Prediction of Extravascular Burden of Carbon Monoxide (CO) in the Human Heart

Abstract

Clinically significant myocardial abnormalities (e.g., arrhythmias, S-T elevation) occur in patients with mild-to-severe carbon monoxide (CO) poisoning. We enhanced our previous whole body model [Bruce, E. N., M. C. Bruce, and K. Erupaka. Prediction of the rate of uptake of carbon monoxide from blood by extravascular tissues. Respir. Physiol. Neurobiol. 161(2):142–159, 2008] by adding a cardiac compartment (containing three vascular and two tissue subcompartments differing in capillary density) to predict myocardial carboxymyoglobin (MbCO) and oxygen tensions (P_cO₂) for several CO exposure regimens at rest and during exercise. Model predictions were validated with experimental data in normoxia, hypoxia, and hyperoxia. We simulated exposure at rest to 6462 ppm CO (10 min) and to 265 ppm CO (480 min), and during three levels of exercise at 20% HbCO. We compared responses of carboxyhemoglobin (HbCO), MbCO and P_cO₂ to estimate the potential for myocardial injury due to CO hypoxia. Simulation results predict that during CO exposures and subsequent therapies, cardiac tissue has higher MbCO levels and lower P_cO₂’s than skeletal muscle. CO exposure during exercise further decreases P_cO₂ from resting levels. We conclude that in rest and moderate exercise, the myocardium is at greater risk for hypoxic injury than skeletal muscle during the course of CO exposure and washout. Because the model can predict CO uptake and distribution in human myocardium, it could be a tool to estimate the potential for hypoxic myocardial injury and facilitate therapeutic intervention.

Appendices

Appendix A: Glossary of Model Parameters and Variables

This section contains tables of symbols, parameter values, steady state values and initial conditions.

Table A1 Primary compartmental variables and parameters

Table A2 Definition of symbols related to diffusion coefficients and permeability surface area products

Table A3 Definitions of symbols and subject-specific values used in simulation of experiments of Benignus et al.7; Burge et al.,11 Kizakevich et al.50 (— Indicates “not applicable”)

Table A4 Parameters and their default values

Table A5 Initial values^a

Table A6 Steady state values of concentrations (mL/mL) of O₂ (C _kO₂) and CO (C _kCO) for different subjects, whose responses were simulated using values from Table A3

Appendix B: Mass Balance Equations for Oxygen (O₂) and Carbon Monoxide (CO)

Lung (Alveolar (L) and End-Pulmonary (ep)) Compartments

$$ V_{\rm L} {\frac{{dC_{\rm A} {\text {O}}_{2} (t)}}{dt}} = \left( {{\text {P}}_{\rm I} {\text {O}}_{2} (t) - {\text {P}}_{\rm A} {\text {O}}_{2} (t)} \right) \cdot {\frac{{\dot{V}_{\rm A} }}{{{\text {P}}_{\rm B} }}} - {\text {O}}_{2} {\text {flux}}_{\rm LB} (t) $$

(B.1)

$$ V_{\rm L} {\frac{{dC_{\rm A} {\text {CO}}(t)}}{dt}} = \left( {{\text {P}}_{\rm I} {\text {CO}}(t) - {\text {P}}_{\rm A} {\text {CO}}(t)} \right) \cdot {\frac{{\dot{V}_{\rm A} }}{{{\text {P}}_{\rm B} }}} - {\text {COflux}}_{\rm LB} (t) $$

(B.2)

Auxiliary Equations for Lung Compartment

$$ {\text {P}}_{\rm I} {\text {O}}_{2} = {\text {F}}_{\rm I} {\text {O}}_{2} \cdot \left( {{\text {P}}_{\rm B} - 47} \right) \cdot \left( {{\frac{310}{{{\text {CO}}_{\rm temp} }}}} \right) $$

$${\hbox{P}}_{\rm I}{\hbox{CO}}(t)=\left\{\begin{array}{ll} {0}, & 0 \le t < T_{\rm CO} \\({\hbox{P}}_{\rm B} - 47)\cdot \left(\frac{310}{{\hbox{CO}}_{\rm temp}}\right)\cdot \left(\frac{{\hbox{CO}}_{\rm ppm}}{10^{6}}\right), & T_{\rm CO}\le t < T {\hbox{th}} \\ {0}, & t \ge T{\hbox{th}} \\ \end{array} \right\}$$

$$ {\text {O}}_{2} {\text {flux}}_{\rm LB} (t) = \dot{Q}(t) \cdot \left( {1 - SF} \right) \cdot \left( {C_{\rm ep} {\text {O}}_{2} (t) - C_{\rm mx} {\text {O}}_{2} (t - d_{\rm v} )} \right) $$

$$ \dot{Q}(t) = \dot{Q}_{0} \cdot \left[ {1 + 0.572\left( {\% {\text {COHb}}(t)} \right)} \right] $$

If $\dot{Q}_{0}$ is not measured by the investigators then regression equations (Eqs. C.3 or C.4) are used to estimate $\dot{Q}_{0}.$

$$ C_{\rm ep} {\text {O}}_{2} (t) = {\text {O}}_{2} {\text {Hb}}_{\rm ep} (t) + {\text {S}}_{{{\text {O}}_{2} }} \cdot {\text {P}}_{\rm ep} {\text {O}}_{2} (t) $$

$$ {\text {O}}_{2} {\text {Hb}}_{\rm ep} (t) = f_{\rm odc} ({\text{P}}_{\text{ep}} {\text{O}}_{2} (t),{\text {P}}_{50} (t),n(t),C_{\text{maxep}} (t)) $$

[See section “Special Functions” (F.1) for description of f _odc]

$$ {\text {P}}_{\rm ep} {\text {O}}_{2} (t) = {\text {P}}_{\rm A} {\text {O}}_{2} (t) $$

$$ {\text {P}}_{\rm A} {\text {O}}_{2} (t) = C_{\rm A} {\text {O}}_{2} (t) \cdot {\text {P}}_{\rm B} $$

$$ {\text {P}}_{50} (t) = 50.4219\left[ {1 + { \exp }\left\{ {0.0215\left( {\% {\text {COHb}}(t)} \right)} \right\}} \right]^{ - 1} $$

$$ n(t) = 1.7493 + 0.5909\left[ {{ \exp }\left\{ { - 0.025\left( {\% {\text {COHb}}(t)} \right)} \right\}} \right] $$

$$ \% {\text {COHb}}(t) = \left( {{\frac{100 \cdot {\text {COHb}}(t)}{{{\text {O}}_{2} {\text {Hb}}_{ \max } }}}} \right) $$

$$ {\text {COHb}}(t) = C_{\rm ar} {\text {CO}}(t) - \left( {{\text {S}}_{\rm CO} \cdot {\text {P}}_{\rm ar} {\text {CO}}(t)} \right) $$

$$ C_{{{ \max }{\rm ep}}} (t) = {\text {O}}_{2} {\text {Hb}}_{ \max } - {\text {COHb}}_{\rm ep} (t) $$

$$ {\text {O}}_{2} {\text {Hb}}_{ \max } = K_{\rm O_{2}/{\rm gHb}} (C_{\rm Hgb} ) $$

$$ {\text {COHb}}_{\rm ep} (t) = \, C_{\rm ep} {\text {CO}}(t) - \left( {{\text {S}}_{\rm CO} \cdot {\text {P}}_{\rm ep} {\text {CO}}(t)} \right) $$

$$ C_{\rm ep} {\text {CO}}(t) = C_{\rm mx} {\text {CO}}(t - d_{\rm v} ) + \left( {{\frac{{{\text {COflux}}_{\rm LB} (t)}}{{(1 - SF) \cdot \dot{Q}(t)}}}} \right) $$

$$ C_{\rm mx} {\text {CO}}(t) = \text{COHb}_{\rm mx}(t) + (S_{\text{CO}}\cdot \text{P}_{\text{mx}}\text{CO}(t)) $$

[See section “Mixed Venous Blood Compartment, (mx)” for definition of COHb_mx(t)]

$$ {\text {COflux}}_{\rm LB} (t) = \left[ {{\text {P}}_{\rm A} {\text {CO}} - 0.5\left( {{\text {P}}_{\rm ep} {\text {CO}}(t) + {\text {P}}_{\rm mx} {\text {CO}}(t - d_{\rm v} )} \right)} \right] \cdot D_{\rm L} {\text {CO}} $$

$$ {\text {P}}_{\rm A} {\text {CO}}(t) = C_{\rm A} {\text {CO}}(t) \cdot {\text {P}}_{\rm B} $$

$$ {\text {P}}_{\rm ep} {\text {CO}}(t) = \left( {{\frac{{{\text {P}}_{\rm ep} {\text {O}}_{2} (t)}}{{M_{\rm H} }}}} \right) \cdot \left( {{\frac{{{\text {COHb}}_{\rm ep} (t)}}{{{\text {O}}_{2} {\text {Hb}}_{\rm ep} (t)}}}} \right) $$

Lung (End-Capillary) Compartment, (ec)

$$ C_{\rm ec} {\text {O}}_{2} (t) = SF \cdot C_{\rm mx} {\text {O}}_{2} (t - d_{\rm v} ) + (1 - SF) \cdot C_{\rm ep} {\text {O}}_{2} (t) $$

$$ {\text {O}}_{2} {\text {Hb}}_{\rm ec} (t) = f_{\rm odc} ({\text {P}}_{\rm ec} {\text {O}}_{2} (t),{\text {P}}_{50} (t),n(t),C_{{{ \max }\rm ec}} (t)) $$

[See section “Special Functions” (F.1) for description of f _odc]

$$ {\text {P}}_{\rm ec} {\text {O}}_{2} (t) = f_{\rm imp} ({\text {S}}_{{{\text {O}}_{2} }} ,C_{{{ \max }\rm ec}} (t),{\text {P}}_{50} (t),n(t),C_{\rm ec} {\text {O}}_{2} (t)) $$

