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

Automatic in-line quantitative myocardial perfusion mapping: Processing algorithm and implementation

PURPOSE

Quantitative myocardial perfusion mapping has advantages over qualitative assessment, including the ability to detect global flow reduction. However, it is not clinically available and remains a research tool. Building upon the previously described imaging sequence, this study presents algorithm and implementation of an automated solution for inline perfusion flow mapping with step by step performance characterization.

METHODS

Proposed workflow consists of motion correction (MOCO), arterial input function blood detection, intensity to gadolinium concentration conversion, and pixel-wise mapping. A distributed kinetics model, blood-tissue exchange model, is implemented, computing pixel-wise maps of myocardial blood flow (mL/min/g), permeability-surface-area product (mL/min/g), blood volume (mL/g), and interstitial volume (mL/g).

RESULTS

Thirty healthy subjects (11 men; 26.4 ± 10.4 years) were recruited and underwent adenosine stress perfusion cardiovascular MR. Mean MOCO quality score was 3.6 ± 0.4 for stress and 3.7 ± 0.4 for rest. Myocardial Dice similarity coefficients after MOCO were significantly improved (P < 1e-6), 0.87 ± 0.05 for stress and 0.86 ± 0.06 for rest. Arterial input function peak gadolinium concentration was 4.4 ± 1.3 mmol/L at stress and 5.2 ± 1.5 mmol/L at rest. Mean myocardial blood flow at stress and rest were 2.82 ± 0.47 mL/min/g and 0.68 ± 0.16 mL/min/g, respectively. The permeability-surface-area product was 1.32 ± 0.26 mL/min/g at stress and 1.09 ± 0.21 mL/min/g at rest (P < 1e-3). Blood volume was 12.0 ± 0.8 mL/100 g at stress and 9.7 ± 1.0 mL/100 g at rest (P < 1e-9), indicating good adenosine vasodilation response. Interstitial volume was 20.8 ± 2.5 mL/100 g at stress and 20.3 ± 2.9 mL/100 g at rest (P = 0.50).

CONCLUSIONS

An inline perfusion flow mapping workflow is proposed and demonstrated on normal volunteers. Initial evaluation demonstrates this fully automated solution for the respiratory MOCO, arterial input function left ventricle mask detection, and pixel-wise mapping, from free-breathing myocardial perfusion imaging.

Extended quantitative dynamic contrast-enhanced cardiac perfusion imaging in mice using accelerated data acquisition and spatially distributed, two-compartment exchange modeling

The objective of the present work was to improve data acquisition and quantification of dynamic contrast-enhanced perfusion imaging in the in vivo murine heart. Four-fold undersampled data were acquired in 14 mice and reconstructed using k-t SPARSE. A two-compartment exchange model was employed to provide additional characterization of myocardial tissue based on compartment volumes and the permeability surface area product. The feasibility of the proposed method was tested using compartment-based analysis of contrast-enhanced perfusion data acquired with intravascular and extracellular contrast agents. A significantly different permeability surface area product was measured for the intravascular versus extracellular contrast agent (0.13-0.15 ml/g/min vs 0.86-0.88 ml/g/min). The reduced extravasation also resulted in significantly smaller interstitial volumes of the intravascular versus extracellular agent (9.8-11% vs 45-47%). No difference was found for myocardial blood flow (6.5-7.2 ml/g/min vs 6.0-7.0 ml/g/min). The results presented here show that two-compartment exchange modeling in the in vivo murine heart is feasible and gives access to tissue parameters beyond myocardial blood flow.

A Factor Increasing Venous Contamination on Bolus Chase Three-dimensional Magnetic Resonance Imaging: Charcot Neuroarthropathy

Background

The study aimed to evaluate the ratio of venous contamination in diabetic cases without foot lesion, with foot lesion and with Charcot neuroarthropathy (CN).

Materials and Methods

Bolus-chase three-dimensional magnetic resonance (MR) of 396 extremities of patients with diabetes mellitus was analyzed, retrospectively. Extremities were divided into three groups as follows: diabetic patients without foot ulcer or Charcot arthropathy (Group A), patients with diabetic foot ulcers (Group B) and patients with CN accompanying diabetic foot ulcers (Group C). Furthermore, amount of venous contamination classified as no venous contamination, mild venous contamination, and severe venous contamination. The relationship between venous contamination and extremity groups was investigated.

Results

Severe venous contamination was seen in Group A, Group B, and Group C, 5.6%, 15.2%, and 34.1%, respectively. Statistically significant difference was seen between groups with regard to venous contamination.

Conclusion

Venous contamination following bolus chase MR was higher in patients with CN.

The Pathway for Oxygen: Tutorial Modelling on Oxygen Transport from Air to Mitochondrion: The Pathway for Oxygen

The 'Pathway for Oxygen' is captured in a set of models describing quantitative relationships between fluxes and driving forces for the flux of oxygen from the external air source to the mitochondrial sink at cytochrome oxidase. The intervening processes involve convection, membrane permeation, diffusion of free and heme-bound O2 and enzymatic reactions. While this system's basic elements are simple: ventilation, alveolar gas exchange with blood, circulation of the blood, perfusion of an organ, uptake by tissue, and consumption by chemical reaction, integration of these pieces quickly becomes complex. This complexity led us to construct a tutorial on the ideas and principles; these first PathwayO2 models are simple but quantitative and cover: (1) a 'one-alveolus lung' with airway resistance, lung volume compliance, (2) bidirectional transport of solute gasses like O2 and CO2, (3) gas exchange between alveolar air and lung capillary blood, (4) gas solubility in blood, and circulation of blood through the capillary syncytium and back to the lung, and (5) blood-tissue gas exchange in capillaries. These open-source models are at Physiome.org and provide background for the many respiratory models there.

