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Revisiting the evolutionary origin of allometric metabolic scaling in biologyApol, M. Emile F.; Etienne, Rampal; Olff, Han; Clarke, Andrew

Published in:Functional Ecology

DOI:10.1111/j.1365-2435.2008.01458.x

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Citation for published version (APA):Apol, M. E. F., Etienne, R. S., Olff, H., & Clarke, A. (Ed.) (2008). Revisiting the evolutionary origin ofallometric metabolic scaling in biology. Functional Ecology, 22(6), 1070-1080. DOI: 10.1111/j.1365-2435.2008.01458.x

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Supplementary Material

�Revisiting the evolutionary origin of allometric metabolic scaling in biology�

M. Emile F. Apol, Rampal S. Etienne & Han Olff

Appendix A. Network model and metabolic rate

In this Appendix we derive the relation between the vessel dimensions of the transport network (see Figure S2)

and the metabolic rate. We use blood volume Vb � �PCk=1Nkr

2klk as a �vehicle� (West & Brown 2005) or

intermediate quantity to link the number of capillaries to body mass (West et al. 1997, Etienne et al. 2006). We

can in general express Nk, rk, lk and hk in terms of the capillary properties and the scaling ratios (Etienne et al.

2006),

Nk =NCQC�1i=k �i

(A-1)

rk =rCQC�1i=k �i

(A-2)

lk =lCQC�1i=k �i

(A-3)

hk =hCQC�1i=k �i

(A-4)

so that

Vb = �CXk=1

Nk r2k lk

= �NC r2C lC

CXk=1

1QC�1i=k �i�2i�i

= �NC r2C lC

CXk=1

1�QC�1i=k �i

�1�2cr�cl= �N2cr+cl

C r2C lC

CXk=1

1

(Nk)2cr+cl�1

= �N2cr+clC r2C lC S3(2cr + cl � 1) (A-5)

with

S3(q) �CXk=1

N�qk (A-6)

q � 2cr + cl � 1 (A-7)Apol, Etienne & Olff - Revisiting the evolutionary origin of allometric metabolic scaling in biology 1

Functional Ecology

a generalization of the S3-property as de�ned in (Etienne et al. 2006). Since in general Nk � 1 and increases

with increasing k, the S3-function will converge or saturate to a value approximately independent of the number

of capillaries C, and hence independent of body massM if q > 0 (Etienne et al. 2006), see also Figure S1. From

the �gure it is clear that S3 converges to a �xed plateau value for a network of at least �10 levels. In humans

the number of levels is about 30�35 for dichotomous branching. WBE (West et al. 1997) derived for their fractal

network that the number of levels C scales with body massM as C / ln(M). From Eq. A-5 it therefore follows

that the number of capillaries is given by

NC =

�Vb

�r2C lC S3

� 12cr+cl

(A-8)

Furthermore we assume that Vb scales allometrically with body mass,

Vb = Vb0Mb (A-9)

so that the total volume �ow is given by

Qtot = �r2CuCNC = �r2CuC

�Vb0

�r2C lC S3

� 12cr+cl

Mb

2cr+cl (A-10)

and the metabolic rate by

B = f0Qtot (A-11)

= f0�r2CuC

�Vb0

�r2C lC S3

� 12cr+2cl

Mb

2cr+cl

with metabolic exponent

a � b

2cr + cl(A-12)

Note that it is easy to see from Eqs. A-1 - A-4 that, since the aortic radius can be written as

r1 =rCQC�1i=1 �i

=rCQC�1

i=1 ��cri

=rC�QC�1

i=1 �i

��cr = rCNcrC (A-13)

it should scale with body mass as r1 / Macr , assuming that the capillary properties are size-invariant and that

N1 = 1. Likewise, the aortic length should scale as l1 /Macl and the aortic wall thickness as h1 /Mach .Apol, Etienne & Olff - Revisiting the evolutionary origin of allometric metabolic scaling in biology 2

Functional Ecology

Figure S1. Behavior of the S3 property as a function of network level k for several values of the quantity

q � 2cr + cl � 1: q = 12 , q =

25 and q =

13 (as in the WBE model) assuming a branching ratio �k = 2 for all

levels. These values of q correspond to a metabolic exponent a = 23 , a =

57 and a =

34 , respectively. Note that for

�xed capillary properties rC and lC , and given branching ratio �k and number of levels C, S3 is proportional to

the blood volume Vb.

kν

kN

k

klkr

431

1 6

2 3

2

23

18

Figure S2. Schematic representation of the transport network (after Etienne et al. 2006).

Apol, Etienne & Olff - Revisiting the evolutionary origin of allometric metabolic scaling in biology 3

Functional Ecology

Appendix B. Pulsatile �ow

In this Appendix we describe the mathematical formulation of pulsatile �ow in an elastic vessel with �tethered�

vessel walls, mimicking the surrounding tissue. We largely follow the derivation and notation of Womersley

(1955a,b, 1957, 1958) with some alterations, see also Table 1. Moreover we discuss the description of pulsatile

�ow in a network, the dissipation of energy, and the description of combined steady and pulsatile �ow in a network

and the concept of impedance matching.

Complex notation

For a complex variable ez we have ez = z + iz0 = jezjei� in which i � p�1 is the imaginary unit; jezj = pz2 + z02is the amplitude; � = arctan(z0=z) is the phase; z and z0 are the real and imaginary parts of ez; ei� = cos � +

i sin �, and the tilde indicates a complex variable. In general it is mathematically convenient to express oscillating

properties in such a complex form, i.e., we de�ne a complex variable eI(t) = eI0ei!t = Imaxei ei!t. The physical

property is the real part of the complex variable: I(t) � Re[eI(t)] = Imax cos(!t + ). The pressure for instance

may therefore also be written in complex notation epk(t) = ePkei!t where ePk may be complex too (containing aphase factor). Because of the oscillating nature, instantaneous values of a property I(t) are not very interesting. A

more useful concept, that also clearly connects with steady �ow, is its magnitude � or better � its root-mean-square

(rms) value, averaged over one period of oscillation T = 2�=!:

Irms �

vuuut 1

T

TZ0

I2(t)dt =

vuuut !

2�

2�=!Z0

�Re[eI(t)]�2 dt =r1

2Re[eI(t)eI�(t)] (B-1)

where eI� is the complex conjugated of eI . For harmonic oscillations it is easy to show that Irms = Imax=p2. The

use of rms values is the most natural way to apply concepts of steady �ow to oscillating �ow, similar to the case

of AC vs. DC electrical circuits (Grob 1985). For steady �ow, Irms converges to the steady value I . The cosine

of the phase shift between two oscillating quantities eI1(t) and eI2(t) with the same angular frequency is in generalgiven by

cos � =ReheI1(t)eI�2 (t)iqeI1(t)eI�1 (t)qeI2(t)eI�2 (t) =

ReheI1(t)eI�2 (t)i

I1;maxI2;max=

12 Re

heI1(t)eI�2 (t)iIrms1 Irms2

(B-2)

If eI1 and eI2 also depend on the spatial position z over a distance l (e.g. in a blood vessel), the rms values mustApol, Etienne & Olff - Revisiting the evolutionary origin of allometric metabolic scaling in biology 4

Functional Ecology

also be averaged over z,

Irmsi �

s1

l

Z l

0

1

2Re[eIi(t; z)eI�i (t; z)] dz (B-3)

and Eq. B-2 generalizes to

cos � =

1l

R l012 Re

heI1(t; z)eI�2 (t; z)i dzIrms1 Irms2

(B-4)

from which it follows that

Irms1 Irms2 l cos � =1

2

Z l

0

ReheI1(t; z)eI�2 (t; z)i dz (B-5)

a result that is used in the formulation of the dissipation function (Eq. B-66).

Pulsatile �ow in a single vessel

Consider an elastic vessel �lled with an incompressible liquid (blood). The vessel walls are strongly longitudinally

constrained and have an added mass, mimicking the tethering of the surrounding tissue. Womersley (1955a,b,

1957) formulated the corresponding Navier-Stokes equations for the liquid and the coupled dynamic equations for

the constrained elastic vessel walls. He showed that as a good approximation these equations may be linearized,

and gave the solution for the volume �ow in an in�nitely long tube as a function of an oscillating pressure wave.

Perpendicular to the streaming direction, the pressure wave generates an oscillating movement in the vessel wall

with displacements that travel along the vessel with a certain wave velocity. The pressure wave is of the form

ep0k(t; z) = ePkei!(t�z=eck) (B-6)

where the tilde indicates a complex quantity, ! is the angular frequency of the wave, z is the longitudinal position

in the vessel, and eck is the complex wave velocity. Note that ep0k is explicitly a function of position z, as the wavetravels along the vessel. The real part of eck is the velocity c1;k of the wave along the vessel wall, the imaginarypart is related to the damping factor ak of the amplitude while the wave moves through the vessel (attenuation):

1eck � 1

c1;k� iak

!(B-7)

Note that the attenuation length is 1=ak. In this way the pressure wave (Eq. B-6) can be restated as

ep0k(t; z) = ePke�akzei!(t�z=c1;k) (B-8)

The wave velocity eck is in general smaller than the ideal wave velocity of an inviscid liquid, c0;k, given by theApol, Etienne & Olff - Revisiting the evolutionary origin of allometric metabolic scaling in biology 5

Functional Ecology

Moens-Korteweg equation,

c0;k =

sEkhk2�Mrk

(B-9)

where hk, rk and Ek are the thickness and radius of the vessel and the Young's modulus of the wall (Nichols

& O'Rourke 2005). Womersley solved the volume �ow as a function of pressure, which in a slightly different

notation is given by

eQ0k(t; z) = �r2k�Meck ie�k ep0k(t; z) (B-10)

in which the complex function e�k is de�ned ase�k � e�(�k) � i

�1� 2J1(i

3=2�k)

i3=2�kJ0(i3=2�k)

��1(B-11)

= �iJ0(i3=2�k)

J2(i3=2�k)(B-12)

where Jn(x) is the Bessel function of the �rst kind of order n (Abramowitz & Stegun 1972), and

�k � rk

r�M!

