The impact of wall thickness and curvature on wall stress in patient-specific electromechanical models of the left atrium
Abstract
The left atrium (LA) has a complex anatomy with heterogeneous wall thickness and curvature. The anatomy plays an important role in determining local wall stress; however, the relative contribution of wall thickness and curvature in determining wall stress in the LA is unknown. We have developed electromechanical finite element (FE) models of the LA using patient-specific anatomical FE meshes with rule-based myofiber directions. The models of the LA were passively inflated to 10mmHg followed by simulation of the contraction phase of the atrial cardiac cycle. The FE models predicted maximum LA volumes of 156.5 mL, 99.3 mL and 83.4 mL and ejection fractions of 36.9%, 32.0% and 25.2%. The median wall thickness in the 3 cases was calculated as \(1.32\, \pm \,0.78\) mm, \(1.21\, \pm \,0.85\) mm, and \(0.74\,\pm \,0.34\) mm. The median curvature was determined as \(0.159\,\pm \,0.080\) \(\hbox {mm}^{-1}\), \(0.165\,\pm \,0.079\,\hbox {mm}^{-1}\), and \(0.166\,\pm \,0.077\,\hbox {mm}^{-1}\). Following passive inflation, the correlation of wall stress with the inverse of wall thickness and curvature was 0.55–0.62 and 0.20–0.25, respectively. At peak contraction, the correlation of wall stress with the inverse of wall thickness and curvature was 0.38–0.44 and 0.16–0.34, respectively. In the LA, the 1st principal Cauchy stress is more dependent on wall thickness than curvature during passive inflation and both correlations decrease during active contraction. This emphasizes the importance of including the heterogeneous wall thickness in electromechanical FE simulations of the LA. Overall, simulation results and sensitivity analyses show that in complex atrial anatomy it is unlikely that a simple anatomical-based law can be used to estimate local wall stress, demonstrating the importance of FE analyses.
Keywords
Left atrium Cardiac mechanics Finite element simulation Patient-specific modeling Wall stress1 Introduction
Atrial fibrillation (AF) is a prevalent and progressive disease, characterized by chaotic electrical activation of the atria (Kirchhof et al. 2016). Early detection and treatment of AF are associated with improved patient outcome and reduced stroke risk (Keach et al. 2015). While AF is an electrophysiological pathology, the risk of developing AF is markedly increased with hypertension, mitral regurgitation, mitral stenosis and aortic stenosis, which increase the mechanical loading on the atria (Benjamin et al. 1994; Iung et al. 2018; Widgren et al. 2012). The changes in mechanical loading of cardiac tissue can activate fibroblasts, leading to an increased fibrotic burden, which might contribute to the initiation and sustenance of AF (Marrouche et al. 2014; Dzeshka et al. 2015).
In the heart, mechanical quantities, such as stress and strain, have previously been used to drive models of growth and remodeling (Kerckhoffs et al. 2012; Rodriguez et al. 1994). While strain can be measured directly from clinical images (Blume et al. 2011), stress must be calculated using a mathematical model (Yin 1981), accounting for the anatomy, micro-structure and material properties of the atria. In the left ventricle, the wall stress can be approximated using the Law of Laplace (Valentinuzzi and Kohen 2011), in which wall stress is proportional to the radius of curvature (inverse of curvature) and inversely proportional to the wall thickness. The Law of Laplace assumes that the wall of the heart is thin relative to the radius of curvature. Due to the thin wall of the atria, the Law of Laplace may provide a reasonable approximation of atrial stress. In the atria, both the wall thickness (Bishop et al. 2016) and curvature (Ahmed et al. 2006) vary across the surface. However, their relative influence on wall stress remains unknown. In addition, other attributes of the atria including the complex anatomy, fiber structures, boundary conditions and active contraction play a role in determining the wall stress and are not accounted for the in the Law of Laplace, potentially limiting its applicability in the atria.
Previous models of human atrial mechanics have assumed a homogeneous wall thickness (Moyer et al. 2015; Hunter et al. 2012). Models of atrial mechanics that accounted for regional variations in thickness were derived from cadaveric data sets and might not reflect the in-vivo anatomy of the atria (Adeniran et al. 2015). Recent developments in computed tomography (CT) image analysis now allow the generation of anatomically detailed geometric models of the LA that account for varying wall thickness derived from clinical scans (Bishop et al. 2016). To determine if local wall stress analysis in the atria requires patient-specific wall thickness, we investigated if wall thickness is an important factor in determining local atrial stress or if the curvature, i.e., the endocardial surface shape, was the dominant factor.
In this study, we first describe an electromechanical modeling framework for simulating active contraction in the LA. Secondly, we perform representative finite element (FE) simulations of the passive inflation and active contraction in the LA. Thirdly, we calculate the wall thickness and curvature across the endocardial LA surface. Finally, we compare the correlation between the 1st principal stress with wall thickness and curvature to identify the more prominent metric.
2 Methods
2.1 Personalized model generation
We focused our modeling efforts on the LA, which plays a more dominant role in AF compared to the right atrium (Kirchhof et al. 2016). The FE simulations were performed on 3 publicly available LA anatomical models developed from CT angiography images, that include a description of the endocardial and epicardial myofiber distributions (Fastl et al. 2018). Atrial fibers were represented by two distinct layers, consistent with previous DTMRI data which showed a sharp transition in fiber direction between the endocardial and epicardial fiber layers (Pashakhanloo et al. 2016). The atrial anatomies were discretized using tetrahedral elements with a mean edge length of \(\approx 238\,\upmu \hbox {m}\). This ensures at least two FEs across the myocardium of the LA, that can be as thin as \(500\,\upmu \hbox {m}\) (Whitaker et al. 2016) and shows transmural variations in myofiber directions. The resulting FE meshes had 2.7, 1.8 and 1.1 million vertices and 14.8, 9.7 and 5.3 million elements, respectively.
2.2 Biomechanics model
2.3 Electrophysiology model
2.4 Electromechanics model
2.5 Computational model parameters
2.5.1 Passive tissue biomechanics
Bellini et al. (2013) measured passive stiffness in tissue samples taken from anterior and posterior regions of the human LA. For each region, the material samples were studied under biaxial loading. In thin square samples, orthogonal distributed tensions were applied along each edge, where the square lay in the plane defined by directions 1 and 2 and direction 3 was out of plane. The ratio between tensions in the 1 and 2 directions were set to 1:0.5, 0.5:1, 1:0.75 and 0.75:1, as well as equiaxial loading, 1:1. The reference tension was set to 30 \(\hbox {Nm}^{-1}\), such that a ratio of 1:0.5 corresponded to tensions of 30 \(\hbox {Nm}^{-1}\) and 15 \(\hbox {Nm}^{-1}\) in directions 1 and 2, respectively. We used this data to fit the parameters of our strain energy function given in Eq. 4.
2.5.2 Passive biomechanics
The LA anatomy is recorded during diastasis when the ventricle and atria are close to relaxed. However, there is still an atrial pressure that could be as high as 1 kPa (Stefanadis et al. 1998). As the atria are very thin, relative to the ventricles, they will be more compliant and even this low pressure will cause the atria to deform. The thin walls make estimating the reference configuration using unloading techniques applied in the ventricles challenging as the atria are prone to buckling. For this reason, we used the measured anatomy as the reference configuration. This approximation results in the model operating at lower fiber strains where the material properties are more compliant (Nikou et al. 2016). To account for this decreased stretch, we scaled the stiffness by a factor of 2. We have tested the dependence of the results to this parameter in the sensitivity analysis (see Sect. 3.3).
2.5.3 Active biomechanics
Since model parameters scale with organ phenotypes, the parameters for the active contraction model were manually scaled to achieve the desired ejection fraction, peak pressure and contraction duration. The parameters were initialized to ventricular parameters (Niederer et al. 2011b). The final parameter set was \(t_{\mathrm {d}}=10.0\,\text {ms}\), \(T_{\mathrm {peak}}=50.0\,\text {kPa}\), \(\tau _{c}=40.0\,\text {ms}\), \(\tau _{\mathrm {r}}=110.0\,\text {ms}\) and \(t_{\mathrm {t}}=300.0\,\text {ms}\). The membrane potential threshold for defining \(t_{\mathrm {a}}\) was set to \(V_{\mathrm {m,Thresh}}=-\,60.0\,\hbox {mV}\). More advanced techniques to personalize the parameterization, such as gradient, Latin hyper cube or grid search-based simulations, were computationally intractable due to the high computational cost of simulating atrial mechanics (approximately 1000 to 10,000 CPU core hours per simulation).
2.5.4 Electrophysiology
Single cell stimulation on the standard Courtemanche model (Courtemanche et al. 1998) using 1000 beats with a basic cycle length of 1000 ms was performed prior to EM simulations to obtain steady state conditions.
