Fractal 15O-labeled water washout from the heart

Network-driven anomalous transport is a fundamental component of brain microvascular dysfunction

Blood microcirculation supplies neurons with oxygen and nutrients, and contributes to clearing their neurotoxic waste, through a dense capillary network connected to larger tree-like vessels. This complex microvascular architecture results in highly heterogeneous blood flow and travel time distributions, whose origin and consequences on brain pathophysiology are poorly understood. Here, we analyze highly-resolved intracortical blood flow and transport simulations to establish the physical laws governing the macroscopic transport properties in the brain micro-circulation. We show that network-driven anomalous transport leads to the emergence of critical regions, whether hypoxic or with high concentrations of amyloid-β, a waste product centrally involved in Alzheimer's Disease. We develop a Continuous-Time Random Walk theory capturing these dynamics and predicting that such critical regions appear much earlier than anticipated by current empirical models under mild hypoperfusion. These findings provide a framework for understanding and modelling the impact of microvascular dysfunction in brain diseases, including Alzheimer's Disease.

Optical coherence tomography angiography

Optical coherence tomography (OCT) was one of the biggest advances in ophthalmic imaging. Building on that platform, OCT angiography (OCTA) provides depth resolved images of blood flow in the retina and choroid with levels of detail far exceeding that obtained with older forms of imaging. This new modality is challenging because of the need for new equipment and processing techniques, current limitations of imaging capability, and rapid advancements in both imaging and in our understanding of the imaging and applicable pathophysiology of the retina and choroid. These factors lead to a steep learning curve, even for those with a working understanding dye-based ocular angiography. All for a method of imaging that is a little more than 10 years old. This review begins with a historical account of the development of OCTA, and the methods used in OCTA, including signal processing, image generation, and display techniques. This forms the basis to understand what OCTA images show as well as how image artifacts arise. The anatomy and imaging of specific vascular layers of the eye are reviewed. The integration of OCTA in multimodal imaging in the evaluation of retinal vascular occlusive diseases, diabetic retinopathy, uveitis, inherited diseases, age-related macular degeneration, and disorders of the optic nerve is presented. OCTA is an exciting, disruptive technology. Its use is rapidly expanding in clinical practice as well as for research into the pathophysiology of diseases of the posterior pole.

Choriocapillaris Flow Features Follow a Power Law Distribution: Implications for Characterization and Mechanisms of Disease Progression

PURPOSE

To investigate flow characteristics of the choriocapillaris using optical coherence tomography angiography.

DESIGN

Retrospective observational case series.

METHODS

Visualization of flow in individual choriocapillary vessels is below the current resolution limit of optical coherence tomography angiography instruments, but areas of absent flow signal, called flow voids, are resolvable. The central macula was imaged with the Optovue RTVue XR Avanti using a 10-μm slab thickness in 104 eyes of 80 patients who ranged in age from 24 to 99 years of age. Automatic local thresholding of the resultant raw data with the Phansalkar method was analyzed with generalized estimating equations.

RESULTS

The distribution of flow voids vs size of the voids was highly skewed. The data showed a linear log-log plot and goodness-of-fit methods showed the data followed a power law distribution over the relevant range. A slope intercept relationship was also evaluated for the log transform and significant predictors for variables included age, hypertension, pseudodrusen, and the presence of late age-related macular degeneration (AMD) in the fellow eye.

CONCLUSIONS

The pattern of flow voids forms a scale invariant pattern in the choriocapillaris starting at a size much smaller than a choroidal lobule. Age and hypertension affect the choriocapillaris, a flat layer of capillaries that may serve as an observable surrogate for the neural or systemic microvasculature. Significant alterations detectable in the flow pattern in eyes with pseudodrusen and in eyes with late AMD in the fellow eye offer diagnostic possibilities and impact theories of disease pathogenesis.

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.

A model for transit time distributions through organs that accounts for fractal heterogeneity

It has been shown that density functions of organ transit time distributions of vascular markers (washout curves) are characterized by a power-law tail, reflecting the fractal nature of the vascular network. Yet, thus far, no closed-form model is available that can be fitted to such organ outflow data. Here we propose a model that accounts for the existing data. The model is a continuous mixture of inverse Gaussian densities, implying flow heterogeneity in the organ. It has been fitted to outflow data from the rabbit heart and rat liver. The power-law decay with exponent -3 observed in the heart, corresponds to an intra-organ flow distribution with a relative dispersion of about 35%.