[See section “Special Functions” (F.3) for description of f _imp]

$$ C_{{{ \max }\rm ec}} (t) = {\text {O}}_{2} {\text {Hb}}_{ \max } - {\text {COHb}}_{\rm ec} (t) $$

$$ {\text {COHb}}_{\rm ec} (t) = C_{\rm ec} {\text {CO}}(t) - {\text {S}}_{\rm CO} \cdot {\text {P}}_{\rm ec} {\text {CO}}(t) $$

$$ C_{\rm ec} {\text {CO}}(t) = C_{\rm mx} {\text {CO}}(t - d_{\rm v} ) + {\frac{{{\text {COflux}}_{\rm LB} (t)}}{{\dot{Q}(t)}}} $$

$$ {\text {P}}_{\rm ec} {\text {CO}}(t) = \left( {{\frac{{{\text {P}}_{\rm ec} {\text {O}}_{2} (t)}}{{M_{\rm H} }}}} \right) \cdot \left( {{\frac{{{\text {COHb}}_{\rm ec} (t)}}{{{\text {O}}_{2} {\text {Hb}}_{\rm ec} (t)}}}} \right) $$

Arterial Blood Compartment, (ar)

$$ V_{\rm ar} {\frac{{dC_{\rm ar} {\text {O}}_{2} (t)}}{dt}} = \left( {C_{\rm ec} {\text {O}}_{2} (t) - C_{\rm ar} {\text {O}}_{2} (t)} \right) \cdot \dot{Q}(t) $$

(B.3)

$$ V_{\rm ar} {\frac{{dC_{\rm ar} {\text {CO}}(t)}}{dt}} = \left( {C_{\rm ec} {\text {CO}}(t) - C_{\rm ar} {\text {CO}}(t)} \right) \cdot \dot{Q}(t) $$

(B.4)

Auxiliary Equations for Arterial Blood Compartment

$$ \text{P}_{\rm ar} {\text {O}}_{2} (t) = f_{\rm imp} ({\text {S}}_{{{\text {O}}_{2} }} ,C_{{{ \max }\rm ar}} (t),{\text {P}}_{50} (t),n(t),C_{\rm ar} {\text {O}}_{2} (t)) $$

[See section “Special Functions” (F.3) for description of f _imp]

$$ C_{{{ \max }\rm ar}} (t) = {\text {O}}_{2} {\text {Hb}}_{ \max } - {\text {COHb}}(t) $$

$$ {\text {O}}_{2} {\text {Hb}}_{\rm ar} (t) = f_{\rm odc} ({\text {P}}_{\rm ar} {\text {O}}_{2} (t),{\text {P}}_{50} (t),n(t),C_{{{ \max }\rm ar}} (t)) $$

[See section “Special Functions” (F.1) for description of f _odc]

$$ {\text {P}}_{\rm ar} {\text {CO}}(t) = \left( {{\frac{{{\text {P}}_{\rm ar} {\text {O}}_{2} (t)}}{{M_{\rm H} }}}} \right) \cdot \left( {{\frac{{\text {COHb}}(t)}{{{\text {O}}_{2} {\text {Hb}}_{\rm ar} (t)}}}} \right) $$

$$ V_{\rm ar} = 0.25\left( {V_{\rm b} - V_{\rm bm} - V_{\rm bot} } \right) $$

Skeletal Muscle Compartment

Skeletal Muscle Tissue Compartment 1, (m ₁ )

$$ {\frac{{dC_{\rm m1} {\text {O}}_{2} (t)}}{dt}} = {\frac{{{\text {Flux}}_{\rm m1} {\text {O}}_{2} (t)}}{{V_{\rm m1} }}} + {\frac{{D_{\rm m}^{'} {\text {O}}_{2} \cdot \left( {C_{\rm m2}^{d} {\text {O}}_{2} (t) - C_{\rm m1}^{d} {\text {O}}_{2} (t)} \right)}}{{D_{\rm xm} }}} - {\frac{{{{MR}}_{\rm m1} {\text {O}}_{2} (t)}}{{V_{\rm m1} }}} $$

(B.5)

$$ {\frac{{dC_{\rm m1} {\text {CO}}(t)}}{dt}} = {\frac{{{\text {Flux}}_{\rm m1} {\text {CO}}(t)}}{{V_{\rm m1} }}} + {\frac{{D_{\rm m}^{'} {\text {CO}} \cdot \left( {C_{\rm m2}^{d} {\text {CO}}(t) - C_{\rm m1}^{d} {\text {CO}}(t)} \right)}}{{D_{\rm xm} }}} $$

(B.6)

Skeletal Muscle Tissue compartment 2, (m ₂ )

$$ {\frac{{dC_{\rm m2} {\text {O}}_{2} (t)}}{dt}} = {\frac{{{\text {Flux}}_{\rm m2} {\text {O}}_{2} (t)}}{{V_{\rm m2} }}} + {\frac{{D_{\rm m}^{'} {\text {O}}_{2} \cdot \left( {C_{\rm m1}^{d} {\text {O}}_{2} (t) - C_{\rm m2}^{d} {\text {O}}_{2} (t)} \right)}}{{D_{\rm xm} \cdot \left( {{\raise0.7ex\hbox{${V_{\rm m2} }$} \!\mathord{\left/ {\vphantom {{V_{\rm m2} } {V_{\rm m1} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${V_{\rm m1} }$}}} \right)}}} - {\frac{{{{MR}}_{\rm m2} {\text {O}}_{2} (t)}}{{V_{\rm m2} }}} $$

(B.7)

$$ {\frac{{dC_{\rm m2} {\text {CO}}(t)}}{dt}} = {\frac{{{\text {Flux}}_{\rm m2} {\text {CO}}(t)}}{{V_{\rm m2} }}} + {\frac{{D_{\rm m}^{'} {\text {CO}} \cdot \left( {C_{\rm m1}^{d} {\text {CO}}(t) - C_{\rm m2}^{d} {\text {CO}}(t)} \right)}}{{D_{\rm xm} \cdot \left( {{\raise0.7ex\hbox{${V_{\rm m2} }$} \!\mathord{\left/ {\vphantom {{V_{\rm m2} } {V_{\rm m1} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${V_{\rm m1} }$}}} \right)}}} $$

(B.8)

Skeletal Blood compartment 1, (bm ₁ )

$$ V_{\rm bm1} {\frac{{dC_{\rm mv1} {\text {O}}_{2} (t)}}{dt}} = \dot{Q}_{\rm m} (t) \cdot \left( {C_{\rm ar} {\text {O}}_{2} (t) - C_{\rm mv1} {\text {O}}_{2} (t)} \right) - {\text {O}}_{2} {\text {Flux}}_{\rm m1} (t) $$

(B.9)

$$ V_{\rm bm1} {\frac{{dC_{\rm mv1} {\text {CO}}(t)}}{dt}} = \dot{Q}_{\rm m} (t) \cdot \left( {C_{\rm ar} {\text {CO}}(t) - C_{\rm mv1} {\text {CO}}(t)} \right) - {\text {COFlux}}_{\rm m1} (t) $$

(B.10)

Skeletal Blood compartment 2, (bm ₂ ):

$$ V_{\rm bm2} {\frac{{dC_{\rm mv2} {\text {O}}_{2} (t)}}{dt}} = \dot{Q}_{\rm m} (t) \cdot \left( {C_{\rm mv1} {\text {O}}_{2} (t) - C_{\rm mv2} {\text {O}}_{2} (t)} \right) - {\text {O}}_{2} {\text {Flux}}_{\rm m2} (t) $$

(B.11)

$$ V_{\rm bm2} {\frac{{dC_{\rm mv2} {\text {CO}}(t)}}{dt}} = \dot{Q}_{\rm m} (t) \cdot \left( {C_{\rm mv1} {\text {CO}}(t) - C_{\rm mv2} {\text {CO}}(t)} \right) - {\text {COFlux}}_{\rm m2} (t) $$

(B.12)

Skeletal Blood compartment 3, (bm ₃ ):

$$ V_{\rm bm3} {\frac{{dC_{\rm mv3} {\text {O}}_{2} (t)}}{dt}} = \dot{Q}_{\rm m} (t) \cdot \left( {C_{\rm mv2} {\text {O}}_{2}(t) - C_{\rm mv3} {\text {O}}_{2} (t)} \right) - {\text {O}}_{2} {\text {Flux}}_{\rm m3} (t) $$

(B.13)

$$ V_{\rm bm3} {\frac{{dC_{\rm mv3} {\text {CO}}(t)}}{dt}} = \dot{Q}_{\rm m} (t) \cdot \left( {C_{\rm mv2} {\text {CO}}(t) - C_{\rm mv3} {\text {CO}}(t)} \right) - {\text {COFlux}}_{\rm m3} (t) $$

(B.14)

Auxiliary Equations for Skeletal Tissue Compartment 1, m₁

$$ V_{\rm m1} = Fv_{\rm m} \cdot V_{\rm m} $$

$$ {{MR}}_{\rm m1} {\text {O}}_{2} = \left( {{{MR}}_{\rm m} {\text {O}}_{2} } \right) \cdot {\frac{{V_{\rm m1} }}{{V_{\rm m1} + V_{\rm m2} }}} $$

$$ {{MR}}_{\rm m} {\text {O}}_{2} = 1.21\left( {{\frac{{{{MR}}_{\rm lm} {\text {O}}_{2} \cdot V_{\rm lm} + MR_{\rm am} {\text {O}}_{2} \cdot V_{\rm am} + {{MR}}_{\rm tm} {\text {O}}_{2} \cdot V_{\rm tm} }}{\rho }}} \right) $$

$$ {{MR}}_{\rm m1} {\text {O}}_{2} (t) = \left( {{{MR}}_{\rm m} {\text {O}}_{2} } \right) \cdot \left( {{\frac{{V_{\rm m1} }}{{V_{\rm m1} + V_{\rm m2} }}}} \right) \cdot \left( {{\frac{{{\text {P}}_{\rm m1} {\text {O}}_{2} (t)}}{{K_{\rm m} {\text {O}}_{2} + {\text {P}}_{\rm m1} {\text {O}}_{2} (t)}}}} \right) $$

$$ {\text {Flux}}_{\rm m1} {\text {O}}_{2} (t) = {\text {O}}_{2} {\text {Flux}}_{\rm m1} (t) + {\text {O}}_{2} {\text {Flux}}_{\rm m3} (t) $$

$$ {\text {P}}_{\rm m1} {\text {O}}_{2} (t) = f_{\rm imp} \left( {{\text {S}}_{{{\text {O}}_{2} }} ,{\text {O}}_{2} {\text {Mb}}_{\max {\rm m1}} (t),{\text {P}}_{50{\rm m}} ,C_{\rm m1} {\text {O}}_{2}(t),V_{\rm m1} } \right) $$

