On the Fractality of the Biological Tree-like Structures

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On the Fractality ^{of the} Biological ^Tree-like Structures

JAAN KALDA

InstituteofCybernetics, Akadeemiatee21, EEO026 Tallinn, Estonia (Received March1999)

Thefractal tree-likestructurescanbe divided intothreeclasses, according tothevalue of the similaritydimension

Ds: Ds

< D,

D

^Dand

D

^{>D,whereDis the}topologicaldimensionof the embeddingspace. Itisarguedthatmostofthephysiologicaltree-like structures have

D

^>_D.The notion of theself-overlappingexponent^isintroducedtocharacterise thetrees with

D

^>D.Amodelof the human blood-vesselsystemisproposed.Themodel is consistent withtheprocesses governingthegrowth ofthe blood-vessels andyields

Ds

^3.4.The model is used toanalysethetransportofpassivecomponentbyblood.

Keywords." Fractals, Blood-vessels,Tree-like structures,Similarity dimension,Advectivediffusion

I INTRODUCTION

The fractal tree-like systems are very common objects, and have been attracted a lot of atten- tion, c.f. Mandelbrot (1983), West (1990), Bassingthwaighte et al.

(1994).

The most obvious examples are ordinary trees. The physiological tree-likestructures such as ablood-vessel system, alung, nervetissues, alymphatic system arenot exceptions.Unliketheordinary trees, the physiolog- ical"trees"are"hidden"bytissues.Forthisreason, it is quite a complicated task to study the fractal properties ofthem. The constituent parts of these tree-like systems (single blood vessels, neurons and bronchial

tubes)

have been studied for along time and the physical properties of them are known in great details. Meanwhile, the global

297

properties of the respective "trees" have been the object of systematic studies only during the last decade.

Inwhat followsweshow that the fractaltree-like structures can be meaningfully divided into three classes, according to the value of the similarity dimension

Ds: Ds <

^D,

Ds

^{D and}

Ds >

D, where D is the topological dimension of the embedding space. In Section II we argue that most of the physiological tree-like structures belong to the second and third classes

(Ds> D).

^{Section Ill is} devotedto thefractaltreeswith

Ds

^{D andto the}

model of lung in particular. In Section IV we discussthe caseof

Ds >

^{Dontheexampleofblood-} vessel system. SectionVisdevoted to the trees with

Ds ^<

^D and SectionVIto theanalysis ofthe trans- port of passivescalar inthe blood-vessel system.

298 J. KALDA

II THE PHYSIOLOGICAL TREE-LIKE STRUCTURES

A fractal model ofa biological tree-like structure should satisfythefollowingrathergenericcriteria:

(a)

it should be in accordance with the simplest physical laws, such as the flowcontinuity and the Poiseuille law in the case of the blood- vessel system;

(b)

it should satisfycertain physiological require- ments, e.g. ensure a complete (homogeneous) bloodsupplyoftheorganism;

(c)

it should be in accordance with ourknowledge about the processes governing thegrowth and formation ofthe

"trees";

(d)

it shouldbe self-similar within a widerangeof scales;

(e)

the result ofmanyiterationsofageneration-to- generationrelationspecifyingthe modelshould not be very sensitive to the subtleties of the model;

(f)

the model should not contradictempirical data.

Some comments are needed here. First, adopt- ingthe criterion

(d)

we disregardthepossibilityof a multifractal or a non-scale-invariant behaviour.

Asdiscussedabove,in some casesthese effectscan besignificant. The simplest way wouldbe to assume that there is a transition scale between two differ- entself-similarregions. However,in SectionIVwe shallargue thatatleast for a considerable rangeof scales, the self-similarity of the blood-vessels can be a consequence ofthe dynamical growth mech- anisms of the vascular tree. In orderto shedmore lightintotheproblem of scale-invariance, detailed three-dimensional experimental data would be needed.

