An integrative model of coupled water and solute exchange in the heart

A Mathematical Model for Transport in Poroelastic Materials with Variable Volume: Derivation, Lie Symmetry Analysis and Examples—Part 2

The model for perfused tissue undergoing deformation taking into account the local exchange between tissue and blood and lymphatic systems is presented. The Lie symmetry analysis in order to identify its symmetry properties is applied. Several families of steady-state solutions in closed formulae are derived. An analysis of the impact of the parameter values and boundary conditions on the distribution of hydrostatic pressure, osmotic agent concentration and deformation of perfused tissue is provided applying the solutions obtained in examples describing real-world processes.

Model of fluid and solute shifts during hemodialysis with active transport of sodium and potassium

Background

Mathematical models are useful tools to predict fluid shifts between body compartments in patients undergoing hemodialysis (HD). The ability of a model to accurately describe the transport of water between cells and interstitium (J_v,ISIC), and the consequent changes in intracellular volume (ICV), is important for a complete assessment of fluid distribution and plasma refilling. In this study, we propose a model describing transport of fluid in the three main body compartments (intracellular, interstitial and vascular), complemented by transport mechanisms for proteins and small solutes.

Methods

The model was applied to data from 23 patients who underwent standard HD. The substances described in the baseline model were: water, proteins, Na, K, and urea. Small solutes were described with two-compartment kinetics between intracellular and extracellular compartments. Solute transport across the cell membrane took place via passive diffusion and, for Na and K, through the ATPase pump, characterized by the maximum transport rate, Jp_MAX. From the data we estimated Jp_MAX and two other parameters linked to transcapillary transport of fluid and protein: the capillary filtration coefficient Lp and its large pores fraction α_LP. In an Expanded model one more generic solute was included to evaluate the impact of the number of substances appearing in the equation describing J_v,ISIC.

Results

In the baseline model, median values (interquartile range) of estimated parameters were: Lp: 11.63 (7.9, 14.2) mL/min/mmHg, α_LP: 0.056 (0.050, 0.058), and Jp_MAX: 5.52 (3.75, 7.54) mmol/min. These values were significantly different from those obtained by the Expanded model: Lp: 8.14 (6.29, 10.01) mL/min/mmHg, α_LP: 0.046 (0.038, 0.052), and Jp_MAX: 16.7 (11.9, 25.2) mmol/min. The relative RMSE (root mean squared error)averaged between all simulated quantities compared to data was 3.9 (3.1, 5.6) %.

Conclusions

The model was able to accurately reproduce most of the changes observed in HD by tuning only three parameters. While the drop in ICV was overestimated by the model, the difference between simulations and data was less than the measurement error. The biggest change in the estimated parameters in the Expanded model was a marked increase of Jp_MAX indicating that this parameter is highly sensitive to the number of species modeled, and that the value of Jp_MAX should be interpreted only in relation to this factor.

Regulation of the glucose supply from capillary to tissue examined by developing a capillary model

A new glucose transport model relying upon diffusion and convection across the capillary membrane was developed, and supplemented with tissue space and lymph flow. The rate of glucose utilization (J util) in the tissue space was described as a saturation function of glucose concentration in the interstitial fluid (C glu,isf), and was varied by applying a scaling factor f to J max. With f = 0, the glucose diffusion ceased within ~20 min. While, with increasing f, the diffusion was accelerated through a decrease in C glu,isf, but the convective flux remained close to resting level. When the glucose supplying capacity of the capillary was measured with a criterion of J util /J max = 0.5, the capacity increased in proportion to the number of perfused capillaries. A consistent profile of declining C glu,isf along the capillary axis was observed at the criterion of 0.5 irrespective of the capillary number. Increasing blood flow scarcely improved the supplying capacity.

Modelling Transcapillary Transport of Fluid and Proteins in Hemodialysis Patients

Background

The kinetics of protein transport to and from the vascular compartment play a major role in the determination of fluid balance and plasma refilling during hemodialysis (HD) sessions. In this study we propose a whole-body mathematical model describing water and protein shifts across the capillary membrane during HD and compare its output to clinical data while evaluating the impact of choosing specific values for selected parameters.