[See section “Special Functions” (F.3) for description of f _imp]

$$ C_{\rm m1}^{d} {\text {O}}_{2} (t) = C_{\rm m1} {\text {O}}_{2} (t) - {\text {O}}_{2} {\text {Mb}}_{\rm m1} (t) $$

$$ C_{\rm m1}^{d} {\text {CO}}(t) = {\text {S}}_{\rm CO} \cdot {\text {P}}_{\rm m1} {\text {CO}}(t) $$

$$ {\text {O}}_{2} {\text {Mb}}_{{{ \max }\rm m1}} (t) = C{\text {Mb}}_{ \max } - {\text {COMb}}_{\rm m1} (t) $$

$$C{\text {Mb}}_{\max } = {\frac{64500}{17000}}\left({{\frac{{K_{{\text {O}}_{2}}} }{4}} \cdot C_{\rm Mbm} } \right) $$

$$ {\text {COMb}}_{\rm m1} (t) = C_{\rm m1} {\text {CO}}(t) - \left( {{\text {S}}_{\rm CO} \cdot {\text {P}}_{\rm m1} {\text {CO}}(t)} \right) $$

$$ {\text {P}}_{\rm m1} {\text {CO}}(t) = \left( {{\frac{{{\text {P}}_{\rm m1} {\text {O}}_{2} (t)}}{{M_{\rm M} }}}} \right) \cdot \left( {{\frac{{{\text {COMb}}_{\rm m1} (t)}}{{{\text {O}}_{2} {\text {Mb}}_{\rm m1} (t)}}}} \right) $$

$$ {\text {O}}_{2} {\text {Mb}}_{\rm m1} (t) = {\text {O}}_{2} {\text {Mb}}_{{{ \max }\rm m1}} (t) \cdot \left( {{\frac{{{\text {P}}_{\rm m1} {\text {O}}_{2} (t)}}{{{\text {P}}_{50{\rm m}} + {\text {P}}_{\rm m1} {\text {O}}_{2} (t)}}}} \right) $$

$$ D_{\rm m}^{'} {\text {O}}_{2} = 600 \cdot D{\text {O}}_{2} $$

$$ {\text {Flux}}_{\rm m1} {\text {CO}}(t) = {\text {COFlux}}_{\rm m1} (t) + {\text {COFlux}}_{\rm m3} (t) $$

$$ D_{\rm m}^{'} {\text {CO}} = 0.75D_{\rm m}^{'} {\text {O}}_{2} $$

$$ C_{\rm m1} {\text {O}}_{2} (t) = {\text {O}}_{2} {\text {Mb}}_{\rm m1} (t) + {\text {S}}_{{{\text {O}}_{2} }} \cdot {\text {P}}_{\rm m1} {\text {O}}_{2} (t) $$

$$ C_{\rm m1} {\text {CO}}(t) = {\text {COMb}}_{\rm m1} (t) + {\text {S}}_{\rm CO} \cdot {\text {P}}_{\rm m1} {\text {CO}}(t) $$

Auxiliary Equations for Skeletal Tissue Compartment 2, m₂

$$ V_{\rm m2} = \left( {1 - Fv_{\rm m} } \right) \cdot V_{\rm m} $$

$$ {{MR}}_{\rm m2} {\text {O}}_{2} = \left( {{{MR}}_{\rm m} {\text {O}}_{2} } \right) \cdot {\frac{{V_{\rm m2} }}{{V_{\rm m1} + V_{\rm m2} }}} $$

$$ {\text {Flux}}_{\rm m2} {\text {O}}_{2} (t) = {\text {O}}_{2} {\text {Flux}}_{\rm m2} (t) $$

$$ {\text {Flux}}_{\rm m2} {\text {CO}}(t) = {\text {COFlux}}_{\rm m2} (t) $$

$$ C_{\rm m2}^{d} {\text {O}}_{2} (t) = C_{\rm m2} {\text {O}}_{2} (t) - {\text {O}}_{2} {\text {Mb}}_{\rm m2} (t) $$

$$ C_{\rm m2}^{d} {\text {CO}}(t) = {\text {S}}_{\rm CO} \cdot {\text {P}}_{\rm m2} {\text {CO}}(t) $$

All other equations are similar to those of skeletal tissue compartment 1.

Auxiliary Equations for Skeletal Blood Compartment 1, (bm₁)

$$ {\text {O}}_{2} {\text {Flux}}_{\rm m1} (t) = D{\text {b}}_{\rm m1} {\text {O}}_{2} (t) \cdot \left( {{\text {P}}_{\rm a1m} {\text {O}}_{2} (t) - {\text {P}}_{\rm m1} {\text {O}}_{2} (t)} \right) $$

$$ {\text {P}}_{\rm a1m} {\text {O}}_{2} (t) = f_{\rm getPavg} \left( {{\text {O}}_{2} {\text {Hb}}_{{{ \max }\rm m1}} (t),V{\text {b}}_{\rm m1} ,{\text {P}}_{50} (t),n(t),C_{\rm m1av} {\text {O}}_{2} (t),{\text {S}}_{{{\text {O}}_{2} }} ,{\text {P}}_{\rm ar} {\text {O}}_{2} (t),{\text {P}}_{\rm mv1} {\text {O}}_{2} (t)} \right) $$

[See section “Special Functions” (F.5) for description of f _getPavg]

$$ {\text {O}}_{2} {\text {Hb}}_{{{ \max }\rm m1}} (t) = {\text {O}}_{2} {\text {Hb}}_{ \max } - {\text {COHb}}(t) $$

$$ C_{\rm m1av} {\text {O}}_{2} (t) = 0.5\left( {C_{\rm ar} {\text {O}}_{2} (t) + C_{\rm mv1} {\text {O}}_{2} (t)} \right) $$

$$ \dot{Q}_{\rm m} (t) = \dot{Q}_{\rm m0} \cdot \left[ {1 + 0.572\left( {\% {\text {COHb}}(t)} \right)} \right] $$

$$ \dot{Q}_{\rm m0} = \left( {\dot{Q}_{\text{lmg}} \cdot V_{\text{lm}} } \right) + \left( {\dot{Q}_{\text{amg}} \cdot V_{\text{am}} } \right) + \left( {\dot{Q}_{\text{tmg}} \cdot V_{\text{tm}} } \right) $$

$$ V_{\rm bm1} = F_{\rm vm} \cdot V_{\rm bm} $$

$$ V_{\rm bm} = {\text {volfrac}}_{\rm m} \cdot V_{\rm m} $$

$$ \left[ {{\text {P}}_{\rm mv1} {\text {O}}_{2} (t),{\text {O}}_{2} {\text {Hb}}_{\rm mv1} (t)} \right] = f_{{{\text {getPO}}_{2} }} ({\text {O}}_{2} {\text {Hb}}_{\rm ar} (t),{\text {P}}_{50} (t),n(t),{\text {O}}_{2} {\text {Hb}}_{\max {\rm m1}} (t),S_{{{\text {O}}_{2} }} ,{\text {P}}_{\rm ar} {\text {O}}_{2} (t),\dot{Q}_{\rm m} (t)) $$

[See section “Special Functions” (F.4) for description of $f_{{{\rm getPO}_{2} }}$]

$$ \begin{aligned} D{\text {b}}_{\rm m1} {\text {O}}_{2} (t) & = {\frac{{PS_{\rm m1} {\text {O}}_{2} (t) \cdot {\text {S}}_{{{\text {O}}_{2} }} \cdot V_{\rm m1} }}{1.04}} \\ PS_{\rm m1} {\text {O}}_{2} (t) & = PS_{\rm mav\_rest} \cdot {\frac{{\dot{Q}_{\rm m} (t)}}{{\dot{Q}_{\rm m0} }}} \\ {\text {COFlux}}_{\rm m1} (t) & = D{\text {b}}_{\rm m1} {\text {CO}}(t) \cdot \left( {{\text {P}}_{\rm a1m} {\text {CO}}(t) - {\text {P}}_{\rm m1} {\text {CO}}(t)} \right) \\ D_{\rm bm1} {\text {CO}}(t) & = D_{\rm M} {\text {CO}} \cdot \left( {{\frac{{D{\text {b}}_{\rm m1} {\text {O}}_{2} (t)}}{{D{\text {b}}_{\rm m2} {\text {O}}_{2} (t)}}}} \right) \\ {\text {P}}_{\rm a1m} {\text {CO}}(t) & = 0.5\left( {{\text {P}}_{\rm ar} {\text {CO}}(t) \, + \, {\text {P}}_{\rm mv1} {\text {CO}}(t)} \right) \\ {\text {P}}_{\rm mv1} {\text {CO}}(t) & = \left( {{\frac{{{\text {P}}_{\rm mv1} {\text {O}}_{2} (t)}}{{M_{\rm H} }}}} \right) \cdot \left( {{\frac{{{\text {COHb}}_{\rm mv1} (t)}}{{{\text {O}}_{2} {\text {Hb}}_{\rm mv1} (t)}}}} \right) \\ {\text {COHb}}_{\rm mv1} (t) & = C_{\rm mv1} {\text {CO}}(t) - \left( {{\text {S}}_{\rm CO} \cdot {\text {P}}_{\rm mv1} {\text {CO}}(t)} \right) \\ \end{aligned} $$

