a new fundamental bioheat equation for muscle tissue ... · a new fundamental bioheat equation for...

ature

m-Jijiods. Aatomy,eenhingionslocalwasterm

archto the

Liang ZhuDepartment of Mechanical Engineering,

University of Maryland, Baltimore County,1000 Hilltop Circle,

Baltimore, MD 21227e-mail: [email protected]

Lisa X. XuSchool of Mechanical Engineering,

Purdue University,West Lafayette, IN 47907

Qinghong HeDepartment of Mechanical Engineering,

University of Maryland Baltimore County,Baltimore, MD 21227

Sheldon WeinbaumDepartment of Mechanical Engineering,

City College of The City University of New York,New York, NY 10031

A New Fundamental BioheatEquation for Muscle Tissue—PartII: Temperature of SAV VesselsIn this study, a new theoretical framework was developed to investigate tempervariations along countercurrent SAV blood vessels from 300 to 1000mm diameter inskeletal muscle. Vessels of this size lie outside the range of validity of the Weinbaubioheat equation and, heretofore, have been treated using discrete numerical methnew tissue cylinder surrounding these vessel pairs is defined based on vascular anMurray’s law, and the assumption of uniform perfusion. The thermal interaction betwthe blood vessel pair and surrounding tissue is investigated for two vascular brancpatterns, pure branching and pure perfusion. It is shown that temperature variatalong these large vessel pairs strongly depend on the branching pattern and theblood perfusion rate. The arterial supply temperature in different vessel generationsevaluated to estimate the arterial inlet temperature in the modified perfusion sourcefor the s vessels in Part I of this study. In addition, results from the current reseenable one to explore the relative contribution of the SAV vessels and the s vesselsoverall thermal equilibration between blood and tissue.@DOI: 10.1115/1.1431263#

r

fl

o

e

le

n

de

w

r

ys.d byeen

re-ionur-no

on’sr-r

era-eatd tosue

ffi-cent

thetem-turee

ng

uili-we

and

y

m-wellselbe-suea-

selsing

i

1 IntroductionSeveral bioheat transfer models are currently used to exam

the effect of microvascular blood flow and/or vascular geomeon the local heat transfer@1–3#. These continuum models collectively describe the temperature distribution over a representacontrol volume, whose size is much larger than the scale ofthermally significant blood vessels and much smaller thanscale of the whole tissue. Due to the large range in size ofthermally significant vessels, it is difficult to define such a contvolume. The regions of validity of these continuum models habeen extensively discussed and clarified in the past@4–12#. Fur-ther, in applications where point to point temperature nonunimities are required near discrete large vessels, vascular modenecessary to accurately predict the tissue temperature fieldexpected, Crezee and Lagendijk@13# have found that blood flowin large, thermally unequilibrated vessels is the main causetemperature inhomogeneity during hyperthermia treatment.

In this paper, we will examine the thermal equilibrationblood in vessels between 300mm and 1000mm in diameter todetermine their contribution to the total blood-tissue heatchange. Lemons et al.@9# have shown that vessels in this sizrange occur nearly exclusively as countercurrent pairs in skemuscle and have a thermal signature indicating that they undpartial thermal equilibration with the surrounding tissue. Thecountercurrent vessels are often referred to as supply artery-pairs or SAV vessels in muscle tissue. They lie outside the raof validity of the Weinbaum-Jiji equation, which experiments atheory @14,15# have demonstrated, is limited to countercurrevessel pairs of,200 mm diameter. At present, there is no aequate theory for describing the thermal contribution of thlarge blood vessels, and they are too numerous to be individutreated in numerical simulations. This paper is Part II of a twpart investigation in which a new basic model is developedblood-tissue heat transfer in skeletal muscle tissue. Part Iconfined tos vessel tissue cylinders and vessels less than 300mmdiameter.

It is well known that the Pennes perfusion source term ove

Contributed by the Bioengineering Division for publication in the JOURNAL OFBIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Divsion Oct. 17, 2000; revised manuscript received Aug. 16, 2001. Associate EdJ. J. McGrath.

Copyright © 2Journal of Biomechanical Engineering

inetry-tivethethetheolve

or-s are. As

of

f

x-eetalrgoseveinnged

nt-seallyo-for

as

es-

timates the actual blood perfusion effect in tissue in two waFirst, it considers that all the heat leaving the artery is absorbethe local tissue and there is no venous rewarming. It has bsuggested by previous researchers@16,17# that a ‘‘correction co-efficient’’ which is less than unity and accounts for venouswarming should multiply the Pennes perfusion term. A correctcoefficient which is close to zero implies a significant countercrent rewarming of the paired vein and a coefficient of unityrewarming. In Part I of this study@18#, a new model for muscletissue heat transfer was developed using Myrhage and Erikssdescription@19# of a muscle tissue cylinder surrounding countecurrent secondary vessels~s vessels! as the basic heat transfeunit. The thermal equilibration of the returning blood in thes veinwas rigorously analyzed to determine the venous return tempture in the tissue cylinder. This led to a modified Pennes biohequation with a new perfusion source term, which can be usepredict average tissue temperature distribution in muscle tisregions containing vessels less than 300mm in diameter. It hasbeen shown that for most muscle tissues, the ‘‘correction coecient’’ varies between 0.6 to 0.8 suggesting that there is 20 perto 40 percent venous rewarming in thes vessel-tissue cylinder. Asecond limitation of the Pennes perfusion source term is thatarterial temperature is assumed to be equal to the body coreperature. Based on the analysis in Part I this arterial temperashould be the local SAV artery temperature at the inlet of thsvessel tissue cylinder. SAV vessels of 1000 to 300mm diameterachieve only partial thermal equilibration with the surrounditissue which is then completed when the blood enters thes vesseltissue cylinders. It is, thus, necessary to study the thermal eqbration along these partially equilibrated larger blood vessels ifwish to determine the relative importance of the SAV vesselsthe s vessels in the overall heat exchange.

As sketched in Fig. 1, thes vessel tissue cylinder is supplied bthe P vessels which branch off from the SAV vessel pairs. ThePvessels are relatively short and thus, the local arterial inlet teperature appearing in the modified Pennes source term isapproximated by the local arterial temperature in the SAV vespair. This temperature depends not only on the interactiontween the countercurrent SAV vessels but also on the local tistemperature distribution. The SAV vessels bifurcate from the mjor branching arteries that emanate from the central supply vesthat run along the axis of the limb. A central question in analyz

-itor:

002 by ASME FEBRUARY 2002, Vol. 124 Õ 121

122 Õ Vol. 124, FEB

Fig. 1 Proposed vasculature showing the relationship between the muscular arter-ies, the SAV tissue cylinder, and the s tissue cylinder in a human limb

m

r

e

a

ensdccs

a

cedpno

i

r

thisforthat

u-al

im-

ntinu-

useepingoce-pairmi-

fu-n ofnd,

eenwopat-

ible

300ntture

thes aren in

e-si-

eachthe

the countercurrent equilibration in the SAV vessels is to determhow much thermal energy leaving the artery returns to the cotercurrent vein and how much is lost to the surrounding tissThis redistribution of energy is a function of the local tissue teperature distribution and the vascular geometry. The anatostructure of the microvascular bed in thes vessel tissue cylindeand our modified Pennes equation were presented in Part I ofstudy. In Part II we primarily focus on the SAV vessel pairs.