Second, the consequence ofthe item

(b)

^is that in thecase of physiological vascularnetworks, the spatial distribution of the branches of the tree should be quasi-homogeneous, i.e. the tree should be space-filling. Indeed, in the case ofthe blood- vessel system the "holes" are not admittable, since all thecellsneed abloodsupply.Inthecaseoflung the alveoli fill almost all the space of the lung.

Perhaps in a lesser extent this is true for a neural network; however, within distinct regions of the organism, thedistributionof neurons isalso quasi- homogeneous. Ifthere would be a truefractalityof thevascularnetwork, the fractaldimensionscould not belessthanthe topologicaldimension

(three

in mostcasesandtwofor effectivelytwo-dimensional structures), because the fractional values less than thedimensionofthe embedding spacewouldmean thatthe structure issparse, with"holes".

The modelsoftheblood-vessel systemhavebeen developedinseveralpapers

(cf.

Spaan, 199!;Family et al., 1989;

Masters,

1994; VanBeek et al., 1989;

Kalda, 1993);mainsubject hasbeenthe geometrical arrangementof theblood-vessels. Specialattention has been paid to the essentially two-dimensional structures, such as subcutaneous arteriovenous networks(Gazit^etal., 1995),human retinal vessels (Family^etal.,

1989)

and vessels of the avian cho- rioallantoicmembrane

(Kurz

et al.,

1993).

Three- dimensionalanalysishas been attractedmuch less attention;one canmention

NMR-computer

tomog- raphy-based analysis of the pig kidney arteries, where the box-counting fractal dimensions have been calculated

(Sernetz

^etal.,

1992).

There is also a considerable number of papers devotedtothe airwaytreeof alung

(cf.

West, 1990;

Kitaokaand Takahashi, 1993; Weibel,

1991).

They include extensive experimental measurements and concern mainly the geometrical arrangement ofthe bronchial tubes (particularly characterised by the box-counting dimension). Also, mathema- tical models have been proposed to match the experimental data.

Accordingto the arguments given above, these models cannot be applied to the blood-vessel sys- tem

(or

^{tothelung)asa}whole,particularly because the reported fractal dimensions were less than the topologicaldimensionof the embedding space.The fractional values of the box-counting dimension should be attributed to the limited range ofscales and can be treated as an evidence ofthe lack of global self-similarity. In order to avoid misinter- pretations, thesefractionalvalues couldbe referred to asthe local scaling exponents.

III TREESWITH

Ds--D.

^THEMODEL

OF BRONCHIALTREE

A characteristic feature of the fractal trees with

Ds

^{Dis that}

(a)

thestructure isspace-filling,

(b) ^{a distinct} branch (together with its sub-

branches)

forms a compact structure, ^so that overlappingof different branches of the same generationisinsignificant.

Bronchial tree is a typicalexample ofthis kind of trees: the bronchial tubes and alveoli are packed tightlyand therearenomajor regions betweenthem filled bytissues ofother organs. Thetomographic images of lung indicate that the condition

(b)

is satisfied,^aswell. ByKitaokaandTakahashi

(1993),

a simple regular three-dimensional model oflung has been suggested. According to that model, all the bronchial tubes are similar toeach other; each tube branches into two smaller tubes which are perpendicular both to the given tube and to the tube ofthe previous generation. Sometimes a con- fusionhasbeen caused by the fact that the experi- mentaldependenciesarenotpowerlaws

(as

would be

expected

in the caseof self-similarity).Thus,ithas beenpointedoutbyWest

(1990)

thatthe plotof the logarithm of the averagediameterofthebronchial tubes versusthe generationnumber differsnotably from the straight line. It has been shown that the experimentalcurves can be modelled fairlywell, if we takeinto account thepresence of thesmall-scale cut-offatthe alveoli size(Kalda,

1993).

Indeed, for real lung, thetwobranches ofabronchialtube^are always of different sizes. Thus we canmodify the modelof KitaokaandTakahashi

(1993)

byintrodu- cingthe distribution function ofthediameter ratioof thebranches.Dueto such anunequal branching, the generation number of the alveoli

(the

alveoli are assumedto beapproximatelyofthe samesize)can vary severaltimes.Thepowerlaws can beexpected only bythe generationnumbers, smaller thanthe smallest generationnumberamongthe alveoli.