The conduction velocities in the LA were chosen as 1.20 and 0.40 m/s in longitudinal and transverse direction, respectively, leading to an anisotropy ratio of 3/1, well within the range of reported values for healthy patients (Dimitri et al. 2012; Kneller et al. 2002). To match the reported conduction velocities in the EP simulations, the monodomain conductivity tensor \({{\varvec{D}}}_{\mathrm{m}}\) was iteratively fitted using the method described in Mendonca Costa et al. (2013). The conductivity in the longitudinal (\(D_{{\ell }}\)) and transverse (\(D_{\mathrm {t}}\)) directions were set to 0.74 \(\hbox {Sm}^{-1}\) and 0.08 \(\hbox {Sm}^{-1}\), respectively. The membrane surface-to-volume ratio and the membrane capacity were set to standard values of \(\beta _{\mathrm {m}}=1400\,\hbox {cm}^{-1}\) and \(C_{\mathrm {m}}=1.0\,\upmu \hbox {Fcm}^{-2}\), respectively, for all simulations (Niederer et al. 2011a).
2.6 Boundary conditions
2.6.1 Electrophysiology
The atrial electrophysiology model was activated by a stimulation applied on the epicardium in the vicinity of Bachmann’s Bundle for 2 ms with an amplitude of 500 \(\upmu \hbox {Acm}^{-2}\) to approximate the physiological activation from the right atrium (Markides et al. 2003).
2.6.2 Biomechanics
The atrial cardiac cycle can be separated into three phases: first as a reservoir, where the atria stores blood during ventricular contraction. Second, as a conduit where the atria passively lets blood flow from the pulmonary veins to the ventricle. Thirdly as a pump, where the atria contracts to force blood into the ventricles.
Our simulations focus on the active contraction phase, when pressure and stress will be highest. The anatomy is derived from the cardiac CT images recorded during ventricular diastasis when the left atrium is in the conduit phase. We have taken this anatomy as an approximation of the reference anatomy.
The simulation was constrained by applying spring-like boundary conditions (Land and Niederer 2018) at the PVs and the MV, respectively. We initialize the model by inflating the atria from a zero pressure up to a pressure of 10 mmHg that reflects the mid pressure reached during atrial systole (Stefanadis et al. 1998; Ágoston et al. 2015). We have not implemented dynamic pressure volume boundary conditions for the pulmonary veins and the left ventricle.
The atrial mechanical boundary conditions are less sophisticated than the level expected in ventricular simulations and the current challenges and required developments are discussed below in the limitations section.
2.7 Numerical framework
Left atrial electromechanics simulations were performed in CARP (Vigmond et al. 2003, 2008). CARP mechanics were previously verified against the cardiac mechanics N-version benchmark Land et al. (2015). The nonlinear mechanical problem was solved using Newton’s method until the minimum of the relative and the absolute norm of the residual vector reduced to \(\varepsilon < 1\mathrm {e}-6\). For all linear subproblems, we used the generalized minimal residual (GMRES) method with algebraic multigrid (AMG) preconditioning and an error tolerance of \(\varepsilon < 1\mathrm {e}-8\).
For more details on the preconditioned Krylov subspace methods see comprehensive research articles by Neic et al. (2012) and Augustin et al. (2016).
2.8 Post-processing
2.8.1 Wall thickness calculation
Wall thickness in thin complex structures can be challenging to define Jones et al. (2000). For this reason, we previously applied the Laplace based wall thickness calculation method to the atria Bishop et al. (2016). This method was able to calculate wall thickness in the atria but is computationally expensive, especially when using very large meshes. Here, we use a faster method based in the eikonal equation. The eikonal equation was solved over the finite element meshes using a fast iterative method (Fu et al. 2013; Neic et al. 2017) where the wavefront was initiated on all nodes of the epicardium simultaneously given a constant isotropic conduction velocity. Local wall thickness was then computed from wavefront arrival times at the endocardium. This fast and robust approach was verified against results from Bishop et al. (2016).
2.8.2 Wall curvature calculation
To calculate wall curvature, we used a 3D sphere fitting approach similar to Thomas and Chan (1989). The algorithm is based on the minimization of the distance between the points of a endocardial surface patch and the radius of a fitted sphere in a least square sense. The local curvature is then defined as the inverse sphere radius. The endocardial surface patch was defined as a circular patch of elements of approximately 5 mm radius. Due to the fine, anatomically detailed FE meshes and the already very high complexity of the framework, curvature computations were based on the relatively simple but most widely applied Ji et al. (2015) approach using spheres. Note that more advanced geometric fittings using ellipses, hyperbolas, and parabolas, e.g., Ahn et al. (2001) and line integrals, e.g., Lin et al. (2010) are also proposed in the literature. However, these more sophisticated algorithms will also introduce additional complexity and parameters. We demonstrated in Sect. 2.9 that the correlations are robust to changes of the endocardial surface patch size. This suggests that the conclusions are not highly dependent on the curvature computation algorithm.
2.9 Sensitivity analysis
- 1.
Inflation pressure p\(\pm \,25\) %;
- 2.
Peak isometric tension \(T_{\mathrm {peak}}\) in Eq. (10) \(\pm \,25\) %;
- 3.
Isotropic material parameter a in Eq. (6) \(\pm \,25\) %;
- 4.
Anisotropic material parameter \(a_{\mathrm {f}}\) in Eq. (7) \(\pm \,25\) %;
- 5.
Stiffness Scale factor (stiff.). Scaling a in Eq. (6) and \(a_{\mathrm {f}}\) in Eq. (7) each by \(\pm \,25\) %;
- 6.
Dispersion material parameter \(\kappa\) in Eq. (7) \(\pm \,25\) %;
- 7.
- 8.
Fully incompressible model with \(1/\mu = 0\) in Eq. (5) and a block-system formulation;
- 9.
Reduced penalty parameter \(\mu =1000\,\text {kPa}\) and \(\mu ={500}\,\text {kPa}\).
- 1.
Curvature patch size (Sect. 2.8.2) \(\pm \,25\) %;
- 2.
Noised model: Gaussian noise (mean \(\mu =0\,{\upmu \hbox {m}}\), standard deviation \(\varsigma =100\,{\upmu \hbox {m}}\)) was added to the initial geometry of patient case 3 and subsequently smoothed using ParaView (Ayachit 2015);
- 3.
Constant thickness model: a mesh with constant thickness of 0.5 mm was generated based on the endocardial surface of patient case 3, using the software Gmsh (Geuzaine and Remacle 2009). Note, that the thickness related Spearman-\(\rho\) values for this model are deliberately omitted.
- 4.
0 % cutoff: the whole region including boundary domains is used for correlation computations
- 5.
50 % cutoff: only the inner 50 % of the domain, measured as distance from the pulmonary inlets and the mitral valve ring, is used for correlation computations
- 6.
5/25 % cutoff: 5 % of the domain close to the pulmonary outlets and 25 % of the domain close to the mitral valve is not considered for the computations
- 7.
25/5 % cutoff: 25 % of the domain close to the pulmonary outlets and 5 % of the domain close to the mitral valve is not considered for the computations
- 8.
\(\varOmega _t\): the deformed domain is used for curvature and wall thickness computations;
- 9.
stim. \(\Gamma _{\mathrm {endo}}\): the whole endocardial surface is used for stimulation. Since cells are contracting simultaneously peak isometric tension \(T_{\mathrm {peak}}\) in Eq. (10) had to be reduced by 25 %.
- 1.Principal stresses: let \(\lambda _1\), \(\lambda _2\) and \(\lambda _3\) be the eigenvalues of the Cauchy stress tensor \(\varvec{\sigma }\). Thenare the principal stresses.$$\begin{aligned} \sigma ^{\mathrm {1st}}&=\max \{\lambda _1,\lambda _2,\lambda _3\}, \\ \sigma ^{\mathrm {3rd}}&=\min \{\lambda _1,\lambda _2,\lambda _3\}, \text{ and }\\ \sigma ^{\mathrm {2nd}}&={\text {tr}}(\varvec{\sigma }) - \sigma ^{\mathrm {1st}} - \sigma ^{\mathrm {3rd}} \end{aligned}$$
- 2.
Fiber stress \(\sigma ^{\mathrm {f}}=\mathbf{f}_0 \cdot \varvec{\sigma }{} \mathbf{f}_0\)
- 3.
Stress magnitude \(\left| \varvec{\sigma }\right| = \left( \varvec{\sigma }:\varvec{\sigma }\right) ^{1/2}\)
- 4.
von Mises stress \(\sigma ^{\mathrm {M}} = \left( 3/2\;\sigma '_{ij}\sigma '_{ij}\right) ^{1/2}\) and \(\sigma '_{ij} = \sigma _{ij} - {1/3} \, \delta _{ij} \sigma _{kk}\) is the deviatoric stress.
