Trabeculae carneae are irregular structures that cover the endocardial surfaces of both ventricles and account for a significant portion of human ventricular mass. The role of trabeculae carneae in diastolic and systolic functions of the left ventricle (LV) is not well understood. Thus, the objective of this study was to investigate the functional role of trabeculae carneae in the LV. Finite element (FE) analyses of ventricular functions were conducted for three different models of human LV derived from high-resolution magnetic resonance imaging (MRI). The first model comprised trabeculae carneae and papillary muscles, while the second model had papillary muscles and partial trabeculae carneae, and the third model had a smooth endocardial surface. We customized these patient-specific models with myofiber architecture generated with a rule-based algorithm, diastolic material parameters of Fung strain energy function derived from biaxial tests and adjusted with the empirical Klotz relationship, and myocardial contractility constants optimized for average normal ejection fraction (EF) of the human LV. Results showed that the partial trabeculae cutting model had enlarged end-diastolic volume (EDV), reduced wall stiffness, and even increased end-systolic function, indicating that the absence of trabeculae carneae increased the compliance of the LV during diastole, while maintaining systolic function.
Trabeculae carneae are irregular muscular bundles that cover both ventricles and, along with papillary muscles, account for a significant proportion of ventricular mass (12–17%) [1,2]. They are arranged in a complex network structure, mostly arising from the ventricular free wall . Trabeculae carneae are among the first features to arise during the embryonic development of the heart, presumably to increase the surface area to help the myocardial mass grow in the absence of a developed coronary circulation [4–6]. During the compaction process, trabeculae carneae condense to form the myocardial wall, papillary muscles, chordae tendineae, and septum [5–7]. However, many free-running trabeculae carneae remain in the ventricles of mammalian hearts. Our morphological study showed that human heart has the highest number of trabeculae among all common species .
The role of trabeculae carneae in the diastolic and systolic functions of the left ventricle (LV) is not well understood. Some studies have focused on quantifying the size and mass of trabeculae as an indicator of cardiac disorder. Lin et al. showed that trabeculae mass index was higher in heart failure patients , and Jacquier et al. suggested that trabeculae mass index can be used in the diagnosis of left ventricular noncompaction cardiomyopathy . Other studies have proposed that the irregular surfaces of the trabeculae carneae correlate with ventricular end-diastolic volume (EDV), reduce the kinetic energy of the blood flowing into the ventricles during diastole, and reduce the turbulence of blood during systole [11,12]. Also, trabeculae help squeeze blood from the apical region during systole  and are important features in LV electrical stimulations .
Our recent study suggested that trabeculae carneae hypertrophy and fibrosis contribute to the increased LV stiffness in patients with diastolic heart failure, and severing free-running trabeculae carneae may improve diastolic compliance of the LV . However, the role of trabeculae carneae in the diastolic and systolic functions of the LV is not well understood. Therefore, the objective of this study was to determine the roles that trabeculae carneae play in the left ventricular diastolic and systolic functions using anatomically detailed patient-specific models of the human LV.
Materials and Methods
Anatomical Geometry of Human Left Ventricle.
An explanted human heart was collected from a 63-year-old female donor with a history of stroke and congestive heart failure within 24 h postmortem from South Texas Blood and Tissue Center (San Antonio, TX). The heart was de-identified in accordance with Institutional Review Board (IRB) requirements and informed consent for research was obtained from the donor's family. Magnetic resonance imaging (MRI) scanning was conducted on a 3 T (128 MHz) MRI system (TIM Trio, Siemens Medical Solutions, Malvern, PA) comprised of a horizontal, superconducting magnet with a 60 cm diameter accessible bore . The heart was submerged in a saline filled plastic container, placed in a quadrature knee coil, and 3D MRI was performed to obtain slices oriented in the LV transverse plane (parallel to the LV short axis), the frontal plane, and the sagittal plane (slice thickness 4 mm, gap 0 mm) as shown in Fig. 1.
Three distinct LV models were derived from the MR images. The first model was the intact trabeculated model (TM), which contained all trabeculae carneae and papillary muscles. This high-resolution anatomically detailed 3D model of the LV was segmented from 2D MR images in DICOM format in Mimics (Materialise NV, Leuven, Belgium). The second model is a papillary model (PM), in which the papillary muscles remain intact but most of the trabeculae carneae were excluded in the smoothing process. The third model was the smooth model (SM) in which the trabeculae carneae and papillary muscles were excluded during image segmentation.