Another example (though not biological) of the treeswith

Ds

^Distherivernetworks. Thenetwork

is space-filling, ifwe assume that the sources ^are distributed quasi-homogeneously. The compact- nessiscaused bythe limitations of two-dimensional topology:two branches cannot intersect.

IV TREES WITH

Ds >

^{D. THE BLOOD-} VESSEL SYSTEM

To _beginwith,let usmake aroughestimate of the similarity dimension of theblood-vesseltree. Here wecan usethe following empirical data: thelength of the capillaries (i.e. the vessels ofthe last gene- ration) ,A00.5mm (cf. Hoppe ^et al., 1978), the length of the largest vessels

(aorta) l0

^{=0.5 m}

and the total length of capillaries,

LAoN

100,000km. The totalnumber ofcapillaries Ncan beexpressedvia the effective number ofgenerations

neffasN=2n% Being guided by the assumptionof self-similarity, we can expressthe similarity factor a as

a=()o/lo)l/n%

Using the definition of the similaritydimensionwecaneasilyfind

Ds ^-1/log

2a 3.4.

(1)

The seemingly curious fact that the similarity dimension exceeds the topological dimension can be explainedasfollows. Itis easy to showthat the Hausdorff-Besicovitch and box-counting dimen- sions ofa space-filling fractal set

DHB

^{and Db are}

always equal to the topological dimension of the embedding space D.

As

for the similarity dimen- sion,it isgenerally accepted(cf.Mandelbrot,

1983)

that

Ds

coincides with the Hausdorff-Besicovitch dimension

DHB.

^{Thus it may seem that always}

Ds ^_<

^{D.However, theequality}

Ds DHB-- Db

^can

be applied only ifall the dimensions are lessthan the dimension of the embedding space. Indeed, one canimagine thatthefractaltree wasoriginally embedded into a space ofdimensionality

Din > Ds

and thenprojectedintothespaceofdimensionality D

<

^{Ds, seeFig. 1.As}a result ofsuchaprojection, the dimensions

DHB

^and

Ds

^{become equal to the}

new value ofD, whereas the similarity dimension will evidently remain unchanged. The similarity

300 J.KALDA

F1GURE Example of a fractal tree with the similarity dimension exceeding the dimensionality of the embedding space:

Ds ^{2.53, D 2.}

dimension exceeds the topological dimension if the ratio 6,/I, of the average distance between the branchesof nthgeneration

6n

^{andaveragelengthof}

them

In

^{vanishes towards} higher generation num- bers n, i.e. towards smaller values of

l.

^{This is}

possiblein two cases:

(a)

the tree is not self-similar, but instead, self- affine;

(b) the branches of the same generation number havesignificant overlapping regions.

Finally, let us note. that in order to provide an homogeneous blood supply of the organism,the dis- tance between capillaries should be less than the

effective diffusion radius

6/..

¹⁰⁰lam:the relative distance between capillaries is somewhat smaller than betweenlargevessels.So,it israthernatural that thesimilaritydimensionof the blood-vesselsystem does exceed the dimension of theembeddingspace.

The self-similarmodelof thespatial arrangement of the blood-vessels is, infact, a generalisation of the Scheidegger’s model of rivers (Scheidegger,

1967).

The Scheidegger’smodel can beoutlined as follows: tree-like network is generated by trajec- tories ofparticles ("droplets") randomly jumping from site-to-site on n-dimensional square lattice (in ^{our case}n

2).

In then

+

1-dimensional space (time-axis added) the trajectories follow the edges

FIGURE2 Scheidegger’srivers.Thispicturearisesas apatternofparticle trajectories.Ateach time step,aparticleisbornatevery integervalue ofx;theparticlesmovewithconstantvelocity andrandom directionalongthe x-axisthroughaunitlength;the colliding particlescoalesce.

ofthe parallelepiped-shaped grid andcoalesce from time-to-timeforminglarger particles,seeFig. 2.

