Table Of ContentAnomalous properties of heat diffusion in living tissue caused by branching artery
network. Qualitative description
I. A. Lubashevsky,1,2 V. V. Gafiychuk,2 and B. Y. Datsko2
1Theory Department, General Physics Institute, Russian Academy of Sciences
2Institute for Applied Problems in Mechanics and Mathematics, National Academy of Sciences of Ukraine
(Dated: February 1, 2008)
2 Weanalyzetheeffectofbloodflowthroughlargearteriesofperipheralcirculationonheattransfer
0 inlivingtissue. Bloodflowinsucharteriesgivesrisetofastheatpropagationoverlargescales,which
0 is described in terms of heat superdiffusion. The corresponding bioheat heat equation is derived.
2 In particular, we show that under local strong heating of a small tissue domain the temperature
distribution inside the surroundingtissue is affected substantially by heat superdiffusion.
n
a
J
I. INTRODUCTION. LIVING TISSUE AS A
5
HETEROGENEOUS MEDIUM
]
t
f Blood flowing throughvessels forms paths of fast heat
o transport in living tissue and under typical conditions
s
it is blood flow that governs heat propagation on scales
.
t
a aboutorgreaterthanonecentimeter(foranintroduction
m tothisproblemsee,e.g.,[1,2]). Bloodvesselsmakeupa
complex network being practically a fractal. The larger
-
d is a vessel, the faster is the blood motion in it and, so, FIG. 1: Schematic illustration of the effect of blood flow
n the strongeristhe effectofbloodflow inthe givenvessel through the vein tree on heat diffusion imitated by random
o on heat transfer. Blood flow in capillaries practically walks. Thefigureshowstrappingofarandomwalkerbecause
c
does not affect heat propagation whereas blood inside ofgettingtheinternalpointsof alarge veinafter passingthe
[
large vessels moves so fast that its heat interaction with vein node.
1 the surrounding cellular tissue is ignorable [1]. Thus,
v
there should be vessels of a certain length ℓ that are
7 v flow on heat transfer is reduced to the renormalization
the smallest ones among the vessels wherein blood flow
5 of the temperature diffusivity, D Deff, [5] and the
affects heat transfer remarkably. The value of ℓ can be →
0 v appearance of the effective heat sink fj [1, 3, 6] in the
1 estimated as [3] (see also [1, 2]): bioheat equation:
0
2 D ∂T
= (D T) fj(T T )+q . (2)
0 ℓv , (1) ∂t ∇ eff∇ − − a T
/ ∼sjfLn
at Here T(r,t)is the tissue temperature fieldaveragedover
m where D = κ/(cρ) is the temperature diffusivity of the scales about ℓ , the parameter T is the blood temper-
v a
cellular tissue determined by its thermal conductivity κ, ature insider the systemic circulation arteries, and the
-
d specific heat c, and density ρ, the value j is the blood summand q (r,t) called below the temperature gener-
T
n perfusion rate (the volume of blood going through tis- ation rate is specified by the heat generation rate q as
o sue region of unit volume per unit time), and the fac- q = q/(cρ). The renormalization of the temperature
T
:c tor Ln ∼ ln(l/a) is logarithm of the mean ratio of the diffusivity is mainly determined by the blood vessels of
v individual length to radius of blood vessels forming pe- lengthsaboutℓ andduetothefractalstructureofvascu-
v
Xi ripheral circulation. For the vascular networks made up lar networks the renormalization coefficient F = Deff/D
of the paired artery and vein trees where all the vessels is practically a constant of unity order, F &1 [3].
r
a are grouped into the pairs of the closely-spaced arter- Let us imitate the temperature evolution in terms of
ies andveins withopposite bloodcurrentsthe coefficient randomwalkswhoseconcentrationis(T T ). Thenthe
a
f L−1/2 accounts for the counter-current effect [15]. part of the vein tree made up of vessel−s whose lengths
n
∼
Forthevascularnetworkswherethearteryandveintrees exceedorareaboutthescaleℓ formsthesystemoftraps.
