Analysis of pH Gradients Resulting from Mass Transport Limitations in Engineered Heart Tissue

Abstract

Transport limitations of critical nutrients are a major obstacle in the construction of engineered heart tissues (EHTs), and the importance of oxygen in this regard is well-documented throughout the literature. An indirect effect of cellular hypoxia is the shunt to the less-efficient glycolytic metabolism, which is accompanied by a reduction in extracellular pH. Image analysis of phenol red coloration in an experimental model of EHTs demonstrated pH gradients towards the center of the construct, which were dependent on experimental variables. Based on these observations, a four-species, 2-D diffusion–reaction mathematical model was developed to predict pH in a radial-diffusion model. The mathematical model predicted lethal values of pH (<6.5) in EHTs comprised of a nominal cell density of 10⁶ cells/cm³. pH predictions were moderately dependent on O₂ concentration, and strongly dependent on cell density, CO₂ concentration, and diffusion path length. It can be concluded from this study that hypoxia-induced acidosis is an important element in the mass transport problem, and future experiments measuring pH with more sensitive methods is expected to further elucidate the extent of this effect.

Appendix: Mathematical Model Formulation

A mathematical model was constructed to aid in the prediction of pH inside diffusion-limited EHTs. The two-dimensional diffusion–reaction model consists of four diffusing/reacting species within a bounded circle: O₂ (l), CO₂ (l), HCO ⁻₃ , and H⁺. A homogeneous medium is assumed with cells evenly distributed. Cells inside the circle are assumed to produce CO₂ constantly and volumetrically, while they consume O₂ and produce H⁺ in an O₂-dependent manner.

Chemical Reactions

The standard pH buffer for DMEM and most other culture media is the bicarbonate/CO₂ system. In this system, CO₂ overlying culture medium will dissolve at a concentration proportional to the overlying gas partial pressure, which in tissue culture incubators is about 5% dry gas fraction. Dissolved CO₂ can undergo a number of transformations, although at physiological and acidotic (mildly acidic) pH, the hydration of CO₂ and subsequent dissociation of carbonic acid (H₂CO₃) to form bicarbonate (HCO ⁻₃ ) are the most important:

$$ {\text{Carbon\ dioxide: CO}}_{{\text{2}}} ({\text{g}})\overset {{\text{H}}_{{{\text{CO}}_{{\text{2}}} }} } \rightleftharpoons {\text{CO}}_{{\text{2}}} ({\text{l}})$$

(1)

$$ {\text{CO}}_{{\text{2}}} \ {\text{ hydration: CO}}_{{\text{2}}} ({\text{l}}) + {\text{H}}_{{\text{2}}} {\text{O}} \Leftrightarrow {\text{H}}_{{\text{2}}} {\text{CO}}_{{\text{3}}} $$

(2)

$$ {\text{Carbonic\ acid \ dissociation: H}}_{{\text{2}}} {\text{CO}}_{{\text{3}}} \Leftrightarrow {\text{H}}^{ + } + {\text{HCO}}_{{\text{3}}} ^{ - } $$

(3)

This reaction can be catalyzed by carbonic anhydrase, which is not typically added to culture media, or proceed uncatalyzed. Proton transfer to carbonic acid is assumed to be instantaneous, or diffusion-controlled,14 so carbon dioxide and carbonic acid are assumed to always be in equilibrium. The overall reaction is limited by the dissociation of carbonic acid, which can deviate substantially from equilibrium during high proton flux.9 These reactions may be grouped into one reaction with effective forward and reverse rate constants k ₁ and k ₋₁:

$$ {\text{Effective \ bicarbonate\ formation}}:{\text{ CO}}_{{\text{2}}} ({\text{l}})\, + \,{\text{H}}_{{\text{2}}} {\text{O}}\,{\mathop{\leftrightharpoons}\limits_{{k_{-1}}}^{k_1}}\,{\text{H}}^{ + } \, + \,{\text{HCO}}_{3} ^{ - }. $$

(4)

Reaction Kinetics

The rates of formation of carbon dioxide and bicarbonate are given by the products of their respective rate constants and substrate concentrations. Embedded in these equations are cellular reaction terms R _cell for the consumption of O₂ and CO₂, and the production of H⁺. The volumetric cell density is ρ.