�(B-13)

is the dimensionless Womersley number. Eq. B-11 is the form used by Womersley (1955a,b, 1957, 1958), whereas

Eq. B-12 is the form used by West et al. (1997); both forms can be seen to be equivalent by employing the

recurrence relation between Bessel functions J0(x) + J2(x) =�2x

�J1(x). Using the asymptotic properties of the

Bessel functions (Abramowitz & Stegun 1972), it can be shown that the limit of e�k for small �k (i.e., Poiseuille-like �ow in narrow vessels at quasi-steady conditions) is given by

lim�k!0

e�k � 8

�2k

�1 +

�4k1152

+ : : :+ i

��2k6� �6k34560

+ : : :

��(B-14)

For large �k (i.e., strong Womersley �ow with high frequency oscillations and/or in wide vessels), we obtain

lim�k!1

e�k � 1

�k

�p2 +

3

�k+ : : :+ i

��k +

p2 + : : :

��(B-15)

see Figure S3. From Eq. B-6 follows that

�rep0k(t; z) = � i!eck� ep0k(t; z) (B-16)

so that Eq. B-10 can be rewritten as

�rep0k(t; z) = �M!

�r2ke�k eQ0k(t; z) (B-17)


Functional Ecology

This is a general relation between (local) pressure gradient and (local) volume �ow. From this relation, using the

de�nition of the rms values of �rep0k and eQ0k (Eq. B-3) it follows that(�rep0k)rms = �M!

�r2kje�kj eQ0rmsk (B-18)

so that from Eq. B-4 we obtain after some algebra that the phase shift between pressure gradient and volume �ow

is given by

cos �k =Re[e�k]je�kj (B-19)

see Figure S3. Using the properties of the Bessel functions, it can also be shown that

lim�k!0

cos �k = lim�k!0

Rehe�ki��e�k�� 1� �4k

72+ : : : (B-20)

and

lim�k!1

cos �k = lim�k!1

Rehe�ki��e�k��

p2

�k+1

�2k+ : : : (B-21)

Womersley (1957) also derived an expression for the wave velocity eck in terms of the Moens-Korteweg velocityc0;k (see Eq. B-9):

c0;keck =

s(1� �2k)

e�ki� c0;kc1;k

� i c0;k!ak (B-22)

with �k the so-called Poisson ratio of the vessel wall, which is usually set to �k = � = 12 (Milnor 1989), implying

that the volume of the vessel walls remains approximately constant during deformation. Using Eq. B-12 the ratio

c0;k=eck can also be expressed as �c0;keck

�2= �(1� �2k)

J0(i3=2�k)

J2(i3=2�k)(B-23)

which corresponds to the expression given by West et al. (1997, 2000). The extra factor (1 � �2k) is due to the

longitudinal constraint (Womersley 1957). For small Womersley number, the ratio c0;k=eck can be expanded in �karound zero, yielding

lim�k!0

c0;keck �q(1� �2k)

�2

�k+�k6+ : : :+ i

�� 2

�k+�k6+ : : :

��(B-24)

showing that the complex wave velocity correctly goes to zero in the limit of Poiseuille �ow. For large Womersley

numbers, using the asymptotic expressions of the Bessel functions (see Eq. B-15), we obtain

lim�k!1

c0;keck �q(1� �2k)

�1 +

1p2

1

�k+ : : :+ i

�� 1p

2

1

�k� 1

�2k+ : : :

��(B-25)

showing that the wave velocity in this limit approaches an upper limit. Note that the real pulse velocity c1;k does

not converge exactly to c0;k (but is in fact 15% larger for �k = 12 ), as the latter is the limiting velocity for an


Functional Ecology

elastic tube without longitudinal constraint (Womersley 1957). From Eq. B-22 and using the de�nition of the

Moens-Korteweg velocity (Eq. B-9) we can also derive the following important dimensionless quantities, that we

need later on in the expression for the pulsatile energy dissipation, Eq. B-68,

aklk = �!

s2�M (1� �2k)

Ekr1=2k lkh

�1=2k Im

24se�ki

35 (B-26)

and

!lkc1;k

= +!

s2�M (1� �2k)

Ekr1=2k lkh

�1=2k Re

24se�ki

35 (B-27)

Similar to the situation in AC electrical circuits (Alonso & Finn 1983; Grob 1985), the response of the volume

�ow generally lags behind the pressure gradient (reactive response), and the phase shift �k between the two depends

on the value of the Womersley number (Nichols & O'Rourke 2005), see also Eqs. B-19, B-20 and B-21. For

very high frequency (�k ! 1) the phase shift tends to 90�, for quasi-steady �ow (�k ! 0) the phase shift

approaches 0�, i.e., pressure gradient and �ow are in phase (Ku 1997), see also Figure S3. The phase shift manifests

itself via a complex relation between pressure gradient and �ow, i.e., a complex longitudinal impedance eZL ��rep0k(t; z)= eQ0k(t; z) that depends only on the local properties of the vessel and the blood within it (Nichols &O'Rourke 2005). It is the pulsatile analog of the steady longitudinal resistance RL = (�pk=Qk)=lk.

It is instructive to see that the Womersley �ow expressions converge to those of Poiseuille �ow (Poiseuille

1846; Milnor 1989; Nichols & O'Rourke 2005) in the limit of small Womersley number. Using Eq. B-17 the

longitudinal impedance eZL;k iseZL;k � �rep0k(t; z)eQ0k(t; z) =

�M!

�r2ke�k (B-28)

With the help of Eq. B-14, in the limit of �k ! 0 this converges to the longitudinal Poiseuille resistance,

lim�k!0

eZL;k � �M!

�r2k

8

�2k=8�

�

1

r4k= RL;k (B-29)

and the usual resistance is simply Rk = lkRL;k, having a 1=r4k dependence (see also Figure S3). For strongly

pulsatile �ow (�k !1) the longitudinal impedance becomes with the help of Eq. B-15

lim�k!1

eZL;k � p2��M!

�

1

r3k

�1 + i rk

r�M!

2�

�(B-30)

so that the (longitudinal) resistance (the real part of the longitudinal impedance) has changed from a 1=r4k to a 1=r3k

dependence. Note that the time-averaged oscillating volume �ow and pressure gradient are zero.Apol, Etienne & Olff - Revisiting the evolutionary origin of allometric metabolic scaling in biology 8

Functional Ecology

Lastly, the so-called characteristic impedance eZC;k is de�ned as (Milnor 1989; Nichols & O'Rourke 2005)eZC;k � ep0k(t; z)eQ0k(t; z) = �Meck

�r2k

e�ki=�M�r2k

c0;kp1� �2k

se�ki

(B-31)

=�Mc

20;k

�(1� �2k)r2keck (B-32)

where we used Eqs. B-10 and B-22. The last form (Eq. B-32) is � apart from the factor (1� �2k) � identical to the

expression given by West et al. (1997, 2000). From Eqs. B-22, B-24 and B-25 it follows that

lim�k!0

eZC;k =2

�

r�M�

!

c0;kp1� �2k

1

r3k(B-33)

lim�k!1

eZC;k =�M�r2k

c0;kp1� �2k

(B-34)

so the characteristic impedance changes from a r�3k to a r�2k dependence, plus an additional radius-dependence in

the Moens-Korteweg velocity c0;k (Eq. B-9), so in total to a r�5=2k dependence.

Figure S3. Behavior of the complex �viscous� function e�k (Eq. B-11) as a function of the Womersleynumber �k: A. real part of e�k, B. imaginary part of e�k, C. real part of �2ke�k=8 showing the convergenceto Poiseuille behavior at small �k, and D. the cosine of the phase-shift �k between pressure gradient and �ow,Apol, Etienne & Olff - Revisiting the evolutionary origin of allometric metabolic scaling in biology 9

Functional Ecology

cos �k = Re[e�k]=je�kj, showing that for small �k the pressure gradient and �ow are in phase.

Pulsatile �ow in a branching network

Consider a part of a branching network as in Figure 1. If a pressure wave ep0k(t; z) in a vessel of level k encountersat position zk a point where the network branches into �k smaller vessels of level k + 1 with radius rk+1, wall

thickness hk+1, etc., part of the wave will in general be re�ected back because of a difference in impedance before

and after the branching point (Milnor 1989; Nichols & O'Rourke 2005). Here we will outline a modi�cation of

the description of Womersley (1958) that provides the re�ection factor e�k (the ratio of the backwards and forwardspressure wave at the branching point) from the equations of continuity of pressure and �ow across the branching,

and thus the net pressure and �ow waves which are the sum of the forward and backward waves.

