Mathematical modeling of the response of a resistive vessel to pressure

The response of small arterial vessels to internal pressure makes an essential contribution to autoregulation in the vascular bed. It is believed that free cytosolic Ca²⁺ concentration plays a pivotal role in the regulation of smooth muscle contractility and hence of the vascular lumen. A simple mathematical model of blood flow in a resistive vessel is suggested. The model is based on the experimental data obtained for cerebral arteries, but may be used for any other resistive vessel. The model not only describes the regulation of the vascular lumen by transmural pressure but also shows realistic behavior of the vessel radius and cytosolic [Ca²⁺] at different rates of pressure change. Possible variations in the radius along the vessel due to the Bayliss effect are considered.     

Pressure and flow in the umbilical cord

A fluid dynamic study of blood flow within the umbilical vessels of the human maternal-fetal circulatory system is considered. It is found that the umbilical coiling index (UCI) is unable to distinguish between cords of significantly varying pressure and flow characteristics, which are typically determined by the vessel curvature, torsion and length. Larger scale geometric non-uniformities superposed over the inherent coiling, including cords exhibiting width and/or local UCI variations as well as loose true knots, typically produce a small effect on the total pressure drop. Crucially, this implies that a helical geometry of mean coiling may be used to determine the steady vessel pressure drop through a more complex cord. The presence of vessel constriction, however, drastically increases the steady pressure drop and alters the flow profile. For pulsatile-flow within the arteries, the steady pressure approximates the time-averaged value with high accuracy over a wide range of cords. Furthermore, the relative peak systolic pressure measured over the period is virtually constant and approximately 25% below the equivalent straight-pipe value for a large range of non-straight vessels. Interestingly, this suggests that the presence of vessel helicity dampens extreme pressures within the arterial cycle and may provide another possible evolutionary benefit to the coiled structure of the cord.     

Transmission characteristics of axial waves in blood vessels

The elastic behavior of blood vessels can be quantitatively examined by measuring the propagation characteristics of waves transmitted by them. In addition, specific information regarding the viscoelastic properties of the vessel wall can be deduced by comparing the observed wave transmission data with theoretical predictions. The relevance of these deductions is directly dependent on the validity of the mathematical model for the mechanical behavior of blood vessels used in the theoretical analysis. Previous experimental investigations of waves in blood vessels have been restricted to pressure waves even though theoretical studies predict three types of waves with distinctly different transmission characteristics. These waves can be distinguished by the dominant displacement component of the vessel wall and are accordingly referred to as radial, axial and circumferential waves. The radial waves are also referred to as pressure waves since they exhibit pronounced pressure fluctuations. For a thorough evaluation of the mathematical models used in the analysis it is necessary to measure also the dispersion and attenuation of the axial and circumferential (torsion) waves.

To this end a method has been developed to determine the phase velocities and damping of sinusoidal axial waves in the carotid artery of anesthetized dogs with the aid of an electro-optical tracking system. For frequencies between 25 and 150 Hz the speed of the axial waves was between 20 and 40 m/sec and generally increased with frequency, while the natural pressure wave travelled at a speed of about 10 m/sec. On the basis of an isotropic wall model the axial wave speed should however be approximately 5 times higher than the pressure wave speed. This discrepancy can be interpreted as an indication for an anisotropic behavior of the carotid wall. The carotid artery appears to be more elastic in the axial than in the circumferential direction.     