Methods

The model follows a two-compartment structure (vascular and interstitial space) and is based on balance equations of protein mass and water volume in each compartment. The capillary membrane was described according to the three-pore theory. Two transport parameters, the fractional contribution of large pores (α_LP) and the total hydraulic conductivity (LpS) of the capillary membrane, were estimated from patient data. Changes in the intensity and direction of individual fluid and solute flows through each part of the transport system were analyzed in relation to the choice of different values of small pores radius and fractional conductivity, lymphatic sensitivity to hydraulic pressure, and steady-state interstitial-to-plasma protein concentration ratio.

Results

The estimated values of LpS and α_LP were respectively 10.0 ± 8.4 mL/min/mmHg (mean ± standard deviation) and 0.062 ± 0.041. The model was able to predict with good accuracy the profiles of plasma volume and serum total protein concentration in most of the patients (average root-mean-square deviation < 2% of the measured value).

Conclusions

The applied model provides a mechanistic interpretation of fluid transport processes induced by ultrafiltration during HD, using a minimum of tuned parameters and assumptions. The simulated values of individual flows through each kind of pore and lymphatic absorption rate yielded by the model may suggest answers to unsolved questions on the relative impact of these not-measurable quantities on total vascular refilling and fluid balance.

Adaptations in the treatment of congenital lymphedema centered on the quality of life

Case Description. This report describes the evolution, necessary adaptations, and complications in the treatment of a 9-year-old child with primary congenital lymphedema. Description of Intervention. The clinical treatment of lymphedema was started in the first year of the patient's life and for five years she was only treated using the Godoy & Godoy technique of cervical stimulation. Three years ago the patient was prescribed a compression stocking made from a cotton-polyester fabric (grosgrain) because of a sudden increase in the lymphedema after she started to take growth hormones. Outcome and Conclusion. The combination of cervical stimulation and a compression stocking was effective to keep the child's life relatively normal, performing all day-to-day and recreational activities.

The cardioprotection of simvastatin in reperfused swine hearts relates to the inhibition of myocardial edema by modulating aquaporins via the PKA pathway

BACKGROUND AND OBJECTIVE

Myocardial edema plays a role in myocardial no-reflow and infarction during ischemia and reperfusion. The effects of statins against no-reflow and infarction may relate to the inhibition of myocardial edema. The current study investigated the role of protein kinase A (PKA) in statin-reduced myocardial edema in reperfused swine hearts.

METHODS AND RESULTS

Minipigs were treated with simvastatin (SIM, 2mg/kg), SIM+H-89 (a PKA inhibitor, 1.0 μg/kg/min), or H-89 alone 1h before 90-min ischemia and 3-h reperfusion or sham operation. Ischemia or ischemia-reperfusion induced severe myocardial edema, PKA activation, and up-regulation of aquaporin-1, -4, -8, and -9 in the reflow and no-reflow myocardium. SIM pretreatment reduced the sizes of no-reflow and infarct areas by 18.5% and 11.1% (P<0.01), decreased water content in the left ventricle, reflow and no-reflow myocardium by 1.4%, 5.3%, and 4.3% (P<0.05), inhibited cardiomyocytes swelling in the reflow and no-reflow areas by 19.8% and 13.1% (P<0.01), suppressed mitochondrial water accumulation in the reflow and no-reflow areas by 49.0% and 35.9% (P<0.01), increased PKA activity (P<0.01), and blocked the up-regulation of aquaporin-1, -4, -8, and -9 in the reflow and no-reflow myocardium. However, these beneficial effects of SIM were partially abolished by inhibiting PKA with H-89.

CONCLUSIONS

The cardioprotective effects of acute SIM therapy against myocardial no-reflow and infarction relate to the reduction of myocardial edema by suppressing the expression of aquaporin-1, -4, -8, and -9 in a partially PKA-dependent manner.

