Modeling of unstable heat-mass air exchange in the lungs

A mathematical model of the unsteady-state heat and mass exchange of expired air in the bronchial tree is suggested. The model includes heat and mass exchange between air and bronchial walls, and heat exchange between blood circulation and bronchial tree. A problem has been numerically solved as a unidimensional one in the quasi-steady-state formulation. It is shown that air conditioning occurs through the whole length of a respiratory tract. During inspiration bronchial walls are cooled, that in its turn induces a decrease of air temperature and water vapour content in time. That process depends on the intensity of lung blood circulation and character of air velocity changes during inspiration.     

Steady-state pleural fluid flow and pressure and the effects of lung buoyancy

Both theoretical and experimental studies of pleural fluid dynamics and lung buoyancy during steady-state, apneic conditions are presented. The theory shows that steady-state, top-to-bottom pleural-liquid flow creates a pressure distribution that opposes lung buoyancy. These two forces may balance, permitting dynamic lung floating, but when they do not, pleural-pleural contact is required. The animal experiments examine pleural-liquid pressure distributions in response to simulated reduced gravity, achieved by lung inflation with perfluorocarbon liquid as compared to air. The resulting decrease in lung buoyancy modifies the force balance in the pleural fluid, which is reflected in its vertical pressure gradient. The data and model show that the decrease in buoyancy with perfluorocarbon inflation causes the vertical pressure gradient to approach hydrostatic. In the microgravity analogue, the pleural pressures would be toward a more uniform distribution, consistent with ventilation studies during space flight. The pleural liquid turnover predicted by the model is computed and found to be comparable to experimental values from the literature. The model provides the flow field, which can be used to develop a full transport theory for molecular and cellular constituents that are found in pleural fluid.     

Transport of a fluid in the lung interstitial space. A model of porous medium with distributed parameters

Velocity and behaviour of the fluid stream into lung interstitium was experimentally evaluated during and after hydrodynamic loading. A mathematical model of unsteady fluid movement in interstitium able to relaxation was proposed.     

A mathematical model of aortic valve vibration

It has been shown experimentally that the second heart sound is produced by diastolic vibrations of the closed aortic valve. In the present paper a mathematical model of this vibration is developed from first principles. The model assumes a one dimensional but non-linear fluid behavior. The problem is coupled through a non-linear, planar valve. A solution is obtained using the method of characteristics developed in finite difference form. The resulting valve frequency and amplitude are in good agreement with patient data. The model predicts a strong dependency of response on the valve forcing function and valve stiffness; and a weaker dependency of response on valve mass.     

Oxytocin-induced labor augments IL-1beta-stimulated lung fluid absorption in fetal guinea pig lungs

We tested the hypothesis that oxytocin-induced labor augmented IL-1beta-induced/-stimulated lung fluid absorption in preterm guinea pig fetuses. IL-1beta was administered subcutaneously daily to timed-pregnant guinea pigs for 3 days with and without simultaneous cortisol synthesis inhibition by metyrapone. At day 3, oxytocin was administered, and fetuses were delivered by abdominal hysterotomy at 61 and by oxytocin-induced birth at 68 days gestation. Delivered fetuses were instilled with isosmolar 5% albumin into the lungs, and lung fluid movement was measured over 1 h by mass balance. Lung fluid absorption was induced in 61-day and stimulated in 68-day gestation lungs by IL-1beta. Labor induction by oxytocin augmented IL-1beta-induced/-stimulated lung fluid absorption. Metyrapone pretreatment did not affect oxytocin-induced/-stimulated lung fluid absorption, while completely blocking IL-1beta-induced/-stimulated fluid absorption. Fetal lung fluid absorption, when present, was always propranolol and amiloride sensitive, suggesting that beta-adrenoceptor stimulation and amiloride-sensitive sodium channels were critical for fluid absorption. Epithelial sodium channel and Na-K-ATPase subunit expressions were both increased by IL-1beta, but not further by oxytocin. Our results indicate that IL-1beta release into the maternal blood circulation positively affects lung maturation due to the IL-1beta-induced release of cortisol and thus prepares the lungs for the epinephrine surge associated with labor.     

