Some controversies in endovenous laser ablation of varicose veins addressed by optical–thermal mathematical modeling

Abstract

Minimally invasive treatment of varicose veins by endovenous laser ablation (EVLA) becomes more and more popular. However, despite significant research efforts performed during the last years, there is still a lack of agreement regarding EVLA mechanisms and therapeutic strategies. The aim of this article is to address some of these controversies by utilizing optical–thermal mathematical modeling. Our model combines Mordon's light absorption-based optical–thermal model with the thermal consequences of the thin carbonized blood layer on the laser fiber tip that is heated up to temperatures of around 1,000 °C due to the absorption of about 45 % of the laser light. Computations were made in MATLAB. Laser wavelengths included were 810, 840, 940, 980, 1,064, 1,320, 1,470, and 1,950 nm. We addressed (a) the effect of direct light absorption by the vein wall on temperature behavior, comparing computations by using normal and zero wall absorption; (b) the prediction of the influence of wavelength on the temperature behavior; (c) the effect of the hot carbonized blood layer surrounding the fiber tip on temperature behavior, comparing wall temperatures from using a hot fiber tip and one kept at room temperature; (d) the effect of blood emptying the vein, simulated by reducing the inside vein diameter from 3 down to 0.8 mm; (e) the contribution of absorbed light energy to the increase in total energy at the inner vein wall in the time period where the highest inner wall temperature was reached; (f) the effect of laser power and pullback velocity on wall temperature of a 2-mm inner diameter vein, at a power/velocity ratio of 30 J/cm at 1,470 nm; (g) a comparison of model outcomes and clinical findings of EVLA procedures at 810 nm, 11 W, and 1.25 mm/s, and 1,470 nm, 6 W, and 1 mm/s, respectively. Interestingly, our model predicts that the dominating mechanism for heating up the vein wall is not direct absorption of the laser light by the vein wall but, rather, heat flow to the vein wall and its subsequent temperature increase from two independent heat sources. The first is the exceedingly hot carbonized layer covering the fiber tip; the second is the hot blood surrounding the fiber tip, heated up by direct absorption of the laser light. Both mechanisms are about equally effective for all laser wavelengths. Therefore, our model concurs the finding of Vuylsteke and Mordon (Ann Vasc Surg 26:424–433, 2012) of more circumferential vein wall injury in veins (nearly) devoid of blood, but it does not support their proposed explanation of direct light absorption by the vein wall. Furthermore, EVLA appears to be a more efficient therapy by the combination of higher laser power and faster pullback velocity than by the inverse combination. Our findings suggest that 1,470 nm achieves the highest EVLA efficacy compared to the shorter wavelengths at all vein diameters considered. However, 1,950 nm of EVLA is more efficacious than 1,470 nm albeit only at very small inner vein diameters (smaller than about 1 mm, i.e., veins quite devoid of blood). Our model confirms the efficacy of both clinical procedures at 810 and 1,470 nm. In conclusion, our model simulations suggest that direct light absorption by the vein wall is relatively unimportant, despite being the supposed mechanism of action of EVLA that drove the introduction of new lasers with different wavelengths. Consequently, the presumed advantage of wavelengths targeting water rather than hemoglobin is flawed. Finally, the model predicts that EVLA therapy may be optimized by using 1,470 nm of laser light, emptying of the vein before treatment, and combining a higher laser power with a greater fiber tip pullback velocity.

Appendices

Appendix 1

The mathematical formulation of the heat conduction governed by Fourier's law supplemented with appropriate boundary and initial conditions is given by the following:

$$ \left\{\begin{array}{cc}\hfill \rho c\frac{\partial T}{\partial t}=\nabla \cdot \left(k\nabla T\right)+q\left(r,z,t\right)\kern0.6em \mathrm{at}\left(r,z\right)\in \varOmega, t\ge 0,\hfill & \hfill (1)\hfill \\ {}\hfill q=\frac{0.55\cdot {P}_{\mu_a}}{4\pi {D}_{\mathrm{dif}}\sqrt{\left({r}^2+{\left(z-{z}_0-\mathrm{v}\cdot t\right)}^2\right)}} \exp \left(-{\mu}_{\mathrm{eff}}\sqrt{\left({r}^2+{\left(z-{z}_0-\mathrm{v}\cdot t\right)}^2\right)}\right),\hfill & \hfill (2)\hfill \\ {}\hfill q\left(z,0\le r\le {r}_{\mathrm{fiber}}\right)={q}_{\mathrm{tip}}=\frac{0.45\cdot P}{V_{\mathrm{tip}}},\hfill & \hfill (3)\hfill \\ {}\hfill T\left({r}_{\operatorname{int}}^{-},z,t\right)=T\left({r}_{\operatorname{int}}^{+},z,t\right)\kern0.6em \mathrm{if}\;0\le z\le L,t\ge 0,\hfill & \hfill (4)\hfill \\ {}\hfill \left(k\frac{\partial T}{\partial r}\right)\left({r}_{\operatorname{int}}^{-},z,t\right)=\left(k\frac{\partial T}{\partial r}\right)\left({r}_{\operatorname{int}}^{+},z,t\right)\kern0.6em \mathrm{if}\;0\le z\le L,\kern0.6em t\ge 0\hfill & \hfill (5)\hfill \\ {}\hfill \frac{\partial T}{\partial t}\left(r,z,t\right)=0\kern0.6em \mathrm{if}\;r=0R,0\le z\le L,\kern0.6em t\ge 0,\hfill & \hfill (6)\hfill \\ {}\hfill \frac{\partial T}{\partial z}\left(r,z,t\right)=0\kern0.6em \mathrm{if}\;0\le r\le R,z=0L,\kern0.6em t\ge 0,\hfill & \hfill (7)\hfill \\ {}\hfill T\left(r,z,0\right)={T}_{\mathrm{initial}}=293\left[K\right]\kern0.6em \mathrm{at}\left(r,z\right)\in \varOmega \hfill & \hfill (8)\hfill \end{array}\right. $$

