Linking Pulmonary Oxygen Uptake, Muscle Oxygen Utilization

and Cellular Metabolism during Exercise

NICOLA LAI,1,2,3 MARCO CAMESASCA,2,4 GERALD M. SAIDEL,1,3 RANJAN K. DASH,2,3 and MARCO

E. CABRERA1,2,3,4,5

1Biomedical Engineering, Case Western Reserve University, Cleveland, OH USA; 2Department of Pediatrics, Case WesternReserve University, Cleveland, OH USA; 3Center for Modeling Integrated Metabolic Systems, Case Western Reserve University,Cleveland, OH USA; 4Rainbow Babies and Children’s Hospital, Cleveland, OH USA; and 5Pediatric Cardiology, MS-6011, Case

Western Reserve University, 11100 Euclid Avenue, RBC 389, Cleveland, OH 44106-6011, USA

(Received 2 November 2006; accepted 25 January 2007; published online 23 March 2007)

Abstract—The energy demand imposed by physical exerciseon the components of the oxygen transport and utilizationsystem requires a close link between cellular and externalrespiration in order to maintain ATP homeostasis. Invasiveand non-invasive experimental approaches have been used toelucidate mechanisms regulating the balance between oxygensupply and consumption during exercise. Such approachessuggest that the mechanism controlling the various subsys-tems coupling internal to external respiration are part of ahighly redundant and hierarchical multi-scale system. In thiswork, we present a ‘‘systems biology’’ framework thatintegrates experimental and theoretical approaches able toprovide simultaneously reliable information on the oxygentransport and utilization processes occurring at the varioussteps in the pathway of oxygen from air to mitochondria,particularly at the onset of exercise. This multi-disciplinaryframework provides insights into the relationship betweencellular oxygen consumption derived from measurements ofmuscle oxygenation during exercise and pulmonary oxygenuptake by indirect calorimetry. With a validated model,muscle oxygen dynamic responses is simulated and quanti-tatively related to cellular metabolism under a variety ofconditions.

Keywords—Cellular metabolism, Modeling, Multi-scale ap-

proach, Oxygen transport, Oxidative phosphorylation, Sys-

tems biology.

LIST OF SYMBOL

CA,tot Total concentration of ADP and ATP(mM)

CADP Concentration of ADP in tissue (mM)CATP Concentration of ATP in tissue (mM)

CCr,tot Total Concentration of PCr and Cr(mM)

CCr Concentration of Cr in tissue (mM)CPCr Concentration of PCr in tissue (mM)Crbc,Hb Concentration of Hb in the red blood

cell (mM)Cmc,Mb Concentration of Mb in myocyte (mM)CxB Bound oxygen concentration in artery,

capillary and tissue (mM)CxF Free oxygen concentration in artery,

capillary and tissue (mM)CxT Total oxygen concentration in artery,

capillary and tissue (mM)Hct Hematocrit (fraction of red blood cells

in blood) (–)kATPase ATPase rate constant (min)1)KHb Hill constant at which Hb is 50% satu-

rated by O2 (mM)n)KMb Hill constant at which Mb is 50% sat-

urated by O2 (mM)1)KADP CK constant (mM)Kb CK constant (mM)Kia CK constant (mM)Kib CK constant (mM)Kiq CK constant (mM)Km Michaelis Menten constant (mM)Kp CK constant (mM)n Hill coefficient (–)PS Permeability-surface area product

(L min)1)PSE Maximal value of permeability-surface

area product (L min)1)PSR Permeability-surface area product at

rest (L min)1)QC Muscle blood flow constant in Eq. (13)

(L min)1)Qm Muscle blood flow (L min)1)

Address correspondence to Marco

E. Cabrera, Pediatric Cardiology, MS-6011, Case Western Reserve

University, 11100 Euclid Avenue, RBC 389, Cleveland, OH 44106-

6011, USA. Electronic mail: mec6@cwru.edu

Annals of Biomedical Engineering, Vol. 35, No. 6, June 2007 (� 2007) pp. 956–969

DOI: 10.1007/s10439-007-9271-4

0090-6964/07/0600-0956/0 � 2007 Biomedical Engineering Society

956

Scap,Hb Oxygen hemoglobin saturation in bloodcapillary (–)

Stis,Mb Oxygen myoglobin saturation in muscletissue (–)

