Tube Law Research Articles

Sound generation mechanisms in a collapsible tube.

Collapsible tubes can be employed to study the sound generation mechanism in the human respiratory system. The goals of this work are (a) to determine the airflow characteristics connected to three different collapse states of a physiological tube and (b) to find a relation between the sound power radiated by the tube and its collapse state. The methodology is based on the implementation of computational fluid dynamics simulation on experimentally validated geometries. The flow is characterized by a radical change of behavior before and after the contact of the lumen. The maximum of the sound power radiated corresponds to the post-buckling configuration. The idea of an acoustic tube law is proposed. The presented results are relevant to the study of self-excited oscillations and wheezing sounds in the lungs.

Tube law parametrization using in vitro data for one-dimensional blood flow in arteries and veins: TUBE LAW PARAMETRIZATION IN ARTERIES AND VEINS.

The deformability of blood vessels in one-dimensional blood flow models is typically described through a pressure-area relation, known as the tube law. The most used tube laws take into account the elastic and viscous components of the tension of the vessel wall. Accurately parametrizing the tube laws is vital for replicating pressure and flow wave propagation phenomena. Here, we present a novel mathematical-property-preserving approach for the estimation of the parameters of the elastic and viscoelastic tube laws. Our goal was to estimate the parameters by using ovine and human in vitro data, while constraining them to meet prescribed mathematical properties. Results show that both elastic and viscoelastic tube laws accurately describe experimental pressure-area data concerning both quantitative and qualitative aspects. Additionally, the viscoelastic tube law can provide a qualitative explanation for the observed hysteresis cycles. The two models were evaluated using two approaches: (i) allowing all parameters to freely vary within their respective ranges and (ii) fixing some of the parameters. The former approach was found to be the most suitable for reproducing pressure-area curves.

Flux vector splitting schemes applied to a conservative 1D blood flow model with transport for arteries and veins

We present novel flux splitting-based numerical schemes for the 1D blood flow equations with an advection equation for a passive scalar, considering tube laws that allow to model blood flow in arteries and veins. Our schemes are inspired by the original flux vector splitting approach of Toro and Vázquez-Cendón (2012) and represent an extension of the work proposed by Toro et al. (2024), which addressed tube laws suitable for describing blood flow in arteries. Our schemes separate advection terms and pressure terms, generating two different systems of PDEs: the advection system and the pressure system, both of which have a very simple eigenstructure compared to that of the full system. We propose discretization schemes of the Godunov type that are simple and efficient. These qualities are evaluated on a suite of test problems with exact solution. A detailed efficiency analysis is performed in order to illustrate situations in which the proposed methodology results advantageous with respect to standard approaches.

A catalog of pressure and deformation profile for thin walled hyperelastic tubes conveying inertialess flow and undergoing large deformation

Slender tubes constituted of hyperelastic materials undergoing large deformations and conveying inertialess flow of Newtonian fluids are a model representation of several systems including those in soft robotics, biology and in industry. In this paper, we undertake a systematic study of this system. The tube’s hyperelastic behavior is modeled by five (5) different constitutive laws: Neo-Hookean, Mooney–Rivlin, Fung, Gent and Ogden. Invoking the principles of finite elasticity, we delineate the local pressure — deformation relationship for the tube for each of the hyperelastic models. The structural mechanical field of the tube is then coupled with fluid flow field, which is simplified using the principles of lubrication approximation. The resultant fluid–structure interaction problem then throws up a set of interesting results including the deformation and pressure profile across the tube, and tube laws for five (5) hyperelastic models. Analyzing these results, we posit that the behavior of Neo-Hookean and Mooney–Rivlin tubes is converse of Fung’s and Gent’s tubes; the latter show a strain hardening behavior whilst the former exhibit a strain softening one. A sensitivity analysis which underscores the relative importance of geometrical nonlinearity in the problem then follows.

MRI-MECH: mechanics-informed MRI to estimate esophageal health.

