Pulsatile Driving Stabilizes Loops in Elastic Flow Networks Purba Chatterjee Sean Fancher and Eleni Katifori Department of Physics and Astronomy University of Pennsylvania Philadelphia Pennsylvania 19104 USA

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Pulsatile Driving Stabilizes Loops in Elastic Flow Networks

Purba Chatterjee, Sean Fancher, and Eleni Katifori

Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA

Existing models of adaptation in biological ﬂow networks consider their constituent vessels (e.g.

veins and arteries) to be rigid, thus predicting a non physiological response when the drive (e.g

the heart) is dynamic. Here we show that incorporating pulsatile driving and properties such as

ﬂuid inertia and vessel compliance into a general adaptation framework fundamentally changes

the expected structure at steady state of a minimal one-loop network. In particular, pulsatility is

observed to give rise to resonances which can stabilize loops for a much broader class of metabolic

cost functions than predicted by existing theories. Our work points to the need for a more realistic

treatment of adaptation in biological ﬂow networks, especially those driven by a pulsatile source, and

provides insights into pathologies that emerge when such pulsatility is disrupted in human beings.

I. INTRODUCTION

The structure of physiological transport networks such

as animal vasculature and leaf venation, has important

consequences for biological functionality, and as such has

elicited considerable scientiﬁc interest over the years. In

particular, looped network architectures, ubiquitous in

biology, are beneﬁcial for mitigating vessel damage and

optimizing responses to source ﬂuctuations [1–3]. Net-

work remodeling [4, 5], also known as adaptation, is now

understood to proceed by optimizing the total energy dis-

sipation in the network, subject to some metabolic cost

[6–9]. Such metabolic costs can generally be described

by a power law (Kσ), where Kis vessel conductivity and

σis a system speciﬁc parameter. Existing theories of

adaptation in ﬂow networks predict a critical transition

at σ= 1, with a structure with many loops for σ < 1 and

one which is a loop-less tree for σ > 1 [1, 2, 10–12]. How-

ever the scope and generality of this prediction remains

to be investigated in the light of biologically relevant dy-

namical considerations.

Previous studies on adaptation consider vessels to be

rigid, leading to the assumption that modulations in ﬂow

boundary conditions are instantaneously propagated to

individual network elements at all times. However, vessel

compliance and ﬂuid inertia have been shown to gener-

ate a ﬁnite timescale of information transfer from the

sources to the bulk [13, 14], leading to trade-oﬀs between

energy eﬃciency on one hand and mechanical response

to sudden changes in the steady state dynamics on the

other [13]. Moreover, while ﬂuctuating sources and sinks

have been implemented within the framework of network

adaptation [1, 6, 15, 16], the eﬀect of deterministic pul-

satile driving at the source is largely unexplored . This is

an important consideration for biological transport net-

works, many of which rely on pulsatility to maintain ﬂuid

pressure. The most prominent example of this is mam-

malian vasculature, with the periodic beating of the heart

muscle introducing pulsatile components into blood ﬂow.

In this Letter, we investigate the eﬀect of both pul-

satile driving and the internal spatio-temporal dynamics

of elastic vessels on the adaptation of a simple one-loop

ﬂow network. Depending on the lengths of the vessels in

FIG. 1. (color online) (a) Minimal model with L1=L2=L,

and the vessel mean-squared current in the absence of ﬂuid

inertia and vessel compliance. (b,c) Flow diagrams for AE

in R1−R2space, a=b=L= 1. Insets show radii at

steady state for the chosen initial condition (green solid circle)

depicted in (a). Sinks at steady state denoted by solid orange

circles, and saddle points by hollow orange circles.

relation to each other and to a characteristic length scale

over which pulsatility is damped, resonant frequencies

are shown to exist, which amplify energy dissipation and

stabilize loops for a much broader class of metabolic cost

functions than predicted by existing theories of adap-

tation. Our results emphasize the need for a more so-

phisticated treatment of adaptation in order to correctly

predict the steady state structure of more complicated

biological transport networks and might be key in ex-

plaining the development of vascular malformations in

patients with artiﬁcial hearts [17, 18].