Anatomically, theP, s, andt vessels are the first three branchof the final generations of a branching network that was initiadescribed in peripheral muscle tissue in Weinbaum et al.@3#. Thisnetwork is part of a space filling structure for all muscle tissuetypical circulatory system in a human limb is fed mainly by onetwo major pairs of countercurrent arteries and veins that runally along the limb. As schematically illustrated in Fig. 1, thblood vessels, which branch off from the main axial supply vsels, are defined as the muscular branches. These large brarun obliquely toward the skin surface and supply the large mubundles in skeletal muscle. These large bundles are suppliethe SAV vessel pairs which run along the length of the musSeveral hundred smalls vessel tissue cylinders comprise ealarge muscle bundle. The muscular branches and the SAV vedecrease in size from approximately 1000mm diameter and ap-proach 300mm diameter when they encounter theP vessels undernormal conditions in a human limb. A single SAV vessel psupplies numerousP vessels and thus a multitude ofs vesseltissue cylinders, as previously discussed in Weinbaum et al.@18#.The s vessels, theP vessels, and the SAV vessels form a spafilling countercurrent network characteristic of all muscle tissu

The temperature distribution in the SAV vessels is mainlytermined by the thermal interaction between the SAV vesseland its surrounding tissue, and by the temperature distributiothe muscular branches. Like thes vessels, the SAV vessels are nperfect countercurrent heat exchangers. If these vessels wereied in infinitely deep tissue all the energy lost from the SAV artewould enter the SAV vein. The presence of thermal interactbetween SAV pairs and the free surface can cause an asymmof this energy transfer. Some of the thermal energy leavingSAV artery does not enter the SAV vein. Because of their lasize, the thermal equilibration in the SAV vessels is at the monly partial and the equilibration length will increase at high

RUARY 2002

ineun-ue.

-mic

this

slly

. Aorxi-es-chescle

byle.hsels

ir

e-s.e-airin

tbur-ryonetry

thegeoster

flow rates in an exercising muscle. We intend to characterizeequilibration for the first time and obtain realistic estimatesTab as a function of branching distance from the axial vesselsrun along the length of the limb.

Strictly speaking, an investigation of the temperature distribtion in the SAV vessels requires the knowledge of the therminteraction between all the blood vessels and tissue, which ispractical. In this paper, contributions of theP vessels and theirsvessel branches to the total heat exchange are treated as a coous bleed-off from the SAV vessel pair~see Fig. 1!. The spacingof the large SAV vessels was never fully described in previoinvestigations that attempted to develop a model for the dmuscle tissue of the limb. Our solution approach for determinthe temperature decay in the SAV vessel pair is a two-step prdure. First, a large tissue cylinder surrounding the SAV vesselis defined using a new scaling analysis which combines anatocal analysis, Murray’s law and the assumption of uniform persion. This analysis leads to a generalized anatomic descriptioboth the SAV vessel pair and its tissue cylinder geometry. Secothe boundary value problem for the thermal interaction betwthe SAV vessels and the tissue cylinder is solved for the tvascular branching patterns shown in Fig. 2. These branchingterns, which we describe next, represent two limits of possthermal interaction between blood vessels and tissue.

2 Description of the SAV Tissue CylinderUnlike the vascular arrangements of blood vessels less than

mm in diameter, which have been studied in detail for differemuscle tissues, there is no systematic description in the literaof vessel branching patterns for vessels between 300 and 1000mmin diameter. Generally speaking, the branching patterns formuscular branches and SAV vessels in humans and mammaleither one or a combination of the two general patterns showFig. 2. The axial variation of the blood vessel radius ‘‘a’’ and theSAV tissue cylinder radiusRt surrounding the vessels can be dtermined by introducing flow continuity and several other phyological constraints as follows.

The first pattern~Fig. 2~a!! is called ‘‘pure branching.’’ In thispattern one vessel branches to form two equal vessels insuccessive generation and there is no capillary bleedoff from

Transactions of the ASME

Journal of Biomechanic

Fig. 2 Two vascular branching patterns for the SAV vessel pairs. Note thateither the countercurrent SAV artery or vein is shown.

nhnrpn

eoo

na,

s

ves-

byate.

theternthech-

eneings

fliresn

the

vessels. This pattern has been observed in skeletal muscle irat @20# and is also prevalent in the first few bifurcations from tmain supply vessels in a muscle that does not have an exteaxial structure. During exercise, the blood flow can be redistuted and directly transported to the cutaneous layer withoutfusing the surrounding tissue. The SAV vessels in the pure braing pattern can be viewed as a tapered vessel pair consisting ovessel branches from all previous generations which containsbranch in each vessel generation. If one assumes that the vbifurcates into two equal-sized branches, the ratio of the blflow rate in the mother vessel to that in the daughter vessel shbe a constant based on mass conservation; thus, 2Qi 115Qi .Mathematical rearrangement of this relationship leads toQi 112Qi /Li52Qi 11 /Li , whereLi is the length of vessel generatioi. If one assumes that the left side of the above relationship caapproximated asdQ/dz, the above equation can be rewrittendQ/dz52Q/Li . Combining this relationship with Murray’s lawQ}a3, one finds that 3a2(da/dz)52a3/Li . The vessel lengthLi , in general, decreases with the vessel radiusa. Previous ex-perimental measurements of the blood vessel lengthLi and thevessel radiusa by Song et al.@14# led to an empirical expressionLi(mm)50.067a1.102(mm) for blood vessels less than 200mm indiameter in skeletal muscle. Here we extrapolate this expresto larger blood vessels ranging from 300 to 1000mm in diameter.Whitmore’s data for a 20 kg dog@21# shows that this relationshipprovides a reasonable fit for all vessels up to 1 mm diameTherefore, substitutingLi into the above relationship, one obtain

da

dz52

a20.102

0.02012. (1)

The axial variation of vessel radius ‘‘a’’ can then be determinedby integrating Eq.~1! with an initial radiusa0 at z50.

Flow continuity requires that nQ5constant, where n51,2,4,8, etc. is the number of artery-vein pairs in each succeing generation. When it is combined with Murray’s law,Q}a3, nwill scale asn}1/a3. In the pure branching pattern we require ththe total tissue cross-sectional area surrounding all vessels ingeneration remain constant. Thus, ifRt is the local radius of thetissue cylinder surrounding a vessel pair andpRt

2 is its cross-

al Engineering

theededib-er-ch-f theonesseloduld

nbe

s

,

ion

ter.s

ed-

ateach

sectional area, this cross-sectional area is halved each time asel divides as shown in Fig. 2~a!. A single vessel in generation 1will have its tissue cylinder cross-sectional area first sharedtwo, then four vessels, and so forth, as it continues to bifurcThus, nRt

2 is a constant. Combining the requirementsnRt2

5constant andn}1/a3, one finds thatRt2}a3 and the axial varia-

tion of Rt can be determined froma. The integrated Eq.~1! andthese scaling relations require that the vessel radiusa, blood ve-locity u, and tissue cylinder radiusRt satisfy

a51.102A25.5z1a01.102, u5u0~a/a0! , Rt5Rt0~a/a0!3/2

(2)

where the subscript 0 represents values at the entrance (z50) ofthe tissue cylinder.

