A concurrent flow model for extraction during transcapillary passage

Review of tracer kinetic models in evaluation of gliomas using dynamic contrast-enhanced imaging

Glioma is the most common type of primary malignant tumor of the central nervous system (CNS), and is characterized by high malignancy, high recurrence rate and poor survival. Conventional imaging techniques only provide information regarding the anatomical location, morphological characteristics, and enhancement patterns. In contrast, advanced imaging techniques such as dynamic contrast-enhanced (DCE) MRI or DCE CT can reflect tissue microcirculation, including tumor vascular hyperplasia and vessel permeability. Although several studies have used DCE imaging to evaluate gliomas, the results of data analysis using conventional tracer kinetic models (TKMs) such as Tofts or extended-Tofts model (ETM) have been ambiguous. More advanced models such as Brix's conventional two-compartment model (Brix), tissue homogeneity model (TH) and distributed parameter (DP) model have been developed, but their application in clinical trials has been limited. This review attempts to appraise issues on glioma studies using conventional TKMs, such as Tofts or ETM model, highlight advancement of DCE imaging techniques and provides insights on the clinical value of glioma management using more advanced TKMs.

Myocardial Oedema as a Consequence of Viral Infection and Persistence-A Narrative Review with Focus on COVID-19 and Post COVID Sequelae

Microvascular integrity is a critical factor in myocardial fluid homeostasis. The subtle equilibrium between capillary filtration and lymphatic fluid removal is disturbed during pathological processes leading to inflammation, but also in hypoxia or due to alterations in vascular perfusion and coagulability. The degradation of the glycocalyx as the main component of the endothelial filtration barrier as well as pericyte disintegration results in the accumulation of interstitial and intracellular water. Moreover, lymphatic dysfunction evokes an increase in metabolic waste products, cytokines and inflammatory cells in the interstitial space contributing to myocardial oedema formation. This leads to myocardial stiffness and impaired contractility, eventually resulting in cardiomyocyte apoptosis, myocardial remodelling and fibrosis. The following article reviews pathophysiological inflammatory processes leading to myocardial oedema including myocarditis, ischaemia-reperfusion injury and viral infections with a special focus on the pathomechanisms evoked by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) infection. In addition, clinical implications including potential long-term effects due to viral persistence (long COVID), as well as treatment options, are discussed.

Beta Cell Imaging-From Pre-Clinical Validation to First in Man Testing

There are presently no reliable ways to quantify human pancreatic beta cell mass (BCM) in vivo, which prevents an accurate understanding of the progressive beta cell loss in diabetes or following islet transplantation. Furthermore, the lack of beta cell imaging hampers the evaluation of the impact of new drugs aiming to prevent beta cell loss or to restore BCM in diabetes. We presently discuss the potential value of BCM determination as a cornerstone for individualized therapies in diabetes, describe the presently available probes for human BCM evaluation, and discuss our approach for the discovery of novel beta cell biomarkers, based on the determination of specific splice variants present in human beta cells. This has already led to the identification of DPP6 and FXYD2ga as two promising targets for human BCM imaging, and is followed by a discussion of potential safety issues, the role for radiochemistry in the improvement of BCM imaging, and concludes with an overview of the different steps from pre-clinical validation to a first-in-man trial for novel tracers.

Noninvasive estimation of quantitative myocardial blood flow with Tc-99m MIBI by a compartment model analysis in rat

BACKGROUND

We aimed to investigate the use of dynamic cardiac planar images to estimate myocardial blood flow (MBF) by a compartment model analysis using time-to-peak (TP) map and compared it by the microsphere technique in rat. Positron emission tomography is considered the gold standard method, but is not available everywhere. By contrast, although myocardial perfusion imaging (MPI) with single-photon tracers is more widely available, it may be difficult to obtain adequate region of interest (ROI) settings. We proposed using the TP map to set the ROI, and hypothesized that this method could facilitate the measurement of absolute MBF by MPI in rat.

METHODS

Twenty-one normal rats were studied. Dynamic planar images with Tc-99m MIBI were obtained, and input function and cardiac ROIs were set using the obtained TP map. MBF was estimated by a one-compartment model analysis with the Renkin-Crone model and by the microsphere technique.

RESULTS

The MBFs from these two methods were significantly correlated. A negative proportional bias was observed, but no significant difference was observed between the mean MBFs calculated with each method.

CONCLUSIONS

MBF estimation by a compartment model analysis using TP map could facilitate absolute MBF measurement in rats.

Krogh’s capillary recruitment hypothesis, 100 years on: Is the opening of previously closed capillaries necessary to ensure muscle oxygenation during exercise?

In 1919, August Krogh published his seminal work on skeletal muscle oxygenation. Krogh’s observations indicated that muscle capillary diameter is actively regulated, rather than a passive result of arterial blood flow regulation. Indeed, combining a mathematical model with the number of ink-filled capillaries he observed in muscle cross sections taken at different workloads, Krogh was able to account for muscle tissue’s remarkably efficient oxygen extraction during exercise in terms of passive diffusion from nearby capillaries. Krogh was awarded the 1920 Nobel Prize for his account of muscle oxygenation. Today, his observations are engrained in the notion of capillary recruitment: the opening of previously closed capillaries. While the binary distinction between “closed” and “open” was key to Krogh’s model argument, he did in fact report a continuum of capillary diameters, degrees of erythrocyte deformation, and perfusion states. Indeed, modern observations question the presence of closed muscle capillaries. We therefore examined whether changes in capillary flow patterns and hematocrit among open capillaries can account for oxygen extraction in muscle across orders-of-magnitude changes in blood flow. Our four-compartment model of oxygen extraction in muscle confirms this notion and provides a framework for quantifying the impact of changes in microvascular function on muscle oxygenation in health and disease. Our results underscore the importance of capillary function for oxygen extraction in muscle tissue as first proposed by Krogh. While Krogh’s model calculations still hold, our model predictions support that capillary recruitment can be viewed in the context of continuous, rather than binary, erythrocyte distributions among capillaries.

NEW & NOTEWORTHY Oxygen extraction in working muscle is extremely efficient in view of single capillaries properties. The underlying mechanisms have been widely debated. Here, we develop a four-compartment model to quantify the influence of each of the hypothesized mechanisms on muscle oxygenation. Our results show that changes in capillary flow pattern and hematocrit can account for the high oxygen extraction observed in working muscle, while capillary recruitment is not required to account for these extraction properties.

Myocardial edema in paroxysmal permeability disorders: The paradigm of Clarkson's disease

PURPOSE

Paroxysmal Permeability Disorders (PPDs) comprise a variety of diseases characterized by recurrent and transitory increase of endothelial permeability. Idiopathic Systemic Capillary Leak Syndrome (ISCLS) is a rare PPD that leads to an abrupt massive shift of fluids and proteins from the intravascular to the interstitial compartment. In some cases, tissue edema may involve the myocardium, but its role in the development of shock has not been elucidated so far.

MATERIALS AND METHODS

Assessment of cardiac involvement during ten life-threatening ISCLS episodes admitted to ICU.

RESULTS

Transthoracic echocardiographic examination was performed in eight episodes, whereas a poor acoustic window prevented cardiac ultrasound assessment in two episodes. Myocardial edema was detected by echocardiography in eight episodes and marked pericardial effusion in one-episode. Cardiac magnetic resonance showed diffuse myocardial edema in another episode. In one case, myocardial edema caused fulminant left ventricular dysfunction, which required extracorporeal life support. The mean septum thickness was higher during the shock phase compared to the recovery phase [15.5 mm (13.1-21 mm) vs. 9.9 mm (9-11.3 mm), p = .0003]. Myocardial edema resolved within 72 h.

