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

Abstract

A detailed nonlinear four-region (red blood cell, plasma, interstitial fluid, and parenchymal cell) axially distributed convection-diffusion-permeation-reaction-binding computational model is developed to study the simultaneous transport and exchange of oxygen (O$_{2}$) and carbon dioxide (CO$_{2}$) in the blood–tissue exchange system of the heart. Since the pH variation in blood and tissue influences the transport and exchange of O$_{2}$ and CO$_{2}$ (Bohr and Haldane effects), and since most CO$_{2}$ is transported as HCO$_{3}^{-}$ (bicarbonate) via the CO$_{2}$ hydration (buffering) reaction, the transport and exchange of HCO$_{3}^{-}$ and H$^{+}$ are also simulated along with that of O$_{2}$ and CO$_{2}$. Furthermore, the model accounts for the competitive nonlinear binding of O$_{2}$ and CO$_{2}$ with the hemoglobin inside the red blood cells (nonlinear ${\rm O_{2}\hbox{--}CO_{2}}$ interactions, Bohr and Haldane effects), and myoglobin-facilitated transport of O$_{2}$ inside the parenchymal cells. The consumption of O$_{2}$ through cytochrome-c oxidase reaction inside the parenchymal cells is based on Michaelis–Menten kinetics. The corresponding production of CO$_{2}$ is determined by respiratory quotient (RQ), depending on the relative consumption of carbohydrate, protein, and fat. The model gives a physiologically realistic description of O$_{2}$ transport and metabolism in the microcirculation of the heart. Furthermore, because model solutions for tracer transients and steady states can be computed highly efficiently, this model may be the preferred vehicle for routine data analysis where repetitive solutions and parameter optimization are required, as is the case in PET imaging for estimating myocardial O$_{2}$ consumption.

APPENDIX: ERRATA IN THE 2004 PUBLICATION OF THE HbO2 AND HbCo2 DISSOCIATION CURVES

Dash and Bassingthwaighte (2004): The article by Dash and Bassingthwaighte (Ann Biomed Eng 32: 1676–1693, 2004) appeared in print without galley proofs having been sent to either of the authors for review and correction. Errors in print setting were such that the equations essential to reproducing the scientific work need to be reprinted in order to make the present paper understandable and the original work reproducible. This appendix corrects the errors in the 2004 printing by providing the original, correct equations and text.

Correction 1: On page 1678 under MATHEMATICAL FORMULATIONS, Beginning at the end of paragraph 1, are a series of corrections to the equations 2a–f, where an upper case K is to be used instead of an erroneous lower case k:

“The governing reactions are:

CO₂ hydration reaction – ${\rm HCO}_3^ -$ buffering of CO₂:

$${\rm CO}_2 + {\rm H}_2 {\rm O}\begin{array}{c} {\hbox{\scriptsize{$kf'_1$}}}\\ \longrightarrow\\ \longleftarrow\\ {\hbox{\scriptsize{$kb'_1$}}} \end{array} {\rm H}_2 {\rm CO}_3 \begin{array}{c} {\hbox{\scriptsize{$K''_1$}}}\\ \longrightarrow\\ \longleftarrow\\ {\hbox{\scriptsize{$\vphantom{p}$}}} \end{array} {\rm HCO}_3^ - + {\rm H}^ + .$$

(2a)

CO₂ binding to HmNH₂ chains – HmNHCOO^- buffering of CO₂:

$${\rm CO}_2\,{+}\,{\rm HmNH}_2 \begin{array}{c} {\hbox{\scriptsize{$kf'_2$}}}\\ \longrightarrow\\ \longleftarrow\\ {\hbox{\scriptsize{$kb'_2$}}} \end{array} {\rm HmNHCOOH} \begin{array}{c} {\hbox{\scriptsize{$K''_2$}}}\\ \longrightarrow\\ \longleftarrow\\ {\hbox{\scriptsize{$\vphantom{p}$}}} \end{array} {\rm HmNHCOO}^ -\,{+}\,{\rm H}^ +\!.$$

(2b)

