Modeling of the deformation of flexible tubes using a single law: application to veins of the lower limb in man

Numerical coupling of 0D and 1D models in networks of vessels including transonic flow conditions. Application to short-term transient and stationary hemodynamic simulation of postural changes

When modeling complex fluid networks using one-dimensional (1D) approaches, boundary conditions can be imposed using zero-dimensional (0D) models. An application case is the modeling of the entire human circulation using closed-loop models. These models can be considered as a tool to investigate short-term transient and stationary hemodynamic responses to postural changes. The first shortcoming of existing 1D modeling methods in simulating these sudden maneuvers is their inability to deal with rapid variations in flow conditions, as they are limited to the subsonic case. On the other hand, numerical modeling of 0D models representing microvascular beds, venous valves or heart chambers is also currently modeled assuming subsonic flow conditions in 1D connecting vessels, failing when transonic and supersonic flow conditions appear. Therefore, if numerical simulation of sudden maneuvers is a goal in closed-loop models, it is necessary to reformulate the current methodologies used when coupling 0D and 1D models, allowing the correct handling of flow evolution for both subsonic and transonic conditions. This work focuses on the extension of the general methodology for the Junction Riemann Problem (JRP) when coupling 0D and 1D models. As an example of application, the short-term transient response to head-up tilt (HUT) from supine to upright position of a closed-loop model is shown, demonstrating the potential, capability and necessity of the presented numerical models when dealing with sudden maneuvers.

A Solution of the Junction Riemann Problem for 1D Hyperbolic Balance Laws in Networks including Supersonic Flow Conditions on Elastic Collapsible Tubes

The numerical modeling of one-dimensional (1D) domains joined by symmetric or asymmetric bifurcations or arbitrary junctions is still a challenge in the context of hyperbolic balance laws with application to flow in pipes, open channels or blood vessels, among others. The formulation of the Junction Riemann Problem (JRP) under subsonic conditions in 1D flow is clearly defined and solved by current methods, but they fail when sonic or supersonic conditions appear. Formulations coupling the 1D model for the vessels or pipes with other container-like formulations for junctions have been presented, requiring extra information such as assumed bulk mechanical properties and geometrical properties or the extension to more dimensions. To the best of our knowledge, in this work, the JRP is solved for the first time allowing solutions for all types of transitions and for any number of vessels, without requiring the definition of any extra information. The resulting JRP solver is theoretically well-founded, robust and simple, and returns the evolving state for the conserved variables in all vessels, allowing the use of any numerical method in the resolution of the inner cells used for the space-discretization of the vessels. The methodology of the proposed solver is presented in detail. The JRP solver is directly applicable if energy losses at the junctions are defined. Straightforward extension to other 1D hyperbolic flows can be performed.

Seven Mathematical Models of Hemorrhagic Shock

Although mathematical modelling of pressure-flow dynamics in the cardiocirculatory system has a lengthy history, readily finding the appropriate model for the experimental situation at hand is often a challenge in and of itself. An ideal model would be relatively easy to use and reliable, besides being ethically acceptable. Furthermore, it would address the pathogenic features of the cardiovascular disease that one seeks to investigate. No universally valid model has been identified, even though a host of models have been developed. The object of this review is to describe several of the most relevant mathematical models of the cardiovascular system: the physiological features of circulatory dynamics are explained, and their mathematical formulations are compared. The focus is on the whole-body scale mathematical models that portray the subject's responses to hypovolemic shock. The models contained in this review differ from one another, both in the mathematical methodology adopted and in the physiological or pathological aspects described. Each model, in fact, mimics different aspects of cardiocirculatory physiology and pathophysiology to varying degrees: some of these models are geared to better understand the mechanisms of vascular hemodynamics, whereas others focus more on disease states so as to develop therapeutic standards of care or to test novel approaches. We will elucidate key issues involved in the modeling of cardiovascular system and its control by reviewing seven of these models developed to address these specific purposes.

Large vessels as a tree of transmission lines incorporated in the CircAdapt whole-heart model: A computational tool to examine heart-vessel interaction

