Hierarchical Loop Stabilization in Periodically Driven Elastic Networks

TL;DR

This work addresses how periodic forcing influences long-term remodeling of elastic transport networks by deriving a pulsatile energy-driven remodeling rule in which the growth term depends on the cycle-averaged dissipative signal $\langle \sigma_e Q_e\rangle$ and a metabolic exponent $\gamma$. A simplified flow model with scales $\lambda$, $\tau$, $\alpha$ and a Womersley framework provides exact solutions for $Q$ and $P$ under periodic drive, enabling computation of $\langle \sigma Q\rangle$ across network paths. Across two-, five-, and eight-vessel toy networks, the authors show that resonant frequencies (where dissipation is amplified) preferentially stabilize higher-level loops near the source, while anti-resonances favor loopless configurations or preservation of lower-level distal loops; importantly, loop stabilization occurs for a broad range of $\gamma>\tfrac{1}{2}$, even when previous steady-flow theories predicted loop loss. The findings suggest a general mechanism by which pulsatility shapes loopy architectural diversity in biological networks and highlight how geometry and driving frequency jointly constrain long-term topology, with implications for understanding vascular remodeling and the design of programmable flow networks.

Abstract

Network remodeling, or adaptation, in the presence of periodically driven forcings has hereto remained largely unexplored, despite the fact that a broad class of biological transport networks, e.g. animal vasculature, depends on periodic driving (pulsatility of the heart) to maintain flow. Short-term pulsatile dynamics of compliant vessels affects the long-term structures of adapting networks; however, what the correct adaptation rule is for pulsatile flows still remains an open question. Here we propose a new adaptation rule for periodically driven complex elastic networks that accounts for the effect of short-term pulsatile dynamics on the remodeling signal at long time-scales. Using this rule to adapt hierarchical elastic networks with multiple levels of looping, we show that very different network architectures are possible at steady-state depending on the driving frequency of the pulsatile source and the geometric asymmetry of the paths between the externally driven nodes of the network. Specifically resonant frequencies are shown to prioritize the stabilization of fully looped structures or higher level loops proximal to the source, whereas anti-resonant frequencies predominantly stabilize loop-less structures or lower-level loops distal to the source. Thus, this model offers a mechanism that can explain the stabilization of phenotypically diverse loopy network architectures in response to source pulsatility under physiologically relevant conditions and in the absence of other known loop stabilization mechanisms, such as random fluctuations in the load or perfusion homogenization.

Figures (10)