Dynamic magnetic resonance imaging (MRI) is a popular medical imaging technique that generates image sequences of the flow of a contrast material inside tissues and organs. However, its application to imaging bolus movement through the esophagus has only been demonstrated in few feasibility studies and is relatively unexplored. In this work, we present a computational framework called mechanics-informed MRI (MRI-MECH) that enhances that capability, thereby increasing the applicability of dynamic MRI for diagnosing esophageal disorders. Pineapple juice was used as the swallowed contrast material for the dynamic MRI, and the MRI image sequence was used as input to the MRI-MECH. The MRI-MECH modeled the esophagus as a flexible one-dimensional tube, and the elastic tube walls followed a linear tube law. Flow through the esophagus was governed by one-dimensional mass and momentum conservation equations. These equations were solved using a physics-informed neural network. The physics-informed neural network minimized the difference between the measurements from the MRI and model predictions and ensured that the physics of the fluid flow problem was always followed. MRI-MECH calculated the fluid velocity and pressure during esophageal transit and estimated the mechanical health of the esophagus by calculating wall stiffness and active relaxation. Additionally, MRI-MECH predicted missing information about the lower esophageal sphincter during the emptying process, demonstrating its applicability to scenarios with missing data or poor image resolution. In addition to potentially improving clinical decisions based on quantitative estimates of the mechanical health of the esophagus, MRI-MECH can also be adapted for application to other medical imaging modalities to enhance their functionality.

A new solution for the deformations of an initially elliptical elastic-walled tube

SummaryWe investigate the small-amplitude deformations of a long thin-walled elastic tube having an initially axially uniform elliptical cross-section. The tube is deformed by a (possibly non-uniform) transmural pressure. At leading order, its deformations are shown to be governed by a single partial differential equation (PDE) for the azimuthal displacement as a function of the axial and azimuthal co-ordinates and time. Previous authors have obtained solutions to this PDE by making ad hoc approximations based on truncating an approximate Fourier representation. In this article, we instead write the azimuthal displacement as a sum over the azimuthal eigenfunctions of a generalised eigenvalue problem and show that we are able to derive an uncoupled system of linear PDEs with constant coefficients for the amplitude of the azimuthal modes as a function of the axial co-ordinate and time. This results in a formal solution of the whole system being found as a sum over the azimuthal modes. We show that the $n$th mode’s contribution to the tube’s relative area change is governed by a simplified second-order PDE and examine the case in which the tube’s deformations are driven by a uniform transmural pressure. The relative errors induced by truncating the series solution after the first and second terms are then evaluated as a function of both the ellipticity and pre-stress of the tube. After comparing our results with Whittaker et al. (A rational derivation of a tube law from shell theory, Q. J. Mech. Appl. Math. 63 (2010) 465–496), we find that this new method leads to a significant simplification when calculating contributions from the higher-order azimuthal modes, which in turn makes a more accurate solution easier to obtain.

A mathematical framework for the dynamic interaction of pulsatile blood, brain, and cerebrospinal fluid

BackgroundShedding light on less-known aspects of intracranial fluid dynamics may be helpful to understand the hydrocephalus mechanism. The present study suggests a mathematical framework based on in vivo inputs to compare the dynamic interaction of pulsatile blood, brain, and cerebrospinal fluid (CSF) between the healthy subject and the hydrocephalus patient. MethodThe input data for the mathematical formulations was pulsatile blood velocity, which was measured using cine PC-MRI. Tube law was used to transfer the created deformation by blood pulsation in the vessel circumference to the brain domain. The pulsatile deformation of brain tissue with respect to time was calculated and considered to be inlet velocity in the CSF domain. The governing equations in all three domains were continuity, Navier-Stokes, and concentration. We used Darcy law with defined permeability and diffusivity values to define the material properties in the brain. ResultsWe validated the preciseness of the CSF velocity and pressure through the mathematical formulations with cine PC-MRI velocity, experimental ICP, and FSI simulated velocity and pressure. We used the analysis of dimensionless numbers including Reynolds, Womersley, Hartmann, and Peclet to evaluate the characteristics of the intracranial fluid flow. In the mid-systole phase of a cardiac cycle, CSF velocity had the maximum value and CSF pressure had the minimum value. The maximum and amplitude of CSF pressure, as well as CSF stroke volume, were calculated and compared between the healthy subject and the hydrocephalus patient. ConclusionThe present in vivo-based mathematical framework has the potential to gain insight into the less-known points in the physiological function of intracranial fluid dynamics and the hydrocephalus mechanism.

