The interpretation of mean transit time measurements for multiphase tissue systems

A model-independent proof of the central-volume principle for multiphase tissue systems is presented. This derivation emphasizes the transport processes which occur within the system, and the physical constraints which the system must satisfy for valid application of the principle. These constraints include: (i) the fluid flowing into the system must be equivalentlylabelled; (ii) the system under study must be part of a larger system which has no diffusive inlets or outlets. The derivation shows that the definition of the volume of distribution and the choice of appropriate partition coefficients for evaluation of this quantity, are independent of any assumption concerning tracer equilibrium between phases as a whole. A less restrictive definition of equivalent labelling also results from the proof.The relationship between the mean transit times measured by global residue detection and by “snapshot” outflow detection is derived. If diffusion of tracer across the boundaries of the monitored system is significant compared to convective transport, then these two transit times will not have identical values.The conditions under which valid measurements of regional, i.e. local, physiological variables can be performed by residue detection are also discussed, and it is shown that regional residue-detection measurements may be used to assess variations in anatomical structure throughout a larger region. However, the use of the height/area method for calculating regional perfusion can lead to error when tracer enters the detector field by diffusion, as opposed to convection, or when a significant amount of diffusion occurs before tracer enters the observed region.     

The variance of the distribution of traversal times in a capillary bed

A random walk model of capillary tracer transit times is developed that treats simulataneously: plug flow in the capillary, radial and axial diffusion in the capillary cylinder and tissue annulus, and endothelial barriers to solute transport. The mean transit time is simply the volume of distribution divided by blood flow. Variance of transit times has additive terms for radial, axial, and barrier influences that are reduceable to variances of simpler models of capillary exchange. The dependence of variance on the solute diffusion coefficient is not monotonic, but has a minimum near 0·5 × 10?6 cm²/s for reasonable parameters and no barrier, Small molecules like inert gases are expected to have larger variances with higher diffusion coefficients, while larger molecules and barrier limited solutes will have the reverse dependence. Available literature data indicates that capillary heterogeneity will have a major influence on whole-body variance of transit times.     

A permeation-diffusion-reaction model of gas transport in cellular tissue of plant materials

Gas transport in fruit tissue is governed by both diffusion and permeation. The latter phenomenon is caused by overall pressure gradients which may develop due to the large difference in O(2) and CO(2) diffusivity during controlled atmosphere storage of the fruit. A measurement set-up for tissue permeation based on unsteady-state gas exchange was developed. The gas permeability of pear tissue was determined based on an analytical gas transport model. The overall gas transport in pear tissue samples was validated using a finite element model describing simultaneous O(2), CO(2), and N(2) gas transport, taking into account O(2) consumption and CO(2) production due to respiration. The results showed that the model described the experimentally determined permeability of N(2) very well. The average experimentally determined values for permeation of skin, cortex samples, and the vascular bundle samples were (2.17+/-1.71)x10(-19) m(2), (2.35+/-1.96)x10(-19) m(2), and (4.51+/-3.12)x10(-17) m(2), respectively. The permeation-diffusion-reaction model can be applied to study gas transport in intact pears in relation to product quality.     

Water transport through plant tissue: the apoplasm and symplasm pathways

A detailed quantitative analysis of water flow through the apoplasm and symplasm of plant tissue is presented. The analysis results in two coupled diffusion equations which describe water transport in the two pathways. Various parameters entering the analysis identify the physical properties of the tissue which control the transport process as the resistance to water flow per cell in the two parallel pathways, the resistance per cell between pathways, and the water capacity per cell in the two pathways. Values for the several resistances and water capacities are estimated from available data, and a model problem is solved wherein a sheet of tissue at an initial water potential of — δ bars is immersed in a container of water. The resulting solutions show that depending on the values assigned to the controlling parameters, local water potential equilibrium between each cell and its cell wall may or may not obtain. In the special case of local equilibrium (water potential in the symplasm and apoplasm pathways essentially equal), the transport process can be described by a single diffusion equation which is derived along with an expression for the tissue diffusivity. It is concluded that the weakest link in the analysis is the estimated value for the permeability of the plasmodesma membrane, and that a logical extension of the theory would be to include the effects of a diffusable solute.     

