A TWO-DIMENSIONAL THERMODYNAMIC MODEL TO PREDICT HEART THERMAL RESPONSE DURING OPEN CHEST PROCEDURES

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1 A TWO-DIMENSIONAL THERMODYNAMIC MODEL TO PREDICT HEART THERMAL RESPONSE DURING OPEN CHEST PROCEDURES F. G. Dias, J. V. C. Vargas, and M. L. Brioschi Universidade Federal do Paraná Departaento de Engenharia Mecânica Centro Politécnico Bairro Jardi das Aéricas CP , Curitiba, Paraná, Brasil ufpr.br Received: April 06, 2016 Revised: May 16, 2016 Accepted: June 16, 2016 ABSTRACT In this work, the teperature distribution of the heart in an open chest surgery scenario is studied. It is also evaluated the cardiac theral effects of the injection of a cooling liquid in the aorta root, which is used in infrared therography. The finite eleent ethod was used to develop a odel that predicts the teperature distribution odification in a 2-diensional slice of the heart. This therodynaic odel allows the coputational siulation of the theral cardiac response to open chest procedures, which are required by cardiac surgery. The influence of several operating paraeters (e.g., coronary flow rate, teperature) on the resulting theral distribution is analyzed. Therefore, this analysis allows the identification of paraeters that could be controlled to iniize the loss of energy, and consequently, avoiding the hazardous theral distribution that could put the heart in danger during cardiac surgery. Keywords: heart therodynaic odel, heart theral distribution, heart therography, finite eleent heart odel, cardiac surgery NOMENCLATURE c p fluid specific heat at constant pressure, J/(kg.K) K fluid theral conductivity, W/(.K) total ass, kg diensionless total ass p pressure, N/ 2 P diensionless pressure Pr fluid Prandtl nuber, ν/α diensionless overall theral conductance, q * Q overall heat transfer rate, W Re Reynolds nuber, u 2b/ν T average fluid teperature, K u 1,u 2,u velocity coponents, /s U 1,U 2,U diensionless velocity coponents x,y,z cartesian coordinates, X,Y,Z diensionless cartesian coordinates Greek sybols α theral diffusivity, 2 /s γ Penalty factor θ diensionless teperature ν fluid kineatic viscosity, 2 /s ρ density, kg/ INTRODUCTION Brazil has one the highest rates of death by heart disease in the world (4 %). One third of those cases are caused by coronary artery disease which is the ain reason of sudden death, infarction, angina and cardio pathology, according to the Brazilian Society of Cardiology (2000). In cases of coronary occlusions, the ost coon solution is the revascularization procedure by using either arterial or venous grafts. An inadequate or insufficient anastoosis results in another surgery or even leads the patient to death by infarction either during or after surgery. As surgery procedure is highly recoended in these cases, it inevitably exposes the ediastinal space, a closed space never eant to have direct contact with the exterior environent, directly to the environent s hazardous agents, such as viruses, bacterial colonies, and the extreely low teperature of the operating roo. The present work focus the theral distribution of the heart in an open chest surgery scenario, and, as ost of open chest surgeries concerns revascularization, a feasibility correlation with the left coronary flow is conducted based on the change of the teperature along the pericardia tissue surrounding the left anterior descending (LAD) branch of the left coronary artery. The approach consists on a scale analysis of the ass, energy and oentu balance equations of the yocardiu, pericardiu tissue and of the blood flow using finite eleents, letting the user to change the ain paraeters of the esh, thus calculating the new equilibriu for the altered siulating environent. To ensure the convergence of the steady-state solution and the proper variation of the physiological and environental paraeters, appropriate nondiensional groups are defined and the results 44 Engenharia Térica (Theral Engineering), Vol. 15 No. 1 June 2016 p

2 reported in diensionless charts. THEORY Figure 1 shows a D odel of the heart. This odel is very precise and will help the reader to understand what the present study is engaged to accoplish. Note that this is the anterior view of the heart in cardioplegic status, with the patient in supine position, and with his thoracic cavity opened. objective of this article is the odeling of the heart theral response during open chest surgery, the heart is supposedly half iersed into the pericardia doain, represented by the (A) portion. The (B) portion represents the area of the heart that is exposed into the operating roo hazardous environent. Portion (C) represents the blood flowing into the ventricles and into the capillaries fro the left anterior descending branch of the left coronary to the vein tributary of the coronary sinus. Portion (D) represent the yocardiu itself and Portion (E) represents the blood that has a known