CONCLUSIONS

During early phases of ISCLS, myocardial edema commonly occurs and can induce transient myocardial dysfunction, potentially contributing to the pathogenesis of shock.

Novel morphometric analysis of higher order structure of human radial peri-papillary capillaries: relevance to retinal perfusion efficiency and age

We apply novel analyses to images of superficial capillaries that are located near and around the optic disc of the human retina: the radial peri-papillary capillaries (RPCs). Due to their unique perfusion of the nerve fibre layer the RPCs are particularly significant for optic-neuropathies. The inputs to the analysis were z-stacks from 3D confocal fluorescence microscopy from 62 human retinas aged 9 to 84 years. Our aim was to find morphometric correlates of age. The retinas had no ophthalmic history. The analysis was undertaken in two stages: (1) converting the z-stacks to 3D tubular networks of vessels, and (2) characterizing the tubular networks using features derived from the Minkowski functionals (MFs). The MFs measure: the capillary volume, surface area, mean breadth, and Euler number. The mean breadth is related to tortuosity, wall shear stress and resistance to flow, and the Euler number is related to the density of loops (collaterals). Features derived from the surface area, mean breadth and Euler number were most related to age (all p ≤ 0.006). The results indicate the importance of pressure-equalizing loops and tortuosity as quantitative measures related to perfusion efficiency. The novel morphometric analysis could quantify disease-related accelerated aging and vessel malformation.

Measurement of absolute myocardial blood flow in humans using dynamic cardiac SPECT and 99mTc-tetrofosmin: Method and validation

BACKGROUND

The objective of this study was to measure myocardial blood flow (MBF) in humans using 99mTc-tetrofosmin and dynamic single-photon emission computed tomography (SPECT).

METHODS

Dynamic SPECT using 99mTc-tetrofosmin and dynamic positron emission tomography (PET) was performed on a group of 16 patients. The SPECT data were reconstructed using a 4D-spatiotemporal iterative reconstruction method. The data corresponding to 9 patients were used to determine the flow-extraction curve for 99mTc-tefrofosmin while data from the remaining 7 patients were used for method validation. The nonlinear tracer correction parameters A and B for 99mTc-tefrofosmin were estimated for the 9 patients by fitting the flow-extraction curve [Formula: see text] for K 1 values estimated with 99mTc-tefrofosmin using SPECT and MBF values estimated with 13N-NH3 using PET. These parameters were then used to calculate MBF and coronary flow reserve (CFR) in three coronary territories (LAD, RCA, and LCX) using SPECT for an independent cohort of 7 patients. The results were then compared with that estimated with 13N-NH3 PET. The flow-dependent permeability surface-area product (PS) for 99mTc-tefrofosmin was also estimated.

RESULTS

The estimated flow-extraction parameters for 99mTc-tefrofosmin were found to be A = 0.91 ± 0.11, B = 0.34 ± 0.20 (R 2 = 0.49). The range of MBF in LAD, RCA, and LCX was 0.44-3.81 mL/min/g. The MBF between PET and SPECT in the group of independent cohort of 7 patients showed statistically significant correlation, r = 0.71 (P < .001). However, the corresponding CFR correlation was moderate r = 0.39 yet statistically significant (P = .037). The PS for 99mTc-tefrofosmin was (0.019 ± 0.10)*MBF + (0.32 ± 0.16).

CONCLUSIONS

Dynamic cardiac SPECT using 99mTc-tetrofosmin and a clinical two-headed SPECT/CT scanner can be a useful tool for estimation of MBF.

Krogh-cylinder and infinite-domain models for washout of an inert diffusible solute from tissue

OBJECTIVE

Models based on the Krogh-cylinder concept are developed to analyze the washout from tissue by blood flow of an inert diffusible solute that permeates blood vessel walls. During the late phase of washout, the outflowing solute concentration decays exponentially with time. This washout decay rate is predicted for a range of conditions.

METHODS

A single capillary is assumed to lie on the axis of a cylindrical tissue region. In the classic "Krogh-cylinder" approach, a no-flux boundary condition is applied on the outside of the cylinder. An alternative "infinite-domain" approach is proposed that allows for solute exchange across the boundary, but with zero net exchange. Both models are analyzed, using finite-element and analytical methods.

RESULTS

The washout decay rate depends on blood flow rate, tissue diffusivity and vessel permeability of solute, and assumed boundary conditions. At low blood flow rates, the washout rate can exceed the value for a single well-mixed compartment. The infinite-domain approach predicts slower washout decay rates than the Krogh-cylinder approach.

CONCLUSIONS

The infinite-domain approach overcomes a significant limitation of the Krogh-cylinder approach, while retaining its simplicity. It provides a basis for developing methods to deduce transport properties of inert solutes from observations of washout decay rates.

JSim, an open-source modeling system for data analysis

JSim is a simulation system for developing models, designing experiments, and evaluating hypotheses on physiological and pharmacological systems through the testing of model solutions against data. It is designed for interactive, iterative manipulation of the model code, handling of multiple data sets and parameter sets, and for making comparisons among different models running simultaneously or separately. Interactive use is supported by a large collection of graphical user interfaces for model writing and compilation diagnostics, defining input functions, model runs, selection of algorithms solving ordinary and partial differential equations, run-time multidimensional graphics, parameter optimization (8 methods), sensitivity analysis, and Monte Carlo simulation for defining confidence ranges. JSim uses Mathematical Modeling Language (MML) a declarative syntax specifying algebraic and differential equations. Imperative constructs written in other languages (MATLAB, FORTRAN, C++, etc.) are accessed through procedure calls. MML syntax is simple, basically defining the parameters and variables, then writing the equations in a straightforward, easily read and understood mathematical form. This makes JSim good for teaching modeling as well as for model analysis for research.   For high throughput applications, JSim can be run as a batch job.  JSim can automatically translate models from the repositories for Systems Biology Markup Language (SBML) and CellML models. Stochastic modeling is supported. MML supports assigning physical units to constants and variables and automates checking dimensional balance as the first step in verification testing. Automatic unit scaling follows, e.g. seconds to minutes, if needed. The JSim Project File sets a standard for reproducible modeling analysis: it includes in one file everything for analyzing a set of experiments: the data, the models, the data fitting, and evaluation of parameter confidence ranges. JSim is open source; it and about 400 human readable open source physiological/biophysical models are available at http://www.physiome.org/jsim/.

