THE DISCRETE MODEL FOR THE SYSTEM OF THE MYOCARDIUM AND CORONARY VESSELS

Background. The numerical heat transfer model for a system of myocardium coronary vessels is considered. Objective. The goal is to develop a discrete model for the physical system of myocardium and coronary vessels that would make it possible to explore the process of hypoand hyperthermia with cardiopulmonary bypass. Methods. To solve the differential equation of heat conduction in the MSC Sinda thermal system the network method (TNM — Thermal Network Method) is used, in which system of heat equations is presented in the form of cellular-centered nodes and resistances between the nodes using the finite difference method. In constructing the model of myocardial in the MSC Sinda system the thermal contact between three-dimensional bodies is implemented — the myocardium, coronary arteries, a liquid cooling of heart. Results. Implementation of the model of heat exchange in the MSC Sinda system for infarction cooling process gives on the final process step in establishing the heat balance the temperature difference at the boundary between the myocardium and coronary vessels not more than 0,5 C. However, in the areas of the myocardium that are removed from the coronary vessels the temperature difference exceeds 1,0 C. The use of additional cooling for hearts allows for the cooling of myocardium with using of ice surface, that provides the unevenness reduction of the heart temperature during its cooling with cardiopulmonary bypass. This result allows exploring the dynamics of the process of hypoand hyperthermia with cardiopulmonary bypass. Conclusions. The discrete 3D-model of heat transfer in the layer structure of the myocardium and coronary vessels allows us to investigate the process of hypoand hyperthermia with cardiopulmonary bypass. The simulation results also make it possible to perform the analysis of the temperature distribution on the surface of the myocardium provided free convection of heat between the layers.


Introduction
The heart is a complex of pump-muscular system, the functions of which depend on the contractile properties of the myocardium material.The myocardium has a large functional reserve (especially the left ventricle), or the ability to maintain a stable pumping function and a high ability to adapt to stress.The most important question of qualitative information regarding coronary flow was not so easy to solve from the only visible side of the heart in open chest conditions of surgery.In the same time, modern devices and information technologies are giving new solutions, which can be useful during extracorporeal cardiopulmonary bypass (CPB).The existing methods of comparative analysis for thermograms of myocardium allows to obtain the thermal picture for open heart in ranges from 4 C up to 37 C in conditions of CPB in ranges 28 -37 C.
The aim was the expansion of available information about protection of the myocardium and the state of its vascular bed using heat transfer model based on the pericardial temperature propagation in circumstances of CPB.The heat transfer during extracorporeal cardiopulmonary bypass (CPB) is due to the heat exchange between the blood and the water in the heat exchanger device of cardiopulmonary bypass (DCB) and to the heat exchange between the blood and the body of the patient's in the circulatory system.The physical model system of myocardium-coronary vessels consists of 2Dlayer infarction and 3D-structures of the coronary vessels, which are located within the bulk layer.
The numerical model of heat transfer allows estimating the parameters of temperature propagation and temperature gradients at the surface of the myocardium at the time of registration of thermal images of the heart.To solve the differential equation of heat conduction in the MSC Sinda thermal system the network method (TNM -Thermal Network Method) is used.It allows to estimating the parameters of temperature propagation and temperature gradients at the surface of the myo-cardium at the time of registration of thermal images of the heart.

Problem statement
The purpose is to create the discrete model for the physical system of myocardium and coronary vessels which is realized on the basis of RCthermal network in the MSC Sinda system for the heat transfer model to 2D-layer myocardium and 3D-structure of the coronary vessels.