flow out fro the left anterior descending artery branch. The duct that is connecting the left anterior descending branch and the vein tributary of the coronary sinus corresponds to the array of capillaries of the plane of the slice analyzed, so, in the adjacent slices is found the sae siilar structures. Thus, as noticed before, the derivate of the velocities and teperatures fro one slice into other is zero, which allows the bidiensional odeling of the given tridiensional proble by the analysis of only one slice, once the sae picture repeats approxiately in every adjacent slices (or planes). Figure 1. D odel of the heart [Yale edicine website Figure 2 shows, using radioactive arked radiograph, the chosen 2D slice of the heart and on its right is shown with the yellow line where the slice is located in the heart. As the odel is built using a 2D slice of the heart, it is crucial for the precise odeling and siulation that this slice is representative of the hole D heart, so in every slice in the sae axial line, there ust exist the sae structures represented, resulting derivates of velocities and teperatures between the adjacent slices to be zero. Figure 2. Axial slice of the heart to be odeled and its location in the real heart [Yale edicine website Figure. Proble sketch and coputational doain. In the present work, blood is considered a Newtonian fluid. Perhaps the earliest attepts to study the flow properties of blood were ade by Young and Poiseuille in the 17th century and siulating blood flow in coplex corporeal and clinical environent is still a difficult proble. The coplexities of siulating blood flow were realized quite early (Fox and Saibel, 196). It is well established now that blood is a non-newtonian fluid. However, ost analysis are perfored considering blood as a Newtonian fluid which works well in large and ediu arteries with siple geoetries, as is the case of the left anterior descendent branch of the left coronary and the ventricles. For ost other interesting and challenging probles the non- Newtonian effects have to be considered. Figure shows the general sketch of the proble configuration of Fig. 2. As the ain u v + = 0 x (1) Engenharia Térica (Theral Engineering), Vol. 15 No. 1 June 2016 p

3 u u 1 p u u u + v = + ν + (2) x ρ x x v v 1 p v v + v = + ν + gβt 2 x y y + x y ρ u 2 () T T T T u + v = α + (4) x x In diensionless for, Eqs. (1)-(4) can be written as: Ra 1/ 2 Pr Ra 1/ 2 Pr U V + = 0 (5) U U U + V P U U = + + (6) V V U + V P V V = Ra Ra 1/ 2 1/ 2 θ θ θ θ U + V = + θ (7) (8) Where diensionless variables have been defined as follows: (, Y) ( x, y), p X = P, L ρu = ( U, V) = ( u, v) Ra, 2 T T gβ(tw T )L θ =, Ra =, T T αν w where (, y) L α 1/ 2 (9) ν Pr = (10) α x are the Cartesian coordinates, ; p the pressure, N/ 2 ; ρ the fluid density, Kg/ ; g the gravity acceleration, /s 2 ; T the heart teperature, K; T the operation roo teperature, K; T w the ventricular blood teperature, K; β the coefficient of theral expansion, K -1 ; L the heart slice length, ; ν the blood kineatics viscosity, 2 /s and α is the blood theral diffusivity, 2 /s; cp is the specific heat capacity, J/kg.K; µ is the dynaic viscosity, Pa.s; k is the theral conductivity, W/.K. To coplete the proble forulation, the following boundary conditions are specified for the extended coputational doain of the slice odel of Fig., where the teperature of the operation roo was set to 25ºC ( θ = 0 ), inner chest pericardiu teperature to 5.5 ºC ( θ = ), blood teperature to 7ºC ( θ = 1) and the flow in the Left Anterior Descending branch of the Left Coronary was set to 8.8 c/s, ( U = and V = 5.10 ): (A) 0, U = V = 0, θ = 0.875, (11) (B) U = 0, V = 0, θ = 0, (12) (C) θ = 1. 0 (1) (D) U = 0, V = 0, (14) (E) U = , V = 5.10, θ = 1.0, (15) Once the geoetry of the coputational doain defined by the slice of Fig. is specified, Eqs. (5)- (15) deliver the resulting velocities, pressure and teperature fields in the doain. NUMERICAL METHOD The nuerical solution of Eqs (5)-(15) was obtained utilizing the finite eleent ethod (Zienkiewicz and Taylor, 1989). This way, the velocities and teperature fields in the odel of Fig. were deterined. The first step in the nuerical ipleentation was the eliination of the pressure variable fro Eqs. (6) and (7), by using a penalty odel, approxiating Eq. (5) as follows (Hughes et al., 1979): U V P + =, γ (16) where γ is the penalty factor, which ust be assued large enough in order to satisfy ass conservation approxiately. The ipleentation of the finite eleent ethod for the solution of Eqs. (5)-(15) starts fro obtaining the variational (weak) for of the proble. Next, the weak for is discretized, by a suitable ethod. Due to the characteristics of the flow inside the arrangeent of Fig., physically, it is seen that the solution at a specified location in the odel depends preferably on what happens upstrea, considering the direction of the forced flow. It is well known that the Galerkin ethod, which is the analogue of a central differences schee in the finite differences ethod, does not capture this physical aspect of in its discretized equations. Therefore, the discretization used in this work is an upwind schee proposed by Hughes (1978), where it is possible to adequate the discrete for of the proble to the physical characteristics of the flow. After developing the discrete for of the proble, the resulting algebraic equations are arranged in a atricial for for the steady-state 2-D 46 Engenharia Térica (Theral Engineering), Vol. 15 No. 1 June 2016 p