We developed a whole-circulation computational model by integrating a transmission line (TL) model describing vascular wave transmission into the established CircAdapt platform of whole-heart mechanics. In the present paper, we verify the numerical framework of our TL model by benchmark comparison to a previously validated pulse wave propagation (PWP) model. Additionally, we showcase the integrated CircAdapt-TL model, which now includes the heart as well as extensive arterial and venous trees with terminal impedances. We present CircAdapt-TL haemodynamics simulations of: 1) a systemic normotensive situation and 2) a systemic hypertensive situation. In the TL-PWP benchmark comparison we found good agreement regarding pressure and flow waveforms (relative errors ≤ 2.9% for pressure, and ≤ 5.6% for flow). CircAdapt-TL simulations reproduced the typically observed haemodynamic changes with hypertension, expressed by increases in mean and pulsatile blood pressures, and increased arterial pulse wave velocity. We observed a change in the timing of pressure augmentation (defined as a late-systolic boost in aortic pressure) from occurring after time of peak systolic pressure in the normotensive situation, to occurring prior to time of peak pressure in the hypertensive situation. The pressure augmentation could not be observed when the systemic circulation was lumped into a (non-linear) three-element windkessel model, instead of using our TL model. Wave intensity analysis at the carotid artery indicated earlier arrival of reflected waves with hypertension as compared to normotension, in good qualitative agreement with findings in patients. In conclusion, we successfully embedded a TL model as a vascular module into the CircAdapt platform. The integrated CircAdapt-TL model allows detailed studies on mechanistic studies on heart-vessel interaction.

Peripheral Venous Pressure as an Indicator of Preload Responsiveness During Volume Resuscitation from Hemorrhage

BACKGROUND

Fluid resuscitation of hypovolemia presumes that peripheral venous pressure (PVP) increases more than right atrial pressure (RAP), so the net pressure gradient for venous return (PVP-RAP) rises. However, the heart and peripheral venous system function under different compliances that could affect their respective pressures during fluid infusion. In a porcine model of hemorrhage resuscitation, we examined whether RAP increases more than PVP, thereby reducing the venous return pressure gradient and blood flow.

METHODS

Anesthetized pigs (n = 8) were bled to a mean arterial blood pressure of 40 mm Hg and resuscitated with stored blood and albumin for pulmonary artery occlusion pressures (PAOPs) of 5, 10, 15, and 20 mm Hg. Venous pressures, inferior vena cava blood flow (ultrasonic flowprobe), and left ventricular diastolic compliance (Doppler echocardiography) were measured. Stroke volume variability was calculated.

RESULTS

With volume resuscitation, the slope of RAP exceeded PVP (P ≤ 0.0001) when PAOP is 10 to 20 mm Hg, causing the pressure gradient for venous return to progressively decrease. Inferior vena cava blood flow did not further increase after PAOP > 10 mm Hg. The E/e' ratio increased (P = 0.001) during resuscitation indicating reduced diastolic compliance. A significant curvilinear relationship was found between PVP and stroke volume variability (R = 0.62; P < 0.001), where fluid responders had PVP < 15 mm Hg.

CONCLUSIONS

Fluid resuscitation above a PAOP 10 mm Hg reduces myocardial compliance and reduces the venous return pressure gradient. The hemodynamic response to fluid resuscitation becomes limited by diastolic properties of the heart. PVP measurement during hemorrhage resuscitation may predict fluid responsiveness and nonresponsiveness.

A 1D pulse wave propagation model of the hemodynamics of calf muscle pump function

The calf muscle pump is a mechanism which increases venous return and thereby compensates for the fluid shift towards the lower body during standing. During a muscle contraction, the embedded deep veins collapse and venous return increases. In the subsequent relaxation phase, muscle perfusion increases due to increased perfusion pressure, as the proximal venous valves temporarily reduce the distal venous pressure (shielding). The superficial and deep veins are connected via perforators, which contain valves allowing flow in the superficial-to-deep direction. The aim of this study is to investigate and quantify the physiological mechanisms of the calf muscle pump, including the effect of venous valves, hydrostatic pressure, and the superficial venous system. Using a one-dimensional pulse wave propagation model, a muscle contraction is simulated by increasing the extravascular pressure in the deep venous segments. The hemodynamics are studied in three different configurations: a single artery-vein configuration with and without valves and a more detailed configuration including a superficial vein. Proximal venous valves increase effective venous return by 53% by preventing reflux. Furthermore, the proximal valves shielding function increases perfusion following contraction. Finally, the superficial system aids in maintaining the perfusion during the contraction phase and reduces the refilling time by 37%.

A new hemodynamic model for the study of cerebral venous outflow

We developed a mathematical model of the cerebral venous outflow for the simulation of the average blood flows and pressures in the main drainage vessels of the brain. The main features of the model are that it includes a validated model for the simulation of the intracranial circulation and it accounts for the dependence of the hydraulic properties of the jugular veins with respect to the gravity field, which makes it an useful tool for the study of the correlations between extracranial blood redistributions and changes in the intracranial environment. The model is able to simulate the average pressures and flows in different points of the jugular ducts, taking into account the amount of blood coming from the anastomotic connections; simulate how the blood redistribution due to change of posture affects flows and pressures in specific points of the system; and simulate redistributions due to stenotic patterns. Sensitivity analysis to check the robustness of the model was performed. The model reproduces average physiologic behavior of the jugular, vertebral, and cerebral ducts in terms of pressures and flows. In fact, jugular flow drops from ∼11.7 to ∼1.4 ml/s in the passage from supine to standing. At the same time, vertebral flow increases from 0.8 to 3.4 ml/s, while cerebral blood flow, venous sinuses pressure, and intracranial pressure are constant around the average value of 12.5 ml/s, 6 mmHg, and 10 mmHg, respectively. All these values are in agreement with literature data.