2.10 Comparison to laplace estimates
3 Results
3.1 Reference anatomies and simulations
Summary attributes of patient attributes
Index | Sex | Age | Comorbidities |
---|---|---|---|
1 | M | 35 | HLD |
2 | F | 48 | |
3 | F | 54 | PAF, SSS |
Summary LA volume changes during simulated atrial contraction
i | \(V_{0}\) (ml) | \(V_{\mathrm {infl}}\) (ml) | \(V_{\mathrm {cont}}\) (ml) | IF (%) | EF (%) |
---|---|---|---|---|---|
1 | 101.03 | 156.47 | 98.81 | 154.9 | 36.9 |
2 | 61.04 | 99.31 | 67.57 | 162.2 | 32.0 |
3 | 51.11 | 83.36 | 62.32 | 163.1 | 25.2 |
3.2 Correlation of wall stress with local anatomy
Correlations are calculated for each plot in Fig. 6. When comparing curvature and wall stress we find a weak but consistent correlation during inflation (0.20–0.25) and maximal contraction (0.16–0.34). A stronger correlation is found between wall thickness and stress during inflation (0.54–0.62), however, this also decreases with maximal contraction (0.38–0.44). As wall stress is proportional to the ratio of the radius of curvature and wall thickness in the Law of Laplace it seemed possible that the wall stress may have a higher correlation with the inverse product of wall thickness and curvature. However, the correlation of the inverse of wall thickness and curvature was only slightly different from the correlation between wall stress and wall thickness during inflation (0.51–0.60) and contraction (0.40–0.42).
Summary, Spearman’s correlations between the principle wall Cauchy stress and the curvature and wall thickness on the reference grid
i | \(\rho ^{\textsf {Wt}}_{\mathrm {infl}}\) | \(\rho ^{\textsf {Wt}}_{\mathrm {cont}}\) | \(\rho ^{\textsf {C}}_{\mathrm {infl}}\) | \(\rho ^{\textsf {C}}_{\mathrm {cont}}\) | \(\rho ^{\textsf {WtC}}_{\mathrm {infl}}\) | \(\rho ^{\textsf {WtC}}_{\mathrm {cont}}\) | \(\rho ^{\textsf {Wt}}_{\mathrm {C}}\) |
---|---|---|---|---|---|---|---|
1 | 0.556 | 0.437 | 0.227 | 0.186 | 0.525 | 0.412 | 0.005 |
2 | 0.615 | 0.384 | 0.199 | 0.162 | 0.603 | 0.402 | − 0.027 |
3 | 0.536 | 0.438 | 0.248 | 0.343 | 0.506 | 0.424 | 0.117 |
3.3 Dependence of wall stress and local anatomy correlations on model parameters
Sensitivity to parameter modifications
Parameter | \(V_{\mathrm {infl}}\) (mL) | \(V_{\mathrm {cont}}\) (mL) | \(t_{\mathrm {cont}}\) (s) | IF (%) | EF (%) | \(\rho ^{\textsf {Wt}}_{\mathrm {infl}}\) | \(\rho ^{\textsf {Wt}}_{\mathrm {cont}}\) | \(\rho ^{\textsf {C}}_{\mathrm {infl}}\) | \(\rho ^{\textsf {C}}_{\mathrm {cont}}\) | \(\rho ^{\textsf {WtC}}_{\mathrm {infl}}\) | \(\rho ^{\textsf {WtC}}_{\mathrm {cont}}\) |
---|---|---|---|---|---|---|---|---|---|---|---|
Control | 83.4 | 62.3 | 142 | 163.1 | 25.2 | 0.536 | 0.438 | 0.248 | 0.343 | 0.506 | 0.522 |
\(p_{\mathrm {infl}}+25\%\) | 85.7 | 69.7 | 144 | 167.6 | 18.6 | 0.540 | 0.488 | 0.250 | 0.346 | 0.510 | 0.553 |
\(p_{\mathrm {infl}}-25\%\) | 80.4 | 52.9 | 141 | 157.3 | 34.2 | 0.528 | 0.337 | 0.243 | 0.320 | 0.498 | 0.446 |
\(T_{\mathrm {peak}}+25\%\) | 83.4 | 54.6 | 140 | 163.1 | 34.4 | 0.536 | 0.372 | 0.248 | 0.334 | 0.506 | 0.477 |
\(T_{\mathrm {peak}}-25\%\) | 83.4 | 69.8 | 144 | 163.1 | 16.2 | 0.536 | 0.486 | 0.248 | 0.333 | 0.506 | 0.542 |
\(a+25\%\) | 81.9 | 61.6 | 141 | 160.3 | 24.8 | 0.541 | 0.440 | 0.245 | 0.342 | 0.506 | 0.523 |
\(a-25\%\) | 85.1 | 63.3 | 142 | 166.5 | 25.7 | 0.530 | 0.436 | 0.251 | 0.343 | 0.505 | 0.521 |
\(a_{\mathrm {f}}+25\%\) | 82.6 | 62.3 | 142 | 161.6 | 24.6 | 0.535 | 0.436 | 0.252 | 0.342 | 0.508 | 0.520 |
\(a_{\mathrm {f}}-25\%\) | 84.3 | 62.4 | 141 | 164.9 | 26.0 | 0.538 | 0.441 | 0.243 | 0.343 | 0.504 | 0.523 |
stiff. \(+25\%\) | 81.2 | 61.6 | 142 | 158.9 | 24.2 | 0.539 | 0.438 | 0.248 | 0.342 | 0.508 | 0.521 |
stiff. \(-25\%\) | 86.1 | 63.4 | 141 | 168.5 | 26.4 | 0.531 | 0.438 | 0.247 | 0.343 | 0.503 | 0.522 |
\(\kappa +25\%\) | 85.7 | 67.0 | 143 | 167.7 | 21.8 | 0.547 | 0.474 | 0.232 | 0.339 | 0.501 | 0.540 |
\(\kappa -25\%\) | 81.3 | 59.8 | 141 | 159.2 | 26.5 | 0.525 | 0.408 | 0.261 | 0.341 | 0.510 | 0.502 |
\(\kappa =0\) for \(\mathbf{S}_{\mathrm {a}}\) | 83.4 | 52.4 | 140 | 163.1 | 37.2 | 0.537 | 0.346 | 0.248 | 0.327 | 0.507 | 0.457 |
\(\kappa =1/3\) | 90.4 | 89.1 | 144 | 176.8 | 1.4 | 0.560 | 0.560 | 0.203 | 0.201 | 0.487 | 0.486 |
\(1/\mu =0\) | 83.5 | 62.5 | 142 | 163.3 | 25.1 | 0.553 | 0.457 | 0.251 | 0.348 | 0.519 | 0.538 |
\(\mu =1000\) | 83.9 | 62.3 | 142 | 164.1 | 25.8 | 0.558 | 0.475 | 0.258 | 0.375 | 0.526 | 0.569 |
\(\mu =500\) | 84.5 | 62.1 | 142 | 165.3 | 26.5 | 0.562 | 0.482 | 0.261 | 0.382 | 0.530 | 0.579 |
To test if the degree of deformation affected the anatomical wall stress correlations, we recalculated the correlations in simulations with altered endocardial pressure and altered active contraction, that will alter deformation. Changes in deformation due to pressure and active contraction caused small (< 0.01) changes in the correlation of wall thickness or curvature with wall stress during inflation. During contraction, changes in deformation due to changes in pressure and active contraction caused small (\(<\,0.03\)) changes in the correlation of curvature with wall stress, but caused larger (\(>0.1\)) changes in the correlation of wall thickness and wall stress. We then tested if the isotropic stiffness, anisotropic stiffness, stiffness scaling factor or degree of anisotropy affected the anatomical wall stress correlations. None of the correlations experienced large changes with the greatest change being from 0.44 to 0.47. We tested limit cases of fiber dispersion: active tension acting only in the fiber direction and isotropic fiber distribution. In the active contraction case, the isotropic fiber distribution acts as a hydro-static pressure, so there is limited deformation; this made wall thickness and wall curvature have the same correlation with stress in the inflation and contraction cases and results in a limited ejection fraction. Active stress only acting in the fiber direction had no affect, as expected, on the inflation correlation and decreased the correlations in the contraction case. Changes in incompressibility caused minor changes in the inflation correlations but increasing the degree of incompressibility caused a decrease in the contraction correlations.