Consider a forward pressure wave with corresponding volume �ow, written for convenience in terms of the real

wave velocity c1;k and damping factor ak (cf. Eq. B-8)

ep0k(t; z) = ePkei!te�ak(z�zk�1)e�i!(z�zk�1)=c1;k eQ0k(t; z) = �r2k�Meck ie�k ep0k(t; z) (B-35)

The re�ected and transmitted pressure and �ow waves will be

ep0�k(t; z) = eP�kei!teak(z�zk�1)ei!(z�zk�1)=c1;k eQ0�k(t; z) = � �r2k�Meck ie�k ep0�k(t; z) (B-36)

ep0k+1(t; z) = ePk+1ei!te�ak+1(z�zk)ei!(z�zk)=c1;k+1 eQ0k+1(t; z) = �r2k+1�Meck+1 ie�k+1 ep0k+1(t; z) (B-37)

where zk�1 and zk are the locations of the begin and end of vessel k, and the amplitudes ePk and eP�k are measuredat the beginning of vessel k at z = zk�1. The pressure must be continuous across the branching point at z = zk,

so we can write

ep0k(t; zk) + ep0�k(t; zk) = ep0k+1(t; zk) (B-38)

Using the expressions of the pressure waves (Eqs. B-35 to B-37) and realizing that zk � zk�1 = lk, we easily

arrive at the relation

ePke�aklke�i!lk=c1;k + eP�keaklkei!lk=c1;k = ePk+1 (B-39)

On the other hand, the volume �ow must also be continuous across the branching point at z = zk, so remembering

that there are �k vessels at level k + 1, we haveApol, Etienne & Olff - Revisiting the evolutionary origin of allometric metabolic scaling in biology 10

Functional Ecology

eQ0k(t; zk) + eQ0�k(t; zk) = �k eQ0k+1(t; zk) (B-40)

Using the full expressions of the �ow waves (Eqs. B-35 to B-37), we �nd

r2kecke�k� ePk � eP�k e2aklke2i!lk=c1;k� = �k

r2k+1eck+1e�k+1 ePk+1 = �kr2k+1eck+1e�k+1

� ePk + eP�k e2aklke2i!lk=c1;k�For convenience, we de�ne the ratio

e k � ePk � eP�k e2aklke2i!lk=c1;kePk + eP�k e2aklke2i!lk=c1;k = �k�2k

eckeck+1 e�ke�k+1 (B-41)

and so the re�ection factor, the ratio of the backward and the forward pressure wave at the junction at zk, is given

by

e�k � ep0�k(t; zk)ep0k(t; zk) =eP�kePk e2aklke2i!lk=c1;k =

1� e k1 + e k (B-42)

so that

eP�k = ePke�k e�2aklke�2i!lk=c1;k (B-43)

Using Eqs. B-22 and B-9, the expression of e k can also be written ase k = �k�

2k

c0;kc0;k+1

q(1� �2k+1)

e�k+1iq

(1� �2k)e�ki

e�ke�k+1= �k�

2k

pEkhk=2�Mrkp

Ek+1hk+1=2�Mrk+1

q1� �2k+1p1� �2k

s e�ke�k+1= �k�

5=2k �

�1=2k

s e�ke�k+1 (B-44)

where we assumed that the Young's moduli E and Poisson ratios � of the vessel walls of both levels k and k + 1

are equal. Using the asymptotic expressions of the Bessel functions in e�k and e�k+1, and realizing that since theangular frequency is constant throughout the network �k+1 = rk+1

p�M!=� = (rk+1=rk)rk

p�M!=� = �k�k,

we can evaluate the small �k behavior ofqe�k=e�k+1 as

lim�k!0

s e�ke�k+1 � �k

�1 +

(1� �2k)(9 + 25�2k)2304

�4k + : : :

+i

�(1� �2k)12

�2k �(1� �2k)(47 + 92�2k + 217�4k)

138240�6k + : : :

��(B-45)

so that

lim�k!0

e k = �k�7=2k �

�1=2k (B-46)


Functional Ecology

On the other hand, using the asymptotic expansion of the Bessel function for large �k (see also Eq. B-15), we

obtain

lim�k!1

s e�ke�k+1 � 1� 1� �kp2�k

1

�k+ : : :+ i

�1� �kp2�k

1

�k+1� �k�k

1

�2k+ : : :

�(B-47)

so that

lim�k!1

e k = �k�5=2k �

�1=2k (B-48)

By summing the forward and backward waves, the total pressure wave epk(t; z) (denoted without the prime) istherefore given by

epk(t; z) = ep0k(t; z) + ep0�k(t; z)= ePkei!te�ak(z�zk�1)e�i!(z�zk�1)=c1;k h1 + e�ke2ak(z�zk�1�lk)e2i!(z�zk�1�lk)=c1;ki (B-49)

from which it follows after some algebra that the pressure gradient is

�repk(t; z) = � i!eck� ePkei!te�ak(z�zk�1)e�i!(z�zk�1)=c1;k h1� e�ke2ak(z�zk�1�lk)e2i!(z�zk�1�lk)=c1;ki

(B-50)

where we used Eq. B-7. The total volume �ow is given by

eQk(t; z) = eQ0k(t; z) + eQ0�k(t; z)=

�r2k�Meck ie�k ePkei!te�ak(z�zk�1)e�i!(z�zk�1)=c1;k

h1� e�ke2ak(z�zk�1�lk)e2i!(z�zk�1�lk)=c1;ki

(B-51)

Combining these two relations gives the relation between total pressure gradient and volume �ow:

�repk(t; z) = �M!

�r2ke�k eQk(t; z) (B-52)

It has the same form as Eq. B-18, so that also here we �nd (cf. Eq. B-19) that the phase shift between total pressure

gradient and total volume �ow is given by cos �k = Re[e�k]=je�kj.

Wave re�ections and impedance matching

A quick way to demonstrate that minimizing wave re�ections, je�kj = 0, corresponds to impedance matching, is

the following. From the de�nition of the characteristic impedance (Eq. B-31),

eZC;k � ep0k(t; z)eQ0k(t; z) = �ep0�k(t; z)eQ0�k(t; z) (B-53)


Functional Ecology

it follows that

ep0k(t; z) = eZC;k eQ0k(t; z) (B-54)

ep0�k(t; z) = � eZC;k eQ0�k(t; z) (B-55)

ep0k+1(t; z) = eZC;k+1 eQ0k+1(t; z) (B-56)

so that continuity of pressure across the junction at zk (Eq. B-38) gives

eZC;k � eQ0k � eQ0�k� = eZC;k+1 eQ0k+1 @zk (B-57)

and continuity of �ow at zk (Eq. B-40) yields

eQ0k+1 = 1

�k

� eQ0k + eQ0�k� @zk (B-58)

Combining these two equation, eliminating eQ0k+1, solving for eQ0�k, and inserting this into the de�nition of there�ection factor gives

e�k �ep0�kep0k = �

eQ0�keQ0k @zk

=eZC;k+1 � �k eZC;keZC;k+1 + �k eZC;k (B-59)

Hence, for zero wave re�ections the impedances before ( eZC;k) and after the branching ( eZC;k+1=�k) must match.Note that if we would also assume that the pressure gradient is continuous at the junction,

�rep0k(t; zk)�rep0�k(t; zk) = �rep0k+1(t; zk) (B-60)

this would change the expression of the re�ection factor in terms of network properties. Using Eq. B-16 we can

write

�rep0k(t; z) =

�i!eck� eZC;k eQ0k(t; z) (B-61)

�rep0�k(t; z) =

�i!eck� eZC;k eQ0�k(t; z) (B-62)

�rep0k+1(t; z) =

�i!eck+1

� eZC;k+1 eQ0k+1(t; z) (B-63)

so that Eq. B-60 reads eZC;keck� eQ0k + eQ0�k� = eZC;k+1eck+1 eQ0k+1 @zk (B-64)

Combining this with Eqs. B-57 and B-58 we get

e�k = eck+1 � eckeck+1 + eck (B-65)


Functional Ecology

so that the condition of zero wave re�ections implies that � besides impedance matching � the complex wave

velocities before (eck) and after the branching (eck+1) must be equal. In the limit of large Womersley number whereeck �! p

1� �2c0;k (see Eq. B-25) this would imply that on both sides of the junction the Moens-Korteweg

velocity is equal: c0;k = c0;k+1. However, the constraint of continuity of the pressure gradient, besides continuity

of �ow and pressure, is a very strong and unusual condition, not encountered anywhere in the hemodynamics

literature (Milnor 1989; Nichols & O'Rourke 2005), and is physically very implausible.

Energy dissipation of pulsatile �ow

The dissipationWtot (energy per unit of time) is in general given by the product of driving force (pressure differ-

ence) and �ux (volume �ow) per vessel times the number of vessels (Milnor 1989): Wtot =PCk=1Nk(�pk)Qk. If

gradient and �ow depend on the spatial position z in the vessel, we have per vessel to sum (integrate) �rpk(z)dz

times Qk(z) over the total vessel length: Wtot =PCk=1Nk

R zk�1+lkzk�1

(�rpk(z))Qk(z)dz, where the integration

limits are from zk�1 to zk�1+lk with the branching points to levels k and k+1 located at zk�1 and zk = zk�1+lk

(see Figure 1). If, in addition, the gradient and �ow periodically depend on time as well, we also have to average

over one oscillation period T = 2�=!:

W ptot =

CXk=1

Nk

zk�1+lkZzk�1

0B@ 1

2�=!