The second branching pattern shown in Fig. 2~b! is called ‘‘pureperfusion.’’ In this branching pattern the SAV vessel tapers assmaller vessels branch from the main axial vessels. This patdescribes the SAV vessel branching pattern shown in Fig. 1 forlong axial muscles in the human limb. The smaller vessels braning from the SAV vessels are theP vessels, which then supply thsubsequents vessel tissue cylinders. In this case there is only oSAV vessel pair and the corresponding tissue cylinder surroundit is of uniform radius in the axial direction. Murray’s law requirethat the local blood flow rateQ is proportional to the cube ovessel radius,Q}a3. If uniform perfusion is assumed in the locatissue cylinder surrounding the SAV vessels, then one requthat dQ/dz}Rt

2. Combining the uniform perfusion assumptioand Murray’s law, one obtains

a2~da/dz!}Rt2. (3)

For a long muscle of constant radius,Rt5Rt0 , Eq. ~3! leads toda/dz}a22. In this case, the axial variations ofa, u, andRt areexpressed as

a5a0A3 12z/C, u5u0A3 12z/C, Rt5Rt0 (4)

where C is a constant determined by the vessel length andchange in vessel radius betweenz50 and z5L which will becalculated later.

FEBRUARY 2002, Vol. 124 Õ 123

a

T

kessiscity

tesed.er

ap-res,

as

thetnd-ion

tionst al.

he

tot ex-hethe

ulti-e-of

3 Mathematic FormulationOne tapered countercurrent SAV vessel pair~SAV vessels! is

considered in each tissue cylinder. The axial variation of theometry is determined either by Eqs.~2! or ~4!. Referring to Fig. 3,the principal simplifying assumptions of the model are~i! both theflow and temperature fields are steady;~ii ! the SAV artery andvein are located symmetrically in the cross-sectional plane.simplify the analysis in the cross-sectional plane, the arteryvein are assumed to be of the same size, the same bloodvelocity, and the same vessel eccentricity. Thus, one has

aa* 5av* 5a* ~z!, aa* uz505av* uz505a0*

ua* 5uv* 5u* ~z!, ua* uz505uv* uz505u0* (5)

sa* 5sv* 5s* ~z!, l a2v* ~z!52s* .

Here the subscriptsa, v refer to the SAV artery and vein, respectively. Asterisks denote dimensional variables or parameters.s* isthe vessel eccentricity, andl * is the vessel center to center spaing; ~iii ! the effect of branching from the SAV vessel in Fig. 2~b!is treated as a continuous bleedoff from the SAV pairs;~iv! Weassume that the vessel eccentricity and the vessel center to cspacing follow the same axial variation as the vessel radius.requirement of uniform perfusion provides a relationship betwea0* , L* , andRt0* . These assumptions, when combined with E~2! and Eq.~4!, lead to the following expressions:

pure branching pattern

a* ~z* !51.102A25.5z* 1a0*1.102, u* 5u0* ~a* /a0* !,

Rt* 5Rt0* ~a* /a0* !3/2 (6)

s* ~z* !5s0* ~a* /a0* !, l * ~z* !5 l 0* ~a* /a0* !

and

pure perfusion pattern

a* 5a0* A3 12z* /C* , u* 5u0* A3 12z* /C* , Rt* 5Rt0* (7)

C* 5L* /@12~ae* /a0* !3#, s* ~z* !5s0* a* /a0* ,

l * ~z* !5 l 0* a* /a0*

Fig. 3 The geometry of the cross-sectional plane and coordi-nate system in the tissue cylinder

124 Õ Vol. 124, FEBRUARY 2002

ge-

Tond

flow

-

c-

enterheenq.

where L* is the cylinder length,a0* is the vessel radius atz*50 andae* is the vessel radius atz* 5L* ; ~v! the velocity field inthe blood vessels can be obtained by solving the Navier-Stoequations for an incompressible fluid. Similar to the analygiven in Part I, one obtains parabolic expressions for the veloprofile in the axial direction of the vessels;~vi! axial conduction isneglected in the vessel region due to the high blood flow ra@22#. However, axial conduction in the tissue region is consider]2Tt* /]z* 2 is approximated by its value on the tissue cylindsurface,d2Tt local* /dz* 2; ~vii ! the temperature gradient]Ta,v /]z inthe convective terms of the vessel energy equations can beproximated by the axial gradient of the vessel bulk temperatudTab,vb /dz, as previously justified@22#.

The nondimensional parameters and variables are defined

sa,v5sa,v*

a0*, r a,v5

r a,v*

a0*, aa,v5

aa,v*

a0*, l 5

l *

a0*, R5

R*

a0*,

Rt5Rt*

a0*, C5

C*

a0*, z5

z*

a0*, L5

L*

a0*,

(8)

ua,v5ua,v*

u0*, Pe05

2r fCfa0* u0*

kf, Ta,v,t5

Ta,v,t* 2Tt local* ~z!

Tab0* 2Tt local* ~L !,

F~z!5Tt local* ~z!

Tab0* 2Tt local* ~z!.

The dimensionless tissue temperatureTt is defined such thatTt50 at the cylinder surface. All length variables are scaled byvessel radiusa0* at z* 50. Tab0* is the artery bulk temperature az* 50. The tissue temperature on the outer tissue cylinder bouary Tt local* is prescribed and need not be constant. The functF(z) is determined by the prescribedTt local* . Employing theseassumptions, one can rewrite the dimensionless energy equafor the blood vessels and the tissue as given in Weinbaum e@18#

1

r a,v

]

]r a,vS r a,v

]Ta,v

]r a,vD1

1

r a,v2

]2Ta,v

]fa,v2

5 f Pe0ua,vS 12r a,v

2

aa,v2 D S dTab,vb

dz1

dF

dzD ,

r a,v<aa,v~z!, z<L(9)

f 51 for artery; f 521 for vein

1

R

]

]R S R]Tt

]R D11

R2

]2Tt

]f2 52d2F~z!

dz2 1BP@Tab~z!2F~z!#,

R<Rt~z!, r a,v>aa,v~z!(10)

BP50 for pure branching pattern;

BP50.7vr fCfaa0*

2

ktfor pure perfusion pattern

wherev is the local blood perfusion rate due to the bleedoff in tpure perfusion pattern. The Pennes-like source term@18# on theright side of Eq.~10! for the pure perfusion pattern takes inconsideration the net heat release due to countercurrent heachange in thes vessel tissue cylinder, as described in Part I. Toverall heat released by the bleedoff should be proportional totemperature difference between the SAV artery and tissue, mplied by a ‘‘correction coefficient’’ to account for the venous rwarming in thes vessel tissue cylinder. 0.7 is an average valuethe correction coefficient calculated by Weinbaum et al.@18# forvarious muscle tissues.


s

r

h

no

o

n

b

r

h

hingaal.

de-

s to

epled

The continuity of temperature and heat flux on the vesselfaces and the convection boundary condition on the tissue cyder surface require that

H Ta,v5Tt , for r a,v5aa,v~z!

]Ta,v

]r a,v5

]Tt

]r a,v, for r a,v5aa,v~z!

(11)

Tt50, for R5Rt (12)

In Eq. ~9!, Tab andTvb are the artery and vein bulk temperaturerespectively. These bulk temperatures are defined as

Tab,vb52

paa,vE

2p

p E0

aa,v

Ta,vS 12r a,v

2

aa,v2 D r a,vdra,vdfa,v .