Passive and Active Material Behavior.
where , , , and are material constants and is the Green strain tensor with corresponding to fiber, radial, and cross-fiber directions, respectively. The material constants , and were obtained from our previous biaxial tests of human LV myocardium [18,19]. The value of was chosen to be equal to half of the since we expect the material to be softer in the fiber-radial and fiber-cross-fiber coordinate planes. The material constant, , was adjusted by an optimization process so that the FE-predicted EDV of TM matched the estimated EDV of the empirical Klotz relation at the specified end-diastolic pressure (EDP) [20,21]. Klotz et al. showed that all volume-normalized end-diastolic pressure–volume relations (EDPVRs) have a common shape, despite of etiology and species.
where = 4.35 μM is the maximum peak intracellular calcium concentration, = 4.75 μm−1 governs the shape of the peak isometric tension–sarcomere length relation, = 1.58 μm is the sarcomere length at which no active tension develops, and is the sarcomere length, which is the product of the fiber stretch and the sarcomere unloaded length = 2.04 μm .
The contractile parameter was determined and calibrated so that the FE predicted ejection fraction (EF) of TM matched the EF of a normal human heart at the specified end-systolic pressure (ESP) [25,26].
Three sets of prescribed values of EDP and ESP were used for calibration of passive () and active () material parameters of TM to assess the sensitivity of pressure–volume relationships to the choice of EDP and ESP: MAT1: EDP = 4 mmHg, ESP =108 mmHg; MAT2: EDP = 12 mmHg, ESP = 120 mmHg; and MAT3: EDP = 20 mmHg, ESP = 132 mmHg . The same passive and active material parameters were then used in the PM and SM FE models.
Ventricular Fiber Architecture.
A rule-based myocardial fiber algorithm was adopted in this study to generate the myofiber directions [14,28]. Rule-based models have been shown to agree well with diffusion tensor MRI measurements [14,29]. Fiber orientation was included in the model by setting the direction of the fiber as a property of each tetrahedral mesh element using the local coordinate system () as shown in Fig. 3(a). The -axis is parallel to the global apical-basal direction, -axis points transmurally from the endocardium toward the epicardium, and -axis is parallel to the equatorial plane and perpendicular to both and .
where is a sign function, is the maximum absolute value of the helix angle, and is an empirically determined constant that regulates the transmural variation of [29,30]. Thus, the fiber direction changed from at the epicardium to at the endocardium (Fig. 4). In papillary muscles and trabeculae carneae, the fiber direction was set parallel to the muscle axis [14,31]. The rule-based fiber algorithm was developed in the R programing environment.
Effect of Fiber Architecture on Left Ventricle Systole.
Apical and equatorial twists around the long axis of the LV during systole were defined as the maximum difference between basal rotation and apical and equatorial rotations, respectively. Counterclockwise rotation was defined as positive, as seen from the apex .
Calculation of Pressure–Volume Relationship.
To characterize the role of trabeculae carneae in diastolic and systolic functions of the LV, pressure–volume relationships, which are often used in clinical description of LV function, were constructed from FE analysis of TM, PM, and SM models in the open-source finite element package FEBio in the website link2 . Indices derived from pressure–volume relationships were used to assess the mechanical function of trabeculae carneae in the LV. In all models, the rigid body motion was suppressed by constraining the base from moving in all directions .
where and are constants. Diastolic functions of TM, PM, and SM models were quantified by comparing the values of and . An improvement in diastolic function was characterized by a decrease in the slope of the EDPVR at EDV .
Global Left Ventricle Performance.
Results from FE analysis using three different tissue properties (MAT1, MAT2, and MAT3) demonstrated that the EDV was 114.8±9.37 ml in the TM. The EDV was higher in the PM (134.92±9.57 ml) and was the highest (150.80±10.67 ml) in the SM. The estimated ESV was 61.04±7.89 ml in TM, 58.25±3.47 ml in PM, and 74.79±3.95 ml in SM.