The Scheidegger’s model has been chosen as a starting point because the underlying parallelo- piped-shaped grid gives us a convenient way to ensure a quasi-homogeneous distribution of the capillary vessels (i.e. ^{ofthe vessels of the smallest} size).Alternatively,wecouldusethe generatorsof fractalstoconstruct trees similar tothatofdepicted onFig. ^1.The lattermethodcanalso be randomised (cf. Mandelbrot,

1983)

and as it can be seen on Fig. 1, the resultant distribution of the capillary vessels is quasi-homogeneous, too.

However,

the model based on Scheidegger’s rivers has still the advantagethat ithasmore controlparameterswhich canbe usedtoadjust the modeltothe empirical data.

ThemodifiedScheidegger’smodel isgivenby the following rules of the dynamics ofthe"droplets":

(i) Thetime is discrete with the unit timestep.

(ii) At^{each timestep,}each coordinate of aparticle changes randomly and independently of the other coordinates and other particles but correlates with itsprevious history so that the resulting motion is fractional Brownian with the Hurst exponent H (the average rms dis- placement ofa particleis proportionalto

t/-/);

the coordinate is increased ordecreasedbyone.

(iii)Colliding particles coalesce with the probabil- ityof ifbothofthe following conditions are fulfilled:

(a)

the mass ratio lies between

1/2

and2;

(b)

^theinteraction cores of the particles

302 J.KALDA

overlap. Otherwise, theparticlescontinue their motion without interaction. The radius of the interaction core is defined by the mass of the particleas follows:

rm ^rorn’,

^p

>

^0.

(2)

(iv) Ateach timestepand at each latticesite,a new particleof unitmassisadded.

As

comparedwith theoriginal Scheidegger’smodel,in item(ii)^now

anon-localityin time isassumed;in item(iii)^now twoparticles^cancoalesce atanon-zero distance and the coalescence ofparticleswhicharevery different in size isprohibited.

Further we have to tie the parameters of the model with the physical observable quantities of the blood-vessel tree. Evidently, the mass of the

"droplets" correspondstothe flux ofbloodthrough the vessels. In order to derive the expression for the average length of vessels via the average flux through them, it is convenient to introduce the massdoublingtime

tm

^{of the}"droplets" theaver- age time needed for^aparticleofmass mtodoubleits mass. Besides, let

lm

^{denote theaverage spatialdis-}

tancebetweenthe"m’-particles(whichwe define as theparticlesof masses within the interval[m,

2m]).

The problem offindingthe mass doublingtime ofa "droplet" is similar to theproblem of finding the kinetic constants of a chemical reaction. There are two possibilities: first, the anti-correlation in time is weak enough, so that the "m"-particles behaveas agas. Then the^areacoveredby theinter- action core of an "m"-particle during a typical coalescence time

tm

should be equaltotheaverage areaperone "m"-particle:

rmtm

12m,

^ifrmlm

< 12m ^{I-I. (3)}

The other possibility is that process becomes

"diffusion-limited":

Eq. (3)

is no longervalid since theself-overlappingof theparticle’s trajectorycan- notbeneglected. Instead,the timetm^{canbeassessed} asthe timeneeded foraparticleto "diffuse" tothe distance

lm:

2H 2 2H

lm,

^ifrmtm

> (4)

Suppose the system of "droplets" has been evolved for a long time, then a stationary regime should have been established. Specifically, the spectral flux ofmass per unit area (towards ^large

masses)

has to be constantoverthewhole spectrum of^masses:

m

]2m

tm

co,st.

(5)

Solving the system

(2)-(5)

for

tm

^{andlm,we find}

lm

^rn

,

tm ^rn

, (6)

where

+p

1-p

A

-1

_{r- if2H>}

4 4

A H r if 2H

<

2H+ 1’ 2H+ 1’

l+p l+p

(7)

Notethat ourarguments ^can berepeated without changingtheoriginal Scheidegger’smodel,aswell.