Three-dimensional T1 mapping of the mouse heart using variable flip angle steady-state MR imaging

Cardiac MR T(1) mapping is a promising quantitative imaging tool for the diagnosis and evaluation of cardiomyopathy. Here, we present a new preclinical cardiac MRI method enabling three-dimensional T(1) mapping of the mouse heart. The method is based on a variable flip angle analysis of steady-state MR imaging data. A retrospectively triggered three-dimensional FLASH (fast low-angle shot) sequence (3D IntraGate) enables a constant repetition time and maintains steady-state conditions. 3D T(1) mapping of the complete mouse heart could be achieved in 20 min. High-quality, bright-blood T(1) maps were obtained with homogeneous T(1) values (1764 ± 172 ms) throughout the myocardium. The repeatability coefficient of R(1) (1/T(1) ) in a specific region of the mouse heart was between 0.14 and 0.20 s(-1) , depending on the number of flip angles. The feasibility for detecting regional differences in ΔR(1) was shown with pre- and post-contrast T(1) mapping in mice with surgically induced myocardial infarction, for which ΔR(1) values up to 0.83 s(-1) were found in the infarct zone. The sequence was also investigated in black-blood mode, which, interestingly, showed a strong decrease in the apparent mean T(1) of healthy myocardium (905 ± 110 ms). This study shows that 3D T(1) mapping in the mouse heart is feasible and can be used to monitor regional changes in myocardial T(1), particularly in relation to pathology and in contrast-enhanced experiments to estimate local concentrations of (targeted) contrast agent.

The transcriptional programme of human heart valves reveals the natural history of infective endocarditis

Infective endocarditis (IE) is an infectious disease that is mainly caused by Staphylococcus aureus and Streptococcus sp. It usually leads to valvular destruction and vegetation formation. Its pathophysiology is badly understood and likely involves immune and coagulation systems with close interactions with the microorganism. Our objective was to evaluate host response by comparing transcriptional profiles of cardiac valves from IE patients with controls. Hierarchical clustering revealed a signature of IE consisting of 146 genes. Among the 89 up-regulated genes, we identified two genes strongly associated with IE: metalloproteinase 12 (MMP-12) and aquaporin-9, a member of the aquaglyceroporin membrane channel family. The up-regulation of MMP-12 gene is strengthened by the down-modulation of the gene encoding its inhibitor TIMP3. In addition, MMP-12 was expressed in macrophages infiltrating EI valves. We also found that aquaporin-9 was expressed in endothelial cells lining neo-vessel lumen, suggesting that aquaporin-9 might be associated with neovascularization of infected valves leading to tissue oedema secondary to the inflammatory process. The Gene Ontology annotation and the resulting functional classification showed that most up-regulated genes account for recruitment of inflammatory cells in vegetations, angiogenesis and remodelling of endocardium tissue. A network analysis confirmed the involvement of molecules related to the remodelling of endocardium tissue and angiogenesis in IE. It also evidenced the role of caspases, especially that of caspase-9 and intrinsic apoptotic pathway in IE. Based on this study we propose a scenario for the natural history of IE in humans. Some parameters identified in this work could become tools for measuring the disease activity and should be tested as biomarkers for diagnosis or prognosis assessment in future studies.

A physiologically based pharmacokinetic model of vascular-extravascular exchanges during liver carcinogenesis: application to MRI contrast agents

The extraction of physiological parameters by non-invasive imaging techniques such as dynamic magnetic resonance imaging (MRI) or positron emission tomography requires a knowledge of molecular distribution and exchange between microvascularization and extravascular tissues. These phenomena not only depend on the physicochemical characteristics of the injected molecules but also the pathophysiological state of the targeted organ. We developed a five-compartment physiologically based pharmacokinetic model focused on hepatic carcinogenesis and MRI contrast agents. This model includes physical characteristics of the contrast agent, dual specific liver supply, microvessel wall properties and transport parameters that are compatible with hepatocarcinoma development. The evolution of concentrations in the five compartments showed significant differences in the distribution of three molecules (differentiated by their diameters and diffusion coefficients ranging, respectively, from 0.9 to 62 nm and from 68.10(-9) to 47.10(-7) cm(2) s(-1)) in simulated regeneration nodules and dysplastic nodules, as well as in medium- and poorly differentiated hepatocarcinoma. These results are in agreement with known vascular modifications such as arterialization that occur during hepatocarcinogenesis. This model can be used to study the pharmacokinetics of contrast agents and consequently to extract parameters that are characteristic of the tumor development (like permeability), after fitting simulated to in vivo data.