with q the volumetric heat generation given by the product of absorption coefficient and fluence rate (watts per cubic meter) [2]; μ _a the absorption coefficient (1/m); μ _s the scattering coefficient (1/m) and g the average cosine of the scattering angle (the anisotropy factor); r _fiber the fiber radius (meters); and V _tip the volume of the carbonized layer (cubic meters), which thickness is about 0.1 mm and does not change during the computation. We assumed that 45 % of the light propagating towards the fiber tip is absorbed by the carbonized layer [3].

The rectangular computational domain is described by Ω = {(r,z) R²|, 0 ≤ r ≤ R, 0 ≤ z ≤ L}, where R is the radial size of the computational domain (meters), and L the axial size of the computational domain (meters).

The optical parameters such as reduced scattering coefficient, μ ^' _s , the optical diffusion constant, D _dif, the effective attenuation coefficient, μ _eff, are defined by the following:

$$ {\mu}_s^{\hbox{'}}={\mu}_s\left(1-g\right) $$

(9)

$$ {D}_{\mathrm{dif}}=\frac{1}{3\left({\mu}_s^{\hbox{'}}+{\mu}_a\right)} $$

(10)

$$ {\mu}_{\mathrm{eff}}=\sqrt{3{\mu}_a\left({\mu}_s^{\hbox{'}}+{\mu}_a\right)} $$

(11)

Heat transfer due to boiling bubbles is not explicitly simulated, but it is approximately taken into account by increasing the coefficient of heat conductivity by a factor of 200 in the region where temperature exceeds the threshold of 95 °C. This approach limits the maximum temperatures of the blood to about 100 °C. This procedure assures energy conservation.

This model is discussed in detail elsewhere [13].

Appendix 2

The amount of volumetric light energy absorbed at the inner vein wall at z = 40 mm and r = r _i, in the time period between t = 0 and t = t _end, where t _end is the time at which the inner wall temperature reaches its maximum value, E _light(r _i,t _end) , is given by the following:

$$ {E}_{\mathrm{light}}\left({r}_i,{t}_{\mathrm{end}}\right)=\underset{o}{\overset{t_{\mathrm{end}}}{{\displaystyle \int }}}{\mu}_a\left({r}_i\right){\phi}^{*}\left({r}_i,t\right) dt\kern0.5em \mathrm{where}\kern0.5em {\phi}^{*}\left(r,t\right)=\kern0.5em \phi \left(r,t\right)\mathrm{if}\kern0.5em \frac{ dT}{ dt}\kern0.5em \ge \kern0.5em 0;=0\kern0.5em \operatorname{if}\kern0.5em \frac{ dT}{ dt}<0 $$

(12)

μ _a (r _i) is the light absorption coefficient and ϕ(r _i,t) the fluence rate (watts per square meter) of the vein wall at point r _i and time t and product μ _a (r _i)ϕ(r _i,t) is the absorbed power in an infinitesimally small volume at r _i, t. In the same period of time, from t = 0 to t = t _end, the total volumetric increase in energy, E _total(r _i,t _end), is given by

$$ {E}_{\mathrm{total}}\left({r}_{\mathrm{i}},{t}_{\mathrm{end}}\right)=\underset{0}{\overset{t_{\mathrm{end}}}{{\displaystyle \int }}}\rho {c}_{\mathrm{p}}{\left.\frac{ dT\left({r}_{\mathrm{i}},t\right)}{ dt}\right|}^{+} dt=\rho {c}_{\mathrm{p}}\left[T\left({r}_{\mathrm{i}},{t}_{\mathrm{end}}\right)-T\left({r}_{\mathrm{i}},t=0\right)\right] $$

(13)

${\left.\frac{ dT}{ dt}\right|}^{+}$ indicates the positive value of the derivative, which by definition is zero if the time rate of change of T is negative. Ratio E _light/E _total has been computed for all wavelengths and vein diameters used.