Auxiliary Equations for Skeletal Blood Compartment 2, (bm₂)

$$ \begin{aligned} {\text {O}}_{2} {\text {Flux}}_{\rm m2} (t) & = D{\text {b}}_{\rm m2} {\text {O}}_{2} (t) \cdot \left( {{\text {P}}_{\rm a2m} {\text {O}}_{2} (t) - {\text {P}}_{\rm m2} {\text {O}}_{2} (t)} \right) \\ V_{\rm bm2} & = \left( {1 - F_{\rm vm} } \right) \cdot V_{\rm bm} \\ D{\text {b}}_{\rm m2} {\text {O}}_{2} (t) & = {\frac{{PS_{\rm m2} {\text {O}}_{2} (t) \cdot {\text {S}}_{{{\text {O}}_{2} }} \cdot V_{\rm m2} }}{1.04}} \\ PS_{\rm m2} {\text {O}}_{2} (t) & = PS_{\rm mcap\_rest} \cdot {\frac{{\dot{Q}_{\rm m} (t)}}{{\dot{Q}_{\rm m0} }}} \\ {\text {COFlux}}_{\rm m2} (t) & = D{\text {b}}_{\rm m2} {\text {CO}}(t) \cdot \left( {{\text {P}}_{\rm a2m} {\text {CO}}(t) - {\text {P}}_{\rm m2} {\text {CO}}(t)} \right) \\ D{\text {b}}_{\rm m2} {\text {CO}}(t) & = D_{\rm M} {\text {CO}} \\ \end{aligned} $$

All other equations are similar to those of skeletal blood compartment 1.

Auxiliary Equations for Skeletal Blood Compartment 3, (bm₃)

$$ \begin{aligned} {\text {O}}_{2} {\text {Flux}}_{\rm m3} (t) & = D{\text {b}}_{\rm m3} {\text {O}}_{2} (t) \cdot \left( {{\text {P}}_{\rm a3m} {\text {O}}_{2} (t) - {\text {P}}_{\rm m1} {\text {O}}_{2} (t)} \right) \\ V_{\rm bm3} & = D_{\rm bvm\_on} \cdot V_{\rm bm1} \\ D{\text {b}}_{\rm m3} {\text {O}}_{2} (t) & = D{\text {b}}_{\rm m1} {\text {O}}_{2} (t) \cdot D_{\rm bvm\_on} \\ {\text {COFlux}}_{\rm m3} (t) & = D_{\rm bm3} {\text {CO}}(t) \cdot \left( {{\text {P}}_{\rm a3m} {\text {CO}}(t) - {\text {P}}_{\rm m1} {\text {CO}}(t)} \right) \\ D_{\rm bm3} {\text {CO}}(t) & = D_{\rm M} {\text {CO}} \cdot \left( {{\frac{{D{\text {b}}_{\rm m3} {\text {O}}_{2}(t)}}{{D{\text {b}}_{\rm m2} {\text {O}}_{2}(t)}}}} \right) \\ \end{aligned} $$

All other equations are similar to those of skeletal blood compartment 1.

Cardiac Muscle Compartment

Cardiac Muscle Tissue Compartment 1, (c ₁ )

$$ {\frac{{dC_{\rm c1} {\text {O}}_{2} (t)}}{dt}} = {\frac{{{\text {Flux}}_{\rm c1} {\text {O}}_{2} (t)}}{{V_{\rm c1} }}} + {\frac{{D_{\rm c}^{'} {\text {O}}_{2} \cdot \left[ {C_{\rm c2}^{d} {\text {O}}_{2} (t) - C_{\rm c1}^{d} {\text {O}}_{2} (t)} \right]}}{{D_{\rm xc} }}} - {\frac{{{{MR}}_{\rm c1} {\text {O}}_{2} (t)}}{{V_{\rm c1} }}} $$

(B.15)

$$ {\frac{{dC_{\rm c1} {\text {CO}}(t)}}{dt}} = {\frac{{{\text {Flux}}_{\rm c1} {\text {CO}}(t)}}{{V_{\rm c1} }}} + {\frac{{D_{\rm c}^{'} {\text {CO}} \cdot \left[ {C_{\rm c2}^{d} {\text {CO}}(t) - C_{\rm c1}^{d} {\text {CO}}(t)} \right]}}{{D_{\rm xc} }}} $$

(B.16)

Cardiac Muscle Tissue Compartment 2, (c ₂ )

$$ {\frac{{dC_{\rm c2} {\text {O}}_{2} (t)}}{dt}} = {\frac{{{\text {Flux}}_{\rm c2} {\text {O}}_{2} (t)}}{{V_{\rm c2} }}} + {\frac{{D_{\rm c}^{'} {\text {O}}_{2} \cdot \left( {C_{\rm c1}^{d} {\text {O}}_{2} (t) - C_{\rm c2}^{d} {\text {O}}_{2} (t)} \right)}}{{D_{\rm xc} \cdot \left( {{\raise0.7ex\hbox{${V_{\rm c2} }$} \!\mathord{\left/ {\vphantom {{V_{\rm c2} } {V_{\rm c1} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${V_{\rm c1} }$}}} \right)}}} - {\frac{{{{MR}}_{\rm c2} {\text {O}}_{2} (t)}}{{V_{\rm c2} }}} $$

(B.17)

$$ {\frac{{dC_{\rm c2} {\text {CO}}(t)}}{dt}} = {\frac{{{\text {Flux}}_{\rm c2} {\text {CO}}(t)}}{{V_{\rm c2} }}} + {\frac{{D_{\rm c}^{'} {\text {CO}} \cdot \left( {C_{\rm c1}^{d} {\text {CO}}(t) - C_{\rm c2}^{d} {\text {CO}}(t)} \right)}}{{D_{\rm xc} \cdot \left( {{\raise0.7ex\hbox{${V_{\rm c2} }$} \!\mathord{\left/ {\vphantom {{V_{\rm c2} } {V_{\rm c1} }}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${V_{\rm c1} }$}}} \right)}}} $$

(B.18)

Cardiac Blood compartment 1, (bc ₁ )

$$ V_{\rm bc1} {\frac{{dC_{\rm cv1} {\text {O}}_{2} (t)}}{dt}} = \dot{Q}_{\rm c} (t) \cdot \left( {C_{\rm ar} {\text {O}}_{2} (t) - C_{\rm cv1} {\text {O}}_{2} (t)} \right) - {\text {O}}_{2} {\text {Flux}}_{\rm c1} (t) $$

(B.19)

$$ V_{\rm bc1} {\frac{{dC_{\rm cv1} {\text {CO}}(t)}}{dt}} = \dot{Q}_{\rm c} (t) \cdot \left( {C_{\rm ar} {\text {CO}}(t) - C_{\rm cv1} {\text {CO}}(t)} \right) - {\text {COFlux}}_{\rm c1} (t) $$

(B.20)

Cardiac Blood compartment 2, (bc ₂ )

$$ V_{\rm bc2} {\frac{{dC_{\rm cv2} {\text {O}}_{2} (t)}}{dt}} = \dot{Q}_{\rm c} (t) \cdot \left( {C_{\rm cv1} {\text {O}}_{2} (t) - C_{\rm cv2} {\text {O}}_{2} (t)} \right) - {\text {O}}_{2} {\text {Flux}}_{\rm c2} (t) $$

(B.21)

$$ V_{\rm bc2} {\frac{{dC_{\rm cv2} {\text {CO}}(t)}}{dt}} = \dot{Q}_{\rm c} (t) \cdot \left( {C_{\rm cv1} {\text {CO}}(t) - C_{\rm cv2} {\text {CO}}(t)} \right) - {\text {COFlux}}_{\rm c2} (t) $$

(B.22)

Cardiac Blood compartment 3, (c ₃ )

$$ V_{\rm bc3} {\frac{{dC_{\rm cv3} {\text {O}}_{2} (t)}}{dt}} = \dot{Q}_{\rm c} (t) \cdot \left( {C_{\rm cv2}} {\text {O}_{2}(t) - C_{\rm cv3} {\text {O}}_{2} (t)} \right) - {\text {O}}_{2} {\text {Flux}}_{\rm c3} (t) $$

(B.23)

$$ V_{\rm bc3} {\frac{{dC_{\rm cv3} {\text {CO}}(t)}}{dt}} = \dot{Q}_{\rm c} (t) \cdot \left( {C_{\rm cv2} {\text {CO}}(t) - C_{\rm cv3} {\text {CO}}(t)} \right) - {\text {COFlux}}_{\rm c3} (t) $$

(B.24)

Auxiliary Equations Cardiac Tissue Compartment 1, c₁

$$ \begin{aligned} V_{\rm c1} & = Fv_{\rm c} \cdot V_{\rm c} \\ {{MR}}_{\rm c1} {\text {O}}_{2} & = \left( {{{MR}}_{\rm c} {\text {O}}_{2} } \right) \cdot {\frac{{V_{{{\text{c}}1}} }}{{V_{{{\text{c}}1}} + V_{{{\text{c}}2}} }}} \\ {{MR}}_{\rm c} {\text {O}}_{2} & = 1.21\left( {{\frac{{M_{\rm c} \cdot V_{\rm c} }}{100\rho }}} \right) \\ M_{\rm c} & = {\frac{46.6}{{1 + \left( {{\frac{122.7}{HR}}} \right)^{4.85} }}} + 9.76 \\ {{MR}}_{\rm c1} {\text {O}}_{2} (t) & = \left( {{{MR}}_{\rm c} {\text {O}}_{2} } \right) \cdot \left( {{\frac{{V_{\rm c1} }}{{V_{\rm c1} + V_{\rm c2} }}}} \right) \cdot \left( {{\frac{{{\text {P}}_{\rm c1} {\text {O}}_{2} (t)}}{{K_{\rm c} {\text {O}}_{2} + {\text {P}}_{\rm c1} {\text {O}}_{2} (t)}}}} \right) \\ {\text {Flux}}_{\rm c1} {\text {O}}_{2} (t) & = {\text {O}}_{2} {\text {Flux}}_{\rm c1} (t) + {\text {O}}_{2} {\text {Flux}}_{\rm c3} (t) \\ {\text {P}}_{\rm c1} {\text {O}}_{2} (t) & = f_{\rm imp} ({\text {S}}_{{{\text {O}}_{2} }} ,{\text {O}}_{2} {\text {Mb}}_{{{ \max }\rm c1}} (t),{\text {P}}_{50{\rm c}} ,C_{\rm c1} {\text {O}}_{2}(t),V_{\rm c1} ) \\ \end{aligned} $$