2�=!Z0

(�rpk(t; z))Qk(t; z) dt

1CA dz

=CXk=1

Nk

zk�1+lkZzk�1

0B@ 1

2�=!

2�=!Z0

Re [�repk(t; z)] Re h eQk(t; z)i dt1CA dz

=1

2

CXk=1

Nk

zk�1+lkZzk�1

Reh�repk(t; z) eQ�k(t; z)i dz (B-66)

=��M!

�

� CXk=1

Nkr2kRe[e�k] 1

2

zk�1+lkZzk�1

j eQkj2 dz (B-67)

where we used Eqs. B-5 and B-52. Using the explicit expression of the volume �ow, Eq. B-51, and writingApol, Etienne & Olff - Revisiting the evolutionary origin of allometric metabolic scaling in biology 14

Functional Ecologye�k � je�kjei��;k as modulus je�kj and phase ��;k, we obtain from Eq. B-67 after quite some algebraW ptot =

��M!�

� CXk=1

Nkr2kRe[e�k] 1

2

zk�1+lkZzk�1

eQk(t; z) eQ�k(t; z) dz=

CXk=1

1

2Nk

��r2k�M!

�!2j ePkj2jeckj2 je�kj Re[

e�k]je�kj �

zk�1+lkZzk�1

dz e�2ak(z�zk�1)�1� je�kjei��;k e2ak(z�zk�1�lk) e2i!(z�zk�1�lk)=c1;k��

�1� je�kje�i��;k e2ak(z�zk�1�lk) e�2i!(z�zk�1�lk)=c1;k�

=CXk=1

1

2Nk

��r2k�M!


e�k]je�kj �

zk�1+lkZzk�1

dz e�2ak(z�zk�1) + e�4aklk je�kj2 e2ak(z�zk�1)�e�2aklk je�kjei��;k e2i!(z�zk�1�lk)=c1;k�e�2aklk je�kje�i��;k e�2i!(z�zk�1�lk)=c1;k

=

CXk=1

1

2Nk

��r2k�M!


e�k]je�kj ��

1� e�2aklk2ak

�+ e�4aklk je�kj2�e2aklk � 1

2ak

��e�2aklk je�kjc1;k

!

�sin (��;k) + sin

�2!lkc1;k

� ��;k��

(B-68)

Moreover, from Eq. B-22 we have c20;k= jeckj2 = (1� �2k)je�kj, so that with the help of Eq. B-9 we get1

jeckj2 je�kj = 2�M (1� �2k)Ek

rkhk

(B-69)

Furthermore, from Eqs. B-39 and B-43 we obtain a relation between the pressure amplitudes at level k and k + 1:

ePk+1 = ePk e�aklke�i!lk=c1;k(1 + e�k)so that we can express every amplitude in terms of the initial amplitude at the aorta, eP1, as

ePk = eP1 k�1Ym=1

e�amlme�i!lm=c1;m(1 + e�m)j ePkj2 = j eP1j2 k�1Y

m=1

e�2amlm��1 + e�m��2

= j eP1j2 k�1Ym=1

e�2amlm�1 + 2 cos (��;m) je�mj+ je�mj2� (B-70)


Functional Ecology

or, alternatively, in terms of the capillary amplitude ePC :ePk = ePC,C�1Y

m=k

e�amlme�i!lm=c1;m(1 + e�m)j ePkj2 = j ePC j2,C�1Y

m=k

e�2amlm��1 + e�m��2

= j ePC j2,C�1Ym=k

e�2amlm�1 + 2 cos (��;m) je�mj+ je�mj2� (B-71)

Inserting Eqs. B-69 and B-71 into the dissipation function (Eq. B-68) we obtain the �nal and exact expression for

the pulsatile energy dissipation,

W ptot =

CXk=1

�(1� �2k)!j ePC j2Ek

Nkr3klk

hk

Re[e�k]je�kj �(C)k (B-72)

with

(C)k �

24 C�1Ym=k

e�2amlm�1 + 2 cos (��;m) je�mj+ je�mj2�!�1�

��1� e�2aklk2aklk


2aklk

��e�2aklk je�kjc1;k

!lk


�2!lkc1;k

� ��;k��

(B-73)

Alternatively,

W ptot =

CXk=1

�(1� �2k)!j eP1j2Ek

Nkr3klk

hk

Re[e�k]je�kj �(1)k (B-74)

with

(1)k �

" k�1Ym=1

e�2amlm�1 + 2 cos (��;m) je�mj+ je�mj2�!� (B-75)��

1� e�2aklk2aklk


2aklk

�(B-76)

�e�2aklk je�kjc1;k!lk


�2!lkc1;k

� ��;k��

(B-77)

Note that the dissipation is now linearly dependent on the vessel length lk, just as for Poiseuille �ow (Eq. 10).

However, by using some approximations these rather complicated expressions can be simpli�ed in the following

way. For large Womersley number we �nd from Eq. B-25

lim�k!1

2aklk = lim�k!1

�2!lk Im�1eck�� lim�k!1

2!lk

p1� �2kc0;k

2

�k! 0

and since the Womersley solutions based on the linearized Navier-Stokes equations are valid for long wavelength

(Womersley 1955a,b, 1957; Taylor 1957)

lim�k!1

!lkc1;k

= lim�k!1

!lk Re

�1eck�� !lk

p1� �2kc0;k

� 1


Functional Ecology

Hence for relatively large �k we have 2aklk � 1 and 2!lkc1;k

� 1. This is also con�rmed for smaller �k (down

to �k � 0:005) by experimental data for different levels in the canine arterial system (aorta, artery, arteriole

and capillary), see Table S1. This implies that e�2aklk � 1 � 2aklk, e2aklk � 1 + 2aklk, sin�2!lkc1;k

��

2!lkc1;k

and cos�2!lkc1;k

�� 1, so that after some algebra and using the trigonometric identity sin

�2!lkc1;k

� ��;k�=

sin�2!lkc1;k

�cos (��;k)� cos

�2!lkc1;k

�sin (��;k), we arrive at the much simpler approximate expression for the k-

functions of the pulsatile dissipation that we will use in this paper,

(C)k � 1� 2 cos (��;k) e�2aklk je�kj+ e�4aklk je�kj2

C�1Ym=k

e�2amlm�1 + 2 cos (��;m) je�mj+ je�mj2�

(B-78)

(1)k �

k�1Ym=1

e�2amlm�1 + 2 cos (��;m) je�mj+ je�mj2� h1� 2 cos (��;k) e�2aklk je�kj+ e�4aklk je�kj2i

(B-79)

For later reference, also note that (within these approximations) the rms value of the oscillatory volume �ow at

level k is given by

(Qp;rmsk )2=

�2(1� �2k)j ePC j2�MEk

r5khk

(C)k

je�kj (B-80)

=�2(1� �2k)j eP1j2

�MEk

r5khk

(1)k

je�kj (B-81)


Functional Ecology

Table S1. Normal cardiovascular parameters of four vessel types corresponding to a typical dog, and several

derived quantities.

Vessel type

Property Unit abdominal aorta carotid artery arteriole capillary Ref.

vessel length lk m 15�10�2 15�10�2 0.15�10�2 6�10�4 a

vessel radius rk m 0.45�10�2 0.25�10�2 0.0025�10�2 3�10�6 a

vessel wall thickness hk m 0.05�10�2 0.03�10�2 0.002�10�2 1�10�6 a

Young's modulus wall Ek Pa 10�105 9�105 �9�105 �9�105 a

Poisson ratio wall �k - 12

12

12

12 b

blood density �M kg m�3 1050 1050 1050 1050 c

blood viscosity � Pa s 4�10�3 4�10�3 4�10�3 4�10�3 c

heart frequency f Hz 2 2 2 2 a

angular frequency ! rad s�1 4� 4� 4� 4�

pulsatile pressure amplitude j ePkj mm Hg �20 �10 �2 �0.2 a

Pa 2670 1335 267 27

Womersley number �k - 8.2 4.5 0.045 0.0054

Moens-Korteweg velocity c0;k m s�1 7.3 7.2 18.5 12.0

real pulse velocity c1;k m s�1 7.7 7.0 0.24 0.019

damping constant ak m�1 0.15 0.32 25.9 334

dimensionless group 1 aklk - 0.023 0.048 0.039 0.20

dimensionless group 2 !lk=c1;k - 0.24 0.26 0.039 0.20

References. a: Caro et al. 1978; b: Milnor 1990; c: Womersley 1955b.


Functional Ecology

Dissipation of combined steady and pulsatile �ow

For a combination of steady and pulsatile �ow, the dissipation is given by

Wtot =CXk=1

Nk

lkZ0

0B@ 1

2�=!

2�=!Z0

Re[�repk(t; z)] Re[ eQk(t; z)] dt1CA dz

=CXk=1

Nk

lkZ0

0B@ 1

2�=!