ADER scheme with a simplified solver for the generalized Riemann problem and an average ENO reconstruction procedure. Application to blood flow

We present numerical schemes for solving hyperbolic balance laws, admitting stiff source terms, to high-order of accuracy in both space and time. The schemes belong to the ADER family of methods and as such rest on two building blocks, namely a non-linear spatial reconstruction procedure and the solution of a generalized Riemann problem (GRP), in which the data is piece-wise smooth and the equations include source terms. This paper presents contributions on both building blocks. Concerning spatial reconstruction we present various versions of a new Essentially Non-Oscillatory (ENO) type reconstruction, called here Averaged ENO (AENO). As to the generalized Riemann problem, we present an improved solver based on two basic ingredients, namely the implicit Taylor series expansion reported in Toro and Montecino (2015) and the simplified Cauchy–Kowalewskaya procedure reported in Montecinos and Balsar (2020).The resulting ADER schemes are implemented and systematically assessed for the linear advection equation and for non-linear hyperbolic system that governs blood flow, with a tube law admitting arteries or veins. For the linear advection equation we implement the new schemes to accuracy ranging from first to seventh order; convergence rate studies confirm that the theoretically expected accuracy is achieved for most versions of the various schemes studied. For the non-linear system we implement the new schemes to accuracy ranging from first to fifth order. Convergence rate studies based on problems with smooth solutions confirm again that the methods attain the theoretically expected accuracy. For the non-linear system the methods are also assessed for Riemann problems for blood flow in arteries with exact solution containing smooth parts and discontinuities, elastic jumps (shocks), and contact discontinuities. Furthermore, for the non-linear system the methods are also assessed for blood flow in veins for a steady problem with smooth analytical solution. As a final assessment of the potential applicability of the methods for realistic simulations in haemodynamics, we apply the methods on a network of 37 blood vessels, for which experimental data is available. The new schemes presented in this paper are simpler than existing versions of ADER methods, involving implicit Taylor series and the Cauchy–Kowalewskaya procedure, published in the literature and, overall, the computational results demonstrate that the new schemes give comparable or superior results, with respect to existing high-order schemes.

A fast tube model predictive control scheme based on sliding mode control for underwater vehicle-manipulator system

This paper presents a novel robust fast tube model predictive control (FTMPC) scheme for controlling the multi-input multi-output (MIMO) underwater vehicle-manipulator system (UVMS) in the case of system unmodeled uncertainties and external disturbances. The structure of the FTMPC is designed by a nominal fast MPC and an ancillary tube sliding mode control (SMC) law. The nominal fast MPC uses the approximated prediction control law for achieving the fast online convex optimization. The ancillary tube law uses the explicit SMC for strong robustness and makes the state error between the actual system and the nominal system converge to zero. Finally, the effectiveness of controlling UVMS is verified by the proposed SMC-based FTMPC scheme through the tracking simulations.

Five years of cardio-ankle vascular index (CAVI) and CAVI0: how close are we to a pressure-independent index of arterial stiffness?