A global multiscale mathematical model for the human circulation with emphasis on the venous system

We present a global, closed-loop, multiscale mathematical model for the human circulation including the arterial system, the venous system, the heart, the pulmonary circulation and the microcirculation. A distinctive feature of our model is the detailed description of the venous system, particularly for intracranial and extracranial veins. Medium to large vessels are described by one-dimensional hyperbolic systems while the rest of the components are described by zero-dimensional models represented by differential-algebraic equations. Robust, high-order accurate numerical methodology is implemented for solving the hyperbolic equations, which are adopted from a recent reformulation that includes variable material properties. Because of the large intersubject variability of the venous system, we perform a patient-specific characterization of major veins of the head and neck using MRI data. Computational results are carefully validated using published data for the arterial system and most regions of the venous system. For head and neck veins, validation is carried out through a detailed comparison of simulation results against patient-specific phase-contrast MRI flow quantification data. A merit of our model is its global, closed-loop character; the imposition of highly artificial boundary conditions is avoided. Applications in mind include a vast range of medical conditions. Of particular interest is the study of some neurodegenerative diseases, whose venous haemodynamic connection has recently been identified by medical researchers.

Compression of the brachial arteryin vivo

Stiffening of the brachial artery is implicated in pseudo-hypertension. To date, a reliable clinical predictor of the condition has not been developed. This paper describes the development of prototype instrumentation and methodology for measurement of the brachial artery transmural pressure/cross-sectional area relationship in vivo. The methodology has been validated using a model of an arm and a thin-walled rubber tube. Application of the technique to a healthy subject shows that the technique is viable and gives good reproducibility. Closure of the brachial artery with reducing transmural pressure is observed and recorded. The new technique has some important advantages over existing methodologies.

Pseudo-hypertension and arterial stiffness: a review

Hypertension is a condition of persistently elevated blood pressure, associated with increased cardiovascular risk. Non-invasive BP measurement using Korotkoff sounds is the most common method of screening for the condition. The possibility of inaccurate readings leading to a false diagnosis of hypertension (pseudo-hypertension) is of concern. Stiffened arteries in the elderly have been proposed as being the primary cause of pseudo-hypertension. Non-invasive detection of pseudo-hypertension remains problematic. This paper reviews clinical literature on pseudo hypertension and approaches to measuring the compressive stiffness of arteries, as well as biomechanical literature regarding models of arterial stiffness and the origin of Korotkoff sounds. Models of the latter show the importance of the relationship between transmural pressure and cross-sectional area (P1/Csa curve) of the brachial artery as it closes under the influence of the pressure cuff. The review concludes that future research on pseudo-hypertension should include development of new instrumentation to measure the P1/Csa curve of the brachial artery in vivo using non-invasive techniques suitable for application to an elderly population.

Is axial wall stress compressive in certain arteries?

In mathematical-physical models of blood vessels, the "zero-stress state" of the vessel wall is usually defined with reference to the atmospheric pressure (pa approximately 750 mmHg = 100 kPa). Due to this conventional choice, axial and circumferential stresses generated by the (positive) transmural pressure over the radial wall depth can only be positive (in absence of residual stresses) and thus, by definition, only tensile. If the zero-stress state were defined "unconventionally" with reference to vacuum pressure (= 0 mm Hg), the isotropic compressive stress--pa generated by the atmospheric pressure everywhere in the wall would, however, be included in the stress values, and negative (= compressive) stresses would become formally possible. Since materials submitted only to compressions do not need to have the same resistive properties as materials which may also experience tractions, the question whether axial stress (and perhaps also circumferential stress) might be permanently compressive in vessels under physiologic conditions may therefore be important for investigations of the relationship between wall stresses on one side and wall structures, vessel growth, vessel damages, or vessel adaptation processes on the other side. In the present study, radial, circumferential, and axial wall stresses were calculated conventionally and "unconventionally" for three representative "vessel examples." The results clearly suggest that axial wall stress might well be compressive in many vessels. Furthermore, relative differences between conventional and unconventional stress values are quite considerable, and ratios between stresses calculated in the same manner appear to be strongly dependent on the chosen zero-stress state definition.