We found that independent of the material model, the EDPVR curve shifts to the right in PM and SM compared to TM (see Fig. 5). However, the ESPVR curve may shift to right (MAT1) or left (MAT2 and MAT3) in PM compared to TM, while shifting to the right in SM for all material models. EDPVR was steeper in TM compared to PM and SM; however, ESPVR was found to be steeper in PM than TM and SM in MAT2 and MAT3 (Fig. 5). The predicted parameters of EDPVR and ESPVR are presented in Table 1. The lower average values of and in PM and SM compared to TM indicate a significant improvement in the compliance and global diastolic function of less trabeculated LV models (p < 0.01). Similarly, the higher average values of elastance and lower volume intersect in PM compared to TM, suggesting that mild cutting of trabeculae carneae slightly improves the global systolic function of the LV (p = 0.89). However, cutting all trabeculae carneae and papillary muscles in SM had a significant adverse effect on the global systolic function (p < 0.01).
Myofiber Stress Distribution.
Our computational simulations demonstrated that the presence of trabeculae carneae decreased apical, equatorial, and basal peak end-diastolic myofiber stress as the TM showed the lowest end-diastolic myofiber stress among all three models. However, trabeculae carneae increased peak end-systolic myofiber stress in the LV (Fig. 6). The peak end-diastolic myofiber stress was near the endocardial apex in all models. In the TM and PM, high stress concentration (both ED and ES) was also found in free running trabeculae carneae and papillary muscles. The maximum end-systolic myofiber stress was located near the apical midwall in all models. The color-coded maps of myofiber stress distribution in a long-axis section at end-diastole and end-systole are compared among TM, PM, and SM in Fig. 7.
Effect of Myofiber Architecture.
Fiber architecture was generated by the rule-based algorithm and implemented in the finite element code. Figure 8 shows the orientation of fibers in the LV midwall where the helix angle is nearly zero (α ∼ 0 deg).
The variation in end-systolic wall thickening, longitudinal shortening, and twist in TM, PM, and SM are important indicators of LV systolic function. Our results showed that SM had the highest apical, equatorial, and basal wall thickening (Figs. 9(a)–9(c)) and twist (Figs. 9(d) and 9(e)) among all models. The longitudinal shortening was found to be higher in TM than in PM and SM (Fig. 9(f)).
Wall thickening, longitudinal shortening, and twist were affected by different values of the fiber helix angle. Increasing the value of helix angle () from 60 deg to 70 deg, led to 24.73 ± 0.51% (TM), 16.23±1.36% (PM), and 6.72±1.25% (SM) decreases in apical wall thickening, 16.83±1.06% (TM), 6.60±2.31% (PM), and 10.41±0.37% (SM) increases in equatorial wall thickening, and 14.23±3.52% (TM), 13.36±2.12% (PM), and16.62±1.77% (SM) increases in basal wall thickening. Accordingly, the apical twists decreased 41.73±3.06% (TM), 42.52±2.23% (PM), and 38.29±3.11% (SM), and equatorial twists decreased 46.03±1.86% (TM), 42.84±2.11% (PM), and 40.39±1.97% (SM). Meanwhile, there were also considerable increases of 106.71±4.10% (TM), 107.10±3.26% (PM), and 99.83±4.21% (SM) in the longitudinal shortening. The longitudinal shortening from the apical endo-cardial surfaces to the base in the 70 deg fiber model were 13.23±1.19%,(TM) 12.82±1.16% (PM), and 11.71±1.85% (SM). These values are comparable with experimental measurements of longitudinal shortening in healthy adult hearts [35,36], suggesting that the 70 deg helix angle is more realistic for our models (TM, PM, and SM).
The resultant normalized EDPVR curves of TM, PM, and SM after optimization of the passive material parameters were in good agreement with the Klotz curve (Fig. 10). It can be seen that the anisotropic characteristics of the models and adjusted material parameters were able to reproduce the Klotz empirical EDPVR curve .
To further verify the accuracy of the FE models, the average end-systolic peak principal strains measured from TM were compared with reported experimental strain measurements of 31 healthy volunteers using tagged MRI . The FEA estimated and experimentally measured average end-systolic peak principal strains in basal, equatorial, and apical regions of the LV are summarized in Table 2. No significant differences between estimated and measured first (E1), second (E2), and third (E3) principal strains were found (p = 0.90).