StO2m Muscle oxygen saturation (–)Tm Total amount of Hb and Mb (mmol)t Time (min)tW Time at the onset of the exercise (min)UO2m Muscle oxygen utilization (mmol min)1)VCKf Maximal forward flux of CK reaction

(mM min)1)VCKr Maximal reverse flux of CK reaction

(mM min)1)Vmax Maximal flux of oxidative Phosphory-

lation (mM min)1)Vcap, Vtis Anatomical volume of capillary and

tissue (L)VO2m, VO2p Muscle and pulmonary oxygen uptake

(mmol min)1)WR Work rate (watt)Wmc Myocyte volume fraction (–)DPS Amplitude of permeability-surface area

(L min)1)DQm Amplitude of response of muscle blood

flow rate (L min)1)

Greek letters

/ATPase ATPase metabolic flux (mM min)1)/CKf CKase forward metabolic flux

(mM min)1)/CKr CKase reverse metabolic flux

(mM min)1)/OxPhos Oxydative phosphorylation metabolic

flux (mM min)1)sPCr Time constant of the PCr kinetics (s)sPower Time constant of the mechanical power

(s)sQm

Time constant of the muscle blood flow(s)

sUO2mTime constant of the muscle oxygenutilization (s)

sVO2mMean response time of the muscle oxy-gen uptake (s)

sVO2pMean response time of the pulmonaryoxygen uptake (s)

Superscript

B Bound oxygen concentrationF Free oxygen concentrationH Heavy conditionj Exercise intensityM Moderate conditionR Resting condition

T Total oxygen concentrationV Very heavy conditionW Warm-up condition

INTRODUCTION

The energy demand imposed by physical exercise onthe components of the oxygen transport and utilizationsystem requires a close link between cellular andexternal respiration in order to maintain adenosinetriphosphate (ATP) homeostasis. At the onset ofexercise, an immediate increase in the rate of produc-tion of (ATP) in active skeletal muscle fibers is requiredto meet the increased rate of ATP utilization (meta-bolic demand). Mechanical power output of muscle –measured on a cycle ergometer – responds quickly(sPower < 2 s to a step change in work rate, indicatingan even faster response of the ATP production rate bycontracting muscles. In contrast, the correspondingdynamic responses of pulmonary oxygen uptake(VO2p) and muscle oxygen uptake (VO2m)

29,38,47,50,55

are much slower ðsVO2� 30 sÞ. This slower response

reflects the ATP production rate from oxidation ofreducing equivalents in active muscle. While the steadystates for ATP demand and ATP supply through oxi-dative metabolism match perfectly after �2–3 min atsubmaximal work rates, their transient responses differby at least an order of magnitude.

Noninvasive measurements of VO2p dynamics cancharacterize differences in onset kinetics of healthy andchronically ill individuals.7,8,49 These measurements,however, not only reflect cellular metabolism, but alsothe effect of components of the delivery system thattransports oxygen from the external environment tothe mitochondria (e.g., pulmonary ventilation, diffu-sion from alveoli to blood, transport in blood – dis-solved and bound to hemoglobin – to tissues, diffusionfrom blood to cells). Because measurements of VO2p

dynamics do not accurately indicate the cellular O2

availability and utilization rate in contracting muscle,they are insufficient for deducing mechanisms of met-abolic control at the cellular level.

To investigate factors controlling the rate of pul-monary oxygen uptake at exercise onset, oxygen up-take has been measured directly across the femoral bedduring cycling in humans29 and across isolated wholeskeletal muscle preparations in dogs.24–27,29 Thesewhole-muscle studies have provided valuable insightsabout the effect of altering convective oxygen deliveryand peripheral oxygen diffusion on the dynamics ofmuscle oxygen uptake during moderate and heavyintensity exercise. Nevertheless, they also have someinherent limitations. When a whole muscle is stimu-lated in lieu of voluntary muscle contraction, the fiber-

Model of Respiration during Exercise 957

type recruitment pattern is different. Furthermore,controlling bulk oxygen delivery does not necessarilycontrol the matching of perfusion to metabolism.Measurements of the oxygen content in the arterialand venous blood do not reflect the actual amount ofoxygen available at the mitochondrial level. Localoxygen availability (i.e., cellular PO2) in intact muscletissue is difficult to assess during exercise in thesewhole-tissue preparations.