Assessment of fluid balance in isolated sheep lungs

In this study we demonstrate the validity and utility of an isolated lung preparation developed for the study of pulmonary fluid balance. Lungs of 2- to 3-mo-old sheep were perfused in situ with autologous blood treated with indomethacin (20 micrograms/ml). Lung lymph flow (QL), uncontaminated by systemic lymph, was measured from either the efferent duct of the caudomediastinal lymph node or the thoracic duct in the superior mediastinum. Lung weight change (delta W) was measured as the opposite of the change in weight of the extracorporeal blood reservoir. A unique feature of this experimental model is the ability to assess lung fluid balance from simultaneous measurements of delta W and QL. In addition, hemodynamic and blood gas variables can be tightly controlled. Our results show that changes in QL and the lymph-to-plasma oncotic pressure ratio caused by an increase in microvascular pressure were comparable with those seen previously in intact sheep. When microvascular pressure was returned to control levels, QL fell despite a sustained increase in the amount of extravascular lung water, suggesting compartmentalization of the filtrate and/or effects of intravascular volume on lymph-driving pressure or resistance. Lymph flow was directly proportional to respiratory frequency over the range of 0-30 min-1 when the change in frequency was maintained for periods as long as 30 min. This preparation should prove useful in the study of lung fluid balance, particularly when it is desired to use interventions which are precluded or difficult in intact animals.     

Rigorous model for spherical cell-support aggregate

The activity of immobilized cell-support particle aggregates is influenced by physical and biochemical elements, mass transfer, and physiology. Accordingly, the mathematical model discussed in this study is capable of predicting the steady state and transient concentration profiles of the cell mass and substrate, plus the effects of the substrate and product inhibition in an immobilized cell-support aggregate. The overall mathematical model is comprised of material balance equations for the cell mass, major carbon source, dissolved oxygen, and non-biomass products in a bulk suspension along with a single particle model. A smaller bead size and higher substrate concentration at the surface of the particle, resulted in a higher supply of the substrate into the aggregate and consequently a higher biocatalyst activity.     

Component inertia modeling of segmental wobbling and rigid masses

A modification to an existing mathematical model is described, which permits the determination of subject-specific inertia parameters for wobbling and rigid masses of female body segments. The model comprises segment-specific soft tissue, bone, and lung components. A total of 59 geometric solids (40 soft tissue, 17 bone, 2 lung) were used to represent the body components. Ninety-five anthropometric measurements were collected from 7 female participants and were used to develop and evaluate the model. The success of the model is evaluated using predicted mass and mass distribution. The overall absolute accuracy in predicted whole body mass was better than 3.0%, with a maximum error of 4.9%. The appropriateness of the cadaver-based density values used in the model is addressed and the accuracy of the component inertia model in relation to uniform density models is discussed. The model offers a novel approach for determining component inertia parameters, which have been used successfully in a wobbling mass model to produce realistic kinetic analyses of drop-landings.     

Gas absorption in pulmonary airways at low Peclet number

A mathematical model is presented that investigates the mass transport of a diffusible and soluble gas contaminant through a liquid-lined tube when the Peclet number is small. The transport is determined by four dimensionless parameters: lambda, the tube aspect ratio; d, the relative difference in end concentrations; gamma, the radial transport coefficient; and Pe, the Peclet number. The problem is formulated for arbitrary gamma, but in the case of ozone and nitrous oxides the value of gamma is small. An asymptotic analysis for Pe much less than 1 and gamma much less than 1 is presented which yields the concentration field and transport characteristics we seek. It also provides a low Peclet number analysis for the conjugate problem of mass and heat transfer that is not currently available in the literature. The application to transport in the small airways of the lung is discussed, particularly the radial absorption differences in inspiratory and expiratory flow. Depending on the relative sizes of gamma and Pe, fractional uptake decreases with increasing Pe during inspiration but can increase during expiration.     

General tissue mass transfer model for cryopreservation applications

Successful cryopreservation of complex specimens, such as tissues and organs, would greatly benefit both the medical and scientific research fields. Vitrification is one of the most promising techniques for complex specimen cryopreservation, but toxicity remains a major challenge because of the high concentration of cryoprotectants (CPAs) needed to vitrify. Our group has approached this problem using mathematical optimization to design less toxic CPA equilibration methods for cells. To extend this approach to tissues, an appropriate mass transfer model is required. Fick’s law is commonly used, but this simple modeling framework does not account for the complexity of mass transfer in tissues, such as the effects of fixed charges, tissue size changes, and the interplay between cell membrane transport and transport through the extracellular fluid. Here, we propose a general model for mass transfer in tissues that accounts for all of these phenomena. To create this model, we augmented a previously published acellular model of mass transfer in articular cartilage to account for the effects of cells. We show that the model can accurately predict changes in CPA concentration and tissue size for both articular cartilage and pancreatic islets, tissue types with vastly different properties.     