Modeling to link regional myocardial work, metabolism and blood flows

Given the mono-functional, highly coordinated processes of cardiac excitation and contraction, the observations that regional myocardial blood flows, rMBF, are broadly heterogeneous has provoked much attention, but a clear explanation has not emerged. In isolated and in vivo heart studies the total coronary flow is found to be proportional to the rate-pressure product (systolic mean blood pressure times heart rate), a measure of external cardiac work. The same relationship might be expected on a local basis: more work requires more flow. The validity of this expectation has never been demonstrated experimentally. In this article we review the concepts linking cellular excitation and contractile work to cellular energetics and ATP demand, substrate utilization, oxygen demand, vasoregulation, and local blood flow. Mathematical models of these processes are now rather well developed. We propose that the construction of an integrated model encompassing the biophysics, biochemistry and physiology of cardiomyocyte contraction, then combined with a detailed three-dimensional structuring of the fiber bundle and sheet arrangements of the heart as a whole will frame an hypothesis that can be quantitatively evaluated to settle the prime issue: Does local work drive local flow in a predictable fashion that explains the heterogeneity? While in one sense one can feel content that work drives flow is irrefutable, the are no cardiac contractile models that demonstrate the required heterogeneity in local strain-stress-work; quite the contrary, cardiac contraction models have tended toward trying to show that work should be uniform. The object of this review is to argue that uniformity of work does not occur, and is impossible in any case, and that further experimentation and analysis are necessary to test the hypothesis.

Comprehensive mechanism-based antibody pharmacokinetic modeling

Pharmacokinetic models of antibody distribution and dynamics are useful for predicting and optimizing therapeutic behavior. Targeted antigens are produced and distributed in various tissues in specific patterns in disease phenotypes. Existing models leave out significant mechanistic detail which would enable an understanding of how to modify therapeutics in an optimal manner to allow appropriate tissue penetration in either a healthy or diseased state. The model presented here incorporates additional complexity such as diffusion through endothelial barriers, differential transcytosis properties, FcRn-mediated recycling, and incorporates these properties in an organ-specific manner. This creates a platform which can be expanded upon to include understanding of the effect of target on therapeutic distribution and clearance, differences in dynamics during a diseased versus healthy state, differential dose strategies, and mechanistic translation between animal models and human disease state. This model represents a superior alternative to typical and potentially over-simplified scaling strategies utilized in most existing physiologically-based pharmacokinetic models. Ultimately, this will enable better therapeutic design and greater pharmacological effects.

Compartmental modeling in the analysis of biological systems

Compartmental models are composed of sets of interconnected mixing chambers or stirred tanks. Each component of the system is considered to be homogeneous, instantly mixed, with uniform concentration. The state variables are concentrations or molar amounts of chemical species. Chemical reactions, transmembrane transport, and binding processes, determined in reality by electrochemical driving forces and constrained by thermodynamic laws, are generally treated using first-order rate equations. This fundamental simplicity makes them easy to compute since ordinary differential equations (ODEs) are readily solved numerically and often analytically. While compartmental systems have a reputation for being merely descriptive they can be developed to levels providing realistic mechanistic features through refining the kinetics. Generally, one is considering multi-compartmental systems for realistic modeling. Compartments can be used as "black" box operators without explicit internal structure, but in pharmacokinetics compartments are considered as homogeneous pools of particular solutes, with inputs and outputs defined as flows or solute fluxes, and transformations expressed as rate equations.Descriptive models providing no explanation of mechanism are nevertheless useful in modeling of many systems. In pharmacokinetics (PK), compartmental models are in widespread use for describing the concentration-time curves of a drug concentration following administration. This gives a description of how long it remains available in the body, and is a guide to defining dosage regimens, method of delivery, and expectations for its effects. Pharmacodynamics (PD) requires more depth since it focuses on the physiological response to the drug or toxin, and therefore stimulates a demand to understand how the drug works on the biological system; having to understand drug response mechanisms then folds back on the delivery mechanism (the PK part) since PK and PD are going on simultaneously (PKPD).Many systems have been developed over the years to aid in modeling PKPD systems. Almost all have solved only ODEs, while allowing considerable conceptual complexity in the descriptions of chemical transformations, methods of solving the equations, displaying results, and analyzing systems behavior. Systems for compartmental analysis include Simulation and Applied Mathematics, CoPasi (enzymatic reactions), Berkeley Madonna (physiological systems), XPPaut (dynamical system behavioral analysis), and a good many others. JSim, a system allowing the use of both ODEs and partial differential equations (that describe spatial distributions), is used here. It is an open source system, meaning that it is available for free and can be modified by users. It offers a set of features unique in breadth of capability that make model verification surer and easier, and produces models that can be shared on all standard computer platforms.

Tracer kinetic modelling in MRI: estimating perfusion and capillary permeability

The tracer-kinetic models developed in the early 1990s for dynamic contrast-enhanced MRI (DCE-MRI) have since become a standard in numerous applications. At the same time, the development of MRI hardware has led to increases in image quality and temporal resolution that reveal the limitations of the early models. This in turn has stimulated an interest in the development and application of a second generation of modelling approaches. They are designed to overcome these limitations and produce additional and more accurate information on tissue status. In particular, models of the second generation enable separate estimates of perfusion and capillary permeability rather than a single parameter K(trans) that represents a combination of the two. A variety of such models has been proposed in the literature, and development in the field has been constrained by a lack of transparency regarding terminology, notations and physiological assumptions. In this review, we provide an overview of these models in a manner that is both physically intuitive and mathematically rigourous. All are derived from common first principles, using concepts and notations from general tracer-kinetic theory. Explicit links to their historical origins are included to allow for a transfer of experience obtained in other fields (PET, SPECT, CT). A classification is presented that reveals the links between all models, and with the models of the first generation. Detailed formulae for all solutions are provided to facilitate implementation. Our aim is to encourage the application of these tools to DCE-MRI by offering researchers a clearer understanding of their assumptions and requirements.

Quantitative imaging of coronary blood flow

Positron emission tomography (PET) is a nuclear medicine imaging modality based on the administration of a positron-emitting radiotracer, the imaging of the distribution and kinetics of the tracer, and the interpretation of the physiological events and their meaning with respect to health and disease. PET imaging was introduced in the 1970s and numerous advances in radiotracers and detection systems have enabled this modality to address a wide variety of clinical tasks, such as the detection of cancer, staging of Alzheimer's disease, and assessment of coronary artery disease (CAD). This review provides a description of the logic and the logistics of the processes required for PET imaging and a discussion of its use in guiding the treatment of CAD. Finally, we outline prospects and limitations of nanoparticles as agents for PET imaging.

GENTEX, a general multiscale model for in vivo tissue exchanges and intraorgan metabolism

Endothelial cells lining myocardial capillaries not only impede transport of blood solutes to the contractile cells, but also take up and release substrates, competing with myocytes. Solutes permeating this barrier exhibit concentration gradients along the capillary. This paper introduces a generic model, GENTEX, to characterize blood-tissue exchanges. GENTEX is a whole organ model of the vascular network providing intraorgan flow heterogeneity and accounts for substrate transmembrane transport, binding and metabolism in erythrocytes, plasma, endothelial cells, interstitial space and cardiomyocytes. The model is tested here for the analysis of multiple tracer indicator dilution data on purine nucleoside metabolism in the isolated Krebs-Henseleit-perfused non-working hearts. It has been also used for analysing NMR contrast data for regional myocardial flows and for positron emission tomographic studies of cardiac receptor kinetics. The facilitating transporters, binding sites and enzymatic reactions are nonlinear elements and allow competition between substrates and a reaction sequence of up to five substrate-product reactions in a metabolic network. Strategies for application start with experiment designs incorporating inert reference tracers. For the estimation of endothelial and sarcolemmal permeability-surface area products and metabolism of the substrates and products, model solutions were optimized to fit the data from pairs of tracer injections (of either inosine or adenosine, plus the reference tracers) injected under the same circumstances a few minutes later. The results provide a self-consistent description of nucleoside metabolism in a beating well-perfused rabbit heart, and illustrate the power of the model to fit multiple datasets simultaneously.