3.4 Dependence of wall stress and local anatomy correlations on model creation and analysis parameters
Sensitivity to geometrical modifications
Parameter | \(V_{\mathrm {infl}}\) (mL) | \(V_{\mathrm {cont}}\) (mL) | \(t_{\mathrm {cont}}\) (s) | IF (%) | EF (%) | \(\rho ^{\textsf {Wt}}_{\mathrm {infl}}\) | \(\rho ^{\textsf {Wt}}_{\mathrm {cont}}\) | \(\rho ^{\textsf {C}}_{\mathrm {infl}}\) | \(\rho ^{\textsf {C}}_{\mathrm {cont}}\) | \(\rho ^{\textsf {WtC}}_{\mathrm {infl}}\) | \(\rho ^{\textsf {WtC}}_{\mathrm {cont}}\) |
---|---|---|---|---|---|---|---|---|---|---|---|
Control | 83.4 | 62.3 | 142 | 163.1 | 25.2 | 0.536 | 0.438 | 0.248 | 0.343 | 0.506 | 0.522 |
Patch \(+25\%\) | 83.4 | 62.3 | 142 | 163.1 | 25.2 | 0.536 | 0.439 | 0.248 | 0.352 | 0.522 | 0.535 |
Patch \(-25\%\) | 83.4 | 62.3 | 142 | 163.1 | 25.2 | 0.536 | 0.438 | 0.250 | 0.339 | 0.491 | 0.511 |
Noised | 80.6 | 60.9 | 141 | 157.7 | 24.5 | 0.512 | 0.432 | 0.293 | 0.380 | 0.481 | 0.510 |
Constant | 82.7 | 68.4 | 145 | 161.8 | 17.3 | – | – | 0.343 | 0.467 | – | – |
0% cutoff | 83.4 | 62.3 | 142 | 163.1 | 25.2 | 0.495 | 0.410 | 0.202 | 0.330 | 0.464 | 0.505 |
50% cutoff | 83.4 | 62.3 | 142 | 163.1 | 25.2 | 0.582 | 0.523 | 0.344 | 0.400 | 0.584 | 0.587 |
25/5% cutoff | 83.4 | 62.3 | 142 | 163.1 | 25.2 | 0.510 | 0.453 | 0.312 | 0.345 | 0.541 | 0.530 |
5/25% cutoff | 83.4 | 62.3 | 142 | 163.1 | 25.2 | 0.606 | 0.506 | 0.260 | 0.369 | 0.536 | 0.565 |
\(\varOmega _t(\mathbf {x})\) | 83.4 | 62.315 | 142 | 163.1 | 25.2 | 0.561 | 0.518 | 0.390 | 0.446 | 0.525 | 0.600 |
stim. \(\Gamma _{\mathrm {endo}}\) | 83.358 | 72.176 | 162 | 163.1 | 13.4 | 0.536 | 0.498 | 0.238 | 0.326 | 0.506 | 0.544 |
In the analysis presented above, we have excluded regions where we applied boundary constraints; when these are included the correlations all decrease. To test if boundary conditions play a role in the correlations, we only considered the middle 50% of the anatomy that is remote from regions where boundary conditions are applied (pulmonary veins and mitral valve). The local wall thickness and curvature to wall stress correlations in these regions were all higher than correlations measured across the whole atria, suggesting boundary conditions decrease the correlations between local anatomy and wall stress. Excluding tissue preferentially near the mitral valve or the pulmonary veins shows that the mitral valve has the greater impact on the correlation. We find that using the deformed, as opposed to the reference, anatomy for calculating wall thickness and curvature improves all correlations. Finally, we demonstrated that the activation pattern did not play a large role in the correlations. Stimulating the entire endocardium causes the correlations to change by − 0.01 to 0.06.
3.5 Dependence of wall stress and local anatomy correlations on stress definition
Sensitivity to stress measurements
Stress type | Stress part | \(\rho ^{\textsf {Wt}}_{\mathrm {infl}}\) | \(\rho ^{\textsf {Wt}}_{\mathrm {cont}}\) | \(\rho ^{\textsf {C}}_{\mathrm {infl}}\) | \(\rho ^{\textsf {C}}_{\mathrm {cont}}\) | \(\rho ^{\textsf {WtC}}_{\mathrm {infl}}\) | \(\rho ^{\textsf {WtC}}_{\mathrm {cont}}\) |
---|---|---|---|---|---|---|---|
1st principal stress | \(\varvec{\sigma }_{\mathrm {p}}+\varvec{\sigma }_{\mathrm {a}}\) | 0.536 | 0.438 | 0.248 | 0.343 | 0.506 | 0.522 |
2nd principal stress | \(\varvec{\sigma }_{\mathrm {p}}+\varvec{\sigma }_{\mathrm {a}}\) | 0.415 | 0.290 | 0.323 | 0.320 | 0.495 | 0.419 |
3rd principal stress | \(\varvec{\sigma }_{\mathrm {p}}+\varvec{\sigma }_{\mathrm {a}}\) | 0.041 | 0.089 | 0.192 | 0.178 | 0.170 | 0.186 |
fiber stress | \(\varvec{\sigma }_{\mathrm {p}}+\varvec{\sigma }_{\mathrm {a}}\) | 0.522 | 0.410 | 0.252 | 0.366 | 0.492 | 0.517 |
Stress magnitude | \(\varvec{\sigma }_{\mathrm {p}}+\varvec{\sigma }_{\mathrm {a}}\) | 0.578 | 0.480 | 0.266 | 0.370 | 0.547 | 0.574 |
Von Mises stress | \(\varvec{\sigma }_{\mathrm {p}}+\varvec{\sigma }_{\mathrm {a}}\) | 0.552 | 0.448 | 0.207 | 0.290 | 0.483 | 0.484 |
1st principal stress | \(\varvec{\sigma }_{\mathrm {p}}\) | 0.536 | 0.310 | 0.248 | 0.307 | 0.506 | 0.424 |
2nd principal stress | \(\varvec{\sigma }_{\mathrm {p}}\) | 0.415 | 0.288 | 0.323 | 0.251 | 0.495 | 0.352 |
3rd principal stress | \(\varvec{\sigma }_{\mathrm {p}}\) | 0.041 | 0.273 | 0.192 | 0.266 | 0.170 | 0.378 |
Fiber stress | \(\varvec{\sigma }_{\mathrm {p}}\) | 0.522 | 0.402 | 0.252 | 0.299 | 0.492 | 0.471 |
Stress magnitude | \(\varvec{\sigma }_{\mathrm {p}}\) | 0.578 | − 0.244 | 0.266 | − 0.206 | 0.547 | − 0.322 |
Von Mises stress | \(\varvec{\sigma }_{\mathrm {p}}\) | 0.552 | 0.126 | 0.207 | 0.135 | 0.483 | 0.171 |
1st principal stress | \(\varvec{\sigma }_{\mathrm {a}}\) | – | 0.222 | – | 0.172 | – | 0.259 |
2nd principal stress | \(\varvec{\sigma }_{\mathrm {a}}\) | – | 0.086 | – | 0.063 | – | 0.077 |
3rd principal stress | \(\varvec{\sigma }_{\mathrm {a}}\) | – | − 0.096 | – | − 0.034 | – | \(-\)0.067 |
Fiber stress | \(\varvec{\sigma }_{\mathrm {a}}\) | – | 0.214 | – | 0.269 | – | 0.314 |
Stress magnitude | \(\varvec{\sigma }_{\mathrm {a}}\) | – | 0.401 | – | 0.317 | – | 0.462 |
Von Mises stress | \(\varvec{\sigma }_{\mathrm {a}}\) | – | 0.285 | – | 0.206 | – | 0.318 |
1st principal stress | \(\varvec{S}_{\mathrm {p}}+\varvec{S}_{\mathrm {a}}\) | 0.553 | 0.374 | 0.249 | 0.333 | 0.517 | 0.483 |
2nd principal stress | \(\varvec{S}_{\mathrm {p}}+\varvec{S}_{\mathrm {a}}\) | 0.388 | 0.315 | 0.308 | 0.280 | 0.469 | 0.399 |
3rd principal stress | \(\varvec{S}_{\mathrm {p}}+\varvec{S}_{\mathrm {a}}\) | 0.041 | 0.145 | 0.149 | 0.266 | 0.133 | 0.289 |
Fiber stress | \(\varvec{S}_{\mathrm {p}}+\varvec{S}_{\mathrm {a}}\) | 0.532 | 0.390 | 0.272 | 0.358 | 0.515 | 0.504 |
Stress magnitude | \(\varvec{S}_{\mathrm {p}}+\varvec{S}_{\mathrm {a}}\) | 0.595 | 0.456 | 0.282 | 0.393 | 0.569 | 0.578 |
Von Mises stress | \(\varvec{S}_{\mathrm {p}}+\varvec{S}_{\mathrm {a}}\) | 0.576 | 0.375 | 0.174 | 0.185 | 0.474 | 0.365 |
3.6 Testing the impact of geometric complexity on the correlation of wall stress with local anatomy
Laplace law