[See section “Special Functions” (F.3) for description of f _imp]

$$ \begin{aligned} C_{\rm c1}^{d} {\text {O}}_{2} (t) & = C_{\rm c1} {\text {O}}_{2} (t) - {\text {O}}_{2} {\text {Mb}}_{\rm c1} (t) \\ C_{\rm c1}^{d} {\text {CO}}(t) & = {\text {S}}_{{\text {CO}}} \cdot {\text {P}}_{{\text {c1}}} {\text {CO}}(t) \\ {\text {O}}_{2} {\text {Mb}}_{{{ \max }\rm c1}} (t) & = {\text {CMb}}_{{{ \max }{\text {c}}}} - {\text {COMb}}_{{\text {c1}}} (t) \\ {\text {CMb}}_{{{ \max }{\text {c}}}} & = {\frac{64500}{17000}}\left( {{\frac{{K_{{{\text {O}}_{2}}} }}{4}} \cdot C_{{\text {Mbc}}} } \right) \\ {\text {COMb}}_{{\text {c1}}} (t) & = C_{{\text {c1}}} {\text {CO}}(t) - \left( {{\text {S}}_{{\text {CO}}} \cdot {\text {P}}_{{\text {c1}}} {\text {CO}}(t)} \right) \\ {\text {P}}_{{\text {c1}}} {\text {CO}}(t) & = \left( {{\frac{{{\text {P}}_{{\text {c1}}} {\text {O}}_{2} (t)}}{{M_{{\text {M}}} }}}} \right) \cdot \left( {{\frac{{{\text {COMb}}_{{\text {c1}}} (t)}}{{{\text {O}}_{2} {\text {Mb}}_{{\text {c1}}} (t)}}}} \right) \\ {\text {O}}_{2} {\text {Mb}}_{{\text {c1}}} (t) & = {\text {O}}_{2} {\text {Mb}}_{{{ \max }{\text {c1}}}} (t) \cdot \left( {{\frac{{{\text {P}}_{{\text {c1}}} {\text {O}}_{2} (t)}}{{{\text {P}}_{50{\text {c}}} + {\text {P}}_{{\text {c1}}} {\text {O}}_{2} (t)}}}} \right) \\ D_{{\text {c}}}^{'} {\text {O}}_{2} & = 600 \cdot D{\text {O}}_{2} \\ {\text {Flux}}_{{\text {c1}}} {\text {CO}}(t) & = {\text {COFlux}}_{{\text {c1}}} (t) + {\text {COFlux}}_{{\text {c3}}} (t) \\ D_{{\text {c}}}^{'} {\text {CO}} & = 0.75D_{{\text {c}}}^{'} {\text {O}}_{2} \\ C_{{\text {c1}}} {\text {O}}_{2} (t) & = {\text {O}}_{2} {\text {Mb}}_{{\text {c1}}} (t) + {\text {S}}_{{{\text {O}}_{2} }} \cdot {\text {P}}_{{\text {c1}}} {\text {O}}_{2} (t) \\ C_{{\text {c1}}} {\text {CO}}(t) & = {\text {COMb}}_{{\text {c1}}} (t) + {\text {S}}_{{\text {CO}}} \cdot {\text {P}}_{{\text {c1}}} {\text {CO}}(t) \\ \end{aligned} $$

Auxiliary Equations Cardiac Tissue Compartment 2, c₂

$$ \begin{aligned} V_{\rm c2} & = \left( {1 - Fv_{\rm c} } \right) \cdot V_{\rm c} \\ {{MR}}_{\rm c2} {\text {O}}_{2} & = \left( {{{MR}}_{\rm c} {\text {O}}_{2} } \right) \cdot {\frac{{V_{{{\text{c}}2}} }}{{V_{{{\text{c}}1}} + V_{{{\text{c}}2}} }}} \\ C_{\rm c2}^{d} {\text {O}}_{2} (t) & = C_{\rm c2} {\text {O}}_{2} (t) - {\text {O}}_{2} {\text {Mb}}_{\rm c2} (t) \\ C_{\rm c2}^{d} {\text {CO}}(t) & = {\text {S}}_{{\text {CO}}} \cdot {\text {P}}_{\rm c2} {\text {CO}}(t) \\ {\text {Flux}}_{\rm c2} {\text {O}}_{2} (t) & = {\text {O}}_{2} {\text {Flux}}_{\rm c2} (t) \\ {\text {Flux}}_{\rm c2} {\text {CO}}(t) & = {\text {COFlux}}_{\rm c2} (t) \\ \end{aligned} $$

All other equations are similar to those of cardiac tissue compartment 1.

Auxiliary Equations Cardiac Blood Compartment 1, bc₁

$$ \begin{gathered} {\text {O}}_{2} {\text {Flux}}_{\rm c1} (t) = D{\text {b}}_{\rm c1} {\text {O}}_{2} (t) \cdot \left( {{\text {P}}_{\rm a1c} {\text {O}}_{2} (t) - {\text {P}}_{\rm c1} {\text {O}}_{2} (t)} \right) \hfill \\ {\text {P}}_{\rm a1c} {\text {O}}_{2} (t) = f_{\rm getPavg} \left( {{\text {O}}_{2} {\text {Hb}}_{{{ \max }\rm c1}} (t),V{\text {b}}_{\rm c1} ,{\text {P}}_{50} (t),n(t),C_{\rm c1av} {\text {O}}_{2} (t),{\text {S}}_{{{\text {O}}_{2} }} ,{\text {P}}_{\rm ar} {\text {O}}_{2} (t),{\text {P}}_{\rm cv1} {\text {O}}_{2} (t)} \right) \hfill \\ \end{gathered} $$

[See section “Special Functions” (F.5) for description of f _getPavg]

$$ \begin{aligned} {\text {O}}_{2} {\text {Hb}}_{{{ \max }\rm c1}} (t) & = {\text {O}}_{2} {\text {Hb}}_{ \max } - {\text {COHb}}(t) \\ C_{\rm c1av} {\text {O}}_{2} (t) & = 0.5\left( {C_{\rm ar} {\text {O}}_{2} (t) + C_{\rm cv1} {\text {O}}_{2} (t)} \right) \\ \dot{Q}_{\rm c} (t) & = \dot{Q}_{\rm c0} \cdot \left[ {1 + 0.572\left( {\% {\text {COHb}}(t)} \right)} \right] \\ \dot{Q}_{\rm c0} & = \left( {{\frac{{\left[ {2.18\left( {HR} \right) - 27.3} \right]}}{100}}} \right) \cdot V_{\text{c}} \\ V_{\rm bc1} & = F_{\rm vc} \cdot V_{\rm bc} \\ V{\text {b}}_{\rm c} & = {\text {volfrac}}_{\rm c} \cdot V_{\rm c} \\ \end{aligned} $$

$$ \left[ {{\text {P}}_{\rm cv1} {\text {O}}_{2} (t),{\text {O}}_{2} {\text {Hb}}_{\rm cv1} (t)} \right] = f_{{\rm getPO_{2} }} ({\text {O}}_{2} {\text {Hb}}_{\rm ar} (t),{\text {P}}_{50} (t),n(t),{\text {O}}_{2} {\text {Hb}}_{{{ \max }\rm c1}} (t),{\text {S}}_{{{\text {O}}_{2} }} ,{\text {P}}_{\rm ar} {\text {O}}_{2} (t),\dot{Q}_{\rm c} (t)) $$

[See section “Special Functions” (F.4) for description of $f_{{\rm getPO_{2} }}$]

$$ \begin{aligned} D{\text {b}}_{\rm c1} {\text {O}}_{2} (t) & = {\frac{{PS_{\rm c1} {\text {O}}_{2} (t) \cdot {\text {S}}_{{{\text {O}}_{2} }} \cdot V_{\rm c1} }}{1.04}} \\ PS_{\rm c1} {\text {O}}_{2} (t) & = PS_{\rm cav\_rest} \cdot {\frac{{\dot{Q}_{\rm c} (t)}}{{\dot{Q}_{\rm c0} }}} \\ {\text {COFlux}}_{\rm c1} (t) & = D{\text {b}}_{\rm c1} {\text {CO}}(t) \cdot \left( {{\text {P}}_{\rm a1c} {\text {CO}}(t) - {\text {P}}_{\rm c1} {\text {CO}}(t)} \right) \\ D{\text {b}}_{\rm c1} {\text {CO}}(t) & = D_{\rm c} {\text {CO}}(t) \cdot \left( {{\frac{{D{\text {b}}_{\rm c1} {\text {O}}_{2}(t)}}{{D{\text {b}}_{\rm c2} {\text {O}}_{2}(t)}}}} \right) \\ D_{\rm c} {\text {CO}}(t) & = D_{\rm M} {\text {CO}} \cdot \left( {{\frac{{D{\text {b}}_{\rm c2} {\text {O}}_{2} (t)}}{{D{\text {b}}_{\rm m2} {\text {O}}_{2} (t)}}}} \right) \\ {\text {P}}_{\rm a1c} {\text {CO}}(t) & = 0.5\left( {{\text {P}}_{\rm ar} {\text {CO}}(t) + {\text {P}}_{\rm cv1} {\text {CO}}(t)} \right) \\ {\text {P}}_{\rm cv1} {\text {CO}}(t) & = \left( {{\frac{{{\text {P}}_{\rm cv1} {\text {O}}_{2} (t)}}{{{\text {M}}_{\rm H} }}}} \right) \cdot \left( {{\frac{{{\text {COHb}}_{\rm cv1} (t)}}{{{\text {O}}_{2} {\text {Hb}}_{\rm cv1} (t)}}}} \right) \\ {\text {COHb}}_{\rm cv1} (t) & = C_{\rm cv1} {\text {CO}}(t) - \left( {{\text {S}}_{\rm CO} \cdot {\text {P}}_{\rm cv1} {\text {CO}}(t)} \right) \\ \end{aligned} $$