Simultaneous blood-tissue exchange of oxygen, carbon dioxide, bicarbonate, and hydrogen ion

A detailed nonlinear four-region (red blood cell, plasma, interstitial fluid, and parenchymal cell) axially distributed convection-diffusion-permeation-reaction-binding computational model is developed to study the simultaneous transport and exchange of oxygen (O2) and carbon dioxide (CO2) in the blood-tissue exchange system of the heart. Since the pH variation in blood and tissue influences the transport and exchange of O2 and CO2 (Bohr and Haldane effects), and since most CO2 is transported as HCO3(-) (bicarbonate) via the CO2 hydration (buffering) reaction, the transport and exchange of HCO3(-) and H+ are also simulated along with that of O2 and CO2. Furthermore, the model accounts for the competitive nonlinear binding of O2 and CO2 with the hemoglobin inside the red blood cells (nonlinear O2-CO2 interactions, Bohr and Haldane effects), and myoglobin-facilitated transport of O2 inside the parenchymal cells. The consumption of O2 through cytochrome-c oxidase reaction inside the parenchymal cells is based on Michaelis-Menten kinetics. The corresponding production of CO2 is determined by respiratory quotient (RQ), depending on the relative consumption of carbohydrate, protein, and fat. The model gives a physiologically realistic description of O2 transport and metabolism in the microcirculation of the heart. Furthermore, because model solutions for tracer transients and steady states can be computed highly efficiently, this model may be the preferred vehicle for routine data analysis where repetitive solutions and parameter optimization are required, as is the case in PET imaging for estimating myocardial O2 consumption.

Strategies and Tactics in Multiscale Modeling of Cell-to-Organ Systems

Modeling is essential to integrating knowledge of human physiology. Comprehensive self-consistent descriptions expressed in quantitative mathematical form define working hypotheses in testable and reproducible form, and though such models are always "wrong" in the sense of being incomplete or partly incorrect, they provide a means of understanding a system and improving that understanding. Physiological systems, and models of them, encompass different levels of complexity. The lowest levels concern gene signaling and the regulation of transcription and translation, then biophysical and biochemical events at the protein level, and extend through the levels of cells, tissues and organs all the way to descriptions of integrated systems behavior. The highest levels of organization represent the dynamically varying interactions of billions of cells. Models of such systems are necessarily simplified to minimize computation and to emphasize the key factors defining system behavior; different model forms are thus often used to represent a system in different ways. Each simplification of lower level complicated function reduces the range of accurate operability at the higher level model, reducing robustness, the ability to respond correctly to dynamic changes in conditions. When conditions change so that the complexity reduction has resulted in the solution departing from the range of validity, detecting the deviation is critical, and requires special methods to enforce adapting the model formulation to alternative reduced-form modules or decomposing the reduced-form aggregates to the more detailed lower level modules to maintain appropriate behavior. The processes of error recognition, and of mapping between different levels of model complexity and shifting the levels of complexity of models in response to changing conditions, are essential for adaptive modeling and computer simulation of large-scale systems in reasonable time.

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.

Four-phase single-capillary stepwise model for kinetics in arterial spin labeling MRI

An extended model for extracting measures of brain perfusion from pulsed arterial spin labeling (ASL) data while considering transit effects and restricted permeability of capillaries to blood water is proposed. We divided the time course of the signal difference between control and labeled images into four phases with respect to the arrival time of labeled blood water at the voxel of interest (t(A)), transit time through the arteries in the voxel (t(ex)), and duration of the bolus of labeled spins (tau). Dividing the labeled slab of blood water into many discrete segments, and adapting numerical integration methods allowed us to conveniently model restricted capillary-tissue exchange based on a modified distributed parameter model. We compared this four-phase single-capillary stepwise (FPSCS) model with models that treat water as a freely diffusible tracer, using both simulations and experimental ASL brain imaging data at 1.5T from eight healthy subjects (24-80 years old). The FPSCS model yielded less errors in the least-squares sense in fitting brain ASL data in comparison with freely diffusible tracer models of water (P = 0.055). These results imply that restricted permeability of capillaries to water should be considered when brain ASL data are analyzed.

Microcirculation and microvasculature in breast tumors: pharmacokinetic analysis of dynamic MR image series

The purpose of this study was to quantify microcirculation and microvasculature in breast lesions by pharmacokinetic analysis of Gd-DTPA-enhanced MRI series. Strongly T1-weighted MR images were acquired in 18 patients with breast lesions using a saturation-recovery-TurboFLASH sequence. Concentration-time courses were determined for blood, pectoral muscle, and breast masses and subsequently analyzed by a two-compartment model to estimate plasma flow and the capillary transfer coefficient per unit of plasma volume (F/VP, KPS/VP) as well as fractional volumes of the plasma and interstitial space (fP, fI). Tissue parameters determined for pectoral muscle (fP = 0.04 +/- 0.01, fI = 0.09 +/- 0.01, F/VP = 2.4 +/- 1.3 min(-1), and KPS/VP = 1.2 +/- 0.5 min(-1)) and 10 histologically proven carcinomas (fP = 0.20 +/- 0.07, fI = 0.34 +/- 0.16, F/VP = 2.4 +/- 0.7 min(-1), and KPS/VP = 0.86 +/- 0.62 min(-1)) agreed reasonable well with literature data. Best separation between malignant and benign lesions was obtained by the ratio KPS/F (0.35 +/- 0.17 vs. 1.23 +/- 0.65). The functional imaging technique presented appears promising to quantitatively characterize tumor pathophysiology. Its impact on diagnosis and therapy management of breast tumors, however, has to be evaluated in larger patient studies.

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

Physiologists have devised many models for interpreting water and solute exchange data in whole organs, but the models have typically neglected key aspects of the underlying physiology to present the simplest possible model for a given experimental situation. We have developed a physiologically realistic model of microcirculatory water and solute exchange and applied it to diverse observations of water and solute exchange in the heart. Model simulations are consistent with the results of osmotic weight transient, tracer indicator dilution, and steady-state lymph sampling experiments. The key model features that permit this unification are the use of an axially distributed blood-tissue exchange region, inclusion of a lymphatic drain in the interstitium, and the independent computation of transcapillary solute and solvent fluxes through three different pathways.

Myocardial fluid balance

Fluid accumulation in the cardiac interstitium or myocardial edema is a common manifestation of many clinical states. Specifically, cardiac surgery includes various interventions and pathophysiological conditions that cause or worsen myocardial edema including cardiopulmonary bypass and cardioplegic arrest. Myocardial edema should be a concern for clinicians as it has been demonstrated to produce cardiac dysfunction. This article will briefly discuss the factors governing myocardial fluid balance and review the evidence of myocardial edema in various pathological conditions. In particular, myocardial microvascular, interstitial, and lymphatic interactions relevant to the field of cardiac surgery will be emphasized.

