in Fig. 4.
Example maps of blood velocity and corresponding relative pressure distribution at three diastolic time points are shown in Fig. 5. Since a uniform offset of pressure could not be determined solely from the flow data, the pressure maps are relative and not absolute and represent the spatial variation of pressure.
Transient diastolic stress and strain gradients were calculated for the three wall segments including the papillary muscle column, the lateral wall, and the interventricular septum, an example of which is shown in Fig. 6. In all segments, the strain gradient was observed to conform to the stress gradient as described under Materials and Methods, “Basic Princples.”
In the canine heart, the MRI data set exclusively covered the early diastolic filling phase, prior to onset of atrial contraction. The myocardium exhibited low internal viscosity during this period (VDTC = 4 ± 6 ms in the septum, 3 ± 9 ms in the lateral LV segment), consistent with previous direct measurements of rat trabeculae (VDTC = 6–12 ms (26)) (P > 0.08). In Fig. 7 the EM and VDTC of the MRI group of dogs are graphed along with postmortem values from the strain gauge group. The strain gauge we used was not applicable to the papillary muscle due to the small size of the muscle itself. The detailed statistics of both measurements are tabulated in Table 1. There were no statistical differences between in vivo MRI values and postmortem strain gauge measures of EM and VDTC in the septal and lateral walls, although the postmortem EMs tended to be slightly higher than in vivo counterparts.
Regional disparities in EM consistent with regional anisotropic fiber arrangement were seen between segments. The P values of paired t tests within MRI and strain gauge groups are summarized in Table 2. MRI-determined EM of the papillary was significantly higher then that of the lateral wall and septal segments (P = 3.6 × 10−5 and 8.7 × 10−5, respectively). MRI-determined EM of the lateral wall segment was slightly higher than that of the septum (P = 0.0209). A similar trend was observed for strain gauge measurements (P = 0.0507).
We believe that the papillary muscle had a higher elastic modulus because it consists of parallel myofibrils and perimysial collagen, and in vitro measurements on muscle samples observed that the EM is markedly greater in the fiber direction than in the perpendicular (27,28). The septal and lateral wall segments are composites of elastic sheets (29,30) with 150° dispersion of fiber angles (31) and therefore were reasonably expected to have lower EM values than the papillary muscle. The strain gauge we used was not applicable to the papillary muscle due to the small size of the muscle itself, which made the measurement unreliable.
Both MRI- and strain-gauge-determined VDTC values were uniformly low in all segments: MRI-determined VDTCs were −1.15 ± 12.37 ms for the papillary column, 3.04 ± 7.25 ms for the lateral wall, and 4.17 ± 5.76 ms for the septal segments; strain-gauge VDTC values were 3.09 ± 0.40 ms for the lateral wall and 4.57 ± 1.86 ms for the septal segments. No significant differences existed either between MRI and strain gauge group or between different segments within the same group (Table 2). Since in diastole the time scale of mechanical events is on the order of 50 ms, these low levels of VDTC indicate that muscle viscosity has minimal influence on diastolic function in the normal heart.
In dog hearts, Usyk et al. estimated myofiber elastic constants based on a best fit to the intraventricular pressure and wall strain data, the latter incorporating a finite-element model of ventricular geometry (32). They used an exponential relationship between tension and fiber strain which led to EM values that are dependent on strain. Figure 8 shows the comparison between their myofiber EM in the physiologic range of fiber strain (0–20%) and our MRI value from the papillary column. The agreement between the two mean values suggests that MR elastic modulus is consistent with global chamber compliance measured with intraventricular pressure transducers.
Stress and strain gradient data determined in the septum of one normal volunteer are graphed in Fig. 9a. The EM values of all six volunteers, which are graphed in Fig. 9b, indicated good compliance when using the range established in the normal dogs as a reference (Fig. 7). The highest EM value occurred in the lateral wall segment of a 58-year-old male with a resting heart rate of 88–90 bpm; however, this limited group of data is not sufficient to determine the correlation between elasticity values and such factors as age and resting heart rate.
Ventricular diastolic wall motion in early diastole results from a complex and active process: metabolically it is active as ATP consumption is required to detach actin–myosin crossbridges, thereby permitting sarcomeric re-lengthening. Mechanically it is active since the elastic energy stored in the structural constituents of the muscle during systolic contraction is released during diastole to drive muscle relaxation, much like a compressed spring that actively relengthens. Nevertheless, a spring in the process of relengthening still responds to external, hemodynamic forces according to its elastic properties which, in turn, are affected by a host of factors such as neurohormonal effects on Ca2+ transport and weakly binding cross-bridges present in diastole. Our results suggest that MRI-determined EM and VDTC of the ventricular wall, through incorporation of DENSE and phase velocity encoding techniques, can accurately assess myocardial viscoelastic constants in early diastole, which are the result of these physiologic events. While validated here in a canine model, our preliminary tests suggest that the method is feasible in humans. Although the current approach is simplified and the estimates are only for large segments of the wall without revealing transmural gradients, the results are consistent with more invasive measurements and demonstrate that it is possible to quantify myocardial material properties with noninvasive imaging.