Pulse wave velocity, a common metric of arterial stiffness, is an established predictor for cardiovascular events and mortality. However, its intrinsic pressure-dependency complicates the discrimination of acute and chronic impacts of increased blood pressure on arterial stiffness. Cardio-ankle vascular index (CAVI) represented a significant step towards the development of a pressure-independent arterial stiffness metric. However, some potential limitations of CAVI might render this arterial stiffness metric less pressure-independent than originally thought. For this reason, we later introduced CAVI0. Nevertheless, advantages of one approach over the other are left debated. This review aims to shed light on the pressure (in)dependency of both CAVI and CAVI0. By critically reviewing results from studies reporting both CAVI and CAVI0 and using simple analytical methods, we show that CAVI0 may enhance the pressure-independent assessment of arterial stiffness, especially in the presence of large inter-individual differences in blood pressure.

A framework for incorporating 3D hyperelastic vascular wall models in 1D blood flow simulations

We present a novel framework for investigating the role of vascular structure on arterial haemodynamics in large vessels, with a special focus on the human common carotid artery (CCA). The analysis is carried out by adopting a three-dimensional (3D) derived, fibre-reinforced, hyperelastic structural model, which is coupled with an axisymmetric, reduced order model describing blood flow. The vessel transmural pressure and lumen area are related via a Holzapfel–Ogden type of law, and the residual stresses along the thickness and length of the vessel are also accounted for. After a structural characterization of the adopted hyperelastic model, we investigate the link underlying the vascular wall response and blood-flow dynamics by comparing the proposed framework results against a popular tube law. The comparison shows that the behaviour of the model can be captured by the simpler linear surrogate only if a representative value of compliance is applied. Sobol’s multi-variable sensitivity analysis is then carried out in order to identify the extent to which the structural parameters have an impact on the CCA haemodynamics. In this case, the local pulse wave velocity (PWV) is used as index for representing the arterial transmission capacity of blood pressure waveforms. The sensitivity analysis suggests that some geometrical factors, such as the stress-free inner radius and opening angle, play a major role on the system’s haemodynamics. Subsequently, we quantified the differences in haemodynamic variables obtained from different virtual CCAs, tube laws and flow conditions. Although each artery presents a distinct vascular response, the differences obtained across different flow regimes are not significant. As expected, the linear tube law is unable to accurately capture all the haemodynamic features characterizing the current model. The findings from the sensitivity analysis are further confirmed by investigating the axial stretching effect on the CCA fluid dynamics. This factor does not seem to alter the pressure and flow waveforms. On the contrary, it is shown that, for an axially stretched vessel, the vascular wall exhibits an attenuation in absolute distension and an increase in circumferential stress, corroborating the findings of previous studies. This analysis shows that the new model offers a good balance between computational complexity and physics captured, making it an ideal framework for studies aiming to investigate the profound link between vascular mechanobiology and blood flow.

An experimental investigation to model wheezing in lungs.

A quarter of the world's population experience wheezing. These sounds have been used for diagnosis since the time of the Ebers Papyrus (ca 1500 BC). We know that wheezing is a result of the oscillations of the airways that make up the lung. However, the physical mechanisms for the onset of wheezing remain poorly understood, and we do not have a quantitative model to predict when wheezing occurs. We address these issues in this paper. We model the airways of the lungs by a modified Starling resistor in which airflow is driven through thin, stretched elastic tubes. By completing systematic experiments, we find a generalized ‘tube law’ that describes how the cross-sectional area of the tubes change in response to the transmural pressure difference across them. We find the necessary conditions for the onset of oscillations that represent wheezing and propose a flutter-like instability model for it about a heavily deformed state of the tube. Our findings allow for a predictive tool for wheezing in lungs, which could lead to better diagnosis and treatment of lung diseases.

Spatially averaged haemodynamic models for different parts of cardiovascular system

AbstractThis paper revisits the usage of spatially averaged haemodynamic models such as non-stationary 1D/0D in space and stationary 0D in space models. Conditions of equivalence between different 1D model formulations are considered. The impact of circular and elliptic shapes of the tube cross-section on the friction term and the tube law is analyzed. Finally, the relationship between 0D lumped and 1D models is revealed.

