Numerical modeling of residual type B aortic dissection: longitudinal analysis of favorable and unfavorable evolution

Abstract

Residual type B aortic dissection was numerically investigated to highlight the contribution of biomechanical parameters to the pathology’s evolution. Patient-specific geometries from cases involving both favorable and unfavorable evolution were modeled to assess their hemodynamic features. This original approach was supported by a longitudinal study confirming the association between morphological changes, hemodynamic features, adverse clinical outcomes, and CT-angioscan observations on the same patient. Comparing one patient with unfavorable evolution with one with favorable one, we identify potential biomechanical indicators predictive of unfavorable evolution: (i) a patent false lumen with a flow rate above 50% of inlet flow rate; (ii) high wall shear stress above 18 Pa at entry tears, and above 10 Pa at some regions of the false lumen wall; (iii) low time-averaged wall shear stress in distal false lumen below 0.5 Pa; (iv) vortical structure dynamics. Although these comparisons could only be conducted on 2 patients and need to be confirmed by a larger number of cases, our findings point to these hemodynamic markers as possible candidates for early evaluation of the pathology’s evolution towards an unfavorable scenario.

Appendix: Windkessel model implementation

This reduced model describes hydraulic systems through an electrical mechanical analogy where pressure (P) and flow rate (Q) are respectively analogous to voltage and current intensity. This led to the following equation:

$$ P(t)+R_{2}C\frac{\partial P(t)}{\partial t}=(R_{1}+R_{2})Q(t)+R_{1}R_{2}C\frac{\partial Q(t)}{\partial t} $$

(1)

with R₁ and R₂ being the proximal and distal resistance, respectively, and C being the capacitance.

The instantaneous pressure at each outlet was then expressed by approximating the derivative terms using the Euler backward scheme, as follows:

$$ P_{t} = \left( \frac{1}{1+\frac{R_{2} C}{dt}}\right)\left[\frac{R_{2}C}{dt} P_{t-1} + (R_{1}+R_{2} + \frac{R_{1} R_{2} C}{dt})Q_{t} - \frac{R_{1}R_{2}C}{dt} Q_{t-1}\right] $$

(2)

The model parameters R₁, R₂ and C for each outlet were identified through an optimization procedure using a genetic algorithm and Matlab software. The optimization problem consisted in finding ${R_{1}^{i}}$, ${R_{2}^{i}}$, and Cⁱ for each outlet i that minimize the mismatch between the computed (comp) and the targeted (obj) pressure values. This mismatch M_i is termed the objective function.

$$ M_{i}=\sqrt{(P_{max,comp}-P_{max,obj})^{2}+(P_{min,comp}-P_{min,obj})^{2}+(P_{mean,comp}-P_{mean,obj})^{2}} $$

(3)

Pressure values corresponding to the hypotensive therapy prescribed to the patient were targeted to obtain physiological pressure values P ∈ [80, 120] mmHg with P_max,obj = 120 mmHg, P_min,obj = 80 mmHg and P_mean,obj = 100 mmHg.

Two constraints were added

• 1. Pressure curves must be periodic: P_t= 0 = P_t=T

• 2. Equation 1 must be verified in the frequency domain when $\omega \rightarrow 0$ so $\frac {\bar {P}}{\bar {Q}}=R_{1}+R_{2}$.

The global optimization procedure was inspired by the work of [4]. The first estimation

of flow rate (shapes and values) was obtained through an independent CFD simulation with zero-pressure at each outlet. This has the merit of providing flow rate curves compatible with the morphology of the patients. These curves were then used for a first estimation of the Windkessel parameters using optimization procedure implemented in Matlab software based on genetic algorithm. The targets (pressure values corresponding to anti-hypertensive treatment) were approximated within a tolerance of 10^− 16, the optimization result was also evaluated through the Pearson correlation Coefficient r₂ which always exceeded 0.99 for each optimization and each outlet. A new simulation was then carried out with Windkessel models defined at each outlet with the estimated parameters. This helped update the flow rate curves that were used again in the optimization procedure (new CFD simulation plus genetic algorithm) and so on until the convergence condition was met. At the end of a step, the objective function (Eq. 3) was assessed for each outlet using the computed pressure values. Their sum was then compared to its value from the previous iteration. If it decreased (meaning that there was still room for minimizing the difference between the computed values and the targeted values) a new optimization iteration was made, else we stopped the procedure and the last parameters estimated were used for the numerical simulations.This procedure is summed up in Fig. 9.

Fig. 9

Windkessel implementation procedure

When optimization procedure was ended, that was after several iterations and several CFD simulations, new simulations were carried out, for the current study, with the set of optimal parameters for the Windkessel models at each outlet. Table 2 summarizes the final parameter values of Windkessel model.

Table 2 Estimated Windkessel parameters

Table 3 shows the percentage of flow rate at outlets in relation to inlet one obtained, using Windkessel models at the outlets during simulations. The obtained values are in the range of physiological ones in vivo recorded [6].

Table 3 Percentage of flow rate at outlets in relation to inlet one