Venous steal blood flow simulation

This 0-D Thevenin's equivalent model simulates blood flow distribution between a compartment with external pressure P_ext (modeled as a Starling resistor) and collateral outflow pathways (1-3).

The vascular network configuration can be modified by adjusting the inflow resistance, fractional flow reserve (FFR=P_d/P_a), and the ratio of collateral to Starling conductance (G_collateral/G_starling). Loading conditions are controlled by varying arterial pressure (P_a), compartment pressure (P_ext), and venous pressure (P_v).

The model calculates the post-stenotic pressure (P_d), segmental perfusion pressure (SPP = P_d-P_ext), total inflow (Q_inflow), and its distribution between the compartment and collateral outflow (Q_Starling/Q_coll). Recruitable venous steal is quantified as the percentage reduction in Q_starling when collateral outflow pressure (P_v) drops below the compartment pressure (P_ext).

Due to the collapsibility of veins, the effective outflow pressure in the compartment is determined by P_ext rather than P_v. Similarly, because of arterial stenosis, the effective inflow pressure is governed by P_d rather than systemic arterial pressure P_a. Blood flow through the compartment is modeled using a Starling resistor equivalent:
Q_Starling=SPP × G_starling, where G_starling represents the conductance of the Starling resistor (assumed constant under maximal vasodilation).

A decrease in venous pressure below P_ext increases collateral outflow and raises the pressure drop across the arterial stenosis (P_a-P_d), which in turn lowers P_d, SPP, and flow Q_starling through the Starling resistor. This phenomenon we referred to as venous steal (1-3).

Attached is an interactive model demonstrating the venous steal phenomenon, along with a graph illustrating pressure and flow distribution between the Starling resistor and collateral outflow. For venous steal to occur, both arterial stenosis and blood flow diversion to a collateral pathway with a lower outflow pressure must be present. Increasing venous pressure to match the compartment pressure (P_ext) eliminates venous steal.

1. Pranevicius M, Pranevicius O. Cerebral venous steal: blood flow diversion with increased tissue pressure. Neurosurgery. 2002 Nov;51(5):1267-73; discussion 1273-4. doi: 10.1097/00006123-200211000-00023. PMID: 12383372.
2. Pranevicius M, Pranevicius H, Pranevicius O. Cerebral venous steal equation for intracranial segmental perfusion pressure predicts and quantifies reversible intracranial to extracranial flow diversion. Sci Rep. 2021 Apr 8;11(1):7711. doi: 10.1038/s41598-021-85931-x. PMID: 33833266; PMCID: PMC8032738. https://rdcu.be/eafZa
3. Pranevičius, M.; Makackas, D.; Macas, A.; Petrikonis, K.; Šakalytė, G.; Pranevičius, O.; Benetis, R. The Concept of Venous Steal: The Impact of Vascular Stenosis and Outflow Pressure Gradient on Blood Flow Diversion. Medicina 2025, 61, 672. https://www.mdpi.com/3258046

本零维戴维南等效模型（0-D Thevenin's equivalent model）可模拟带外压的腔室（以斯塔林电阻器（Starling resistor）建模）与侧支流出通路（1-3）之间的血流分配情况。

可通过调整流入阻力、血流储备分数（FFR=P_d/P_a，fractional flow reserve）以及侧支电导与斯塔林电阻电导的比值（G_collateral/G_starling），修改该血管网络的配置。负载条件可通过改变动脉压（P_a）、腔室压（P_ext）与静脉压（P_v）进行调控。

该模型可计算狭窄后压力（P_d）、节段灌注压（SPP = P_d-P_ext，segmental perfusion pressure）、总流入量（Q_inflow），以及总流入量在腔室与侧支流出通路间的分配比例（Q_Starling/Q_coll）。当侧支流出压力（P_v）降至腔室压（P_ext）以下时，可募集性静脉盗血（recruitable venous steal）的量化指标为Q_Starling的降低百分比。

由于静脉具备可塌陷性，腔室内的有效流出压力由P_ext而非P_v决定。同理，因动脉狭窄的存在，有效流入压力由狭窄后压力P_d而非体动脉压P_a决定。流经腔室的血流采用斯塔林电阻器等效模型进行建模：Q_Starling=SPP × G_starling，其中G_starling代表斯塔林电阻器的电导（假定在最大血管舒张（maximal vasodilation）状态下为恒定值）。

当静脉压降至P_ext以下时，侧支流出量会增加，动脉狭窄两端的压力差（P_a-P_d）也会随之增大，进而导致P_d、SPP以及通过斯塔林电阻器的血流量Q_starling降低。我们将该现象命名为静脉盗血（venous steal，1-3）。

附件包含演示静脉盗血现象的交互式模型，以及一张展示斯塔林电阻器与侧支流出通路间压力与血流分配情况的示意图。若要引发静脉盗血，需同时满足动脉狭窄与血流向流出压力更低的侧支通路分流两个条件。将静脉压升高至与腔室压（P_ext）持平即可消除静脉盗血。

1. Pranevicius M, Pranevicius O. Cerebral venous steal: blood flow diversion with increased tissue pressure. Neurosurgery. 2002 Nov;51(5):1267-73; discussion 1273-4. doi: 10.1097/00006123-200211000-00023. PMID: 12383372.
2. Pranevicius M, Pranevicius H, Pranevicius O. Cerebral venous steal equation for intracranial segmental perfusion pressure predicts and quantifies reversible intracranial to extracranial flow diversion. Sci Rep. 2021 Apr 8;11(1):7711. doi: 10.1038/s41598-021-85931-x. PMID: 33833266; PMCID: PMC8032738. https://rdcu.be/eafZa
3. Pranevičius, M.; Makackas, D.; Macas, A.; Petrikonis, K.; Šakalytė, G.; Pranevičius, O.; Benetis, R. The Concept of Venous Steal: The Impact of Vascular Stenosis and Outflow Pressure Gradient on Blood Flow Diversion. Medicina 2025, 61, 672. https://www.mdpi.com/3258046