Auxiliary Equations Cardiac Blood Compartment 2, bc₂

$$ \begin{aligned} {\text {O}}_{2} {\text {Flux}}_{\rm c2} (t) & = D{\text {b}}_{\rm c2} {\text {O}}_{2} \cdot \left( {\text {{P}}_{\rm a2c} {\text {O}}_{2} (t) -{\text { P}}_{\rm c2} {\text {O}}_{2} (t)} \right) \\ V_{\rm bc2} & = \left( {1 - F_{\rm vc} } \right) \cdot V_{\rm bc} \\ D{\text {b}}_{\rm c2} {\text {O}}_{2} (t) & = {\frac{{PS_{\rm c2} {\text {O}}_{2} (t) \cdot {\text {S}}_{{{\text {O}}_{2} }} \cdot V_{\rm c2} }}{1.04}} \\ PS_{\rm c2} {\text {O}}_{2} (t) & = PS_{\rm ccap\_rest} \cdot {\frac{{\dot{Q}_{\rm c} (t)}}{{\dot{Q}_{\rm c0} }}} \\ {\text {COFlux}}_{\rm c2} (t) & = D{\text {b}}_{\rm c2} {\text {CO}}(t) \cdot \left( {{\text {P}}_{\rm a2c} {\text {CO}}(t) - {\text {P}}_{\rm c2} {\text {CO}}(t)} \right) \\ D{\text {b}}_{\rm c2} {\text {CO}}(t) & = D_{\rm c} {\text {CO}}(t) \\ \end{aligned} $$

All other equations are similar to those of cardiac blood compartment 1.

Auxiliary Equations Cardiac Blood Compartment 3, bc₃

$$ \begin{aligned} {\text {O}}_{2} {\text {Flux}}_{\rm c3} (t) & = D{\text {b}}_{\rm c3} {\text {O}}_{2} (t) \cdot \left( {{\text {P}}_{\rm a3c} {\text {O}}_{2} (t) - {\text {P}}_{\rm c1} {\text {O}}_{2} (t)} \right) \\ V_{\rm bc3} & = D_{\rm bvc\_on} \cdot V_{\rm bc1} \\ D{\text {b}}_{\rm c3} {\text {O}}_{2} (t) & = D_{\rm bvc\_on} \cdot D{\text {b}}_{\rm c1} {\text {O}}_{2} (t) \\ {\text {COFlux}}_{\rm c3} (t) & = D_{\rm bc3} {\text {CO}}(t) \cdot \left( {{\text {P}}_{\rm a3c} {\text {CO}}(t) - {\text {P}}_{\rm c1} {\text {CO}}(t)} \right) \\ D{\text {b}}_{\rm c3} {\text {CO}}(t) & = D_{\rm c} {\text {CO}}(t) \cdot \left( {{\frac{{D{\text {b}}_{\rm c3} {\text {O}}_{2} (t)}}{{D{\text {b}}_{\rm c2} {\text {O}}_{2} (t)}}}} \right) \\ \end{aligned} $$

All other equations are similar to those of cardiac blood compartment 1.

Other Tissue Compartment, (ot)

$$ V_{\rm ot} {\frac{{dC_{\rm ot} {\text {O}}_{2} (t)}}{dt}} = {\text {O}}_{2} {\text {flux}}_{\rm ot} (t) - {{MR}}_{\rm ot} {\text {O}}_{2} $$

(B.25)

$$ V_{\rm ot} {\frac{{dC_{\rm ot} {\text {CO}}(t)}}{dt}} = {\text {COflux}}_{\rm ot} (t) $$

(B.26)

Auxiliary Equations for Other Tissue (Nonmuscle) Compartment

$$ \begin{gathered} {{MR}}_{\rm ot} {\text {O}}_{2} = {{MR}{\rm O}}_{2} - \left( {{{MR}}_{\rm m} {\text {O}}_{2} + {{MR}}_{\rm c} {\text {O}}_{2} } \right) \hfill \\ {\text {O}}_{2} {\text {flux}}_{\rm ot} (t) = \left[ {{\text {P}}_{\rm ot\_avg} {\text {O}}_{2} (t) - {\text {P}}_{\rm ot} {\text {O}}_{2} (t)} \right] \cdot D{\text {b}}_{\rm ot} {\text {O}}_{2} \hfill \\ {\text {P}}_{\rm ot\_avg} {\text {O}}_{2} (t) = f_{\rm pinv} \left( {{\text {O}}_{2} {\text {Hb}}_{\rm ot} (t),{\text {P}}_{50} (t),n(t),C_{{{ \max }\rm ot}} (t)} \right) \hfill \\ \end{gathered} $$

[See section “Special Functions” (F.2) for description of f _pinv]

$$ \begin{gathered} {\text {O}}_{2} {\text {Hb}}_{\rm ot} (t) = 0.5({\text {O}}_{2} {\text {Hb}}_{\rm ar} (t) + {\text {O}}_{2} {\text {Hb}}_{\rm vot} (t)) \hfill \\ \tau_{\rm ot} {\frac{{d{\text {O}}_{\rm 2} {\text {Hb}}_{\rm vot} (t)}}{dt}} + {\text {O}}_{2} {\text {Hb}}_{\rm vot} (t) = {\text {O}}_{2} {\text {Hb}}_{\rm votun} (t) \hfill \\ \left[ {{\text {P}}_{\rm vot} {\text {O}}_{2} (t),{\text {O}}_{2} {\text {Hb}}_{\rm votun} (t)} \right] = f_{{\rm getPO_{2} }} ({\text {O}}_{2} {\text {Hb}}_{\rm ar} (t),{\text {P}}_{50} (t),n(t),C_{{{ \max }\rm ot}} (t),{\text {S}}_{{\rm O_{2} }} ,{\text {P}}_{\rm ar} {\text {O}}_{2} (t),\dot{Q}_{\rm ot} (t)) \hfill \\ \end{gathered} $$

[See section “Special Functions” (F.4) for description of $f_{{\rm getPO_{2} }}$]

$$ \begin{gathered} C_{{{ \max }\rm ot}} (t) = {\text {O}}_{2} {\text {Hb}}_{ \max } - {\text {COHb}}_{\rm vot} (t) \hfill \\ \tau_{\rm ot} {\frac{{d{\text {COHb}}_{\rm vot} (t)}}{dt}} + {\text {COHb}}_{\rm vot} (t) = {\text {COHb}}_{\rm votun} (t) \hfill \\ {\text {COHb}}_{\rm votun} (t) = C_{\rm ar} {\text {CO}}(t) - {\text {S}}_{\rm CO} \cdot {\text {P}}_{\rm vot} {\text {CO}}(t) - {\frac{{{\text {COflux}}_{\rm ot} (t)}}{{\dot{Q}_{\rm ot} (t)}}} \hfill \\ \dot{Q}_{\rm ot} (t) = \dot{Q}(t) - \left( {\dot{Q}_{\rm m} (t) + \dot{Q}_{\rm c} (t)} \right) \hfill \\ {\text {P}}_{\rm vot} {\text {CO}}(t) = \left( {{\frac{{{\text {P}}_{\rm vot} {\text {O}}_{2} (t)}}{{M_{\rm H} }}}} \right) \cdot \left( {{\frac{{{\text {COHb}}_{\rm vot} (t)}}{{{\text {O}}_{2} {\text {Hb}}_{\rm vot} (t)}}}} \right) \hfill \\ {\text {COflux}}_{\rm ot} (t) = D{\text {b}}_{\rm ot} {\text {CO}} \cdot \left( {{\text {P}}_{\rm bot} {\text {CO}}(t) - {\text {P}}_{\rm ot} {\text {CO}}(t)} \right) \hfill \\ D{\text {b}}_{\rm ot} {\text {CO}} = D_{\rm M} {\text {CO}} \cdot \left( {{\frac{{V_{\rm ot} }}{{V_{\rm m} }}}} \right) \hfill \\ V_{\rm ot} = V_{\rm m} \cdot \left( {{\frac{{V_{{\rm ot_{0} }} }}{{V_{{\rm m_{0} }} }}}} \right) \hfill \\ {\text {P}}_{\rm bot} {\text {CO}}(t) = 0.5\left( {{\text {P}}_{\rm ar} {\text {CO}}(t) + {\text {P}}_{\rm vot} {\text {CO}}(t)} \right) \hfill \\ {\text {P}}_{\rm ot} {\text {O}}_{2} (t) = {\frac{{C_{\rm ot} {\text {O}}_{2} (t)}}{{{\text {S}}_{{{\text {O}}_{2} }} }}} \hfill \\ C_{\rm vot} {\text {O}}_{2} (t) = {\text {P}}_{\rm vot} {\text {O}}_{2} (t) \cdot {\text {S}}_{{{\text {O}}_{2} }} \hfill \\ {\text {P}}_{\rm ot} {\text {CO}}(t) = {\frac{{C_{\rm ot} {\text {CO}}(t)}}{{{\text {S}}_{\rm CO} }}} \hfill \\ C_{\rm vot} {\text {CO}}(t) = {\text {P}}_{\rm vot} {\text {CO}}(t) \cdot {\text {S}}_{\rm CO} \hfill \\ \tau_{\rm ot} = {\frac{{V_{\rm bot} }}{{\dot{Q}_{\rm ot} (t)}}} \hfill \\ V_{\rm bot} = {\text {Volfrac}}_{\rm ot} \cdot V_{\rm ot} \, \hfill \\ \end{gathered} $$