• Figure 1: Eight vessel toy network with a current $Q$ from the pulsatile source entering and leaving the network through two externally driven nodes (green dots), and with three levels of looping. Arrows point to possible steady states that may be obtained through long-term adaptation of the network, with dashed lines illustrating vessel shunting, i.e. the vessel radius going to zero.
• Figure 2: (a) Two vessel toy network. (b-e) The energy dissipated per unit time in the $e^{th}$ vessel as a function of the re-scaled frequency $\omega\tau_0$, with $D_e=\langle Q_e^2\rangle L_e/R_e^4$ in the steady-flow case in (b,c) and $D_e=\langle \sigma_e Q_e\rangle L_e/R_e$ in the pulsatile-flow case (d,e). Panels (b) and (d) correspond to vessels of equal lengths $L_1=L_2=1.0$, and panels (c) and (e) correspond to a case of unequal vessel lengths with $L_1=1.0, L_2=\sqrt{2}$. In (b-e), the solid curves correspond to vessel $1$ and the dashed curves to vessel $2$, whereas the two colors differentiate between two distinct sets of radii pairs: green for $R_1=1.0$, $R_2=1.0$ and red for $R_1=0.75$, $R_2=1.25$. In both the steady-flow and pulsatile-flow cases, the dissipation is amplified in each vessel for $\omega\tau_0>0$, with maximum amplification occurring at certain resonant frequencies. The two vessels have identical resonant frequencies only when their lengths are equal, as in (b) and (d). The pulsatile-flow case generally shows higher dissipation amplification for all $\omega\tau_0$ than the steady-flow case, and this trend becomes progressively more pronounced as $\omega\tau_0$ is increased. $R_0=\tau_0=Q_s=Q_p=1$, $\lambda_0=2$.
• Figure 3: Phase diagrams in the $\gamma-\omega\tau_0$ phase-space for the average network entropy $\langle S\rangle$ over $144$ initial conditions, where $\langle S\rangle=1$ corresponds to a symmetric loop ($R_1= R_2$), $0<\langle S\rangle<1$ to an asymmetric loop ($R_1\neq R_2$ and $R_1,R_2>0$) and $\langle S\rangle=0$ to shunting or loop destabilization ($R_1=0$ or $R_2=0$) on average. (a-c) correspond to the steady-flow adaptation rule, with (a,b) considering cases of equal vessel lengths while (c) considers unequal vessel lengths. Similarly (d-f) correspond to the pulsatile-flow rule, with equal vessel lengths considered in (d,e), but not in (f). In all cases, loops are stabilized in broad regions of the phase space with $\gamma>1/2$, with $\langle S\rangle>0$. For vessels of equal lengths (a,b,d,e), the critical value $\gamma_c(\omega\tau_0)$ above which loops become unstable, shows periodic modulations as a function of the re-scaled frequency, reaching maximum values at certain resonant frequencies. An overall increase in the vessel length in comparison to the damping length-scale $\lambda_0$ results in shorter and more frequent resonant peaks, as seen by comparing (a) to (b) and (d) to (e). For unequal vessel lengths, $\gamma_c(\omega\tau_0)$ is not strictly periodic, and loop stabilization can only occur in regions where the resonances of each vessel, which are distinct, have significant overlap. While the steady and pulsatile flow rules yield nearly identical phase diagrams in the equal vessel length cases (a,b) and (d,e), for unequal vessel lengths the two rules generate somewhat different steady-state structures especially in regions of the phase-space where symmetric loops cannot be supported (e.g around $\omega\tau_0=6\pi$). This is because of the heightened asymmetry in dissipation amplification of each vessel under the pulsatile-flow rule, which favors the growth of a thick vessel at the expense of the other, thinner vessel, leading to shunting instead of the stabilization of an asymmetric loop. The equal vessel length cases (a,b,d,e) can always stabilize a symmetric loop when the starting radii of the two vessels are exactly equal, unlike the case of unequal vessel lengths (c,f). This leads to the minimum value of $\langle S\rangle$ to be higher in (a,b,d,e) than in (c,f). $R_0=\tau_0=Q_s=Q_p=a=b=1$ and $\lambda_0=2$.
• Figure 4: (a) Five vessel toy network. (b-d) Phase-diagrams in the $\gamma-\omega\tau_0$ phase-space for the average network entropy $\langle S\rangle$ for $250$ initial conditions. The primary path-length $\ell_1$ corresponds to the length of the single vessel connecting nodes $1$ and $4$, whereas the secondary path-lengths $\ell_2$ and $\ell_3$ correspond to the total length of the two paths that connect node $1$ to $2$ to $3$ to $4$. All three cases show robust loop stabilization for $\gamma>1/2$, although the resonant frequencies at which loops are stabilized for the largest range of $\gamma$ values, are periodically spaced only in the case where all three path-lengths are equal, as in (b). For unequal path-lengths, loop stabilization is stronger when the primary path is longer than the secondary path as in (d). $R_0=\tau_0=Q_s=Q_p=a=b=1$ and $\lambda_0=2$.
• Figure 5: (a) All possible steady state ($SS$) architectures allowed for the five vessel network. (b-e) Fraction of each possible structure in $SS1-SS5$ obtained at steady-state for $250$ different initial conditions, for $\gamma=2/3$. The color coding of the steady state fraction follows that of panel (a), e.g. red for $SS1$, orange for $SS2$ etc. When the three path-lengths between the driven nodes are equal, as in (b,d), resonant frequencies are periodically spaced, unlike in the unequal path-length cases in (c,e). In all cases however, resonant frequencies stabilize the fully looped structure ($SS5$) or the primary loop ($SS3$), whereas anti-resonant frequencies lead to loopless structures ($SS1,SS2$), or stabilize the secondary loop ($SS4$). Comparing (c) and (e) we find that when the primary path-length ($\ell_1$) is longer than the secondary path-lengths ($\ell_2,\ell_3$), the probability that the secondary loop ($SS4$) is stabilized close to anti-resonant frequencies, is larger, and the probability of obtaining loopless structures ($SS1,SS2$) is smaller. $R_0=\tau_0=Q_s=Q_p=a=b=1$ and $\lambda_0=2$.