The physical model for the system of the myocardium and coronary vessels
The physical model system of myocardiumcoronary vessels consists of 2D-layer infarction and 3D-structures of the coronary vessels, which are located within the bulk layer.In general, differential equation of the heat transfer, that includes transient processes, respectively, convection, diffuse and source terms, has the form [1]: where p C -the specific heat capacity of the myo- cardium, J ; kg К   -the material density (myo- cardium), temperature change rate per unit volume due to heat irradiation; R qthe heat flux due div( ) , where x, y, zthe space coordinates.
Simultaneously with the heat transfer in the myocardium during the hypo-and hyperthermia of heart heat dissipation is also involved, which defines the heat exchange between the myocardium and the environment layer in contact with itthe structure of the coronary vessels.The heat irradiation in accordance with the law of Newton-Richman is proportional to the temperature gradient between the myocardium and coronary vessels: where  -the heat transfer coefficient, which is introduced into coronary vessels, K.
If the value of myocardial density is constant over the three spatial directions, the convective term of the heat equation has the form: To determine the temperature field in the MSC Sinda system the differential equation of thermal conductivity obtained using the generalized heat equation is used: div( ) , the rate of temperature change per unit myocardial volume; v qthe distribution density of the structures of the coronary vessels in the myocardium.
In a particular case, for areas of the myocardium which is depleted of the coronary vessels, the internal heat sources in the myocardium may not be available, that means 0.
v q  At the same time, for different degrees of non-uniformity of the temperature field, flow velocity vector must satisfy the law of conservation of mass of transferred heat, or the continuity equation, which has the form: Thus, in general the differential equation considering heat conductivity of the convective flow in the myocardium takes on form [2]: In tensor form in a Cartesian coordinate system, the equation will be written as As a result of the convective heat transfer between the moving blood in the coronary vessels and the myocardium surface the heat transfer is performed with an intensity which is characterized by the heat transfer coefficient: where p  -the coefficient of the myocardial thermal conductivity, W ; m К  lthe thickness of the myocardial wall, m; Nu -Nusselt number, which characterizes the similarity of the processes of heat transfer at the boundary between the myocardium and coronary vessels.
For laminar blood flow in the system of artificial circulatory the Nusselt number can be expressed as the equation:

T T T   
the temperature difference between the wall of the myocardium and blood in the coronary vessels, K.

The discrete form of the heat equation of myocardial
To solve the differential equation of heat conduction in the MSC Sinda thermal system the network method (TNM -Thermal Network Method) is used, in which system of heat equations is presented in the form of cellular-centered nodes and resistances between the nodes using the finite difference method (Fig. 1).
Application of the TNM to the heat equation yields the following discrete form of equation [3,4]: ,  ( ) where ( ) p i mC the junction capacitance at node i; Nthe total number of diffuse network nodes; the thermal radiation from the resistance between the nodes i and j; the capacity of linear conductor between nodes i and j; i qthe distributed of heat sources in the myocardium.
In the model the heat radiation from the resistance between the nodes i and j is a member which defines a heat transfer from the materialblood and coronary vessels.Discrete formula has the form: , , , where  -Stefan -Boltzmann constant; n i Sthe radiant heat irradiation area; , i j f the heat factor of the heat radiation, which can be considered the same for constant surgical field with cardiopulmonary bypass.
In the model the linear conductor capacitance between nodes i and j represents a member , i j C that defines the heat transfers to the materialthe myocardium and blood.Its discrete formula has the form: , , , , where , i j


the effective thermal conductivity between nodes i and j; n i Sthe area of heat conduction between nodes i and j; , i j hthe distance between the nodes i and j.
Since most of the heat conductors in a system of myocardial-blood cannot be considered as homogeneous, the calculation of their values .
To solve the heat equation in the MSC Sinda system the modified Dyufort -Frenkel scheme is used.According to this scheme, the heat conduction equation has the form [5]: where Nthe total number of diffusion nodes; , i jthe nodes of thermal RC-network; 1  the previous time of calculation; 1   -next (cur- rent) one time of calculation; i qthe heat source node i; the overall conductivity between nodes i and j.