Fibers | R (mm) | T (mm) | \(V_{\mathrm {wall}}\) (\(\hbox {cm}^{3}\)) | \(V_0\) (ml) | \(V_{\mathrm {infl}}\) (ml) | \(V_{\mathrm {cont}}\) (ml) | IF (%) | EF (%) | \(\overline{\sigma }^{\mathrm {1st}}_{\mathrm {infl}}\) (kPa) | \(\overline{\sigma }^{\mathrm {1st}}_{\mathrm {cont}}\) (kPa) | \(\overline{\sigma }^{\mathrm {circ}}_{\mathrm {infl}}\) (kPa) | \(\overline{\sigma }^{\mathrm {circ}}_{\mathrm {cont}}\) (kPa) | \(\sigma ^{\mathrm {La}}_{\mathrm {infl}}\) (kPa) | \(\sigma ^{\mathrm {La}}_{\mathrm {cont}}\) (kPa) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
isotropic | 20 | 0.5 | 2.6 | 33.5 | 59.7 | 58.6 | 178.0 | 1.8 | 53.3 | 52.8 | 46. 4 | 45.9 | 46. 6 | 46.6 |
tr. iso. | 20 | 0.5 | 2.6 | 33.5 | 57.0 | 43.7 | 170.2 | 23.4 | 52.8 | 42.8 | 48. 4 | 40.8 | 44. 6 | 44.7 |
ortho. | 20 | 0.5 | 2.6 | 33.5 | 56.0 | 49.9 | 167.3 | 11.0 | 50.5 | 44.8 | 44. 0 | 39.0 | 43. 8 | 43.9 |
incomp. | 20 | 0.5 | 2.6 | 33.5 | 55.9 | 50.5 | 166.9 | 9.6 | 50.4 | 47.0 | 43. 9 | 41.0 | 43. 7 | 43.8 |
isotropic | 20 | 2.5 | 14.2 | 33.5 | 48.6 | 47.7 | 145.0 | 1.7 | 8.7 | 7.9 | 7. 5 | 6.7 | 7. 1 | 7.2 |
tr. iso. | 20 | 2.5 | 14.2 | 33.5 | 46.3 | 22.9 | 138.2 | 50.5 | 8.5 | 12.8 | 7. 7 | 8.6 | 6. 8 | 7.1 |
ortho. | 20 | 2.5 | 14.2 | 33.5 | 45.5 | 22.0 | 136.0 | 51.7 | 8.1 | 5.1 | 7. 0 | 3.2 | 6. 7 | 7.0 |
isotropic | 20 | 5.0 | 31.9 | 33.5 | 44.5 | 43.8 | 133.0 | 1.6 | 4.0 | 3.1 | 3. 4 | 2.5 | 3. 1 | 3.1 |
tr. iso. | 20 | 5.0 | 31.9 | 33.5 | 42.5 | 21.7 | 126.9 | 48.9 | 3.9 | 13.0 | 3. 5 | 4.2 | 2. 9 | 3.2 |
ortho. | 20 | 5.0 | 31.9 | 33.5 | 41.8 | 20.7 | 124.8 | 50.5 | 3.7 | 4.3 | 3. 1 | − 1.1 | 2. 9 | 3.1 |
isotropic | 25 | 0.5 | 4.0 | 65.4 | 119.9 | 117.7 | 183.2 | 1.8 | 68.5 | 68.0 | 59. 7 | 59.2 | 60. 2 | 60.2 |
tr. iso. | 25 | 0.5 | 4.0 | 65.4 | 114.5 | 98.4 | 175.0 | 14.1 | 67.9 | 60.7 | 62. 3 | 57.5 | 57. 5 | 57.5 |
ortho. | 25 | 0.5 | 4.0 | 65.4 | 112.4 | 104.2 | 171.7 | 7.3 | 64.9 | 60.1 | 56. 5 | 52.3 | 56. 4 | 56.5 |
incomp. | 25 | 0.5 | 4.0 | 65.4 | 112.0 | 104.8 | 171.2 | 6.4 | 64.8 | 61.9 | 56. 4 | 54.0 | 56. 3 | 56.3 |
isotropic | 25 | 2.5 | 21.7 | 65.4 | 97.5 | 95.8 | 149.0 | 1.7 | 11.1 | 10.3 | 9. 7 | 8.8 | 9. 3 | 9.3 |
tr. iso. | 25 | 2.5 | 21.7 | 65.4 | 93.0 | 45.7 | 142.2 | 50.8 | 10.9 | 13.3 | 10. 0 | 10.3 | 8. 9 | 9.2 |
ortho. | 25 | 2.5 | 21.7 | 65.4 | 91.5 | 43.9 | 139.9 | 52.1 | 10.5 | 5.8 | 9. 0 | 4.6 | 8. 8 | 9.1 |
isotropic | 25 | 5.0 | 47.6 | 65.4 | 89.5 | 87.0 | 136.8 | 1.7 | 5.1 | 4.2 | 4. 4 | 3.5 | 4. 1 | 4.1 |
tr. iso. | 25 | 5.0 | 47.6 | 65.4 | 85.3 | 43.0 | 130.4 | 49.5 | 5.0 | 12.7 | 4. 5 | 5.5 | 3. 9 | 4.1 |
ortho. | 25 | 5.0 | 47.6 | 65.4 | 83.8 | 41.2 | 128.2 | 50.8 | 4.7 | 4.5 | 4. 0 | 0.3 | 3. 8 | 4.1 |
isotropic | 30 | 0.5 | 5.8 | 113.1 | 212.1 | 208.3 | 187.5 | 1.8 | 84.0 | 83.7 | 73. 2 | 72.9 | 74. 1 | 74.1 |
tr. iso. | 30 | 0.5 | 5.8 | 113.1 | 202.3 | 183.3 | 178.9 | 9.4 | 83.4 | 78.0 | 76. 6 | 73.7 | 70. 7 | 70.7 |
ortho. | 30 | 0.5 | 5.8 | 113.1 | 198.3 | 187.5 | 175.3 | 5.4 | 79.5 | 75.2 | 69. 3 | 65.5 | 69. 3 | 69.3 |
incomp. | 30 | 0.5 | 5.8 | 113.1 | 197.5 | 188.2 | 174.7 | 4.7 | 79.5 | 76.9 | 69. 2 | 67.0 | 69. 0 | 69.1 |
isotropic | 30 | 2.5 | 30.7 | 113.0 | 172.2 | 169.2 | 152.4 | 1.8 | 13.7 | 12.9 | 11. 9 | 11.0 | 11. 5 | 11.5 |
tr. iso. | 30 | 2.5 | 30.7 | 113.0 | 164.4 | 80.9 | 145.5 | 50.8 | 13.4 | 14.1 | 12. 2 | 11.9 | 11. 0 | 11.3 |
ortho. | 30 | 2.5 | 30.7 | 113.0 | 161.9 | 77.3 | 143.2 | 52.3 | 12.9 | 6.9 | 11. 1 | 5.9 | 10. 9 | 11.2 |
isotropic | 30 | 5.0 | 66.5 | 113.0 | 158.2 | 155.5 | 139.9 | 1.7 | 6.3 | 5.4 | 5. 4 | 4.5 | 5. 1 | 5.1 |
tr. iso. | 30 | 5.0 | 66.5 | 113.0 | 150.7 | 75.4 | 133.3 | 50.0 | 6.1 | 12.6 | 5. 5 | 6.6 | 4. 8 | 5.1 |
ortho. | 30 | 5.0 | 66.5 | 113.0 | 148.2 | 72.3 | 131.1 | 51.2 | 5.8 | 4.7 | 5. 0 | 1.4 | 4. 8 | 5.0 |
3.7 Testing if the Law of Laplace can be used to estimate mean wall stress in the left atria
Laplace law
Value | Unit | Patient 1 | Patient 2 | Patient 3 | |||
---|---|---|---|---|---|---|---|
infl. | cont. | infl. | cont. | infl. | cont. | ||
Min | (kPa) | − 573.01 | − 336.46 | − 515.86 | − 635.81 | − 416.53 | − 381.09 |
Max | (kPa) | 2304.60 | 422.54 | 869.39 | 749.39 | 389.49 | 357.67 |
Mean | (kPa) | 21.01 | 25.51 | 19.92 | 28.16 | 30.93 | 33.69 |
SD | (kPa) | 20.51 | 17.02 | 24.47 | 19.92 | 25.21 | 20.01 |
\(\sigma ^{\mathrm {La}}\) | (kPa) | 17.67 | 17.85 | 17.19 | 17.34 | 26.30 | 26.40 |
Summary, Spearman’s correlations between the principle wall Cauchy stress and the curvature and wall thickness where we quantified local anatomy in the deformed anatomy, increased endocardial pressure, decreased active tension and only considered the middle 50% of the atria
i | \(\rho ^{\textsf {Wt}}_{\mathrm {infl}}\) | \(\rho ^{\textsf {Wt}}_{\mathrm {cont}}\) | \(\rho ^{\textsf {C}}_{\mathrm {infl}}\) | \(\rho ^{\textsf {C}}_{\mathrm {cont}}\) | \(\rho ^{\textsf {WtC}}_{\mathrm {infl}}\) | \(\rho ^{\textsf {WtC}}_{\mathrm {cont}}\) |
---|---|---|---|---|---|---|
1 | 0.550 | 0.636 | 0.402 | 0.407 | 0.642 | 0.669 |
2 | 0.600 | 0.616 | 0.411 | 0.406 | 0.682 | 0.639 |
3 | 0.598 | 0.644 | 0.367 | 0.480 | 0.608 | 0.699 |
4 Discussion
We have developed a modeling framework for simulating left atrium contraction. We have simulated passive inflation and active contraction of the atria. We have shown that, consistent with the Law of Laplace, the principal wall stress is dependent on LA wall thickness and curvature under conditions of passive inflation, with a higher dependence on wall thickness. Under conditions of active contraction we find a smaller correlation between wall stress and curvature or wall thickness. This finding is replicated in both idealized and complex geometries. A sensitivity analysis demonstrated that the correlations are robust to many model simulation parameters, model creation and analysis parameters and the definition of stress (Tables 4, 5, 6). To maximize the correlation of wall stress and local anatomy required calculations of wall thickness and curvature on the deformed geometry, only consider tissue that is remote from boundary conditions and when deformations are reduced.