Cellular oxygen availability can be assessed inmuscleof contrasting fiber types with phosphorescencequenching9 or in intact workingmuscle with either near-infrared spectroscopy28 or 1H magnetic resonancespectroscopy.36,60 However, these methods lack speci-ficity when measuring the dynamics of cellular oxygenconsumption in a large muscle group during exercise.Nevertheless, intracellular oxygen consumption can beestimated using 31Pmagnetic resonance spectroscopy byusing PCr dynamics as a proxy (within 10%) to muscleoxygen consumption dynamics.3,53,54,65 Amore suitableapproach to investigate mechanisms controlling respi-ration is to conduct simultaneous measurements ofputative intramuscular control mediators (PCr, Cr, Pi,ATP, ADP) dynamics and VO2p dynamics while per-forming exercise involving a large muscle group (e.g.,quadriceps). This approach provides a large change inVO2p at exercise onset and reliable VO2p dynamic re-sponses which can be compared to those of PCr.53,65

Even if measurements at each scale (i.e., organelle,cellular, tissue/organ, and whole organism level) wereexperimentally feasible with the appropriate time res-olution, these must be integrated into a coordinatedsystem for data interpretation and for elucidation ofmechanisms of metabolic control and regulation. Forquantitative understanding, integration of transportand metabolic processes at the cellular, tissue/organ,and whole-body levels requires a formal theoreticalframework. As a complement to experimental studiesof oxygen uptake/consumption at various exerciseintensities and biological levels, we used a mathemat-ical model to integrate data and to simulate responsesat the cellular and tissue level. The models integrated inthis study have been validated previously.42,62

The ‘‘systems biology’’ approach employed here inlinking external to internal respiration uses bothexperimental data from non-invasive methods andcomputational models to understand complex phe-nomena and biological systems underlying physiolog-ical function and ATP homeostasis. For initial modeldevelopment to analyze exercise responses, we useddata at the whole body and tissue level. In addition, wemeasured pulmonary oxygen uptake (indirect calo-rimetry) and muscle oxygenation (near-infrared spec-troscopy) dynamics during exercise in healthyvolunteers.42 Based on these data, we developed a

computational model of oxygen transport and metab-olism in skeletal muscle that can simulate and predictmuscle oxygen consumption dynamics during exer-cise.42 In this study, we integrated our model with amodel of energy balance in myocytes41,62 to provide amore mechanistic expression for oxidative phosphor-ylation,35 which is linked to the ATPase and creatinekinase (CK) reactions.62 With this model, muscleoxygen consumption dynamics at the cellular level canbe estimated during exercise. Finally, model simula-tions are used to provide insight with respect to con-centration and flux rate dynamics of key metabolites(PCr, ATP, ADP, O2) participating in oxidativephosphorylation and ATP homeostasis.

METHODS

Model Development

Oxygen utilization rate in skeletal muscle (UO2m) islinked to oxygen uptake rate in lungs (VO2p) bytransport processes within the tissues and via the car-diovascular system (Fig. 1). To analyze the dynamicrelationship between UO2m and VO2p, a mathematicalmodel is needed. In an earlier approach to developingthis relationship, simulations were based on a model42

that approximated oxygen utilization in working skel-etal muscle as a mono-exponential function of workrate.10 This model was used to analyze dynamicresponses to exercise at several levels of intensity.42

With this simple model, however, distinct metabolicprocesses involved during exercise stimulus cannot beevaluated. Therefore, a more mechanistic model isneeded that describes oxidative phosphorylation byincorporating ATPase and CK reactions. Such a

ADP

Cr PCr

ATP Pi+

CKase-

ATPase

+ PiATP ADP

O2

CKase+

OxidativePhosphorylation

H2O

mQTcapC T

artC

2mUO

PS

Muscle O2 utilization

Pulmonary O2 uptake 2VO

FIGURE 1. Oxygen utilization and transport between lungsand skeletal muscle and cellular metabolism during exercise.

LAI et al.958

metabolic model has been developed previously.62

Here, this model of oxidative phosphorylation inmyocytes is combined with a model of oxygen transportand utilization42 for analysis of responses to exercise.