v
are arranged independently of each other the factor f Infact, bloodstreamsgoingthroughthe veintree merge
shouldbesetequaltounity,f =1. Inparticular,forthe into greater streams at the nodes (Fig. 1). Therefore
typical values of the ratio l/a 20–40 [4], the thermal an effective random walker after reaching the boundary
conductivity κ 7 10−3 W/∼cmK, the heat capacity of one of these veins inevitably will be moved by blood
c 3.5J/g K,a∼ndth·edensityρ ·1 g/cm3 ofthetissue flow into the internal points of large veins. Then, due to
as∼well as s·etting the blood perfu∼sion rate j 0.3min−1 relatively fast blood motion inside these veins it will be
∼
from(1)wegettheestimatesℓ 4mmandL 3 4. carried away from the tissue region under consideration,
v n
∼ ≈ −
In the mean-field approximation the effect of blood which may be described in terms of the walker trapping
2
or, what is the same, the heat sink [3]. Since the mean
distancebetweentheseveinsisdeterminedmainlybythe
shortestones,i.e. bytheveinsoflengthℓ themeantime
v
during which a walker wanders inside the cellular tissue
before being trapped is [3]
ℓ2
τ vL . (3)
n
∼ D
In obtaining the given expression we have assumed the
vascular networkto be embedded uniformly in the cellu-
lar tissue, so the tissue volume ℓ3 falls per one vein (and
v
artery, respectively) of length ℓ . Whence it follows, in FIG. 2: Schematic illustration of the anomalously fast heat
v
propagationcausedbybloodflowthroughlargearterytreein
particular, that the rate at which the walkers are being
termsofrandomwalks. Thefigureshowstheeffectivewalker
trapped by these vein, i.e. the rate of their disappear-
motion with blood from a large artery to a small one where
ance is estimated as 1/τ, leading together with expres-
thewalker leaves it wandering in thecellular tissue.
sion (1) to the heat sink of intensity fj in the bioheat
equation (2). The characteristic spatial scale of walker
diffusion in the cellular tissue before being trapped is
ℓ √Dτ, i.e. example, inside a cellular tissue neighborhoodof a point
T ∼ r can be found during the time τ inside a cellular tissue
neighborhoodofapointr′ atadistancemuchlargerthan
D
ℓ ℓ L 1cm. (4) ℓ ,i.e. r r′ ℓ . Thegiveneffectmayberegardedas
T v n v v
∼ ∼sjf ∼ anomal|ou−sly f|a≫st heat diffusion in living tissue, i.e. heat
p
superdiffusion.
The scale ℓ gives us also the mean penetration depth
T
Dealingwithheattransferinlivingtissuewemaycon-
of heat penetration into the cellular tissue from a point
fineourconsiderationtotheperipheralvascularnetworks
source or,what is the same,the widening of the temper-
typically embedded uniformly into the cellular tissue, at
aturedistributioncausedbyheatdiffusioninthecellular
leastatthe firstapproximation[8]. Thelatterstatement
tissue. It is the result obtained within the mean-field
means, in particular, the fact that for a fixed peripheral
approximation.
vascular network the vessel collection comprising all the
Beyondthescopeofthemean-fieldtheorywemeetsev-
arteries of length l meets the condition of the volume
eralphenomena. Oneofthemisthetemperaturenonuni-
l3 approximately falling per each one of these arteries.
formities caused by the vessel discreteness [7] which can
Therefore as is seen in Fig. 2 a walker going into a large
be described assuming the heat sink in equation (2) to
arteryoflength l atinitialtime during the time τ before
contain a random component [3]. Another is fast heat
beingtrappedby theveinscanbe foundequiprobablyat
transport over scales exceeding substantially the length
each point of the given artery neighborhood of size l. In
ℓ caused by the effect of blood flow through the artery
T
other words, this walker makes a large jump of length l
tree. The latter phenomena is the main subject of the
thatexceedssubstantiallythemean-fielddiffusionlength
present paper.
ℓ .