In that case alogarithmic factor should be added totake into account theeffect of self-overlappingof the trajectories; the relevant expressions of papers by Takayasu

(1989)

andHuber

(1991)

canbe easily recovered.

Nowletusrecall that in thecaseof blood-vessel system, m is the flux ofblood,

l

^{is the average}

distancebetween the vesselsand

tm

^{is thelength of}

vessels. Due to the continuity condition, the total flux through all the vessels ofa given size is con- stant,thusthenumberof"m"-vessels

Nm

^{cvm-1 On}

the other hand, the number ofvessels scales with their lengthas

Nm

^o(

tTnDs.

^{Further, ifwe take into}

account

Eqs. (5)

and

(6)

wefind

2A

liDs. ⁽⁸⁾

ComparingEqs. (1),

(7)

and(8),^{it is}easyto seethat there aresolutions,

0=0.4, H> 1.2,

(9)

p>0.4, H= 1.2.

(10)

Thefirst solutioncorrespondstothe case,whenthe capillaries fed by afixed artery form a sparsesys- tem and in a vicinity ofevery capillary there are capillaries fed by other arteries. We shall show somewhatlater that this situation is more realistic than the opposite one (corresponding to

Eq. (10))

when the capillaries fed by a fixed artery form a densesystem.

Toconclude with themodel,letusdiscuss it from the evolutionary pointof view. The growth of the vascularnetworkiscontrolled by several^chemical mechanisms. The generally accepted model (cf.

Gazitetal., 1995;Nekkaetal., 1996)^ofthisprocess can be outlined as follows.

In

the growing organ- ism, the tissue cells grow at a certain rate and sub-divide when a maximum size is reached. The existing vascular structuregrowswith all the other tissues. The distance between capillaries grows as well;this can causeishemia ofthe most distantcells.

Ishemiccells generatechemicalsubstances angio- genic factors

(AF)

^which lead to angiogenesis.

The particles of AF diffuse in all the directions.

These particles can be captured by blood vessels;

when captured, they cause a new vessel sprouting towards the ishemic cell (actually, towards the higherconcentration of

AF).

Some purelyperfused vessels undergo regression and disappear.

Despite thefact thatdiffusionplaysanimportant roleinsuch amodel,it seemsthat in most cases the growth is not diffusion-limited. Instead, diffusion is faster than the growth of the tissues: the time between the subsequentemergence of two ishemic regions is longer than the characteristic diffusion time. Suchagrowth model leads to aspace-filling statisticallyself-similarvasculartree. Ifweassume that the average distance between the capillaries is constantduring all the growth process and that theregression of vessels is negligible, there would beafractaltree of

Ds

^{3 with}slightly overlapping branches. Besides, the relativedistancebetween the large vessels would be equaltothe relativedistance between capillaries.

Ifwe admit thatthe regression ofvesselscanbe significant,weobtainatree with

Ds ^>

^{3.The higher}

the regression rate is, the higher the similarity

dimension will be. Such ^aninequality hastwo ob- servable consequences. First, the relative distance between vessels increases with the size ofvessels.

Second, there will be a significant overlapping of the same-generation branches. This is rather im- portant from the physiological point ofview: the damage of a vessel will not lead to the complete ceaseof the blood supply,in avicinity of every cell fedbyacapillarybelongingtothedamaged branch therearecapillaries belongingtohealthy branches.

In order to describe this effect quantitatively, we introduce the overlapping exponent of a fractal tree.Letusdrawaround^abranchofsize asphere of diameter 1. We repeat this procedure with all these branches which satisfy the condition L

<

<

^2L.Further,let the maximumnumberofspheres of non-zero intersectionscalewith sizeL^as

Nmax

^o(L-8

(11)

Thenwesay

that/3

^istheoverlapping exponent.It iseasy to see that ifthetree is self-similar(i.e. ^not justself-affine),

/3- Ds

^D.

(12)

Indeed,in the caseof vascular system the average distance betweenthevessels ofsize withL

< <

2L can be calculated as

d[l/(2mL)] ^1/2.