4 proble as follows: B1 0 T C(U ) B2 T 0 C(U ) U U 2 2K11 + K 22 K K12 K11 + 2K 22 U U 2 Kˆ Kˆ F, Kˆ Kˆ U U = (17) F D (U )T L T = G, + (18) where B 1 and B 2 are the buoyancy ters; C (U ) is the capacity atrix that contains the advective ters of the oentu equations, which depends nonlinearly on the solution U (bipartitioned vector: U 1 - direction X and U 2 - direction Y, each one with a nuber of coponents equal to the nuber of unknowns in the esh); Kˆ 11, Kˆ 12, Kˆ 21 and Kˆ 22 are the penalty of the pressure variable fro the oentu equations by using Eq. (11), which were coputed with reduced integration (one point in each direction with linear shape functions) to avoid locking; D (U ) is the capacity atrix that contain the advective ters of the energy equation; L is the diffusive atrix with constant coefficients (constant theal conductivity), that contain the diffusive ters fro the energy equation; F 1 and F2 are the force vectors of the oentu equations that contain the field forces and the velocity boundary conditions, and G is the heat source ters of the energy equation and the teperature boundary conditions. For the sake of brevity, the atheatical details of the coponents of the above-described atrices are not presented. However, the reader is directed to the work of Reddy and Gartling (1994), which was the basis for the present study. A Fortran subroutine was written to ipleent the Navier-Stokes and Energy equations in two diensions as an isoparaetric, four-noded, linear eleent, which was the aggregated to the open code called finite eleent analysis progra, FEAP, originally written by Zienkiewicz and Taylor (1989). Material 2 Pericardiu: Kg J W ρ = 900, cp = 2650, k = Kg.K.K Material Blood: 4 Kg µ = Pa.s, ρ = 1060, cp = 889 β = J Kg.K 1,g K,k = = s W,.K Initially, it was analyzed the heart theral distribution at usual conditions for operating roo, i.e., teperature of the operation roo was set to 25ºC ( θ = 0 ), inner chest pericardiu teperature to 5.5 ºC ( θ = ) and blood teperature to 7ºC ( θ = 1 ). The flow in the Left Anterior Descending branch of the Left Coronary was set to U = and V = 5.10 Figure 4 shows the obtained results of the noralized teperature distribution over the heart slice. Figure 5 shows the obtained vectors representing the results for the internal flows in the capillary duct linking the arterial feeding source to the venous draining syste and the flows internal to the ventricular cavities, as this last one is only due to the teperature gradient inside the heart. Notice that the center of the vortex of each ventricle s flow vectors is located near the region of ost intense teperature gradient, i.e., the transition area between the heart iersed into the pericardia doain and the heart exposed to the environent. As a atter of fact, this intense flow near the higher gradient area is explained by the buoyancy ter in the oentu Eq. (2), gβ T, which directs the distribution of the velocities derivates according to the teperature gradient. RESULTS AND DISCUSSION For the present study, the esh odel was divided in three different aterials, each one with the following properties: Material 1 Myocardiu: Kg J ρ = 1125, cp = 475 Kg.K, k = 0.49 W.K Figure 4. Cardiac theral distribution at coon conditions for operating roo. Next, it was analyzed the correlation of the LAD flow with the change of the teperature distribution along the pericardia tissue surrounding the left anterior descending (LAD) branch of the left Engenharia Térica (Theral Engineering), Vol. 15 No. 1 June 2016 p

5 coronary artery and in the yocardial tissue near the left ventricle. aneuver is very iportant to estiate the LAD arterial flow using Infrared Iaging (Brioschi et al., 2002), thus allowing the edical staff to obtain a fair qualitative estiative of the flow based on the new teperature gradients trough tie of the LAD surrounding tissue (pericardial fat), seconds after the liquid injection. Notice that the ethod is quite invasive and hazardous to the heart, so the period of injection ust be shorter as possible, thus avoiding unnecessary teperature and isqueic daage to the anterior portion of the left ventricle. Figure 5. Internal heart blood flow vectors. Figure 6 shows the correlation between LAD flow and the left anterior pericardial teperature, where the probed site was above the