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.

Capillary-wall collagen as a biophysical marker of nanotherapeutic permeability into the tumor microenvironment

The capillary wall is the chief barrier to tissue entry of therapeutic nanoparticles, thereby dictating their efficacy. Collagen fibers are an important component of capillary walls, affecting leakiness in healthy or tumor vasculature. Using a computational model along with in vivo systems, we compared how collagen structure affects the diffusion flux of a 1-nm chemotherapeutic molecule [doxorubicin (DOX)] and an 80-nm chemotherapy-loaded pegylated liposome (DOX-PLD) in tumor vasculature. We found a direct correlation between the collagen content around a tumor vessel to the permeability of that vessel permeability to DOX-PLD, indicating that collagen content may offer a biophysical marker of extravasation potential of liposomal drug formulations. Our results also suggested that while pharmacokinetics determined the delivery of DOX and DOX-PLD to the same tumor phenotype, collagen content determined the extravasation of DOX-PLD to different tumor phenotypes. Transport physics may provide a deeper view into how nanotherapeutics cross biological barriers, possibly helping explain the balance between biological and physical aspects of drug delivery.

Kinetic modeling in the context of cerebral blood flow quantification by H2(15)O positron emission tomography: the meaning of the permeability coefficient in Renkin-Crone׳s model revisited at capillary scale

One the one hand, capillary permeability to water is a well-defined concept in microvascular physiology, and linearly relates the net convective or diffusive mass fluxes (by unit area) to the differences in pressure or concentration, respectively, that drive them through the vessel wall. On the other hand, the permeability coefficient is a central parameter introduced when modeling diffusible tracers transfer from blood vessels to tissue in the framework of compartmental models, in such a way that it is implicitly considered as being identical to the capillary permeability. Despite their simplifying assumptions, such models are at the basis of blood flow quantification by H2(15)O Positron Emission Tomgraphy. In the present paper, we use fluid dynamic modeling to compute the transfers of H2(15)O between the blood and brain parenchyma at capillary scale. The analysis of the so-obtained kinetic data by the Renkin-Crone model, the archetypal compartmental model, demonstrates that, in this framework, the permeability coefficient is highly dependent on both flow rate and capillary radius, contrarily to the central hypothesis of the model which states that it is a physiological constant. Thus, the permeability coefficient in Renkin-Crone׳s model is not conceptually identical to the physiologic permeability as implicitly stated in the model. If a permeability coefficient is nevertheless arbitrarily chosen in the computed range, the flow rate determined by the Renkin-Crone model can take highly inaccurate quantitative values. The reasons for this failure of compartmental approaches in the framework of brain blood flow quantification are discussed, highlighting the need for a novel approach enabling to fully exploit the wealth of information available from PET data.

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 serotonin uptake in the lung shows endothelial transporters dominate over cleft permeation

A four-region (capillary plasma, endothelium, interstitial fluid, cell) multipath model was configured to describe the kinetics of blood-tissue exchange for small solutes in the lung, accounting for regional flow heterogeneity, permeation of cell membranes and through interendothelial clefts, and intracellular reactions. Serotonin uptake data from the Multiple indicator dilution "bolus sweep" experiments of Rickaby and coworkers (Rickaby DA, Linehan JH, Bronikowski TA, Dawson CA. J Appl Physiol 51: 405-414, 1981; Rickaby DA, Dawson CA, and Linehan JH. J Appl Physiol 56: 1170-1177, 1984) and Malcorps et al. (Malcorps CM, Dawson CA, Linehan JH, Bronikowski TA, Rickaby DA, Herman AG, Will JA. J Appl Physiol 57: 720-730, 1984) were analyzed to distinguish facilitated transport into the endothelial cells (EC) and the inhibition of tracer transport by nontracer serotonin in the bolus of injectate from the free uninhibited permeation through the clefts into the interstitial fluid space. The permeability-surface area products (PS) for serotonin via the inter-EC clefts were ~0.3 ml·g⁻¹·min⁻¹, low compared with the transporter-mediated maximum PS of 13 ml·g⁻¹·min⁻¹ (with Km = ~0.3 μM and Vmax = ~4 nmol·g⁻¹·min⁻¹). The estimates of serotonin PS values for EC transporters from their multiple data sets were similar and were influenced only modestly by accounting for the cleft permeability in parallel. The cleft PS estimates in these Ringer-perfused lungs are less than half of those for anesthetized dogs (Yipintsoi T. Circ Res 39: 523-531, 1976) with normal hematocrits, but are compatible with passive noncarrier-mediated transport observed later in the same laboratory (Dawson CA, Linehan JH, Rickaby DA, Bronikowski TA. Ann Biomed Eng 15: 217-227, 1987; Peeters FAM, Bronikowski TA, Dawson CA, Linehan JH, Bult H, Herman AG. J Appl Physiol 66: 2328-2337, 1989) The identification and quantitation of the cleft pathway conductance from these studies affirms the importance of the cleft permeation.