In this study, the myocardium during the early filling period was surprisingly compliant by both MRI and postmortem strain gauge assessments, both in the normal dog group and in normal volunteers. The MRI values were in agreement both with direct strain gauge measurements immediately postmortem and with estimates from global chamber compliance measurements (32). The myocardium during the early filling period was surprisingly compliant according to both MRI and postmortem strain gauge assessment in the normal canine group as well as normal human volunteers. Of note was the particularly low level of myocardial viscosity assessed both by MRI and strain gauge: the VDTC values were less than 5 ms and, in fact, were below the temporal resolution of the scans (10 ms). This was the reason for the relatively large SD bars in the VDTC measurements (Fig. 7). We interpret this as a null result, meaning that in the normal heart muscle viscosity has little significance in the mechanics of diastolic relaxation and filling.
We should also note that in the normal volunteers, secondary oscillation of the pressure gradient was sometimes present in the early diastolic period, as shown in Fig. 9a. Similar oscillations were also seen in some dogs as illustrated in Fig. 10. These are probably due to mechanical resonant conditions of the entire atrial–ventricular flow system and are variable between individuals.
Last, while we found that the persistence of the DENSE-encoded signal was sufficient to cover the early phases of diastole (Fig. 1), it had deteriorated by the time of atrial transport. Evaluation of later phases of diastole may be important to isolate the effect of material ventricular properties (e.g., collagen remodeling) from earlier, energy-dependent processes. In order to assess the atrial emptying period it would be necessary to perform scans synchronized to the onset of atrial systole. Since current MRI scanners depend on the ECG R-wave as the timing reference, prospective identification of atrial systolic onset is not possible due to the sensistivity of diastasis (the period between the end of early filling and atrial transport) to heart rate shifts. Thus, the time interval between the R-wave and atrial contraction may be quite variable and, for this reason, we only obtained atrial transport elasticity measurements in two unpaced dogs during periods of very consistent heart rates, which was insufficient for statistical analysis. In the future, technical improvements in scan timing that allow assessment of later phases of diastole, following termination of active relaxation, and a piecewise analysis that separates the diastolic phases (30), may better reflect ventricular material properties.
It is important to note that the results reported here were validated in a canine model and, at present, form only a proof of concept. However, these preliminary results suggest that the method may be feasible in humans although these initial data reflect implementation in a limited number of volunteers without known cardiovascular disease. Future clinical validation of this technique will require: (a) assessment of a group of patients with carefully characterized HFNEF and (b) comparison of elasticity parameters in this group to disease-free individuals matched for non-pathologic covariates such as age that could effect ventricular stiffness. If successful, the practical applicability of this technique may be substantial. The ability to routinely measure regional and/or global myocardial viscoelasticity may permit the clinician to better assess the underlying mechanisms of congestive heart failure. For example, the extent and severity of stiff wall segments may indicate progression of disease and risk of elevated filling pressures.
In this report we employed a single slice imaging modality to demonstrate the techniques validity on a regional basis. However, a multislice implementation of the same technique could be employed to create a global map of regional compliance. Integration of regional stress/strain data into a global index could provide a noninvasive measure of ventricular compliance analogous to generation of a catheterization-based pressure/volume relationship. This would permit high-throughput, low-risk assessment of patients with HFNEF so that underlying mechanisms of this syndrome could be better understood and treatment effects better monitored. In humans, under the present protocol, data acquisition required a total scan time of 20 min or 30–40% of a usual MRI exam. This, however, can be shortened with parallel imaging techniques, and a compliance examination covering the LV in six radial segments (three radial slices) could reasonably be undertaken in half an hour.
We should stress that in order to estimate viscoelastic properties without the knowledge of myofiber layout of individual hearts and actual sarcomere length information, we treated the wall segment as a Hookian material of an average set of elastic parameters. It is, therefore, unrevealing of transmural variations. The LV myofibrils are known to be organized into laminar sheets that are oblique to the global ventricular axes (33–37), thereby facilitating shear and strain within the wall. This results in transmural EM gradients. Our technique, which may be clinically sufficient, may be insensitive to some conditions affecting such gradients. Further refinement will be possible if an independent measure of fiber orientation becomes available, for example, a reliable MRI diffusion measurement in vivo. Additionally this technique provides values for relatively large and user-defined segments of the left ventricular wall, and the definition of the region of interest could be a source of uncertainty in the results.
As described in the Appendix, the biaxial treatment of the myocardial wall assumes that shear stress in the ZY plane does not contribute significantly to the elastic energy. To further assess the role of ZY shear stress future work will need to measure three-dimensional strain. One such technique that acquires displacement maps in two adjacent slices simultaneously has been demonstrated in short-axis slices by Reese et al. (38). Adaptation of this or similar methods for multiphase long-axis scans may be possible. Future advances in this area may permit the inclusion of ZY shear stress contributions.
