BIOHEAT EQUATIONS BIOHEAT EQUATIONS. Heat transfer in blood vessels and tissues. Mihir Sen 1/ 36

Size: px

Start display at page:

Download "BIOHEAT EQUATIONS BIOHEAT EQUATIONS. Heat transfer in blood vessels and tissues. Mihir Sen 1/ 36"

Gordon Collins
7 years ago
Views:

1 1/ 36 BIOHEAT EQUATIONS Heat transfer in blood vessels and tissues Mihir Sen March 17, 2013

2 2/ 36 Outline Bioheat transfer problem Large blood vessels Tissues and microvasculature Pennes s equation Other models Transient equations Multiscale problem

3 3/ 36 Bioheat transfer applications Body heat balance Thermoregulation Thermogeneration Heat transfer in muscles and tissues Skin burns Surgical procedures Ablative surgery Cryosurgery Therapeutic hyperthermia Therapeutic hypothermia Cryopreservation Organs for transplant Resuscitation medicine Extracorporeal equipment Measuring instruments

Ablative surgery Cryosurgery Therapeutic hyperthermia Therapeutic hypothermia

4 4/ 36 Properties Thermodynamic properties (density, compressibility, specific heat) Transport properties (thermal conductivity) Properties of frozen tissue Temperature dependence of properties Water and fat content dependence Convective heat transfer coefficient Rate of perfusion

of frozen tissue Temperature dependence of properties Water and fat

5 5/ 36 Metabolism Chemical reactions in living organisms to generate heat Thermodynamically, metabolism maintains order by creating disorder Metabolic regulation

6 Allometry of metabolism Savage et al. (2004): 626 mammalian species BMR = basal metabolic rate [W], M = mass [g] BMR M 3/4 Heart and respiratory rates, stride frequencies M 1/4 Life spans, times to first reproduction M 1/4 6/ 36

7 Summary of heat transfer mechanisms Conduction (Fourier s law) Advection q = k T q = ρcv (T T ref ) Convection (Newton s law of cooling) q = h (T surf T fluid ) Radiation (Stefan-Boltzmann s law) q = ǫσ ( Tsurf 4 T surr 4 ) q 1 R rad (T surf T surr ) 7/ 36

of cooling) q = h (T surf T fluid ) Radiation (Stefan-Boltzmann

8 Large blood vessels z T w q w Wall boundary conditions (inner surface) Constant wall temperature, T w = constant Constant wall heat flux, q w = constant Conjugate heat transfer Continuity of temperature at wall Continuity of heat flux at wall Matching with external heat transfer Bulk (mean, average or mixing cup) temperature A T m = ρvc pt da ṁc p = 2 v m R 2 R 0 v(r)t(r)r dr 8/ 36

Continuity of temperature at wall Continuity of heat flux at wall Matching with external heat

9 Large blood vessels: energy equation Integral approach { q w P dz ṁc dt m = h(t w T m ) P dz known wall heat flux known wall temperature Differential approach (cylindrical coordinates) T t + v T r r + v θ T r θ + v z [ = k ρc p 1 r r T z ( T r r ] ) 1 2 T + r 2 θ T z 2 + Φ Φ = viscous dissipation (conversion from mechanical to thermal energy) Nondimensional: Péclet number = V D/α 9/ 36

r + v θ T r θ + v z [ = k ρc p 1 r r T z ( T r r ] ) 1 2 T + r 2 θ 2 + 2 T z 2 + Φ Φ = viscous

10 10/ 36 Large blood vessels: preliminaries Thermally fully developed if z { Tw (z) T(r, z) T w (z) T m (z) } = 0 Entrance length L thermal = Pr L hydro ; for blood Pr = Nusselt number Nu D = hd/k where q = h(t w T m ) Correlations Nu D = Nu D (Re,Pr)

length L thermal = Pr L hydro ; for blood Pr = 10 25 Nusselt

11 11/ 36 Large blood vessels: laminar flow Constant wall temperature T w At the exit Nu D = 3.66 T w T m (z) T w T m (0) = e z/le L e = ṁc p Ph T w T m (L) T w T m (0) = e L/Le Notice that T m (L) = T w for L L e. (thermal equilibration length)

12 12/ 36 Chen and Holmes (1980) Vessels of diameter 175µm have anatomical length thermal equilibration length (these are called thermally significant blood vessels) Temperature in smaller vessels are quickly equilibrated and do not contribute much to heat transfer. Larger vessels are sparse and do not contribute much to heat transfer.

Temperature in smaller vessels are quickly equilibrated and do not contribute much

13 13/ 36 Large blood vessels: laminar flow Constant wall heat flux q w Nu D = 4.36 T m (z) = T m (0) + q wp ṁc p z

14 14/ 36 Flow in porous media Darcy s law (1856) K = permeability [m 2 ] p = µ K v

15 15/ 36 Modifications Transient Darcy equation p = µ v v + ρc K t Darcy-Brinkman equation Darcy-Forchheimer equation p = µ K v µ eff 2 v p = µ K v + c Fρ v v K Darcy-Brinkman-Forchheimer equation p = µ K v + c Fρ v v K µ eff 2 v