Comparison of four mathematical models to analyze indicator-dilution curves in the coronary circulation

While several models have proven to result in accurate estimations when measuring cardiac output using indicator dilution, the mono-exponential model has primarily been chosen for deriving coronary blood/plasma volume. In this study, we compared four models to derive coronary plasma volume using indicator dilution; the mono-exponential, power-law, gamma-variate, and local density random walk (LDRW) model. In anesthetized goats (N = 14), we determined the distribution volume of high molecular weight (2,000 kDa) dextrans. A bolus injection (1.0 ml, 0.65 mg/ml) was given intracoronary and coronary venous blood samples were taken every 0.5–1.0 s; outflow curves were analyzed using the four aforementioned models. Measurements were done at baseline and during adenosine infusion. Absolute coronary plasma volume estimates varied by ~25% between models, while the relative volume increase during adenosine infusion was similar for all models. The gamma-variate, LDRW, and mono-exponential model resulted in volumes corresponding with literature, whereas the power-model seemed to overestimate the coronary plasma volume. The gamma-variate and LDRW model appear to be suitable alternative models to the mono-exponential model to analyze coronary indicator-dilution curves, particularly since these models are minimally influenced by outliers and do not depend on data of the descending slope of the curve only.

A model of competing saturable kinetic processes with application to the pharmacokinetics of the anticancer drug paclitaxel

A saturable multi-compartment pharmacokinetic model for the anti-cancer drug paclitaxel is proposed based on a meta-analysis of pharmacokinetic data published over the last two decades. We present and classify the results of time series for the drug concentration in the body to uncover the underlying power laws. Two dominant fractional power law exponents were found to characterize the tails of paclitaxel concentration-time curves. Short infusion times led to a power exponent of -1.57 ± 0.14, while long infusion times resulted in tails with roughly twice the exponent. Curves following intermediate infusion times were characterized by two power laws. An initial segment with the larger slope was followed by a long-time tail characterized by the smaller exponent. The area under the curve and the maximum concentration exhibited a power law dependence on dose, both with compatible fractional power exponents. Computer simulations using the proposed model revealed that a two-compartment model with both saturable distribution and elimination can reproduce both the single and crossover power laws. Also, the nonlinear dose-dependence is correlated with the empirical power law tails. The longer the infusion time the better the drug delivery to the tumor compartment is a clinical recommendation we propose.

Scaling rules for diffusive drug delivery in tumor and normal tissues

Delivery of blood-borne molecules and nanoparticles from the vasculature to cells in the tissue differs dramatically between tumor and normal tissues due to differences in their vascular architectures. Here we show that two simple measures of vascular geometry--δ(max) and λ--readily obtained from vascular images, capture these differences and link vascular structure to delivery in both tissue types. The longest time needed to bring materials to their destination scales with the square of δ(max), the maximum distance in the tissue from the nearest blood vessel, whereas λ, a measure of the shape of the spaces between vessels, determines the rate of delivery for shorter times. Our results are useful for evaluating how new therapeutic agents that inhibit or stimulate vascular growth alter the functional efficiency of the vasculature and more broadly for analysis of diffusion in irregularly shaped domains.

Assessment of myocardial ischaemia and viability: role of positron emission tomography

In developed countries, coronary artery disease (CAD) continues to be a major cause of death and disability. Over the past two decades, positron emission tomography (PET) imaging has become more widely accessible for the management of ischemic heart disease. Positron emission tomography has also emerged as an important alternative perfusion imaging modality in the context of recent shortages of molybdenum-99/technetium-99m ((99m)Tc). The clinical application of PET in ischaemic heart disease falls into two main categories: first, it is a well-established modality for evaluation of myocardial blood flow (MBF); second, it enables assessment of myocardial metabolism and viability in patients with ischaemic left ventricular dysfunction. The combined study of MBF and metabolism by PET has led to a better understanding of the pathophysiology of ischaemic heart disease. While there are potential future applications of PET for plaque and molecular imaging, as well as some clinical use in inflammatory conditions, this article provides an overview of the physical and biological principles behind PET imaging and its main clinical applications in cardiology, namely the assessment of MBF and metabolism.

