Nowadays endovenous laser coagulation (ELC) is the priority method of treatment of lower limb varicose vein disease (VVD). It is explained by less traumatic treatment, cosmetic effect and reduction in recovery time. However, the analysis of literary sources shows that ELC has a number of specific complications (phlebitis, hematomas over the coagulase vein, ecchymosis or dermatomelasma, pain syndrome of different intensity, recanalization of coagulase vein resulting in recurrent lower limb VVD). The aforementioned complications of ELC for varicose veins affect its efficiency and clinical course of the postoperative period. According to many authors, the results of the implementation of ELC for varicose veins directly depend on the density of radiation energy, speed of movement of laser light pipe , mode of endovenous laser obliteration and diameter of the vein that is exposed to laser radiation from inside [1, 2, 4-8]. In addition, series of experiments that were conducted in vitro and ex vivo showed that the influence of laser coagulation is realized through direct action of laser radiation, evaporating blood as well as through the impact of warmed-up working part of a laser having the temperature higher than 1000°С. The basic component is the direct action of laser radiation on the wall of the vein. The obtained thermal energy releases gradually into the environment (hypoderm) through the wall of the vein [6]. The inner vascular temperature depends on the amount of energy given to an action zone. The evenness of damage to the venous wall depends on the uniformity of drawing the light pipe and does not depend on the mode of radiation (impulsive or continuous). The noncontact venous wall perforation is impossible. The basic parameter that influences the degree of damage to the vein wall and possibility of carbonization is the amount of energy that is given in the vessel lumen in the process of laser obliteration (linear density of energy). The increase in linear density of energy results in carbonization that does not depend on the mode of radiation, emission type, wavelength [3]. It is very important to calculate the required linear density of laser radiation and speed of reverse traction of the light pipe in every particular case.

Thus, the technology of ELC for varicose veins has a great experience of usage but it is not perfect. Therefore, scientific research which is devoted to study of ELC for varicose veins has certain directions. They include further standardization of its technology as well as a choice of optimal wavelength. The solution of these questions will give us an opportunity to decrease the amount of specific complications that can occur during ELC for varicose veins.

** The objective **of the research was to decrease specific postoperative
complications after endovenous laser coagulation due to the introduction of its mathematical
design.

The main group included 168 patients being operated on during 2014-2015. ELCV was
performed
using the portable highly intensive semiconductor (diode) laser device “Lika-khirurh”
(Photonics Plus, Cherkasy, Ukraine) with the wavelength of 1470 nm and an output power
of
10-12.5 W. There were 64 men and 104 women at the age from 26 to 54 years. There were
operated on 59 patients with C_{II} lower limb VVD, 83 patients with C_{III}
lower limb VVD and 26 patients with C_{IV} lower limb VVD according to the CEAP
classification.

Prior to surgery, the size of the diameter of the vein in the separate areas was determined by ultrasound scan and the speed of reverse traction of the light pipe as well as the density of laser radiation were discretely changed according to the previous calculations of the mathematical formulas.

Thus, the process of heating can be described by the heat balance equation:

where:

W is the power of laser emitter, averaged for 1second of impulse action;

c is the heat capacity of blood;

m is the mass of blood which is heated for1second;

H is the heat transfer coefficient;

S is the surface area of the internal was of the vessel that is exposed to radiation for 1 second;

T1 and T2 are the temperature of normal tissue (36.60 С) and temperature to which it is necessary to warm blood and the wall of the vein.

The mass of blood which is heated for 1 second is determined by the dependence:

where:

ρ is the density of blood;

ν is the speed of moving irradiator;

d^{2} is the diameter of the vein;

d^{2}_{0} is the diameter of the light pipe ;

π is the coefficient pi.

The surface that is exposed to radiation for 1 second is determined by the expression:

When substituting the expressions (2) and (3) in the expression (1), we obtain the expression for radiation power in extended form:

Using the expression (4) the speed of light pipe traction depending on different parameters (construction of irradiator, diameter of the vein, required temperature of heating) is determined.

The analysis of the formula (5) shows that speed of traction is proportional to the power of laser irradiator, inversely proportional to the difference of temperatures and inversely depends on the diameter of the vein. For convenient interpretation of the results graphic dependences (Fig. 1, Fig. 2) being determined by the formulas (4, 5) are presented.

In this group we analyzed specific complications associated with ELC for varicose veins according to which it is possible to estimate safety of its implementation and occurrence of postoperative relapse of varicose vein disease. Due to the intraoperative ultrasonic control technical complications such as the position of laser light pipe in the popliteal or femoral vein were avoided.

Among specific complications of ELC for varicose veins we distinguished the following: ecchymosis or dermatomelasma in the early postoperative period, induration of skin over coagulase great or small saphenous vein, phlebitis, hematomas over coagulase great or small saphenous vein, pain syndrome, paresthesia of the tibia, as well as the postoperative relapses of varicose vein disease associated with insufficient vein coagulation.

The analogical complications and the presence of the postoperative relapse of varicose
vein
disease were analyzed in 146 patients; the speed of reverse traction of the light
pipe and
the density of laser radiation was established in accordance with the instruction
of a laser
device. There were 49 men and 97 women at the age from 21 to 58 years. There were
operated
on 41 patients with C_{II} lower limb VVD, 76 patients with C_{III} lower
limb VVD and 29 patients with C_{IV} lower limb VVD according to the CEAP
classification.

Endovenous laser coagulation for varicose veins was performed under general anaesthesia or spinal anaesthesia according to the protocol proposed by Chernukha LM, et al., which included crossectomy (281 patients), surgical treatment of tributaries of the great and small saphenous veins using minimal access, endovenous laser coagulation of the trunks of the subcutaneous veins, intersection and ligation of perforating veins using minimal access or subfascial dissection of the veins.

All surgical interventions concerning lower limb VVD aimed at the removal of vertical and horizontal reflux. In particular, ELC was combined with endoscopic subfascial dissection of the veins (26 patients). In 103 (36.7%) cases catheter sclerosing of collateral veins and mini-phlebectomy were performed in addition to surgical interventions.

33 patients underwent ELC without crossectomy; the end of the light pipe was 1.0- 1.5 cm from the saphena-femoral embouchement. In all cases ELC for varicose veins was performed under ultrasound control. The criteria for selecting patients for using ELC were patients with the great saphenous vein the diameter of which did not exceed 12 mm.

The analysis of complications in two groups of patients who underwent ELC for varicose veins (Table 1) showed that the complications such as paravasal hematomas over coagulase great and small saphenous veins occurred due to their damage during conducting paravasal tumescent infiltration anaesthesia using the Klein solution.

At the same time, due to the application of the mathematical calculation of linear density of laser radiation and the speed of reverse traction of the light pipe the number of complications such as induration of skin over coagulase great or small saphenous vein, phlebitis in the projection of the great or small saphenous vein, ecchymosis or dermatomelasma, pain syndrome reduced (Table 1). The positive moment in the application of this method is a reduction in the amount of postoperative relapses in patients who were examined 1 year after surgery.

The application of the mathematical model of calculation of linear density of laser radiation and the speed of reverse traction of the light pipe with its transfer to the graphic image allows us to improve the results of endovenous laser coagulation for varicose veins due to the reduction in specific complications.