Revision of the theory of tracer transport and the convolution model of dynamic contrast enhanced magnetic resonance imaging

Counterexamples are used to motivate the revision of the established theory of tracer transport. Then dynamic contrast enhanced magnetic resonance imaging in particular is conceptualized in terms of a fully distributed convection–diffusion model from which a widely used convolution model is derived using, alternatively, compartmental discretizations or semigroup theory. On this basis, applications and limitations of the convolution model are identified. For instance, it is proved that perfusion and tissue exchange states cannot be identified on the basis of a single convolution equation alone. Yet under certain assumptions, particularly that flux is purely convective at the boundary of a tissue region, physiological parameters such as mean transit time, effective volume fraction, and volumetric flow rate per unit tissue volume can be deduced from the kernel.      

Cerebrovascular response after interstitial irradiation.

To characterize the role of the cerebrovascular response in the development of brain injury after focal irradiation, 125I sources were implanted in frontal white matter of the brain of normal dogs; dose was 20 Gy, 7.5 mm from the source. Cerebral blood flow, vascular volume and mean transit time of blood were quantified in irradiated tissues relative to tissues in the contralateral hemisphere and analyzed with respect to previously determined volumetric measurements of damage and the blood-to-brain transfer constant. Blood flow and vascular volume within the radiation-induced focal lesion were maximally reduced 3 weeks after implant, when necrosis volume was maximal. By 6 weeks, vascular volume and mean transit time were increased, suggesting a strong neovascular response. In tissues surrounding the lesion, blood flow and vascular volume were reduced 1-4 weeks after irradiation and approached normal at 6 weeks; average mean transit time was not altered significantly. Alterations in blood flow and mean transit time were significantly related to edema volume and transfer constant, but alterations in vascular volume were not, suggesting that edema-induced vascular compression was not responsible for changes in blood flow. Reductions of radiation-induced permeability of the blood-brain barrier and/or edema might limit radiation-induced changes in blood flow and the extent of tissue injury.     

Site-specific effects of compression on macromolecular diffusion in articular cartilage

Articular cartilage is the connective tissue that lines joints and provides a smooth surface for joint motion. Because cartilage is avascular, molecular transport occurs primarily via diffusion or convection, and cartilage matrix structure and composition may affect diffusive transport. Because of the inhomogeneous compressive properties of articular cartilage, we hypothesized that compression would decrease macromolecular diffusivity and increase diffusional anisotropy in a site-specific manner that depends on local tissue strain. We used two fluorescence photobleaching methods, scanning microphotolysis and fluorescence imaging of continuous point photobleaching, to measure diffusion coefficients and diffusional anisotropy of 70 kDa dextran in cartilage during compression, and measured local tissue strain using texture correlation. For every 10% increase in normal strain, the fractional change in diffusivity decreased by 0.16 in all zones, and diffusional anisotropy increased 1.1-fold in the surface zone and 1.04-fold in the middle zone, and did not change in the deep zone. These results indicate that inhomogeneity in matrix structure and composition may significantly affect local diffusive transport in cartilage, particularly in response to mechanical loading. Our findings suggest that high strains in the surface zone significantly decrease diffusivity and increase anisotropy, which may decrease transport between cartilage and synovial fluid during compression.     

Investigating sensitivity coefficients characterizing the response of a model of tau protein transport in an axon to model parameters

Evaluating the sensitivity of biological models to various model parameters is a critical step towards advancing our understanding of biological systems. In this paper, we investigated sensitivity coefficients for a model simulating transport of tau protein along the axon. This is an important problem due to the relevance of tau transport and agglomeration to Alzheimer’s disease and other tauopathies, such as some forms of parkinsonism. The sensitivity coefficients that we obtained characterize how strongly three observables (the tau concentration, average tau velocity, and the percentage of tau bound to microtubules) depend on model parameters. The fact that the observables strongly depend on a parameter characterizing tau transition from the retrograde to the anterograde kinetic states suggests the importance of motor-driven transport of tau. The observables are sensitive to kinetic constants characterizing tau concentration in the free (cytosolic) state only at small distances from the soma. Cytosolic tau can only be transported by diffusion, suggesting that diffusion-driven transport of tau only plays a role in the proximal axon. Our analysis also shows the location in the axon in which an observable has the greatest sensitivity to a certain parameter. For most parameters, this location is in the proximal axon. This could be useful for designing an experiment aimed at determining the value of this parameter. We also analyzed sensitivity of the average tau velocity, the total tau concentration, and the percentage of microtubule-bound tau to cytosolic diffusivity of tau and diffusivity of bound tau along the MT lattice. The model predicts that at small distances from the soma the effect of these two diffusion processes is comparable.     