Biophysical model of the spatial heterogeneity of myocardial flow

The blood flow in the myocardium has significant spatial heterogeneity. The objective of this study was to develop a biophysical model based on detailed anatomical data to determine the heterogeneity of regional myocardial flow during diastole. The model predictions were compared with experimental measurements in a diastolic porcine heart in the absence of vessel tone using nonradioactive fluorescent microsphere measurements. The results from the model and experimental measurements showed good agreement. The relative flow dispersion in the arrested, vasodilated heart was found to be 44% and 48% numerically and experimentally, respectively. Furthermore, the flow dispersion was found to have fractal characteristics with fractal dimensions (D) of 1.25 and 1.27 predicted by the model and validated by the experiments, respectively. This validated three-dimensional model of normal diastolic heart will play an important role in elucidating the spatial heterogeneity of coronary blood flow, and serve as a foundation for understanding the interplay between cardiac mechanics and coronary hemodynamics.

Fractal michaelis-menten kinetics under steady state conditions: Application to mibefradil

PURPOSE

To provide the first application of fractal kinetics under steady state conditions to pharmacokinetics as a model for the enzymatic elimination of a drug from the body.

MATERIALS AND METHODS

A one-compartment model following fractal Michaelis-Menten kinetics under a steady state is developed and applied to concentration-time data for the cardiac drug mibefradil in dogs. The model predicts a fractal reaction order and a power law asymptotic time-dependence of the drug concentration, therefore a mathematical relationship between the fractal reaction order and the power law exponent is derived. The goodness-of-fit of the model is assessed and compared to that of four other models suggested in the literature.

RESULTS

The proposed model provided the best fit to the data. In addition, it correctly predicted the power law shape of the tail of the concentration-time curve.

CONCLUSION

A simple one-compartment model with steady state fractal Michaelis-Menten kinetics describing drug elimination from the body most accurately describes the pharmacokinetics of mibefradil in dogs. The new fractal reaction order can be explained in terms of the complex geometry of the liver, the organ responsible for eliminating the drug.

Automatic fuzzy classification of the washout curves from magnetic resonance first-pass perfusion imaging after myocardial infarction

OBJECTIVES

We sought to investigate the diagnostic ability of cardiac magnetic resonance imaging (MRI) perfusion in acute reperfused myocardial infarction. The study used fuzzy logic to automatically classify signal intensity-time curves from myocardial segments into 3 categories: normal, hypointense, and Hyperintense.

MATERIALS AND METHODS

Thirty-eight patients with myocardial infarction underwent short-axis cine-MRI and contrast-enhanced MRI to provide data on wall thickening and the transmural extent of infarction. Of these, 17 had a second cardiac MRI to ascertain the functional recovery in each segment.

RESULTS

The fuzzy logic based classification performs well (kappa= 0.87, P < 0.01) in comparison with a visual approach. Segments providing "hypo" curves do not recover (Delta = 0.11 SD = 0.96) whereas segments demonstrating the other curve types recover (Delta = 1 SD = 1.98 for "hyper" curves and Delta = 1.54 SD = 1.77 for "normal" curves).

CONCLUSIONS

The proposed automatic signal intensity-time curve classification has a prognostic value when studying the functional recovery of pathologic segments and clearly identifies the no-reflow phenomenon known to induce poor recovery.

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.

Analysis of blood flow in the entire coronary arterial tree

A hemodynamic analysis of coronary blood flow must be based on the measured branching pattern and vascular geometry of the coronary vasculature. We recently developed a computer reconstruction of the entire coronary arterial tree of the porcine heart based on previously measured morphometric data. In the present study, we carried out an analysis of blood flow distribution through a network of millions of vessels that includes the entire coronary arterial tree down to the first capillary branch. The pressure and flow are computed throughout the coronary arterial tree based on conservation of mass and momentum and appropriate pressure boundary conditions. We found a power law relationship between the diameter and flow of each vessel branch. The exponent is ∼2.2, which deviates from Murray’s prediction of 3.0. Furthermore, we found the total arterial equivalent resistance to be 0.93, 0.77, and 1.28 mmHg·ml−1·s−1·g−1 for the right coronary artery, left anterior descending coronary artery, and left circumflex artery, respectively. The significance of the present study is that it yields a predictive model that incorporates some of the factors controlling coronary blood flow. The model of normal hearts will serve as a physiological reference state. Pathological states can then be studied in relation to changes in model parameters that alter coronary perfusion.