Mass Transport Balances

Oxygen concentration dynamics in skeletal muscleare represented by compartmental mass balances.These balance equations involve total oxygen (Ccap

T ,CtisT ) to free oxygen (Ccap

F , CtisF ) in the capillaries and

tissues:

Vcap

dCTcap

dt¼ QmðtÞ CT

art � CTcap

� �� PSðtÞ CF

cap � CFtis

� �

ð1Þ

VtisdCT

tis

dt¼ PSðtÞ CF

cap � CFtis

� ��UO2mðtÞ ð2Þ

Based on relationships between total, bound, and freeoxygen concentrations, these equations have anequivalent representation with derivatives of free oxy-gen (Appendix I). The muscle blood flow, Qm(t), thecapillary-tissue transport coefficient represented as thepermeability – surface area product, PS(t), and theoxygen utilization, UO2m(t), depend on exerciseintensity. The rate of muscle oxygen uptake is definedas VO2mðtÞ ¼ QmðtÞ CT

art � CTcap

� �(All variables and

symbols are defined in the ‘‘List of Symbols’’).

Metabolic Reaction Balances

The metabolic reaction processes of oxidativephosphorylation during an exercise stimulus have beenreported in Fig. 1 and can be expressed as the con-centration dynamics of ATP and PCr:

dCATP

dt¼ �/ATPase þ b/OxPhos � /r

CK þ /fCK ð3Þ

dCPCr

dt¼ /r

CK � /fCK ð4Þ

where b is the P:O2 ratio in vivo and the reaction fluxes/j are functions of the ATP and PCr concentrations,which must satisfy the mass balances for conservationof adenosine and creatine:

CA;tot ¼ CADP þ CATP ð5Þ

CCr;tot ¼ CCr þ CPCr ð6Þ

The metabolic flux of oxygen is nonlinear related to theADP and oxygen concentrations:

/OxPhos ¼ Vjmax

CADP

KADP þ CADP

� �CF

tis

Km þ CFtis

� �ð7Þ

where the rate coefficient, Vmaxj , depends on exercise

intensity. The metabolic flux for the ATPase reaction isproportional to the ATP concentration:

/ATPase ¼ kjATPaseCATP ð8Þ

where the reaction rate coefficient, kATPasej , depends on

exercise intensity. The forward and reverse reactionfluxes of CK are nonlinearly related to the coupledconcentrations of Cr, PCr, ADP, and ATP:62

/rCK ¼

VrCKðCCrCATP

KiqKpÞ

1þ CADP

Kiaþ CATP

Kiqþ CPCr

Kibþ CADPCPCr

KbKiaþ CCrCATP

KiqKp

ð9Þ

/fCK ¼

VfCK

CADPCPCr

KbKia

1þ CADP

Kiaþ CATP

Kiqþ CPCr

Kibþ CADPCPCr

KbKiaþ CCrCATP

KiqKp

ð10Þ

The contributions of glycogenolysis and glycolysisto ATP synthesis are not included in the mathemati-cal model Eq. (3) at this stage of model develop-ment. Thus, results obtained from simulationswith this incomplete model need to be interpretedaccordingly.

The reaction processes, Eqs. (3) and (4), are relatedto the transport processes, Eqs. (1) and (2), throughthe rates of oxygen metabolism, /OxPhos, and oxygenutilization:

/OxPhosVtis ¼ UOj2m ð11Þ

which involves a tissue volume, Vtis, and depends onthe exercise intensity j. The total muscle volume isdefined as Vm ¼ Vcap þ Vtis where Vm is computed asa fraction of the whole body mass (49%) and Vcap, Vtis

are each computed as a fraction of the Vm.42

Exercise-dependent Functions

In response to a step increase in work rate from asteady-state warm up condition, the dynamic responseof blood flow Qm

j at exercise intensity j is assumed to beexponential:42

QjmðtÞ ¼ QW

m þ DQjm 1� exp tW � t

� �=sjQm

h ið12Þ

where QmW is the steady-state value during warm-up,

DQmj is the increase in blood flow, and sjQm

is the timeconstants of muscle blood flow, and tW is the initialtime. In response to exercise, blood flow increases,which increases the rate of capillary-tissue transport:11

PSjðtÞ ¼ PSR þ DPS 1� exp QRm �Qj

m

� �=QC

� �ð13Þ

where DPS ¼ PSE � PSR, PSR and QC are constants.For a description of their meaning refer to the List ofSymbol.