$$ {\text{O}}_{{\text{2}}} {\text{ reaction\ rate: }}R_{{{\text{O}}_{{\text{2}}} }} = - \rho \cdot R^{{{\text{cell}}}}_{{{\text{O}}_{{\text{2}}} }} $$

(5)

$$ {\text{CO}}_{{\text{2}}} {\text{ reaction\ rate: }}R_{{{\text{CO}}_{{\text{2}}} }} = \rho \cdot R^{{{\text{cell}}}}_{{{\text{CO}}_{{\text{2}}} }} - k_{1} [{\text{CO}}_{{\text{2}}} ] + k_{{ - 1}} [{\text{H}}^{ + } ][{\text{HCO}}_{{\text{3}}} ^{ - } ] $$

(6)

$$ {\text{HCO}}_{{\text{3}}} ^{{{ - }}} {\text{ reaction\ rate: }}R_{{{\text{HCO}}_{{\text{3}}} }} = k_{1} [{\text{CO}}_{{\text{2}}} ] - k_{{ - 1}} [{\text{H}}^{ + } ][{\text{HCO}}_{{\text{3}}} ^{ - } ] $$

(7)

$$ {\text{H}}^{{\text{ + }}} {\text{ reaction\ rate: }}R_{{\text{H}}} = \rho \cdot R^{{{\text{cell}}}}_{{\text{H}}} + k_{1} [{\text{CO}}_{{\text{2}}} ] - k_{{ - 1}} [{\text{H}}^{ + } ][{\text{HCO}}_{{\text{3}}} ^{ - } ] $$

(8)

The cellular consumption of oxygen is given as a concentration-dependent function. Shortly after the onset of hypoxia, neonatal cardiomyocytes reversibly decrease their rates of oxygen consumption to compensate for the reduced oxygen availability.7 Because a large fraction of cells in engineered tissues can be exposed to very low concentrations of oxygen (<0.1%),19 it is important to include a reaction term in mass transport models that is dependent on the local concentration of oxygen. One such model of variable consumption rate takes the form of enzyme kinetics, developed by Michaelis and Menten and modified with a steady-state assumption by Briggs and Haldane.24 In this model, the initial rate of reaction V ₀ is dependent on the substrate concentration [S], the maximum rate of reaction ${\text{V}}_{{{\text{O}}_{{\text{2}}} {\text{,max}}}} ,$ and the half-maximal oxygen concentration $k_{{{\text{O}}_{{\text{2}}} }} :$

$$ R^{{{\text{cell}}}}_{{{\text{O}}_{{\text{2}}} }} = \frac{{V_{{{\text{O}}_{{\text{2}}} {\text{,max}}}} [{\text{O}}_{{\text{2}}} ]}} {{k_{{{\text{O}}_{{\text{2}}} }} + [{\text{O}}_{{\text{2}}} ]}} .$$

(9)

Cellular utilization of oxygen by oxidative phosphorylation takes place at cytochrome oxidase enzymes in the inner mitochondrial membrane, which facilitate the transfer of electrons from cytochrome c to molecular oxygen. Both isolated mitochondria and cardiomyocytes in suspension have been shown to regulate respiration and oxidative phosphorylation in an oxygen-dependent manner, which resemble Michaelis–Menten-type curves.7,11 A previous analysis of oxygen transport demonstrated good congruency with EHT data when $V_{{{\text{O}}_{{\text{2}}} {\text{,max}}}}$ and $k_{{{\text{O}}_{{\text{2}}} }}$ for electrically stimulated adult rat cardiomyocytes were instituted in the model (data from35).

The cellular production of protons is also given as a concentration-dependent function. Although no literature data was found for a specific proton production rate in cardiomyocytes as a function of oxygen concentration, reports do exist for oxygen-dependent lactate production. Two such studies used neonatal cardiomyocytes in suspension,7,8 while another used adult cardiomyocytes in suspension26 (data reproduced in Table 2). The latter study reported the proton production in concentration per cellular protein mass, which can be converted to concentration per cell with the reported value of 0.3 ng protein per cell.10 With this conversion, the values from the cited studies unfortunately differ by a factor of 675. The shape of the curve is similar in both studies: a higher-order inverse polynomial trend that drops rapidly at low oxygen and levels off to a steady-state rate at high oxygen. In an effort to make use of the shape of this curve without knowledge of its y-shift, an equation was formulated that included a variable scaling factor A. The data reporting lower values of cellular proton production26 as a function of oxygen concentration was fit to the following third-order inverse polynomial:

$$ R^{{{\text{cell}}}}_{{\text{H}}} = A{\left( {5.5 \times 10^{{ - 9}} + \frac{{2.9 \times 10^{{ - 6}} }} {{({{C}}_{{{\text{O}}_{{\text{2}}} }} + 4.95)^{3} }} + \frac{{2.5 \times 10^{{ - 7}} }} {{({{C}}_{{{\text{O}}_{{\text{2}}} }} + 4.95)}}} \right)}. $$