In the model, we predicted maximal LA volumes of 83–156 mL following passive inflation. The final pressure was higher than expected under physiological conditions leading to higher volumes but close to reported values of \(80\,\pm \,30\) mL in controls and \(115\,\pm \,33\) mL in AF patients (Stojanovska et al. 2014). The ejection fractions predicted by the model were 25–37%, which are consistent with measurements of \(\approx 30\%\) (Rodevand et al. 1999; Stefanadis et al. 1998). This shows that the model is capable of operating within a physiological range consistent with clinical observations.
A correlation was found between wall stress and both wall thickness and curvature during passive inflation (Fig. 6). This shows that both anatomical attributes are important. Wall thickness does have a stronger correlation (0.54–0.62) than curvature (0.20 to 0.25) emphasizing the importance of accounting for atrial wall thickness in personalized calculations of local wall stress. This may be particularly important when studying how local wall stress is correlated with local tissue remodeling.
In the Law of Laplace, wall stress is proportional to the ratio of the radius of curvature (the inverse of curvature) and the wall thickness. To test if this ratio had a stronger correlation with wall stress than wall thickness or curvature independently, we plotted the inverse product of wall thickness and curvature. This resulted in negligible improvement in the correlation (Fig. 6).
In the ventricles, myocardial wall stress plays a role in regulating growth and oxygen demand (Yin 1981) and was first associated with cardiac shape by Woods (1892). However, these relationships have not been evaluated in the atrium, where measuring wall thickness across the entire atria has only recently become possible (Bishop et al. 2016). We found that wall thickness had a greater impact on determining wall stress than curvature in the three patients cases studied. These three cases span a range of atrial pathologies from hyperlipidemia, which is associated with elevated blood pressure, atrial fibrillation, which is associated with a decrease in atrial mechanical function and a healthy control.
Increased left ventricle pressure is hypothesized to cause cellular hypertrophy and increased wall thickness to bring wall stress back to normal levels (Grossman et al. 1975). This is the first study to show that in the atria wall thickness correlates with local principal wall stress and similar regulatory pathways, that are hypothesized in the ventricle, may be present in the atria. Previous studies have also found that atrial wall stress is correlated with remodeling, in the form of fibrosis (Hunter et al. 2012), although this study was performed with homogeneous wall thickness and may need to be confirmed in models that account for varying wall thickness across the atrial body.
The potential link between curvature, wall thickness, wall stress, growth and fibrosis may have important, albeit complex, interactions with atrial electrophysiology. This becomes particularly important if remodeling in the atria are regulated by wall stress, as proposed in the ventricle (Grossman et al. 1975), and provides a possible link between increased arrhythmia risk and pathological changes in atrial loading. Previous studies have found atrial wall thickness and curvature impact the conduction velocity (Rossi et al. 2018), gradients in thickness are associated with stabilizing re-entrant activation patterns (Yamazaki et al. 2012) and patients with thicker atria are at higher risk of developing arrhythmias (Whitaker et al. 2016). Studying the interaction between wall stress, anatomy and atrial arrhythmias using computational simulations will require large highly detailed complex models and motivates further investment in multi-physics simulators and simulation speed.
To identify the factors that are important in determining the correlation between wall stress with wall thickness or curvature we preformed three sensitivity studies investigating model parameters, variables for creating and analyzing the model geometry and the definition of stress. For the vast majority of model perturbations, there was no to limited changes in the correlations suggesting that these are robust to model assumptions. We identified the choice of reference frame, degree of incompressibility, amount of deformation and boundary conditions as confounding factors in the correlation of local anatomy with wall stress. However, when including all of these factors in the model the maximum correlation was only 0.6–0.7, showing that in complex atrial anatomy it is unlikely that a simple anatomical-based law can be used to estimate local wall stress.
4.1 Limitations
This is the first study of atrial mechanics to account for varying wall thickness derived from clinical images. We have applied our modeling framework to three patient cases, demonstrating that the techniques are applicable beyond a single case study. However, in contrast to work in the ventricles (Nordsletten et al. 2011; Augustin et al. 2016) we have applied a simple active contraction model, static boundary conditions and we have not unloaded the geometry.
The model of active contraction is driven by a phenomenological model of tension development to estimate atrial cellular contraction. We have used a simplified contraction model that would benefit from increased physiological detail. Previous attempts at modeling atrial contraction in organ scale models have adapted human ventricular models (Land and Niederer 2018) or used models initially fitted to rat ventricular data (Moyer et al. 2015; Adeniran et al. 2015). To improve simulations of atrial contraction will require the development of a model of human atrial contraction from detailed human atrial experimental measurements.
The passive mechanics parameters used in the simulations were dependent on two modeling decisions. First, in fitting the passive material properties the value of \(\phi\) was fixed to fall within 0.1 and 0.9, to ensure that the model included both an endocardium and epicardium layer. The final fitted value was 0.1, suggesting that the optimal \(\phi\) value could fall between 0 and 0.1. This constraint may have affected the fitted passive stiffness parameters. Second, the atria was not unloaded at the time point when the reference anatomy was created. As the atria exhibit nonlinear constitutive properties, the loaded reference geometry will results in smaller calculated strains. To compensate for these effects, we scaled the passive mechanics model by a factor of 2. To test if these decisions impacted the calculated correlations we altered the isotropic, fiber or combined stiffness (Table 4). None of these changes caused large differences in the correlations suggesting these model assumptions do not affect the study conclusions.
In the performed simulations, a biophysical cell model was used that simulates the full action potential and calcium dynamics. These models are more complex than may be necessary for our simulations. However, the relatively small cost of using a full cell model allows us to better capture the effects, if any, of wave curvature and activation speed. As electrophysiology was not the focus of the study, we did not investigate what effect the use of a biophysical cell model had on simulation results.
We have simulated a single phase of the atrial cardiac cycle against a fixed pressure boundary condition. To capture wall thickness in our model, we derived the model anatomy from cardiac CT images. While CT gives excellent resolution, to capture motion with cardiac CT requires a higher radiation dose and was not available for our patient cases. This meant that we did not have information on volume transients nor did we have information on mitral valve ring motion. Further, we did not have access to invasive pressure measurements or echocardiogram Doppler flow measurements to allow us to estimate atrial pressure. As a result, we were unable to simulate the reservoir phase of the atrial cycle that is driven by ventricular motion and we were unable to personalize dynamic boundary conditions so used literature values to define the fixed pressure boundary conditions.
Large deformation mechanics models are initiated from a reference unloaded geometry. The heart is never in an unloaded state as it always has a cavity pressure. To account for the cavity pressure, when the heart is imaged in patient-specific ventricular models, the meshes are unloaded to estimate the reference configuration based on the boundary conditions and the imaged deformed anatomy. This results in the myocardium being under strain when the heart is re-inflated back to the imaged volume. Due to the thin shell of the atria, unloading the atria risks buckling that would lead to an unstable simulation. The unloading of the atria may require specific numerical techniques to allow for a stable simulation. The absence of unloading would make the atria more compliant as it would operate at lower strains. To account for this potential underestimation of atrial stiffness, we increased the stiffness in our simulations. Accounting for atrial unloading will be important to enable the mapping of detailed-ex-vivo measurements into simulations of the atria in-vivo.
This study used publicly available patient-specific anatomies for three cases. This small sample does not represent the full variation in atria anatomies and limits the ability to generalize these findings to other patients. However, the correlations that we identified were qualitatively similar across all three cases in the reference simulations and all three cases saw similar increases in correlations under idealized conditions (Table 9). In addition, we were able to replicate the decreased correlation between local anatomy with wall stress during contraction in a sphere model (Table 7) observed in the reference models (Table 3). Two of the patients had known pathologies and were likely taking corresponding medication that have the potential to alter atrial electrophysiology and/or contraction. As models were generated from anonymized data, the drug history of each patient was not available for this study.
5 Conclusion
We have created the first cohort of atrial mechanics models personalized to patients anatomy, including wall thickness. We found that the principal wall stress was determined more by the wall thickness than the curvature, necessitating personalized wall thickness measurements for calculating local wall stress. For the conditions considered here, the Law of Laplace provides a poor estimate of local wall stress in the left atrium. The choice of reference frame, degree of incompressibility, amount of deformation and boundary conditions were the main confounding factors, but did not fully explain the difference between the simulated wall stress and the Law of Laplace. This simulation framework provides a platform for studying the link between local anatomy, mechanics and electrophysiology.
Notes
Acknowledgements
The authors would like to thank Dr Salvatore Federico for providing his expertise on the characterization of the mechanical material behavior. This research was supported by the UK Medical Research Council (MR/ N001877 /1, MR/ N011007 /1), the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska–Curie Grant agreement No 750835, the Austrian Science Fund (F3210-N18, I2760-B30), the UK Engineering and Physical Sciences Research Council (EP/ F043929 /1, EP/ P01268X /1), the British Heart Foundation (PG/ 13/ 37/ 30280), the Wellcome Trust Center for Medical at King’s College London and the Department of Health via the National Institute for Health Research comprehensive Biomedical Research Centre award to Guy’s & St Thomas’ NHS Foundation Trust in partnership with King’s College London and King’s College Hospital NHS Foundation Trust. This work made use of ARCHER, the UK’s national high-performance computing service located at the University of Edinburgh and funded by the Office of Science and Technology through Engineering and Physical Sciences Research Council’s High End Computing Programme.