Kinetic model optimization for characterizing tumour physiology by dynamic contrast-enhanced near-infrared spectroscopy

Dynamic contrast-enhanced (DCE) methods are widely used with magnetic resonance imaging and computed tomography to assess the vascular characteristics of tumours since these properties can affect the response to radiotherapy and chemotherapy. In contrast, there have been far fewer studies using optical-based applications despite the advantages of low cost and safety. This study investigated an appropriate kinetic model for optical applications to characterize tumour haemodynamics (blood flow, F, blood volume, V(b), and vascular heterogeneity) and vascular leakage (permeability surface-area product, PS). DCE data were acquired with two dyes, indocyanine green (ICG) and 800 CW carboxylate (IRD(cbx)), from a human colon tumour xenograph model in rats. Due to the smaller molecular weight of IRD(cbx) (1166 Da) compared to albumin-bound ICG (67 kDa), PS of IRD(cbx) was significantly larger; however, no significant differences in F and V(b) were found between the dyes as expected. Error analysis demonstrated that all parameters could be estimated with an uncertainty less than 5% due to the high temporal resolution and signal-to-noise ratio of the optical measurements. The next step is to adapt this approach to optical imaging to generate haemodynamics and permeability maps, which should enhance the clinical interest in optics for treatment monitoring.

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.

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.

Multiscale modeling of metabolism, flows, and exchanges in heterogeneous organs

Large-scale models accounting for the processes supporting metabolism and function in an organ or tissue with a marked heterogeneity of flows and metabolic rates are computationally complex and tedious to compute. Their use in the analysis of data from positron emission tomography (PET) and magnetic resonance imaging (MRI) requires model reduction since the data are composed of concentration-time curves from hundreds of regions of interest (ROI) within the organ. Within each ROI, one must account for blood flow, intracapillary gradients in concentrations, transmembrane transport, and intracellular reactions. Using modular design, we configured a whole organ model, GENTEX, to allow adaptive usage for multiple reacting molecular species while omitting computation of unused components. The temporal and spatial resolution and the number of species are adaptable and the numerical accuracy and computational speed is adjustable during optimization runs, which increases accuracy and spatial resolution as convergence approaches. An application to the interpretation of PET image sequences after intravenous injection of 13NH3 provides functional image maps of regional myocardial blood flows.

An Improved Algorithm and Its Parallel Implementation for Solving a General Blood-Tissue Transport and Metabolism Model

Fast algorithms for simulating mathematical models of coupled blood-tissue transport and metabolism are critical for the analysis of data on transport and reaction in tissues. Here, by combining the method of characteristics with the standard grid discretization technique, a novel algorithm is introduced for solving a general blood-tissue transport and metabolism model governed by a large system of one-dimensional semilinear first order partial differential equations. The key part of the algorithm is to approximate the model as a group of independent ordinary differential equation (ODE) systems such that each ODE system has the same size as the model and can be integrated independently. Thus the method can be easily implemented in parallel on a large scale multiprocessor computer. The accuracy of the algorithm is demonstrated for solving a simple blood-tissue exchange model introduced by Sangren and Sheppard (Bull. Math. Biophys. 15:387-394, 1953), which has an analytical solution. Numerical experiments made on a distributed-memory parallel computer (an HP Linux cluster) and a shared-memory parallel computer (a SGI Origin 2000) demonstrate the parallel efficiency of the algorithm.

Microcirculation and the physiome projects

The Physiome projects comprise a loosely knit worldwide effort to define the Physiome through databases and theoretical models, with the goal of better understanding the integrative functions of cells, organs, and organisms. The projects involve developing and archiving models, providing centralized databases, and linking experimental information and models from many laboratories into self-consistent frameworks. Increasingly accurate and complete models that embody quantitative biological hypotheses, adhere to high standards, and are publicly available and reproducible, together with refined and curated data, will enable biological scientists to advance integrative, analytical, and predictive approaches to the study of medicine and physiology. This review discusses the rationale and history of the Physiome projects, the role of theoretical models in the development of the Physiome, and the current status of efforts in this area addressing the microcirculation.

Modeling oxygen and carbon dioxide transport and exchange using a closed loop circulatory system

The binding and buffering of O2 and CO2 in the blood influence their exchange in lung and tissues and their transport through the circulation. To investigate the binding and buffering effects, a model of blood-tissue gas exchange is used. The model accounts for hemoglobin saturation, the simultaneous binding of O2, CO2, H+, 2,3-DPG to hemoglobin, and temperature effects. Invertible Hill-type saturation equations facilitate rapid calculation of respiratory gas redistribution among the plasma, red blood cell and tissue that occur along the concentration gradients in the lung and in the capillary-tissue exchange regions. These equations are well-suited to analysis of transients in tissue metabolism and partial pressures of inhaled gas. The modeling illustrates that because red blood cell velocities in the flowing blood are higher than plasma velocities after a transient there can be prolonged differences between RBC and plasma oxygen partial pressures. The blood-tissue gas exchange model has been incorporated into a higher level model of the circulatory system plus pulmonary mechanics and gas exchange using the RBC and plasma equations to account for pH and CO2 buffering in the blood.

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.

Dermal capillary clearance: physiology and modeling

Substances applied to the skin surface may permeate deeper tissue layers and pass into the body's systemic circulation by entering blood or lymphatic vessels in the dermis. The purpose of this review is an in-depth analysis of the dermal clearance/exchange process and its constituents: transport through the interstitium, permeability of the microvascular barrier and removal via the circulation. We adapt an 'engineering' viewpoint with emphasis on quantifying the dermal microcirculatory physiology, providing the theoretical framework for the physics of key transport processes and reviewing the available computational clearance models in a comparative manner. Selected experimental data which may serve as valuable input to modeling attempts are also reported.