Mixed Venous Blood compartment, (mx)

$$ V_{\rm mx} {\frac{{dC_{\rm mx} {\text {O}}_{2} (t)}}{dt}} = \dot{Q}(t) \cdot \left( {C_{\rm mx\_in} {\text {O}}_{2} (t) - C_{\rm mx} {\text {O}}_{2} (t)} \right) $$

(B.27)

$$ V_{\rm mx} {\frac{{dC_{\rm mx} {\text {CO}}(t)}}{dt}} = \dot{Q}(t) \cdot \left( {C_{\rm mx\_in} {\text {CO}}(t) - C_{\rm mx} {\text {CO}}(t)} \right) $$

(B.28)

Auxiliary Equations for Mixed Venous Compartment

$$ C_{\rm mx\_in} {\text {O}}_{2} (t) = C_{\rm vot} {\text {O}}_{2} (t) \cdot \left( {{\frac{{\dot{Q}_{\rm ot} (t)}}{{\dot{Q}(t)}}}} \right) + C_{\rm mv3} {\text {O}}_{2} (t) \cdot \left( {{\frac{{\dot{Q}_{\rm m} (t)}}{{\dot{Q}(t)}}}} \right) + C_{\rm cv3} {\text {O}}_{2} (t) \cdot \left( {{\frac{{\dot{Q}_{\rm c} (t)}}{{\dot{Q}(t)}}}} \right) $$

$$ \tau_{\rm mx} = {\frac{{{\text{V}}_{\rm mx} }}{{\dot{Q}(t)}}} $$

$$ V_{\rm mx} = 0.75 \cdot \left( {V_{\rm b} - V_{\rm bm} - V_{\rm bot} } \right) $$

$$ {\text {P}}_{\rm mx} {\text {O}}_{2} (t) = f_{\rm imp} \left( {{\text {S}}_{{{\text {O}}_{2} }} ,C_{{{ \max }\rm mx}} (t),{\text {P}}_{50} (t),n(t),C_{\text{mx}} {\text {O}}_{2} (t),V_{\text{mx}} } \right) $$

[See section “Special Functions” (F.3) for description of f _imp]

$$ C_{{{ \max }\rm mx}} (t) = {\text {O}}_{2} {\text {Hb}}_{ \max } - {\text {COHb}}_{\rm mx} (t) $$

$$ {\text {COHb}}_{\rm mx} (t) = {\text {COHb}}_{\rm vot} (t) \cdot \left( {{\frac{{\dot{Q}_{\rm ot} (t)}}{{\dot{Q}(t)}}}} \right) + {\text {COHb}}_{\rm mv3} (t) \cdot \left( {{\frac{{\dot{Q}_{\rm m} (t)}}{{\dot{Q}(t)}}}} \right) + {\text {COHb}}_{\rm cv3} (t) \cdot \left( {{\frac{{\dot{Q}_{\rm c} (t)}}{{\dot{Q}(t)}}}} \right) + \left( {{\frac{{\dot{V}{\text {co}}}}{{\dot{Q}(t)}}}} \right) $$

$$ {\text {P}}_{\rm mx} {\text {CO}}(t) = \left( {{\frac{{{\text {P}}_{\rm mx} {\text {O}}_{2} (t)}}{{M_{\rm H} }}}} \right) \cdot \left( {{\frac{{{\text {COHb}}_{\rm mx} (t)}}{{{\text {O}}_{2} {\text {Hb}}_{\rm mx} (t)}}}} \right) $$

$$ {\text {O}}_{2} {\text {Hb}}_{\rm mx} (t) = f_{\rm odc} \left( {{\text {P}}_{\rm mx} {\text {O}}_{2} (t),{\text {P}}_{50} (t),n(t),C_{{{ \max }\rm mx}} (t)} \right) $$

[See section “Special Functions” (F.1) for description of f _odc]

$$ C_{\rm mx\_in} CO(t) = C_{\rm vot} {\text {CO}}(t) \cdot \left( {{\frac{{\dot{Q}_{\rm ot} (t)}}{{\dot{Q}(t)}}}} \right) + C_{\rm mv3} {\text {CO}}(t) \cdot \left( {{\frac{{\dot{Q}_{\rm m} (t)}}{{\dot{Q}(t)}}}} \right) + C_{\rm cv3} {\text {CO}}(t) \cdot \left( {{\frac{{\dot{Q}_{\rm c} (t)}}{{\dot{Q}(t)}}}} \right) $$

Special Functions

$$ f_{\rm odc} :{\text {O}}_{2} {\text {Hb}} = f_{\rm odc} \left( {{\text {PO}}_{2} ,{\text {P}}_{50} ,n,C_{ \max } } \right) $$

(F.1)

f_odc calculates concentration of O₂ bound to hemoglobin, O₂Hb (oxyhemoglobin), from the oxygen dissociation curve (ODC) of Hb as ${\text {O}}_{2} {\text {Hb}} = C_{\max } \cdot \left( {{\frac{{\left( {{\frac{{{\text {PO}}_{2} }}{{{\text {P}}_{50} }}}} \right)^{n} }}{{1 + \left( {{\frac{{{\text {PO}}_{2} }}{{{\text {P}}_{50} }}}} \right)^{n} }}}} \right)\left( {1 + \left( {0.25 + \left( {{\frac{{5{\text {PO}}_{2} }}{{{\text{P}}_{50} }}}} \right)^{3} } \right)^{ - 1} } \right)$ where the values specified for C_max (Maximum concentration of Hb bound to O₂ excluding the Hb bound to CO), P₅₀ (Partial pressure of oxygen at 50% of Hb saturation), and n (Hill exponent for Hb) represent the vascular compartment of interest and are all functions of %COHb(t) (See Appendix B, section “Lung (Alveolar (L) and End-Pulmonary (ep)) Compartments”). The second term in brackets applies only when n > 1.1.

$$ f_{\rm pinv} :{\text {PO}}_{2} = f_{\rm pinv} ({\text {O}}_{2} {\text {Hb}},{\text {P}}_{50} ,n,C_{ \max } ) $$

(F.2)

f_pinv calculates blood PO₂ (Partial pressure of oxygen) corresponding to a given oxyhemoglobin content as $\begin{gathered} {\text{P}}^{*} {\text {O}}_{2} = {\text {P}}_{50} \cdot \left( {{\frac{{{\text {O}}_{2} {\text {Hb}}}}{{C_{ \max } - {\text {O}}_{2} {\text {Hb}}}}}} \right)^{{{\frac{1}{n}}}} \hfill \\ \hfill \\ \end{gathered}$ where the values specified for C_max, P₅₀, and n represent the vascular compartment of interest and are all functions of %COHb(t) (See Appendix B, section “Lung (Alveolar (L) and End-Pulmonary (ep)) Compartments”). If P^*O₂ < 20 Torr, then ${\text {PO}}_{2} \, = \, {\frac{{{\text {P}}^{*} {\text {O}}_{2} (t)}}{{1 + \left( {0.25 + \left( {{\frac{{5{\text {P}}^{*} {\text {O}}_{2} }}{{{\text {P}}_{50} }}}} \right)^{3} } \right)^{ - 1} }}}$.

Otherwise, PO₂ = P^*O₂.

$$ f_{\rm imp} :{\text {PO}}_{{2({\text {P}}_{\rm bi} {\text {O}}_{2}\,{\text{or}}\,{\text {P}}_{\rm ti} {\text {O}}{}_{2})}} = f_{\rm imp} \left( {{\text {S}}_{{{\text {O}}_{2} }} ,C_{{\max (C_{{{ \max }\rm bi}}\,{\text{or}}\,{\text {O}}_{2} {\text {Mb}}_{{{ \max }i}} )}} ,{\text {P}}_{50} ,n,C_{{{\text {O}}_{2} (C_{\rm bi} {\text {O}}{}_{2}\,{\text{or}}\,C_{\rm ti} {\text {O}}{}_{2})}} } \right) $$

(F.3)

f_imp is an ACSL operator which finds an implicit solution to a nonlinear algebraic equation involving a single unknown variable (in our case, blood $\left( {{\text {P}}_{{{\text {bi}}}} {\text {O}}_{2} (t)} \right)$ or tissue $\left( {{\text {P}}_{{\text{t}_{\text{i}} }} {\text {O}}_{2} (t)} \right)$ partial pressure of oxygen). The algebraic expression solved implements the concept that the total concentration of O₂$\left( {C_{{{\text {O}}_{2} }} } \right)$ in a given compartment (blood or tissue) equals to the concentration of O₂ dissolved and the concentration of O₂ bound to heme protein (Hb or Mb) in that compartment. These expressions are:

For blood compartment “bi”at time “t”:

$$ \left( {{\text {P}}_{\rm bi} {\text {O}}_{2} (t) \cdot {\text {S}}_{{{\text {O}}_{2} }} } \right) + \left( {C_{\max {\rm bi}} (t) \cdot \left[ {{\frac{{\left( {{\frac{{{\text {P}}_{\rm bi} {\text {O}}_{2} (t)}}{{{\text {P}}_{50} (t)}}}} \right)^{n(t)} }}{{1 + \left( {{\frac{{{\text {P}}_{\rm bi} {\text {O}}_{2} (t)}}{{{\text {P}}_{50} (t)}}}} \right)^{n(t)} }}}} \right]} \right) - C_{\rm bi} {\text {O}}{}_{2}(t) \, = \, 0 $$

where ${\text {P}}_{\rm bi} {\text {O}}_{2} (t),C_{\rm bi} {\text {O}}_{2} (t)$ are the PO₂ and O₂ concentration in the blood compartment ‘i’

