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CHAPTER 10

CHAPTER 10. HEAT TRANSFER IN LIVING TISSUE. 10.1 Introduction · Examples · Hyperthermia · Cryosurgery · Skin burns · Frost bite · Body thermal regulation · Modeling. Modeling heat transfer in living tissue requires the formulation of a special heat equation. 1. Key features

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CHAPTER 10

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  1. CHAPTER 10 HEAT TRANSFER IN LIVING TISSUE 10.1Introduction ·Examples ·Hyperthermia ·Cryosurgery ·Skin burns ·Frost bite ·Body thermal regulation ·Modeling Modeling heat transfer in living tissue requires the formulation of a special heat equation 1

  2. Key features • (1) Blood perfused tissue • (2) Vascular architecture • (3) Variation in blood flow rate and tissue properties 10.2 Vascular Architecture and Blood Flow • Vessels • Artery/vein • Aorta/vena cava • Supply artery/vein • Primary vessels • Secondary vessels • Arterioles/venules • Capillaries 2

  3. Blood leaves heart at • Blood mixing from various sources brings temperature to 10.3 Blood Temperature Variation • Equilibration with tissue: prior to arterioles and capillaries • Metabolic heat is removed from blood near skin 3

  4. (a) Formulation (1) Equilibration Site: (2) Blood Perfusion: (3) Vascular Architecture: (4) Blood Temperature: 10.4 Mathematical Modeling of Vessels-Tissue Heat Transfer 10.4.1 Pennes Bioheat Equation (1948) Assumptions: Arterioles, capillaries & venules Neglects flow directionality. i.e. isotropic blood flow No influence 4

  5. Let ¢ ¢ ¢ = net rate of energy added by the blood per unit q b volume of tissue ¢ ¢ ¢ = rate of metabolic energy production per unit q m volume of tissue ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ = = + & (a) E q dx dy dz ( q q ) dx dy dz g b m Conservation of energy for the element shown in Fig. 10.3: Treat energy exchange due to blood perfusion as energy generation 5

  6. (10.3) 6

  7. Cartesian coordinates: cylindrical coordinates: spherical coordinates: 7

  8. (10.3) (1) Equilibration Site: Notes on eq. (10.3): • This is known as the Pennes Bioheat equation • The blood perfusion term is mathematically identical to surface convection in fins, eqs. (2.5), (2.19), (2.23) and (2.24) • (3) The same effect is observed in porous fins with coolant flow (see problems 5.12, 5.17, and 5.18) (b) Shortcomings of the Pennes equation • Does not occur in the capillaries 8

  9. · thermally significant Occurs in the pre - arteriole and post - venule vessels (dia. 70 - 500 ) m m L · e Thermally significant vessels > : 1 L · Equilibration length = : distance blood travels for L e its temperatur e to equilibrate with tissue (2) Blood Perfusion: (3) Vascular Architecture : ·  Perfusion in not isotropic ·  Directionality is important in energy interchange • Local vascular geometry not accounted for • Neglects artery-vein countercurrent heat exchange • Neglects influence of nearby large vessels 9

  10. (4) Blood Temperature: · Blood does not reach tissue at body core temperature · Blood does not leave tissue at local temperature T (c) Applicability · Surprisingly successful, wide applications · Reasonable agreement with some experiments 10

  11. Model forearm as a cylinder Blood perfusion rate w & b ¢ ¢ ¢ Metabolic heat production q m Convection at the surface h Heat transfer coefficient is Ambient temperature is T ¥ Use Pennes bioheat equation to determine the 1 - D temperature distribution Example 10.1: Temperature Distribution in the Forearm (1) Observations 11

  12. Arm is modeled as a cylinder with uniform energy • generation • ·   Heat is conduction to skin and removed by convection • ·   In general, temperature distribution is 3-D (2) Origin and Coordinates. See Fig. 10.4 (3) Formulation (i) Assumptions (1)Steady state (2)Forearm is modeled as a constant radius cylinder (3)Bone and tissue have the same uniform properties (4)Uniform metabolic heat (5)Uniform blood perfusion (6)No variation in the angular direction (7)Negligible axial conduction 12

