Fluid dynamic assessment of tracheal flow in infants with congenital tracheal stenosis before and after surgery

Abstract

Tracheal flow in infants with congenital tracheal stenosis (CTS) was numerically investigated using subject-specific airway models before and after reconstructive surgery. We quantified tracheal flow based on airway resistance during inhalation, and compared it between controls and patients before and after surgery. The airway resistance in each subject was assessed using geometrical parameters of the trachea: the minimum cross-sectional area A_min, the minimum cross-sectional area normalized by the standard deviation of the cross-sectional area A_min/σ_A, the area ratio of the minimum and maximum cross-sectional area A_min/A_max, and ratio of the normalized standard deviation of cross-sectional area to the mean cross-sectional area σ_A/A_mean. Our numerical results demonstrated that such geometrical parameters could be used to assess the severity of CTS. Since subjects can be more clearly categorized as controls and most preoperative patients in terms of the airway resistance, a simulation using subject-specific airway models can lead us to a precise understanding of tracheal flow, and also provide knowledge about therapeutic decision. Our numerical results also demonstrated that significant surgical expansion of cross-sectional area did not help recover tracheal flow because of expansion loss. These results will be helpful not only when making therapeutic decisions about surgery but also when assessing quality of life in postoperative patients.

Appendix

1.1 Effects of length of straight tube and mesh resolution

We used various lengths of straight tubes (L_tube = 0, 10, 50, and 100 mm) at Re = 1000 with the normal real airway model of C1, and calculated the error of the maximum velocity at the start of the real airway, U_err, as:

$$ {U}_{err}=\frac{\mid {U}_{0,\mathit{\max}}-{U}_{0,\mathit{\max}}^{ref}\mid }{U_{0,\mathit{\max}}^{ref}} $$

(4)

where U_{0, max} is the maximum velocity at the start of the real airway for different L_tube (= 0, 10, and 50 mm) and ${U}_{0,\mathit{\max}}^{ref}$ is the reference with the maximum straight tube length L_tube = 100 mm. The results of U_err, presented in Fig. 8a, show that there was a less than 2% difference for L_tube = 50 mm and L_tube = 100 mm. Hence, we generally used 50-mm straight tubes in simulation, except for the highest Re (= 2000), where 100-mm straight tubes were used.

Fig. 8

a The error of maximum velocity at the start of the real airway ${U}_{err}=\mid {U}_{0,\mathit{\max}}-{U}_{0,\mathit{\max}}^{ref}\mid /{U}_{0,\mathit{\max}}^{ref}$ at Re = 1000, where U_0,max is the maximum velocity for different L_tube (= 0, 10, and 50 mm), and ${U}_{0,\mathit{\max}}^{ref}$ is the reference with a maximum tube length L_tube = 100 mm. bU_err is also calculated with the results of different mesh resolutions Δx, where ${U}_{0,\mathit{\max}}^{ref}$ is the reference with the finest mesh resolution Δx = 0.125 mm with L_tube = 50 mm. These results were calculated with the normal real airway model of C1

To check the effect of the computational mesh resolution Δx, we performed simulations and calculated U_err for different values of Δx (0.125, 0.25, 0.5, and 1 mm). The results of U_err for different Δx are shown in Fig. 8b, where the reference is the result of the finest mesh resolution Δx = 0.125 mm. Since there was less than 0.4% difference between Δx = 0.25 and 0.125 mm, we defined the mesh size as Δx = 0.25 mm in our simulations.

1.2 Pressure gradient in subject-specific airway model

Assuming that the flow in an ideal straight tube is Poiseuille flow, the pressure gradient ΔP/ΔL can be expressed by:

$$ \frac{\varDelta P}{\varDelta L}=\frac{f}{D}\frac{1}{2}\rho {U}^2 $$

(5)

where f is the coefficient (= 64/Re), U is the mean velocity, and D is the tube diameter. Using D = 2(A/π)^1/2 and U = μRe/(ρD), the Eq. 5 is rewritten as follows:

$$ \frac{\varDelta P}{\varDelta L}=\frac{4{\pi}^{3/2}{\mu}^2\mathit{\operatorname{Re}}}{\rho }{A}^{-3/2} $$

(6)

The numerical result of ΔP/ΔL was calculated as follows:

$$ \frac{\varDelta P}{\varDelta L}=\frac{\varDelta {P}_1^{\mathrm{in}}}{\varDelta {L}_1}+\frac{1}{2}\left(\frac{\varDelta {P}_2^{\mathrm{out}}}{\varDelta {L}_2}+\frac{\varDelta {P}_3^{\mathrm{out}}}{\varDelta {L}_3}\right) $$

(7)

where ΔP₁ⁱⁿ is the pressure difference between the inlet and branch point, and ΔP₂^out and ΔP₃^out are the pressure difference between the branch point and each outlet. ΔL_i (i = 1–3) includes the length of the straight tubes, where ΔL₁ is the length of the trachea before the branch point, and ΔL₂ and ΔL₃ are the lengths of the bronchi after the branch point. Figure 9 shows a comparison of ΔP/ΔL at relatively low Re (= 100 and 500) for the numerical and analytical results given by Eq. 6. As expected, the numerical values for pressure difference increase as the cross-sectional area decreases, independent of subject group. The pressure gradient was minimized when the airway was assumed to be a straight tube, and hence the numerical results of ΔP/ΔL were always plotted above the analytical solutions of Eq. 6 Through these analyses, we confirmed that approximate pressure gradient in a realistic airway model can be estimated by assuming a Poiseuille flow at least for Re ≤ 500.

Fig. 9

Pressure gradient ΔP/ΔL as a function of mean cross-sectional area A_mean for each subject type at Re = 100 (a) and 500 (b). The solid line is determined by Eq. (6)