Model of Respiration during Exercise 959

Model Simulation

For comparison of simulated and experimental re-sponses of oxygen obtained from human exerciseexperiments at several intensities, muscle oxygen sat-uration is computed:42

StO2m ¼ CBcapVcap þ CB

tisVtis

h i=Tm ð14Þ

The parameters Tm (total amount of Hb and Mb), Vcap

and Vtis are constants independent of exercise inten-sity. The initial conditions for the simulations of thedynamic responses are specified at a warm-up steadystate (Appendix II):

t ¼ tW; CFcap ¼ CF;W

cap ; CFtis ¼ CF;W

tis ;

CATP ¼ CWATP; CPCr ¼ CW

PCr

ð15Þ

To simulate responses to exercise, the model equationswere solved numerically using a robust algorithm forstiff ordinary differential equations (DLSODE, http://www.netlib.org/odepack/).31

Parameter Estimation

Values of most parameters in the model equationsabove have been determined previously and reportedin Tables 1A, B and 2.42,62 Others are evaluated di-rectly from steady-state relationships (kATPase

j , PSR)(Table 3) or by optimal estimation (DQm

j , Vmaxj ) (Ta-

ble 4).From experiments, the values of Vtis and UO2m

j aredetermined so that at each exercise intensity we canevaluate /OxPhos from Eq. (11). Under steady-stateconditions and assuming b = 6 (32), Eqs. (3) and (4)simplify to

0 ¼ �/ATPase þ 6/OxPhos ð16Þ

Combining this with Eq. (8) and given a constant valueof CATP, due to the ATP homeostasis, we can evaluatekATPasej (see Table 3). To calculate PSR, we consider

Eqs. (1) and (2) at steady state under resting condi-tions:

0 ¼ QRm CT

art � CTcap

� �� PSR CF

cap � CFtis

� �

0 ¼ PSR CFcap � CF

tis

� ��UOR

2m ð18Þ

From experiments, CartT , Ctis

F,R, Qmj and UO2m

R aredetermined. By simultaneous solution of Eqs. (17) and(18), incorporating the relationship CT;R

cap ðCF;Rcap Þ, we

estimate PSR and CcapF,R.

The parameters DQmj and Vmax

j must be estimatedfor each subject at each exercise intensity. For thispurpose, we find the parameter values that yield thebest fit of the model output dynamics (from numericalsolution of the model equations) to the experimentaldata. Specifically, for each subject at each exerciseintensity, we minimize a least-squares objective func-tion:

CðDQjm;V

jmaxÞ ¼

1

2

XNi¼1

StO2mðtiÞ � StOW2m

StOW2m

!

exp

24

� StO2mðtiÞ � StOW2m

StOW2m

!

mod

#2

ð19Þ

where N is the number of data points. The objectivefunction is minimized by numerical optimization usingadaptive, non-linear algorithm (DN2FB, http://www.netlib.org).19

RESULTS

In response to different exercise intensities, simula-tions represent dynamic responses of muscle ATP andPCr concentrations, oxygen saturation, StO2m, AT-Pase (/ATPase), oxidative phosphorylation (/OxPhos),and net CK ð/r

CK � /fCKÞ flux rates. By fitting of

simulated StO2m to experimental data, optimal esti-mates were obtained for the maximal flux rate of oxi-dative phosphorylation Vmax

j and change in muscle

TABLE 1. (A) Values of muscle oxygen utilization, UO2mj , and pulmonary oxygen uptake, VO2p

j , (mmol min)1) at various exerciseintensities (j = R,W,M,H,V)42 and (B) Values of blood flow Qm

j (L min)1) at various exercise intensities (j = R,W), mean responsetimes sQ_m

j (s) (j = M,H,V) and compartment volumes V(L) for model simulations.42

Panel A

Muscle oxygen utilization UO2mj

Parameter UO2mR UO2m

W UO2mM UO2m

H UO2mV

Mean ± SD 2 ± 0.2 15 ± 4 58 ± 18 83 ± 18 109 ± 13

Pulmonary oxygen uptake VO2pj

Parameter VO2pR VO2p

W VO2pM VO2p

H VO2pV

Mean ± SD 9.8 ± 0.9 23.6 ± 3.5 70.7 ± 17 96 ± 15 122 ± 11.6

Panel B

Parameter QmR Qm

W sMQm

sHQm

sVQm

Vcap Vtis

Mean ± SD 0.8 ± 0.1 3.4 ± 0.4 21.3 ± 2 24 ± 1 24 ± 1 2.3 ± 0.3 31 ± 5

LAI et al.960

blood flow DQmj of seven subjects at exercise intensities

j = M,H,V. From model simulations, mean responsetimes ðsVO2m

; sUO2m; sPCrÞ were computed for muscle O2

uptake, muscle O2 utilization, and PCr, respectively.Typical StO2m step responses to a change in exercise

