A computational model of oxygen delivery by hemoglobin-based oxygen carriers in three-dimensional microvascular networks

https://doi.org/10.1016/j.jtbi.2007.06.012Get rights and content

Abstract

A detailed computational model is developed to simulate oxygen transport from a three-dimensional (3D) microvascular network to the surrounding tissue in the presence of hemoglobin-based oxygen carriers. The model accounts for nonlinear O2 consumption, myoglobin-facilitated diffusion and nonlinear oxyhemoglobin dissociation in the RBCs and plasma. It also includes a detailed description of intravascular resistance to O2 transport and is capable of incorporating realistic 3D microvascular network geometries. Simulations in this study were performed using a computer-generated microvascular architecture that mimics morphometric parameters for the hamster cheek pouch retractor muscle. Theoretical results are presented next to corresponding experimental data. Phosphorescence quenching microscopy provided PO2 measurements at the arteriolar and venular ends of capillaries in the hamster retractor muscle before and after isovolemic hemodilution with three different hemodilutents: a non-oxygen-carrying plasma expander and two hemoglobin solutions with different oxygen affinities. Sample results in a microvascular network show an enhancement of diffusive shunting between arterioles, venules and capillaries and a decrease in hemoglobin's effectiveness for tissue oxygenation when its affinity for O2 is decreased. Model simulations suggest that microvascular network anatomy can affect the optimal hemoglobin affinity for reducing tissue hypoxia. O2 transport simulations in realistic representations of microvascular networks should provide a theoretical framework for choosing optimal parameter values in the development of hemoglobin-based blood substitutes.

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.

References (60)

  • T.W. Secomb et al.

    Analysis of oxygen transport to tumor tissue by microvascular networks

    Int. J. Radiat. Oncol. Biol. Phys.

    (1993)
  • Altman, P.L., Dittmer, D.S., 1971. Respiration and Circulation. Federation of the American Society of Experimental...
  • D.A. Beard et al.

    Modeling advection and diffusion of oxygen in complex vascular networks

    Ann. Biomed. Eng.

    (2001)
  • D.A. Beard et al.

    Myocardial oxygenation in isolated hearts predicted by an anatomically realistic microvascular transport model

    Am. J. Physiol. Heart Circ. Physiol.

    (2003)
  • R.A. Bennett et al.

    Capillary spatial pattern and muscle fiber geometry in three hamster striated muscles

    Am. J. Physiol.

    (1991)
  • R.W. Benodekar et al.

    Finite difference procedure for solution of Poisson equation over complex domains with Neumann boundary conditions

    Comput. Fluids

    (1977)
  • T.B. Bentley et al.

    Temperature dependence of oxygen diffusion and consumption in mammalian striated muscle

    Am. J. Physiol.

    (1993)
  • B.R. Berg et al.

    Functional capillary organization in striated muscle

    Am. J. Physiol.

    (1995)
  • P. Cabrales et al.

    Oxygen transport by low and normal oxygen affinity hemoglobin vesicles in extreme hemodilution

    Am. J. Physiol. Heart Circ. Physiol.

    (2005)
  • C. Christoforides et al.

    Effect of temperature on solubility of O2 in human plasma

    J. Appl. Physiol.

    (1969)
  • A.A. Constantinescu et al.

    Elevated capillary tube hematocrit reflects degradation of endothelial cell glycocalyx by oxidized LDL

    Am. J. Physiol. Heart Circ. Physiol.

    (2001)
  • C. Desjardins et al.

    Heparinase treatment suggests a role for the endothelial cell glycocalyx in regulation of capillary hematocrit

    Am. J. Physiol.

    (1990)
  • M. Dong

    Influence of aging on oxygen transport in the microcirculation of skeletal muscle

    (1997)
  • J.H. Ferziger et al.

    Computational Methods for Fluid Dynamics

    (1999)
  • D. Goldman et al.

    Computational modeling of oxygen transport from complex capillary networks. Relation to the microcirculation physiome

    Adv. Exp. Med. Biol.

    (1999)
  • D. Goldman et al.

    Effect of sepsis on skeletal muscle oxygen consumption and tissue oxygenation: interpreting capillary oxygen transport data using a mathematical model

    Am. J. Physiol. Heart Circ. Physiol

    (2004)
  • J.D. Hellums et al.

    Simulation of intraluminal gas transport processes in the microcirculation

    Ann. Biomed. Eng.

    (1996)
  • C.R. Honig et al.

    Correlation of O2 transport on the micro and macro scale

    Int. J. Microcirc. Clin. Exp.

    (1982)
  • L. Hoofd et al.

    Realistic modelling of capillary spacing in dog gracilis muscle greatly influences the heterogeneity of calculated tissue oxygen pressures

    Adv. Exp. Med. Biol.

    (1996)
  • M. Intaglietta et al.

    Microvascular and tissue oxygen distribution

    Cardiovasc. Res.

    (1996)
  • Cited by (50)

    • Selective induction of sprouting and intussusception is associated with the concentration distributions of oxygen and hypoxia-induced VEGF

      2020, Microvascular Research
      Citation Excerpt :

      The values of the parameters for calculation are presented in Table 2. Given the drop in oxygen pressure along the capillary in the direction of flow (Tsoukias et al., 2007), the different u0 values on the longer side of the rectangular unit are used for calculation. As shown in Fig. 3A, the oxygen concentration at the wall of the left capillary which is assumed to be close to arteriole side, is taken as 1 × 10−4 mLmL−1, while that of the right capillary close to venule side is taken as 0.9 × 10−4 mLmL−1, with linear variation between them.

    • Artificial oxygen carriers

      2020, Current Trends and Future Developments on (Bio-) Membranes: Membrane Applications in Artificial Organs and Tissue Engineering
    • Oxygen transport in a cross section of the rat inner medulla: Impact of heterogeneous distribution of nephrons and vessels

      2014, Mathematical Biosciences
      Citation Excerpt :

      A similar approach to this skeletal muscle analysis was used in the present study to examine oxygen transport in the inner medulla of the kidney, including a finite-difference method for spatial discretization. 3D models of skeletal muscle oxygen transport from Goldman et al. [27] and Tsoukias et al. [29] used morphological parameters to construct distributions of vessels within a tissue block, but did not use explicit image data to get at heterogeneity of vascular structure. More recently, models of metabolic blood flow regulation in the skeletal muscle microcirculation from Roy et al. [36] and Fry et al. [26] used high-resolution images of microvascular networks to explicitly represent positions of vessels within a volume of tissue.

    View all citing articles on Scopus
    View full text