Advection and diffusion of substances in biological tissues with complex vascular networks

For highly diffusive solutes the kinetics of blood-tissue exchange is only poorly represented by a model consisting of sets of independent parallel capillary-tissue units. We constructed a more realistic multicapillary network model conforming statistically to morphometric data. Flows through the tortuous paths in the network were calculated based on constant resistance per unit length throughout the network and the resulting advective intracapillary velocity field was used as a framework for describing the extravascular diffusion of a substance for which there is no barrier or permeability limitation. Simulated impulse responses from the system, analogous to tracer water outflow dilution curves, showed flow-limited behavior over a range of flows from about 2 to 5 ml min(-1) g(-1), as is observed for water in the heart in vivo. The present model serves as a reference standard against which to evaluate computationally simpler, less physically realistic models. The simulated outflow curves from the network model, like experimental water curves, were matched to outflow curves from the commonly used axially distributed models only by setting the capillary wall permeability-surface area (PS) to a value so artifactually low that it is incompatible with the experimental observations that transport is flow limited. However, simple axially distributed models with appropriately high PSs will fit water outflow dilution curves if axial diffusion coefficients are set at high enough values to account for enhanced dispersion due to the complex geometry of the capillary network. Without incorporating this enhanced dispersion, when applied to experimental curves over a range of flows, the simpler models give a false inference that there is recruitment of capillary surface area with increasing flow. Thus distributed models must account for diffusional as well as permeation processes to provide physiologically appropriate parameter estimates.

A network thermodynamic model of kidney perfusion with a cryoprotective agent

A network thermodynamic model has been devised to describe the coupled movement of water and a permeable additive within a kidney during perfusion under the combined action of diffusive, hydrodynamic, and mechanical processes. The model has been validated by simulating perfusions with Me2SO, glycerol, and sucrose and comparing predicted weight and vascular resistance with experimental results obtained by Pegg (1993). The flows of CPA, water, colloid, and cellular impermeants are governed by a combination of the individual osmotic potential and pressure differences between compartments of the kidney, the viscoelastic behavior of the tissue, and the momentum transferred between the flows. The model developed in this study presents an analytical tool for understanding the dynamics of the perfused kidney system and for modifying perfusion protocols to minimize the changes in cell volume, internal pressure build-up, and increases in vascular resistance that currently present barriers to the successful perfusion of organs.

Pulmonary inflammation alters the lung disposition of lipophilic amine indicators

Many lipophilic amine compounds are rapidly extracted from the blood on passage through the pulmonary circulation. The extent of their extraction in normal lungs depends on their physical-chemical properties, which affect their degree of ionization, lipophilicity, and propensity for interacting with blood and tissue constituents. The hypothesis of the present study was that changes in the tissue composition that occur during pulmonary inflammation would have a differential effect on the pulmonary extraction of lipophilic amines having different properties. If so, measurement of the extraction patterns for a group of lipophilic amines, having different physical-chemical properties, might provide a means for detecting and identifying lung tissue abnormalities. To evaluate this hypothesis, we measured the pulmonary extraction patterns for four lipophilic amines, [(14)C]diazepam, [(3)H]alfentanil, [(14)C]lidocaine, and [(14)C]codeine, along with two hydrophilic compounds, (3)HOH and [(14)C]phenylethylamine, after the bolus injection of these indicators into the pulmonary artery of isolated lungs from normal rabbits and from rabbits with pulmonary inflammation induced by an intravenous injection of complete Freund's adjuvant. The pulmonary extraction patterns, parameterized using a previously developed mathematical model, were, in fact, differentially altered by the inflammatory response. For example, the tissue sequestration rate, k(seq) (ml/s), per unit (3)HOH accessible extravascular lung water volume significantly increased for diazepam and lidocaine, but not for codeine and alfentanil. The results are consistent with the above hypothesis and suggest the potential for using lipophilic amines as indicators for detection and quantification of changes in lung tissue composition associated with lung injury and disease.

A model for the regulation of cerebral oxygen delivery

On the basis of the assumption that oxygen delivery across the endothelium is proportional to capillary plasma PO2, a model is presented that links cerebral metabolic rate of oxygen utilization (CMRO2) to cerebral blood flow (CBF) through an effective diffusivity for oxygen (D) of the capillary bed. On the basis of in vivo evidence that the oxygen diffusivity properties of the capillary bed may be altered by changes in capillary PO2, hematocrit, and/or blood volume, the model allows changes in D with changes in CBF. Choice in the model of the appropriate ratio of Omega identical with (DeltaD/D)/(DeltaCBF/CBF) determines the dependence of tissue oxygen delivery on perfusion. Buxton and Frank (J. Cereb. Blood Flow. Metab. 17: 64-72, 1997) recently presented a limiting case of the present model in which Omega = 0. In contrast to the trends predicted by the model of Buxton and Frank, in the current model when Omega > 0, the proportionality between changes in CBF and CMRO2 becomes more linear, and similar degrees of proportionality can exist at different basal values of oxygen extraction fraction. The model is able to fit the observed proportionalities between CBF and CMRO2 for a large range of physiological data. Although the model does not validate any particular observed proportionality between CBF and CMRO2, generally values of (DeltaCMRO2/CMRO2)/(DeltaCBF/CBF) close to unity have been observed across ranges of graded anesthesia in rats and humans and for particular functional activations in humans. The model's capacity to fit the wide range of data indicates that the oxygen diffusivity properties of the capillary bed, which can be modified in relation to perfusion, play an important role in regulating cerebral oxygen delivery in vivo.

Application of the dispersion model for description of the outflow dilution profiles of noneliminated reference indicators in rat liver perfusion studies

The dispersion model (DM) is a stochastic model describing the distribution of blood-borne substances within organ vascular beds. It is based on assumptions of concurrent convective and random-walk (pseudodiffusive) movements in the direction of flow, and is characterized by the mean transit time (t) and the dispersion number (inverse Peclet number), DN. The model is used with either closed (reflective) boundary conditions at the inflow and the outflow point (Danckwerts conditions) or a closed condition at the inflow and an open (transparent) condition at the outflow (mixed conditions). The appropriateness of DM was assessed with outflow data from single-pass perfused rat liver multiple indicator dilution (MID) experiments, with varying lengths of the inflow and outflow catheters. The studies were performed by injection, of bolus doses of 51Crlabeled red blood cells (vascular indicator), 125I-labeled albumin and [14C] sucrose (interstitual indicators), and [3H]2O (whole tissue indicator) into the portal vein at a perfusion rate of 12 ml/ min. The outflow profiles based on the DM were convolved with the transport function of the catheters, then fitted to the data. A fairly good fit was obtained for most of the MID curve, with the exception of the late-in-time data (prolonged tail) beyond 3 x [symbol: see text]. The fitted DNS were found to differ among the indicators, and not with the length of the inflow and outflow catheters. But the differences disappeared when a delay parameter, t0 = 4.1 +/- 0.7 sec (x +/- SD), was included as an additional fitted parameter for all of the indicators except water. Using the short catheters, the average DN for the model with delay was 0.31 +/- 0.13 for closed and 0.22 +/- 0.07 for mixed boundary conditions, for all reference indicators. Mean transit times and the variances of the fitted distributions were always smaller than the experimental ones (on average, by 6.8 +/- 3.7% and 58 +/- 19%, respectively). In conclusion, the DM is a reasonable descriptor of dispersion for the early-in-time data and not the late-in-time data. The existence of a common DN for all noneliminated reference indicators suggests that intrahepatic dispersion depends only on the geometry of the vasculature rather than the diffusional processes. The role of the nonsinusoidal ("large") vessels can be partly represented by a simple delay.

