## ABSTRACT

By virtue of their cardiovascular anatomy, reptiles and amphibians can shunt blood away from the pulmonary or systemic circuits, but the functional role of this characteristic trait remains unclear. It has been suggested that right-to-left (R–L) shunt (recirculation of systemic blood within the body) fuels the gastric mucosa with acidified and CO_{2}-rich blood to facilitate gastric acid secretion during digestion. However, in addition to elevating *P*_{CO2}, R–L shunt also reduces arterial O_{2} levels and would compromise O_{2} delivery during the increased metabolic state of digestion. Conversely, arterial *P*_{CO2} can also be elevated by lowering ventilation relative to metabolism (i.e. reducing the air convection requirement, ACR). Based on a mathematical analysis of the relative roles of ACR and R–L shunt on O_{2} and CO_{2} levels, we predict that ventilatory modifications are much more effective for gastric CO_{2} supply with only modest effects on O_{2} delivery. Conversely, elevating CO_{2} levels by means of R–L shunt would come at a cost of significant reductions in O_{2} levels. The different effects of altering ACR and R–L shunt on O_{2} and CO_{2} levels are explained by the differences in the effective blood capacitance coefficients.

## INTRODUCTION

The ability to shunt blood away from the pulmonary or systemic circulations is a defining character of the reptilian and amphibian cardiovascular systems (Hicks, 1998). However, whilst much is known about the anatomical basis for central vascular shunts and their autonomic regulation, the functional role of bypassing one or the other circulation remains as mysterious as it is debated (Hicks and Wang, 2012). Thus, it remains uncertain as to whether this cardiovascular design is an exquisite adaptation to low ectothermic metabolism and intermittent pulmonary ventilation, or merely an atavistic relict with no particular functional benefits (Hicks and Wang, 2012).

In several species of reptiles and amphibians, the right-to-left (R–L) shunts (i.e. the direct recirculation of systemic venous blood into the arterial systemic circulation) decrease whenever oxygen demands are elevated (Hicks and Wang, 2012). However, in crocodilians, an elevated oxygen consumption associated with digestion may be an exception. A combination of unique anatomical features of the crocodilian cardiovascular system (Hicks, 1998) combined with physiological measurements fostered the idea that increased R–L shunts serve to fuel the gastric mucosa with acidic proton-rich blood during digestion in alligators (Farmer et al., 2008; Gardner et al., 2011; Jones and Shelton, 1993). Central to this proposal is the observation that the crocodilian coeliac artery appears as a continuation of the left aortic arch, which indicates that the stomach is preferentially perfused with CO_{2}-rich blood from the right ventricle (e.g. Jones, 1996; Webb, 1979). In support for elevated (systemic) arterial partial pressure of CO_{2} (*P*_{CO2}) governing acid secretion, Farmer et al. (2008) reported slower digestion after surgical removal of the left aorta in alligators. However, a number of other studies show that growth is not affected by similar procedures (Eme et al., 2009, 2010), and it is possible that the slower digestion stems from reduced perfusion of the gastrointestinal organs after occlusion of the left aortic arch (Hicks and Wang, 2012).

Although the cardiovascular system must simultaneously provide for O_{2} delivery and CO_{2} removal, the proposition that R–L shunts assist gastric acid secretion has not included considerations of the inexorable reduction in O_{2} delivery. R–L shunts cause large reduction in arterial O_{2} levels – whether expressed as partial pressure, O_{2} concentration or haemoglobin saturation (Wang and Hicks, 1996) – while the effects on arterial *P*_{CO2} are predicted to be considerably smaller given the high capacitance coefficient for CO_{2} in blood. An increased R–L shunt during digestion would therefore also compromise O_{2} delivery, which seems undesirable given the fourfold elevation in O_{2} demands during digestion (Busk et al., 2000). In this context, it may be more prudent to elevate arterial *P*_{CO2} by means of ventilation [i.e. a lowering of the air convection requirement (ACR) for CO_{2}], a response that has been suggested to compensate for the rise in plasma bicarbonate during digestion (the so-called ‘alkaline tide’; Hicks et al., 2000; Hicks and White, 1992; Wang et al., 2001b). However, decreasing the ACR to elevate CO_{2} levels will simultaneously lower the lung *P*_{O2} and could negatively impact O_{2} delivery.

To address the compromise between adequate O_{2} delivery and arterial acid–base status, we developed an integrated numerical model that can be applied to amphibians and reptiles, to provide a quantitative comparison of the effects of R–L shunting and altered ventilation on blood O_{2} and CO_{2} levels.