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.

Principles of quantitative positron emission tomography

The central distinguishing feature of positron emission tomography (PET) is its ability to investigate quantitatively regional cellular and molecular transport processes in vivo with good spatial resolution. This review wants to provide a concise overview of the established principles underlying quantitative data evaluations of the acquired PET images. Especially, the compartment modelling framework is discussed on which virtually all quantification methods utilized in PET are based. The aim of the review is twofold: first, to provide the reader with an idea of the theoretical framework and mathematical tools and second, to enable an intuitive grasp of the possibilities and limitations of a quantitative approach to PET data evaluation. This should facilitate an understanding of how PET measurements translate into quantities such as regional blood flow, volume of distribution, and metabolic rates of specific substrates.

Capillaries within compartments: microvascular interpretation of dynamic positron emission tomography data

Measurement of exchange of substances between blood and tissue has been a long-lasting challenge to physiologists, and considerable theoretical and experimental accomplishments were achieved before the development of the positron emission tomography (PET). Today, when modeling data from modern PET scanners, little use is made of earlier microvascular research in the compartmental models, which have become the standard model by which the vast majority of dynamic PET data are analysed. However, modern PET scanners provide data with a sufficient temporal resolution and good counting statistics to allow estimation of parameters in models with more physiological realism. We explore the standard compartmental model and find that incorporation of blood flow leads to paradoxes, such as kinetic rate constants being time-dependent, and tracers being cleared from a capillary faster than they can be supplied by blood flow. The inability of the standard model to incorporate blood flow consequently raises a need for models that include more physiology, and we develop microvascular models which remove the inconsistencies. The microvascular models can be regarded as a revision of the input function. Whereas the standard model uses the organ inlet concentration as the concentration throughout the vascular compartment, we consider models that make use of spatial averaging of the concentrations in the capillary volume, which is what the PET scanner actually registers. The microvascular models are developed for both single- and multi-capillary systems and include effects of non-exchanging vessels. They are suitable for analysing dynamic PET data from any capillary bed using either intravascular or diffusible tracers, in terms of physiological parameters which include regional blood flow.

Analysis of myocardial perfusion MRI

Rapid MR imaging (MRI) during the first pass of an injected tracer is used to assess myocardial perfusion with a spatial resolution of 2-3 mm, and to detect any regional impairments of myocardial blood flow (MBF) that may lead to ischemia. The spatial resolution is sufficient to detect flow reductions that are limited to the subendocardial layer. The capacity of the coronary system to increase MBF severalfold in response to vasodilation can be quantified by analysis of the myocardial contrast enhancement. The myocardial perfusion reserve (MPR) is a useful concept for quantifying the vasodilator response. The perfusion reserve can be estimated from the ratio of MBFs during vasodilation and at baseline, in units identical to those used for invasive measurements with labeled microspheres, or from dimensionless flow indices normalized by their value for autoregulated flow at rest. The perfusion reserve can be reduced as a result of a blunted hyperemic response and/or an abnormal resting blood flow. The absolute quantification of MBF removes uncertainties in the evaluation of the vasodilator response, and can be achieved without the use of complex tracer kinetic models; therefore, its application to clinical studies is feasible.

Distributed versus compartment models for PET receptor studies

Although distributed models are generally accepted as being more realistic than compartment models, use of simpler compartment models is pervasive in nuclear medicine applications, particularly in positron emission tomography (PET). Here, we report on comparisons made between distributed and compartment model outputs to address the question of whether differences between them are sufficient to justify distributed models for analysis of PET receptor experiments. For both two- and three-injection experiments, "data" sets were obtained by simulation using a distributed model and a wide range of parameter values. Optimal fits of the compartment model output to these "data" were achieved with three strategies in which values of different groups of parameter were estimated. Compartment model outputs yielded good fits to all the distributed model outputs and the values of the corresponding parameters were in close agreement. Given the temporal resolution typically available with PET, the use of a distributed model has no advantage over a compartment model for PET receptor quantification.

Bolus arterial-venous transit in the lower extremity and venous contamination in bolus chase three-dimensional magnetic resonance angiography

RATIONALE AND OBJECTIVES

To investigate the phenomena and causes for undesired venous signal in the distal station of bolus chase 3D MRA.

METHODS

Consecutive patients (in 8 months) undergoing peripheral MRA consisting of 2D projection MRA of the tibial trifurcation and 3D bolus chase MRA were retrospectively evaluated. Venous contamination in mid-calf in bolus chase 3D MRA was correlated to the arterial phase duration, the time between the contrast bolus arrival and venous return measured on time resolved 2D images. Statistical analyses were performed to identify the clinical parameters indicative of venous contamination.

RESULTS

The arterial phase durations at the mid-calf were 49 +/- 8 seconds on 101 legs without venous signal in the bolus chase 3D MRA, 35 +/- 9 seconds on 13 legs with moderate venous signal, and 20 +/- 4 seconds on 40 legs with substantial venous signal; the differences were significant among different venous signal levels (P < 0.001 for all pairs). Legs with cellulitis had shorter arterial phase and more venous contamination than legs without cellulitis (P < 0.05). Patients with myocardial infarction had longer arterial phase and less venous contamination than patients without myocardial infarction (P < 0.01).

CONCLUSION

Venous signal in the distal calf station of bolus chase 3D peripheral MRA is caused by fast arterial-venous transit. It is worse in legs with cellulitis and less in patients with a history of myocardial infarction.