A constrained mixture model for developing mouse aorta

Mechanical stresses influence the structure and function of adult and developing blood vessels. When these stresses are perturbed, the vessel wall remodels to return the stresses to homeostatic levels. Constrained mixture models have been used to predict remodeling of adult vessels in response to step changes in blood pressure, axial length and blood flow, but have not yet been applied to developing vessels. Models of developing blood vessels are complicated by continuous and simultaneous changes in the mechanical forces. Understanding developmental growth and remodeling is important for treating human diseases and designing tissue-engineered blood vessels. This study presents a constrained mixture model for postnatal development of mouse aorta with multiple step increases in pressure, length and flow. The baseline model assumes that smooth muscle cells (SMCs) in the vessel wall immediately constrict or dilate the inner radius after a perturbation to maintain the shear stress and then remodel the wall thickness to maintain the circumferential stress. The elastin, collagen and SMCs have homeostatic stretch ratios and passive material constants that do not change with developmental age. The baseline model does not predict previously published experimental data. To approximate the experimental data, it must be assumed that the SMCs dilate a constant amount, regardless of the step change in mechanical forces. It must also be assumed that the homeostatic stretch ratios and passive material constants change with age. With these alterations, the model approximates experimental data on the mechanical properties and dimensions of aorta from 3- to 30-day-old mice.     

Diffusion model of tumor vascularization and growth

A diffusion model of tumor growth, vascularization and necrosis is used to analyze experimental data describing the temporal changes in tumor cell and blood vessel radial distributions in a host-tissue field transplanted with a fibrosarcoma. The experimental results showed a peak density of vessels occurring at the advancing migration front of the tumor and a decline in the vessel surface area at the tumor center with time. The peak density of tumor cells shifts away from the tumor center with time. These dynamic changes can be explained by a mathematical model which views the process as one of diffusion and proliferation in time and space. Coupled diffusion equations with nonlinear source and sink terms describe the proliferation, death, and migration of tumor cells and vascular surface area. The concept of an angiogenic factor elaborated by tumor cells is incorporated.     

A simple model of a vessel with a wall sensitive to mechanical stimuli

Smooth muscles in the walls of small blood vessels under normal conditions are always moderately active, so that there is a certain reserve for blood flow adaptation to changed conditions by either narrowing or expanding of the vessel lumen. The previous studies of small vessel hydrodynamics have shown that the activity can cause specific instability of vessel steady states. In order to trace qualitatively the influence of numerous factors on the active state of the vessel, a simplified model of the vessel was proposed, which is reduced to a nonlinear ordinary equation of the first order with delayed argument.     

Homogenization modeling for the mechanics of perfused myocardium

Altered coronary perfusion can change the apparent diastolic stiffness of ventricular myocardium--the ‘garden hose’ effect. Our recent findings showed that myocardial strains are reduced during ventricular filling, primarily along the directions transverse to the coronary microvessels. In this article, we review hypotheses and theoretical models regarding the role that regional wall stress plays in the mechanical interaction between myocardium and coronary circulation. Various mechanisms have been used to explain the effects of the tissue stress on coronary flow, as well as the effect of coronary dynamics on myocardial mechanics. Many models of coronary pressure-flow relations using lumped parameter circuit analogs. Poroelasticity and swelling theories have been used to model the mechanics of perfused muscle. Here, we describe a new mathematical model of the mechanics of perfused myocardium derived using homogenization theory. In this model, perfused myocardium is treated as a nonlinear anisotropic elastic solid embedded with cylindrical vessels of known distensibility. The solid compartment is incompressible but the vascular compartment may change volume according to a simple relation between vessel diameter and perfusion pressure. The work done by the perfusion pressure in changing vascular volume contributes to the macroscopic strain energy and hence affects the stress and stiffness of the composite. Conversely, the stress in the tissue affects microvessel diameter and volume, since tractions transverse to the vessel axis oppose the internal blood pressure. Finite element simulations of passive filling show good agreement of model with experimental results.     