In this study, we presented an anatomically detailed model of the human LV, derived from high-resolution MR scans, and investigated the role of endocardial trabeculation in the diastolic and systolic functions of human LV. By comparing the anatomically detailed model with models either removed most of the trabecular alone or removed both trabecular and papillary muscle of the same LV, it is shown that removing trabeculae improves the diastolic compliance without adverse effect on the systolic function.
Most reported studies in the literature have used the smoothed ventricular geometries due to limitations in image acquisition, image segmentation, fiber map registration, and meshing complexity [27,38–40]. Our results demonstrated the importance of considering endocardial structures, i.e., papillary muscles and trabeculae carneae, in the assessment of LV global function in patient-specific computational LV models.
Global Diastolic and Systolic Functions.
Our FE simulations demonstrated that trabeculae carneae serve a functional role of providing mechanical support to the LV during diastolic filling and systolic ejection. Our results demonstrated that removing trabeculae carneae decreased left ventricular diastolic stiffness. The EDPVR was found to be less steep in PM and SM than in TM, which indicated that severing trabeculae carneae could improve compliance of the LV. This improvement depends on the material property of trabeculae carneae and would be greater in hypertrophic hearts because trabeculae carneae would be more hypertrophic and fibrotic . This conclusion is consistent with the findings from our previous ex vivo experiments of human hearts , which demonstrated an increase in LV compliance after trabecular cutting. Although FE prediction of is larger than the measurement results by Halaney et al. , the values and relative decrease are comparable.
In terms of ESPVR, aggressive cutting of trabeculae carneae and papillary muscles in SM adversely affected the systolic phase by reducing the average value of and elevating the average value of . Evidently, cutting papillary muscles is prohibited in order to maintain normal mitral valve function. However, it is notable that mild cutting of trabeculae carneae (PM versus TM) could be sufficient to increase the average value of and decreased the average value of , indicating an improvement in systolic function. These simulation results are in good agreement with our recent experimental observation of the systolic function changes due to trabecular cutting in the rabbit heart .
Myofiber Stress Distribution.
We compared the diastolic and systolic myofiber stress distribution in the presence and absence of trabeculae carneae. Our computational simulations demonstrated that the increase in EDV in the absence of trabeculae carneae was accompanied by a decrease in LV wall thickness, which led to elevated end-diastolic myofiber stress in SM and PM compared to TM. The results also showed that during systole, the LV wall stress decreased in the absence of trabeculae carneae due to the larger end-systolic wall thickening in SM and PM compared to TM. Systolic wall stress is the primary determinant of myocardium oxygen consumption . Thus, its reduction after severing trabeculae carneae could reduce myocardium oxygen consumption, having a beneficial effect in hearts compromised by LV hypertrophy .
Systolic Parameters and Effects of Fiber Architecture.
Our computational model was able to reproduce realistic values of systolic wall thickening, longitudinal shortening, and twist angle in human LV. Our model's predicted values of equatorial and basal systolic wall thickening were in good agreement with the in vivo measurements reported by Shen et al.  in 40 patients. The reported average wall thickening was 25–30% at the apex, 19–24% in the midventricle, and 13–20% at the base of the LV . Our computational model underestimated the apical wall thickening (9–13%), but predicted the equatorial and basal wall thickening well.
During systole, the percentage of apical wall thickening was found to be higher in SM and PM compared to TM. This increase in wall thickness led to larger radial differences between endocardium and epicardium, resulting in increased twist in SM and PM [44,45]. The reason is that the epicardial myofibers (compared with the endocardial myofibers) produce larger torque due to their larger radius, and determine the overall direction of the LV rotation . Accordingly, an increase in relative wall thickness produces larger radius differences between endocardium and epicardium and results in an augmentation of twist . Similarly, our simulation results showed that in the fiber model with 60 deg helix angle, the apical wall thickening and twist angle are larger than in the model with 70 deg helix angle. Nakatani also reported that increasing the myofiber rotation angle leads to smaller twist . Our FE predicted peak apical systolic twist angle matched with the reported in vivo MRI measurements (18 deg) and speckle tracking echocardiographic measurements (12 deg) [44,49]. In addition, our results showed that models with the 70 deg helix angle reproduced more realistic values of longitudinal shortening, compared to the 60 deg helix angle, and matched well with experimental measurements (18%) using speckle tracking echocardiography .