arXiv:2210.06557v2 [nlin.AO] 19 Oct 2022

II. THE ADAPTATION EQUATION

Our minimal network consists of two vessels connecting

the nodes N1and N2(Fig. 1(a)), subject to pulsatile

driving at the frequency ω. The vessel radii Rµ,µ∈[1,2],

change in response to the local current Qµover time t0,

according to the Adaptation Equation (AE)

dRµ

dt0=ahQ2

µiγ

−bRµ,(1)

where aand bare constants, and γis a parameter as-

sociated with the metabolic cost. Assuming Poiseuille

ﬂow (Kµ∝R4

µ), this AE is identical to a general adap-

tation rule that has been used to model the dynamics of

hydraulic vessel conductivities (Kµ) during animal vas-

cular development and the slime mold Physarum poly-

cephalum [6, 18–23]. Each vessel adapts through a local

positive feedback, expanding in radius when the current

through it is large, and shrinking at the characteristic

timescale b−1when it is small. The steady states of the

AE correspond to the critical points of the optimization

functional

E=X

Lµ

Kµ

+β X

LµKσ

µ−C!,(2)

where, Lµis the vessel length, βis a Lagrange multiplier

and Cis a constant [6]. The ﬁrst term corresponds to

the total power dissipated in the network, and the second

term imposes a metabolic or material cost characterized

by σ= 1/γ −1. For γ= 2/3, this material cost is equal

to the total volume of ﬂow in the network, which is an

important constraint for animal vasculature.

We deﬁne the vessel mean-squared current at a given

time t0of adaptation as

hQ2

µi=1

TZT

dt h1

LµZLµ

dz Qµ(z, t)2i,(3)

where T= 2π/ω is the time-period of pulsatility. Note

that we distinguish the adaptation time t0in the AE

(Eq. 1) from the time used to calculate the vessel mean-

squared current t, because adaptation typically occurs on

much longer timescales than that of local modulations of

ﬂow in individual vessels (i.e. t0t).

When ﬂuid inertia and vessel compliance are neglected,

the current at node N1splits proportionally between the

two vessels depending on their conductance, and the ves-

sel mean-squared current has the form given in Fig. 1(a).

The critical transition of the AE for this two-vessel net-

work ∀ωcan be analytically shown to occur at γAE

c= 1/2

(see supplemental materials). This is illustrated by the

steady state ﬂow-diagrams in R1−R2space (Fig. 1(b,c)).

For γ < γAE

c, the diagonal has a stable ﬁxed point (sink),

which corresponds to a stabilized loop with vessels of

equal radius at steady steady, irrespective of their initial

sizes (Fig. 1(b)). For γ > γAE

c, there exist two stable

ﬁxed points at the boundaries with large basins of at-

traction, indicating that for most initial conditions, one

or the other vessel is lost. The diagonal has an unsta-

ble ﬁxed point (saddle), indicating that the loop is stable

only for a narrow range of initial conditions correspond-

ing to exactly equal starting radii (Fig. 1(b)).

III. COMPLIANT VESSELS

The treatment above does not consider the opposi-

tion to changes in ﬂow pressure due to ﬂuid mass and

the resulting ﬂuid inertia. Moreover, biological networks

are composed of compliant vessels, which can change

in radius reversibly at short timescales to accommodate

changes in ﬂow volume. As shown in [13, 24], inertia and

compliance generates a ﬁnite time lag in ﬂow propagation

from the sources to the bulk of the transport network.