- ACR
- air convection requirement
*C*_{PaCO2},*C*_{PaO2}- concentration of CO
_{2}or O_{2}in the pulmonary artery *C*_{PvCO2},*C*_{PvO2}- concentration of CO
_{2}or O_{2}in pulmonary venous return (i.e. left atrium) *C*_{SaCO2},*C*_{SaO2}- concentration of CO
_{2}or O_{2}in the systemic arterial blood *C*_{SvCO2},*C*_{SvO2}- concentration of CO
_{2}or O_{2}in systemic venous return (i.e. right atrium) - Hb
- haemoglobin
*L*_{shunt}- gas exchange limitation imposed by shunts
*p*- number of Bohr-groups of haemoglobin
*P*_{ACO2},*P*_{AO2}- partial pressure of CO
_{2}or O_{2}in the lung gas *P*_{CO2},*P*_{O2}- partial pressure of CO
_{2}or O_{2}in a given compartment *P*_{ICO2},*P*_{IO2}- inspired partial pressure of CO
_{2}or O_{2} *Q̇*_{LR}- left-to-right shunt flow
- pulmonary blood flow
- right-to-left shunt flow
- systemic blood flow
- total cardiac output
- R–L
- right-to-left shunt
*R*_{perf}- blood convective/perfusive resistance
- RQ
- respiratory quotient
*R*_{tot}- total resistance imposed to transport between tissues and the environment
*R*_{vent}- air convective/ventilatory resistance
*S*_{H}- fractional saturation of haemoglobin with protons
*S*_{O2}- HbO
_{2}saturation - λ
- blood/gas partitioning coefficient

## MATERIALS AND METHODS

Fig. 1A illustrates the model of gas exchange for O_{2} and CO_{2} based on mass balances and relationships that express electro-neutrality in blood compartments. The model does not include diffusion limitations or spatial heterogeneities at tissues or lungs, and incorporates a thermodynamically correct description of the Bohr–Haldane effect.

### Mass balances

For O_{2}:
(1)
(2)
(3)
(4)

For CO_{2}:
(5)
(6)
(7)
(8)

See Table 1 and the list of symbols and abbreviations for parameter definitions.

### Concentrations and partial pressures in blood

The concentration of O_{2} in each blood compartment (*C*_{bO2}) is the sum of haemoglobin (Hb)-bound O_{2} [product of blood Hb concentration (*C*_{Hb}), number of O_{2} binding sites (*q*=4) and saturation (*S*_{O2})] and the physically dissolved O_{2} [product of physical solubility (α_{O2}) and *P*_{O2}]:
(9)

To quantify the saturation of Hb with O_{2} and protons, the Monod–Wyman–Changeux two-state model (Monod et al., 1965) was incorporated where saturation is a function of both *P*_{O2} and proton concentration to include the Bohr–Haldane effect.

The total concentration of CO_{2} in blood (*C*_{bCO2}) is the sum of the physically dissolved CO_{2} (α_{CO2}*P*_{CO2}) and the bicarbonate and carbonate concentration, as quantified by the equilibrium constants of CO_{2} hydration (*K*_{1} and *K*_{2}) and the proton concentration ([H^{+}], which is related to *S*_{O2}):
(10)

### Electro-neutrality in blood

Equations that express electro-neutrality were derived by conservation of charge, where electro-neutrality in a given blood compartment (subscript *i*) is given below:
(11)

where SID is the strong-ion difference (Stewart, 1978), *K*_{w} is the ionic product of water, β_{NB} is the non-bicarbonate buffer capacity, pH_{iso} is the pH of zero net charge of the buffer groups, *S*_{H} is the fractional saturation of haemoglobin with protons and *p* is the number of Bohr-groups of haemoglobin.

### Shunt fractions and blood flows

Total cardiac output () is the sum of pulmonary and systemic flows ( and , respectively) and the shunt flows (and ) are given by total blood flow and the shunt fractions ( and ). Given the desired general applicability of the model to reptiles with (both R–L and L–R) intra-cardiac shunts, and not just crocodilians with central vascular (R–L) shunts, we derived the following expressions by mass balance, assuming uniformly well-stirred compartments with constant volume where bi-directional shunts can occur independently: (12) (13)

However, given the present purpose we only considered unidirectional R–L shunts.

### Numerical and analytical solutions

Owing to the simplifying assumptions of the model, at steady-state the pulmonary venous partial pressures of O_{2} and CO_{2} (*P*_{PvO2} and *P*_{PvCO2}) are equal to the partial pressures in the lung (*P*_{AO2} and *P*_{ACO2}). The total system of 12 equations that express mass balance and electro-neutrality with 12 dependent variables (i.e. partial pressures and proton concentrations in the systemic and pulmonary arterial and venous system for O_{2} and CO_{2}) was solved numerically in Mathematica (v.10.3, Wolfram Research).