Impulse-response function of splanchnic circulation with model-independent constraints: theory and experimental validation

Modeling physiological processes using tracer kinetic methods requires knowledge of the time course of the tracer concentration in blood supplying the organ. For liver studies, however, inaccessibility of the portal vein makes direct measurement of the hepatic dual-input function impossible in humans. We want to develop a method to predict the portal venous time-activity curve from measurements of an arterial time-activity curve. An impulse-response function based on a continuous distribution of washout constants is developed and validated for the gut. Experiments with simultaneous blood sampling in aorta and portal vein were made in 13 anesthetized pigs following inhalation of intravascular [15O]CO or injections of diffusible 3-O-[11C]methylglucose (MG). The parameters of the impulse-response function have a physiological interpretation in terms of the distribution of washout constants and are mathematically equivalent to the mean transit time (T) and standard deviation of transit times. The results include estimates of mean transit times from the aorta to the portal vein in pigs: T = 0.35 +/- 0.05 min for CO and 1.7 +/- 0.1 min for MG. The prediction of the portal venous time-activity curve benefits from constraining the regression fits by parameters estimated independently. This is strong evidence for the physiological relevance of the impulse-response function, which includes asymptotically, and thereby justifies kinetically, a useful and simple power law. Similarity between our parameter estimates in pigs and parameter estimates in normal humans suggests that the proposed model can be adapted for use in humans.

In vivo measurement of coronary circulation angiotensin-converting enzyme activity in humans

Angiotensin-converting enzyme (ACE) is present on the luminal surface of the coronary vessels, mostly on capillary endothelium. ACE is also expressed on coronary smooth muscle cells and on plaque lipid-laden macrophages. Excessive coronary circulation (CC)-ACE activity might be linked to plaque progression. Here we used the biologically inactive ACE substrate (3)H-labeled benzoyl-Phe-Ala-Pro ([(3)H]BPAP) to quantify CC-ACE activity in 10 patients by means of the indicator-dilution technique. The results were compared with atherosclerotic burden determined by coronary angiography. There was a wide range of CC-ACE activity as revealed by percent [(3)H]BPAP hydrolysis (30-74%). The atherosclerotic extent scores ranged from 0.0 to 66.97, and the plaque area scores ranged from 0 to 80 mm(2). CC-ACE activity per unit extracellular space (V(max)/K(m)V(i)), an index of metabolically active vascular surface area, was correlated with myocardial blood flow (r = 0.738; P = 0.03) but not with measures of the atherosclerotic burden. These results show that CC-ACE activity can be safely measured in humans and that it is a good marker of the vascular area of the perfused myocardium. It does not, however, reflect epicardial atherosclerotic burden, suggesting that local tissue ACE may be more important in plaque development.

Multivariate statistics of disposition pharmacokinetic parameters for structurally unrelated drugs used in therapeutics

PURPOSE

To explore the quantitative structure pharmacokinetic relationships of the disposition parameters: clearance (CL), apparent volume of drug distribution (V(ap)), fractal clearance (CL(f)), and fractal volume (v(f)) for 272 structurally unrelated drugs used in therapeutics.

METHODS

Literature data were used for CL and V(ap) whereas CL(f) and v(f) were estimated as described previously (Pharm. Res. 18, 1056, 2001 and 19, 697, 2002). A variety of molecular descriptors expressing lipophilicity, ionization, molecular size and hydrogen bonding capacity were estimated using computer packages. The data were analyzed using multivariate statistics. For each disposition parameter (CL, V(ap) CL(f), v(f)) PCA (principal component analysis) and PLS (projection to latent structures) were applied to the total set of data as well as to subsets of data.