Thus, in addition to the ﬂow resistance (or conductance),

the combined contribution of ﬂuid inertia and compliance

can be expected to alter the vessel mean-squared current

on the timescale of adaptation. We follow the treatment

of compliant vessels in [13], and assuming an incompress-

ible, laminar ﬂow with rotational symmetry, the axial

current Q(z, t) and pressure P(z, t) in each vessel satisfy

∂Q

∂z +c∂P

∂t = 0,(4)

∂P

∂z +l∂Q

∂t +rQ = 0.(5)

The cross-section of the vessel changes as A(z, t) =

A0+cP (z, t) in response to wall pressure, where cis

the vessel compliance. We assume such changes in cross-

section to be small in magnitude, i.e. A0cP (z, t).

These vessel parameters can be combined to construct

the characteristic length (λ), time (τ) and admittance (α)

scales, which all vary proportional to the area of cross-

section, as

λ=λ0(R/R0)2=2

rrl

τ=τ0(R/R0)2=2l

α=α0(R/R0)2=rc

l,(6)

where R0is a typical radius. In particular, increasing λ0

at constant radius, with αλ and τheld ﬁxed, reﬂects a de-

crease in the compliance cof the vessel. Generalizing the

single compliant vessel to a network of compliant vessels

is straightforward, and following [13], the vessel mean-

squared current hQ2

µi(Eq. 3) can be calculated for each

vessel. For the two-vessel network, this mean-squared

current has a more complex dependence (see supplemen-

tal materials) on the vessel radius than the form given

in Fig. 1(a), and when used to drive the AE, results in a

more realistic description of the evolution of the network

structure, as we show below. For convenience, we will

refer to the new framework of adaptation with ﬂuid iner-

tia and vessel compliance taken into consideration as the

Modiﬁed Adaptation Equation (MAE), to distinguish it

from the AE.

IV. RESULTS

The consequences of periodic driving of the MAE for

the steady state structure of the two-vessel network are

signiﬁcant. As an example, Fig. 2 shows the steady state

ﬂow diagrams in R1−R2space with vessels of equal

but small eﬀective lengths (L/λ0<1), for which com-

pliance is relatively low and pulsatile components of the

ﬂow are non-negligible. Like in the AE, the stable ﬁxed

point for γ= 1/3 and all values of ω, lies on the di-

agonal (R1=R2), indicating a stable symmetric loop.

For γ= 2/3, the non-pulsatile (ω= 0) ﬂow diagram

has sinks on the boundaries, meaning the loss of one or

more vessel, once again similar to the AE result shown in

Fig. 1(c).For ω= 1.8πhowever, while stable steady states

exist on both the diagonal as well as the boundaries, the

former has a much broader basin of attraction than the

latter. This implies that unlike the AE, for most initial

conditions of the MAE loops can be stabilized for pul-

satile driving at this frequency. Even more interestingly,

for ω= 2.6π, new steady states with vessels of ﬁnite but

unequal radii emerge and are stable, with basins of at-

traction larger than that of the boundary sinks. Clearly,

at this frequency loops exist at steady state for a broad

range of initial conditions, albeit with asymmetric ﬂow

distribution between the two vessels.

In general with pulsatility, the energy dissipation

through vessel µis ampliﬁed at special resonant frequen-

cies depending on the value of Lµ/λ0, causing it to ex-

pand even for γ > γAE

c. For each initial condition, we can

calculate the quantity Z= min hR1/R2, R2/R1it→∞, the

minimum of the ratio of the radii of the vessels at steady

state. For Z= 0, the steady state is loopless, and for

0< Z ≤1 the steady state is looped, with Z= 1 corre-

sponding to vessels of equal radius. The critical transi-

tion from a looped to a loop-less structure in the MAE

as a function of the driving frequency can then be under-

pinned by the order parameter hZi, averaged over many

initial conditions. Fig. 3(a) shows the phase-diagram of

hZiin the γ−ωphase space, for the low compliance case

depicted in Fig. 2. The AE in this case would yield loops

(hZi>0) below γAE

c= 1/2 and no loops (hZi= 0) above

it (dashed red line). In contrast, the phase boundary

between looped and loop-less steady states in the MAE

has periodic modulations with respect to ω, with loops

stabilized for all physiologically relevant values of γat

resonant frequencies (dashed blue lines). Moreover there

is a signiﬁcant spread of frequencies around the resonant

values, for which loops are stable above γAE

c. Also, stable

loops tend to be increasingly more asymmetric in radius

for higher values of γ, even at resonant frequencies.