When blood capacitances of O_{2} and CO_{2} are assumed constant (approximately true for CO_{2} and applicable to O_{2} during hypoxia), the system of equations can be solved analytically, leading to the following solutions:
(14)
(15)

where *R*_{tot} is the total resistance imposed to transport from the blood/tissues to the environment equal to the sum of the resistances associated with blood convective/perfusive transport (*R*_{perf}) and ventilation (*R*_{vent}):
(16)

When only considering unidirectional R–L shunts, the total resistance simplifies to: (17)

where β_{b} is the blood capacitance coefficient for O_{2} or CO_{2}. The left part on the right-hand side of Eqn 17 corresponds to *R*_{perf} and simplifies to the normal perfusive resistance [] when there are no shunts, whereas the right part is *R*_{vent}. The perfusive resistance (*R*_{perf}) can be expressed as the normal resistance without shunts (*R*_{perf,FRL=0}) multiplied by a function of the shunt fraction [i.e. *f*(*F*_{RL})=½(2−*F*_{RL})/(1−*F*_{RL})]:
(18)

While *R*_{vent} is the same for O_{2} and CO_{2}, *R*_{perf} and hence *R*_{tot} differ given different β_{b}. The gas exchange limitation (Piiper and Scheid, 1972, 1981) imposed by R–L shunts (*L*_{shunt}) is given by 1 minus the total resistance without shunts (*R*_{tot}, where *F*_{RL}=0) divided by the total resistance with shunts (i.e. *R*_{tot}):
(19)

This can also be expressed by the dimensionless ratio of the normal perfusive to ventilatory resistance without shunts (): (20)

where is given by the ventilation to perfusion ratio and the blood gas partitioning coefficient (λ=β_{b}/β_{g}) as follows:
(21)

From Eqn 20 it is given that the transport limitation imposed by shunts approaches zero when approaches zero (i.e. infinitely high blood flow and partitioning coefficient relative to ventilation). Conversely, the limitation approaches *F*_{RL}/(2−*F*_{RL}) when approaches infinity (i.e. infinitely high ventilation and low partitioning coefficient relative to blood flow).

## RESULTS AND DISCUSSION

The isolated and combined effects of R–L shunts and ACR are illustrated in 3D plots in Fig. 1B–D, where arterial *P*_{CO2}, *P*_{O2} and HbO_{2} saturation (*S*_{O2}) are shown as functions of both R–L shunt fraction (*F*_{RL}) and alveolar ventilation. It is immediately clear that arterial *P*_{CO2} increases most steeply when alveolar ventilation is reduced (i.e. reduced ACR), but only moderately when *F*_{RL} is increased (Fig. 1B). Conversely, both arterial *P*_{O2} and *S*_{O2} are markedly reduced as the R–L shunt increases, whilst reductions in alveolar ventilation only moderately reduce *S*_{O2} (Fig. 1C,D). Thus, our theoretical analysis reveals substantial differences on the influence of R–L shunt and ACR on arterial blood gases, and predicts that ventilatory compensations are much more effective in altering arterial *P*_{CO2} than cardiac shunt patterns.

The differences in the behaviours of O_{2} and CO_{2} upon changing shunt pattern or ACR are also illustrated in Fig. 2A–D, which shows *P*_{O2}–*P*_{CO2} diagrams and similar plots that relate *P*_{CO2} and *S*_{O2}. In Fig. 2B, the dashed line describes steady-state solutions for lung gases and hence also the arterial blood gases in the absence of cardiac shunts (i.e. the mammalian condition). In this case, reductions in ACR cause similar, but reciprocal changes in arterial *P*_{O2} and *P*_{CO2} as predicted by the respiratory quotient (RQ; set to 1 in the simulations). Conversely, an introduction of R–L shunt at a given ACR causes large reductions in arterial *P*_{O2} while arterial *P*_{CO2} only increases moderately (full green curve in Fig. 2B). Thus, to produce the same elevation in arterial *P*_{CO2} by means of a R–L shunt as by a moderate reduction in ACR (e.g. a reduction from 28 to 20 ml air ml^{–1} CO_{2}; Fig. 2B), the shunt fraction would have to increase to 0.8, meaning that 80% of the systemic venous return bypasses the lungs (Fig. 2B). Such a large shunt fraction would concomitantly reduce arterial *P*_{O2} from more than 120 mmHg to less than 30 mmHg (Fig. 2B) and reduce *S*_{O2} from approximately 1.0 to less than 0.5 (Fig. 2A).