(13)

The fundamental differences between the boundary value plem in the foregoing formulation and that in Part I are~i! axialconduction is considered in the tissue region;~ii ! the cross-sectional area of the tissue cylinder, the vessel radius, andblood flow velocity can change in the axial direction;~iii ! there isan axial temperature variation on the tissue cylinder surface;~iv! the effect of blood bleedoff from the SAV vessels on ttissue temperature field is described by a Pennes-like sourcedeveloped in Part I. These will substantially change the boundvalue problem in the axial direction without significantly changithe boundary value problem in the cross-sectional plane. The ltissue temperature on the outer boundary of the tissue cylinTt local* (z) is prescribed. This determinesF(z).

4 Solutions for the Countercurrent FlowA theoretical solution to the complicated boundary value pr

lem summarized in Section 3 can be obtained by modifyingsolution procedure outlined in Wu et al.@23#. The simplificationsintroduced in the governing equations and the boundary cotions enable us to separate the variables and solve the bounvalue problem in the cross-sectional plane independent of thathe axial direction. Using this approach, the axial interactiontween vessels is reduced to a coupled system of two ordindifferential equations for the variation of the axial bulk tempetures in the SAV vessel pair.

The solutions forTa,v andTt are decomposed into a particulasolution and a homogeneous solution. These solutions whichisfy Eqs.~9!–~10! and matching conditions Eq.~11! are

F Ta

Tv

Tt

G5FTh

Th

Th

G1@M #332F d~Tab1F !

dzd~Tvb1F !

dz

G1H 1

4 F2d2F

dz2 2~Tab2F !BPG J

3F 2aa2 lnS r a

aaD1aa

21sa222r asa cosfa2Rt

2

2av2 lnS r v

avD1av

21sv222r vsv cosfv2Rt

2

R22Rt2

G .

(14)

The third matrix on the right side of Eq.~14! arises from the axialconduction in the tissue region and the source term. The tgroup of terms in the expressions forTa andTv are harmonic andsatisfy the matching conditions on the vessel surfaces, whilethird group of terms in the expression forTt satisfies the outsideboundary condition atR5Rt . @M #332 is a 3 by 2 matrix definedas

Journal of Biomechanical Engineering

ur-lin-

s,

ob-

the

ande

termarygcalder

b-the

di-daryt ine-arya-

rsat-

ird

the

@M #332

53uaS r a

221

4

r a4

aa22

3

4aa

2D 2uaav2 ln

r v

av

uvaa2 ln

r a

aauvS 2r v

211

4

r v4

av2 1

3

4av

2Duvaa

2 lnr a

aa2uaav

2 lnr v

av

4 Pe0

4

(15)

and Th is the homogeneous solution in the three regions.Th isgiven by

Th5uvaa2 Pe0H ba01(

n51

`

banRn cos@n~f2p!#J d~Tab1F !

dz

1uaav2 Pe0H 2bv02(

n51

`

bvnRn cos~nf!J d~Tvb1F !

dz.

(16)

The solutions for the blood vessels and tissue satisfy the matcboundary conditions~Eq. ~11!! on the vessel surfaces. Followingderivation similar to that presented in the Appendix of Wu et@23#, one can solve for the coefficientsban and bvn by applyingthe boundary condition atR5Rt

ba05bv05b0520.25 ln@Rt~z!/a~z!# (17)

ban5bvn5bn50.25sn~z!/@nRt2n~z!#, n51,2,3, . . . . (18)

The temperature solutions in the artery and vein are used totermine the bulk temperatures~Tab andTvb!. Substituting the ves-sel temperature expressions into Eq.~13! and evaluating thedouble integrals, one can relate the vessel bulk temperaturetheir gradients:

Tab5A11

d~Tab1F !

dz1A12

d~Tvb1F !

dz

1A13Fd2F

dz2 1BP~Tab2F !G (19)

Tvb5A21

d~Tab1F !

dz1A22

d~Tvb1F !

dz

1A23Fd2F

dz2 1BP~Tab2F !G (20)

where the coefficientsA11– 23 which depend onz and the localvascular geometry, are given by

A1152A225F(n50

`

bnsn211

96GPe0 ua2

52Pe0

ua2

4 H lnFRt

a S 12s2

Rt2D G1

11

24JA1252A215

1

4Pe0 ua2F(

n50

`

bnsn cos~np!1ln~ l /a!

4 G5Pe0

ua2

4 H lnFRt /a

l S 11s2

Rt2D G J (21)

A135A2350.25~Rt22s210.5!.

It is relatively straightforward to apply the new model. Thoriginal boundary value problem has been reduced to two coufirst-order ordinary differential Eqs.~19! and ~20! for the vesselbulk temperatures. The coefficients in~19! and ~20! are given by


126 Õ Vol. 124

Table 1 Tissue temperature distribution

Temperature Distribution at z* 50 at z* 5L*

Arterial temperatureTab* (z* ) model predicted 37~°C!

Venous temperatureTvb* (z* ) model predicted Tvb(L)50.3Tab(0)ConstantTt local* (z* ) any value between 30–36~°C! 30–36~°C! 30–36~°C!

Linearly decreasedTt local* (z* ) (30236)z* /L* 136 (°C) 36~°C! 30~°C!

Nonlinearly distributedTt local* (z* ) (30236)z* 2/L* 2136 (°C) 36~°C! 30~°C!

,e

l

uv

n

rr

a

ve

n.omasre

.4–9

al tof the

s inthela-

is-

m-

owsels.forew,ry

ental-tery

on-de-

the algebraic expressions in Eqs.~17!, ~18!, and ~21!, which de-scribe the thermal interaction in the cross-sectional plane. In pciple, the axial dependence of the vessel bulk temperatures caobtained once the appropriate boundary conditions in the adirection as well as the functionF(z) are specified. One of theboundary conditions is the artery inlet temperatureTab0* 537 °C atz50 and its dimensionless value can be determined accordinthe dimensionless temperature definition, while the other isrelationship between vessel bulk temperatures atz5L which isdetermined in Part I@18#. The difference betweenTab* (L* ) andTvb* (L* ) is equal to 0.6;0.8 timesTab* (L* )2Tt local* (L* ). Con-verting this relationship into the definitions of the dimensionlevariables, one obtainsTvb(L)50.2;0.4 Tab(L). In this paper anaverage value of 0.3 is used forTvb(L).

5 Results

5.1 Temperature Decay in SAV Vessels for Two BranchingPatterns. We considered three special cases where constantearly decreasing or quadratically distributed local tissue tempture distributions are used as the input forF(z) in the model.These three temperature variations are summarized in TabThe arterial temperature at the entrance,Tab0* , is equal to the bodycore temperature~37°C!. Under hyperemic conditions, which mabe induced by vasoactive drugs, it is reasonable to assumeonly small temperature variations exist within the limb and thTt local* (z* ) should be close to the body core temperature. Howelarge tissue temperature variations may occur under normal phological conditions. The tissue temperature is assumed to decrfrom 36°C to approximately 30°C as one proceeds from the ceof the limb to the periphery.