2�=!Z0

Re[�(rpsk +reppk(t; z))] Re[Qsk + eQpk(t; z)] dt1CA dz

=CXk=1

Nk

lkZ0

0B@�rpskQsk + 1

2�=!

2�=!Z0

Re[�reppk(t; z)] Re[ eQpk(t; z)] dt1CA dz

+CXk=1

Nk

lkZ0

1

2�=!

0B@�repsk2�=!Z0

Re[ eQpk(t; z)] dt+Qsk2�=!Z0

Re[�reppk(t; z)] dt1CA dz

=CXk=1

Nk�pskQ

sk +

1

2

CXk=1

Nk

lkZ0

Re[�reppk(t; z) (Qpk(t; z))�] dz= W s

tot +Wptot (B-82)

The time average of the two cross terms in the third line, Re[�reppk(t; z)]Qsk and �rpsk Re[ eQpk(t; z)], is identicalzero, because the time average over a periodic (sine) function is zero. The dissipation is therefore the sum of a

steady and an oscillatory contribution. Since the Fourier components are all independent within the linearized

Navier-Stokes equations, this result can be simply extended to a pressure gradient and volume �ow that include

more than one Fourier component: Wtot =W stot +W

ptot;1 +W

ptot;2 +W

ptot;3 + : : :

Drag force of pulsatile �ow

The �uid moving with respect to the immobile vessel wall imposes a drag force in the plane tangential to the wall.

This wall shear force must be supplied by the pressure gradient acting on the cross-sectional area Ak. Per unit of

vessel length this force is thus given by (Milnor 1989)

efpk (t; z) = �r2k(�reppk(t; z)) (B-83)

However, since this force is oscillating in time and space, a better measure for the strength is its rms value,

fp;rmsk = �r2k(�rpp;rmsk ) (B-84)


Functional Ecology

The total (rms) drag force of a vessel of length lk is therefore

F p;rmsk = �r2klk(�rpp;rmsk ) (B-85)

and the drag force on the total network is simply F p;rmstot =PCk=1NkF

p;rmsk . Using Eqs. B-52 and B-80 or B-81

we �nd

F p;rmstot =CXk=1

�!j ePC js�M (1� �2k)

Ek

Nkr5=2k lk

h1=2k

(C)k

je�kj!1=2

(B-86)

=CXk=1

�!j eP1js�M (1� �2k)

Ek

Nkr5=2k lk

h1=2k

(1)k

je�kj!1=2

(B-87)

Appendix C. Optimization with constraints

In this Appendix we discuss the method of undetermined Lagrange multipliers. We want to optimize the transport

network, i.e., adjust number Nk, radius rk, length lk and thickness hk of the vessels of each level k, for an

organism in such a way that the transport costs are minimal. Moreover, we impose several constraints during

minimization that relate the vessel properties Nk, rk, lk and hk: we assume an organism with �xed mass Mobs,

�xed blood volume V obsb and a �xed quantity Xobsk (in general a �service volume�) for each level k = 1 : : : C

in the network. Note that these ng = C + 2 constraints can be written in the form gM = M � Mobs = 0,

gb = Vb � V obsb =PCk=1Nk�r

2klk � V obsb = 0 and gk = Xk � Xobs

k = NdNk rdrk l

dlk h

dhk � Xobs = 0 for

k = 1 � � �C.

If we were to optimize the dissipation without taking the constraints into consideration, we would simply set

all partial derivatives @Wtot=@Nj = @Wtot=@rj = @Wtot=@lj = @Wtot=@hj to zero for all j = 1 � � �C, and

solve for all Nj , rj , lj and hj . However, because of the extra ng constraints the variables Nk, rk, lk and hj cannot

be varied independently. One way to approach this problem is to solve ng variables from the constraints and to put

these explicit expressions in the original expression ofWtot. Then all the partial derivatives ofWtot except for the

eliminated ng variables can be set to zero again. However, this is in general a very involved and clumsy method.

Optimization of the total dissipationWtot with additional constraints of this type can be achieved much more

elegantly and simply by the method of undetermined Lagrange multipliers (Arfken 1985). A Lagrange function L

is being de�ned by adding toWtot all the extra ng constraints g (which are by de�nition all zero at the optimum)Apol, Etienne & Olff - Revisiting the evolutionary origin of allometric metabolic scaling in biology 20

Functional Ecology

multiplied by a Lagrange multiplier:

L = Wtot(fNkg; frkg; flkg; fhkg) + �b

"CXk=1

Nk�r2klk � V obsb

#

+CXk=1

�k

hNdNk rdrk l

dlk h

dhk �Xobs

i+ �M

�M �Mobs

�(C-1)

Here �b, �k and �M are the still undetermined Lagrange multipliers. The network is to be optimized with respect

to all number of vessels, radii, lengths etc., by setting all partial derivatives ofL to zero, i.e., @L=@Nj = @L=@rj =

@L=@lj = @L=@hj = 0 for all j = 1 � � �C, as well as @L=@M = 0. The resulting set of equations can be solved

to obtain the optimal values of Nk, rk, lk, and hk, as well as the optimal values of the Lagrange multipliers �b,

�k and �M . The partial derivatives @L=@�b = @L=@�k = @L=@�M = 0 simply yield the original constraints,

and provide no new information.

Note that the observed and �xed quantitiesMobs, V obsb andXobs can also be removed from the formulation of

the Lagrangian L (as for example in West et al. 1997; Dodds et al. 2001), since they do not depend on Nk, rk, lk,

hk andM , and therefore their partial derivatives are zero:

L =Wtot + �b

CXk=1

Nk�r2klk +

CXk=1

�k NdNk rdrk l

dlk h

dhk + �M M (C-2)

There is also a nice geometrical interpretation of the method of Lagrange multipliers. If we move (in multi-

dimensional space) along one constraint curve gk = 0, we cross many contours of the goal function Wtot, but

the optimum of Wtot at the curve gk = 0 is when the gradient of Wtot is perpendicular to the constraint gk,

or in other words when the gradients of Wtot and gk are parallel, i.e., when they are a multiple of each other.

This multiple is the Lagrange multiplier �k. Mathematically, the simultaneous requirements that gk = 0 and

rWtot = �krgk are equivalent to de�ning the function L =Wtot +�kgk and settingrL = 0. Physically, since

�k = @L=@gk = @W opttot =@gk, the Lagrange multiplier can also be thought of as the force of the constraint, i.e.,

how hard the constraint gk has to �pull� to establish a balance.

Appendix D. Minimization of steady dissipation (Poiseuille�ow)

In this Appendix we describe the minimization of transport costs (energy dissipation) in a branching network with

dominant steady Poiseuille �ow. The total dissipated energy in the network due to viscous friction W stot is given

by Eq. 10. We minimize this work under the assumptions of constant volume �ow Qtot, constant blood volumeApol, Etienne & Olff - Revisiting the evolutionary origin of allometric metabolic scaling in biology 21

Functional Ecology

Vb, �xed body mass M and preservation of the quantity Xk = NdNk rdrk l

dlk across all levels. Note that for the

WBE model, Xk = Vs;k = Nkl3k is the service volume. Since the volume �ow is kept constant, minimizing the

total work is identical to minimizing the hydrodynamic resistance Rtot. The constraints Vb = constant andM =

constant, together with the C constraints NdNk rdrk l

dlk = constant can be incorporated into the minimization via the

help of Lagrange multipliers �b, �M and �k (see Arfken 1985). The Lagrangian to be minimized is

L =�8�

�

�Q2tot

CXk=1

lkNkr4k

+ �b�CXk=1

Nkr2klk +

CXk=1

�kNdNk rdrk l

dlk + �M M (D-1)

and this is accomplished as usual by setting the derivatives @L=@Nj = @L=@rj = @L=@lj = 0 for all j = 1 : : : C,

as well as @L=@M = 0. The derivative in Nj gives

@L@Nj

= 0 = ��8�

�

�Q2tot

ljN2j r4j

+ �b�r2j lj + �jdNN

dN�1j rdrj l

dlj

for all j = 1 : : : C, yielding for the Lagrange multiplier �b

�b =

�8�

�2

�Q2tot

1

N2j r6j

� dN�j�NdN�1j rdr�2j ldl�1j (D-2)

The derivative in lj gives

@L@lj

= 0 = +

�8�

�

�Q2tot

1

Njr4j+ �b�Njr

2j + �jdlN

dNj rdrj l

dl�1j

for all j = 1 : : : C, resulting in

�b = ��8�

�2

�Q2tot

1

N2j r6j

� dl�j�NdN�1j ldl�1j rdr�2j (D-3)

The derivative in rj , �nally, gives

@L@rj

= 0 = �4�8�

�

�1

Njr5j+ 2�b�Njrj ll + �jdrN

dNj ldlj r

dr�1j

for all j = 1 : : : C, resulting in

�b = 2

�8�

�2

�Q2tot

1

N2j r6j

� dr2

�j�NdN�1j rdr�2j ldl�1j (D-4)

We now have three expressions for the Lagrange multiplier �b (Eqs. D-2, D-3 and D-4) that must be simultane-

ously true. This means that we can solve this set of three equations in three unknown variables: �b, �j and dN ,

say. The result, after some algebra, is

dN =dl + dr3

(D-5)

�b =

�4dl + dr2dl � dr

�8�

�2Q2tot

1

N2j r6j

(D-6)

�j =

�6

dr � 2dl

�8�

�Q2tot

1

NdN+1j rdr+4j ldl�1j

(D-7)


Functional Ecology

The �rst equation directly gives the condition

dr + dl = 3dN (D-8)

with dr 6= 2dl, which is consistent with the �nding of Bengtsson & Edén (2003) that dr + dl = 3 for dN = 1. The

situation dr = 2dl (i.e., dl = dN and dr = 2dN ) means that the quantity�Nkr

2klk�dN is preserved, or equivalently

that the blood volume per level is preserved. Since this immediately implies that also the total blood volume is

constant, adding C such constraints is redundant and gives therefore no solution.

