Predicting Nitric Oxide transport in Microvascular Networks Using the Green's Function Method

Introduction: Nitric oxide (NO) is an important regulatory molecule within the vascular system, influencing a wide range of physiological processes, from maintaining vascular health to blood flow control. NO production in endothelial cells by endothelial NO synthase (eNOS) depends on intracellular Ca2+ levels. The endothelium-derived NO can diffuse across cell membranes and reach arteriolar smooth muscle cells, promoting relaxation and increased blood perfusion. NO, however, is scavenged rapidly by the hemoglobin in flowing red blood cells (RBCs) limiting its bioavailability in the vasculature. Intriguingly, a recent study has shown the presence of robust Ca2+ transients in cerebral capillary endothelial cells (cECs) [1]. It was suggested that these Ca2+ events contribute to the regulation of local blood supply by generating localized bursts of NO affecting tone in nearby capillary Pericytes (PCs) and/or arteriolar Smooth Muscle Cells (SMCs). In this study, we examine the potential of this Ca2+ activity for NO-dependent control of cerebral blood flow. Previous mathematical models have investigated NO production and transport around isolated blood vessel identifying diffusional barriers that can reduce uptake by RBCs and modulate NO bioavailability. Here, we extend the earlier models and examine NO release, transport, and uptake in realistic reconstructed microvascular networks. We employ a Green's Function approach to efficiently simulate the transport of NO in a large complex three-dimensional geometry.

Materials and

Methods: We develop a mathematical model of NO biotransport in the cerebral microvasculature. We reconstructed a microvascular network comprising 15,000 vascular segments with cECs. For each cEC we use a model of stochastic Ca2+ dynamics to simulate the observed sporadic Ca2+ events and corresponding increases in NO release. Hemodynamics simulations in the vascular network predict the distribution of RBCs flowing within the capillaries.
The diffusion-reaction equation for NO in the three-dimensional geometry is solved using Green’s function method, treating active cECs as point sources and RBCs as points of consumption [2].
The analytical steady-state solution provided by Green’s Functions Method (G(x)) predicts the NO concentration around each discrete production and consumption point (x*):
G(x-x*)=1/4πD_NO|x-x*|
Where G(x-x^* ) describes the concentration at location x, due to the presence of a singular point source at x*. The overall NO concentration distribution (CNO(x)) within the examined tissue volume is obtained by superposition of all positive and negative contributions:
C_NO (x)=∫G(x,x*)q(x*)dx*
Where q(x*) is the distribution of source and sink strengths. The time-dependent implementation of the method was adapted as previously described [3].
We aim to quantify parameters and conditions that affect NO bioavailability within the examined brain volume, and the potential role of cEC Ca2+ signaling for blood flow control.

Results, Conclusions, and Discussions: We tested the proposed steady-state solution against Finite Element Method simulations for a simplified problem with two-point sources representing capillaries and two sinks representing RBCs. We tested the feasibility of the approach for simulations in macroscopic tissue volumes. Simulations were performed in a small vascular network containing a capillary network its feeding arteriole and draining venule. The steady-state solution is compared against the transient implementation of Green’s function approach after sufficient time. Simulations predict NO levels as a function of the frequency of Ca2+ events. We identify conditions where capillary/arteriole tone and perfusion rate are determined by the frequency of Ca2+ events in cECs.

Acknowledgements (Optional): This work is supported by the National Institute of Neurological Disorders and Stroke and the National Institute of Aging of the NIH under award number R01NS19971.

References:
[1] Longden, T. A., Mughal, A., Hennig, G. W., Harraz, O. F., Shui, B., Lee, F. K., ... & Nelson, M. T. (2021). Local IP3 receptor–mediated Ca2+ signals compound to direct blood flow in brain capillaries. Science advances, 7(30), eabh0101.
[2] Secomb, T. W., Hsu, R., Park, E. Y., & Dewhirst, M. W. (2004). Green's function methods for analysis of oxygen delivery to tissue by microvascular networks. Annals of biomedical engineering, 32, 1519-1529.
[3] Secomb, T. W. (2016). A Green's function method for simulation of time-dependent solute transport and reaction in realistic microvascular geometries. Mathematical medicine and biology: a journal of the IMA, 33(4), 475-494.