All the other parameters and thermal properties are taken fthe literature and are listed in Table 2. The blood velocity is pportional to the vessel radius as required by Murray’s law. Tradial size of the tissue cylinderRt0* was determined by requiringthat the blood perfusion rate in the pure perfusion pattern fornormal conditions be equal tov53 ml/100g/min, i.e., v5pa0*

2u0 /pRt0*2L* . This blood perfusion value is equivalent t

0.0005 s21 if the blood density is equal to 1000 kg/m3. Previousexperimental studies by Song et al.@15# suggested an eightfoldincrease in the blood flow rate when the vessel was manipul

, FEBRUARY 2002

rin-n bexial

g tothe

ss

lin-ra-

e 1.

ythats,er,ysi-easeter

omo-he

the

o

ted

from a constricted to a fully dilated condition using vasoactidrugs. Two physiological conditions were considered:~1! a nor-mal condition:v53 ml/100 g/min; and~2! a hyperemic condition:v524 ml/100g/min corresponding to drug induced vasodilatioL* is the distance over which the SAV vessel pair bifurcates fr1000 to 300mm in diameter under normal conditions and wcalculated from Eq.~6!. We shall examine how blood temperatudecays in the countercurrent vessel pair over a distance ofL*512 cm for the two vessel branching patterns shown in Fig. 2

For convenience the dimensionless temperatures in Figs.are defined as

Tab,vb,t,t local~z!5Tab,vb,t,t local* ~z* !2Tt local* ~L* !

Tab0* 2Tt local* ~L* !. (22)

Accordingly, the dimensionless arterial temperature atz50 is al-ways equal to 1. A calculated dimensionless temperature equzero implies a temperature equal to the surface temperature otissue cylinder atz* 512 cm.

Figure 4 shows the dimensionless temperature distributionthe SAV artery and vein for the pure branching pattern for bothnormal and hyperemic conditions. Note that in this first calcution the tissue temperature at the cylinder surfaceTt local* (z* ) isuniform in the axial direction. At the end of the cylinder (z*512 cm), the arterial blood nearly equilibrates with the local tsue temperature under normal conditions~thin solid line!. How-ever, for hyperemic conditions, the dimensionless arterial teperature decays only 10 percent atz* 512 cm ~heavy solid line!.This is expected since for hyperemic conditions the blood flrate is higher and results in less heat loss from the blood vesNotice also that the venous rewarming of the blood is smallerthe hyperemic conditions. From an energy balance point of viheat lost from the SAV artery by conduction through the artewall must enter either the tissue cylinder or its countercurrvein. Therefore, the temperature increase in the SAV vein isways smaller than the temperature decrease in the SAV arunless the tissue cylinder is infinitely large.

For the pure perfusion pattern shown in Fig. 5, the dimensiless arterial temperature defined in Eq.~22! decreases to 0.23 an0.92 atz* 512 cm for normal and hyperthermia conditions, rspectively. Note thatRt* is axially uniform in the pure perfusion

rn

Table 2 Thermal properties and geometric parameters

Pure Branching Pattern Pure Perfusion Patte

SAV vessel radiusa0* normal:a* (0)5500mm, a* (L* )5150mmhyperemic:a* (0)51000mm, a* (L* )5300mm

SAV vessel lengthL* 12 cmBlood perfusion ratev~for pure branching pattern!

normal: 3 ml/100 g/min or 0.0005 s21;hyperemic: 24 ml/100 g/min or 0.004 s21;

Vessel eccentricitys0* normal:s0* 5625mm; hyperemic:s0* 51250mmBlood flow velocityu0* normal:u0* 5100 mm/s; hyperemic:u0* 5200 mm/sTissue cylinder radiusRt0* Rt0* 520.14 mmConstantC* ~Eq. ~7!! C* 5123.3 mmDensityr f 1000 kg/m3

Specific heatCf 3600 J/kg KThermal conductivitykf 0.5 J/mK


Journal of Biomechan

Fig. 4 Axial temperature distributions in the SAV artery and vein for the purebranching pattern under both normal and hyperemic conditions

i

u

o

rn

ff is

xialforthesolid

xialerialat

rial

uret

pattern while it decays in the axial direction in the pure branchpattern. If one compares the results given in Figs. 4 and 5, itbe found that less heat is lost from the SAV artery in the pperfusion pattern and, thus, a smaller temperature decay isserved than that in the pure branching pattern. Another interesobservation from Figs. 4 and 5 is that, unlike the pure branchpattern, the temperature increase in the vein in the pure perfupattern can be either larger or smaller than the temperaturecrease in the artery depending on the blood perfusion rate.fundamental difference between the two branching patterns isbleedoff from the SAV vessels in the pure perfusion pattern. Ttotal energy lost in the arterial side is not directly proportionalthe temperature difference between two axial locations. Eneleaving the SAV artery consists of both the conduction loss frits surface and the advection due to the bleed off. Both conducand advection contribute to the rewarming of the countercurSAV vein. The venous temperature increase can be larger tha

ical Engineering

ngcanreob-

tingingsionde-

Thethehetorgym

tionentthe

artery temperature decay, ifTvb(L) is smaller thanTab0 . This ismore pronounced for the hyperemic condition since the bleedosignificant.

The effect of the axial tissue temperature variation on the aarterial temperature decay is shown in Figs. 6–9. Figure 6 isthe pure branching pattern under normal conditions. Herecurves represent three tissue temperature profiles. The heavyline is for a uniformTt local , a linearly decreasingTt local is denotedby the thin solid line, while a nonlinearly distributedTt local isrepresented by the dashed line. For the normal condition, the atissue temperature variation has a modest effect on the arttemperature profile; while its effect is smaller for the total heloss from the artery for the pure branching pattern atx5L. Incontrast, for hyperemic conditions shown in Fig. 7, the artetemperature profiles vary little~,5 percent! with the tissue tem-perature variation. Similar conclusions can be drawn for the pperfusion pattern~Figs. 8 and 9!. Variations in the temperature a

Fig. 5 Axial temperature distributions in the SAV artery and vein for the pureperfusion pattern under both normal and hyperemic conditions


128 Õ Vol. 124, FEBR

Fig. 6 Effect of the temperature variation at the tissue cylinder surface on thetemperature decay in the SAV artery under normal conditions for the purebranching pattern

l

t

1

stotalcon-then-

eatforre-

ons.tionnge.the

the tissue cylinder surface result in less than 10 percent anpercent changes in the arterial temperature atz* 512 cm for thispattern under normal and hyperemic conditions, respectively.

The present model for the SAV vessels makes it possibleevaluate the contribution of various size vessels to the total blotissue heat exchange under different physiological conditions.the pure branching pattern, the total heat released from the bvessels to the tissue is proportional to the temperature differebetweenTab and Tvb at the SAV vessel entrance. Similarly, thheat released from thes vessels to the tissue is proportionalTab2Tvb at the SAV vessel exit. The percentage of heat releafrom the blood vessels prior to thes vessels is represented by2@Tab* (L* )2Tvb* (L* )#/@Tab* (0)2Tvb* (0)#. The larger this value,the greater the fractional heat release by the SAV vessels c

UARY 2002

d 2

tood-Foroodnceeosed

om-

pared to the smaller downstreams vessels. Table 3 summarizethe percentage of heat released in the SAV vessels to thevessel-tissue heat exchange. One observes that for hyperemicditions, far less heat is released in the SAV vessels than insmaller downstreams vessels and their subsequent branches. Uder normal conditions, more than 48 percent of the total hexchange occurs in the region containing large SAV vesselsthe pure branching pattern. However, less than 10 percent isleased in the SAV vessels for the assumed hyperemic conditiThe local tissue temperature variation does affect the contribuof different sized vessels to the total blood-tissue heat exchaIts effect is more pronounced for the normal condition than forhyperemic condition.