Coupling texture analysis and physiological modeling for liver dynamic MRI interpretation

We coupled our physiological model of the liver, to a MRI simulator (SIMRI) in order to find image markers of the tumor growth. Some pathological modifications related to the development of Hepatocellular carcinoma are simulated (flows, permeability, vascular density). Corresponding images simulated at typical acquisition phases (arterial, portal) are compared to real images. The evolution of some textural features with arterial flow is also presented.

Myocardial water handling and the role of aquaporins

Cardiac surgery is performed in approximately 770,000 adults and 30,000 children in the United States of America annually. In this review we outline the mechanistic links between post-operative myocardial stunning and the development of myocardial edema. These interrelated processes cause a decline in myocardial performance that account for significant morbidity and mortality after cardiac surgery. Factors leading to myocardial edema include hemodilution, ischemia and reperfusion as well as osmotic gradients arising from pathological change. Several members of the aquaporin family of water transport proteins have been described in the myocardium although their role in the pathogenesis and resolution of cardiac edema is not established. This review examines evidence for the involvement of aquaporins in myocardial water handling during normal and pathological conditions.

Cardiac aquaporin expression in humans, rats, and mice

Water accumulation in the heart is important in ischemia-reperfusion injury and operations performed by using cardiopulmonary bypass, with cardiac dysfunction associated with myocardial edema being the principal determinant of clinical outcome. As an initial step in determining the role of aquaporin (AQP) water channels in myocardial edema, we have assessed the myocardial expression of AQPs in humans, rats, and mice. RT-PCR revealed expression of AQP-1, -4, -6, -7, -8, and -11 transcripts in the mouse heart. AQP-1, -6, -7, and -11 mRNAs were found in the rat heart as well as low levels of AQP-4 and -9. Human hearts contained AQP-1, -3, -4, -5, -7, -9, -10, and -11 mRNAs. AQP-1 protein expression was confirmed by Western blot analysis in all three species. AQP-4 protein was detected in the mouse heart but not in the rat or human heart. To determine the potential functional consequences of myocardial AQP expression, water permeability was measured in plasma membrane vesicles from myocardial cells of wild-type versus various AQP knockout mice. Water permeability was reduced by AQP-1 knockout but not by AQP-4 or AQP-8 knockout. With the use of a model of isolated rat heart perfusion, it was found that osmotic and ischemic stresses are not associated with changes in AQP-1 or AQP-4 expression. These studies support a possible functional role of AQP-1 in myocardium but indicate that early adaptations to osmotic and ischemic stress do not involve transcriptional or posttranslational AQP-1 regulation.

A practical extension of hydrodynamic theory of porous transport for hydrophilic solutes

OBJECTIVE

The equations for transport of hydrophilic solutes through aqueous pores provide a fundamental basis for examining capillary-tissue exchange and water and solute flux through transmembrane channels, but the theory remains incomplete for ratios, alpha, of sphere diameters to pore diameters greater than 0.4. Values for permeabilities, P, and reflection coefficients, sigma, from Lewellen, working with Lightfoot et al., at alpha = 0.5 and 0.95, were combined with earlier values for alpha < 0.4, and the physically required values at alpha = 1.0, to provide accurate expressions over the whole range of 0 < alpha < 1.

METHODS

The "data" were the long-accepted theory for alpha < 0.2 and the computational results from Lewellen and Lightfoot et al. on hard spheres (of 5 different alpha's) moving by convection and diffusion through a tight cylindrical pore, accounting for molecular exclusion, viscous forces, pressure drop, torque and rotation of spheres off the center line (averaging across all accessible radial positions), and the asymptotic values at alpha = 1.0. Coefficients for frictional hindrance to diffusion, F(alpha), and drag, G(alpha), and functions for sigma(alpha) and P(alpha), were represented by power law functions and the parameters optimized to give best fits to the combined "data."

