Computational fluid dynamics modelling of left valvular heart diseases during atrial fibrillation

Cardiovascular model, variables and parameters definition

The cardiovascular model used here, first proposed by Korakianitis & Shi (2006) for healthy and diseased valves, has then been validated over more than 30 clinical measurements regarding AF (Scarsoglio et al., 2014). It has been recently adopted to evaluate, from a computational point of view, the impact of higher HR during AF at rest (Anselmino et al., 2015), as well as the role of AF in the fluid dynamics of healthy heart valves (Scarsoglio et al., 2016).

The model relies on a lumped parameterization of the four heart chambers, together with the systemic and pulmonary circulation. Cardiac and circulatory regions are described using electrical terminology, such as compliance (accounting for the elastic properties), resistance (simulating the viscous effects) and inductance (approximating inertial terms). The resulting ordinary differential system is expressed in terms of pressure, P [mmHg], volume, V [ml], flow rate, Q [ml/s], and valve opening angle, ϑ [°]. Each of the four heart chambers is active and governed by an equation for mass conservation (considering the volume variation), a constitutive equation (for the pressure-volume relation through a time-varying elastance, E), an orifice model equation (relating pressure and flow rate), and an equation for the valve motion mechanisms. Both systemic and pulmonary circuits are partitioned into four arterial and one venous sections. Each circulatory compartment is ruled by an equation for mass conservation (in terms of pressure variation), an equation of motion (flow rate variation) and a constitutive linear equation between pressure and volume. The elastic vessel properties are in general dependent on the pressure level. However, a linear relation between pressures and volumes can be assumed in the range of physiological values (Ottesen, Olufsen & Larsen, 2004). The complete system was numerically solved through an adaptive multistep scheme implemented in Matlab. Since the cardiovascular dynamics present stiff features, i.e. rapid and abrupt variations in time, a stiff solver implemented in the ode15s Matlab function was adopted (all the modeling and computational details are given in Scarsoglio et al. (2014)).

We focused here on the left heart dynamics by means of pressure (P) and volume (V) variables, also evaluating end-diastolic (ed) and end-systolic (es) values: left atrial pressure and volume (P_la and V_la, respectively), left ventricle pressure (P_lv) and volume (V_lv, V_lved, V_lves), systemic arterial pressure (P_sas, P_sas,syst, P_sas,dias), pulmonary arterial (P_pas) and venous (P_pvn) pressures. End-systole is the instant defined by the closure of the aortic valve, while end-diastole corresponds to the closure of the mitral valve. We introduce RR [s] as the temporal range between two consecutive heart beats, while HR [bpm] is the heart rate, i.e., the number of heart beats per minute. Performance indexes are computed as well:

• stroke volume, SV = V_lved − V_lves [ml];

• ejection fraction, EF = SV/V_lved × 100 [%];

• cardiac output, CO = (FV_ao + RV_ao) × HR [l/min], where FV [ml/beat] and RV [ml/beat] are the forward and regurgitant volumes, respectively. The forward volume (1) $FV={\displaystyle \underset{RR}{\int }{Q}^{+}}(t)dt,$ is the volume of blood per beat flowing forward through the valve (the symbol Q⁺ indicates the positive flow rate outgoing from the valve), while the regurgitant volume (2) $RV={\displaystyle \underset{RR}{\int }{Q}^{-}}\left(t\right)dt,$ is the volume of blood per beat which regurgitates backward through the valve, with the symbol Q⁻ representing the negative flow rate going backward through the valve (RV < 0 by definition). As FV and RV are here computed for the aortic valve, FV_ao + RV_ao is the net volume per beat [ml/beat] across the aortic valve (Scarsoglio et al., 2016).