Fig. 7 Effect of the temperature variation at the tissue cylinder surface on thetemperature decay in the SAV artery under hyperemic conditions for the purebranching pattern


Journal of Biomechani

Fig. 8 Effect of the temperature variation at the tissue cylinder surface on thetemperature decay in the SAV artery under normal conditions for the pure per-fursion pattern

me

a

c

i

ma-allthe

ationecay

o beec-romormur-

ves-

5.2 Temperature Decay in the SAV Vessels in HumanLimb. Based on results obtained in Section 5.1 for the two liiting vessel branching patterns, we shall now consider the tperature decay in the SAV vessels of a human limb. Unfortunatno detailed anatomic data are available for blood vessels lathan 300mm diameter in the human limb, equivalent to the Myrhage and Eriksson data for thes vessel tissue cylinder in Part@18#. We assume that for the blood vessels to be space filling,vascular structure reproduces itself on a larger scale. As sketin Fig. 1, the SAV vessels bifurcate from the large muscubranches and then run roughly parallel to the skin surface. ESAV vessel pair supplies a large tissue cylinder of uniform radvia severalP vessels and their subsequents vessels andt vessels.

cal Engineering

-m-

ely,rger-

Ithehed

larachus,

The temperature variation in the muscular branches from thejor supply artery and vein for the limb is assumed to be smcompared to that in the SAV vessels. This assumption is due tolarge size of the muscular branches and is based on our calculin Section 5.1 where less than 10 percent of the temperature doccurs in vessels between 1000 and 800mm in diameter. Thetemperature at the SAV tissue cylinder surface is assumed tuniform since it is parallel to the skin surface. Thus, in this stion, the temperature decay in the SAV vessels is determined fthe thermal interaction between a tapered vessel pair with unifbleed off, and a uniform radius tissue cylinder with constant sface temperature.

Several parameters such as the tissue cylinder radius, SAV

Fig. 9 Effect of the temperature variation at the tissue cylinder surface on thetemperature decay in the SAV artery under hyperemic conditions for the pureperfusion pattern


rle

sl

enl

o

h

n

qual

ves-AVandtwoe ofes.

s-odethees

theer-ed

turean

n-g/beiongthili-in.

per-em-n the

eatAVrfu-

.

theat

sel size at the entrance and exit, and the tissue cylinder lengthneeded to simulate the temperature field. In Myrahage and Eson’s@19# description of thes vessel tissue cylinder, many muscfibers ~100 or more! supplied by thet vessels combine to creatones vessel tissue cylinder. In the same manner, we proposemanys vessel tissue cylinders combine to provide a larger tiscylinder supplied by the SAV vessel pair. The SAV vessels pthe same role as thes vessels in our smallers vessel tissue cylin-der except that they are one order of magnitude larger. Similathe P vessels are equivalent to thet vessels except that thePvessels are one order of magnitude larger. The average radii oP and t vessels in the different muscle tissues listed in Table 1Part I @18# are 110mm and 12mm, respectively. Applying a simi-lar scaling law, one finds that the SAV vessel at its entrancapproximately 350mm in radius if thes vessel radius has aaverage value of 38mm. The SAV tissue cylinder length is hathe distance between any two adjacent large muscular brancGray’s anatomy text indicates that there are typically three or flarge muscular branches in the human upper limb@24#. The dis-tance between any two adjacent muscular branches will beproximately 16 cm if the length of the upper limb is 48 cm. Tblood flow in the SAV artery supplies a tissue cylinder 8 cmlength. The total blood flow rateQ in the SAV artery is thencalculated as 1.61 ml/min. Thus, the tissue cylinder radius ca

Table 3 Percentage of total heat loss prior to the s vessels

Conditions Pattern

UniformTt local

~percent!

LinearlyDecreased

Tt local~precent!

NonlinearTt local

~precent!

Normal pure branching 83.8 64.8 48.5

Hyperemic pure branching 9.0 4.1 2.2

Table 4 Vascular parameters and blood flow in human limb

a0*~mm!

ae*~mm!

Rt*~mm!

C*~mm!

L*~mm!

Q0~ml/min…

v~ml/100g/min!

Normal 350 150 14.6 86.8 80 1.61 3Hyperemic 700 300 14.6 86.8 80 12.9 24


, areiks-e

thatueay

rly,

f thein

is

fhes.ur

ap-ein

be

determined by requiring the average blood perfusion rate be eto 3 ml/100g/min under normal conditions. This will lead toRt0514.6 mm.

Another parameter that needs to be determined is the SAVsel size at the midpoint of the tissue cylinder where two Svessel pairs running in opposite directions meet. MyrahageEriksson@19# observed that the average distance between anyadjacentP vessels is about 12 mm; thus, there are an averag13 ~16 cm/12 mm! P vessels between major muscular branchThe blood flow rate in the centralP vessel at the midpoint of theSAV tissue cylinder should be equal toQe , the blood flow rate inthe SAV at the juncture of the two SAV vessel pairs. If one asumes that all theP vessels have the same blood supply, the blosupply in eachP vessel will be 1/13 of the blood supply to thSAV artery at its entrance. Using Murray’s law, one finds thatblood velocityu is proportional to the vessel radius. This requirthat the SAV vessels be approximately 300mm in diameter at theend of the large tissue cylinder. Under hyperemic conditionsSAV vessels will dilate 100 percent to achieve a local blood pfusion rate of 24 ml/100g/min. Table 4 lists all the estimatparameters.

Figures 10 and 11 show model predictions for the temperadecay along the SAV artery and vein, respectively, in a humlimb for different blood perfusion rates ranging from normal coditions of 3 ml/100g/min to hyperemic conditions of 24 ml/100min. It is shown that up to 90 percent thermal equilibration canachieved in SAV arterial blood depending on the blood perfusrate. The SAV artery has a longer thermal equilibration lenwhen the blood flow is higher and only 20 percent thermal equbration is reached under hyperemic conditions of 24 ml/100g/mIn contrast to the arterial temperature decay, the local bloodfusion rate seems to play a minor role in the venous return tperature since the dimensionless venous return temperature iSAV vein is ;0.4 independent ofv, as shown in Fig. 11. Themodel predicts that in the human limb, 40 percent of the hreleased from the artery is recaptured by its countercurrent Svein. The independence of the venous return on the blood pesion rate is similar to that found for thes vessel tissue cylinder inpart I @18#, and will be discussed more fully in the next section

6 DiscussionIn this study a vascular model is developed to simulate

thermal equilibration in the SAV artery and vein. It is shown th

Fig. 10 Temperature decay in the SAV artery in a human limb


Journal of Biomechani

Fig. 11 Temperature decay in the SAV vein in a human limb

i

mut

e

on

r.

eTm

n

o

ss

pwt

t

tdy,ernsAVrea-

thatiscent

iti-fers inis aredm-ch-

em-as a

e as

ularom-

cha-he

dbe-

ees-ctng-teur-

40s is

tionter-ther-era-ure.ud-ue

m-

both the vessel branching pattern and the physiological conditare the main factors affecting the temperature decay in the Svessels. Under normal or hyperemic conditions without exterheating, the local tissue temperature variation seems to plaminor role in determining the temperature decay in the SAVtery. The current results provide two limits for the possible teperature variation in the SAV artery. This SAV arterial temperatdecay, as well as the correction coefficient defined in Part I ofstudy, can be used in the modified perfusion source term to maccurately estimate the net heat release from the countercurrsvessel pair in thes tissue cylinder. One of the shortcomings of athe continuum models is that they are incapable of accurapredicting point-to-point temperature variation in the vicinitylarge blood vessels. This study shows that for hyperemic cotions encountered in hyperthermia it is a reasonable assumptiouse the body core temperature as the arterial inlet temperatuthe modified perfusion source term for thes vessel tissue cylinderThe lack of thermal equilibration will create large tissue tempeture gradients in the vicinity of 300;1000mm diameter vessels.