RESULTS

Drugs were divided into two classes (I and II) according to their v(f)/V(ap) ratio. Class I comprises 131 drugs with v(f)/V(ap) > 1, whereas class 11141 drugs with v(f)/V(ap) < 1. After initial PLS analysis, class I was subdivided in subclusters I(a) (30 drugs) and I(b) (101 drugs). It was found that I(a) included mostly acidic drugs with high protein binding, whereas class II comprises mainly basic, lipophilic compounds. No correlation was found between CL, V(ap), CL(f) and the used descriptors. Adequate PLS models were derived for v(f) considering subclusters I(a), I(b) and class II separately. The low v(f) values of class I(a) drugs were affected negatively from molecular size descriptors and non-polar surface area. For class I(b) drugs with intermediate v(f) values, apparent lipophilicity contributed positively, although molecular size descriptors and polarity were inhibitory factors. The high v(f) values of class II drugs were positively dependent on intrinsic lipophilicity and increased basicity, whereas polarity entered with negative contribution.

CONCLUSIONS

The parameters V(ap), CL, and CL(f) fail to reflect the physicochemical properties of drugs. The transformation of V(ap) values to v(f) is the underlying cause for the valid models for v(f). These models allow a global consideration of molecular properties (lipophilicity, ionization, molecular size, polar surface area) which govern the distribution of drugs in the human body. The present study provides additional evidence for the physiologically sound concept of v(f).

Fractal or biologically variable delivery of cardioplegic solution prevents diastolic dysfunction after cardiopulmonary bypass

OBJECTIVE

To determine whether myocardial protection is improved by restoring physiologic variability to the cardioplegia pressure signal during cardiopulmonary bypass, we compared cardiac function in pigs in the first hour after either conventional cold-blood cardioplegia (group CC) or computer-controlled biologically variable pulsatile cardioplegia (group BVC).

METHODS

Invasive monitors and sonomicrometry crystals were placed, and cardiopulmonary bypass was initiated. The aorta was crossclamped, and cold blood cardioplegic solution was infused intermittently through the aortic root with either conventional cardioplegia (n = 8) or biologically variable pulsatile cardioplegia (n = 8; mean pressure, 75 mm Hg for 85 minutes). The crossclamp was released, cardiac function was restored, and separation from cardiopulmonary bypass was completed. With stable temperature and arterial blood gases, hemodynamics and systolic and diastolic indices were compared at 15, 30, and 60 minutes after cardiopulmonary bypass.

RESULTS

Diastolic stiffness doubled from 0.027 +/- 0.016 mm Hg/mm (mean +/- SD) at baseline to 0.055 +/- 0.036 mm Hg/mm (P =.003) at 1 hour after bypass in group CC, associated with increased left ventricular end-diastolic pressure from 9 +/- 2 to 11 +/- 2 mm Hg (P =.001), mean pulmonary artery pressure from 14 +/- 2 to 20 +/- 3 mm Hg (P =.003), and serum lactate levels from 2.0 +/- 0.5 to 5.6 +/- 2.3 mmol/L (P =.008). Systolic function was not affected. In group BVC diastolic stiffness, left ventricular end-diastolic pressure, and pulmonary artery pressure values were not different from control values at any time after bypass, and serum lactate levels were significantly less than with conventional cold blood cardioplegia. Peak pressure variability with biologically variable pulsatile cardioplegia fit a power-law equation (exponent = -3.0; R(2) = 0.97), indicating fractal behavior.

CONCLUSION

Diastolic cardiac function is better preserved after cardiopulmonary bypass with biologically variable pulsatile cardioplegia and fractal perfusion. This may be attributed to enhanced microcirculatory perfusion with improved myocardial protection. A model supporting these results is presented.

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.

A stochastic model for the self-similar heterogeneity of regional organ blood flow

The theory of exponential dispersion models was applied to construct a stochastic model for heterogeneities in regional organ blood flow as inferred from the deposition of labeled microspheres. The requirements that the dispersion model be additive (or reproductive), scale invariant, and represent a compound Poisson distribution, implied that the relative dispersion (RD = standard deviation/mean) of blood flow should exhibit self-similar scaling in macroscopic tissue samples of masses m and m(ref) such that RD(m) = RD(m(ref)). (m/m(ref))(1-D), where D was a constant. Under these circumstances this empirical relationship was a consequence of a compound Poisson-gamma distribution that represented macroscopic blood flow. The model also predicted that blood flow, at the microcirculatory level, should also be heterogeneous but obey a gamma distribution-a prediction supported by observation.