^{Here m}

denotes the effective generation number of the vessels ofgiven size; it can beeliminated using the expression for the similarity factor

a-(L/lo)/"=

2

/D.

Fin.a11.y, ^theJumbe.rofOlezrJO.j.7),

tPhCU,y

canbeassessed as

Nmax L3/(d2L) (lolL)

^D

3.

Due to the lack of experimental data, it is impossible to check directly the applicability of our model. In fact, it is a very difficult technical task to make three-dimensional measurements of vascular tree and cover a wide range of scales.

However,

detailed data are available concerning the correspondence between blood pressure, flux ofbloodand diametersof the vessels. Particularly, the diameter of the vessels scales with the flux w of blood as do( w

/,

^{c 2.7}

(see

Takayasu,

1990).

Accordingto ourmodel andPoiseuille law, this scaling law corresponds to the dependence

304 J.KALDA

p(d)_=P0-cd

"

^{with -y -0.5 (theexponent-y can}

be expressedviacand Ds),wherep(d)denotesthe average bloodpressureinthe vessels ofdiameterd

(for

details seeKalda,

1993).

^Thisdependenceisin accordance withthe experimental data

(cf.

Hoppe

et al.,

1978)

and ^can be considered as an indirect argument supportingourmodel.

It should be emphasised thatthe model described above cannot be usedequally wellfor allthe scale- lengths. The experimental data

(Kurz

et al., 1993;

Sernetz et al.,

1992)

^indicate that for some scale- lengths the vascular tree can be notably non-self- similar:theexponent of thelocal fit to apower-law revealed a significant dependence on space-scale.

Further experimental dataareneededtodetermine the range of applicability ofthemodel.

V TREES WITH

Ds<D

Most ofthe ordinary trees fall into this category.

Typically, the fractal dimension ofthemis some- thing betweentwoand three. Thetrees with

Ds <

2 are very

"transparent"

the shade of such a tree

(even

withleaves)has significantholes.Onthe other hand, the trees with

Ds ^_>

^{3 are very} thick: it is impossible to climb on these trees, because all the spaceoftheheadsof the trees is filled withbranches.

Despite the fact that the ordinary trees can be easily accessed and measured,it is rather difficult to calculate the fractaldimension ofthem. This is caused by the three-dimensional geometry. One possible solution is to measure the length

li

^and

mass

Mi

of each branch and find the similarity dimension astheminimumof thefunction

)2/12Os

Fzs- (/1/s-+ ^/2’Dis-n

^t-

^/3Di

³ⁱ

⁽¹³⁾

In fact,we cando themeasurements even ontwo- dimensional photographic images, assumed that treeshave losttheirleavesand all the branchescan be distinguished

(the

othermethods wouldfailhere and yield D

2). For

instance, using the images of several birch trees we obtainedD 2.6.

VI CONVECTION OF PASSIVE COMPONENT BY BLOOD

Inthis section we outline asimple implementation ofthe model ofvascular tree (Kalda,

1996).

We consider the transport of a passive admixture throughtheblood-vessel system. Itisassumed that theadmixturehasbeen injectedintotissuesandfills a certain region between the vessels. Besides, the following assumptionsaremade:

(a)

^outside the vessels, the propagation ofthe ad- mixture isdiffusive,ofmolecular diffusivityDo;

(b)

the admixtureparticlescanpenetrate the walls ofthevessels;

(c)

thepresence ofthe admixturearoundand inside the vessels does not affect substantially the blood flow in these vessels. However, a small change (byafactorofthe order oftwo) ⁱⁿthe rateof the blood flowisadmitted;

(d)

avessel iscalledto be ofsizeL, if itslength is betweenLand2L.The vesselsof sizeL forman homogeneous network;

(e)

the transportisaccomplishedinthevenous half oftheblood-vesseltree.Infactthe admixture is convected also by the arterial flow, but this is the convection towards the capillaries and the transportdistance in thearterialtree is limited by thesizeof thevesselwheretheinjectionwas made;

(f)

the blood flowin vessels is laminar

(cf.