References
- Adeniran I, MacIver DH, Garratt CJ, Ye J, Hancox JC, Zhang H (2015) Effects of persistent atrial fibrillation-induced electrical remodeling on atrial electro-mechanics–insights from a 3D model of the human atria. PLoS ONE 10(11):e0142397CrossRefGoogle Scholar
- Ágoston G, Szilágyi J, Bencsik G, Tutuianu C, Klausz G, Sághy L, Varga A, Forster T, Pap R (2015) Impaired adaptation to left atrial pressure increase in patients with atrial fibrillation. J Interv Card Electrophysiol 44(2):113–118CrossRefGoogle Scholar
- Ahmed J, Sohal S, Malchano ZJ, Holmvang G, Ruskin JN, Reddy VY (2006) Three-dimensional analysis of pulmonary venous ostial and antral anatomy: Implications for balloon catheter-based pulmonary vein isolation. J Cardiovasc Electrophysiol 17(3):251–255CrossRefGoogle Scholar
- Ahn SJ, Rauh W, Warnecke HJ (2001) Least-squares orthogonal distances fitting of circle, sphere, elipse, hyperbola, and parabola. Pattern Recogn 34(12):2283–2303. https://doi.org/10.1016/S0031-3203(00)00152-7 ISSN 00313203CrossRefzbMATHGoogle Scholar
- Augustin CM, Neic A, Liebmann M, Prassl AJ, Niederer SA, Haase G, Plank G (2016) Anatomically accurate high resolution modeling of human whole heart electromechanics: a strongly scalable algebraic multigrid solver method for nonlinear deformation. J Comput Phys 305:622–646MathSciNetzbMATHCrossRefGoogle Scholar
- Ayachit U (2015) The paraview guide: a parallel visualization application. Kitware Inc, New YorkGoogle Scholar
- Bellini C, Di Martino ES, Federico S (2013) Mechanical behaviour of the human atria. Ann Biomed Eng 41(7):1478–1490CrossRefGoogle Scholar
- Benjamin E, Levy D, Vaziri S, D’Agostino R, Belanger A, Wolf P (1994) Independent risk factors for atrial fibrillation in a population-based cohort: the framingham heart study. JAMA 271(11):840–844CrossRefGoogle Scholar
- Bishop M, Rajani R, Plank G, Gaddum N, Carr-White G, Wright M, O’Neill M, Niederer S (2016) Three-dimensional atrial wall thickness maps to inform catheter ablation procedures for atrial fibrillation. Europace 18(3):376–383CrossRefGoogle Scholar
- Blume GG, Mcleod CJ, Barnes ME, Seward JB, Pellikka PA, Bastiansen PM, Tsang TS (2011) Left atrial function: physiology, assessment, and clinical implications. Eur J Echocardiogr 12(6):421–430CrossRefGoogle Scholar
- Cherry EM, Evans SJ (2008) Properties of two human atrial cell models in tissue: Restitution, memory, propagation, and reentry. J Theor Biol 254(3):674–690CrossRefGoogle Scholar
- Cherry EM, Hastings HM, Evans SJ (2008) Dynamics of human atrial cell models: Restitution, memory, and intracellular calcium dynamics in single cells. Prog Biophys Mol Biol 98(1):24–37CrossRefGoogle Scholar
- Courtemanche M, Ramirez RJ, Nattel S (1998) Ionic mechanisms underlying human atrial action potential properties: Insights from a mathematical model. Am J Physiol Heart Circ Physiol 275(44):H301–H321CrossRefGoogle Scholar
- DeBotton G (2005) Transversely isotropic sequentially laminated composites in finite elasticity. J Mech Phys Solids 53(6):1334–1361MathSciNetzbMATHCrossRefGoogle Scholar
- DeBotton G, Shmuel G (2009) Mechanics of composites with two families of finitely extensible fibers undergoing large deformations. J Mech Phys Solids 57(8):1165–1181zbMATHCrossRefGoogle Scholar
- Demiray H (1972) A note on the elasticity of soft biological tissues. J Biomech 5(3):309–311CrossRefGoogle Scholar
- Dimitri H, Ng M, Brooks AG, Kuklik P, Stiles MK, Lau DH, Antic N, Thornton A, Saint DA, McEvoy D, Antic R, Kalman JM, Sanders P (2012) Atrial remodeling in obstructive sleep apnea: implications for atrial fibrillation. Heart Rhythm 9(3):321–327CrossRefGoogle Scholar
- Dzeshka MS, Lip GY, Snezhitskiy V, Shantsila E (2015) Cardiac fibrosis in patients with atrial fibrillation: mechanisms and clinical implications. J Am Coll Cardiol 66(8):943–959CrossRefGoogle Scholar
- Eriksson TSE, Prassl AJ, Plank G, Holzapfel GA (2013) Modeling the dispersion in electromechanically coupled myocardium. Int J Numer Methods Biomed Eng 29(11):1267–1284MathSciNetCrossRefGoogle Scholar
- Fastl TE, Tobon-Gomez C, Crozier A, Whitaker J, Rajani R, McCarthy KP, Sanchez-Quintana D, Ho SY, O’Neill MD, Plank G, Bishop MJ, Niederer SA (2018) Personalized computational modeling of left atrial geometry and transmural myofiber architecture. Med Image Anal 47(5):180–190CrossRefGoogle Scholar
- Flory P (1961) Thermodynamic relations for high elastic materials. Trans Faraday Soc 57:829–838MathSciNetCrossRefGoogle Scholar
- Fu Z, Kirby RM, Whitaker RT (2013) A fast iterative method for solving the eikonal equation on tetrahedral domains. SIAM J Sci Comput 35(5):C473–C494MathSciNetzbMATHCrossRefGoogle Scholar
- Gasser TC, Ogden RW, Holzapfel GA (2006) Hyperelastic modelling of arterial layers with distributed collagen fibre orientations. J R Soc Interface 3(6):15–35CrossRefGoogle Scholar
- Geuzaine C, Remacle J-F (2009) Gmsh: a 3-d finite element mesh generator with built-in pre-and post-processing facilities. Int J Numer Meth Eng 79(11):1309–1331MathSciNetzbMATHCrossRefGoogle Scholar
- Grossman W, Jones D, McLaurin L (1975) Wall stress and patterns of hypertrophy in the human left ventricle. J Clin Investig 56(1):56–64CrossRefGoogle Scholar
- Gsell MAF, Augustin CM, Prassl AJ, Karabelas E, Fernandes JF, Kelm M, Goubergrits L, Kuehne T, Plank G (2018) Assessment of wall stresses and mechanical heart power in the left ventricle: finite element modeling versus Laplace analysis. Int J Numer Methods Biomed Eng 34(12):e3147MathSciNetCrossRefGoogle Scholar
- Gültekin O, Dal H, Holzapfel GA (2019) On the quasi-incompressible finite element analysis of anisotropic hyperelastic materials. Comput Mech 63(3):443–453MathSciNetzbMATHCrossRefGoogle Scholar
- Helfenstein J, Jabareen M, Mazza E, Govindjee S (2010) On non-physical response in models for fiber-reinforced hyperelastic materials. Int J Solids Struct 47(16):2056–2061zbMATHCrossRefGoogle Scholar
- Hunter RJ, Liu Y, Lu Y, Wang W, Schilling RJ (2012) Left atrial wall stress distribution and its relationship to electrophysiologic remodeling in persistent atrial fibrillation. Circ Arrhythm Electrophysiol 5(2):351–360CrossRefGoogle Scholar
- Iung B, Leenhardt A, Extramiana F (2018) Management of atrial fibrillation in patients with rheumatic mitral stenosis. Heart 104(13):1062–1068CrossRefGoogle Scholar
- Ji C, Yu J, Li T, Tian L, Huang Y, Wang Y, Zheng Y (2015) Dynamic curvature topography for evaluating the anterior corneal surface change with corvis st. BioMed Eng OnLine 14(1):53. https://doi.org/10.1186/s12938-015-0036-2 ISSN 1475-925XCrossRefGoogle Scholar
- Jones SE, Buchbinder BR, Aharon I (2000) Three-dimensional mapping of cortical thickness using laplace’s equation. Hum Brain Mapp 11(1):12–32CrossRefGoogle Scholar
- Keach JW, Bradley SM, Turakhia MP, Maddox TM (2015) Early detection of occult atrial fibrillation and stroke prevention. Heart 101(14):1097–1102CrossRefGoogle Scholar
- Kerckhoffs RC, Omens JH, McCulloch AD (2012) A single strain-based growth law predicts concentric and eccentric cardiac growth during pressure and volume overload. Mech Res Commun 42:40–50 Recent Advances in the Biomechanics of Growth and RemodelingCrossRefGoogle Scholar
- Kirchhof P, Benussi S, Kotecha D, Ahlsson A, Atar D, Casadei B, Castella M, Diener H-C, Heidbuchel H, Hendriks J, Hindricks G, Manolis AS, Oldgren J, Popescu BA, Schotten U, Van Putte B, Vardas P, Group ESD (2016) 2016 esc guidelines for the management of atrial fibrillation developed in collaboration with eacts. Eur Heart J 37(38):2893–2962CrossRefGoogle Scholar