A physiologically based pharmacokinetic model incorporating dispersion principles to describe solute distribution in the perfused rat hindlimb preparation

A physiologically based pharmacokinetic model incorporating dispersion principles has been developed to describe outflow data from the isolated perfused rat hindlimb preparation, for the three reference markers 14C-sucrose, 14C-urea, and 3H-water and three 14C-labeled 5-n-alkyl-5-ethyl barbiturates; the methyl, butyl, and nonyl homologues. Also 51Cr-RBC and 125I-albumin were studied. The model consists of four parallel components representing each of the tissues comprising the hindlimb: skeletal muscle, skin, bone, and adipose. Attempts to simplify the model by using the principle of tissue lumping were made by examining the tissue equilibration rate constant k tau for each of respective tissues for each compound. It was found that simplification was only possible in the case of 3H-water data. The model took into account a possible shunting component in the skin tissue and incomplete mass but not volumetric recovery from the system. The dispersion model characterizes the relative spreading of solute on transit through a tissue bed by a dimension-less parameter DN. The estimated dispersion numbers (DN) obtained were in the region of 2.7-4.72, 8.39-15.54, 0.61-2.74, and 6.02-14.0 for skeletal muscle, skin, bone, and adipose, respectively, and were independent of the compound studied. These values are much larger than the range reported in the literature for hepatic outflow data, DN = 0.2-0.5, and suggest a greater heterogeneity of vascular flow in the different component tissues of the rat hindlimb.

Potassium and thallium uptake in dog myocardium

We sought to ascertain the rates and mechanisms of uptake of markers for regional myocardial blood flows.

METHODS

The rates of exchange of potassium and thallium across capillary walls and cell membranes in isolated blood-perfused dog hearts were estimated from multiple indicator dilution curves recorded for 131I-albumin, 42K and 201Tl from the coronary sinus outflow following injection into arterial inflow. Analysis involved fitting the observed dilution curves with a model composed of a capillary-interstitial fluid-cell exchange region and nonexchanging larger vessels.

RESULTS

Capillary permeability surface products (PSc) for potassium and thallium were similar, 0.82 +/- 0.33 (mean +/- s.d., n = 19) and 0.87 +/- 0.32 ml min-1 g-1 (n = 24) with a ratio for simultaneous pairs of 1.02 +/- 0.27 (n = 19). For the myocardial cells, PSpc averaged 3.7 +/- 3.1 ml min-1 g-1 (n = 19) for K+ and 9.5 +/- 3.9 (n = 24) for Tl+; the ratio of potassium to thallium averaged 0.40 +/- 0.19 (n = 18), thereby omitting a single high value for potassium. This high cellular influx for thallium is interpreted as due to its passage through ionic channels for both Na+ and K+.

CONCLUSION

The high permeabilities and large volumes of distribution make thallium and potassium among the best ionic deposition markers for regional flow. Their utility for this purpose is compromised by significant capillary barrier limitation retarding uptake; so regional flow is underestimated modestly in high-flow regions particularly.

Effect of glucose on uptake of radiolabeled glucose, 2-DG, and 3-O-MG by the perfused rat liver

In the transition from the fasting to the fed state, plasma glucose levels rise, and the liver converts from an organ producing glucose to one of storage. To determine the effect of glucose on hepatic glucose uptake, radiolabeled glucose, 2-deoxyglucose, and 3-O-methylglucose were injected into perfused rat livers during different nontracer glucose levels, and the concentrations in the outflow were measured. A mathematical model was developed that described the behavior of the injected compounds as they traveled through the liver and was used to simulate and fit the experimental results. The rates of membrane transport, glucokinase, glucose-6-phosphatase, and the consumption of glucose 6-phosphate were estimated. Membrane transport for all of the tracers decreased as nontracer glucose increased, demonstrating competitive inhibition of the glucose transporter. In contrast, the consumption of injected [2-14C]glucose increased when glucose was elevated, demonstrating that glucose caused an activation of enzyme activity that overcame the competitive inhibition of transport and phosphorylation. When glucose was elevated, the rate coefficient of glucokinase did not decrease, indicating that glucokinase was stimulated by glucose. Both changes would lead to the increased glycogen synthesis and decreased glucose production rate observed in vivo during the fasted-to-fed transition.

Graphical lung analysis and simulation environment

The study of transport across the pulmonary vasculature is an important aspect of the study of the lung. Models have often been used in conjunction with experimental work to further the information which can be obtained from experimental work alone. GLANSE was developed as an environment to carry out such analysis on microcomputers. The main model employed is a three region, homogeneous model which includes provisions for tracer diffusion in the extravascular region, hydrophilic and lipophyilic tracers as well as physiological parameters such as blood flow. Several heterogeneous models based on simplified versions of the three region model as well as two models which are not related to the three region model are also included. Computationally efficient routines for model simulations are used so as to enable their execution on microcomputers with large data sets. In addition, several methods for models analysis, such as parameter sensitivity and curve-fitting, as well as statistical analysis of results are also included. GLANSE has been tested and has been in use for several years for routine analysis of experimental data.

Modeling membrane transport

Many substrates cross cell membranes by processes other than passive diffusion. When the transport is carrier-mediated, e.g., facilitated diffusion, active transport, and exchange diffusion, the carrier modifies the conductance of the membrane and may either increase or decrease the flux of the substrate across the membrane. A common characteristic of all carrier-mediated transport is its saturability, as only a finite amount of carrier is available to bind with the substrate; even the simplest one-site carrier model exhibits saturation. Inclusion of carrier-mediated transport adds additional model parameters that describe the transporter. In addition, the model must account for both labeled (tracer) and unlabeled (mother) substrate, but this introduces no new parameters. There are many possible models for a membrane carrier. The applicability of these models must be examined for the specific substrate of interest. Many experiments aimed at measuring carrier parameters are carried out on isolated cells or cell fragments. Experiments in intact organs (either in vivo and in vitro) are also possible. Of particular note is the "bolus sweep" method described by Rickaby et al. (1981) and Malcorps et al. (1984). The increasing sophistication of experimental procedures, data collection techniques, and computers available to investigators continues to extend the depth to which we can probe biological systems. With this increased sophistication comes increased costs in time and equipment. It behooves us then to extract the maximum amount of information from each experimental procedure. Mathematical models assist in doing so, and sophistication in model analysis should parallel that in other phases of the experiment. Increased realism brings several advantages. Simplification of a model to increase its ease of usage and speed in routine data analysis is a desirable goal, and comparing a simplified model against a more realistic model under the conditions specific to a given experiment is one way to test the simplifying assumptions. Additionally, increased model realism can bring new insight into unknown aspects of the system. All models, no matter how realistic, are always "wrong" in that they are less complex than the real system. Failure of the model to explain observed results forces us to further refine the model and teaches us something more about the system.

A theoretical approach to the estimation of tissue flows using tritiated water as indicator

An alternative method for the determination of flow rates to individual organs or tissues that collectively comprises experimental isolated preparations is described. The isolated hindlimb constitutes a good example of such an experimental preparation. After injecting a bolus dose of tritiated water to the arterial side, the amount of marker (tritiated water) remaining in different tissues at various time is determined. From these experimental data, the regional flow rate to individual tissues can be estimated, applying either a physiological model approach or statistical moment theory.