In this study the MRI results were compared with estimates from catheter-based global chamber compliance and with strain gauge measurements in intact postmortem hearts examined within 30 min of KCl arrest. While there is probably some effect of the immediate postmortem state on ventricular viscoelastic properties there were no significant differences in either EM or VDTC in this group of dogs. Further validation studies assessing in vivo, near-simultaneous comparisons of MRI-determined myocardial and global chamber compliance possibly involving high-fidelity micromanometry are being developed.
These data provide evidence that regional myocardial viscoelastic parameters can be accurately and noninvasively acquired. Future development of methods to extend the sample period into late diastole and to incorporate a multislice technique for the creation of global ventricular assessment could result in a clinical tool for enhanced appreciation of compliance abnormalities.
We thank Dr. Frank Rademakers of University Hospital Gasthuisberg, Leuven, for insightful discussions, and Drs. Hiroshi Ashikaga and Elliot McVeigh of NHLBI/NIH for allowing us to scan their ischemic dog model and providing the Gd-DTPA contrast images.
The basis of the MR elasticity measurement is the relation between stress and strain gradients in the myocardial wall. In diastole nonuniform pressure decline leads to transient intraventricular pressure gradients. This phenomenon can be regarded as pressure impulses bouncing within the elastic boundaries of the myocardial wall, causing the wall to conform to the pressure waves, i.e., a pulse-wave phenomenon. A simple mathematical description makes the following assumption: (a) a segment of the myocardial wall at the equatorial level can be regarded as having uniform thickness; (b) the wall segment is Hookian and the material property can be represented as the mean over its thickness, i.e., without resolving transmural differences.
Figure 11 illustrates such a wall segment. Its equation of motion in the longitudinal (z) direction is
Boundary conditions at the inner and outer surfaces:
where Dz is the z displacement of the wall segment; Pm is the intramyocardial pressure; σzz, σrr, σzy, σzr are, respectively, the normal longitudinal stress, the normal radial stress, shear stress in the zy plane, and shear stress in the zr plane; P1 and P2 are the pressures on the surfaces of the myocardial wall; fb is the shear force exerted by flowing blood on the wall surface.
The shear force of flowing blood on the ventricular surface is negligible in this context: given a relative blood viscosity of 4.0 (39), peak flow velocity of 500 cm/s, and ventricle radius of 1.5 cm, the shear force amounts to 0.54 Pa. For comparison, a typical diastolic pressure gradient in the LV is 4 kPa/m (0.3 mm Hg/cm) (21,40), which corresponds to 20 Pa force on a 5-mm-thick wall. Therefore, the shear force of blood is less than 3% of the net force. The corresponding shear force in the zr plane, σzr, is therefore negligible.
The shear stress in the zy plane σzy comes from shear strain in the zy plane. Geometrically this shear strain appears as warping of a transverse slice during diastolic filling. The canine data showed a typical shear strain of 3–7%, lower than the normal strain of approximately 10% in all anatomic directions. There is also evidence that the shear moduli are significantly lower than normal elastic moduli (41). For these reasons, the shear strain in the zy plane is also omitted, although it is an interesting topic for future more detailed studies.
When averaged over the thickness of the wall, taking into account the boundary conditions and the above approximations, the equation of motion Eq.  is now
where <……> denotes the average over the wall thickness.
From here on two separate assumptions were considered.
It is further assumed that the elasticity of the myofibrils along their length is the primary component of the elasticity of the myocardium as a whole. Although in a thin sheet of myocardium there is one dominant fiber orientation, across the thickness of the myocardial wall the fiber orientation rotates approximately 180° and evenly covers all directions in the plane. Therefore, when averaged over its thickness, the myocardial wall appears isotropic in the plane normal to its thickness, or the zy plane. Under this assumption, the elastic energy function is the biaxial symmetric function (42)
The stress–strain relationship is then
where E1 is the elastic’s modulus in the zy plane, E2 is the elastic’s modulus in the radial direction, ν1 is the Poisson ratio in the zy plane, ν2 is the Poisson ratio in the zr and yr plane. For an incompressible material,
The equation of motion is then
This equation describes the relationship between stress gradient and strain gradient.
An LV papillary column contains myofibrils that are uniformly aligned along its axis. It is assumed that the elasticity of the papillary column comes from the elasticity of myofibrils along their length, which act like springs aligned along the axial direction. The elastic energy function is a one-dimensional function,
where εzz is the strain along its axis. The stress–strain relationship in a papillary column is
The equation of motion of a papillary column is therefore
where P(z) is the ventricular pressure at the level z along the papillary axis. Note that the chordae attached to the end of the papillary column also exert forces onto the column, but these forces act as a terminal boundary condition and do not enter the gradient relationships expressed in Eq. .