- Kneller J, Ramirez RJ, Chartier D, Courtemanche M, Nattel S (2002) Time-dependent transients in an ionically based mathematical model of the canine atrial action potential. Am J Physiol Heart Circ Physiol 282(4):H1437–H1451CrossRefGoogle Scholar
- Land S, Niederer SA (2018) Influence of atrial contraction dynamics on cardiac function. Int J Numer Methods Biomed Eng 34(3):e2931 e2931 cnm.2931MathSciNetCrossRefGoogle Scholar
- Land S, Gurev V, Arens S, Augustin CM, Baron L, Blake R, Bradley C, Castro S, Crozier A, Favino M, Fastl TE, Fritz T, Gao H, Gizzi A, Griffith BE, Hurtado DE, Krause R, Luo X, Nash MP, Pezzuto S, Plank G, Rossi S, Ruprecht D, Seemann G, Smith NP, Sundnes J, Rice JJ, Trayanova N, Wang D, Jenny Wang Z, Niederer SA (2015) Verification of cardiac mechanics software: benchmark problems and solutions for testing active and passive material behaviour. Proc R Soc A Math Phys Eng Sci 471(2184):20150641CrossRefGoogle Scholar
- Lin W-Y, Chiu Y-L, Widder K, Hu Y, Boston N (2010) Robust and accurate curvature estimation using adaptive line integrals. EURASIP J Adv Signal Process 2010(1):240309. https://doi.org/10.1155/2010/240309 ISSN 1687-6180CrossRefGoogle Scholar
- Markides V, Schilling RJ, Yen Ho S, Chow AW, Davies DW, Peters NS (2003) Characterization of left atrial activation in the intact human heart. Circulation 107(5):733–739CrossRefGoogle Scholar
- Marrouche NF, Wilber D, Hindricks G, Jais P, Akoum N, Marchlinski F, Kholmovski E, Burgon N, Hu N, Mont L et al (2014) Association of atrial tissue fibrosis identified by delayed enhancement mri and atrial fibrillation catheter ablation: the decaaf study. JAMA 311(5):498–506CrossRefGoogle Scholar
- Mendonca Costa C, Hoetzl E, Martins Rocha B, Prassl AJ, Plank G (2013) Automatic parameterization strategy for cardiac electrophysiology simulations. Comput Cardiol 40:373–376Google Scholar
- Mirsky I, Parmley WW (1973) Assessment of passive elastic stiffness for isolated heart muscle and the intact heart. Circul Res. ISSN 00097330Google Scholar
- Moyer CB, Norton PT, Ferguson JD, Holmes JW (2015) Changes in global and regional mechanics due to atrial fibrillation: insights from a coupled finite-element and circulation model. Ann Biomed Eng 43(7):1600–1613CrossRefGoogle Scholar
- Nash MP, Hunter PJ (2000) Computational mechanics of the heart. From tissue structure to ventricular function. J Elast. https://doi.org/10.1023/A:1011084330767 CrossRefGoogle Scholar
- Neic A, Liebmann M, Hoetzl E, Mitchell L, Vigmond EJ, Haase G, Plank G (2012) Accelerating cardiac bidomain simulations using graphics processing units. IEEE Trans Biomed Eng 59(8):2281–90CrossRefGoogle Scholar
- Neic A, Campos FO, Prassl AJ, Niederer SA, Bishop MJ, Vigmond EJ, Plank G (2017) Efficient computation of electrograms and ecgs in human whole heart simulations using a reaction-eikonal model. J Comput Phys 346:191–211MathSciNetCrossRefGoogle Scholar
- Niederer SA, Kerfoot E, Benson AP, Bernabeu MO, Bernus O, Bradley C, Cherry EM, Clayton R, Fenton FH, Garny A, Heidenreich E, Land S, Maleckar M, Pathmanathan P, Plank G, Rodríguez JF, Roy I, Sachse FB, Seemann G, Skavhaug O, Smith NP (2011a) Verification of cardiac tissue electrophysiology simulators using an n-version benchmark. Philos Trans R Soc A Math Phys Eng Sci 369(1954):4331–4351. https://doi.org/10.1098/rsta.2011.0139 MathSciNetCrossRefGoogle Scholar
- Niederer SA, Plank G, Chinchapatnam P, Ginks M, Lamata P, Rhode KS, Rinaldi CA, Razavi R, Smith NP (2011b) Length-dependent tension in the failing heart and the efficacy of cardiac resynchronization therapy. Cardiovasc Res 89(2):336–343CrossRefGoogle Scholar
- Nikou A, Dorsey SM, McGarvey JR, Gorman JH III, Burdick JA, Pilla JJ, Gorman RC, Wenk JF (2016) Effects of using the unloaded configuration in predicting the in vivo diastolic properties of the heart. Comput Methods Biomech Biomed Eng 19(16):1714–1720CrossRefGoogle Scholar
- Nordsletten D, Niederer S, Nash M, Hunter P, Smith N (2011) Coupling multi-physics models to cardiac mechanics. Prog Biophys Mol Biol 104(1):77–88 Cardiac Physiome project: Mathematical and Modelling FoundationsCrossRefGoogle Scholar
- Pashakhanloo F, Herzka DA, Ashikaga H, Mori S, Gai N, Bluemke DA, Trayanova NA, McVeigh ER (2016) Myofiber architecture of the human atria as revealed by submillimeter diffusion tensor imaging. Circ Arrhythm Electrophysiol 9(4):e004133CrossRefGoogle Scholar
- Rodevand O, Bjornerheim R, Ljosland M, Maehle J, Smith H, Ihlen H (1999) Left atrial volumes assessed by three- and two-dimensional echocardiography compared to mri estimates. Int J Card Imag 15(5):397–410CrossRefGoogle Scholar
- Rodriguez EK, Hoger A, McCulloch AD (1994) Stress-dependent finite growth in soft elastic tissues. J Biomech 27(4):455–467CrossRefGoogle Scholar
- Rossi S, Gaeta S, Griffith BE, Henriquez CS (2018) Muscle thickness and curvature influence atrial conduction velocities. Front Physiol 9:1344CrossRefGoogle Scholar
- Sansour C (2008) On the physical assumptions underlying the volumetric-isochoric split and the case of anisotropy. Eur J Mech A Solids 27(1):28–39MathSciNetzbMATHCrossRefGoogle Scholar
- Stefanadis C, Dernellis J, Stratos C, Tsiamis E, Tsioufis C, Toutouzas K, Vlachopoulos C, Pitsavos C, Toutouzas P (1998) Assessment of left atrial pressure-area relation in humans by means of retrograde left atrial catheterization and echocardiographic automatic boundary detection: Effects of dobutamine. J Am Coll Cardiol 31(2):426–436CrossRefGoogle Scholar
- Stojanovska J, Cronin P, Gross BH, Kazerooni EA, Tsodikov A, Frank L, Oral H (2014) Left atrial function and maximum volume as determined by mdct are independently associated with atrial fibrillation. Acad Radiol 21(9):1162–1171CrossRefGoogle Scholar
- Thomas SM, Chan YT (1989) A simple approach for the estimation of circular arc center and its radius. Comput Vis Gr Image Process 45(3):362–370CrossRefGoogle Scholar
- Valentinuzzi ME, Kohen AK (2011) Laplace’s law: what it is about, where it comes from, and how it is often applied in physiology. IEEE Pulse 2(4):74–84CrossRefGoogle Scholar
- Vigmond EJ, Hughes M, Plank G, Leon LJ (2003) Computational tools for modeling electrical activity in cardiac tissue. J Electrocardiol 36:69–74CrossRefGoogle Scholar
- Vigmond EJ, Weber dos Santos R, Prassl AJ, Deo M, Plank G (2008) Solvers for the cardiac bidomain equations. Prog Biophys Mol Biol 96(1–3):3–18CrossRefGoogle Scholar
- Whitaker J, Rajani R, Chubb H, Gabrawi M, Varela M, Wright M, Niederer S, O’Neill MD (2016) The role of myocardial wall thickness in atrial arrhythmogenesis. EP Europace 18(12):1758–1772Google Scholar
- Widgren V, Dencker M, Juhlin T, Platonov P, Willenheimer R (2012) Aortic stenosis and mitral regurgitation as predictors of atrial fibrillation during 11 years of follow-up. BMC Cardiovasc Disord 12(1):92CrossRefGoogle Scholar
- Woods RH (1892) A few applications of a physical theorem to membranes in the human body in a state of tension. Trans R Acad Med Irel 10(1):417CrossRefGoogle Scholar
- Yamazaki M, Mironov S, Taravant C, Brec J, Vaquero LM, Bandaru K, Avula UMR, Honjo H, Kodama I, Berenfeld O, Kalifa J (2012) Heterogeneous atrial wall thickness and stretch promote scroll waves anchoring during atrial fibrillation. Cardiovasc Res 94(1):48–57CrossRefGoogle Scholar
- Yin F (1981) Ventricular wall stress. Circ Res 49(4):829–842CrossRefGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.