Computational Approaches to Solving Equations Arising from Wound Healing

In the wound healing process, the cell movement associated with chemotaxis generally outweighs the movement associated with random motion, leading to advection-dominated mathematical models of wound healing. The equations in these models must be solved with care, but often inappropriate approaches are adopted. Two one-dimensional test problems arising from advection-dominated models of wound healing are solved using four algorithms—MATLAB’s inbuilt routine pdepe.m, the Numerical Algorithms Group routine d03pcf.f, and two finite volume methods. The first finite volume method is based on a first-order upwinding treatment of chemotaxis terms and the second on a flux limiting approach. The first test problem admits an analytic solution which can be used to validate the numerical results by analyzing two measures of the error for each method: the average absolute difference and a mass balance error. These criteria as well as the visual comparison between the numerical methods and the exact solution lead us to conclude that flux limiting is the best approach to solving advection-dominated wound healing problems numerically in one dimension. The second test problem is a coupled nonlinear three species model of wound healing angiogenesis. Measurement of the mass balance error for this test problem further confirms our hypothesis that flux limiting is the most appropriate method for solving advection-dominated governing equations in wound healing models. We also consider two two-dimensional test problems arising from wound healing, one that admits an analytic solution and a more complicated problem of blood vessels growth into a devascularized wound bed. The results from the two-dimensional test problems also demonstrate that the flux limiting treatment of advective terms is ideal for an advection-dominated problem.     

Some problems and solutions for modeling overall cardiovascular regulation

A brief history of the development of mathematical models of the cardiovascular system is presented. Until the advent of computers, very little modeling of transient physiological phenomena was done, but this is now commonplace. The problem of stability in complex physiological models fortunately is averted by the fact that the physiological controls are themselves highly stable. The reason for this is that evolution has eliminated unstable feedback loops because they are lethal. Indeed, enough safety factor has been provided in the design of the body so that even poor mathematical models are often quite stable. An especially important use of complex cardiovascular models has been to derive new concepts of cardiovascular function. One such concept is the “principle of infinite gain” for long-term control of arterial pressure, which states that the long-term level of arterial pressure is controlled by a balance between the fluid intake and the output of fluid by the kidneys, not by the level of total peripheral resistance as has been a long-standing misconception based on acute rather than chronic animal experiments.     

The role of stress in the growth of a multicell spheroid

Rather recent experimental results demonstrate the non–negligible role of mechanical stress in the growth of a multicell spheroid. In this paper we discuss a theoretical framework for volumetric growth suitable for modeling the growth of soft tissues exhibiting the properties of a solid. After a proper kinematic decomposition, balance equations for mass, momentum and energy are discussed together with constitutive relationships. The mathematical model is then applied to avascular tumor growth. We show by numerical simulation that, under assumption of spherical symmetry, the mathematical model is able to reproduce the experimental data with a satisfying qualitative agreement.     

Integrated control of lung fluid balance

This review summarizes the highlights of the EB2004 symposium that dealt with the integrated aspects of the lung fluid balance. It is apparent that maintenance of lung fluid balance requires the proper functioning of vascular endothelial and alveolar epithelial barriers. Under physiological conditions, the transcytotic pathway requiring repeated fission-fusion events of the caveolar membrane with other caveolae solely transports albumin. Caveolin-1, which forms caveolae, and albumin-binding proteins play a central role in signaling the transcytosis of albumin. Signals responsible for increasing endothelial permeability in lung microvessels in response to inflammatory mediators were also described. These studies in gene knockout mouse models revealed the importance of Ca(2+) signaling via store-operated transient receptor channel 4 and the activation of endothelial myosin light chain kinase isoform in mediating the increase in microvessel permeability. Increases in the cytosolic Ca(2+) in situ in microvessel endothelia can occur by mitochondria-dependent as well as mitochondria-independent pathways (such as the endoplasmic reticulum). Both these pathways, by triggering endothelial cell activation, may result in lung microvascular injury. The resolution of alveolar edema, requiring clearance of fluid from the air space, is another area of intense investigation in animal models. Although beta-adrenergic agonists can activate alveolar fluid clearance, signaling pathways regulating these events in intact alveoli remain to be established. Development of mouse models in which the function of regulatory proteins (identified in cell culture studies) can be systematically analyzed will provide a better and more integrated picture of lung fluid balance. In vivo veritas!     