Since the value of the Lagrange multiplier �b is independent of the value of j, it follows from Eq. D-6 that

N2j r6j is a constant, yielding the relation

�j�3j = 1 (D-9)

The constraints NdNk rdrk l

dlk = constant immediately give the relation �dNk �drk �

dlk = 1. Combining this with Eqs.

D-8 and D-9 gives a relation for �j :

�j�3j = 1 (D-10)

Using Eqs. D-8, D-9 and D-10 in Eq. D-7 it is easy to demonstrate that �j = �j+1, a result also obtained by

Dodds et al. (2001) for a more restricted minimization.

If we use the expression of the optimal Lagrange multipliers �k (Eq. D-7) in the original Lagrangian L (Eq.

D-1) that must be optimized, we see that L now becomes

L =�8�

�

�Q2tot

CXk=1

lkNkr4k

+ �bVb +

�8�

�

�Q2tot

�6

dr � 2dl

� CXk=1

lkNkr4k

+ �M M (D-11)

It is clear that the �rst and third term cancel if 6=(dr � 2dl) = �1 or 2dl � dr = 6, which results with the help of

Eq. D-8 in the following relations:

dl = dN + 2 and dr = 2dN � 2 (D-12)

Therefore, if the preserved quantity Xk = NdNk rdrk l

dlk is such that dl = dN + 2, the partial derivative of L inM

now simply becomes

@L@M

= 0 = �b@Vb@M

+ �M

from which it follows that

Vb(M) =

Z M

0

��M�b

dM 0 = ��M�b

M � Vb0M (D-13)


Functional Ecology

proving that the allometric blood volume exponent is

b = 1 (D-14)

Note that if we also use the optimal value of �b (Eq. D-6) the Lagrangian becomes

L = ��

6dldr � 2dl

��8�

�

�Q2tot

CXk=1

lkNkr4k

+ �M M

Therefore, if dl = 0 there is no mass dependence any more, so we must have

dl 6= 0 (D-15)

The space-�lling constraint of West et al. (1997), i.e.,Xk = Nkl3k or dl = 3 and dN = 1, is indeed consistent with

Eqs. D-12 and D-15 so optimizing a network with regarding Poiseuille dissipation implies that the blood volume

scales isometrically.

Appendix E. Minimization of pulsatile dissipation (Womersley�ow)

In this Appendix we describe the minimization of transport costs (energy dissipation) in a branching network with

dominant pulsatile Womersley �ow. The expression for the transport costs within a network with pulsatile �ow, the

pulsatile dissipation functionW ptot (Eqs. B-72 and B-78), is a very complicated function. The dissipation per level

k not only is a function of the network parameters Nk, rk, lk and hk of the same level k via the e�k-function (Eq.B-11), the Womersley number �k (Eq. B-13) and the damping constants ak (Eqs. B-22 and B-9), but at the same

time also of the parameters of level k + 1 via the re�ection factors e�k (Eqs. B-42 and B-44), so thus a functionof the scaling ratios �k, �k, �k and �k. This is different from the situation of Poiseuille dissipation, where the

dissipation per level only depends on the vessel properties of the same level (see Eq. 10). Since this complicates

the optimization procedure very much, we therefore perform the optimization in two stages. The �rst stage yields

a relation between the vessel properties (Eq. E-9) that is used as extra constraint during the optimization process

of the second stage. In this way the interaction between both stages is taken into account.

First stage: minimize re�ections

Based on insight from electronic transmission-line theory (Taylor 1957; Milnor 1989; West et al. 1997; West 1999;Apol, Etienne & Olff - Revisiting the evolutionary origin of allometric metabolic scaling in biology 24

Functional Ecology

West & Brown 2005), we will �rst minimize the dissipation function with respect to the wave re�ection factors

e�k = je�kjei��;k that combine parameters of both levels k and k+1 (so are dependent on the scaling ratios �k, �k,�k and �k), i.e., we optimize their amplitude je�kj and phase ��;k such that the dissipation is minimal. This meansoptimizing the (C)k -functions (Eq. B-78) with respect to je�kj and ��;k. Since W p

tot is a sum of (C)k -functions

over all levels k, we have to minimize each level separately. We focus initially on the dissipation within the vessels

of level C � 1, i.e., at the capillary side. The extremes of

(C)C�1 =

1� 2 cos (��;C�1) e�2aC�1lC�1 je�C�1j+ e�4aC�1lC�1 je�C�1j2e�2aC�1lC�1

�1 + 2 cos (��;C�1) je�C�1j+ je�C�1j2� (E-1)

can be found by setting@

(C)C�1

@je�C�1j = 0 and@

(C)C�1

@��;k= 0 (E-2)

and solving for je�kj and ��;k. There are in fact several extremes, but the physically allowed ones must have0 � je�kj � 1 real with (C)minC�1 > 0. It follows that the allowed minimum value of the (C)C�1-function is

(C)minC�1 = e2aC�1lC�1 (E-3)

for

je�C�1j = 0 and ��;C�1 = ��

2(E-4)

Focussing now on the next level, C � 2, and using Eq. E-4 we see that

(C)C�2 =

1� 2 cos (��;C�2) e�2aC�2lC�2 je�C�2j+ e�4aC�2lC�2 je�C�2j2e�2aC�2lC�2

�1 + 2 cos (��;C�2) je�C�2j+ je�C�2j2� � e2aC�1lC�1 (E-5)

has the same mathematical form as Eq. E-1. Therefore, its minimum is

(C)minC�2 = e2aC�2lC�2e2aC�1lC�1 with je�C�2j = 0 and ��;C�2 = �

�

2(E-6)

By repeating the same argument for levels C � 3, C � 4 etc., down to the aorta, we �nd in general that

(C)mink =

C�1Ym=k

e2amlm with je�kj = 0 and ��;k = ��

2(E-7)

This con�rms the general notion from transmission-line theory that by minimizing the wave re�ections, the dissi-

pation is (partly) minimized.

Note that the condition je�kj � j(1 � e k)=(1 + e k)j = 0 (see Eq. B-42), or equivalently je�kj2 = 0, implies

with e k � je kjei� that1� 2je kj cos � + je kj21 + 2je kj cos � + je kj2 = 0


Functional Ecology

so that we get the set of conditions 1� 2je kj cos � + je kj2 = 0, 1+ 2je kj cos � + je kj2 6= 0 for je kj � 0 and � both real. The solution of this set of conditions is that for zero wave re�ections we must have

je kj = 1 and � = 0 (mod 2�) (E-8)

Using Eq. B-44, we get

je kj = �k�5=2k �

�1=2k

qje�kjqje�k+1j = 1

so that the condition of zero re�ections in fact implies preservation of the quantity

Xyk � Nkr

5=2k h

�1=2k je�kj�1=2 = constant (E-9)

across the network.

Second stage: optimize the remaining network parameters for �k !1

Variation 1: general space �lling and hk=rk constant

In the �rst stage of optimization we have shown that part of the pulsatile dissipation is minimized if there exist

a preserved quantity Xyk of the form of Eq. E-9, i.e., a speci�c relation between the network variables Nk, rk

and hk. In the second optimization stage we will further optimize the network parameters Nk, rk, lk and hk to

fully minimize the pulsatile dissipation in the limit of large Womersley number (�k ! 1), i.e., for the network

close to the aorta. As constraints we consider an organism of �xed body mass M , �xed blood volume Vb, �xed

total net volume �ow Qtot, and general preserved quantity Xk = NdNk rdrk l

dlk h

dhk (a general �service volume� or

�general space �lling�) across all levels, with of course the extra constraint that the wave re�ections are zero, so

Xyk is preserved too. Moreover, we assume that the vessel wall thickness is a constant fraction of the radius, so

Xzk � rk=hk is preserved.

In the limit �k !1 we have (Eq. B-21)

lim�k!1

Re[e�k]��e�k�� p2

�k=

s2�

�M!r�1k (E-10)

For the damping factors ak we �nd from Eqs. B-25 and B-9

lim�k!1

ak = lim�k!1

�! Im[ 1eck ] = lim�k!1

!

p1� �2kc0;k

1p2

1

�k= limrk!1

s�!(1� �2k)Ekhk

r�1=2k ! 0


Functional Ecology

so that

(C)mink � 1 (E-11)

(Note that this is the same level of approximation as already used to obtain Eq. B-78 from Eq. B-73.) Hence for

�k !1 the pulsatile dissipation is

W ptot = x

CXk=1

Nkr2klk

hk(E-12)

with

x �s2�!