Hoppe

etal.,

1978).

The analysis is based on two "integrals of motion". The first one is the expression for the totalvolume of the wholebody:

V

N(L)LA(L) :, (14)

where

(L)

denotestheaveragedistancebetweenthe neighbouring vessels of sizeLand

N(L)

thetotal number of vesselsof sizeL.

Thesecondone istheestimateofthe total flux of blood through the heart:

Q N(L)v(L)d(L) 2. ₍₁₅₎

Here

v(L)

denotesthe characteristicvelocityof the bloodin avesselof sizeLandd(L) the diameter of the vessel of size L. These equations are valid for anyvalue ofL.Sometimes it is more convenientto usethe combined and hence a dependent"integral of motion""

V/ (L)J(L)

^1000s.

(16)

Here the numerical value 1000s was obtained by substituting V-70 dm and 70

cm/s.

Letus assumethatinside the tissuesthereis aspot ofpassiveadmixture which diffuses into theblood vessels and will be carried into the other parts of the organism by blood. The admixture canbe an injection,avenomofaninsectorofasnakeor some- thing else. The character ofpropagation depends onthe seed diffusivity

Do

^{and onthe}initial size of the spot r. It can be shown that there are four qualitativelydifferent regimes of propagation.

Theadmixturepropagatesinthe form of^a"saus-

age"

aroundthe vessel stretching outof the initial spot.The diameterofit canbeassessedas

/Dot

^and

the"stretching" velocityof the

"sausage"

as

d(L)

²

Vef

v(L). (17)

Dot

If the spot is large and diffusivity low, the admixture fills the vascular system approximately during one rotational cycle of blood,

- ^W/

Q min, W beingthe total volume of theblood.

Otherwise the convection is slowed down by diffusion inside the tissues. It can be shown that in this case thecharacteristic time ofinvading the whole organismisgiven by

--

^{V/Q O0s.}

VII CONCLUSION

The values ofthe similaritydimension

Ds

^{and the}

dimension ofthe embedding space D canbe used to divide the fractal tree-like structures into three classes. Most of the ordinary trees belong to the class of sparse trees with

Ds<D.

^{Most of the}

physiologicaltree-like structuresare quasi-homog- eneous with Ds>_D. The compact self-similar structures withnon-overlapping branches

(such

as a lung) have

Ds-D.

^{The dense structures with}

Ds>D ^(such

^as a blood-vessel system) can be additionally characterised by the self-overlapping exponent. Ifthe tree is self-similar(i.e.^notjustself- affine), theexponent

,- _Ds

^D

>

0; this implies a significant overlappingofbranches.

We have constructed a self-similar model of blood-vessel system, which is in agreement with themodernunderstanding of the processes govern- ing thegrowth of thevascularnetwork. Ourbasic assumptions were:

(a)

the tree can be considered to be self-similar;

(b)

variations ofthe blood con- sumptionrate ofthebody cells are notsignificant.

The similaritydimensionofthe model

Ds

^{3.4. On}

thebasisofthismodel,wehaveanalysedthe trans- portofpassive component byblood.Dependingon the diffusivityof thepassive component, the char- acteristic time ofinvading the whole vascular tree canvary fromoneto twentyminutes.

Itshouldbe stressedthat the applicability ofthe assumptionofself-similarityitself is not quite clear and deservesfurther studies. On theonehand, it is supported by the dynamical growth model ofthe blood-vessel system; on the other hand, several papers have been reported that the fractal box- counting dimension ofthe effectively two-dimen- sional structures retinaland subcutaneousvascu- lar networks is close to Db,- 1.7 (Family et al., 1989; Masters, 1994; Gazit et al.,

1995).

These results couldbe treated^{as an}evidenceofthe lack of self-similarity for wider inertial range of scales.

However,

it should be noted that the effectively two-dimensional structures constitute onlyanegli- gible part ofthewholebody. Oneshouldalso bear in mindthat theuncertainties of the box-counting dimensions can beeasilyunderestimated, especially if the availablerange of scalesislimited.

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