A stochastic model for the self-similar heterogeneity of regional organ blood flow

The theory of exponential dispersion models was applied to construct a stochastic model for heterogeneities in regional organ blood flow as inferred from the deposition of labeled microspheres. The requirements that the dispersion model be additive (or reproductive), scale invariant, and represent a compound Poisson distribution, implied that the relative dispersion (RD = standard deviation/mean) of blood flow should exhibit self-similar scaling in macroscopic tissue samples of masses m and m(ref) such that RD(m) = RD(m(ref)). (m/m(ref))(1-D), where D was a constant. Under these circumstances this empirical relationship was a consequence of a compound Poisson-gamma distribution that represented macroscopic blood flow. The model also predicted that blood flow, at the microcirculatory level, should also be heterogeneous but obey a gamma distribution-a prediction supported by observation.

Endothelial adenosine transporter characterization in perfused guinea pig hearts

Adenosine (Ado), a smooth muscle vasodilator and modulator of cardiac function, is taken up by many cell types via a saturable transporter, blockable by dipyridamole. To quantitate the influences of endothelial cells in governing the blood-tissue exchange of Ado and its concentration in the interstitial fluid, one must define the permeability-surface area products (PS) for Ado via passive transport through interendothelial gaps [PS(g)(Ado)] and across the endothelial cell luminal membrane (PS(ecl)) in their normal in vivo setting. With the use of the multiple-indicator dilution (MID) technique in Krebs-Ringer perfused, isolated guinea pig hearts (preserving endothelial myocyte geometry) and by separating Ado metabolites by HPLC, we found permeability-surface area products for an extracellular solute, sucrose, via passive transport through interendothelial gaps [PS(g)(Suc)] to be 1.9 +/- 0.6 ml. g(-1). min(-1) (n = 16 MID curves in 4 hearts) and took PS(g)(Ado) to be 1. 2 times PS(g)(Suc). MID curves were obtained with background nontracer Ado concentrations up to 800 micrometer, partially saturating the transporter and reducing its effective PS(ecl) for Ado. The estimated maximum value for PS(ecl) in the absence of background adenosine was 1.1 +/- 0.1 ml. g(-1). min(-1) [maximum rate of transporter conformational change to move the substrate from one side of the membrane to the other (maximal velocity; V(max)) times surface area of 125 +/- 11 nmol. g(-1). min(-1)], and the Michaelis-Menten constant (K(m)) was 114 +/- 12 microM, where +/- indicates 95% confidence limits. Physiologically, only high Ado release with hypoxia or ischemia will partially saturate the transporter.

Strategies for the physiome project

The physiome is the quantitative description of the functioning organism in normal and pathophysiological states. The human physiome can be regarded as the virtual human. It is built upon the morphome, the quantitative description of anatomical structure, chemical and biochemical composition, and material properties of an intact organism, including its genome, proteome, cell, tissue, and organ structures up to those of the whole intact being. The Physiome Project is a multicentric integrated program to design, develop, implement, test and document, archive and disseminate quantitative information, and integrative models of the functional behavior of molecules, organelles, cells, tissues, organs, and intact organisms from bacteria to man. A fundamental and major feature of the project is the databasing of experimental observations for retrieval and evaluation. Technologies allowing many groups to work together are being rapidly developed. Internet II will facilitate this immensely. When problems are huge and complex, a particular working group can be expert in only a small part of the overall project. The strategies to be worked out must therefore include how to pull models composed of many submodules together even when the expertise in each is scattered amongst diverse institutions. The technologies of bioinformatics will contribute greatly to this effort. Developing and implementing code for large-scale systems has many problems. Most of the submodules are complex, requiring consideration of spatial and temporal events and processes. Submodules have to be linked to one another in a way that preserves mass balance and gives an accurate representation of variables in nonlinear complex biochemical networks with many signaling and controlling pathways. Microcompartmentalization vitiates the use of simplified model structures. The stiffness of the systems of equations is computationally costly. Faster computation is needed when using models as thinking tools and for iterative data analysis. Perhaps the most serious problem is the current lack of definitive information on kinetics and dynamics of systems, due in part to the almost total lack of databased observations, but also because, though we are nearly drowning in new information being published each day, either the information required for the modeling cannot be found or has never been obtained. "Simple" things like tissue composition, material properties, and mechanical behavior of cells and tissues are not generally available. The development of comprehensive models of biological systems is a key to pharmaceutics and drug design, for the models will become gradually better predictors of the results of interventions, both genomic and pharmaceutic. Good models will be useful in predicting the side effects and long term effects of drugs and toxins, and when the models are really good, to predict where genomic intervention will be effective and where the multiple redundancies in our biological systems will render a proposed intervention useless. The Physiome Project will provide the integrating scientific basis for the Genes to Health initiative, and make physiological genomics a reality applicable to whole organisms, from bacteria to man.