Determination of erythrocyte transit times through micropores. I--Basic operational principles

To study the transit times of each red blood cell passing through cylindrical micropores and in order to evaluate sub-population of cells with regard to their deformability, we have developed a new system called the cell transit time analyser (CTTA). By using an AC voltage (100 KHz) across a special filter, we measure the electrical conductance change produced by the cells passing through the pores under a known driving pressure. This computer based device provides the distribution of transit times tau for 2000 cells in 1 minute and as a result the mean transit time [tau]. Experiments with red cells were designed to evaluate the flow behavior of both normal cells and cells whose mechanical properties were artificially altered. Cell volume was changed by use of non-isotonic media. Cell shape and cell volume were modified by varying the pH of the suspending buffer. Results of these experiments are: 1) a skew distribution of transit times towards high tau values for both control cells and artificially altered cells is observed: 2) [tau] is minimum for isotonic conditions and increases sharply for either hypotonic or hypertonic media: 3) [tau] is minimum at physiological pH and increases for either acid or alcaline changes of pH.     

Use of diazepam for interpreting changes in extravascular lung water.

Estimates of extravascular lung water volume (Qew) by use of the multiple indicator-dilution method with a hydrophilic indicator such as tritiated water, along with a vascular reference indicator, depend not only on tissue hydration but also on tissue perfusion. Separation of these effects might be facilitated if both hydrophilic and lipophilic indicators were used, with the assumption that the extravascular volume accessible to the lipophilic indicator would be independent of hydration. We found that in isolated perfused dog lung lobes the extravascular volume accessible to the lipophilic amine [14C]diazepam (Qed) was inversely proportional to the albumin concentration of the perfusate. This suggested that while the bolus was in the lungs, only a small fraction of the diazepam was in the aqueous phase of either lung tissue or perfusate. Changing the flow rate over a fairly wide range had little influence on the pattern of the tritiated water or [14C]diazepam effluent concentration curves when time was normalized to the lobar mean transit time. This suggests that the association of the diazepam with both the plasma albumin and the lipoid fraction of the tissue was in very rapid equilibrium on the time scale of a single pass through the lung lobe and that there was little barrier to its diffusion to and from the tissue. When the extravascular water volume was increased by either raising the hydrostatic pressure or instilling saline into the airways, both Qew and Qew/Qed increased.(ABSTRACT TRUNCATED AT 250 WORDS)     

Use of norepinephrine uptake to measure lung capillary recruitment with exercise

We used the multiple indicator-dilution technique with norepinephrine, a vascular endothelium surface marker, to study the pulmonary vascular changes in awake exercising dogs. The vascular space tracers, labeled erythrocytes and albumin, and a water space tracer, 1,8-octanediol, were injected with the norepinephrine, and right atrium-aortic root dilution curves were obtained in nine dogs, at rest and at two increasing levels of exercise. Extravascular lung water multiple tracer dilutional estimates increased with flow and rapidly approached a maximal asymptotic value representing 75% of the postmortem lung weight. The ratio of the extravascular lung water measured in this way to that measured gravimetrically also increased, to reach an asymptotic proportion of close to 100%. The transit time-defined central vascular space increased linearly with flow; the ratio of lung tissue space to lung vascular space, therefore, decreased with increasing flow. The mean tracer upslope norepinephrine extractions at rest and at the two levels of exercise were 17 +/- 1.2, 14 +/- 0.8, and 15 +/- 0.8% (SE). With the use of the Crone approximation, we computed permeability-surface area products for norepinephrine; these increased linearly with flow. If permeability does not change, the increase in the permeability-surface area product with flow can be attributed to capillary recruitment. We conclude that when all lung tissue has become accessible to 1,8-octanediol delivered via the perfused vascular space, there is nevertheless further recruitment, with increase in flow, of vascular surface that can extract norepinephrine.     

