A computational model of oxygen delivery by hemoglobin-based oxygen carriers in three-dimensional microvascular networks
Introduction
The microvasculature is the site of oxygen transport to tissue and regulation of local blood flow, and therefore has been studied extensively. Motivated by experimental observations in skeletal muscle, Krogh (1919) presented a simple mathematical model for oxygen transport in capillary-perfused tissue. The model assumed uniformly spaced parallel capillaries, each receiving the same convective O2 supply and delivering O2 to the same amount of tissue. The uniformity of the capillary/tissue configuration allowed a single capillary and the surrounding ‘tissue cylinder’ to be considered; several other simplifying assumptions then made an exact solution possible. The Krogh model has provided many valuable insights into O2 transport; however, over the last two decades it has been substantially extended to include many physiologically important aspects of microvascular O2 delivery. In particular, it is now known that the complexity of microvascular geometry and hemodynamics (Pittman, 1995), as well as blood transport properties (Hellums et al., 1996; Popel et al., 2003), can significantly affect O2 delivery to tissue.
Given the physiological importance of microvascular O2 delivery, it is of interest to obtain a better quantitative understanding than is possible with the Krogh model. However, the complex nature of microvascular oxygen transport has posed difficulties. Experimentally, it has been difficult to measure the main quantity of interest, the tissue O2 concentration (or partial pressure, PO2), in three dimensions with a micron resolution. This has motivated theoretical work to enable calculation of tissue PO2 distributions (Popel, 1989). Modeling studies in skeletal muscle have shown the importance of many features neglected in the Krogh model, including heterogeneity of parallel capillary spacing (Hoofd and Turek, 1996), heterogeneity of capillary convective O2 supply (Ellsworth et al., 1988; Popel et al., 1986), diffusive shunting between capillaries (Ellsworth et al., 1988; Wieringa et al., 1993), capillary tortuosity and anastomoses (Goldman and Popel, 2000), interactions between capillaries and arterioles (Secomb and Hsu, 1994), and intravascular transport resistance (Federspiel and Popel, 1986). In addition, it is known that O2 transport from pre- and post-capillary vessels (arterioles and venules) can be significant in resting muscle (Kuo and Pittman, 1988; Swain and Pittman, 1989). Therefore, these features are desirable for realistic modeling of O2 transport in skeletal muscle, as well as in other tissues (e.g., brain (Hudetz, 1999; Kislyakov and Ivanov, 1986), heart (Beard and Bassingthwaighte, 2001; Beard et al., 2003; Wieringa et al., 1993), tumors (Secomb et al., 1993, Secomb et al., 2004)).
This need for a high degree of realism is particularly great when situations of relatively low O2 supply are considered, which is generally the case for applications of blood substitutes (Winslow, 2002). Hemoglobin-based oxygen carriers (HBOC) with different properties (i.e., oxygen affinity, molecular size, NO reactivity) have been developed and hold promise as blood substitutes. Diaspirin cross-linked hemoglobin (DCLHb), for example, is a first-generation artificial oxygen carrier that has O2 affinity similar to the erythrocytic hemoglobin (P50=32 mmHg; Hill coefficient=2.4). 3261BR on the other hand, is a genetically cross-linked human hemoglobin that was made by recombinant methods to have a higher O2 affinity (P50=14.6 mmHg; Hill coefficient=2.15). However, at this point the optimal values for the design parameters of these products (including their affinity for O2) have not been established and theoretical studies can assist in this effort.
The purpose of this paper is to extend a previously described mathematical/computational model (Goldman and Popel, 1999, Goldman and Popel, 2000, Goldman and Popel, 2001; Goldman et al., 2004; Popel et al., 2003) so that it can describe oxygen delivery to tissue in the presence and absence of plasma-based hemoglobin. The current work modifies the original model by the addition of arterioles and venules to the geometric component and the addition of plasma hemoglobin to the blood flow and O2 transport components. This work also contains an approximate derivation of intravascular O2 transport resistance in the presence and absence of blood substitutes that agrees with, but is much simpler to use than, full-scale intravascular transport calculations. Thus, in this study, we present the methodology for the development of a computational model that can describe O2 transport in macroscopic tissue volumes after transfusion of HBOC. The study also presents sample results of blood flow and O2 transport in muscle. Representative theoretical simulations are presented next to corresponding experimental data in three hemodilution scenarios from previous studies. Experimental measurements of PO2 in the arteriolar and venular end of capillaries from hamster cheek pouch retractor muscle are reported. Sample simulations are also presented at increased O2 consumption rate that yields hypoxic tissue regions. The model represents a significant advance in theoretical capabilities for studying microvascular O2 transport, especially when blood substitutes are involved.
Section snippets
Methods
Microvascular network: Three-dimensional (3D) microvascular networks from different tissues have been reconstructed using a number of different methods such as scanning electron micrographs of corrosion casts or intravital confocal microscopy (Secomb et al., 2004). In skeletal muscles, most capillaries run approximately parallel to muscle fibers, allowing the construction of a computer-generated approximation of the vascular network by random placement of capillaries around cylindrical muscle
Isovolemic hemodilution study
Simulations were performed for exchange transfusion scenarios using three different hemodiluents for which experimental data of blood PO2 are available (Pittman et al., 2003). Simulation parameters for each hemodilution scenario are summarized in Table 2 and were chosen to resemble the experimental conditions. Experimental measurements of the blood flow rate and detailed description of network geometry were not available to complement the experimental data. Thus, a direct comparison between
Discussion
This paper presents a detailed mathematical model that describes oxygen delivery to tissue in the presence and absence of plasma-based hemoglobin. The study extends a computational model described previously for studying O2 transport to tissue (Goldman and Popel, 1999, Goldman and Popel, 2000, Goldman and Popel, 2001). In the current study, (1) we present the development of a theoretical framework for studying O2 delivery by HBOC, (2) evaluate representative computer simulations against
Acknowledgments
This project was supported by the National Institutes of Health Grants NHLBI HL18292 and HL079087 and by the American Heart Association Grant N0435067.
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