Valve dynamics

The valve dynamics introduced by Korakianitis & Shi (2006) include several mechanisms, such as the pressure difference across the valve, the dynamic motion effect of the blood acting on the valve leaflet, the frictional effects from neighboring tissue resistance and the action of the vortex downstream of the valve. Only the shear stress on the leaflet, considered negligible, has not been taken into account. The described fluid dynamics, based on 2D or 3D CFD studies on local flow conditions, was modelled by means of a lumped parameterization, which leads to a second-order differential equation for each opening angle, ϑ. Even though the adopted model for the valve motion is lumped, the equation for the dynamics of the opening angle, ϑ, accounts for different physical mechanisms. Thus, global variations are modeled and in great part captured through the temporal variations of the valve area, A, and the opening angle, ϑ. Fine details of the local dynamics–which are mostly influenced by the shape of the valve area–are not caught, thereby falling outside the goal of the present work. The angle ϑ reaches values in the range [ϑ_min, ϑ_max], where in healthy conditions ϑ_min = ϑ_min,h = 0° (closed valve) and ϑ_max = ϑ_max,h = 75° (fully open valve).

We related the valve area, A [cm²], to the opening angle, ϑ, by means of the following law (Korakianitis & Shi, 2006): (3) $A=\frac{{(1-\cos \vartheta )}^2}{{(1-\cos {\vartheta }_{max,h})}^2}{A}_h,$ where A_h is the reference valve area value for an healthy adult. Only left-sided valvulopathies were investigated here, thus we set A_h = 5 cm² for the mitral valve and A_h = 4 cm² for the aortic valve (Baumgartner et al., 2009; Lancellotti et al., 2010a; Lancellotti et al., 2010b). In normal conditions, A varies between 0 and A_h, with a quadratic dependence on ϑ, as reported in Fig. 1 for the mitral (panel A) and aortic (panel B) valves.

Grading left-sided valve disease severity

For each of the four left valvulopathies (AS, aortic stenosis, MS, mitral stenosis, AR, aortic regurgitation, MR, mitral regurgitation), we considered three valve area values, corresponding to different grades of severity (Baumgartner et al., 2009; Lancellotti et al., 2010a; Lancellotti et al., 2010b):

• AS: A_s [cm²] = 2 (mild), 1.25 (moderate), 0.90 (severe);

• MS: A_s [cm²] = 2 (mild), 1.25 (moderate), 0.90 (severe);

• AR: A_r [cm²] = 0.07 (mild), 0.20 (moderate), 0.33 (severe);

• MR: A_r [cm²] = 0.13 (mild), 0.30 (moderate), 0.44 (severe).

Observing the dependence between A and ϑ introduced through Eq. (3), we expect lower ϑ_max values for increasing stenosis severity, and higher ϑ_min values for growing regurgitation grades.

For stenosis conditions, to find the maximum opening angle (ϑ_max,s) corresponding to the stenotic area, A_s, we exploited Eq. (3) for each grade of severity as follows: (4) $A_s=\frac{{(1-\cos {\vartheta }_{max,s})}^2}{{(1-\cos {\vartheta }_{max,h})}^2}{A}_h.$

In regurgitant conditions, the minimum opening angle (ϑ_min,r) corresponding to the regurgitant orifice area, A_r, was found reformulating Eq. (3) as reported below: (5) $A_r=\frac{{(1-\cos {\vartheta }_{min,r})}^2}{{(1-\cos {\vartheta }_{max,h})}^2}{A}_h.$

From Eqs. (4) and (5) we were able to easily extract the opening angles ϑ_max,s and ϑ_min,r related to each grade of stenosis and regurgitation, respectively. A scheme summarizing the ϑ_min and ϑ_max values used in the model for the healthy and the twelve valve diseased configurations is provided in Table S1. Both stenosis and regurgitation were modelled in a simplified manner through geometrical variations of the opening angles ϑ, accounting for the mechanical dysfunctions of the valve opening/closure failure. Because of the lack of clear data, during stenosis the increased stiffness of the leaflets is neglected, thus these latter were assumed as in healthy conditions. Altered valvular functions–due to valve prolapse, rheumatic disorders, congenital heart defects or endocarditis, and usually associated with regurgitation–were also not taken into account.