The search for a suitable energy equation that accuratelyflects the heat released or removed by the blood has been a ctheme in bioheat transfer research for nearly half a century.individual s vessel muscle tissue cylinder defined in the anatocal studies of Myrhage and Eriksson@19# is the largest repetitiveunit that is common to nearly all muscle tissue whether the tisis deep or peripheral. This suggested that a single bioheat traequation could be derived and applied to all muscle tissue whwas based on this anatomical unit. In Part I of this study, a mofied Pennes source term was derived in which a correction cficient was defined to account for the heat loss recaptured bycountercurrents vein. Roemer and Dutton@25# have also proposeda modified Pennes-like perfusion term based on a generic timodel for convective energy exchange. Although it is nontisspecific, the correction coefficient is rather complicated and hto evaluate. This coefficient depends on the total heat trancoefficients between tissue and blood in artery and vein, restively, which vary from generation to generation with the floThe correction coefficient also depends on the local venous reand the thermally significant arteries and veins. In comparison,model developed in this study gives a simple correction coecient for the Pennes source term, which ranges from 0.6 to 0.8muscle tissues. This correction is much easier to apply in pracIn the present paper the temperature distribution in the SAV art

cal Engineering

onsAV

naly aar-

-rehisorentlltelyfdi-n toe in

ra-

re-ntralhei-

suesferichdi-ef-the

sueueardsferec-.

urn,theffi-for

ice.ery,

or the entrance temperature for thes vessel tissue cylinder thaappears in the new modified source term in Part I of this stuwas calculated for two vascular branching patterns. These pattrepresent two limits of possible thermal interaction between Sblood vessels and tissue. Thus, these two cases provide thesonable range of temperature variation along the SAV arteryone would anticipate for different physiological conditions. Itfound that for both vessel branching patterns, at least 77 perof the thermal equilibration is achieved in the SAV artery beforeenters thes vessel tissue cylinder for normal physiological condtions. However, for hyperemic conditions most of the heat transfrom the countercurrent blood vessels to the local tissue occurthe tissue region containing smaller blood vessels. Theresmaller temperature variation in the SAV arterial blood compato that under normal conditions. The dimensionless arterial teperature in the SAV vessels is almost a constant for both braning patterns during hyperemia. Thus, for this case the arterial tperature appearing in the modified source term can be treatedconstant whose value is close to the body core temperaturnoted earlier.

The heat exchange between countercurrent microvascartery-vein pairs has attracted widespread attention since the cbined theoretical and experimental studies by the authors@3# firstsuggested that this might be the dominant heat transfer menism in local microvascular blood-tissue heat transfer. TWeinbaum-Jiji bioheat transfer equation@26# was developed baseon the incomplete countercurrent exchange that took placetween the paired vessels. However, subsequent studies@14,15,27#showed that the limit for the validity of the expression for theffective conductivity describing these vessels was limited to vsel pairs of,200 mm diameter. The current study can prediaxial temperature distributions in countercurrent vessel pairs raing from 300 to 1000mm in diameter and is also able to evaluathe deviation of the thermal interaction from perfect countercrent heat exchange. The model predicts that for a human limbpercent of the heat lost from the SAV arteries and their brancherecaptured by their countercurrent vein. The thermal interacbetween the SAV vessels is thus, far from being perfect councurrent heat exchange. We also observe that for these largermally significant vessels the gradient of the mean tissue tempture does not follow the gradient of the mean blood temperatThis simulation confirms the conclusion of our experimental sties @15,27# that the Weinbaum-Jiji equation breaks down in tissregions with blood vessels.200 mm diameter.

The 40 percent venous rewarming in a human limb is a co


t

-i

s

n

a

f

kr

s

R

to

in

ntsParth.

ortch.

m-E

das-

hes,’’

. J.,in

.,er,’’

oferi-

hef-

lheatl

io-arge

t inula-

lar

s-

eir

n-,’’

in

tal0 to.

d-io-

ren-

g

n-h.

rJ.

al

bined venous rewarming that includes both the SAV ands vesseltissue cylinders, since the thermal interaction in thes vessel tissuecylinder was included in the boundary condition applied atx5L, the end of the SAV vessel tissue cylinder. The fact thatvenous return temperature does not depend onv is not a surpris-ing result. This was previously observed for thes vessel tissuecylinders in Weinbaum et al.@18#. This behavior is the consequence of several assumptions used in the model. First, thereaxial temperature gradient in the SAV tissue cylinder. Thus,countercurrent heat exchange is the only thermal driving forcethe blood-tissue heat transfer. Second, the same mass flow isumed in each vessel of the SAV vessel pair. Finally, if one cobines the boundary value problems for both the SAV ands vesseltissue cylinders, the flow must vanish at the end of the combitissue cylinder where both the artery and vein temperaturesequal to the local tissue temperature. In contrast, if there isaxial thermal gradient in the tissue cylinder, the blood flow rcan play a significant role in the venous rewarming. As shownTable 3, the local tissue temperature profile has a significant eon the redistribution of the thermal interaction in different tissregions. The percentage of the total heat loss prior to thes vesselsis decreased by 35 percent when the uniform tissue temperadistribution is replaced by a quadratically distributed local tisstemperature for the normal perfusion rate.

In summary, this study presents a new theoretical frameworinvestigate axial temperature variations along countercurblood vessels from 300 to 1000mm diameter in skeletal muscleThe thermal interaction between the blood vessel pair andsurrounding tissue was investigated for two vascular branchpatterns. It was shown that temperature variations along thlarger thermally significant vessels depend strongly on the valar geometry and local blood perfusion rate. The arterial suptemperature to thes vessel tissue cylinders was evaluated aused in the modified perfusion source term in Part I of this stuto evaluate the relative contribution of the SAV vessels and thsvessels to the overall blood-tissue heat exchange.

AcknowledgmentThis research was supported by a Whitaker Biomedical

search Grant to Liang Zhu.

Nomenclature

a 5 vessel radiusC 5 constant in Eq.~4!