A class of flow bifurcation models with lognormal distribution and fractal dispersion

We report a quantitative analysis of a simple dichotomous branching tree model for blood flow in vascular networks. Using the method of moment-generating function and geometric Brownian motion from stochastic mathematics, our analysis shows that a vascular network with asymmetric branching and random variation at each bifurcating point gives rise to an asymptotic lognormal flow distribution with a positive skewness. The model exhibits a fractal scaling in the dispersion of the regional flow in the branches. Experimentally measurable fractal dimension of the relative dispersion in regional flow is analytically calculated in terms of the asymmetry and the variance at local bifurcation; hence the model suggests a powerful method to obtain the physiological information on local flow bifurcation in terms of flow dispersion analysis. Both the fractal behavior and the lognormal distribution are intimately related to the fact that it is the logarithm of flow, rather than flow itself, which is the natural variable in the tree models. The kinetics of tracer washout is also discussed in terms of the lognormal distribution.

The anomalous pharmacokinetics of amiodarone explained by nonexponential tissue trapping

Conventional pharmacokinetic (PK) concepts fail to describe the long-term pharmacokinetics of the extremely cationic amphiphilic drug amiodarone. A nonclassical model based on the phenomenon of trapping at tissue binding sites with very long release times is presented, which implies that a volume of distribution and a steady-state level cannot be defined. In agreement with clinical PK data available in the literature, the model well describes not only single-dose disposition curves but also the persistently increasing plasma concentration-time curve during long-term treatment (up to 5 years) and the washout curve following cessation of therapy. The novel aspect is a long-tailed tissue residence time distribution which is incorporated into a recirculatory model leaving the initial distribution process and the clearance concept unchanged. The underlying theoretical approach, which is known as "strange or anomalous" kinetics in physical sciences, and the fractal scaling property of the model may enhance our understanding of the PK of extremely hydrophobic xenobiotics.

Heterogeneities in myocardial flow and metabolism: exacerbation with abnormal excitation

Because regional myocardial blood flows are remarkably heterogeneous-with a 6- to 10-fold range of flows in normal hearts-and because the spatial profiles of the flows are stable over long periods and over a range of conditions, the relation between flows and other physiologic functions has been explored. Local fatty acid uptake and oxygen consumption are almost linearly related to the flows. Coronary network structure and hydrodynamic resistances give suitable flow heterogeneity but are thought to be a response to local needs rather than being causative. Presumably the cause is the need for adenosine triphosphate (ATP) synthesis locally, and therefore flows, substrate delivery, and oxygen utilization are driven primarily by local rates of ATP hydrolysis, mainly by contractile proteins. This hypothesis is by no means fully tested. Data on pacing dog hearts from different sites, on patients with left bundle branch block, and on unloading transplanted rat hearts, all point in the same direction: unloading ventricular muscle leads to diminished flow and exaggeratedly diminished glucose uptake. The mechanism is likely to be that discovered by Taegtmeyer and colleagues, namely, the expression of fetal genes in regions where the muscle is unloaded and particular metabolic enzymes and transporters are downregulated.