Transit time dispersion in pulmonary and systemic circulation: effects of cardiac output and solute diffusivity

We present an in vivo method for analyzing the distribution kinetics of physiological markers into their respective distribution volumes utilizing information provided by the relative dispersion of transit times. Arterial concentration-time curves of markers of the vascular space [indocyanine green (ICG)], extracellular fluid (inulin), and total body water (antipyrine) measured in awake dogs under control conditions and during phenylephrine or isoproterenol infusion were analyzed by a recirculatory model to estimate the relative dispersions of transit times across the systemic and pulmonary circulation. The transit time dispersion in the systemic circulation was used to calculate the whole body distribution clearance, and an interpretation is given in terms of a lumped organ model of blood-tissue exchange. As predicted by theory, this relative dispersion increased linearly with cardiac output, with a slope that was inversely related to solute diffusivity. The relative dispersion of the flow-limited indicator antipyrine exceeded that of ICG (as a measure of intravascular mixing) only slightly and was consistent with a diffusional equilibration time in the extravascular space of approximately 10 min, except during phenylephrine infusion, which led to an anomalously high relative dispersion. A change in cardiac output did not alter the heterogeneity of capillary transit times of ICG. The results support the view that the relative dispersions of transit times in the systemic and pulmonary circulation estimated from solute disposition data in vivo are useful measures of whole body distribution kinetics of indicators and endogenous substances. This is the first model that explains the effect of flow and capillary permeability on whole body distribution of solutes without assuming well-mixed compartments.     

Measurement of Gd-DTPA diffusion through PVA hydrogel using a novel magnetic resonance imaging method.

Polyvinyl alcohol-cryogel (PVA-C) is a hydrogel that is an excellent tissue mimic. In order to characterize mass transfer in this material, as well as to demonstrate in principle the ability to noninvasively measure solute diffusion in tissue, we measured the diffusion coefficient of the magnetic resonance (MR) contrast agent gadolinium diethylene triaminopentaacetic acid (Gd-DTPA) through PVA-C using a clinical MR imager. The method involved filling thick-walled rectangular PVA-C "cups" with known concentrations of Gd-DTPA solutions. Then by using a fast inversion recovery spin echo MR imaging protocol, a signal "null" contour was created in the MR image that corresponded to a second, known concentration of Gd-DTPA. By collecting a series of MR images through the PVA-C wall as a function of time, the displacement of this second known isoconcentration contour could be tracked. Application of Fick's second law of diffusion yielded the diffusion coefficient. Seven separate experiments were performed using various combinations of initial concentrations of Gd-DTPA within the PVA-C cups (3.2, 25.6, or 125 mM) and tracked isoconcentrations contours (0.096, 0.182, or 0.435 mM Gd-DTPA). The experimental results and the predictions of Fick's law were in excellent agreement. The diffusivity of Gd-DTPA through 10% PVA hydrogel was found to be (2.6 +/- 0.04) x 10(-10) m(2)/s (mean +/- s.e.m.). Separate permeability studies showed that the diffusion coefficient of Gd-DTPA through this hydrogel did not change with an applied pressure of up to 7.1 kPa. Accurate measurements could be made within 30 min if suitable Gd-DTPA concentrations were selected. Due to the excellent repeatability and fast data acquisition time, this technique is very promising for future in vivo studies of species transport in tissue.     

O2 exchange between blood and brain tissues studied with 18O2 indicator-dilution technique

A technique has been developed to record 18O2 dilution curves of an organ in vivo by use of 51Cr-labeled erythrocytes as a reference tracer. The technique employs anaerobic sampling of venous outflow following an intraarterial injection of tracer-laden blood and off-line determination of [18O2] and [51Cr] profiles in the venous outflow. O2 and reference indicator-dilution curves of cerebral circulation were recorded in eight experiments with six halothane-anesthetized dogs. Autologous blood labeled with the tracers was injected into a carotid artery, and brain venous outflow was sampled from the sagittal sinus. The total net extraction of O2 tracer was equal to the extraction of elemental O2. Instantaneous extraction of 18O2 along the outflow curve fell linearly with time, from an initial value of 0.6-0.7 to very small or even negative values toward the end of a pulse. This indicates that O2 undergoes a flow-limited distribution. In all experiments, the mean transit time of unmetabolized 18O2 was longer than the mean transit time of the Cr tracer. An index of the tissue O2 dilution space, hence the mean tissue PO2, is calculated from this data with the use of a modified central volume principle. This estimate of mean tissue PO2 increases as a linear function of sagittal sinus PO2 with a slope of 0.97. The method may provide an index of the critical PO2 of venous blood, the PO2 below which O2 diffusion from blood to tissue may limit its rate of metabolic uptake.     