A computational model of oxygen transport from red blood cells to mitochondria

A computational model of oxygen transport from red blood cells to mitochondria with subsequent reaction to water is presented. This computational model consists of a five region convection-diffusion-reaction mathematical model which is solved using a standard numerical time-split method. The unique feature of this mathematical model is the treatment of the red blood cells and the plasma as two separate flows. The numerical method is second order accurate overall. This computational model is useful for analyzing residue data from positron emission tomography or data from multiple indicator dilution curves.

The mechanical and metabolic basis of myocardial blood flow heterogeneity

Precise measurements of regional myocardial blood flow heterogeneity had to be developed before one could seek causation for the heterogeneity. Deposition techniques (particles or molecular microspheres) are the most precise, but imaging techniques have begun to provide high enough resolution to allow in vivo studies. Assigning causation has been difficult. There is no apparent association with the regional concentrations of energy-related enzymes or substrates, but these are measures of status, not of metabolism. There is statistical correlation between flow and regional substrate uptake and utilization. Attribution of regional flow variation to vascular anatomy or to vasomotor control appears not to be causative on a long-term basis. The closest relationships appear to be with mechanical function, but one cannot say for sure whether this is related to ATP hydrolysis at the crossbridge or associated metabolic reactions such as calcium uptake by the sarcoplasmic reticulum.

Heterogeneity of local myocardial flow and oxidative metabolism

In mammalian hearts, local myocardial flow (LMF) varies between 20 and 200% of the mean. It is not clear whether oxidative metabolism has a similar degree of heterogeneity. Therefore, we investigated the relation between LMF and local oxidative metabolism in isolated rabbit hearts. Buffer oxygenation with (18)O(2) resulted in labeled myocardial oxidation water (H(2)(18)O). In four hearts, myocardial oxygen consumption (MVO(2)) was calculated from the H(2)(18)O production and compared with that calculated according to Fick. In eight additional hearts, LMF was measured using microspheres. Coronary venous H(2)(18)O kinetics and local H(2)(18)O residues were determined and analyzed by mathematical modeling. MVO(2) recovery from H(2)(18)O was >93% compared with that according to Fick. LMF ranged from 1.91 to 11.24 ml. min(-1). g(-1), and local H(2)(18)O residue ranged from 0.41 to 1.04 micromol/g. Both variables correlated (r = 0.62, n = 64, P < 0.001). Measurements in nine hearts were fitted by modeling using capillary permeability-surface area products (PS(c)) from 2 to 10 ml. min(-1). g(-1). With flow-proportional PS(c), a 3.33-fold difference in LMF was associated with a 6.45-fold difference in local MVO(2). Both LMF and local oxidative metabolism are spatially heterogeneous, and they correlate to one another.

Muscle as a consumer of lactate

Historically, muscle has been viewed primarily as a producer of lactate but is now considered also to be a primary consumer of lactate. Among the most important factors that regulate net lactate uptake and consumption are metabolic rate, blood flow, lactate concentration ([La]), hydrogen ion concentration ([H+]), fiber type, and exercise training. Muscles probably consume more lactate during steady state exercise or contractions because of increased lactate oxidation since enhancements in lactate transport due to acute activity are small. For optimal lactate consumption, blood flow should be adequate to maintain ideal [La] and [H+] gradients from outside to inside muscles. However, it is not clear that greater than normal blood flow will enhance lactate exchange. A widening of the [La] gradient from outside to inside muscle cells along with an increase in muscle [La] enhances both lactate utilization and sarcolemmal lactate transport. Similarly, a significant outside to inside [H+] gradient will stimulate sarcolemmal lactate influx, whereas an increased intramuscular [H+] may stimulate exogenous lactate utilization by inhibiting endogenous lactate production. Oxidative muscle fibers are metabolically suited for lactate oxidation, and they have a greater capacity for sarcolemmal lactate transport than do glycolytic muscle fibers. Endurance training improves muscle capacity for lactate utilization and increases membrane transport of lactate probably via an increase in Type I monocarboxylate transport protein (MCT1) and perhaps other MCT isoforms as well. The future challenge is to understand the regulatory roles of both lactate metabolism and membrane transport of lactate.

Cardiac endothelial transport and metabolism of adenosine and inosine

The influence of transmembrane flux limitations on cellular metabolism of purine nucleosides was assessed in whole organ studies. Transcapillary transport of the purine nucleosides adenosine (Ado) and inosine (Ino) via paracellular diffusion through interendothelial clefts in parallel with carrier-mediated transendothelial fluxes was studied in isolated, Krebs-Henseleit-perfused rabbit and guinea pig hearts. After injection into coronary inflow, multiple-indicator dilution curves were obtained from coronary outflow for 90 s for 131I-labeled albumin (intravascular reference tracer), [3H]arabinofuranosyl hypoxanthine (AraH; extracellular reference tracer and nonreactive adenosine analog), and either [14C]Ado or [14C]Ino. Ado or Ino was separated from their degradative products, hypoxanthine, xanthine, and uric acid, in each outflow sample by HPLC and radioisotope counting. Ado and Ino, but not AraH, permeate the luminal membrane of endothelial cells via a saturable transporter with permeability-surface area product PS(ecl) and also diffuse passively through interendothelial clefts with the same conductance (PSg) as AraH. These parallel conductances were estimated via fitting with an axially distributed, multi-pathway, four-region blood-tissue exchange model. PSg for AraH were approximately 4 and 2.5 ml. g(-1). min(-1) in rabbits and guinea pigs, respectively. In contrast, transplasmalemmal conductances (endothelial PS(ecl)) were approximately 0.2 ml. g(-1). min(-1) for both Ado and Ino in rabbit hearts but approximately 2 ml. g(-1). min(-1) in guinea pig hearts, an order of magnitude different. Purine nucleoside metabolism also differs between guinea pig and rabbit cardiac endothelium. In guinea pig heart, 50% of the tracer Ado bolus was retained, 35% was washed out as Ado, and 15% was lost as effluent metabolites; 25% of Ino was retained, 50% washed out, and 25% was lost as metabolites. In rabbit heart, 45% of Ado was retained and 5% lost as metabolites, and 7% of Ino was retained and 3% lost as metabolites. We conclude that endothelial transport of Ado and Ino is a prime determinant of their metabolic fates: where transport rates are high, metabolic transformation is high.