The proposed algorithm was used to simulate a specific grade of valvulopathy, once the corresponding reference valve area value is given. To double check the validity of this procedure, besides the hemodynamic parameters introduced at the beginning of this section, we also evaluated as post-processing parameters the regurgitant volumes, RV [ml/beat] (for regurgitations), and the mean pressure gradients, MPG [mmHg] (for stenosis), to evaluate the indexes recommended by current clinical guidelines to grade regurgitation and stenosis severity (Baumgartner et al., 2009; Lancellotti et al., 2010a; Lancellotti et al., 2010b). Recall that RV for both left valves was calculated as defined in Eq. (2). For MPG we used the velocity across the valve, v = Q/A [m/s], and the Bernoulli equation, defining the transvalvular pressure gradient, ΔP = 4v² [mmHg]. The mean pressure gradient, MPG, was calculated by averaging the instantaneous gradients, ΔP, over the systolic phase (i.e., when there is forward flow Q⁺) (Baumgartner et al., 2009). Mean pressure gradient, MPG, for stenosis and regurgitant volume, RV (as absolute values), for regurgitation, are reported in Table S2, as averaged over 5,000 cardiac periods.

Simulations

To mimic AF conditions, both atria were assumed to be passive, i.e. atrial elastances were kept constant. A condition of lone AF was first simulated as reference baseline. Then, twelve simulations reproducing AF together with a specific grade of left valvulopathy were run. A ventricular contractile dysfunction has been described in both stenosis and regurgitation (Maganti et al., 2010), though without definitive results (Shikano et al., 2003). Given the lack of clear data (Scarsoglio et al., 2014) during heart valve diseases in AF, the reduced left ventricular inotropy was not modelled here and a normal left ventricular contractility was assumed for all the configurations. For each simulation, the transient dynamics were exceeded after 20 periods (Scarsoglio et al., 2014). Afterwards, 5,000 cardiac cycles were computed and recorded to account for a period lasting about one hour. This choice allowed the statistical stationarity of the results to be achieved. For all the cardiovascular variables and hemodynamic parameters, mean and standard deviation values were calculated.

AF beating features were approximated extracting uncorrelated RR from an Exponentially Gaussian Modified distribution (mean μ = 0.67 s, standard deviation σ = 0.16 s, rate parameter γ = 8.47 Hz), which is unimodal and describes the majority of AF cases (Hennig et al., 2006; Scarsoglio et al., 2014). The twelve AF with left-valvular disease simulations present the same AF beating features of the lone AF case. The defective valve opening/closure was added by varying ϑ_max and ϑ_min values according to the criteria discussed in the previous Section.

Stenosis

During AS, data dispersion remained practically unvaried with respect to lone AF, with the only exception of P_lv, presenting more dispersion. An increased mean P_lv value is a consequence of the higher aortic resistance during AS and is necessary to guarantee an adequate CO. Moreover, volume time series (Figs. 2A and 2B) and probability density functions (Figs. 2C and 2D) preserved the same behavior and shape as observed during lone AF, thereby confirming the modest hemodynamic impact of AS already evidenced by data dispersion.

The scenario was different for MS. With respect to lone AF, dispersion of data decreased for atrial variables (P_la and V_la), P_pvn e P_pas, while performance indexes experienced more dispersion (SV, CO, EF). Atrial overload is detectable by the increased mean V_la and P_pvn values, as well as by the different shape assumed by the V_la time series and the P_pvn probability density function with respect to lone AF (Figs. 2A and 2C). Changes at ventricular level were less pronounced, but largely imputable to inefficient atrial ejection. This latter in turn reduced V_lved values, leading to an overall SV reduction. The cardiac efficiency, CO, was weakened as a result of the decreased mean net volume available to be ejected from ventricle to the aorta.

Regurgitation

Both aortic and mitral regurgitation similarly increased the mean atrial volume. However, MR induced the highest peak values (up to 110 ml) and substantially changed the temporal dynamics with respect to lone AF (Fig. 3A). The enlarged atrial volume led for both regurgitations to an increase of P_pvn, with an accentuated right tail for the probability density function of MR (Fig. 3C).