Invited review: Active fluid clearance from the distal air spaces of the lung.

Active ion transport drives iso-osmolar alveolar fluid clearance, a hypothesis originally suggested by in vivo studies in sheep 20 yr ago. Over the last two decades, remarkable progress has been made in establishing a critical role for active sodium transport as a primary mechanism that drives fluid clearance from the distal air spaces of the lung. The rate of fluid transport can be increased in most species, including the human lung, by cAMP stimulation. Catecholamine-independent mechanisms, including hormones, growth factors, and cytokines, can also upregulate epithelial fluid clearance in the lung. The new insights into the role of the distal lung epithelium in actively regulating lung fluid balance has important implications for the resolution of clinical pulmonary edema.     

Oleic acid lung injury in sheep

Intravenous infusion of oleic acid into experimental animals causes acute lung injury resulting in pulmonary edema. We investigated the mechanism of oleic acid lung injury in sheep. In experiments with anesthetized and unanesthetized sheep with lung lymph fistulas, we measured pulmonary arterial and left atrial pressures, cardiac output, lung lymph flow, and lymph and plasma protein concentrations. We injured the lungs with intravenous infusions of oleic acid at doses ranging from 0.015 to 0.120 ml/kg. We found that oleic acid caused reproducible dose-related increases in pulmonary arterial pressure and pulmonary vascular resistance, arterial hypoxemia, and increased protein-rich lung lymph flow and extravascular lung water. The lung fluid balance changes were characteristic of increased permeability pulmonary edema. Infusion of the esterified fat triolein had no hemodynamic or lung fluid balance effects. Depletion of leukocytes with a nitrogen mustard or platelets with an antiplatelet serum had no effect on oleic acid lung injury. Treatment of sheep before injury with methylprednisolone 30 mg/kg or ibuprofen 12.5-15.0 mg/kg also had no effects. Unlike other well-characterized sheep lung injuries, injury caused by oleic acid does not require participation of leukocytes.     

Impaired cortical processing of inspiratory loads in children with chronic respiratory defects

Background

We tested the hypothesis that maternal interleukin-1β (IL-1β) pretreatment and induction of fetal cortisol synthesis activates MAP kinases and thereby affects lung fluid absorption in preterm guinea pigs.

Methods

IL-1β was administered subcutaneously daily to timed-pregnant guinea pigs for three days. Fetuses were obtained by abdominal hysterotomy and instilled with isosmolar 5% albumin into the lungs and lung fluid movement was measured over 1 h by mass balance. MAP kinase expression was measured by western blot.

Results

Lung fluid absorption was induced at 61 days (D) gestation and stimulated at 68D gestation by IL-1β. Maternal IL-1β pretreatment upregulated ERK and upstream MEK expression at both 61 and 68D gestation, albeit being much more pronounced at 61D gestation. U0126 instillation completely blocked IL-1β-induced lung fluid absorption as well as IL-1β-induced/stimulated ERK expression. Cortisol synthesis inhibition by metyrapone attenuated ERK expression and lung fluid absorption in IL-1β-pretreated fetal lungs. JNK expression after maternal IL-1β pretreatment remained unaffected at either gestation age.

Conclusion

These data implicate the ERK MAP kinase pathway as being important for IL-1β induction/stimulation of lung fluid absorption in fetal guinea pigs.     

Inert gas diffusion in heterogeneous tissue I: Without perfusion

A mathematical model is proposed to describe transient gas diffusion into a block of heterogenous tissue placed on an impermeable base. The corresponding asymptotic sultion of mass uptake of the gas is derived on the assumption that the diffusion constant is very much smaller in the cellular phase. It is expected that this will be useful in evaluating the diffusion constant in cellular material, and the volume fraction of extracellular fluid, providing the partition coefficient is known. The phenomenon of mutual interaction and multiple feedback between cellular and extracellular fluid is clearly seen in the overall response of the tissue. In this regard it is shown that the extraction of the two least dominant time constants, by backward projection of the experimental data curve of gas uptake, is likely to confuse the numerical evaluation of the physical parameters of the system. In an appendix, the problem of diffusion straight through a tissue slice is solved at the asymptotic stage, before steady state is reached. The resulting expression predicts the by-passing of cells by the diffusing gas and shows how the parameters cannot reliably be, determined.