intensity relative to a warm-up steady state, StO2mW , for

one subject are presented in Fig. 2. The decrease inStO2m is greater for a higher exercise intensity. Refer-ring to the same subject, model simulations of UO2m

j

responses to increased exercise intensities (j = M,H,V)are compared to measured VOj

2p (Fig. 3). Dynamicresponses of UOj

2m are faster than the dynamic re-sponses of VOj

2m for all subjects. The correspondingmean response times are compared in Table 5.

From optimal estimation of Vmaxj from seven sub-

jects, their mean values are independent of exerciseintensity (Table 4). In contrast, the mean change inmuscle blood flow DQm

j increased by about 80% withhigher exercise intensity.

Model simulated metabolic flux rates /OxPhos and/rCK � /f

CK in response to a step change in /ATPase

(Table 3), corresponding to a given exercise intensity,are presented in Fig. 4. As expected (Eq. 3), at allexercise intensities, 6/OxPhos reaches the same steady as

/ ATPase. Accordingly to the direct proportionality of/OxPhos and UOj

2m (Eq. 11), the mean response time of/OxPhos given by sUO2m

(Table 5) increases by about

TABLE 2. Model parameters values for simulation of all experiments.42,62

Transport model Metabolic model

Notation Unit Value Notation Unit Value

KHb (mM))2.7 7800.7 CCr,tot (mM) 42

KMb (mM))1 308.6 CATPR (mM) 8.2

CartF (mM) 0.135 VCK

r (mM min)1) 3008.6

CtisF,R (mM) 3.375 � 10)2 30 VCK

f (mM min)1) 6000

Crbc,Hb (mM) 5.18 Kb (mM) 1.11

Cmc,Mb (mM) 0.5 Kp (mM) 3.8

Hct (–) 0.45 Kia (mM) 0.135

QC (L min)1) 8a Kib (mM) 3.9

Vcap (L) 7% Vm Kiq (mM) 3.5

Vtis (L) 93% Vm KADP (mM) 0.058

Wmc (–) 0.75 Km (mM) 7 � 10)4 17

PSE (L min)1) 20,000

a This work.

TABLE 3. Means (n = 7) of reaction rate coefficients for ATPase kATPasej (min)1) at different exercise intensities (j = R,W,M,H,V)

and of permeability-surface area at rest (L min)1).

Parameter kATPaseR kATPase

W kATPaseM kATPase

H kATPaseV PSR

Mean ± SD 0.048 ± 0.008 0.36 ± 0.04 1.4 ± 0.3 2 ± 0.3 2.6 ± 0.4 113 ± 26

TABLE 4. Mean (n = 7) parameter values for exercise intensity j = M,H,V: change in muscle blood flow DQmj (L min)1) from warm-

up; maximal rate of oxidative phosphorylation Vmaxj (mM min)1).

Parameter DQmM DQm

H DQmV Vmax

M VmaxH Vmax

V

Mean ± SD 5.6 ± 2.2 7.6 ± 2 9.7 ± 2 45 ± 15 45 ± 15 44 ± 16

0 1 2 3 4 5 6 7 8 9 10-50

-40

-30

-20

-10

0

10

20

30tW

(StO

J 2m-S

tOW 2m

)/S

tOW 2m

[%]

Time [min]

Simulations moderate heavy very heavy

Experimental data moderate heavy very heavy

FIGURE 2. Relative oxygen saturation in muscle, StO2mj : for

representative subject responses to step changes from awarm-up steady-state condition (W) to a steady state duringmoderate, heavy, and very heavy exercise (j = M,H,V). Modeloutput compared with experimental data.42

Model of Respiration during Exercise 961

35% from moderate to very high intensity exercise.While /OxPhos increases monotonically to the steadystate, /r

CK � /fCK shows an initial step decrease and

then increases in exponential manner towards zero.The dynamic responses of the /r

CK � /fCK and /OxPhos

fluxes are closely balanced to guarantee ATP homeo-stasis.