In case of AR, data dispersion decreased for atrial variables, P_pvn, P_pas, P_lv, EF, with respect to lone AF, while data were sparser for P_pas, CO, V_lv. The failed closure of the aortic valve during diastole caused substantial regurgitant flow from the aorta back to the ventricle. This regurgitation on the one hand promoted ventricular overfilling, with elevated V_lved values (Fig. 3B), which in turn partially inhibited the normal atrial emptying. On the other hand, the regurgitant flow reduced the net antegrade CO, into the aorta (Fig. 3D).

Comparing MR with respect to lone AF, data dispersion was lower for P_lv, P_pas, SV and EF, while it increased for atrial variables, P_pvn, V_lv, and CO. The defective closure of the mitral valve during systole resulted in regurgitant flow from ventricle towards the atrium, causing high V_la peaks and abnormally emptying of the ventricle after ejection (i.e., decrease of V_lves, Fig. 3B). As a consequence, the net forward CO, was reduced (Fig. 3D). At the end of systole, the atrium was overfilled and ejected a greater amount of blood into the ventricle during diastole, leading eventually to an increase of V_lved.

Comparative framework of valvular heart disease

Recall that dispersion of data is mainly produced by irregular beating. Changes in the dispersion of the results–with respect to lone AF–can be interpreted as the (more or less) pronounced ability of the valvulopathy to modify AF hemodynamics. From this point of view, AS had the least impact since dispersion remains basically unaltered, while both MR and AR acted to substantially vary the cardiovascular response.

In order to compare the relative effects of each valvular disease by grade, the percentage variation of every averaged hemodynamic variable compared to the control, lone AF simulation, was evaluated. Figure 4 shows the most significant percentage variations, involving atrial and upstream pulmonary venous return (A), ventricular dynamics (B and C), performance indexes (D and F), and systemic arterial pressure (E). In the pulmonary circulation, although mean pulmonary arterial pressure (P_pas) did not undergo substantial changes, mean pulmonary vein pressure (P_pvn) increased by 31.4, 27.7, and 23.2%, in case of severe MR, AR, and MS, respectively (Fig. 4A). Similarly, mean left atrial pressure (P_la), increased by 34.4, 30.7 and 25.2% in the cases of severe MR, AR and MS, respectively. In the left ventricle, an increase in mean left ventricular pressure (P_lv) was seen in severe AS (+9.0%), while there was a decrease in severe MS (−13.3%) and MR (−18.5%) (Fig. 4B); mean left ventricular volume (V_lv) increased due to severe AR (+28.8%) and MR (+7.4%), and decreased in case of severe MS (−18.7%) (Fig. 4C). Concomitantly, stroke volume (SV) showed an upsurge in severe AR (+102.7%) and MR (+88.0%), and a decrease due to severe MS (−12.9%) (Fig. 4D). Finally, mean systemic arterial pressure (P_sas) declined in severe AR (−24.1%), MR (−22.8%) and MS (−12.4%) (Fig. 4E), with an analogous decrease in CO in severe MR (−22.5%), AR (−20.5%) and MS (−13.8%) simulations (Fig. 4F).

Limitations

In addition to the previously stated lack of autonomic nervous system regulation, some other limitations of the present modelling study should be considered. First, AF conditions were set the same for all simulations in the attempt to quantify the “net impact” of the specific valve disease during the arrhythmia, regardless of other differential compensatory mechanisms that may, in fact, be present in clinical practice. Second, coronary circulation was not taken into account, since its peculiar features (e.g., diastolic flow) makes the modelling challenging; therefore, the effect of AF and different valve diseases on pressures and volumes in that circulation was not accounted for by the present model. Third, the model predicted hemodynamic effects of valvular disease during AF, without considering other pathological conditions, such as hypertension or heart failure, that could themselves affect cardiovascular variables. Moreover, linear relations are assumed for the pressure-volume constitutive equations in the vasculature, which can lead to an underestimation of diastolic pressures in severe stenosis conditions. In the end, AF beating features were limited to the unimodal distribution only, while multimodal RR distributions were not analyzed.