Computational hemodynamics in arteries with the one-dimensional augmented fluid-structure interaction system: viscoelastic parameters estimation and comparison with in-vivo data

Mathematical models are widely recognized as a valuable tool for cardiovascular diagnosis and the study of circulatory diseases, especially to obtain data that require otherwise invasive measurements. To correctly simulate body hemodynamics, the viscoelastic properties of vessels walls are a key aspect to be taken into account as they play an essential role in cardiovascular behavior. The present work aims to apply the augmented fluid-structure interaction system of blood flow to real case studies to assess the validity of the model as a valuable resource to improve cardiovascular diagnostics and the treatment of pathologies. Main contributions of the paper include the evaluation of viscoelastic tube laws, estimation of viscoelastic parameters and comparison of models with literature results and in-vivo experiments. The ability of the model to correctly simulate pulse waveforms in single arterial segments is verified using literature benchmark test cases, designed taking into account a simple elastic behavior of the wall in the upper thoracic aorta and in the common carotid artery. Furthermore, in-vivo pressure waveforms, extracted from tonometric measurements performed on four human common carotid arteries and two common femoral arteries, are compared to numerical solutions. It is highlighted that the viscoelastic damping effect of arterial walls is required to avoid an overestimation of pressure peaks. Finally, an effective procedure to estimate the viscoelastic parameters of the model is herein proposed, which returns hysteresis curves of the common carotid arteries dissipating energy fractions in line with values calculated from literature hysteresis loops in the same vessel.

Modeling blood flow in viscoelastic vessels: the 1D augmented fluid–structure interaction system

Nowadays mathematical models and numerical simulations are widely used in the field of hemodynamics, representing a valuable resource to better understand physiological and pathological processes in different medical sectors. The theory behind blood flow modeling is closely related to the study of incompressible flow through compliant thin-walled tubes, starting from the incompressible Navier–Stokes equations. Furthermore, the mechanical interaction between blood flow and vessels wall must be properly described by the model. Recent works showed the benefits of characterizing the rheology of the vessel wall through a viscoelastic law. Taking into account the viscous contribution of the wall material and not simply the elastic one leads to a more realistic representation of the vessel behavior, which manifests not only an instantaneous elastic strain but also a viscous damping effect on pulse pressure waves, coupled to energy losses. In this context, the aim of this work is to propose an easily extensible one-dimensional mathematical model able to accurately capture fluid–structure interactions. The originality of the model lies in the introduction of a viscoelastic tube law in PDE form, valid for both arterial and venous networks, leading to an augmented fluid–structure interaction system. In contrast to well established mathematical models, the proposed one is natively hyperbolic. The model is solved with an efficient and robust second-order numerical scheme; the time integration is based on an Implicit–Explicit Runge–Kutta scheme conceived for applications to hyperbolic systems with stiff relaxation terms. The validation of the proposed model is performed on several different test cases. Results obtained in Riemann problems, adopting a simple elastic tube law for the characterization of the vessel wall, are compared with available exact solutions. To validate the contribution given by the viscoelastic term, the Method of Manufactured Solutions has been applied. Specific tests have been designed to verify the well-balancing with respect to fluid-at-rest condition and the accuracy-preserving property of the scheme. Finally, a specific test case with an inlet pulse pressure wave has been designed to assess the effects of viscoelasticity with respect to a simple elastic behavior of the vessel wall. The complete code, written in MATLAB (MathWorks Inc.) language, with the implemented test cases, is made available in Mendeley Data repository.

Effect of base-state curvature on self-excited high-frequency oscillations in flow through an elastic-walled channel

We derive a new ``tube law'' for elastic-walled channels, which takes into account the axial stretching that arises because of axial curvature in the base state. We quantify the effect of the new law on oscillatory fluid-structure-interaction modes in the channel and their stability.

Test tube baby technology, surrogacy arrangement and Law: Some human right concerns

Test tube baby technology, surrogacy arrangement and Law: Some human right concerns

Choke point physiology in airway stenting: A case presentation and discussion.