Incorporation of first-order uptake rate constants from simple mammillary models into blood-flow limited physiological pharmacokinetic models via extraction efficiencies

Incorporation of First-Order Uptake Rate Constants from Simple Mammillary Models into Blood-Flow Limited Physiological Pharmacokinetic Models via Extraction Efficiencies. W. L. Roth, L. W. D. Weber, and K. Rozman (1995). Pharm. Res. 263-269. First-order rate constants obtained from classical pharmacokinetic models correspond to mammillary systems in which all of the blood (or plasma) is assumed to be located in a central compartment. In such models the rate at which chemicals are transported out of this pool and into another compartment is the product of the mass of chemical in the central compartment multiplied by a rate constant, which is not limited in magnitude by the blood flow, or the rate at which chemicals from the blood are delivered to the peripheral compartment. Most of the physiologically-based models published to date dispense with some of the information available from mammillary models by assuming that all of the chemical delivered by the flow of blood rapidly equilibrates and can be taken up by the tissue under the control of a "partition coefficient" (Rij = Cj/Ci). We show that the partition coefficient alone does not retain the uptake rate (kji) information available from a classical mammillary model, but that the uptake rate information can be incorporated via unitless extraction efficiency parameters, epsilon j.

Magnetic resonance imaging verification of a multi-compartment perfusion model for a chromatography gel phantom

A model of MRI signal intensity was developed for chromatography column phantoms containing Sephadex gels, which were used to simulate tissue perfusion and the exchange of protons between extravascular and intravascular tissue compartments. Computer simulations of two-compartment and three-compartment models were compared in experiments that used two chromatography columns. One column contained a Sephadex G-25 gel that had a smaller pore size and a slower rate of exchange between extrabead and intrabead compartments than did a second column, which contained a G-50 gel with a larger pore size. In both columns the beads were of approximately the same size to simulate the same ratio of intrabead volume to extrabead volume. Slice-selective spin-echo experiments were performed with the columns oriented parallel to each other in a 1.5 T imager and parallel to the magnetic field, with water flowing through each column perpendicular to the slice at mean flow velocities ranging between 0 and 0.66 mm/s. The results of the experiments agreed with computer simulations, which showed that the MRI signal intensity in the perfused columns is a function of the rate of exchange between extrabead and intrabead compartments. The exchange process modifies the transit time of protons passing through an excited region. The simulations also showed that both two-compartment and three-compartment models could be used to fit the experimental data. Experiments with chromatography gel columns are a preliminary investigation into the potential of using MRI to measure the exchange between extravascular and intravascular compartments of endogenous protons during blood perfusion of biological tissue.

Effects of blood flow on [11C]raclopride binding in the brain: model simulations and kinetic analysis of PET data

To assess the stability of different measures of receptor occupancy from [11C]raclopride (a D2 antagonist) studies with positron emission tomography, we analyze data from five test/retest studies in normal volunteers in terms of individual model parameters from a three-compartment model, the distribution volume (DV) and the ratio of DVs from a receptor-containing region of interest to a non-receptor-containing region. Large variations were found in the individual model parameters, limiting their usefulness as an indicator of change in receptor systems. The DV ratio showed the smallest variation. Individual differences were reflected in the greater intersubject variation in DV than intrasubject variation. The potential effects of blood flow on these measurements were addressed both experimentally and by simulation studies using three models that explicitly incorporate blood flow into a compartmental model that also includes receptor-ligand binding. None of the models showed any variation in the DV with changes in blood flow as long as flow was held constant during the simulation. Experimentally, blood flow was significantly reduced by hyperventilation in a human subject. The DV was found to be reduced relative to baseline in the hyperventilation study, but the DV ratio remained unchanged. The effect of elevated and reduced flow was also tested in two baboon experiments in which PCO2 was varied. Some variability in the DV ratio was observed but was not correlated with changes in blood flow. This raises the possibility that other factors indirectly related to changes in blood flow (or PCO2) may cause changes in DV, and these effects need to be considered when evaluating experimental results.

Changes in myocardial capillary diffusion capacity during infusion of vasoactive drugs

The present study investigated how variations in coronary vascular resistance and metabolic demand affected myocardial capillary diffusion capacity. Hearts from Wistar rats were perfused with Krebs-Henseleit-albumin buffer in a Langendorff preparation, where heart rate (HR), contractility (dP/dtmax) and myocardial oxygen consumption (MVO2) were recorded continuously. Myocardial capillary diffusion capacity was measured as the permeability surface area product (PS) for Cr-EDTA and vitamin B12 by the single injection colorimetric indicator dilution method. After base-line recordings without drugs, angiotensin II+Arginine-vasopressin was infused, which increased coronary vascular resistance by 90%, stimulated HR by 11%, decreased dP/dtmax by 21% and reduced MVO2 by 4%. PSCr-EDTA and PSB12 decreased by 24 and 27%, respectively, leaving the ratio PSCr-EDTA/PSB12 unchanged indicating unaltered capillary permeability. Moreover, the reductions in MVO2 and PS correlated significantly. During vasodilation: (1) nitroprusside-NA stimulated HR by 7% and decreased dP/dtmax by 14%; (2) adenosine reduced dP/dtmax by 37% and decreased MVO2 by 9%; and (3) isoproterenol increased HR, dP/dtmax and MVO2 by 53, 76 and 9%, respectively. However, all three vasodilators reduced PSCr-EDTA and PSB12 in parallel by 7-25% leaving PSCR-EDTA/PSB12 unchanged. Thus, maximal estimated diffusion capacities were obtained during spontaneous coronary vascular tone, most likely reflecting maximal capillary recruitment in the Krebs-Henseleit-albumin perfused heart. The derecruiting effects of the vasoconstrictors were partly overridden by metabolic factors, while the reductions of PS after vasodilation more likely were due to increased heterogeneity in coronary flow.

Computationally efficient algorithms for convection-permeation-diffusion models for blood-tissue exchange

Analysis of data on tissue depositions obtained by positron tomographic or NMR imaging, or of multiple tracer outflow dilution curves, requires fitting data with models composed of aggregates of capillary-tissue units. These units account for heterogeneities of flows and multisolute exchanges between longitudinally distributed regions across capillary and cell barriers within an organ. Because the analytic solutions to the partial differential equations require convolution integration, solutions are obtained relatively efficiently by a fast numerical method. Our approach centers on the use of a sliding fluid element algorithm for capillary convection, with the time step set equal to the length step divided by the fluid velocity. Radial fluxes by permeation between plasma, interstitial fluid, and cells and axial diffusion exchanges within each time step are calculated analytically. The method enforces mass conservation unless there is regional consumption. Solution for a 2-barrier, 3-region model, accurate to within 0.5%, are 100 to 1000 times faster than the corresponding, purely analytic solution, and over 10,000 times for a 4-region model. Applications include multiple indicator dilution studies of kinetics of transcapillary exchange and positron emission tomographic studies of the mechanisms of substrate transport into cells of organs in vivo.