The complete solutions for different combinations of *F*_{RL} (varied 0–0.8) and ACR (varied 12.5–50) for arterial blood are given in Fig. 2C,D. The colour coding indicates increasing *P*_{CO2} and the thicker lines originating from the air-line (the black dashed line/curve) depict how *P*_{O2}, *P*_{CO2} and *S*_{O2} change as *F*_{RL} is altered at several constant levels of ACR. The thinner curves, originating from the thicker blood curves, represent solutions when ACR is altered at a given constant *F*_{RL}. By combining the horizontal axes of Fig. 2C,D, the possible solutions are summarized as a 3D diagram with *P*_{CO2} on the vertical *z*-axis and *S*_{O2} and *P*_{O2} on the horizontal *x*- and *y*-axes (Fig. 2E). In this representation, the horizontal *x*–*y* plane reflects the effective O_{2} equilibrium curve. Fig. 2E illustrates that increasing *F*_{RL} causes large reductions in *P*_{O2} of the arterial and venous blood along the O_{2} equilibrium curve, leading to pronounced *S*_{O2} reduction with only moderate elevation of *P*_{CO2}. Conversely, reducing ACR at a given *F*_{RL} leads to a large elevation of *P*_{CO2} with only moderate reductions in *S*_{O2} (thinner upwards-bending curves in Fig. 2E).

The different effects of altering ACR and R–L shunt on O_{2} and CO_{2} is explained by the differences in blood capacitance coefficients (β_{b}) (alternatively expressed as differences in blood gas partitioning coefficients, λ). This is illustrated in Fig. 3, showing the limitation imposed on gas exchange by *F*_{RL} (Eqn 21). Here, the ratio of the normal perfusive to ventilatory resistance without shunts () is varied from a physiologically relevant range for O_{2} and CO_{2} (colour coded), and the asymptotic relationship between the limitation and *F*_{RL} for approaching infinity and =0 is given by the black curve and the horizontal axis. Fig. 3 emphasizes that at a given shunt fraction, the gas species mostly limited by the shunt is the one with the highest blood to air convective resistance () and hence the lowest λ (i.e. lowest β_{b}). Therefore, owing to the high β_{b}, CO_{2} is less limited by shunts than O_{2}, although the differences may become less distinct in deep hypoxia where the effective β_{b} for O_{2} increases. The same conclusion was made by Wagner (1979) when considering the effects of lung shunts on O_{2} versus CO_{2} exchange. Besides the differences in effects of shunts on O_{2} and CO_{2}, Fig. 3 also illustrates that the limitation in general is predicted to increase when overall is high and vice versa.

If digestion is facilitated by supplying the gut with blood with higher CO_{2} levels, our model predicts that this is best mediated by reducing ACR instead of increasing R–L shunt. Elevating CO_{2} levels by increasing R–L shunt would come at the cost of pronounced reductions in O_{2} levels, producing hypoxemia at a time at which O_{2} demand may be elevated fourfold above resting (e.g. Busk et al., 2000). Conversely, reductions of ACR entail much smaller reductions in O_{2} delivery, but provide for an effective elevation of *P*_{CO2} that compensates for the alkaline tide during digestion (Wang et al., 2001a). Furthermore, these postprandial reductions in ACR are well known in reptiles (Hicks et al., 2000; Overgaard et al., 1999; Secor et al., 2000) and *P*_{O2} remains high during digestion in all animals studied, including alligators (Busk et al., 2000; Hartzler et al., 2006; Overgaard et al., 1999).

For many reptiles and amphibians, digestion is associated with large elevations in oxygen demands and an increased need to secrete gastric acid with resulting challenges to blood acid–base balance. Our theoretical approach clearly demonstrates that reliance on R–L shunting to meet the digestive demands conflicts significantly with increased metabolic demands of the digestive organs, and cannot provide adequate compensation for the alkaline tide. In contrast, ventilatory regulation, through reductions in ACR, addresses all the physiological challenges simultaneously, i.e. blood acid–base regulation, increased CO_{2} delivery to the gastric mucosa without sacrificing O_{2} delivery. Thus, while our theoretical model obviously does not provide information on the actual physiological responses of living animals, it would certainly seem that natural selection should favour efficient ventilatory regulation on arterial *P*_{CO2} rather than the ineffective mean of regulation by central vascular shunts.

## FOOTNOTES

**Competing interests**The authors declare no competing or financial interests.

**Author contributions**This analysis results from numerous discussions over the past decade involving all the authors. C.L.M. constructed the model used in the manuscript on the basis of previous simpler attempts. The manuscript was written by H.M. and T.W. with continuous input and final approval of all co-authors.

**Funding**This study was funded by the Danish Research Council (Natur og Univers, Det Frie Forskningsråd).

- Received September 12, 2016.
- Accepted December 6, 2016.

- © 2017. Published by The Company of Biologists Ltd