Network Scale Modeling of Lymph Transport and Its Effective Pumping Parameters

The lymphatic system is an open-ended network of vessels that run in parallel to the blood circulation system. These vessels are present in almost all of the tissues of the body to remove excess fluid. Similar to blood vessels, lymphatic vessels are found in branched arrangements. Due to the complexity of experiments on lymphatic networks and the difficulty to control the important functional parameters in these setups, computational modeling becomes an effective and essential means of understanding lymphatic network pumping dynamics. Here we aimed to determine the effect of pumping coordination in branched network structures on the regulation of lymph flow. Lymphatic vessel networks were created by building upon our previous lumped-parameter model of lymphangions in series. In our network model, each vessel is itself divided into multiple lymphangions by lymphatic valves that help maintain forward flow. Vessel junctions are modeled by equating the pressures and balancing mass flows. Our results demonstrated that a 1.5 s rest-period between contractions optimizes the flow rate. A time delay between contractions of lymphangions at the junction of branches provided an advantage over synchronous pumping, but additional time delays within individual vessels only increased the flow rate for adverse pressure differences greater than 10.5 cmH₂O. Additionally, we quantified the pumping capability of the system under increasing levels of steady transmural pressure and outflow pressure for different network sizes. We observed that peak flow rates normally occurred under transmural pressures between 2 to 4 cmH₂O (for multiple pressure differences and network sizes). Networks with 10 lymphangions per vessel had the highest pumping capability under a wide range of adverse pressure differences. For favorable pressure differences, pumping was more efficient with fewer lymphangions. These findings are valuable for translating experimental measurements from the single lymphangion level to tissue and organ scales.     

Steady state and (bi-) stability evaluation of simple protease signalling networks

Signal transduction networks are complex, as are their mathematical models. Gaining a deeper understanding requires a system analysis. Important aspects are the number, location and stability of steady states. In particular, bistability has been recognised as an important feature to achieve molecular switching. This paper compares different model structures and analysis methods particularly useful for bistability analysis.

The biological applications include proteolytic cascades as, for example, encountered in the apoptotic signalling pathway or in the blood clotting system. We compare three model structures containing zero-order, inhibitor and cooperative ultrasensitive reactions, all known to achieve bistability. The combination of phase plane and bifurcation analysis provides an illustrative and comprehensive understanding of how bistability can be achieved and indicates how robust this behaviour is.

Experimentally, some so-called “inactive” components were shown to have a residual activity. This has been mostly ignored in mathematical models. Our analysis reveals that bistability is only mildly affected in the case of zero-order or inhibitor ultrasensitivity. However, the case where bistability is achieved by cooperative ultrasensitivity is severely affected by this perturbation.     

Viscoelastic properties of microvessels in rat spinotrapezius muscle

In order to establish a quantitative model of blood flow in skeletal muscle, the mechanical properties of the blood vessels need to be measured. We present measurements of the viscoelastic properties of arterioles, venules, and capillaries in exteriorized rat spinotrapezius muscle. Muscles were perfused with an inert silicone polymer and a uniform static pressure was established by occlusion of the venous outflow. Vessel diameters were then measured as a function of the static pressure. This study provides the first measurements of the viscoelastic properties of microvessels in skeletal muscle in situ. Over a pressure range of 20-200 mmHg, the transverse arterioles are the most distensible vessels, while the arcade venules are the stiffest. In response to a step change in pressure, all vessels show an initial elastic deformation, followed by a nonlinear creep. Based on the experimental results for different pressure histories a constitutive equation relating vessel diameter to the local transmural pressure is proposed. Diameter changes are expressed in the form of a diameter strain, analogous to a Green's strain, and are related to the local transmural pressure using a standard linear solid model. This model has only three empirical coefficients and could be fitted to all experimental results for all vessels within error of measurement.     

Blood flow regulation in the cerebral microvasculature with an arcadal network: a numerical simulation

Blood flow regulation in the cerebral microvasculature with an arcadal network was investigated using a numerical simulation. A mathematical model for blood flow in the arcadal network, based on in vivo data of cat cerebral microvasculature and flow velocity was developed. The network model consists of 45 vessel segments and 25 branching points. To simulate microvascular response to blood flow, non-reactive (solid), cerebral arteriole-like, or skeletal muscle arteriole-like responses to wall shear stress were taken into account. Numerical calculation was carried out in the flow condition where the inlet (arterial) pressure was changed from 60 to 120 mmHg. Flow-rate in each efferent vessel and the mean flow-rate over all efferent vessels were evaluated for assessment of blood supply to the local area of cerebral tissue. The simulation demonstrated the wall shear stress-induced vasodilation in the arcadal network worked to maintain the blood flow at a constant level with pressure variable in a wide range. It is suggested that an individual microvessel (segment) should join in the regulatory process of flow, interacting with other microvessels (cooperative regulation).     