The CADP at steady-state, during exercise at allintensities, reached in our simulations (i.e., Figs. 2–5),

consistently increases from moderate to very heavyintensity of exercise (M, 2.32 Æ 10)3 mM; H,5.26 Æ 10)3 mM; V, 1.45 Æ 10)2 mM).

The ATP concentration change is small compare tothe PCr concentration change which decreases in re-sponse to a change in exercise intensity (Fig. 5). Themean response time of PCr, sPCr, shows a change withexercise that is similar to the change of sUO2m

. Modelsimulations were also performed assuming that theMichaelis-Menten parameter Km is sufficiently small in

(b)109876543210

-30

0

30

60

90

120

150

Time [min]

(U

OJ 2m

2m-U

OW

’) (V

OJ 2p-V

OW 2p )

[mm

olm

in-1

]

UO2m

-UOWJ

J

2m’ Simulations

moderate heavy very heavy

VO -VOW

2p’2p Experimental Data

moderate heavy very heavy

tW

0

30

60

90

120

150U

OJ 2m

’ VO

J 2p

[mm

ol m

in-1

]UO

J

2m

Simulations moderate heavy

very heavy

VOJ

2p

Experimental Data

moderate heavy very heavy

tW (a)

FIGURE 3. Comparison between dynamic responses ofexperimental pulmonary oxygen uptake and simulated muscleoxygen consumption for a representative subject.42 Forcingfunctions are step changes from a warm-up steady-statecondition (W) to a work rate of moderate, heavy, and veryheavy intensity exercise (j = M,H,V). Muscle oxygen con-sumption and pulmonary oxygen uptake are represented (a) inabsolute terms as VO2p

j and UO2mj , respectively and (b) as

absolute changes from warm-up steady-state values, i.e.,VOj

2p � VOW2p and UOj

2m � UOW2m, respectively.

TABLE 5. Mean (N = 7) response times for exercise intensityj = M,H,V of muscle oxygen uptake s

jVO2m

(s), muscle oxygenutilization s

jUO2m

(s) and PCr dynamics sPCrj (s).

Mean ± SD Mean ± SD Mean ± SD

sMVO2m

18 ± 3 sHVO2m

18 ± 2 sVVO2m

20 ± 3

sMUO2m

14 ± 3 sHUO2m

15 ± 4 sVUO2m

18 ± 4

sPCrM 14 ± 3 sPCr

H 15 ± 4 sPCrV 18 ± 4

0 1 2 3 4 5 6 7 8 9-30

-20

-10

0

10

20

30

φjATPase

6 φj

OxPhos

φr,jCK−φf,j

CK

very heavy

heavy

moderate

nim

Mm[ xul

F cilobateM

-1]

Time [min]

moderate

heavy

very heavy

tW

FIGURE 4. Simulation results for representative subjectfor the fluxes of ATPase, /ATPase

j , oxygen phosphorylation,/OxPhos

j and creatine kinase, /r;jCK � /

f;jCK. The simulations rep-

resent the dynamic between the warm-up steady state and thethree levels of exercise intensity: moderate, heavy, and veryheavy (j = M,H,V).

0 1 2 3 4 5 6 7 8 9 100

10

20

30

40

50

60

PCr

ATP

moderate heavy very heavyM

etab

olite

[mM

]

Time [min]

tW

FIGURE 5. Simulation results for representative subject.Comparison of absolute variation between the PCr break-down and the ATP consumption. The simulations are carriedout starting from the warm-up steady state to the three levelsof exercise intensity: moderate (continuous), heavy (dashed)and very heavy (dotted).