16 16/ 36 Porous medium heat transfer: single-temperature model T (ρc) m + (ρc) f v T } {{ t} } {{ } advection accumulation = (k m T) } {{ } conduction + q m }{{} generation (ρc) m = (1 α)(ρc) s + α(ρc) f k m = (1 α)k s + αk f q m = (1 α) q s + α q f heat generation per unit volume

conduction + q m }{{} generation (ρc) m = (1 α)(ρc) s + α(ρc) f k m =

17 17/ 36 Porous medium heat transfer: two-temperature model T s (1 α)(ρc) s t = (1 α) (k s T s ) + h(t f T s ) +(1 α) q s } {{ } ( (ρc) f α T ) f t + v T f α = porosity (fraction of fluid by volume) q s,f = heat generation per unit volume s = solid f = fluid fluid to solid = α (k f T f ) + h(t s T f ) } {{ } solid to fluid +α q f

$porosity (fraction of fluid by volume) q s,f = heat generation per unit volume s =$

18 18/ 36 Circulation tree Branching R n+1 < R n P in P out Womersley number α = R ωρ/µ α 0 as R 0, so flow is steady (not pulsating)

19 19/ 36 Branching in human cerebral cortex Francis et al. (2009) Typical arteriole (right) and venule (left), scale bar = 1 mm

20 20/ 36 Murray s law (1926) Applied to Circulatory system Respiratory system Water transport system in plants (xylem) Obtained from Minimization of energy expenditure by an organism. One parent branch of radius r with n daughter branches of radii r i r 3 = n i=1 r 3 i

Obtained from Minimization of energy expenditure by an organism.

21 21/ 36 Pennes s bioheat equation (1948) tissue q x artery T a T q x + q x x dx ρc T t = (k T) + q m + ωρ b c b (T a T) } {{ } perfusion T = tissue temperature q m = metabolic heat source rate [W/m 3 ] ω = perfusion rate, volumetric flow rate of blood per volume of tissue [s 1 ] T a = arterial blood temperature

22 22/ 36 Experiments Wissler s (1998) analysis of Pennes s data (resting human forearm) Experimental data and Pennes s theoretical values

23 23/ 36 Whole-body heat transfer Ferreira and Yanagihara (2009)

24 24/ 36 Heat transfer by conduction inside the body ρc T = (k T) + q m t Heat transfer to outside by radiation and convection q rad + q rad = C rc (T skin T outside ) Heat transfer to outside by evaporation (p = partial pressure) q evap = C e (p skin p outside ) Heat transfer between blood and tissue (Pennes s bioheat equation) ρc T t = (k T) + ωρ bc b (T a T) + q m Thermoregulatory system ω skin = K 1 (T h Th 0 } {{ } ) ( +K 2 Tskin T 0 ) skin hypothalamus

25 25/ 36

26 26/ 36 Berkeley Comfort Model (2001) Huizenga et al. (2001)

27 27/ 36 Counterflow model for extremities

28 28/ 36 Thermal resistance For electrical resistance Thermal resistance resistance = voltage difference electric current or R = Ṫ q q = T R

29 Thermal resistance representation of human body Top: exposed skin Second: clothed skin Third: clothed skin with conductive contact Fourth: bare skin 29/ 36

30 30/ 36 Steady state (left) and transient from 28 to 4.7 C (right)

31 Tumor detection Thermography of skin (Agnelli et al., 2011) Healthy tissue and tumor region in three-dimensional domain. Temperature distribution (left), and temperature profile on the skin surface (right). 31/ 36

32 32/ 36 Limitations of Pennes s model Assumption: leaving temperature = tissue temperature. Ignores Directional dependence of perfusion heat source. Different diameters of blood vessels (µm to mm range). Sharply varying material properties. Heat generation by necrosis. Vasculature geometry. Transvascular transport of energy and mass.

33 33/ 36 Continuum models: Chen-Holmes Single-temperature model ρc T t = (k T) + ωρ bc b (T a T) ρ b c b V T + (k p T) + q m Two-temperature model T s (1 α)ρ s c s t = (1 α) (k s T s ) + ha V (T f T s ) + (1 α) q s αρ f c f T f t = α (k f T f ) + ha V (T s T f ) + α q f

34 Vasculature-based models Weinbaum-Jiji-Lemons ρ b c b πr 2 b V dt a ds = q a bc b πr 2 b V dt v ds = q v c T t = (k T) + ngρ bc b (T a T v ) n πrb 2 ρ bc b V d ds (T a T v ) + q m Simplified Weinbaum-Jiji ρc T t = (k eff T) + q m { k eff = k 1 + n ( ρ b c b πrb 2 V cos γ) } 2 σ k 2 34/ 36

35 Non-Fourier heat transfer Conservation of energy ρc T t + q = 0 Fourier = parabolic heat conduction q = k T = ρc T t = (k T) Cattaneo-Vernotte = hyperbolic heat conduction τ q t + q = k T = τρc 2 T t 2 + ρc T t = (k T) Dual phase lag q τ q t + 2 T q = k T kτ T t x = τ q ρc 2 T t 2 + ρc T t = (k T) + ( kτ T 2 T t x ) 35/ 36

36 36/ 36 Experiments with processed meat Mitra et al. (1995): Hyperbolic heat conduction Antaki (2005): Dual phase lag

Basic Equations, Boundary Conditions and Dimensionless Parameters

Chapter 2 Basic Equations, Boundary Conditions and Dimensionless Parameters In the foregoing chapter, many basic concepts related to the present investigation and the associated literature survey were