RESULTS

The reflection coefficient sigma = {1 - [1 - (1 - phi)2]G'(alpha)} + 2alpha2 phi F'(alpha), and the relative permeability P/Pmax = phi F '(alpha)[1 + 9alpha5.5 x (1.0 - alpha5)0.02], where phi is the partition coefficient or volume fraction of the pore available to solute. The new expression for the diffusive hindrance is F'(alpha) = (1 - alpha2)(3/2) phi/[1 + 0.2 x alpha2 x (1 - alpha2)16], and for the drag factor is G'(alpha) = (1 - 2alpha(2)/3 - 0.20217 alpha5)/(1 - 0.75851 alpha5) - 0.0431[1 - (1 - alpha10)]. All of these converge monotonically to the correct limits at alpha = 1.

CONCLUSIONS

These are the first expressions providing hydrodynamically based estimates of sigma(alpha) and P(alpha) over 0 < alpha < 1 They should be accurate to within 1-2%.

Hemofiltration for cytokine-driven illnesses: the mediator delivery hypothesis

Hemofiltration is evolving as an adjunctive therapy for sepsis and other forms of systemic inflammation. Designed as a substitute for lost renal function, it is sometimes employed prior to the onset of renal failure to facilitate the nonspecific clearance of pro-inflammatory mediators. Prevailing theories suggest that hemofiltration attenuates the immune response when a threshold amount of excess cytokine is removed at the semi-permeable membrane. In this article we introduce an alternative hypothesis, in which hemofiltration exerts its effect by reinvigorating lymphatic flow and function. Crystalloid "replacement" solution, as much as 48 to 72 liters daily, is infused to restore intravascular volume lost through production of ultrafiltrate. Partial redistribution into interstitium and lymph mobilizes inflammatory mediators and other proteins, cellular byproducts, excessive ground matrix, fragments of apoptotic cells and free DNA. These substances are then metabolized, scavenged or cleared at multiple sites, including the reticuloendothelial system, liver, kidney, erythrocyte, and hemofilter.

Multiscale modeling of cardiac cellular energetics

Multiscale modeling is essential to integrating knowledge of human physiology starting from genomics, molecular biology, and the environment through the levels of cells, tissues, and organs all the way to integrated systems behavior. The lowest levels concern biophysical and biochemical events. The higher levels of organization in tissues, organs, and organism are complex, representing the dynamically varying behavior of billions of cells interacting together. Models integrating cellular events into tissue and organ behavior are forced to resort to simplifications to minimize computational complexity, thus reducing the model's ability to respond correctly to dynamic changes in external conditions. Adjustments at protein and gene regulatory levels shortchange the simplified higher-level representations. Our cell primitive is composed of a set of subcellular modules, each defining an intracellular function (action potential, tricarboxylic acid cycle, oxidative phosphorylation, glycolysis, calcium cycling, contraction, etc.), composing what we call the "eternal cell," which assumes that there is neither proteolysis nor protein synthesis. Within the modules are elements describing each particular component (i.e., enzymatic reactions of assorted types, transporters, ionic channels, binding sites, etc.). Cell subregions are stirred tanks, linked by diffusional or transporter-mediated exchange. The modeling uses ordinary differential equations rather than stochastic or partial differential equations. This basic model is regarded as a primitive upon which to build models encompassing gene regulation, signaling, and long-term adaptations in structure and function. During simulation, simpler forms of the model are used, when possible, to reduce computation. However, when this results in error, the more complex and detailed modules and elements need to be employed to improve model realism. The processes of error recognition and of mapping between different levels of model form complexity are challenging but are essential for successful modeling of large-scale systems in reasonable time. Currently there is to this end no established methodology from computational sciences.