Pharmacokinetics in organs and the intact body: model validation and reduction

Physiological pharmacokinetic models are based on the structure of the circulatory system reflecting the convective transport of drug by blood flow to the various organs and tissues. Distribution kinetics at the organ level is mostly simplified as transfer between well-stirred compartments neglecting a priori the effects of intravascular dispersion and diffusion within tissue parenchyma. Recirculatory models based on residence time theory overcome these structural limitations since they allow in a most general way the decomposition of the body into its natural subsystems. Because of the unidentifiability of the global multi-organ model on the basis of plasma concentration-time curves the following methods/experimental designs will be discussed which provide quantitative information regarding the subsystems under in vivo conditions: (i) determination of tissue concentration-time profiles (destructive sampling), (ii) estimation of the organ transit time density from input/output profiles and (iii) application of a recirculatory model with reduced complexity to clinical pharmacokinetic data.

Assessing myocardial perfusion in coronary artery disease with magnetic resonance first-pass imaging

MRFP perfusion imaging can now be used clinically on most MR scanner systems (1.0 to 1.5 T). The current experimental data demonstrate that MRFP imaging allows the quantitative assessment of myocardial blood flow changes and accurate measurements of collateral flow, including changes in the collateral dependent zones. Certain protocols, however, as outlined here have to be followed to obtain all the possible diagnostic information. Based on the current data on MRFP imaging, it is realistic to anticipate that MRFP imaging in combination with cine or tagging MR imaging will provide clinicians with better methods to distinguish stunned and hibernating, from nonviable myocardium and obtain better outcome data. Dedicated MR scanners are now being designed to meet the needs for MR imaging of patients with coronary artery disease. These scanners, small in size and with better patient access, make placement near the coronary care unit or catheterization laboratory feasible. This is a major step toward enhancing the utility of this new technique by providing the necessary infrastructure for scanning large numbers of patients. The main obstacle to wider use of these new diagnostic tools to assess perfusion is the lack of a large clinical database because there have not yet been major multicenter trials. With the development of novel intravascular contrast agents, however, larger trials are planned that should provide the clinical data mandatory for full integration of MRFP imaging into clinical practice. In particular, the development of dedicated and user-friendly perfusion analysis software will create the means to evaluate MR perfusion data accurately in large patient populations. These studies need to be conducted in a collaborative fashion by cardiologists, heart surgeons, and radiologists to be fully accepted by health care providers in an increasingly cost-averse and competitive health care environment.

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.

Parameter estimation in distributed models of blood-tissue exchange: a Monte Carlo strategy to assess precision

Distributed parameter models of blood-tissue exchange are increasingly used to interpret multiple tracer dilution data in regional kinetic studies. To derive a measure of the precision with which the model parameters are estimated is therefore of paramount importance. The standard approach to deriving precision of estimates does not take into account the fact that some of the model parameters are fixed. Thus, the precision of parameter estimates is not realistic and, in all likelihood, it is overestimated. The aim of this study is to describe a Monte Carlo method devised to obtain a theoretically sound measure of the precision of estimates, which takes into account both measurement error and the uncertainty associated with the fixed parameters. The fixed parameter values are taken from a probability distribution. By letting the fixed parameters vary according to their distribution, a large number of synthetic datasets is generated. Noise is then added. Estimating the parameters in each of these synthetic datasets allows the derivation of a Monte Carlo mean and standard deviation, which provides a realistic measure of precision. The methodology is illustrated for a simulated data case study dealing with the estimation of the capillary permeability-surface area product in a two tracer experiment.

Nonlinear model for capillary-tissue oxygen transport and metabolism

Oxygen consumption in small tissue regions cannot be measured directly, but assessment of oxygen transport and metabolism at the regional level is possible with imaging techniques using tracer 15O-oxygen for positron emission tomography. On the premise that mathematical modeling of tracer kinetics is the key to the interpretation of regional concentration-time curves, an axially-distributed capillary-tissue model was developed that accounts for oxygen convection in red blood cells and plasma, nonlinear binding to hemoglobin and myoglobin, transmembrane transport among red blood cells, plasma, interstitial fluid and parenchymal cells, axial dispersion, transformation to water in the tissue, and carriage of the reaction product into venous effluent. Computational speed was maximized to make the model useful for routine analysis of experimental data. The steady-state solution of a parent model for nontracer oxygen governs the solutions for parallel-linked models for tracer oxygen and tracer water. The set of models provides estimates of oxygen consumption, extraction, and venous pO2 by fitting model solutions to experimental tracer curves of the regional tissue content or venous outflow. The estimated myocardial oxygen consumption for the whole heart was in good agreement with that measured directly by the Fick method and was relatively insensitive to noise. General features incorporated in the model make it widely applicable to estimating oxygen consumption in other organs from data obtained by external detection methods such as positron emission tomography.

