Mathematical modeling at the level of the full cardiovascular system requires the numerical approximation of solutions to a one-dimensional nonlinear hyperbolic system describing flow in a single vessel. This model is often simulated by computationally intensive methods like finite elements and discontinuous Galerkin, while some recent applications require more efficient approaches (e.g. for real-time clinical decision support, phenomena occurring over multiple cardiac cycles, iterative solutions to optimization/inverse problems, and uncertainty quantification). Further, the high speed of pressure waves in blood vessels greatly restricts the time step needed for stability in explicit schemes. We address both cost and stability by presenting an efficient and unconditionally stable method for approximating solutions to diagonal nonlinear hyperbolic systems. Theoretical analysis of the algorithm is given along with a comparison of our method to a discontinuous Galerkin implementation. Lastly, we demonstrate the utility of the proposed method by implementing it on small and large arterial networks of vessels whose elastic and geometrical parameters are physiologically relevant.
One–dimensional blood flow models take the general form of nonlinear hyperbolic systems but differ in their formulation. One class of models considers the physically conserved quantities of mass and momentum, while another class describes mass and velocity. Further, the averaging process employed in the model derivation requires the specification of the axial velocity profile; this choice differentiates models within each class. Discrepancies among differing models have yet to be investigated. In this paper, we comment on some theoretical differences among models and systematically compare them for physiologically relevant vessel parameters, network topology, and boundary data. In particular, the effect of the velocity profile is investigated in the cases of both smooth and discontinuous solutions, and a recommendation for a physiological model is provided. The models are discretized by a class of Runge–Kutta discontinuous Galerkin methods.
Key points Inflammation in response to bacterial endotoxin challenge impacts physiological functions, including cardiovascular, thermal and pain dynamics, although the mechanisms are poorly understood. We develop an innovative mathematical model incorporating interaction pathways between inflammation and physiological processes observed in response to an endotoxin challenge. We calibrate the model to individual data from 20 subjects in an experimental study of the human inflammatory and physiological responses to endotoxin, and we validate the model against human data from an independent study. Using the model to simulate patient responses to different treatment modalities reveals that a multimodal treatment combining several therapeutic strategies gives the best recovery outcome. Abstract Uncontrolled, excessive production of pro‐inflammatory mediators from immune cells and traumatized tissues can cause systemic inflammatory conditions such as sepsis, one of the ten leading causes of death in the USA, and one of the three leading causes of death in the intensive care unit. Understanding how inflammation affects physiological processes, including cardiovascular, thermal and pain dynamics, can improve a patient's chance of recovery after an inflammatory event caused by surgery or a severe infection. Although the effects of the autonomic response on the inflammatory system are well‐known, knowledge about the reverse interaction is lacking. The present study develops a mathematical model analyzing the inflammatory system's interactions with thermal, pain and cardiovascular dynamics in response to a bacterial endotoxin challenge. We calibrate the model with individual data from an experimental study of the inflammatory and physiological responses to a one‐time administration of endotoxin in 20 healthy young men and validate it against data from an independent endotoxin study. We use simulation to explore how various treatments help patients exposed to a sustained pathological input. The treatments explored include bacterial endotoxin adsorption, antipyretics and vasopressors, as well as combinations of these. Our findings suggest that the most favourable recovery outcome is achieved by a multimodal strategy, combining all three interventions to simultaneously remove endotoxin from the body and alleviate symptoms caused by the immune system as it fights the infection.
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