  13. (8) Skin layer is neglected (9) Pennes bioheat equation is applicable (ii) Governing Equations Pennes equation (10.3) for 1-D steady state radial heat transfer (iii) Boundary Conditions: (4) Solution 13

  14. Rewrite (a) in dimensionless form. Define (d) into (a) Define (f) and (g) into (e) 14

  15. The boundary conditions become Bi is the Biot number Homogeneous part of (h) is a Bessel differential equation. The solution is 15

  16. Boundary conditions give (m) into (k) (5) Checking Dimensional check:Bi,and are dimensionless. The arguments of the Bessel functions are dimensionless. Dimensional check: Limiting check: If no heat is removed (),arm reaches a uniform temperature . All metabolic heat is transferred to the blood. Conservation of energy for the blood: Limiting check: 16

  17. Solve for which agrees with (o) (6) Comments 17

  18. 10.4.2 Chen-Holmes Equation • First to show that equilibration occurs prior to reaching • the arterioles • Accounts for blood directionality • Accounts for vascular geometry • The Pennes equation is modified to: 18

  19. NOTE: 19

  20. < m (1) Vessel diameter 300 m L < e (2) 0 . 6 L (3) Requires detailed knowledge of the vascular network and blood perfusion Limitations 10.4.3 Three-Temperature Model for Peripheral Tissue Rigorous Approach • Accounts for vasculature and blood flow directionality 20

  21. (1) Arterial temperature (2)Venous temperature • Assign three temperature variables: (3) Tissue temperature T • Identify three layers: • Intermediate layer: • porous media • Cutaneous layer: thin, • independently supplied by counter-current artery-vein • vessels called cutaneous plexus • Regulates surface heat flux 21

  22. Consists of two regions: • (i) Thin layer near skin with negligible blood flow • (ii) Uniformly blood perfused layer (Pennes model) Formulation • Seven equation: • 3 for the deep layer • 2 for the intermediate layer • 2 for the cutaneous layer • Model is complex • Simplified form for the deep layer is presented in the next • section • Attention is focused on the cutaneous layer: • (i) Region 1, blood perfused. For 1-D steady state: 22

  23. 2 r & d T c w + - = 1 b b cb (10.5) ( T T ) 0 c 0 1 2 k dx = temperature variable in the lower layer T 1 = temperature of blood supplying the cutaneous pelxus T c 0 & = cutane ous layer blood perfusion rate w cb = coordinate normal to skin surface x The 3 eqs. for and are replaced by one equation T 2 d T = 2 (10.6) 0 2 dx T , T a v (ii) Region 2, pure conduction , for 1-D steady state: 10.4.3 Weinbaum-Jiji Simplified Bioheat Equation for Peripheral Tissue 23

  24. · Contains artery - vein pairs · ¹ Countercurrent flow, T T a v · Includes capillaries, arterioles and venules (1) Uniformly distributed blood bleed - off leaving artery is equal to that returning to vein T T (2) Bleed - off blood leaves artery at and enters the vein at a v • Effect of vasculature and heat exchange between artery, • vein, and tissue are retained • Added simplification narrows applicability of result Control Volume (a) Assumptions 24

  25. (3) Artery and vein have the same radius (4) Negligible axial conduction through vessels << (5) Equilibration length ratio L / L 1 e (6) Tissue temperature is approximated by T (7) One-dimensional: blood vessels and temperature gradient are in the same direction (b) Formulation Conservation of energy for tissue in control volume takes into consideration: (1) Conduction through tissue (2) Energy exchange between vessels and tissue due to capillary blood bleed-off from artery to vein 25