Increasing vessel lengths to be greater than λ0, but

still equal, decreases the looped hZi>0 phase in area,

as shown in Fig. 3(b). The periodic modulations with

frequency in the critical transition are however retained,

with shorter and more frequent peaks. Thus, even in

the case of high relative compliance (L/λ0>1), where

the eﬀect of pulsatile driving is signiﬁcantly damped, the

MAE is able to stabilize loops for a larger range of γ

values than the AE.

For L1=L2=L, an on-diagonal steady state always

exists (Fig. 2), but is stable for each ωvalue only be-

low a critical value γ=γMAE

c(ω). Above γMAE

c(ω) the

MAE generates either a loop-less network or an asym-

metric loop at steady state (0 <hZi<1). Fig 3(c)

shows the analytical phase diagram of γMAE

c(see supple-

mental materials for derivation) in the ωτ −L/λ space.

Here τand λare the full radius dependent time and

length scales (Eq. 6) corresponding to the on-diagonal

ﬁxed point for the given ωand for γ=γM AE

c. Clearly,

the value of γMAE

coscillates with the frequency, with

γAE

c< γM AE

c≤1. The trajectories in black and red

in Fig 3(c) correspond to the trajectories of the critical

transition in Fig 3(a,b), showing excellent agreement be-

tween the predictions of the numerical phase-diagram of

hZi= 1 and the analytical phase-diagram of γMAE

c(ω).

The diﬀerences between the steady sate structures of the

MAE and the AE are most pronounced for shorter vessels

with L/λ0<1, and the upper bound of oscillations in

γMAE

c(ω) decreases monotonically with increasing L/λ0.

Lastly, Fig. 4(a,b) shows two cases with L16=L2, one

where both vessels are shorter, and another where only

one vessel is shorter than λ0. Here, unlike in the L1=L2

case, the on-diagonal steady state does not exist for all

values of γand ω. This explains the decrease in area of

the hZi= 1 phase from Fig. 3 to Fig. 4. However, while

the AE would predict hZi= 0 for all γ > γAE

c(dashed

green line), clearly the MAE supports resonance frequen-

cies (dashed blue lines) for which asymmetric loops are

stabilized for a broad range of γvalues (see supplemental

materials for a more detailed discussion).

V. DISCUSSION

In summary, our simple model conﬁrms that through

the interplay between the time and length scales of pul-

satile driving and those generated by network properties

such as ﬂuid inertia and vessel compliance, the modi-

ﬁed adaptation framework (MAE) displays rich behavior

that would be missed if the bulk ﬂow was considered to

instantaneously reﬂect modulations at the source, as in

the AE. Loops (symmetric or asymmetric) are stable in

the MAE for a much broader ranger of γvalues than

in the AE, more so for less compliant vessels, that are

shorter than the characteristic length scale λ0. This is

true even when the two vessels comprising the loop have

unequal lengths, and importantly also when only one is

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PulsatileDrivingStabilizesLoopsinElasticFlowNetworksPurbaChatterjee,SeanFancher,andEleniKatiforiDepartmentofPhysicsandAstronomy,UniversityofPennsylvania,Philadelphia,Pennsylvania19104,USAExistingmodelsofadaptationinbiologicalownetworksconsidertheirconstituentvessels(e.g.veinsandarteries)toberigid,th...

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Pulsatile Driving Stabilizes Loops in Elastic Flow Networks Purba Chatterjee Sean Fancher and Eleni Katifori Department of Physics and Astronomy University of Pennsylvania Philadelphia Pennsylvania 19104 USA

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