Cf 5 specific heat of bloodk 5 thermal conductivityl 5 vessel center to center spacing

Li 5 vessel length in vessel generationiL 5 total length of SAV vessels

Pe 5 blood flow Peclet numberr 5 radial coordinateR 5 radial coordinate

Rt 5 radius of the tissue cylinders 5 vessel eccentricity

Tab0 5 arterial blood temperature at its entranceTt local 5 temperature at the tissue cylinder surface

u 5 flow velocity component in the axial direction in theblood vessels

x,y 5 Cartesian coordinatesr 5 densityf 5 polar angle in cylindrical coordinate

Subscripts

a 5 arteryb 5 bulkf 5 fluid in vesselsh 5 homogeneous solutiont 5 tissue

v 5 vein0 5 z50


he

s notheforas-

m-

edareantein

fectue

tureue

toent.theingesecu-plynddy,e

e-

Superscripts

• 5 dimensional parameters and variables

References@1# Chen, M. M., and Holmes, K. K., 1980, ‘‘Microvascular Contributions

Tissue Heat Transfer,’’ Ann. N.Y. Acad. Sci.,335, pp. 137–150.@2# Pennes, H. H., 1948, ‘‘Analysis of Tissue and Arterial Blood Temperatures

the Resting Human Forearm,’’ J. Appl. Physiol.,1, pp. 93–122.@3# Weinbaum, S., Jiji, L. M., and Lemons, D. E., 1984, ‘‘Theory and Experime

for the Effects of Vascular Microstructure on Surface Tissue Heat Transfer-I: Anatomical Foundation and Model Conceptualization,’’ ASME J. BiomecEng.,106, pp. 321–330.

@4# Baish, J. W., Ayyaswamy, P. S., and Foster, K. R., 1986, ‘‘Heat TranspMechanism in Vascular Tissues: a Model Comparison,’’ ASME J. BiomeEng.,108, pp. 324–331.

@5# Baish, J. W., Ayyaswamy, P. S., and Foster, K. R., 1986, ‘‘Small-Scale Teperature Fluctuations in Perfused Tissue During Local Hyperthermia,’’ ASMJ. Biomech. Eng.,108, pp. 246–250.

@6# Brinck, H., and Werner, J., 1994, ‘‘Estimation of the Thermal Effect of BlooFlow in a Branching Countercurrent Network Using a Three-Dimensional Vcular Model,’’ ASME J. Biomech. Eng.,116, pp. 324–330.

@7# Charny, C. K., Weinbaum, S., and Levin, R. L., 1990, ‘‘An Evaluation of tWeinbaum-Jiji Bioheat Equation for Normal and Hyperthermic ConditionASME J. Biomech. Eng.,112, pp. 80–87.

@8# Crezee, J., Mooibroek, J., Lagendijk, J. J. W., and Van Leneuwen, G. M1994, ‘‘The Theoretical and Experimental Evaluation of the Heat BalancePerfused Tissue,’’ Phys. Med. Biol.,39, pp. 813–832.

@9# Lemons, D. E., Chien, S., Crawshaw, L. I., Weinbaum, S., and Jiji, L. M1987, ‘‘The Significance of Vessel Size and Type in Vascular Heat TransfAm. J. Physiol.,253, pp. R128–R135.

@10# Roemer, R. B., Moros, E. G., and Hynynen, K., 1989, ‘‘A ComparisonBioheat Transfer and Effective Conductivity Equation Predictions to Expmental Hyperthermia Data,’’Proceedings of ASME Winter Annual Meeting,pp. 11–15.

@11# Valvano, J. W., Nho, S., and Anderson, G. T., 1994, ‘‘Analysis of tWeinbaum-Jiji Model of Blood Flow in the Canine Kidney Cortex for SelHeated Thermistors,’’ ASME J. Biomech. Eng.,116, pp. 201–207.

@12# Xu, L. X., Chen, M. M., Holmes, K. R., and Arkin, H., 1991, ‘‘The TheoreticaEvaluation of the Pennes, the Chen-Holmes and the Weinbaum-Jiji BioTransfer Models in the Pig Renal Cortex,’’Proceeding of ASME Winter AnnuaMeeting, Atlanta,189, pp. 15–22.

@13# Crezee, J., and Lagendijk, J. J. W., 1990, ‘‘Experimental Verification of Bheat Transfer Theories: Measurement of Temperature Profiles Around LArtificial Vessels in Perfused Tissue,’’ Phys. Med. Biol.,35, No. 7, pp. 905–923.

@14# Song, J., Xu, L. X., Weinbaum, S., and Lemons, D. E., 1997, ‘‘Enhancementhe Effective Thermal Conductivity in Rat Spinotrapezius due to Vasoregtion,’’ ASME J. Biomech. Eng.,119, pp. 461–468.

@15# Song, J., Xu, L. X., Weinbaum, S., and Lemons, D. E., 1999, ‘‘MicrovascuThermal Equilibration in Rat Spinotrapezius Muscle,’’ Ann. Biomed. Eng.,27,pp. 56–66.

@16# Brinck, H., and Werner, J., 1994, ‘‘Efficiency Function: Improvement of Clasical Bioheat Approach,’’ J. Appl. Physiol.,77, No. 4, pp. 1617–1622.

@17# Wissler, E. H., 1988, ‘‘Comments on Weinbaum-Jiji’s Discussion of ThProposed Bioheat Equation,’’ ASME J. Biomech. Eng.,108, pp. 355–356.

@18# Weinbaum, S., Xu, L. X., Zhu, L., and Ekpene, A., 1997, ‘‘A New Fundametal Bioheat Equation for Muscle Tissue: Part I—Blood Perfusion TermASME J. Biomech. Eng.,119, pp. 278–288.

@19# Myrhage, R., and Eriksson, E., 1984, ‘‘Arrangement of the Vascular BedDifferent Types of Skeletal Muscles,’’ Progress in Applied Microcirculation,5,pp. 1–14.

@20# He, Q., Zhu, L., Weinbaum, S., and Lemons, D. E., 2001, ‘‘ExperimenMeasurements of Temperature Variations Along Paired Vessels from 201000mm in Diameter in Rat Hind Leg,’’ ASME J. Biomech. Eng., in review

@21# Whitmore, R. L., 1968,Rheology of Circulation, Pergamon Press, London.@22# Zhu, L., and Weinbaum, S., 1995, ‘‘A Model for Heat Transfer from Embe

ded Blood Vessels in Two-Dimensional Tissue Preparations,’’ ASME J. Bmech. Eng.,117, pp. 64–73.

@23# Wu, Y. L., Weinbaum, S., Jiji, L. M., and Lemons, D. E., 1993, ‘‘A NewAnalytic Technique for 3-d Heat Transfer from a Cylinder With Two or MoAxially Interacting Eccentrically Embedded Vessels With Application to Coutercurrent Blood Flow,’’ Int. J. Heat Mass Transf.,36, pp. 1073–1083.

@24# Gray, H., 1974, ‘‘Anatomy, Descriptive and Surgical,’’ 1901 Ed., RunninPress, Philadelphia.

@25# Roemer, R. B., and Dutton, A. W., 1998, ‘‘A Generic Tissue Convective Eergy Balance Equation: Part I—Theory and Derivation,’’ ASME J. BiomecEng.,120, pp. 395–404.

@26# Weinbaum, S., and Jiji, L. M., 1985, ‘‘A New Simplified Bioheat Equation fothe Effect of Blood Flow on Local Average Tissue Temperature,’’ ASMEBiomech. Eng.,107, pp. 131–139.

@27# Zhu, L., Weinbaum, S., and Lemons, D. E., 1996, ‘‘Microvascular ThermEquilibration in Rat Cremaster Muscle,’’Ann. Biomed. Eng.,24, pp. 109–123.


a new fundamental bioheat equation for muscle tissue ... · a new fundamental bioheat equation for...

Documents