For tissue compartment “ti” at time “t”:

$$ \left( {{\text {P}}_{\rm ti} {\text {O}}_{2} (t) \cdot {\text {S}}_{{{\text {O}}_{2} }} \cdot V_{\rm ti} } \right) + \left( {{\text {O}}_{2} {\text {Mb}}_{\max i} (t) \cdot V_{\rm ti} \cdot \left[ {{\frac{{{\text {P}}_{\rm ti} (\text{O}_{2} (t)}}{{{\text {P}}_{50m} + {\text {P}}_{\rm ti} {\text {O}}_{2} (t)}}}} \right]} \right) - \left( {C_{\rm ti} {\text {O}}_{2}(t) \cdot V_{\rm ti} } \right) = 0 $$

where, ${\text {P}}_{\rm ti} {\text {O}}_{2} (t),C_{\rm ti} {\text {O}}_{2} (t)$ are the PO₂ and O₂ concentration in the tissue compartment ‘i’.

$$ f_{{\rm getPO_{2} }} :\left[ {{\text {PO}}_{2} ,{\text {O}}_{2} {\text {Hb}}} \right] = f_{{\rm getPO_{2} }} ({\text {O}}_{2} {\text {Hb}}_{\text{b(in)}} ,{\text {P}}_{50} ,n,{\text{O}}_{2} {\text {Hb}}_{ \max } ,{\text {S}}_{{{\text {O}}_{2} }} ,{\text {P}}_{{\text {b}}(\text{in})} {\text {O}}_{2} ,\dot{Q}_{i} ) $$

(F.4)

$f_{{\rm getPO_{2} }}$ calculates P_biO₂(t) and O₂Hb_bi(t) in the venous outflow from blood subcompartments of muscle compartment, with inflowing blood described by P_b(in)O₂ and O₂Hb_in, while accounting for tissue exchange of oxygen with blood, oxygen combined with Hb, and oxygen dissolved in plasma. Two (total O₂ content, Haldane equation) simultaneous algebraic equations of PO₂ are solved iteratively under the constraint that change in total oxygen content of the blood, multiplied by blood flow, must equal the blood-to-tissue oxygen flux (predetermined on the basis of mean partial pressure gradients). $\dot{Q}_{i}$ is the blood flow to the compartment i.

Total O₂ content: ${\text{C}_{{\rm O}_{2}}} = {\frac{{{\text{C}_{{\rm O}_{2(\max )}}} \cdot \left( {{\frac{{{\text {PO}}_{2} }}{{{\text {P}}_{50} }}}} \right)^{n} }}{{1 + \left( {{\frac{{{\text {PO}}_{2} }}{{{\text {P}}_{50} }}}} \right)^{n} }}} + {\text {S}}_{{{\text {O}}_{2} }} \cdot {\text{PO}}_{2};$

Haldane equation: ${\frac{{M_{{\text {H}}} \cdot {\text {PCO}}}}{{\text {COHb}}}} = {\frac{{{\text {PO}}_{2} }}{{{\text {O}}_{2} {\text {Hb}}}}}$

A gradient search method is used. The iterations terminate when a solution for PO₂ is reached within the limits of a specified tolerance (0.01 Torr). An error flag is set if the maximum iteration number (10,000) is reached. O₂Hb is calculated from the determined PO₂ using the f _odc function.

$$ f_{\rm getPavg} :{\text {P}}_{{\rm avgi}} = f_{\rm getPavg} \left( {{\text {O}}_{2} {\text {Hb}}_{\max \text{i}} ,V{\text {b}}_{\rm i} ,{\text {P}}_{50} ,n,C_{{\rm iav}} {\text {O}}_{2} ,{\text {S}}_{{{\text {O}}_{2} }} ,{\text {P}}_{\rm b(i - 1)} {\text {O}}_{2} ,{\text {P}}_{\rm bi} {\text {O}}_{2} } \right) $$

(F.5)

f_getPavg calculates average vascular PO₂, P_avgi(t), in the blood subcompartments of the tissue. P_avgi(t) is the partial pressure of O₂ corresponding to the concentration of O₂, C_iavO₂(t), halfway between the inlet (subscript “i − 1”) and outlet (subscript “i”) concentrations of the vascular subcompartment ‘i’. P_avgi(t) takes into account O₂ dissolved as well as O₂ bound to hemoglobin. A dichotomous search method is used to determine P_avgi(t). Vb_i is the volume of blood in compartment i.

Appendix C: Parameter Estimates Based On Regression Equations

Eqs. (C.3) to (C.4) and Eq. (C.6) are used to estimate parameter values only when their values are not provided by the experimental study being simulated.

This section contains the regression equations used in the model to estimate certain parameter values, when those values are not reported in the experimental studies.

Volume of muscle tissue, V _m (mL):

$$ V_{\text{m}} = \left[ {1000\left( {0.244\,BW + 7.80\,HT + 6.6G - 0.098\,A - 3.3} \right)} \right]/1.04 $$

(C.1)

$$ V_{{{\text{m}}\_{\text{obese}}}} = \left( {V_{\text{m}} - 8706.9} \right)/0.805 $$

BW = Body Weight in kg; HT = Height in cm; G = 0: Female; 1: Male; A = Age in years

• Volume of leg muscle tissue, V _lm (ml):

  $$ {\text {For Male,}} \, V_{\text {lm}}=0.4663(V_{\rm m}); \, {\text {For Female,}} \, V_{\text {lm}}=0.4953(V_{\rm m}) $$

  (C.1.1)

• Volume of arm muscle tissue, V _am (ml):

  $$ {\text {For Male,}} \, V_{\text {am}}=(0.1564*V_{\rm m}); \, {\text {For Female,}} \, V_{\text {am}}=(0.1396 * V_{\rm m})$$

  (C.1.2)

• Volume of trunk muscle tissue, V _tm (ml):

  $$ {\text {For Male,}} \, V_{\text {tm}}=(0.3773 * V_{\rm m}); \, {\text {For Female,}} \, V_{\text {tm}} = (0.3651 * V_{\rm m}) $$

  (C.1.3)

Total body volume of blood, V _b (mL):

$$ V_{\rm b} = 70\left( {\sqrt {{\frac{BMI\_p}{{22(HT)^{2} }}}} } \right)^{ - 1} $$

(C.2)

$$ BMI\_p = \left( {{\frac{delta\_IBW}{100}}} \right)22 + 22 $$

$$ {\text{delta}}\_IBW = \left( {{\frac{BMI}{22}}} \right)100 $$

$$ BMI = {\frac{BW}{{(HT)^{2} }}} $$

Cardiac Output, $\dot{Q}_{0}$ (L/min):

$$ \dot{Q}_{0} = (54.1 + 7.9G) \cdot BW + 1400 - 200 \cdot G $$

(C.3)

During exercise;

$$ \dot{Q}_{0} = 3. 1 8 6 + 7. 3 4 6\left( {{{MR}{\rm O}}_{ 2} } \right) - 0. 5 3 5\left( {{{MR}{\rm O}}_{ 2} } \right)^{ 2} $$

(C.4)

MRO₂ = Total Body Metabolic Rate in STPD, L/min

The changes in cardiac output $\left( {\dot{Q}} \right)$ with CO exposure were implemented in our previous model.8 Our objective in this study was to simulate changes in $\dot{Q}$ in response to physical activity. In order to account for change in cardiac output with physical activity, we developed a predictive equation for estimating cardiac output, $\dot{Q},$ as a function of total body oxygen consumption, MRO₂. Rest and exercise data for cardiac output and MRO₂ from various published papers3,4,6,18,21,34,37,72 from human subjects are shown in Fig. 14. The developed prediction equation, based on a nonlinear, least squares polynomial fit, is Eq. (C.4). Cardiac output was measured either by dye dilution, transthoracic electric bioimpedance, or Fick’s method. The data were obtained from healthy, non-smoking, untrained individual subjects. As shown in Fig. 14, cardiac output reaches a plateau as maximal body O₂ consumption is reached. Eq. (C.3) was used in simulation of Burge and Skinner 11 and Eq. (C.4) was not used in any of the simulations of the manuscript.

Figure 14

Cardiac output ($\dot{Q}$) vs. body oxygen consumption (MRO₂). The data (○) used to build the relationship3,4,6,18,21,34,37,72 and the regression line. The regression equation is given by $\dot{Q} = 3.186 \, + 7.346\left( {{{MR}{\rm O}}_{2} } \right) - 0.535\left( {{{MR}{\rm O}}_{2} } \right)^{2}.$ See Appendix C for details

Percent change in Cardiac Output with CO exposure:

$$ \% \Updelta \dot{Q} = 0.572 \, \left( { \, \% {\text{COHb}}} \right) $$

(C.5)

Heart Rate, HR (beats/min):

$$ {{HR}} = 42.819 + 68.884\left( {{{MR}{\rm O}}_{2} } \right) - 8.26\left( {{{MR}{\rm O}}_{2} } \right)^{2} $$

(C.6)

In our enhanced model, MBF and MOC for the cardiac compartment were estimated from heart rate. In order to develop a predictive equation for estimating HR, we collected data from the literature14,50,83 for HR and total body metabolic rate of human subjects during rest and exercise. The polynomial fit to experimental data (Fig. 15) is Eq. (C.6). In most of our simulations, HR was provided to us by the investigators. However, in experiments where heart rate was not measured, it was estimated from the prediction equation. An heart rate of 66 beats/min was assumed for simulations7,11 when MRO₂ was not measured.

Figure 15

Heart rate (HR) vs. body oxygen consumption (MRO₂). The data (○) used to build the relationship.14,50,83 The regression equation is given by ${{HR}} = 42.819 + 68.884\left( {{{MR}{\rm O}}_{2} } \right) - 8.26 \, \left( {{{MR}{\rm O}}_{2} } \right)^{2}.$ See Appendix C for details

Myocardial Blood Flow, MBF (mL min⁻¹g⁻¹):

$$ \text{MBF} = 2.18(HR) - 27.3 $$

(C.7)

Myocardial Oxygen Consumption, MOC ( mL min ⁻¹ g ⁻¹ ):

$$ \text{MOC} = {\frac{46.6}{{1 + \left( {{\frac{122.7}{HR}}} \right)^{4.85} }}} + 9.76 $$

(C.8)