T
Inwhatfollowswewillanalyzethetemperaturedistri-
II. FAST HEAT TRANSPORT WITH BLOOD bution averaged over all the possible realizations of the
FLOW THROUGH LARGE ARTERY TREE vascular network embedding. This enables us to regard
a walker entering a large artery of length l as a random
Let at a certain time a random walker wandering in eventwhoseprobabilityis independent ofthe walkerini-
the cellular tissue get a boundary of a large artery, i.e. tial position. Thereby, the probability Pl for a walker to
an artery of length exceeding ℓ . It should be noted make a large jump over the distance l is also indepen-
v
that such an event is of low probability and cannot be dentofthespatialcoordinatesr. Itshouldbenotedthat
consideredwithinthestandardmean-fieldapproximation forafixedrealizationofthe vascularnetworkembedding
because the relative number of large arteries is small. the probability Pl depends essentially on the spatial co-
Due to the direction of the blood motion from larger ordinatesrandthe heattransferinliving tissueonlarge
arteries to smaller ones as well as the high blood flow scales has to exhibit substantial dependence on the spe-
rate in the large arteries the walker will be transported cific position of the heat sources.
fasttooneofthearteriesoflengthℓ (Fig.2). Theblood Now we estimate the value of P assuming the heat
v l
flow rate in smallvesselsis nothigh enoughto affect the sourcestobelocalizedinsideadomainQLofsize . Two
L
walker motion essentially and it has inevitably to leave differentfactorsdeterminethevalueofP . First,itisthe
l
this artery and wander in the cellular tissue until being process of walker trapping by an artery of length l >
L
trapped by the veins of length ℓv. Thereby a certain goingthroughthedomainQL. Ifl< bloodflowinthis
L
not too large number of random walkers generated, for arteryhas practically no effect onheat diffusion. On the
3
average a random walker during the time τ travels the
distance ℓ in the cellular tissue until being trapped by
T
theveinsoflengthℓ . Soforawalkertoenterthisartery
v
and, thus, to leave the domain QL with blood flow in it
thewalker,ononehand,shouldbelocatedatinitialtime
inside a cylindrical neighborhood Q of the given artery
l
whose radius is about ℓ andthe volume is ℓ2. Onthe
T L T
other hand, it has to avoid being trapped by the veins
of length ℓ . The probability of the latter event is about
v
(ℓ /ℓ )2. Indeed, aveinoflengthℓ maybe treatedasa
v T v
trap of cylindrical form. Thereby in qualitative analysis FIG. 3: Models of the peripheral artery network embedding
the walker trapping can be described in terms of two- into the cellular tissue, (a) four-fold node model used in the
dimensional random walks in the plane perpendicular to present analysis and (b) a more realistic dichotomic artery
treeuniformlyembeddedintoacellulartissuedomainM. In
the artery under consideration where the trapping veins
the qualitative description of heat transfer both the models
are represented by small circular regions. Their density
lead to thesame result.
is about 1/ℓ2 which directly leads to the latter estimate.
v
Thereforethetotalnumberofwalkersleavingthedomain
QL with blood flow through the given artery per unit For the infinitely long cylinder
time is
P(2e) L
1 ℓ 2 D l ∼ l
v ℓ2 (T T )= (T T ). (5) (cid:18) n(cid:19)
τ · ℓ ·L T · − a L L − a
(cid:18) T(cid:19) n and for the plane layer P(1e) 1. Multiplying g (r) by
l ∼ l
In obtaining (5) we have taken into account expres- the corresponding values of P(e) we get the result of av-
l
sion(3). Sincethetrappedwalkersspreaduniformlyover eraging the walker transition rate g (r) over the possible
l
aregionofsize l the resultingdensity ofthe walkertran- realizationsof the vascularnetwork embedding. The ob-
sitionrateto apoint rspacedata distanceaboutl from tained result is written as
the domain QL is
D d
g (r) L (T T ), (7)
g (r) D L(T T ). (6) h l i∼ Lnln2+d − a
l ∼ Lnl3 − a wherethevaluedactuallyplaystheroleofthedimension
of the space inside which the temperature field can be
It should be noted that the transition rate gl(r), as it considered. Atthenextstepweshouldsumtheterms(7)
must, does not depend onthe localvalue of bloodperfu- over all the levels of the large artery tree. However due
sion rate. to the strongincreaseofthe terms (7), g (r) 2n(2+d),
l
h i∝
At the second step we should average the obtained as the level number n increases the arteries of length
transition rate gl(r) over the possible realizations of the l r mainly contribute to the value of gl(r) . So the
∼ h i
vascular network embedding. Let us adopt a simplified term describing the fast heat transport with blood flow
model for the vascular network shown in Fig. 3a where throughlargearteriesfromthe domainQL (locatednear
the vessel lengths ln and ln+1 of the neighboring hierar- the origin of the coordinate system) can be written as
chylevelsnandn+1arerelatedasl =2l . Figure3b
n n+1
demonstrates a more adequate model for the peripheral g(r) D dr′ T(r′) , (8)
aprrteesreynttrqeueawlithaicthiv,ehaonwaelvyesris, wmitahyinbethreedfruacmedewtoorkthoefftohre- ∼ Ln ZM |r−r′|5
mer one by combining three sequent two-fold nodes into where is the region containing the peripheral vascu-
M
one effective four-fold node at all the levels. In this case lar network as a whole and the integration in the three
the cubic domain of volume l3 falls per each artery of dimensional space over the domain QL allows for all its
n
level n. Let us now consider individually three charac- three considered types.
