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

10.2 Vascular Architecture and Blood Flow 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

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

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

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

(10.3) 6

Cartesian coordinates: cylindrical coordinates: spherical coordinates: 7

(b) Shortcomings of the Pennes equation Notes on eq. (10.3): (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 Equilibration Site: (1) Does not occur in the capillaries 8

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

(c) Applicability (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

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

(2) Origin and Coordinates. See Fig. 10.4 (3) Formulation 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

(ii) Governing Equations (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

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

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

(5) Checking Boundary conditions give (m) into (k) 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

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

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

NOTE: 19

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

Assign three temperature variables: (1) Arterial temperature (2)Venous temperature (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

Formulation 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

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

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

(b) Formulation (3) Artery and vein have the same radius (4) Negligible axial conduction through vessels (5) Equilibration length ratio 1 / << L e (6) Tissue temperature T is approximated by (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

(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 ú û ù ê ë é + = 2 ) ( 1 u a c k n b eff r p s ( 10.9) = a vessel radius = n number of vessel pairs crossing surface of control volume per unit area = u average blood velocity in countercurrent artery or vein 26

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

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

Cutaneous layer: Use eqs. (10.5) and (10.6) ) ( = - + T k w c dx d & r 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) ) ( 1 2 = - + T k w c dx d cb b & r (10.12) 29

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

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

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

(1) Observations (iii) Plot showing effect of blood flow & metabolic heat (1) Observations · Variation of k with distance is known · Tissue can be modeled as a single layer with variable eff k 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

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

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

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

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

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

39

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

10.4. 5 The s-Vessel Tissue Cylinder Model (iii) The governing parameters are Pe and g . The two are 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

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

Diameter and typically 10-15 mm long 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

(c) Formulation Capillaries, arterioles and venules are essentially in (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 Capillaries, arterioles and venules are essentially in local thermal equilibrium with the surrounding tissue Three temperature variables are needed: T , a and v Three governing equations are formulated 44

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

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

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

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

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

Artery supply temperature (1) body core temperature T a T 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

(2) is a correction coefficient defined in (10.18) (a) It depends only on the vascular geometry of the tissue cylinder (b) It is independent of blood flow rate (c) Its value for most muscle tissues ranges from 0.6 to 0.8 (d) This vascular structure parameter is much simpler than that required by Chen-Holmes and Weinbaum- Jiji equations (3) The model analytically determines the venous return temperature (4) Accounts for contribution of countercurrent heat exchange in the thermally significant vessels. 51

Example 10.3: Surface Heat Loss from Peripheral Tissue (a)        It is approximated by the body core temperature in the Pennes bioheat equation (b)        Its determination involves countercurrent heat exchange in SAV vessels (6)    While equations (10.5) and (10.6) apply to the cutaneous layer of peripheral tissue, eq. 10.23 applies to the region below the cutaneous layer. Example 10.3: Surface Heat Loss from Peripheral Tissue Peripheral tissue of thickness L 52

(2) Origin and Coordinates. See Fig. 10.12 Cutaneous plexus: Perfusion rate bc w & (uniformly distributed), blood supply temperature cb T Skin temperature s T Metabolic heat Specified correction coefficient * T D Use the s-vessel tissue cylinder model, determine surface flux (1)      Observations Temperature distribution gives surface flux This is a two layer problem: tissue and cutaneous (2) Origin and Coordinates. See Fig. 10.12 53

(ii) Governing Equations ((3) Formulation (i)     Assumptions (1) Apply all assumptions leading to (10.5) and (10.27) (2) Steady state (2) One-dimensional (3) Constant properties (4) Uniform metabolic heat in tissue layer (5) Negligible metabolic heat in cutaneous layer (7) Uniform blood perfusion in cutaneous layer (ii) Governing Equations Fourier’s law at surface: 54

(iii) Boundary Conditions Need 2 equations: one for tissue layer and one for cutaneous Tissue layer temperature T: eq. (10.27): (iii) Boundary Conditions 55

(4) Solution Let (b) and (c) become 56

Dimensionless parameters: (k) into (i) and (j) 57

Solutions to (m) and (n): Boundary conditions Solutions to (m) and (n): Boundary conditions (p)-(s) give constants 58

59

Surface heat flux: (5) Checking Dimensional check: Limiting check: 60

(5) Comments 61