Dynamics of water transport and storage in conifers studied with deuterium and heat tracing techniques

The volume and complexity of their vascular systems make the dynamics of long-distance water transport in large trees difficult to study. We used heat and deuterated water (D2)) as tracers to characterize whole-tree water transport and storage properties in individual trees belonging to the coniferous species Pseudotsuga menziesii (Mirb.) Franco and Tsuga heterophylla (Raf.) Sarg. The trees used in this study spanned a broad range of height (13.5-58 m) and diameter (0.14-1.43 m). Sap flow was monitored continuously with heat dissipation probes near the base of the trunk prior to, during and following injection of D2O. The transit time for D2O transport from the base of the trunk to the upper crown and the tracer residence time were determined by measuring hydrogen isotope ratios in water extracted from leaves sampled at regular intervals. Transit times for arrival of D2O in the upper crown ranged from 2.5 to 21 d and residence times ranged from 36 to 79 d. Estimates of maximum sap velocity derived from tracer transit times and path length ranged from 2.4 to 5.4 m d(-1). Tracer residence time and half-life increased as tree diameter increased, independent of species. Species-independent scaling of tracer velocity with sapwood-specific conductivity was also observed. When data from this study were combined with similar data from an earlier study of four tropical angiosperm trees, species-independent scaling of tracer velocity and residence time with sapwood hydraulic capacitance was observed. Sapwood capacitance is an intrinsic tissue-level property that appears to govern whole-tree water transport in a similar manner among both tracheid- and vessel-bearing species.     

A transient diffusion model yields unitary gap junctional permeabilities from images of cell-to-cell fluorescent dye transfer between Xenopus oocytes

As ubiquitous conduits for intercellular transport and communication, gap junctional pores have been the subject of numerous investigations aimed at elucidating the molecular mechanisms underlying permeability and selectivity. Dye transfer studies provide a broadly useful means of detecting coupling and assessing these properties. However, given evidence for selective permeability of gap junctions and some anomalous correlations between junctional electrical conductance and dye permeability by passive diffusion, the need exists to give such studies a more quantitative basis. This article develops a detailed diffusion model describing experiments (reported separately) involving transport of fluorescent dye from a "donor" region to an "acceptor" region within a pair of Xenopus oocytes coupled by gap junctions. Analysis of transport within a single oocyte is used to determine the diffusion and binding characteristics of the cellular cytoplasm. Subsequent double-cell calculations then yield the intercellular junction permeability, which is translated into a single-channel permeability using concomitant measurements of intercellular conductance, and known single-channel conductances of gap junctions made up of specific connexins, to count channels. The preceding strategy, combined with use of a graded size series of Alexa dyes, permits a determination of absolute values of gap junctional permeability as a function of dye size and connexin type. Interpretation of the results in terms of pore theory suggests significant levels of dye-pore affinity consistent with the expected order of magnitude of typical (e.g., van der Waals) intermolecular attractions.     

Optimizing Detection of Tissue Anisotropy by Fluorescence Recovery after Photobleaching

Fluorescence recovery after photobleaching (FRAP) has been widely used to measure fluid flow and diffusion in gels and tissues. It has not been widely used in detection of tissue anisotropy. This may be due to a lack of applicable theory, or due to inherent limitations of the method. We discuss theoretical aspects of the relationship between anisotropy of tissue structure and anisotropy of diffusion coefficients, with special regard to the size of the tracer molecule used. We derive a semi-mechanistic formula relating the fiber volume fraction and ratio of fiber and tracer molecule diameters to the expected anisotropy of the diffusion coefficients. This formula and others are tested on simulated random walks through random simulated and natural media. We determine bounds on the applicability of FRAP for detection of tissue anisotropy, and suggest minimum tracer sizes for detection of anisotropy in tissues of different composition (fiber volume fraction and fiber diameter). We find that it will be easier to detect anisotropy in monodisperse materials than in polydisperse materials. To detect mild anisotropy in a tissue, such as cartilage, which has a low fiber fraction would require a tracer molecule so large that it would be difficult to deliver to the tissue. We conclude that FRAP can be used to detect tissue anisotropy when the tracer molecule is sufficiently large relative to the fiber diameter, volume fraction, and degree of polydispersivity, and when the anisotropy is sufficiently pronounced.     