teristic forms of the domain QL, a ball or a cube of size Expression (8) together with the mean-filed bioheat
(d=3), aninfinitely longcylinder ofradius (d=2), equation (2) enables us to write the following equation
Land a plane layer of thickness (d=1). For tLhe ball or governing the anomalous heat transfer in living tissue
L
the cube, i.e. a region bounded in three dimensions the
∂T D T(r′) T(r)
probability that an artery of level n passes through the =D 2T + dr′ − fj(T T )+q ,
domain QL is about ∂t ∇ Ln ZM |r−r′|5+ℓ5v − − a T
(9)
2
P(3e) L . where we have added directly the value ℓv in order to
l ∼(cid:18)ln(cid:19) cut off the spatial scales smaller than the length ℓv and
4
ignored the difference between the effective temperature givingusalsotheminimalsize oftheregionwherein
mim
L
diffusivity and the true one of the cellular tissue. Equa- the tissue temperature increase(T T ) is mainly lo-
max a
−
tion (9) is the desired governing equation of the anoma- calized. In the neighboring tissue the blood perfusion
lous fast heat diffusion in living tissue for the averaged ratekeepsupasufficientlylowvaluej ,whichmakesthe
0
temperature field. It should be noted that the second heat propagation with blood flow through large arteries
term on the right-hand side of equation (9) depends considerable. Indeed, let us estimate the temperature
weakly on the blood perfusion rate. Therefore, for the increase caused by this effect using the obtained equa-
nonuniformdistributionofthebloodperfusionratej(r,t) tion (9). The temperature increase T(r) T at a point
a
−
equation (9) holds also. spaced at the distance r > from the region (of size
L
) affected directly, i.e. the tail of the temperature dis-
L
tribution is mainly determined by the anomalous heat
Anomalous temperature distribution under local diffusion and, so, is estimated by the expression
strong heating
Hyperthermia treatment as well as thermotherapy of
smalltumors ofsize aboutorless1 cmisrelatedto local
strong heating of living tissue up to temperatures about
45 ◦C or higher values. In this case the tissue region T(r) T jmax (ℓ∗T)2L3 (T T ) . (11)
heated directly, for example, by laser irradiation is also − a ∼ Lnj0 r5 max− a
ofasimilarsize. Duetothetissueresponsetosuchstrong
heatingthe bloodperfusionratecangrowtenfoldlocally
whereas in the neighboring regions the blood perfusion
rate remains practically unchanged [9]. The feasibility
of such nonuniform distribution of the blood perfusion
Asseenfrom(11)forasufficientlylocalandstrongheat-
ratemaybeexplainedapplyingtothecooperativemech-
ing of the tissue, i.e. when ℓ∗ and j j the
anismofself-regulationinhierarchicallyorganizedactive L ∼ T max ≫ 0
temperature increase at not too distant points such that
media [3, 10]. Therefore in the region affected directly
r & can be considerable. Otherwise the anomalous
the blood perfusion rate jmax can exceed the blood per- heat Ldiffusion is ignorable.
fusion rate j in the surrounding tissue substantially. In
0
Acknowledgments
this case the characteristic length of heat diffusion into
the surrounding tissue is about
D
∗
ℓ , (10) This work was supported by STCU grant #1675 and
T ∼sfjmax INTAS grant #00-0847.
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