Intracoronary administration of FGF-2: a computational model of myocardial deposition and retention

This study uses a computational model to characterize the myocardial deposition and retention of basic fibroblast growth factor (FGF-2) at the cellular level after intracoronary (IC) administration of exogenous FGF-2. The model is applied to the in situ conditions present within the myocardium of a dog for which the plasma pharmacokinetics resulting from IC injection of FGF-2 were recorded. Our estimates show that the processes involved in FGF-2 signaling are not diffusion limited; rather, the response time is determined by the reaction time of FGF-2 binding to cell surface receptors. Additionally, the processes of receptor secretion and internalization are found to play crucial roles in the FGF-2 dynamics; future experiments are required to quantify these processes. The model predictions obtained in this study suggest that IC administration of FGF-2 via either a single bolus or repetitive injections causes a transient increase (time scale of hours) in myocardial FGF-2 concentration if the endogenous level of free interstitial FGF-2 is low enough to allow permeation of FGF-2 molecules from the microvascular to the interstitial spaces. The model shows that the majority (64%) of the extracellular FGF-2 ligands are located within the interstitium, and similar fractions are found in the basement membrane and extracellular matrix. Among the FGF-2 molecules found within the interstitium, 2% are free and 98% are bound to interstitial heparan sulfate proteoglycans. These results support the theory of extracellular control of the bioavailability of FGF-2 via dynamic storage of FGF-2 within the basement membrane and extracellular matrix.

Myocardial density and composition: a basis for calculating intracellular metabolite concentrations

Systems for describing myocardial cellular metabolism with appropriate thermodynamic constraints on reactions have to be on the basis of estimates of intracellular and mitochondrial concentrations of metabolites as driving forces for reactions. This requires that tissue composition itself must be modeled, but there is marked inconsistency in the literature and no full data set on hearts of any species. To formulate a self-consistent set of information on the densities, contents, or concentrations of chemical components and volumes of tissue spaces, we drew on information mostly on rats. From the data on densities, volumes, volume fractions, and mass fractions observed mainly on left ventricular myocardium, cytoplasm, and mitochondria and from morphometric data on cellular components and the vasculature, we constructed a matrix based on conservation laws for density, volume, and constituent composition. The four constituents were water, protein, fat, and solutes (or ash). To take into account the variances in the observed data sets, we used a constrained nonlinear least squares optimization to minimize the differences between the final results and the data sets. The results provide a detailed estimate of cardiac tissue composition, previously unavailable, for the translation of whole tissue concentrations or concentrations per gram protein into estimated local concentrations that are relevant to reaction processes. An example is that the concentrations of phosphocreatine and ATP in cytosolic water space are twice as high as their mean tissue concentrations. This conservation optimization method is applicable to any tissue or organ.

Transient transcapillary exchange of water driven by osmotic forces in the heart

Osmotic transient responses in organ weight after changes in perfusate osmolarity have implied steric hindrance to small-molecule transcapillary exchange, but tracer methods do not. We obtained osmotic weight transient data in isolated, Ringer-perfused rabbit hearts with NaCl, urea, glucose, sucrose, raffinose, inulin, and albumin and analyzed the data with a new anatomically and physicochemically based model accounting for 1) transendothelial water flux, 2) two sizes of porous passages across the capillary wall, 3) axial intracapillary concentration gradients, and 4) water fluxes between myocytes and interstitium. During steady-state conditions approximately 28% of the transcapillary water flux going to form lymph was through the endothelial cell membranes [capillary hydraulic conductivity (Lp) = 1.8 +/- 0.6 x 10-8 cm. s-1. mmHg-1], presumably mainly through aquaporin channels. The interendothelial clefts (with Lp = 4.4 +/- 1.3 x 10-8 cm. s-1. mmHg-1) account for 67% of the water flux; clefts are so wide (equivalent pore radius was 7 +/- 0.2 nm, covering approximately 0.02% of the capillary surface area) that there is no apparent hindrance for molecules as large as raffinose. Infrequent large pores account for the remaining 5% of the flux. During osmotic transients due to 30 mM increases in concentrations of small solutes, the transendothelial water flux was in the opposite direction and almost 800 times as large and was entirely transendothelial because no solute gradient forms across the pores. During albumin transients, gradients persisted for long times because albumin does not permeate small pores; the water fluxes per milliosmolar osmolarity change were 200 times larger than steady-state water flux. The analysis completely reconciles data from osmotic transient, tracer dilution, and lymph sampling techniques.