A hybrid bioregulatory model of angiogenesis during bone fracture healing

Bone fracture healing is a complex process in which angiogenesis or the development of a blood vessel network plays a crucial role. In this paper, a mathematical model is presented that simulates the biological aspects of fracture healing including the formation of individual blood vessels. The model consists of partial differential equations, several of which describe the evolution in density of the most important cell types, growth factors, tissues and nutrients. The other equations determine the growth of blood vessels as a result of the movement of leading endothelial (tip) cells. Branching and anastomoses are accounted for in the model. The model is applied to a normal fracture healing case and subjected to a sensitivity analysis. The spatiotemporal evolution of soft tissues and bone, as well as the development of a blood vessel network are corroborated by comparison with experimental data. Moreover, this study shows that the proposed mathematical framework can be a useful tool in the research of impaired healing and the design of treatment strategies.     

Physical properties of resistance vessel wall in peripheral blood flow regulation--I. Mathematical model

A mathematical model is introduced to investigate the influence of the physical properties of the resistance vessel wall on the metabolic and myogenic mechanisms. The resistance vessel wall is assumed to have an elastic property and the elastic modulus to be a function of pressure (myogenic) and flow (metabolic). Blood is Poiseuille's flow. The resulting mathematical equations for pressure-flow, pressure-diameter, pressure-wall tension and pressure-wall elastic modulus relationships introduced obey Laplace's law. Poiseuille's law and Hooke's law. In comparison with the experimental data (pressure diameter), the mathematical model is confirmed to explain well the dynamic behavior of the resistance vessel wall in vivo.     

Three-dimensional modeling of flow and deformation in idealized mild and moderate arterial vessels

Three-dimensional numerical calculations of mild and moderate stenosed blood vessels have been performed. Large eddy simulation through a dynamic subgrid scale Smagorinsky model is applied to model the transitional and turbulent pulsatile flow. For the compliant stenosed model, fluid-structure interaction is realized through a two-way coupling between the fluid flow and the deforming vessel through the change in the external diameter due to the increment of circumferential pressure via a novel moving boundary approach. Model predictions compare very well against measured and numerical data for the centerline velocities, thickness of the flow separation zones and radial wall displacements.     

Nonlinear modelling of renal vasoaction

The control of blood pressure is a complex mixture of neural, hormonal and intrinsic interactions at the level of the heart, kidney and blood vessels. While experimental approaches to understanding these interactions are useful, it remains difficult to conduct experiments to quantify these interactions as the number of parameters increases. Thus, modelling of such physiological systems can offer considerable assistance. Typical mathematical models which describe the ability of the blood vessels to change their diameter (vasoconstriction) assume linearity of operation. However, due to the interaction of multiple vasocontrictive and vasodilative effectors, there is a significant nonlinear response to the influence of neural factors, particularly at higher levels of nerve activity (often seen in subjects with high blood pressure) which leads to low blood flow rates. This paper proposes a number of nonlinear mathematical models for the relationship between neural influences (sympathetic nerve activity (SNA)) and renal blood flow, using a feedback path to model the predominantly nonlinear effect of local vasoactive modulators such as nitric oxide, which oppose the action of SNA. The model structures are motivated by basic physiological principles, while the model parameters are determined using numerical optimisation techniques using open-loop data collected from rabbits. The models were verified by demonstrating correlation between experimental results and model outputs.     