The Cardiome Project. An integrated view of cardiac metabolism and regional mechanical function

The goal, to develop a functioning three-dimensional computational model of the excitation, metabolism and contraction of the heart within three years, is one of the beginnings for the Cardiome Project. Our first stage will not be likely to provide highly accurate prediction of physiological behavior in general, but will be focussed so that it is adequate for at least three specific purposes: response to regional flow reduction, response to heart rate changes, and response to increased metabolic drive. We would like to make the model visualizable by three-dimensional viewing, with cross-sectional and transparency viewing approaches, illustrate the fiber directions, the arteries, the deformation with contraction and images of regional functions such as oxygen consumption, preejection strain, or lactate concentration. The display techniques developed by Hunter et al. and by McCulloch et al. would be excellent for such demonstration and teaching purposes, and should be attractive enough for public display. The Physiome Project is underway now, with growing government and private support. Now we are going from the era of molecular biology, led by the Genome Project, into a new era of integrative biology. The goal is to understand biology so deeply and so broadly that predictions about interventions can be made. Methods of experimentation and of diagnosis are critical to acquiring the data, and therefore in making the prediction, and so all aspects of our Society's efforts and interests are relevant to undertaking this grand challenge.

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.

Power-law kinetics of tracer washout from physiological systems

Recent studies suggest that the tail of the washout of tracer-labeled substances from physiological systems can exhibit power-law behavior. In this work we develop a theoretical interpretation of the power-law behavior of the flow-limited washout of tracer-labeled water from the myocardium. Using minimal assumptions concerning the complicated structure of the coronary network we show that the washout from a heterogeneous flow system is given by h(t) approximately equal to A x p1 (V/t)(-beta), where beta is close to 3, p1 is the probability density of flows through the system, V is a constant volume associated with each pathway, and A is a constant. This prediction fits observed power-law washout behavior of tracer water in the heart. This theory is general enough to lead us to speculate that close examination of transport in other heterogeneity-perfused systems is likely to reveal similar power-law behavior.

Blood flows and metabolic components of the cardiome

This is a plan for the first stage of The Cardiome Project. The cardiome is the representation, in quantitative, testable form, of the functioning of the normal heart and its responses to intervention. The goal is to integrate the efforts of many years into a comprehensive understandable scheme. Past efforts have spanned the fields of transport within blood vessels, the distributions of regional coronary blood flows, permeation processes through capillary and cell walls, mediated cell membrane transport, extra- and intracellular diffusion, cardiac electrophysiology, the uptake and metabolism of the prime substrates (fatty acid and glucose), the metabolism of the purine nucleosides and nucleotides (mainly adenosine and ATP), the regulation of the ionic currents and of excitation-contraction coupling and finally the regulation of contraction. The central theme is to define the coronary flows and metabolic components of a computer model that will become a part of a three-dimensional heart with appropriate fibre shortening and volume ejection. The components are: (a) coronary flow distributions with appropriate heterogeneity, (b) metabolism of the substrates for energy production, (c) ATP, PCr and energy metabolism and (d) calcium metabolism as it relates to excitation-contraction coupling. The modeling should provide: (1) appropriate responses to regional ischemia induced by constriction of a coronary artery, including tissue contractility loss and aneurysmal dilation of the ischemic region; (2) physiological responses to rate changes such as treppe and changes in metabolic demand and (3) changes in local metabolic needs secondary to changes in the site of pacing stimulation and shortening inactivation or stretch activation of contraction.

Noncompartmental models of whole-body clearance of tracers: a review

Noncompartmental models are defined as models that allow for transport of material through regions of the body that are not necessarily well-mixed or of uniform concentration. The clearance of a substance of interest (metabolite or drug) from a noncompartmental system will not necessarily be governed by a sum of exponentials or even be describable by a set of ordinary differential equations. The model may involve diffusion or other random walk processes, leading to the solution in terms of the partial differential equation of diffusion or in terms of probability distributions. It may use the theory of linear systems to obviate the need for defining any precise anatomical structure. A number of the models reviewed deal with plasma clearance curves that are best described by power functions of time. Circulatory models are reviewed from their inception to the present. Recent studies on clearance as a fractal process are introduced.

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.

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.

The imaging science of positron emission tomography

To meet the goals of converging molecular imaging with molecular biology and molecular medicine, there is a need to define the strategy and structure for perfecting the accuracy of functional images derived using PET. This also relates directly to how clinical research, diagnostic questions and challenges from the pharmaceutical industry are addressed. In order to exploit the sensitivity and specificity of PET, an integrated, multidisciplinary approach is imperative. The structure to provide this needs to been seen in the context of an institutional approach, collaborations within the academic and industrial sectors and the funding needed to meet the challenges of addressing difficult questions.