�M

�(1� �2)j ePC j2E

(E-13)

for short. From Eq. B-48 we obtain that the preserved quantity Xyk converges for large �k to

Xyk � Nkr

5=2k h

�1=2k (E-14)

The Lagrangian therefore reads

L = xCXk=1

Nkr2klk

hk+ �b�

CXk=1

Nkr2klk +

CXk=1

�kNdNk rdrk l

dlk h

dhk

+

CXk=1

�yk Nkr5=2k h

�1=2k +

CXk=1

�zk rkh�1k + �M M (E-15)

Setting as usual the partial derivatives @L=@Nj = @L=@rj = @L=@lj = @L=@hj = 0 for all j = 1 : : : C, we

obtain after rearranging the following set of equations:

�b = �Y1;j � dN�j Y2;j � �yj Y3;j (E-16)

�b = �Y1;j �dr2�jY2;j �

5

4�yjY3;j �

1

2�zjY4;j (E-17)

�b = �Y1;j � dl�j Y2;j (E-18)

0 = �Y1;j + dh�j Y2;j �1

2�yj Y3;j � �

zj Y4;j (E-19)

where

Y1;j ��x�

�h�1j (E-20)

Y2;j ��1

�

�NdN�1j rdr�2j ldl�1j hdhj (E-21)

Y3;j ��1

�

�r1=2j l�1j h

�1=2j (E-22)

Y4;j ��1

�

�N�1j r�1j l�1j h�1j (E-23)

If an optimal solution does exist, these four relations must be simultaneously true, so solving in the four unknown

quantities �b, �j , �yj and �zj gives after some algebra


Functional Ecology

�b = �2dN � dr � dl � dh2dN � dr � dh

�x�

�h�1j (E-24)

�j = � 1

2dN � dr � dh(x)N1�dN

j r2�drj l1�dlj h�1�dhj (E-25)

�yj =dN � dl

2dN � dr � dh(x) r

�1=2j ljh

�1=2j (E-26)

�zj = �12

5dN � 2dr � dl2dN � dr � dh

(x)Njrj lj (E-27)

Since the Lagrange multiplier �b must be independent of the level j, we �nd from Eq. E-24 that hj is a constant,

so that

�k = 1 so ch = 0 (E-28)

However, by combining the initial constraints, �dNk �drk �dlk �

dhk = 1, �k�

5=2k �

�1=2k = 1, and �k�

�1k = 1, we �nd

after some algebra

cr = ch =1

2(E-29)

cl =1

dl

�dN �

1

2(dr + dh)

�(E-30)

To reconcile Eqs. E-28 and E-29, we have to conclude that

�k = 1 (E-31)

This means that the optimal network is not a network anymore, but a single vessel, consisting of C identical

parts (N1 = Nk = Nk+1 = NC = 1) with dimensions r1 = rk = rk+1 = rC , l1 = lk = lk+1 = lC and

h1 = hk = hk+1 = hC , see Figure S4.

Biologically this is possible � a single vessel permeating the whole body and servicing all cells, but the phys-

iological performance is very problematic: 1) the concentration gradient over the vessel wall (and corresponding

metabolite transport) is initially very large, but decreases toward the end of the very long single vessel, and as a

result part of the body does not obtain the metabolites, 2) the �uid velocity is everywhere equal, and must be slow

to allow diffusion across the vessel walls, 3) in case of damage of obstruction, there is no alternative passage for

the blood. Most important, the number of capillariesNC does not scale with body mass (all organisms would have

only a single capillary), so combined with the assumed invariance of the capillary dimensions, we arrive at the

conclusion that such a network predicts

B _ NC _M0 so a = 0 (E-32)Apol, Etienne & Olff - Revisiting the evolutionary origin of allometric metabolic scaling in biology 28

Functional Ecology

kν

k

kN

klkr

1 2 3 4

1 1 1

1 1 1

Figure S4. Degenerate network: a single vessel after optimization according to the WBE model.

Version 2: general space �lling only

If we perform the same optimization procedure as above, but do not constrain the vessel wall thickness to radius

ratio, the Lagrangian is

L = xCXk=1

Nkr2klk

hk+ �b�

CXk=1

Nkr2klk +

CXk=1

�kNdNk rdrk l

dlk h

dhk

+CXk=1

�yk Nkr5=2k h

�1=2k + �M M (E-33)

Evaluating the derivatives in Nj , rj , lj and hj we �nd after some algebra

�b = �Y1;j � dN�j Y2;j � �yj Y3;j (E-34)

�b = �Y1;j �dr2�jY2;j �

5

4�yjY3;j (E-35)

�b = �Y1;j � dl�j Y2;j (E-36)

0 = �Y1;j + dh�j Y2;j �1

2�yj Y3;j (E-37)

If an optimal solution does exist, these four relations must be simultaneously true, so solving in the four unknownApol, Etienne & Olff - Revisiting the evolutionary origin of allometric metabolic scaling in biology 29

Functional Ecology

quantities �b, �j , �yj and dN gives after some algebra

dN =1

5(2dr + dl) (E-38)

�b = ��dr + 3dl + 5dhdr � 2dl + 5dh

�x

�h�1j (E-39)

�j =

�5

dr � 2dl + 5dh

�x N1�dN

j r2�drj l1�dlj h�1�dhj (E-40)

�yj = ��

2dr � 4dldr � 2dl + 5dh

�x r

�1=2j ljh

�1=2j (E-41)

The �rst equation directly gives the condition

2dr + dl = 5dN (E-42)

with dr � 2dl 6= �5dh 6= 0. Since the Lagrange multiplier �b is independent of the network level j, it follows

from Eq. E-39 that h�1j is a constant, so

�k = 1 so ch = 0 (E-43)

Combining Eqs. E-42 and E-43 with the constraint relations �dNk �drk �dlk �

dhk = 1 and �k�

5=2k �

�1=2k = 1 we �nally

obtain

�k = ��2=5k so cr =

2

5(E-44)

�k = ��1=5k so cl =

1

5(E-45)

If we substitute the optimal values of �j and �yj back into the Lagrangian function, we get

L = x

�5� (dr � 2dl � 5dh)dr � 2dl + 5dh

� CXk=1

Nkr2klk

hk+ �bVb + �M M

If the factor between square brackets is zero, the partial derivative in body massM immediately yields isometric

scaling of blood volume withM (see Eq. D-13). Thus isometric scaling of blood volume requires that

dr � 2dl = 5 + 5dh

On the other hand, if we also substitute the optimal value of �b into the Lagrangian, we get

L = x

�5� 2dr � dldr � 2dl + 5dh

� CXk=1

Nkr2klk

hk+ �M M

so that no mass dependence remains if 2dr + dl = 5. For isometric blood volume scaling we must therefore have

2dr + dl = 5dN 6= 5, or with Eq. E-42 that dN 6= 1. So summarizing, an optimal network with minimal pulsatileApol, Etienne & Olff - Revisiting the evolutionary origin of allometric metabolic scaling in biology 30

Functional Ecology

dissipation and no wave re�ections with isometric scaling of blood volume with body mass (b = 1) requires that a

property Xk is preserved throughout the network with

dr = 2dN + dh + 1 dl = dN � 2dh � 2 dN 6= 1 (E-46)

Clearly, the space-�lling constraint of West et al. (1997), Xk = Nkl3k = constant, is not compatible with these

relations, and therefore incompatible with an optimal network.

Variation 3: geometrical space �lling and hk=rk constant

If we apply the de�nition of West et al. (1997) of the service volume, Nkl3k = constant, we can use the equation

of variation 1 with dN = 1, dl = 3 and dr = dh = 0. Hence the Lagrangian reads

L = x

CXk=1

Nkr2klk

hk+ �b�

CXk=1

Nkr2klk +

CXk=1

�kNkl3k

+

CXk=1

�yk Nkr5=2k h

�1=2k +

CXk=1

�zk rkh�1k + �M M (E-47)

and the derivatives in Nj , rj , lj and hj yield after some rearrangement

�b = �Y1;j � �j Y2;j � �yj Y3;j (E-48)

�b = �Y1;j �5

4�yjY3;j �

1

2�zjY4;j (E-49)

�b = �Y1;j � 3�j Y2;j (E-50)

0 = �Y1;j �1

2�yj Y3;j � �

zj Y4;j (E-51)

where

Y1;j ��x�

�h�1j (E-52)

Y2;j ��1

�

�r�2j l2j (E-53)

Y3;j ��1

�

�r1=2j l�1j h

�1=2j (E-54)

Y4;j ��1

�

�N�1j r�1j l�1j h�1j (E-55)

If an optimal solution does exist, these four relations must be simultaneously true, so solving in the four unknownApol, Etienne & Olff - Revisiting the evolutionary origin of allometric metabolic scaling in biology 31

Functional Ecology

quantities �b, �j , �yj and �zj gives after some algebra

�b =1

2

�x�

�h�1j (E-56)

�j = �12(x)N1�dN

j r2�drj l1�dlj h�1�dXhj (E-57)

�yj = � (x) r�1=2j ljh�1=2j (E-58)

�zj = �12(x)Njrj lj (E-59)

Since the Lagrange multiplier �b must be independent of the level j, we �nd from Eq. E-56 that hj is a constant,

so that

�k = 1 so ch = 0 (E-60)

but by combining the initial constraints, �k�3k = 1, �k�5=2k �

�1=2k = 1, and �k�

�1k = 1, we �nd

cr = ch =1

2(E-61)

cl =1

3(E-62)

To reconcile Eqs. E-60 and E-61, we have again to conclude that

�k = 1 (E-63)

meaning that the optimal network is again a single vessel, implying that

a = 0 (E-64)

Variation 4: geometric space �lling only

If we omit the constraint that the vessel wall thickness is a �xed fraction of the vessel radius, the Lagrangian is

L = xCXk=1

Nkr2klk

hk+ �b�

CXk=1

Nkr2klk +

CXk=1

�kNkl3k

+CXk=1

�yk Nkr5=2k h

�1=2k + �M M (E-65)

The derivatives in Nj , rj , lj and hj yield after some rearrangement

�b = �Y1;j � �j Y2;j � �yj Y3;j (E-66)

�b = �Y1;j �5

4�yjY3;j (E-67)

�b = �Y1;j � 3�j Y2;j (E-68)

0 = �Y1;j �1

2�yj Y3;j (E-69)


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with

Y1;j ��x�

�h�1j (E-70)

Y2;j ��1

�

�r�2j l2j (E-71)

Y3;j ��1

�

�r1=2j l�1j h

�1=2j (E-72)

Solving in the three unknown quantities �b, �j and �yj gives no solution, so the constraints are con�icting.