Implication for Treatment of Heart Failure With Preserved Ejection Fraction.
Left ventricular hypertrophy is often associated with heart failure with preserved ejection fraction (HFpEF) . There is no effective treatment for HFpEF, and HFpEF is characterized by impaired diastolic relaxation due to increased LV stiffness . Medications that reduce the myocyte intracellular calcium level or those that slow heart rate to prolong diastole have not been able to improve diastolic compliance [52,53]. Normalization of blood pressure also was not an effective treatment for normalizing the compliance of LV in patients with hypertension . Therefore, there is a pressing clinical need for a new effective treatment for HFpEF.
In hypertrophic hearts, the hypertrophy of the myocardium is associated with changes in trabeculae carneae tissue architecture and fibrosis, which affect compliance of the LV [2,8,55]. Thus, severing trabeculae carneae could improve the LV compliance in HFpEF patients . Our current results are consistent with this observation that the diastolic performance improved after severing trabeculae carneae due to the reduction in LV stiffness. Furthermore, our current results also suggested that severing of trabeculae carneae (without affecting papillary muscle) might improve the LV systolic function. This improvement would be greater in hypertrophic hearts because trabeculae carneae are also hypertrophic and more fibrotic . In addition, the systolic myofiber wall stress decreased after severing of trabeculae carneae, which could attenuate adverse effects of hypertrophy in the LV .
There are a few limitations in this work. Though we used high-quality MRI to capture the endocardial details of the LV, reproducing very fine trabeculae carneae was restricted by the MRI resolution [15,56]. We also did not have access to the pressure–volume data of the donated heart; thus, we adjusted and validated the patient-specific model with the average measurements found in the literature [21,37]. To overcome this limitation, we used the Klotz relationship for diastolic material scaling and validation. For the systolic phase, we compared the FEA strain prediction with reported tagged MRI strain measurements in 31 healthy volunteers . In addition, we did not include the right ventricle in our computational model, which could alter loading at the septum. However, the lack of right ventricle would not affect the assessment of the role trabeculae carneae play in the LV. Finally, we did not explore the long-term effects of trabecular cutting on the LV function, a topic which needs further study in the future.
This work was supported by a National Innovation Award (15IRG23320009) from the American Heart Association and by a seed grant from the Office of the Vice President for Research of the University of Texas at San Antonio. All the authors would like to wish Dr. Fung a very happy 100th Birthday.
HCH would like to thank Dr. Fung's mentoring and support throughout the years and would like to add the following passage tribute to Dr. Fung:
When I was a Ph.D. student at Xi'an Jiaotong University, through the recommendation of my advisor, Professor Zhen-bang Kuang, I was fortunate to join Dr. Fung's lab to work on my thesis. Under the tutelage of Dr. Fung, I worked on a project on residual stress in arteries along with Shu Qian Liu, Ghassan Kassab, and other lab members. I have many great memories of those days and would like to share a few of them.
Even though Dr. Fung is well known and highly accomplished, he is also very caring and approachable to everyone. His hearty laughter are always contagious and brightens the mood of those around him. As a new arrival to America, I felt welcomed by Dr. Fung and appreciated how Dr. Fung would host us at his home to celebrate the holidays and show us the beautiful loquat trees in his backyard.
Dr. Fung has also been a great mentor and role model for many of us. Dr. Fung has always had the gift to concisely and clearly explain complex theory and equations - something well demonstrated in his books and papers. A training that helped my career was writing papers with Dr. Fung. He reviews every manuscript with attention to detail and thoroughness. I would often get many changes, insertions, and sticky notes in the manuscript drafts I submitted to him and, although I may not have realized at the time, each revision was a learning experience for me and something I appreciate greatly today as a professor myself.
Nearly 30 years later and with a few more grey hairs, Dr. Fung's signature laughter still brightens the mood of all those around him. Thank you Dr. Fung and Happy Birthday!
American Heart Association (Funder ID: 10.13039/100000968).
University of California at San Diego (Funder ID: 10.13039/100005522).
University of Texas at San Antonio (Funder ID: 10.13039/100008634).
Xi'an Jiaotong University (Funder ID: 10.13039/501100002412).