CO₂ binding to O₂HmNH₂ chains – O₂HmNHCOO^- buffering of CO₂:

$${\rm CO}_2 + {\rm O}_2 {\rm HmNH}_2 \begin{array}{c} {\hbox{\scriptsize{$kf'_3$}}}\\ \longrightarrow\\ \longleftarrow\\ {\hbox{\scriptsize{$kb'_3$}}} \end{array} {\rm O}_{\rm 2} {\rm HmNHCOOH}\\ \quad \begin{array}{c} {\hbox{\scriptsize{$K''_3$}}}\\ \longrightarrow\\ \longleftarrow\\ {\hbox{\scriptsize{$\vphantom{p}$}}} \end{array} {\rm O}_{\rm 2} {\rm HmNHCOO}^ - + {\rm H}^ + .$$

(2c)

O₂ binding to HmNH₂ chains – one-step kinetics using the $P_{{\rm O}_2 }$-dependent values of the rates of association and dissociation to accounts for the cooperativity.

$$ {\rm O}_2 + {\rm HmNH}_2 \begin{array}{c} {\hbox{\scriptsize{$kf'_4$}}}\\ \longrightarrow\\ \longleftarrow\\ {\hbox{\scriptsize{$kb'_4$}}} \end{array} {\rm O}_{\rm 2} {\rm HmNH}_{\rm 2} . $$

(2d)

Ionization of HmNH₂ chains – pH buffering:

$$ {\rm HmNH}_3^ + \begin{array}{c} {\hbox{\scriptsize{$K''_5$}}}\\ \longrightarrow\\ \longleftarrow\\ {\hbox{\scriptsize{$\vphantom{p}$}}} \end{array} {\rm HmNH}_2 + {\rm H}^ + . $$

(2e)

Ionization of O₂HmNH₂ chains – pH buffering:

$${\rm O}_{\rm 2} {\rm HmNH}_3^ + \begin{array}{c} {\hbox{\scriptsize{$K''_6$}}}\\ \longrightarrow\\ \longleftarrow\\ {\hbox{\scriptsize{$\vphantom{p}$}}} \end{array} {\rm O}_{\rm 2} {\rm HmNH}_2 + {\rm H}^ + .$$

(2f)

Correction 2: On page 1679, in the last sentence of this same section, and ending just above the section heading “Equilibrium Relations” the “$k' _4$” should be replaced by the upper case $K' _4$ in two places:

“To account for this through our one-step kinetic approach in reaction (2d), we use the equilibrium “constant” $K' _4 ( = kf' _4 /kb' _4 )$ a function of O₂ partial pressure $P_{{\rm O}_2 }$; $K' _4$ also depends on the levels of pH, $P_{{\rm CO}_2 }$, 2,3-DPG concentration, and temperature inside the RBCs⁴.”

Correction 3: All of Equations 3 and 4 require corrections to the upper versus lower case Ks and the kfs in the section on Equilibrium Relations:

$$K'_1 \;[{\rm CO}_2 ] = [{\rm H}_2 {\rm CO}_3 ],$$

(3a)

$$K''_1 \;[{\rm H}_{\rm 2} {\rm CO}_2 ] = [{\rm HCO}_{\rm 3}^ - ][{\rm H}^ + ],$$

(3b)

$$K'_2 \;[{\rm CO}_2 ][{\rm HmNH}_2 ] = [{\rm HmNHCOOH}],$$

(3c)

$$K''_2 \;[{\rm HmNHCOOH}] = [{\rm HmNHCOO}^ - ][{\rm H}^ + ],$$

(3d)

$$K'_3 \;[{\rm CO}_2 ][{\rm O}_2 {\rm HmNH}_2 ] = [{\rm O}_2 {\rm HmNHCOOH}],$$

(3e)

$$K''_3 \;[{\rm O}_2 {\rm HmNHCOOH}] = [{\rm O}_2 {\rm HmNHCOO}^ - ][{\rm H}^ + ],$$

(3f)

$$K'_4 \;[{\rm O}_2 ][{\rm HmNH}_2 ] = [{\rm O}_2 {\rm HmNH}_2 ],$$

(3g)

$$K''_5 \;[{\rm HmNH}_3^ + ] = [{\rm HmNH}_2 {\rm ][H}^ + ],$$

(3h)

$$K''_6 \;[{\rm O}_2 {\rm HmNH}_3^ + ] = [{\rm O}_2 {\rm HmNH}_2 {\rm ][H}^ + ],$$