The myocardium model in the MSC Sinda system
In constructing the model of myocardial in the MSC Sinda system the thermal contact between three-dimensional bodies is implementedthe myocardium, coronary arteries, a liquid cooling of heart.The model of heat exchange for the local area of the myocardium for two conditions of heat transfer is built: heat conduction and free convection for the myocardial area, which is depleted of the coronary vessels and the myocardial region with double density distribution of the coronary vessels in the myocardium, relative to the case of depleted distribution.These three-dimensional objects with embedded in them of thermal RC-network are shown in Fig. 2, a and 2, b, respectively.
The heat transfer coefficient under free convection between three-dimensional objectsthe myocardium and coronary vessels corresponds to the natural model of convection laminar flow (model number ID  701 for Convection Correlation Lib) across the surface with the characteristic length L: where 1 10 Gr Pr 5 10 .     Implementation of heat transfer model for myocardium cooling process is executed in the Sind MSC system.On the final step myocardium cooling process in establishing the heat balance the temperature difference at the boundary between the myocardium and coronary vessels does not exceed 0,5 C.However, in the areas of the myocardium that are removed from the coronary vessels the temperature difference exceeds 1,0 C.
The use of additional cooling for heartsthe cooling of myocardium with using of ice surface, which is at a temperature liq 1 C T   provides the unevenness reduction of the heart temperature during its cooling with cardiopulmonary bypass.The example of numerical heat transfer model, which employs the heat convection between ice cube and the surface of the myocardium, is shown in Fig. 3.  From the analysis of the temperature distribution on the surface of the myocardium can be seen that a large temperature gradient T T    at the surface of the myocardium, which is depleted of coronary vessels, is caused by insufficient heat exchange between the blood and the myocardium.It is obvious that the lack of heat exchange between the three-dimensional objects in the modelthe myocardium and coronary vessels is associated with a decrease in area of the critical section of the myocardium heart S which has contact with the coronary vessels that filled with blood.

Experimental verification of the heat transfer model
For experimental verification model of heat transfer that implemented in the MSC Sinda measurements of thermal fields of open heart were performed [6].With the help of thermal imaging Flir i7 and TermoCAM E300 for 15 patients thermographic images of the myocardium during open heart surgery with cardiopulmonary bypass were recordedfor chilled heart at 24 -25 C, as well as prior to and after the hypothermia at temperature 33 -36 C.
At the images of registered thermograms maximum gradient of temperatures on the surface of the myocardium at the beginning of the process of hypothermia is

Conclusions
Thus, a discrete 3D-model of heat transfer in the layer structure of the myocardium and coronary vessels allows us to investigate the process of hypoand hyperthermia with cardiopulmonary bypass.The simulation results also make it possible to perform the analysis of the temperature distribution on the surface of the myocardium provided free convection of heat between the layers.
The 3D numerical heat transfer model in the myocardium takes into account real specific heat of human heart, the initial temperature distribution and free convection mechanism in the myocardium, and also allows calculating the rate of cooling of the myocardium and to determine the presence of ischemic lesions on the surface of the myocardium.Comparison of the model with real patients IRT shows that this method can provide additional important information regarding temperature and vascular uniformity in time of myocardial cooling and heating.Методика реализации.Для решения дифференциального уравнения теплопроводности в системе MSC Sinda использован метод тепловой сети (TNM), в котором система уравнений теплопроводности представлена с помощью конечноразностного метода в виде клеточно-ориентированных узлов и сопротивлений между узлами.При построении модели миокарда в системе MSC Sinda реализован тепловой контакт между трехмерными телами -миокардом, коронарными артериями, жидкостью для охлаждения сердца.

Рекомендована
unit volume due to heat transfer; c qthe heat flux due to heat transfer, temperature of the myocardium, K; liq T the coolant temperature (perfusion fluid),

C
requires to use in the model of thermal resistance.In these cases, the conductivity of the conductor , i j G is calculated as the reciprocal of the contact resistance between the two layers of nodes in the model:

Fig. 1 .
Fig. 1.The discrete model of the RC-thermal network: athe cellular elements of RC-thermal network; bthe structure of the RC-thermal network For the laminar flow of blood in the blood vessels, these values are actually determined by the product Gr Pr  of the indicator of the de-

Fig. 2 .
Fig. 2. Model of local fragment of myocardium, coronary vessels and cooling fluid for the heat RC-network in MSC Sinda: athe myocardium, which is depleted of the coronary vessels; bthe myocardium with twice the density of the coronary vessels.