  26. é ù n 2 2 = + p r ( 10.9) k k 1 ( c a u ) ê ú eff b b 2 ë û s k = vessel radius a = number of vessel pairs crossing surface of control n volume per unit area = average blood velocity in countercurrent artery or vein u (3) Conduction between vessel pairs and tissue Note: Conduction from artery to tissue not equal to conduction from the tissue to the vein (incomplete countercurrent exchange) Conservation of energy for the artery, vein and tissue and conservation of mass for the artery and vein give 26

  27. · NOTE · accounts for the effect of vascular geometry and blood k eff perfusion s · , and depend on the vascular geometry a n u , Conservation of mass gives in terms of inlet velocity to u u o tissue layer and the vascular geometry. Eq. (10.9) becomes 27

  28. = = vessel radius at inlet to tissue layer, x 0 a o x = dimensionless vascular geometry function V ( ) (independent of blood flow) x = = dimensionless distance x / L = tiss ue layer thickness L = = blood velocity at inlet to tissue layer, u x 0 o r is independent of vascular geometry. ( 2 c a u / k ) NOTE: b b o o b Notes on : k eff It represents the inlet Peclet number: Eq. (10.12) into eq. (10.11) 28

  29. 2 r d T c w & + - = 1 b b cb (10.12) ( T T ) 0 c 0 1 2 k dx • For the 3-D case, orientation of vessel pairs relative • to the direction of local tissue temperature gradient gives • rise to a tensor conductivity • (2) The second term on the right hand side of eqs. (10.11) and • (10.13) represents the enhancement in tissue conductivity • due to blood perfusion Cutaneous layer: Use eqs. (10.5) and (10.6) 29

  30. = number of arteries entering tissue layer per unit area n o = rate of blood to the cutaneous layer to the rate R total total of blood to the tissue layer = is the thickness of the cutaneous layer L 1 Eq. (10.12) into eq. (10.14) Define R Eqs. (10.15) and (10.16) into (10.5) 30

  31. · - temperature model of Results are compared with 3 Section 10.4.3 · Accurate tissue temperature prediction for: (1) Vessel diameter < 200 μm (2) Equilibration length ratio < L / L 0 . 2 e (3) Peripheral tissue thickness < 2mm (c) Limitation and Applicability 31

  32. Skin surface at T - 5 ´ 7 10 s = k Blood supply temperature eff 2 + x k [ 1 Pe V ( )] x o V ( ) T a 0 0 x 1 Fig. 10.7 described by x V ( ) 2 x = + x + x V ( ) A B C - - - 5 5 5 = ´ = - ´ = ´ A 6 . 32 10 , B 15 . 9 10 and C 10 10 Example 10.2: Temperature Distribution in Peripheral Tissue Peripheral tissue Neglect blood flow through cutaneous layer vascular geometry is • Use the Weinbaum-Jiji equation determine temperature distribution • (ii) Express results in dimensionless form: . 32

  33. (1) Observations · Variation of k with distance is known · Tissue can be modeled as a single layer with variable k eff (iii) Plot showing effect of blood flow & metabolic heat • Metabolic heat is uniform • Temperature increases as blood perfusion • and/or metabolic heat are increased (2) Origin and Coordinates. See Fig. 10.8 (3) Formulation 33

  34. x (4) T issue temperature at the base = 0 is equal to (5) Skin is maintained at uniform temperature (6) Negligible blood perfusion in the cutaneous layer. (a) (i) Assumptions (1) All assumptions leading to eqs. (10.8) and (10.9) are applicable (2) Steady state (3) One-dimensional (ii) Governing Equations. Obtained from eq. (10.8) 34

  35. = (d) T ( 0 ) T a 0 = (e) T ( L ) T s (iii) Boundary Conditions (4) Solution Define Substituting (b), (c) and (f) into (a) Boundary conditions 35

  36. q = (h) ( 0 ) 1 q = (i) ( 1 ) 0 [ ] q d 2 2 + + x + x = - gx 1 Pe ( A B C ) C 0 1 x d x x x d d ò ò q = - g + C C 1 2 2 2 2 2 + + x + x + + x + x 1 Pe ( A B C ) 1 Pe ( A B C ) 0 0 (j) x x x d d ò ò and (k) 2 2 + x + x + x + x a b c a b c Integrating (g) once integrating again integrals (j) are of the form where 36