However, we could � arbitrarily � decide not to optimize one of the vessel dimensions Nj , rj , lj or hj , and to

only optimizing the other remaining three dimensions. This means dropping one of the four equations E-66 - E-69

and solving in �b, �j and �yj . Independent of the choice which vessel dimension not to optimize, the resulting

solution of �b is always

�b /�x�

�h�1j (E-73)

implying that

ch = 0 (E-74)

Together with the relations arising from the constraints we �nd

cr =2

5(E-75)

cl =1

3(E-76)

predicting a metabolic exponent

a =15

17� 0:88 (E-77)

However, it is rather arbitrary not to optimize one of the vessel characteristics. Note that if we substitute the

optimal values of the Lagrange multipliers �j and �yj back into the Lagrangian, we obtain

L = xCXk=1

Nkr2klkh

�1k + �bVb �

1

2x

CXk=1

Nkr2klkh

�1k � 2x

CXk=1

Nkr2klkh

�1k + �MM

= �32x

CXk=1

Nkr2klkh

�1k + �bVb + �MM (E-78)

which means that within this optimization procedure isometric scaling of blood volume (b = 1) does not follow

from optimization of energy dissipation for pulsatile �ow through the condition @L=@M = 0, as was the case for

Poiseuille dissipation.Apol, Etienne & Olff - Revisiting the evolutionary origin of allometric metabolic scaling in biology 33

Functional Ecology

Appendix F. Incorrect de�nition of pulsatile dissipation

In this Appendix we discuss the consequences of the (inferred) incorrect de�nition of the pulsatile dissipation

function used by West et al. (1997, 2000). They supposedly used the following de�nition of the dissipation due to

pulsatile �ow,

W ptotjWBE =

1

2

CXk=1

Nk1

lk

zk�1+lkZzk�1

Rehepk(t; z) eQ�k(t; z)i dz (F-1)

If we assume for simplicity that there are no wave re�ections (so that Eq. E-9 must be valid), we have epk(t; z) =ep0k(t; z) and eQk = eQ0k, so that with the de�nition of the characteristic impedance (Eq. B-31) and the rms �ow (Eq.B-3) we obtain

W ptotjWBE =

1

2

CXk=1

Nk Reh eZC;ki 1

lk

zk�1+lkZzk�1

�� eQ0k��2 dz=

CXk=1

Nk Reh eZC;ki (Qp;rmsk )

2 (F-2)

=CXk=1

1

NkReh eZC;ki �Qp;rmstot;k

�2(F-3)

where we de�ned the total rms �ow via�Qp;rmstot;k

�2� N2

k (Qp;rmsk )

2. Since it is not clear whether or not West et

al. (1997, 2000) kept this total pulsatile rms �ow constant during optimization, we will investigate both options.

Optimization with �xed rms �ow

We minimize the pulsatile dissipation, Eq. F-3, assuming �xed total rms �ow�Qp;rmstot;k

�2, �xed body mass M ,

�xed blood volume Vb and geometrical space �lling, Xk = Nkl3k in the limit of large Womersley number. In that

limit (see Eq. B-34)

lim�k!1

Reh eZC;ki = 1

�

s�ME

2(1� �2)r�5=2k h

1=2k (F-4)

and using Eq. E-14, i.e., the wave re�ection constraint Nkr5=2k h

�1=2k = constant, the Lagrangian therefore reads

L = xCXk=1

h1=2k

Nkr5=2k

+ �b�

CXk=1

Nkr2klk +

CXk=1

�kNkl3k

+CXk=1

�yk Nkr5=2k h

�1=2k + �M M (F-5)

where

x � 1

�

s�ME

2(1� �2) (F-6)


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Note that the dissipation function has the same form as the wave-re�ection constraint. Therefore, the dissipation

function cannot be further optimized. This also follows from evaluating the derivatives of L in Nj , rj , lj and hj .

We obtain after some algebra

�b = Y1;j � �j Y2;j � �yj Y3;j (F-7)

�b =5

4Y1;j �

5

4�yj Y3;j (F-8)

�b = �3�j Y2;j (F-9)

0 =1

2Y1;j �

1

2�yj Y3;j (F-10)

where

Y1;j ��x�

�N�2j r

�9=2j l�1j h

1=2j (F-11)

Y2;j ��1

�

�r�2j l2j (F-12)

Y3;j ��1

�

�r1=2j h

�1=2j (F-13)

Solving this set of equations in �b, �j and �yj we obtain

�b = 0 (F-14)

�j = 0 (F-15)

�yj = xN�2j r�5j hj (F-16)

Therefore, from optimizing this dissipation function we do not obtain the relation hk=rk = constant, as claimed

by West et al. (1997, 2000).

Optimization with �xed initial pressure amplitude

If we do not assume the total rms �ow to be constant, we can employ Eq. B-80 in Eq. F-2 to obtain with the help

of Eq. B-31

W ptotjWBE =

CXk=1

Nk Re

24 �M�r2k

c0;kp1� �2

se�ki

35 �2(1� �2)j ePC j2�ME

r5khk

(C)k

je�kj=

s(1� �2)2�ME

j ePC j2 CXk=1

Nkr5=2k h

�1=2k

Re[

qe�k=i]je�kj

(C)k (F-17)


Functional Ecology

In the limit of large Womersley number we have

lim�k!1

(C)k = 1 (F-18)

lim�k!1

Re[

qe�k=i]je�kj = 1� 1p

2�k+ � � � ! 1 (F-19)

so that the Lagrangian becomes

L = xCXk=1

Nkr5=2k

h1=2k

+ �b�CXk=1

Nkr2klk +

CXk=1

�kNkl3k

+CXk=1

�yk Nkr5=2k h

�1=2k + �M M (F-20)

where

x �

s(1� �2)2�ME

j ePC j2 (F-21)

Again, the dissipation function cannot be optimized further, because it has the same form as the wave-re�ection

constraint. This also follows from setting the derivatives of L in Nj , rj , lj and hj to zero. We obtain

�b = �Y1;j � �j Y2;j � �yj Y3;j (F-22)

�b = �54Y1;j �

5

4�yj Y3;j (F-23)

�b = �3�j Y2;j (F-24)

0 = �12Y1;j �

1

2�yj Y3;j (F-25)

where

Y1;j ��x�

�r1=2j l�1j h

�1=2j (F-26)

Y2;j ��1

�

�r�2j l2j (F-27)

Y3;j ��1

�

�r1=2j h

�1=2j (F-28)

Solving this set of equations in �b, �j and �yj we obtain

�b = 0 (F-29)

�j = 0 (F-30)

�yj = �xl�1j (F-31)

Therefore, from optimizing this dissipation function we again do not obtain the relation hk=rk = constant, as

claimed by West et al. (1997, 2000).Apol, Etienne & Olff - Revisiting the evolutionary origin of allometric metabolic scaling in biology 36

Functional Ecology

Appendix G. References

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Etienne, R.S., Apol, M.E.F. & Olff, H. (2006) Demystifying the West, Brown & Enquist model of the allometry of

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Grob, B. (1985) Basic electronics (5th ed.) McGraw Hill, New York.

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West, G.B., Brown, J.H. & Enquist, B.J. (2000) Chapter 6 in Brown, J.H. & West, G.B. (eds) Scaling in Biology.


Functional Ecology

Oxford, NY, pp. 87�112.

West, G.B. & Brown, J.H. (2005) The origin of allometric scaling laws in biology from genomes to ecosystems:

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Womersley, J.R. (1955b) On the oscillatory motion of a viscous liquid in a thin-walled elastic tube: I. The linear

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Womersley, J.R. (1958) Oscillatory �ow in arteries. II: the re�ection of the pulse wave at junctions and rigid inserts

in the arterial system. Physics in Medicine and Biology 2, 313�323.

Zamir, M. (1977) Shear force and blood vessel radii in the cardiovascular system. Journal of General Physiology

69, 449�461.


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