Flow-limited tracer oxygen distribution in the isolated perfused rat liver: effects of temperature and hematocrit

We used the multiple-indicator dilution technique to examine the kinetics of tracer oxygen distribution and uptake in the rat liver perfused in a nonrecirculating fashion with blood. 51Cr-labeled 18O2-saturated erythrocytes, labeled albumin, sucrose and water (the tracers for oxygen and vascular, interstitial and cellular references) were injected simultaneously into the portal vein. Timed anaerobic samples were collected from the hepatic vein and analyzed by mass spectrometry for relative 18O2 enrichment and radioactivity. In a set of experiments performed at 32 degrees C, oxygen uptake was substantially diminished; tracer oxygen profiles approached those expected for a completely recovered, flow-limited substance. At 37 degrees C, much larger tracer oxygen sequestration occurred. Experiments were carried out at each temperature at higher and lower hematocrit, and oxygen consumption at each temperature was found to be independent of hematocrit. The tissue space of distribution for tracer oxygen relative to the total sinusoidal vascular content was influenced by the hematocrit: it was smaller at higher hematocrit and larger at lower hematocrit, as expected. The derived partition coefficient of oxygen for liver cells relative to plasma (expressed in terms of the liver and plasma water spaces) was, on average, 2.62 ml/ml; it was independent of the hematocrit. Analysis of the indicator dilution experiments indicates that the tracer oxygen is distributed into tissue in a flow-limited rather than a barrier-limited fashion, and that with this, an ongoing concomitant intracellular sequestration of tracer can be seen.

A distributed model of exchange processes within the osteon

In order to understand various exchange processes within the osteon, a mathematical model to describe the system has been developed which allows for concentration gradients in the axial and radial directions as well as cellular consumption and binding to bone surface. The normal values for the model parameter are discussed and the effects of the model parameters on the behaviour of the model are investigated. This model supports the idea that diffusion alone may be an inefficient mechanism in transport between blood and osteocytes.

Axial tissue diffusion can account for the disparity between current models of hepatic elimination for lipophilic drugs

An assumption of previous models of hepatic elimination is that there is negligible axial diffusion in the liver. We show, by construction of a stochastic model and analysis of published data, that compounds which are readily diffusible and partitioned into hepatocytes may undergo axial tissue diffusion. The compounds most likely to be affected by axial tissue diffusion are the lipophilic drugs for which the cell membranes provide little resistance and which are highly extracted, thereby creating steep concentration gradients along the sinusoid at steady state. This phenomenon greatly modifies the availability of the compound under conditions of altered hepatic blood flow and protein binding. For moderately diffusible compounds, these relationships are similar to those predicted by the simplistic venous-equilibrium model. Hence, the paradoxical ability of the venous-equilibrium model to describe the steady-state kinetics of lipophilic drugs such as lidocaine, meperidine, and propranolol may be finally resolved. The effects of axial tissue diffusion and vascular dispersion on hepatic availability of drugs are compared. Vascular dispersion is of major importance to the availability of poorly diffusible compounds, whereas axial tissue diffusion becomes increasingly dominant for highly diffusive and partitioned substances.

Regional pharmacokinetics. III. Modelling methods

Regional pharmacokinetics is the study of drug concentrations in specific regions of the body due to drug uptake and elution. Mathematical methods of interpreting regional pharmacokinetic data can vary greatly in their complexity depending on their intended use (i.e. to describe or predict), but must reinforce rather than replace experimental pharmacokinetics. 'Black box' analysis provides and empirical method for the study of complex pharmacokinetic systems using either statistical moment or linear systems analysis. However, these methods are only applicable to linear and time-invariant systems, and ignore the large body of information concerning the physiological and physiochemical basis of regional pharmacokinetics. Clearance concepts are suitable for describing linear drug uptake processes, but mass balance principles have wider applications in describing the rate and extent of both drug uptake and elution. Compartmental models of a region can vary from single compartment descriptions based on the concept of venous equilibrium to complex multi-compartmental models of the intravascular, interstitial, and intracellular spaces, in which drug transport between compartments is a function of drug binding and ionization. Ultimately, as more regional pharmacokinetic information is obtained, more complex three dimensional models may be necessary such as those used to describe the uptake of oxygen from capillaries.

Mathematical modelling in nuclear medicine

Modern imaging techniques can provide sequences of images giving signals proportional to the concentrations of tracers (by emission tomography), of X-ray-absorbing contrast materials (fast CT or perhaps NMR contrast), or of native chemical substances (NMR) in tissue regions at identifiable locations in 3D space. Methods for the analysis of the concentration-time curves with mathematical models describing the physiological processes and the appropriate anatomy are now available to give a quantitative portrayal of both structure and function: such is the approach to metabolic or functional imaging. One formulates a model first by defining what it should represent: this is the hypothesis. When translated into a self-consistent set of differential equations, the model becomes a mathematical model, a quantitative version of the hypothesis. This is what one would like to test against data. However, the next step is to reduce the mathematical model to a computable form; anatomically and physiologically realistic models account of the spatial gradients in concentrations within blood-tissue exchange units, while compartmental models simplify the equations by using the average concentrations. The former are known as distributed models and the latter as lumped compartmental or mixing chamber models. Since both are derived from the same ideas, the parameters are usually the same; their differences are in their ability to represent the hypothesis correctly, quantitatively, and sometimes in their computability. In this essay we review the philosophical and practical aspects of such modelling analysis for translating image sequences into physiological terms.

An MRI perfusion model incorporating nonequilibrium exchange between vascular and extravascular compartments

A model of MRI signal intensity which is a function of perfusion is developed based upon the assumption that biological tissue can be represented by a blood and tissue compartment. The longitudinal magnetization is derived from the Bloch equations which are modified to model the magnetization in both the blood and tissue as a function of the following physiological parameters: blood flow velocity; perfusion fraction, which in the model is parameterized in terms of the ratio of the cross-sectional areas of the tissue and blood compartments; diffusion; rate of exchange between the blood and extravascular tissue compartments. Simulations of slice profiles excited by a repetitive sequence of 90 degrees slice-selective pulses show that the signal intensity in the blood and tissue compartments are modulated by the physiological parameters. A key factor in the modulation of the MRI signal is a time-of-flight effect whereby unexcited spins perfuse the excited region and exchange with blood and tissue compartments, thus immediately increasing the slice signal intensity but also delaying the spin exits from the slice, thereby decreasing their contribution to slice signal intensity in future repetitive pulse measurements.

Myocardial transport of hexakis(2-methoxyisobutylisonitrile) and thallium before and after coronary reperfusion

Effects of no-flow ischemia (NFI) and reperfusion (RPF) on myocardial extraction and retention of technetium-99m hexakis(2-methoxyisobutylisonitrile) (sestamibi) and thallium-201 were investigated in 12 isolated, blood-perfused rabbit hearts with isotope dilution studies at constant coronary perfusion. After a control injection of tracers, NFI was induced for 30-60 minutes. After coronary reflow, repeat tracer injections were given at early RPF (5-15 minutes of RPF) and late RPF (40-60 minutes of RPF). After NFI-RPF, maximal fractional extraction and capillary permeability-surface area product increased for sestamibi (from +39% to 69%) and decreased for thallium (from -14% to -68%). Net extraction was 33% lower for sestamibi than for thallium at control, 13% lower at early RPF, and 90% higher than thallium at late RPF. Interstitial-myocyte exchange estimates were always higher for sestamibi than for thallium and increased for both with NFI-RPF (sestamibi, from 57.4 to 122.4 ml/min/g; thallium, from 3.1 to 22.3 ml/min/g). Intramyocyte volumes of distribution were higher for sestamibi than for thallium (greater than 200% at control, 800-1,000% with RPF), and NFI-RPF had opposite effects on the two tracers (late RPF vs. control: +28% for sestamibi, -50% for thallium). Our results suggest that sestamibi and thallium have different transport or sequestering mechanisms and that NFI-RPF had opposite effects on myocardial capillary-tissue exchange and tissue retention of sestamibi and thallium. Therefore, myocardial perfusion might be overestimated with sestamibi and underestimated with thallium during early RPF.