Experimental determination of wall shear rate in canine carotid arteries perfused in vitro

The mathematical model of Hung (Tsai and Hung, 1984) is employed to determine the wall shear rate acting on canine carotid arteries perfused in vitro. Model equations for pulsatile flow in a deformable vessel are coupled with experimental data of dynamic pressure drop, flow rate, vessel radius and radial wall motion. Derived quantities, e.g. velocity profiles and wall shear, are obtained for vessels exposed to 'normotensive' hemodynamics, 'hypertension' simulations and perfusions in which the compliance of the vessel wall is deliberately altered. Our results indicate that wall shear varies markedly as a function of the hemodynamic environment. The effects of vessel radius vs flow rate on the development of wall shear are also demonstrated. It is found that convective processes correlate with the magnitude of wall shear in the 'hypertension' simulations. The present findings and complementary published data may explain, at least in part, the variations in vessel wall transport and endothelial cell biology we observe as a function of the hemodynamic environment. For example we have documented that the exposure of canine carotids to 'hypertensive' (vs 'normotensive') hemodynamics is associated with an increased flux of lipoproteins (LDL) into the intima and luminal media. Alternations in wall compliance, on the other hand, profoundly influence endothelial shape, orientation and cytoskeletal array.     

Deformation-induced endothelin B receptor-mediated smooth muscle cell apoptosis is matrix-dependent

To maintain normal blood flow, pressure overload in both arteries and veins requires a structural adaptation of the vessel wall (remodelling) that involves smooth muscle cell (SMC) hypertrophy and/or hyperplasia. Due to its potent vasoconstrictor and growth-promoting effects, endothelin-1 (ET-1) is a likely candidate to initiate and/or promote remodelling in blood vessels exposed to a chronic increase in blood pressure. To test this hypothesis, isolated segments of the rabbit carotid artery and jugular vein were perfused at different levels of intraluminal pressure. In both types of segments, pressure overload (160 and 20 mmHg, respectively) resulted in an increase in endothelial prepro-ET-1 and SMC endothelin B receptor (ETB-R) expression. Moreover, in pressurised segments from the carotid artery an ETB-R antagonist-sensitive increase in SMC apoptosis in the media was observed, while in the vein medial SMC started to proliferate. Isolated SMC from these rabbit blood vessels as well as from the aorta and vena cava of the rat, when cultured on a collagen or laminin matrix, uniformly revealed an ETB-R-mediated increase in apoptosis upon exposure to mechanical deformation plus exogenous ET-1 (10 nmol/L). However, when grown on a fibronectin matrix, the cultured SMC did not respond with an increase in apoptosis under otherwise identical experimental conditions. These findings suggest that deformation-induced activation of the endothelin system in the vessel wall not only plays a crucial role in remodelling, but that the structural components of the vessel wall, in particular the cell-matrix interaction, determine how SMC respond phenotypically to these changes in gene expression.     

光学成像技术在体研究肿瘤的光动力效应

光动力疗法 (PDT) 已发展成为一种较成熟的肿瘤治疗方法, PDT 诱导的血管损伤是杀死肿瘤的重要机制之一 . 为了在活体肿瘤模型上实时监测 PDT 导致的血管损伤效应,使用稳定高表达绿色荧光蛋白 (GFP) 的人涎腺腺样囊性癌细胞株 (ACC-M-GFP) ,建立了基于鸡胚尿囊膜 (CAM) 的肿瘤模型 . 应用荧光成像技术对肿瘤的生长位置、大小,以及治疗区域进行方便精确的定位;利用激光散斑成像技术,实时监测 CAM 上肿瘤周围血管的血液动力学参数 . 发现不同光动力剂量所导致的血管损伤有显著不同 . 结果表明,荧光标记的鸡胚尿囊膜肿瘤模型为研究 PDT 导致的血管损伤效应提供了良好的实验模型,激光散斑成像技术适用于实时监测 PDT 过程中血管结构、血流速度的变化,由此得出血液灌注率可用以评估 PDT 对肿瘤周围血管的损伤效应 .     

The effect of blood flow velocity on the process of thrombus formation in microvessels

The growth rate constant of platelet thrombi as a function of mean blood flow velocity was studied using mesentery vessels of the white rat. An advanced mathematical model for the kinetics of platelet thrombi was developed. The dependence of platelet activation delay time upon the distance from the damaged vessel wall is pointed out as a result of theoretical and experimental data comparison.