Efficient numerical methods for nonlinear-facilitated transport and exchange in a blood-tissue exchange unit

The analysis of experimental data obtained by the multiple-indicator method requires complex mathematical models for which capillary blood-tissue exchange (BTEX) units are the building blocks. This study presents a new, nonlinear, two-region, axially distributed, single capillary, BTEX model. A facilitated transporter model is used to describe mass transfer between plasma and intracellular spaces. To provide fast and accurate solutions, numerical techniques suited to nonlinear convection-dominated problems are implemented. These techniques are the random choice method, an explicit Euler-Lagrange scheme, and the MacCormack method with and without flux correction. The accuracy of the numerical techniques is demonstrated, and their efficiencies are compared. The random choice, Euler-Lagrange and plain MacCormack method are the best numerical techniques for BTEX modeling. However, the random choice and Euler-Lagrange methods are preferred over the MacCormack method because they allow for the derivation of a heuristic criterion that makes the numerical methods stable without degrading their efficiency. Numerical solutions are also used to illustrate some nonlinear behaviors of the model and to show how the new BTEX model can be used to estimate parameters from experimental data.

High resolution measurement of cerebral blood flow using intravascular tracer bolus passages. Part I: Mathematical approach and statistical analysis

The authors review the theoretical basis of determination of cerebral blood flow (CBF) using dynamic measurements of nondiffusible contrast agents, and demonstrate how parametric and nonparametric deconvolution techniques can be modified for the special requirements of CBF determination using dynamic MRI. Using Monte Carlo modeling, the use of simple, analytical residue models is shown to introduce large errors in flow estimates when actual, underlying vascular characteristics are not sufficiently described by the chosen function. The determination of the shape of the residue function on a regional basis is shown to be possible only at high signal-to-noise ratio. Comparison of several nonparametric deconvolution techniques showed that a nonparametric deconvolution technique (singular value decomposition) allows estimation of flow relatively independent of underlying vascular structure and volume even at low signal-to-noise ratio associated with pixel-by-pixel deconvolution.

Modeling regional myocardial flows from residue functions of an intravascular indicator

The purpose of the present study was to determine the accuracy and the sources of error in estimating regional myocardial blood flow and vascular volume from experimental residue functions obtained by external imaging of an intravascular indicator. For the analysis, a spatially distributed mathematical model was used that describes transport through a multiple-pathway vascular system. Reliability of the parameter estimates was tested by using sensitivity function analysis and by analyzing "pseudodata": realistic model solutions to which random noise was added. Increased uncertainty in the estimates of flow in the pseudodata was observed when flow was near maximal physiological values, when dispersion of the vascular input was more than twice the dispersion of the microvascular system for an impulse input, and when the sampling frequency was < 2 samples/s. Estimates of regional blood volume were more reliable than estimates of flow. Failure to account for normal flow heterogeneity caused systematic underestimates of flow. To illustrate the method used for estimating regional flow, magnetic resonance imaging was used to obtain myocardial residue functions after left atrial injections of polylysine-Gd-diethylenetriaminepentaacetic acid, an intravascular contrast agent, in anesthetized chronically instrumental dogs. To test the increase in dispersion of the vascular input after central venous injections, magnetic resonance imaging data obtained in human subjects were compared with left ventricular blood pool curves obtained in dogs. It is concluded that if coronary flow is in the normal range, when the vascular input is a short bolus, and the heart is imaged at least once per cardiac cycle, then regional myocardial blood flow and vascular volume may be reliably estimated by analyzing residue functions of an intravascular indicator, providing a noninvasive approach with potential clinical application.

Quantitative relation between interstitial adenosine concentration and coronary blood flow

The effect of exogenous and endogenous adenosine in controlling coronary flow was determined using an axially distributed mathematical model of the myocardium to estimate interstitial adenosine concentration from coronary arterial and venous adenosine values. The left main coronary artery was perfused at constant pressure in closed-chest, anesthetized dogs, and exogenous adenosine was infused intracoronary to increase coronary flow. Basal interstitial adenosine was 92 nmol/L, just at the threshold for increasing coronary flow. An increase in interstitial adenosine concentration of only 62% was sufficient to increase coronary flow from 5% to 50% of maximal flow. The possible contribution of an endothelial dilator secondary to activation of adenosine receptors on endothelial cells was tested by comparing the response to exogenous intracoronary adenosine infusion with increases in endogenous adenosine produced by inhibition of adenosine kinase and adenosine deaminase. If adenosine increases coronary flow by an endothelial mechanism, then the interstitial ED50 of exogenous adenosine would be lower than that for endogenous adenosine due to the postulated additional endothelial dilator. The interstitial ED50 for exogenous adenosine was 156 nmol/L, not different from the endogenous ED50 of 150 nmol/L. In conclusion, basal interstitial adenosine concentration is at the threshold of a remarkably steep dose-response curve for increasing coronary blood flow. No evidence was found for an endothelium-mediated vasodilator mechanism secondary to adenosine receptor activation of endothelial cells in vivo. The steep adenosine dose-response curve indicates that measurements of adenosine concentration should be interpreted with caution, because small changes in adenosine concentration cause large changes in coronary flow.

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.

