A functional exchange shunt in the umbilical cord: the role of coiling in solute and heat transfer

TL;DR

The paper investigates whether the coiled umbilical cord forms a functional diffusive shunt between the UV and UA that could influence fetal thermoregulation and cord tissue oxygenation. It develops a helicity-aware diffusion framework that reduces the 3D intervascular problem to a 2D cross-section via a helical coordinate transform, parameterized by $\Omega$, with boundary terms $\eta$, $\alpha$, and $\beta$, and uses Damköhler number $\mathrm{Da}$ to relate diffusive and advective transport. Findings show that cord coiling and close vessel proximity amplify diffusive exchange, yet observed human configurations tend to minimize shunting, while a potential 'virtual' vasa vasorum can augment tissue oxygenation under certain metabolism and geometry. The framework provides a quantitative structure–function basis for cord physiology with implications for fetal thermoregulation and pathology, and offers a platform for cross-species comparisons and future in vivo validation.

Abstract

The umbilical cord plays a critical role in delivering nutrients and oxygen from the placenta to the fetus through the umbilical vein, while the two umbilical arteries carry deoxygenated blood with waste products back to the placenta. Although solute exchange in the placenta has been extensively studied, exchange within the cord tissue has not been investigated. Here, we explore the hypothesis that the coiled structure of the umbilical cord could strengthen diffusive coupling between the arteries and the vein, resulting in a functional shunt. We calculate the diffusion of solutes, such as oxygen, and heat in the umbilical cord to quantify how this shunt is affected by vascular configuration within the cord. We demonstrate that the shunt is enhanced by coiling and vessel proximity. Furthermore, our model predicts that typical vascular configurations of the human cord tend to minimise shunting, which could otherwise disrupt thermal regulation of the fetus. We also show that the exchange, amplified by coiling, can provide additional oxygen supply to the cord tissue surrounding the umbilical vessels.

Figures (4)

• Figure 1: The vascular structure of the umbilical cord. (a) A schematic of the helical structure of the umbilical cord that connects the fetus to the placenta; the pitch $2\pi/\Omega$ represents the average length of a coil relative to the radius of the cord; $\Omega$ is the dimensionless helicity parameter (proportional to the UCI). (b) A histological image of the cross-section of a healthy human cord (adapted from Blanco2011 under CC BY-NC 4.0); the vascular cord configuration is characterised by the angle between the umbilical vein and the first artery ($\theta_1$, measured relative to the $x$-axis that passes through the centre of the cord and the centre of the vein), the angle between the two arteries $\Delta\theta$, and the artery--vein separation distances $d_1,\, d_2$ normalised by the cord radius. (c) Schematic illustrations of hypercoiled and hypocoiled umbilical cords.
• Figure 2: The impact of cord helicity and vascular proximity on solute exchange. (a) Relative exchange flux $N_\mathrm{rel}$ per unit length \ref{['eq:flux_rel_uncoiled']} of the umbilical vein vs. cord helicity $\Omega$ for different UA-UV separation distances $d=0.3$ and $d=0.04$, with $\alpha =0,\, \eta=0$. For weakly coiled cords, the solute (heat) exchange amplification scales as $\Omega^2$ (relative to the uncoiled vessels of the same length). Concentration fields for four different vessel configurations and three-dimensional diagrams show a normocoiled ($\Omega$=0.75) and hypercoiled ($\Omega$=3.77) cord, at parameters highlighted with squares on the graph. Cords with $\Omega<0.4$ are hypocoiled while cords with $\Omega>2$ are hypercoiled. The colour bar shows solute concentration in normalised units (venous concentration is unity and arterial concentration is zero). (b) Computed exchange flux per unit length of the vein $N$ for small separation $d$. In the limit of small $d$, the leading-order flux is $\mathcal{O}(d^{-1/2})$. Here $\alpha =0,\, \eta=0$. (c) Exchange flux $N$ and relative exchange flux $N_\mathrm{rel}$ for different values of the exchange parameter $\eta$ in the cord boundary condition \ref{['eq:outbc']} for $d = 0.33$. Other geometric parameters in (a-c) were fixed: $R_v = 0.25$, $r_v = 0.25$, $r_a = 0.12$, $\theta_1 = 0.8\pi$, $\theta_2 = 1.2\pi$ (see Supplement for more details). The triangles in (a) and (b) show the relative fold-change in the variables.
• Figure 3: Solute exchange flux $N$ for different cord configurations in the ($\theta_1$, $\Delta\theta$) parameter space (see Fig. \ref{['fig:fetus']}b). The geometrical parameters used were $r_v = 0.25, r_a = 0.12, d = 0.04,\eta = 0, \Omega=3.77$ (see Table \ref{['tab:par']}) for (a) $R_v = 0$ (a straight vein, as shown in (c)) and (b) $R_v = 0.21$. Cord configurations predicted by the computational model (e) ($\theta_1 = 0.9\pi$; $\Delta \theta = 0.2\pi$) that minimise and (d) ($\theta_1= 0.5\pi$; $\Delta \theta=\pi$) maximise solute exchange between the vessels (shown in (b)). Solute concentrations are plotted in normalised units, varying from $C=0$ on the arterial surfaces (blue) to $C=1$ on the surface of the vein (red). (f) Magnification of a region in (b) showing estimated angles obtained from histology and from ultrasound images of human cord cross-sections from healthy pregnancies. The line of symmetry (dashed) has a slope of $-2$ and a zero intercept at $\theta_1 = \pi$. (g) and (h) are examples of histology images with irregular shapes, circularity $<1$ (Kurakazu2019, reproduced under CC BY 4.0, and Thomas2020, reproduced by permission of Taylor & Francis Ltd, tandfonline.com). (i) Ultrasound image of a cord with approximately unit circularity (Kurita2009, reproduced by permission of Karger Publishers, © 2009).
• Figure 4: The role of umbilical vascular structure in the cord tissue oxygenation. (a) Uptake flux $N_u$\ref{['eq:Nfluxes']} by the cord tissue (excluding a thin strip of thickness $h = 0.05$ outside the vascular lumens), for varying rate of tissue metabolism $\alpha$ and cord helicity $\Omega$. The solid lines correspond to $\beta = 1$, while the dotted lines show $\beta = 2$. Scaled concentration fields for (b) $\Omega = 0;~\alpha = 1$, (c) $\Omega = 4;~\alpha = 1$ and (d) $\Omega = 4;~\alpha = 10^3$ plotted in normalised units. (e) The contribution of the umbilical vein to the cord tissue oxygenation, quantified by the uptake flux $N_s$ relative to the flux in the case of diffusive oxygen supply by the umbilical arteries alone. Colours indicate different inter-vessel distances $d$. Other geometrical parameters used were: $R_v = 0.25$, $r_v = 0.25$, $r_a = 0.12$, $d = 0.3$, $\theta_1 = 0.8\pi$, $\theta_2 = 1.2\pi$ and $\eta = 0$.