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Table 2 Literature values of oxygen-dependent lactate production rates from neonatal cardiomyocytes in a sealed suspension culture23 or perfused suspension culture;22 or adult cardiomyocytes in a sealed suspension culture14

Diffusion Equations

In static incubator conditions, mass transport in hydrogels such as those that comprise EHTs is regarded as predominantly diffusive. The fibers of collagen, like those of the ECM, are spaced very closely, which produces a large interstitial hypertension of water in the gel and severely limits convection.32 Indeed, the similarity of collagen gels to the tumor extracellular matrix has been exploited by some investigators modeling drug transport by diffusion.25 Four diffusing-reacting species are taken into account in the model. The mass balance on each of these can be described with Fick’s 2nd Law of diffusion in radial coordinates:12

$$ \frac{{\partial [C]}} {{\partial t}} = \frac{1} {r} \cdot \frac{\partial } {{\partial r}}{\left[ {D_{C} \cdot r\frac{{\partial [C]}} {{dr}}} \right]} + R_{C} , $$

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C represents the concentration of CO₂, HCO ⁻₃ , O₂, or H⁺. Measurements of the diffusivity of small molecules through hydrogels are generally similar to the corresponding diffusivities in water.43 Thus the diffusivity values used for the four species in the EHT will be the corresponding diffusivities in water at 37 °C.

Boundary Conditions

Ambient oxygen at sea level constitutes 20.9% of the total dry gas pressure, which is reduced in tissue culture incubators by both the supplemental CO₂ and water vapor. CO₂ at 5% will reduce the dry gas oxygen fraction of ambient air to (100–5%) × 0.209 = 19.9%. Gas partial pressures will be reduced by humidification in the incubator according to the vapor pressure of water at 37 °C (47 Torr).23 The actual humidified partial pressure of gas Y in Torr is a function of the dry gas oxygen fraction (% Y) and the relative humidity (RH) in the incubator:

$$ p{\text{Y}} = \% {\text{Y}} \cdot (760 - 47 \cdot {\text{RH}}). $$

(12)

The cell culture incubators used in this study typically sustain about 80% relative humidity, which yields a pO₂ value of 144 Torr for 19.9% dry gas oxygen fraction, and a pCO₂ value of 36 Torr. Oxygen is slightly less soluble in tissue culture medium than water, with a Henry’s Law constant of 0.78 Torr/μM at 37 °C in DMEM.22 The quotient of this solubility coefficient and pO₂ gives the boundary condition for dissolved O₂ in the model ([O₂] _i  = 185 μM). Gaseous CO₂ is highly soluble in aqueous solutions such as culture medium. The partial pressure of CO₂ in the overlying gas (pCO₂) is related to the concentration of dissolved CO₂ by the Henry’s Law constant, H:

$$ \frac{{p{\text{CO}}_{{\text{2}}} }} {H} = [{\text{CO}}_{{\text{2}}} ]. $$

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The Henry’s Law constant has a known dependence on temperature:34

$$ H\,[{\text{Torr}}\times \mu{\text{M}}^{-1}] = 7.5 \times 10^{-6} \cdot \exp {\left(11.25 - {\frac{{395.9}}{T[{\text{K}}] - 175.9}}\right)}, $$

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H can be calculated as 0.0302 Torr/μM for mammalian cell culture medium at 37 °C, with a corresponding value of [CO₂] _i  = 1196 μM for the bulk solution. The concentration of HCO ⁻₃ in the bulk solution (pH 7.6) is given by the equilibrium equation with [CO₂] _i :

$$ [{\text{HCO}}_{{\text{3}}} ^{ - } ]_{i} = 10^{{{\text{pH}}_{i} }} \cdot K_{B} [{\text{CO}}_{{\text{2}}} ]_{i} . $$

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