(3i)

where the equilibrium constants $K' _1 ,K' _2 ,K' _3$ and $K' _4$ are defined by

$$ K'_1 = \frac{{kf'_1 }}{{kb'_1 }}[{\rm H}_2 {\rm O}], \quad K'_2 = \frac{{kf'_2 }}{{kb'_2 }}, \quad K'_3 = \frac{{kf'_3 }}{{kb'_3 }}, \quad K'_4 = \frac{{kf'_4 }}{{kb'_4 }}\vphantom{\bigg(_{\big(}}.$$

(4)

Correction 4: In Appendix A, Eq. A.2 should be corrected to read:

$$[{\rm CO}_2 ]_{{\rm bl}} = W_{{\rm bl}} \alpha _{{\rm CO}_2 } P_{{\rm CO}_2 } + [{\rm HCO}_3^ - ]_{{\rm bl}} + 4[{\rm Hb}]_{{\rm bl}} S_{{\rm HbCO}_2 },$$

(A.2)

Correction 5: The last sentence in the paragraph below Eq. A.3 should correct R _rbc to read:

“The value of R _rbc is about 0.69^{13–15,31,38}. Multiplying [CO₂]_bl by 22.256 converts the units of molar to ml of CO₂ per ml of blood.”

Correction 6: The first sentence of the paragraph above Eq. A.4 should correct H₂O:

“From the data used in Hill et al.^13–15, [H₂O] $kf' _1 \approx 0.12$ s^-1 and $kb' _1 \approx 89$ s^-1, so $K' _1 = [{\rm H}_2 {\rm O}]\;kf' _1 /kb' _1 \approx 1.35\times 10^{ - 3}$.”

Correction 7: In Appendix B, Eqs. B.1 to B.3 should be corrected re $K' _{{\rm HbO}_2 }$ and K _fact and the latter part of the first paragraph corrected to read:

“

$$S_{{\rm HbO}_2 } = \frac{{K'_{{\rm HbO}_2 } [{\rm O}_2 ]^{1 + n_0 } }}{{1 + K'_{{\rm HbO}_2 } [{\rm O}_2 ]^{1 + n_0 } }},$$

(B.1)

where $K' _{{\rm HbO}_2 }$ (with units $M^{ - (1 + n_0 )}$) is given by Equations 13 and 14:

$$K'_{{\rm HbO}_2 } = \frac{{K''_4 K_{{\rm fact}} }}{{K_{{\rm ratio}} [{\rm O}_2 ]_S^{n_0 } }} = \frac{1}{{( {\alpha _{{\rm O}_2 } P_{50} })^{1 + n_0 } }},$$

(B.2)

The P ₅₀ is defined as before by Eq. (11) and the kinetic terms K _ratio and K _fact are defined by Eqs. (18) and (14b). The Hill exponent in the expression for $S_{{\rm HbO}_2 }$ is now 1+n ₀ and apparent Hill coefficient $K' _{{\rm HbO}_2 }$ is now independent of [O₂]. This makes $S_{{\rm HbO}_2 }$ analytically invertible, when $P_{{\rm CO}_2 }$, pH, [DPG] and T are known. The inverted equation for [O₂] is given by

$$ [{\rm O}_2] = \left[\!\! {\frac{{S_{{\rm HbO}_2 } }}{{K'_{{\rm HbO}_2 } (1 - S_{{\rm HbO}_2 } )}}}\!\! \right]^{\frac{1}{{1 + n_0 }}} = \alpha _{{\rm O}_2 } P_{50} \left[\!\! {\frac{{S_{{\rm HbO}_2 } }}{{{\rm 1} - S_{{\rm HbO}_2 } }}}\! \right]^{\frac{1}{{1 + n_0 }}}.$$

(B.3)

It is clear from Eq. (2) that $K' _{{\rm HbO}_2 }$ can be determined completely using only the P ₅₀=P ₅₀([H⁺], [CO₂], [DPG], T) data which is given by Eq. (11). This avoids the calculations of K _ratio and K _fact which involves the complex computations of the empirical indices n ₁, n ₂, n ₃, and n ₄ using Eqs. (20a) to (20d). This also simplifies the computation of $S_{{\rm HbO}_2 }$ from [O₂] and vice-versa significantly. However, in the computation of $S_{{\rm HbCO}_2 }$ from [CO₂] and vice-versa (not shown in this appendix), the calculations of K _fact and K _ratio are essential as the P ₅₀ data for 50% HbCO₂ saturation is not available in the literature.”