Mathematical modelling of thermal and electrical processes during electrosurgical resection of colorectal polyps

Abstract

This paper is devoted to mathematical modelling of thermal processes in the polyp-colon system during electrosurgical polypectomy. This medical surgery (colonoscopy) is used to remove abnormal growths from the colon (the large intestine) in order to reduce the risk of the colon cancer development.In the electrosurgical polypectomy procedure, a polypectomy snare is passed over the polyp and tightened around the stalk of the polyp. Electric current is then passed for a few seconds through the snare loop to cut off the polyp stalk from the colon wall. The differential equations describing the processes occurring during polypectomy have been proposed. Also, examples of numerical calculations have been presented. Simulation results depending on the geometry (size) of the polyp can be useful for the endoscopist to choice the optimal time and parametrics of the electric current flow by tissues during the endoscopy procedure.

Introduction

Colon polyps are benign, slow-growing tumors arising in the large intestine. Colorectal cancer is an increasing medical problem, as one of the most common cancers with a high mortality rate. Colonoscopic polypectomy is a method for the prevention of colorectal cancer and was performed for the first time in 1969 (Facciorusso & Muscatiello, 2018). All polyps detected by colonoscopic examination can be removed immediately or biopsied. Proper removal of polyps requires knowledge of the characteristics of endoscopic accessories and instruments, the size of the colorectal polyp and the skills of an experienced surgeon. Most polyps can be treated with various polypectomy methods (Engin, 2015; Gordon & Nivatvongs, 2007; Haycock, Cohen, Saunders, Cotton & Williams, 2014). One of the popular methods of examination is the technique of hot snare polypectomy (the electrosurgical polypectomy). During the electrosurgical polypectomy, a polypectomy snare is passed over the polyp and tightened around the stalk of the polyp. An electric current is then passed for a short period of time (a few seconds) through the snare loop to cut off the polyp stalk from the colon wall while providing electrocautery at the same time in order to close the wound and stop the bleeding.

Colonoscopic polypectomy snares are available in a variety of sizes, shapes and materials. These snares (made from braided stainless-steel wire - typically 0.3–0.5 mm in diameter) consist of a continuous wire loop (typically 20–25 mm in diameter) placed within a flexible polymer sheath which is passed through the accessory channel of the colonoscope (a typical channel size is of 3–5 mm and about 2.5 m in length). The wire and sheath are attached to the moving plastic handle at the end of the device controlled by the endoscopist. The handle controls the opening and closing of the wire loop during medical treatment. The snare wire couples to an electrical connector within the handle and is connected to an active cord to an electrosurgical unit. In this work, we considered snares that are designed to be used with monopolar current (the current flows from the snare to a distant return electrode placed on i.e. the back of patient). The scheme of this procedure is illustrated in Fig. 1.

The current flow in tissues causes the generation of local thermal energy and induces a temperature rise in tissue (for cutting and coagulation) (Gupta, Singh, & Rai, 2010; Majchrzak, Dziatkiewicz, & Paruch, 2008; Martínez et al., 2016). Thermal coagulation is the conversion of colloidal systems from a sol to gel state. In the case of biological tissues, the thermal coagulation occurs if the temperature increases to approximately 60 °C, and then the structure of the tissue cell changes. It is important during polypectomy treatment that the temperature rise should only occur within the target tissue (i.e. the polyp base) and any thermal damage to adjacent tissues must be avoided. These conditions are difficult to achieve because hot tissue causes heating within the adjacent tissue. The polypectomy is an invasive procedure and can lead to some side effects, such as a perforation.

The research presented in this work is related to the computation of temperature distribution and thermal damage of tissues in the polyp-colon system. The aim of this paper is the mathematical modelling of these processes in the considered system. Different times of electric current flow during electrosurgical polypectomy are analyzed. The geometry of the considered system is treated as an axially-symmetrical domain.

The mathematical model of heat transfer in the vital tissues is based on the Pennes equations which contain terms (heat sources) associated with the blood perfusion and metabolism. In literature (Arkin, Xu & Holmes, 1994; Charny, 1992; Jasinski, Majchrzak & Turchan, 2016; Mochnacki & Majchrzak, 2017; Pennes, 1948; Wissler, 1998) one can find more details about bio-heat transfer models. The wire loop is one of the electrodes, the second electrode is placed at the skin of patient (usually on the back), but we assume that the ground potential on the other surface of colon is taken into account. During the few seconds of electric current flow through tissues between two electrodes, the stationary electric potential distribution in tissue domains should be determined based on the adequate boundary conditions. The electric field causes the Joule effect. The heat source generation depends on the value of gradient of the electric potential and next this value is introduced as additional term into the Pennes equation. If the electric current stops flowing then the process of heat generation in tissues is finished, the heat is still transferred to the adjacent tissues which can be caused further thermal destruction of tissues. In order to determine the degree of the tissue damage, the values of the Arrhenius integral are used (Jasinski, 2018; Korczak & Jasinski, 2019; Paruch, 2018; Qin, Balasubramanian, Wolkers, Pearce & Bischof, 2014).

The analytical solution of equations of the proposed mathematical models is rather impossible. In this order the numerical methods should be used. In this work, we use the Control Volume Method (CVM), and the considered polyp-colon domain is discretized using the Voronoi tessellation (Ciesielski & Mochnacki, 2014; Ciesielski, Mochnacki & Siedlecki, 2016).