The point in the airway that allows the smallest maximal flow is known as the "choke point". The tube law describes the velocity of the expired air, which cannot exceed the wave-speed. Flow limitation during forced expiration is affected by the relationship between the transmural pressure (Ptm) and cross-sectional area (A) of the airway. Wave speed is dependent on the stiffness of the airway wall, as well as on the cross-section of the airway itself (dA/dPtm). Airway stenting at the wave-speed, flow-limiting segment (choke point) is assessed by using a catheter, via the working channel of a stereoscopic bronchoscope, to measure the difference between lateral pressure and pleural pressure. Based on the wave-speed concept of maximal expiratory flow limitation, stenting at the choke point increased the cross-sectional area and supported the weakened airway wall, thus improving expiratory flow limitation and relieving dyspnea. To ensure correct stent positioning and thus optimal functional benefit, it is important to locate the exact position of tracheobronchial stenosis.

Tube Law of the Pharyngeal Airway in Sleeping Patients with Obstructive Sleep Apnea.

Obstructive sleep apnea (OSA) is characterized by repetitive pharyngeal collapse during sleep. However, the dynamics of pharyngeal narrowing and re-expansion during flow-limited breathing are not well described. The static pharyngeal tube law (end-expiratory area versus luminal pressure) has demonstrated increasing pharyngeal compliance as luminal pressure decreases, indicating that the airway would be sucked closed with sufficient inspiratory effort. On the contrary, the airway is rarely sucked closed during inspiratory flow limitation, suggesting that the airway is getting stiffer. Therefore, we hypothesized that during inspiratory flow limitation, as opposed to static conditions, the pharynx becomes stiffer as luminal pressure decreases. Upper airway endoscopy and simultaneous measurements of airflow and epiglottic pressure were performed during natural nonrapid eye movement sleep. Continuous positive (or negative) airway pressure was used to induce flow limitation. Flow-limited breaths were selected for airway cross-sectional area measurements. Relative airway area was quantified as a percentage of end-expiratory area. Inspiratory airway radial compliance was calculated at each quintile of epiglottic pressure versus airway area plot (tube law). Eighteen subjects (14 males) with OSA (apnea-hypopnea index = 57 ± 27 events/h), aged 49 ± 8 y, with a body mass index of 35 ± 6 kg/m(2) were studied. A total of 163 flow limited breaths were analyzed (9 ± 3 breaths per subject). Compliances at the fourth (2.0 ± 4.7 % area/cmH2O) and fifth (0.0 ± 1.7 % area/cmH2O) quintiles were significantly lower than the first (12.2 ± 5.5 % area/cmH2O) pressure quintile (P < 0.05). The pharyngeal tube law is concave (airway gets stiffer as luminal pressure decreases) during respiratory cycles under inspiratory flow limitation.

Upper Airway Elasticity Estimation in Pediatric Down Syndrome Sleep Apnea Patients Using Collapsible Tube Theory.

Elasticity of the soft tissues surrounding the upper airway lumen is one of the important factors contributing to upper airway disorders such as snoring and obstructive sleep apnea. The objective of this study is to calculate patient specific elasticity of the pharynx from magnetic resonance (MR) images using a 'tube law', i.e., the relationship between airway cross-sectional area and transmural pressure difference. MR imaging was performed under anesthesia in children with Down syndrome (DS) and obstructive sleep apnea (OSA). An airway segmentation algorithm was employed to evaluate changes in airway cross-sectional area dilated by continuous positive airway pressure (CPAP). A pressure-area relation was used to make localized estimates of airway wall stiffness for each patient. Optimized values of patient specific Young's modulus for tissue in the velopharynx and oropharynx, were estimated from finite element simulations of airway collapse. Patient specific deformation of the airway wall under CPAP was found to exhibit either a non-linear 'hardening' or 'softening' behavior. The localized airway and tissue elasticity were found to increase with increasing severity of OSA. Elasticity based patient phenotyping can potentially assist clinicians in decision making on CPAP and airway or tissue elasticity can supplement well-known clinical measures of OSA severity.