Matrix proof of flow,volume and mean transit time theorems for regional and compartmental systems

The relations (inflow) = (dose)/(area under indicator curve), and (volume of distribution) = (throughflow) × (mean transit time) are derived by a matrix method for a system of interconnected subsystems, within which spatial indicator activity gradients may exist, and for compartments, within which the indicator activity is spatially uniform. The inflow theorem, is different from the outflow theorem. Equivalent labeling of multi-input systems reduces them formally to single input systems. Foreign indicator flow-volume kinetics are more general than, and include as a special case, tracer flux-mass (metabolic) kinetics. Volume of distribution in the indicator steady state may be different from the equilibrium volume of distribution.     

Investigation of the Spatiotemporal Responses of Nanoparticles in Tumor Tissues with a Small-Scale Mathematical Model

The transport and accumulation of anticancer nanodrugs in tumor tissues are affected by many factors including particle properties, vascular density and leakiness, and interstitial diffusivity. It is important to understand the effects of these factors on the detailed drug distribution in the entire tumor for an effective treatment. In this study, we developed a small-scale mathematical model to systematically study the spatiotemporal responses and accumulative exposures of macromolecular carriers in localized tumor tissues. We chose various dextrans as model carriers and studied the effects of vascular density, permeability, diffusivity, and half-life of dextrans on their spatiotemporal concentration responses and accumulative exposure distribution to tumor cells. The relevant biological parameters were obtained from experimental results previously reported by the Dreher group. The area under concentration-time response curve (AUC) quantified the extent of tissue exposure to a drug and therefore was considered more reliable in assessing the extent of the overall drug exposure than individual concentrations. The results showed that 1) a small macromolecule can penetrate deep into the tumor interstitium and produce a uniform but low spatial distribution of AUC; 2) large macromolecules produce high AUC in the perivascular region, but low AUC in the distal region away from vessels; 3) medium-sized macromolecules produce a relatively uniform and high AUC in the tumor interstitium between two vessels; 4) enhancement of permeability can elevate the level of AUC, but have little effect on its uniformity while enhancement of diffusivity is able to raise the level of AUC and improve its uniformity; 5) a longer half-life can produce a deeper penetration and a higher level of AUC distribution. The numerical results indicate that a long half-life carrier in plasma and a high interstitial diffusivity are the key factors to produce a high and relatively uniform spatial AUC distribution in the interstitium.     

A Simulation Tool for Dynamic Contrast Enhanced MRI

The quantification of bolus-tracking MRI techniques remains challenging. The acquisition usually relies on one contrast and the analysis on a simplified model of the various phenomena that arise within a voxel, leading to inaccurate perfusion estimates. To evaluate how simplifications in the interstitial model impact perfusion estimates, we propose a numerical tool to simulate the MR signal provided by a dynamic contrast enhanced (DCE) MRI experiment. Our model encompasses the intrinsic and relaxations, the magnetic field perturbations induced by susceptibility interfaces (vessels and cells), the diffusion of the water protons, the blood flow, the permeability of the vessel wall to the the contrast agent (CA) and the constrained diffusion of the CA within the voxel. The blood compartment is modeled as a uniform compartment. The different blocks of the simulation are validated and compared to classical models. The impact of the CA diffusivity on the permeability and blood volume estimates is evaluated. Simulations demonstrate that the CA diffusivity slightly impacts the permeability estimates ( for classical blood flow and CA diffusion). The effect of long echo times is investigated. Simulations show that DCE-MRI performed with an echo time may already lead to significant underestimation of the blood volume (up to 30% lower for brain tumor permeability values). The potential and the versatility of the proposed implementation are evaluated by running the simulation with realistic vascular geometry obtained from two photons microscopy and with impermeable cells in the extravascular environment. In conclusion, the proposed simulation tool describes DCE-MRI experiments and may be used to evaluate and optimize acquisition and processing strategies.