Tissue distribution kinetics as determinant of transit time dispersion of drugs in organs: application of a stochastic model to the rat hindlimb

A stochastic theory of drug transport in a random capillary network with permeation across the endothelial barrier is coupled with a model of tissue residence time of drugs assuming radial intratissue diffusion. Axial diffusion is neglected both in tissue as well as in the radially well-mixed vascular phase. The convective transport through the microcirculatory network is characterized by an experimentally determined transit time distribution of a nonpermeating vascular indicator. This information is used to identify three adjustable model parameters characterizing permeation, diffusion, and steady-state distribution into tissue. Predictions are made for the influence of distribution volume, capillary permeability, and tissue diffusion on transit time distributions. The role of convection (through the random capillary network), permeation, and diffusion as determinants of the relative dispersion of organ transit times has been examined. The relationship to previously proposed models of capillary exchange is discussed. Results obtained for lidocaine in the isolated perfused hindleg in rats indicate that although the contribution of intratissue diffusion to the dispersion process is relatively small in quantitative terms, it has a pronounced influence on the shape of the impulse response curve. The theory suggests that the rate of diffusion in muscle tissue is about two orders of magnitude slower than in water.

Modeling [15O]oxygen tracer data for estimating oxygen consumption

The most direct measure of oxidative tissue metabolism is the conversion rate of oxygen to water via mitochondrial respiration. To calculate oxygen consumption from the analysis of tissue residue curves or outflow dilution curves after injection of labeled oxygen one needs realistic mathematical models that account for convection, diffusion, and transformation in the tissue. A linear, three-region, axially distributed model accounts for intravascular convection, penetration of capillary and parenchymal cell barriers (with the use of appropriate binding spaces to account for oxygen binding to hemoglobin and myoglobin), the metabolism to [15O]water in parenchymal cells, and [15O]water transport into the venous effluent. Model solutions fit residue and outflow dilution data obtained in an isolated, red blood cell-perfused rabbit heart preparation and give estimates of the rate of oxygen consumption similar to those obtained experimentally from the flow times the arteriovenous differences in oxygen contents. The proposed application is for the assessment of regional oxidative metabolism in vivo from tissue 15O-residue curves obtained by positron emission tomography.

Modeling blood flow heterogeneity

It has been known for some time that regional blood flows within an organ are not uniform. Useful measures of heterogeneity of regional blood flows are the standard deviation and coefficient of variation or relative dispersion of the probability density function (PDF) of regional flows obtained from the regional concentrations of tracers that are deposited in proportion to blood flow. When a mathematical model is used to analyze dilution curves after tracer solute administration, for many solutes it is important to account for flow heterogeneity and the wide range of transit times through multiple pathways in parallel. Failure to do so leads to bias in the estimates of volumes of distribution and membrane conductances. Since in practice the number of paths used should be relatively small, the analysis is sensitive to the choice of the individual elements used to approximate the distribution of flows or transit times. Presented here is a method for modeling heterogeneous flow through an organ using a scheme that covers both the high flow and long transit time extremes of the flow distribution. With this method, numerical experiments are performed to determine the errors made in estimating parameters when flow heterogeneity is ignored, in both the absence and presence of noise. The magnitude of the errors in the estimates depends upon the system parameters, the amount of flow heterogeneity present, and whether the shape of the input function is known. In some cases, some parameters may be estimated to within 10% when heterogeneity is ignored (homogeneous model), but errors of 15-20% may result, even when the level of heterogeneity is modest. In repeated trials in the presence of 5% noise, the mean of the estimates was always closer to the true value with the heterogeneous model than when heterogeneity was ignored, but the distributions of the estimates from the homogeneous and heterogeneous models overlapped for some parameters when outflow dilution curves were analyzed. The separation between the distributions was further reduced when tissue content curves were analyzed. It is concluded that multipath models accounting for flow heterogeneity are a vehicle for assessing the effects of flow heterogeneity under the conditions applicable to specific laboratory protocols, that efforts should be made to assess the actual level of flow heterogeneity in the organ being studied, and that the errors in parameter estimates are generally smaller when the input function is known rather than estimated by deconvolution.

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.

Regional myocardial blood volume and flow: first-pass MR imaging with polylysine-Gd-DTPA

The authors investigated the utility of an intravascular magnetic resonance (MR) contrast agent, poly-L-lysine-gadolinium diethylenetriaminepentaacetic acid (DTPA), for differentiating acutely ischemic from normally perfused myocardium with first-pass MR imaging. Hypoperfused regions, identified with microspheres, on the first-pass images displayed significantly decreased signal intensities compared with normally perfused myocardium (P < .0007). Estimates of regional myocardial blood content, obtained by measuring the ratio of areas under the signal intensity-versus-time curves in tissue regions and the left ventricular chamber, averaged 0.12 mL/g +/- 0.04 (n = 35), compared with a value of 0.11 mL/g +/- 0.05 measured with radiolabeled albumin in the same tissue regions. To obtain MR estimates of regional myocardial blood flow, in situ calibration curves were used to transform first-pass intensity-time curves into content-time curves for analysis with a multiple-pathway, axially distributed model. Flow estimates, obtained by automated parameter optimization, averaged 1.2 mL/min/g +/- 0.5 (n = 29), compared with 1.3 mL/min/g +/- 0.3 obtained with tracer microspheres in the same tissue specimens at the same time. The results represent a combination of T1-weighted first-pass imaging, intravascular relaxation agents, and a spatially distributed perfusion model to obtain absolute regional myocardial blood flow and volume.