Nonlinear model for capillary-tissue oxygen transport and metabolism

Abstract

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 15 O-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 axiallydistributed 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 pO 2 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.

Simultaneous Blood–Tissue Exchange of Oxygen, Carbon Dioxide, Bicarbonate, and Hydrogen Ion

“…Given the rapid dissipation of intracapillary radial concentration gradients, the RBC can be well represented as a central column of fluid surrounded by plasma, a situation mathematically similar to Hb being dispersed throughout the plasma. 20,51 (This assumption fails at low hematocrits, Hct, as shown by Hellums 36 ). The RBC and plasma move by plug flow with different velocities; RBC move faster than the plasma, being in the higher velocity central stream.…”

“…The configuration of the blood-tissue exchange (BTEX) unit for the present study of oxygen (O 2 ), carbon dioxide (CO 2 ), bicarbonate ( ), and hydrogen ion (H + ) transport and exchange in the heart is similar to the geometry of a classical Krogh cylinder model, e.g., see Li et al 51 as illustrated in Fig. 1.…”

“…Oxygen (O 2 ) transport and metabolism involves convection, diffusion, permeation, chemical reactions, binding and facilitated transport. 20,51,54 Oxygen's binding to hemoglobin (Hb) is affected directly by carbon dioxide (CO 2 ) levels, pH, temperature, and 2,3-diphosphoglycerate (2,3-DPG) levels, and affected indirectly therefore by bicarbonate ( ) buffering, metabolism and tissue acidification, renal acid and base handling, and the hemoglobin mediated nonlinear O 2 -CO 2 interactions. 26,37,44,63 A decrease in pH or an increase in CO 2 partial pressure (P CO 2 ) in the systemic capillaries reduces the affinity of Hb for O 2 (the Bohr effect) and enhances the delivery of O 2 to tissue.…”

“…The interpretation and understanding of alveoli-blood or blood-tissue gas exchange, whether being assessed from the observations of arterial and venous O 2 concentrations, 20,51,54 or by intra-tissue chemical signals such as BOLD MRI 46,53 or hemoglobin and myoglobin spectra, 59,60 or by external detection of tracer content such as 15 O-oxygen and 15 O-water through PET studies, 27,51,55,62 depends in general on all the factors involved in the dissociation of oxyhemoglobin. To study the dynamics of O 2 transport and metabolism, it is therefore important to account for the transport and exchange of CO 2 , , and H + as well as O 2 -CO 2 interactions.…”

“…Reviews by Hellums et al 37 and Popel 54 and recent studies oriented toward PET 15 O-oxygen image analysis by Beyer et al, 20 Deussen and Bassingthwaighte, 27 and Li et al 51 illustrate the applicability of the geometry that Krogh modeled. Other models based on non-cylindrical geometry have been designed or proposed to reflect the morphological and anatomical structures of the specific tissues.…”