LAD and bellow the portion of the heart in direct contact with the environent, the ideal and ost accessible point to get a direct easure of the teperature in revascularization or valve transplantation surgeries, e.g., using infrared iaging (Brioschi et al., 2000; Siebert et al., 2001; Brioschi et al., 2002). Figure 7 shows the correlation between LAD flow and the yocardial tissue teperature near the left ventricle. This clearly shows that the near presence of the left ventricle is predoinant and sufficient to keep this region warn, therefore reducing the dependence with the heat distribution generated by the LAD flow for this specific site. Figure 7. Correlation between LAD flow and the yocardial tissue teperature. Figure 8. Theral distribution in the injection of a cooling liquid at 28º C in the aorta root. 5. CONCLUSIONS Figure 6. Correlation between LAD diensionless flow and the left anterior pericardial diensionless teperature. After that, the injection of a cooling liquid at 28º C ( θ = ) in the aorta root was siulated by setting the input teperatures of the LAD branch to this chosen teperature. Figure 8 shows the heat distribution over the heart after this procedure. This This study deonstrates that an open chest procedure greatly changes the theral distribution of the heart. It provided a odel that enables the physician to evaluate the teperature distribution of each desirable portion of the heart, thus providing a handful tool to predict possible teperature daages during the exposed heart surgeries. It also provided a feasible correlation of the left coronary flow based on the change of the teperature along the pericardia tissue surrounding the left anterior descending (LAD) branch of the left coronary artery. This correlation is very iportant, especially upon the fact that the practical correlations 48 Engenharia Térica (Theral Engineering), Vol. 15 No. 1 June 2016 p

6 used nowadays cross pointing LAD flow versus pericardiu teperature, e.g., infrared iaging, are just qualitative, providing the surgeon a fair idea of the flow status. Nonetheless, this works shows that it s possible to obtain a quantitative calculation of the LAD flow based on the teperature of the pericardial region surrounding the left anterior descending branch, which can be accoplished in future researches. Ultiately, coputational systes that cobine two-diensional odeling with practical acquired theral distribution paraeters will enable physicians to design therapeutic and visualize procedures that yield iproved surgery safety and certified blood flow to patients with cardiovascular disease. Reddy, J. N., and Gartling, J. N., 1994, The Finite Eleent Method in Heat Transfer and Fluid Dynaics, CRC Press. Rocha, L. A. O., Saboya, F. E. M., and Vargas, J. V. C., 1997, A Coparative Study of Elliptical and Circular Sections in One And Two-Row Tubes and Plate Fin Heat Exchangers, International Journal of Heat Fluid Flow, Vol. 18, No. 2, pp Saboya, S. M., and Saboya, F. E. M., 2001, Experients on Elliptic Sections in One and Two- Row Arrangeents of Plate Fin and Tube Heat Exchangers, Experiental Theral and Fluid Science, Vol. 24, No. 1-2, pp Zienkiewicz, O. C., and Taylor, R. L., 1989, The Finite Eleent Method, Vol. 1, McGraw-Hill. REFERENCES Brazilian Society of Cardiology (SBC), 2000, National Preventive and Epideilogical Progra. Brioschi, M. L., Cibalista, M., Colan, D., Machuca, T. N., and Loures, D. R. R., 2002, Tero- Coronário-Angiografia: Padronização do Método e Prieiras Aplicações Clínicas no Brasil, Revista Brasileira de Cirurgia Cardiovascular, Vol. 17, pp (in Portuguese) Brioschi, M. L., Cibalista Jr., M., Nakagawa, C. R., Colan, D., Silveira, F., Coltro, A., von Bahten, L. C., and Brofan, P. R., 2000, Theral Coronary Angiography for Intraoperative Avaliation of Cardiac Perfusion: Preliinary Experiental Study, Arq Med PUCPR, Vol. 2, pp Fox, E. A., and Saibel, E., 196, Attepts in Matheatical Analysis of Blood Flow, Transactions of Society of Rheology Vol. VII, pp Hughes, T. J. R., 1978, A Siple Schee for Developing Upwind Finite Eleents, International Journal for Nuerical Methods in Engineering, Vol. 12, pp Hughes, T. J. R., Liu, W. K., and Brooks, A., 1979, Review of Finite Eleent Analysis of Incopressible Viscous Flows by Penalty Function Forulation, Journal of Coputational Physics, Vol. 0, pp Reddy, J. N., and Gartling, D. K., 1994, The Finite Eleent Method in Heat Transfer and Fluid Dynaics, CRC Press, Boca Raton, FL (Chapters 4-5). Siebert, J., Keita, L., Kaczarek, M., Anisiowicz, L., Rogowski, J., and Narkiewicz, M., 2001, Application of Intraoperative Therography in Coronary Artery Bypass Grafting without Extracorporeal Circulation - Clinical Report, Medical science Monitor, Vol. 7, pp Zienkiewicz, O. C., and Taylor, R. L., 1989, The Finite Eleent Method, Vol. 1, McGraw-Hill, London, (Chapter 15). Engenharia Térica (Theral Engineering), Vol. 15 No. 1 June 2016 p