  37. 2 2 2 = + = = (m) a 1 APe , b BPe , c CPe 0 0 0 + x 2 b 2 c - 1 q = - C tan 1 d d (n) g + x é ù 1 b b 2 c - 2 1 + x + x - + ln ( a b c ) tan C ê ú 2 ë û c 2 d d 2 = - (o) d 4 ac b Boundary conditions (h) and (i) give the C C constants and 1 2 Evaluate integrals, substitute into (j) 37

  38. a b c d (1) , , and depend on Listed in Table 10.1 Pe . 0 (2) depends on both and g Pe C : o 1 where Note: 38

  39. 39

  40. (i) Enhancement in due to blood perfusion k eff g = = (ii) Temperature distribution for and is 0 . 02 Pe 60 0 g = = nearly linear. At and the 0 . 6 Pe 180 0 temperature is h igher (5) Checking Dimensional check: Boundary conditions check: Boundary conditions (h) and (i) are satisfied Qualitative check: Tissue temperature increases as blood perfusion and metabolic heat are increased (6) Comments 40

  41. g (iii) The governing parameters are and . The two are Pe 0 physiologically related (iv) Neglecting blood perfusion in the cutaneous layer during vigorous exercise is not reasonable 10.4. 5 The s-Vessel Tissue Cylinder Model Model Motivation • Shortcomings of the Pennes equation • The Chen-Holmes equation and the Weinbaum-Jiji equation are • complex and require vascular geometry data (a) Basic Vascular Unit Vascular geometry of skeletal muscles has common features • Main supply artery and vein, SAV 41

  42. Primary pairs, P • Secondary pairs, s • Terminal arterioles and venules, t • Capillary beds, c 42

  43. NOTE: Blood flow in the SAV, P and s is countercurrent Each countercurrent s pair is surrounded by a cylindrical tissue which is approximately 1 mm Diameter and typically 10-15 mm long • The tissue cylinder is a repetitive unit consisting of arterioles, • venules and capillary beds • This basic unit is found in most skeletal muscles • A bioheat equation for the cylinder represents the governing • equation for the aggregate of all muscle cylinders (b) Assumptions (1) Uniformly distributed blood bleed-off leaving artery is equal to that returning to vein of the s vessel pair 43

  44. Capillaries, arterioles and venules are essentially in local thermal equilibrium with the surrounding tissue • Three temperature variables are needed: , and T T T a v (2) Negligible axial conduction through vessels and cylinder (3) Radii of the s vessels do not vary along cylinder (4) Negligible temperature change between inlet to P vessels and inlet to the tissue cylinder (5) Temperature field in cylinder is based on conduction with a heat-source pair representing the s vessels (6) Outer surface of cylinder is at uniform temperature (c) Formulation • Three governing equations are formulated 44

  45. Navier-Stokes equations of motion give the velocity field • in the s vessels (axially changing Poiseuille flow) Boundary Conditions (1)Continuity of temperature at the surfaces of the vessels (2)Continuity of radial flux at the surfaces of the vessels 45

  46. The three eqs. for , and are solved analytically T T T a v • Solution gives , the outlet bulk vein temperature at = x 0 T vb 0 (d) Solution Simplified Case Assume: 46

  47. (1) Artery and vein are equal in size (2) Symmetrically positioned relative = to center of cylinder, i.e., l l a v Results 47

  48. (e) Modification of Pennes Perfusion Term Eq. (10.18) gives (a) into (10.21) Dividing by the volume of cylinder 48

  49. Blood flow energy generation per unit tissue volume: (10.23) and 10.24) into (10.22) (10.25) becomes 49

  50. (1) Artery supply temperature body core temperature T a 0 Use (10.26) to replace the blood perfusion term in the Pennes equation (10.3) NOTE: (1) This is the bioheat equation for the s-vessel cylinder model 50

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