LAI et al.962

Eq. (7) such that /OxPhos is independent of oxygenconcentration. In response to exercise, the mean re-sponse times for UOj

2m and CPCrðsUO2m; sPCrÞ are

nearly the same for moderate, heavy and very heavyexercise (13, 13, 15 s). These response times are about2–3 s less than those for which /OxPhos depends onoxygen concentration (Table 5). Also, for /OxPhos withno oxygen dependence, the mean values of Vmax

j (44(M), 43 (H), 37 (V) mM min)1) compared to thosegiven in Table 4 differ only for the very heavy intensity(j = V). Although Vmax

V decreased from 44 to 37 mMmin)1 without oxygen dependence, the mean responsetime of muscle oxygen uptake sVO2m

did not change.A reduction of the muscle volume, Vm, from 40 to

9 l leads to greater amplitude changes of /ATPase, net/rCK � /f

CK fluxes, CPCr and about a 50% increase insPCr and sUO2m

. Also, the estimated Vmaxj in this range

lies between 45 and 25 mM min)1 and was indepen-dent of exercise intensity.

DISCUSSION

A mathematical model based on mass balances andmetabolic and transport fluxes was successfully devel-oped and applied to study the interacting regulation ofcellular respiration and bioenergetics required tomaintain ATP homeostasis during step transitionsfrom rest to work rate intensities representing moder-ate, heavy, and very heavy exercise on a cycle ergom-eter in humans. This model couples oxygen transportand utilization in skeletal muscle42 to a model of cel-lular metabolism and energetics.62 The latter model,which describes oxidative phosphorylation, ATPaseand CK reactions in finger and wrist flexors during animposed stimulus, was adapted to investigate cellularenergetics in large muscle groups.

In this study, transport and metabolic processes atthe onset of exercise were quantified in the steady stateand during the exercise transition by comparing modelsimulations with measurements of pulmonary oxygenuptake and muscle oxygen saturation at differentexercise intensities on a cycle ergometer. Then, thedynamics of the metabolic flux rates of /OxPhos, /AT-

Pase, /CKf and /CK

r were simulated for each of the sevensubjects at three levels of exercise intensity.

Internal and External Respiration

The regulation of cellular oxygen consumption inskeletal muscle during exercise in vivo depends onelectron flow, proton pumping, metabolic fluxes ofNADH, ADP-dependent feedback control and oxygendelivery.16 At the cellular level, the dynamics of oxi-dative phosphorylation to a step increase in energy

demand is expected to be fast (<450 ms) based onstudies with isolated mitochondria in vitro.12,58 Thisresponse time is at least two orders of magnitude fasterthan in vivo measurements64,65 of oxygen uptakedynamics at the mouth (�45 s). Since direct in vivomeasurements of muscle oxygen consumptiondynamics (UO2m) are difficult to obtain in humansduring exercise, an indirect estimate is obtained frompulmonary oxygen uptake (VO2p) based on measur-able variables at the airway opening.66 Under non-steady state conditions, however, VO2p cannot providea reliable estimate of muscle oxygen consumption be-cause of differences in dynamics at exercise onset.37

Experimentally, Grassi et al.29 found no significantdifferences in response dynamics between muscle andpulmonary uptake, VO2m and VO2p, during the tran-sition from light to moderate intensity exercise. Themethodology used for measuring VO2m dynamicsacross the femoral bed during exercise may not besufficiently accurate because the VO2m response de-pends on lumped measurements of oxygen content inthe femoral vein and bulk measurements of femoralblood flow,61 which may result in having slow muscleoxygen uptake dynamics close to that of pulmonaryoxygen uptake. Several studies have considered whe-ther the observed dynamics of oxygen uptake at theonset of exercise is the manifestation of an ‘‘inertia’’ inthe rate of O2 delivery to the muscle fibers18 or of anintrinsic slowness of the intracellular oxidativemetabolism.23,28

Linking Simulated and Measured Fluxes

The model developed here takes into account thedynamic interplay of oxygen delivery (via convectionand diffusion),42 and mechanisms of cellular energymetabolism occurring in skeletal muscle during exer-cise.62 Specifically, muscle oxygen saturation (Fig. 2)and pulmonary oxygen uptake (Fig. 3) measurementswere linked using a mathematical model to quantifythe metabolic fluxes involved in cellular respiration inskeletal muscle during rhythmic voluntary contrac-tions on a cycle ergometer.

According to the model of cellular energy metabo-lism proposed by Vicini and Kushmerick,62 /OxPhos iscoupled to the metabolic fluxes of /r

CK and /fCK

(Fig. 4). This model describes how the dynamics of netPCr breakdown ð/r

CK � /fCKÞ results in a drop of CPCr

to supply the amount of ATP (Fig. 5) that /OxPhos

cannot synthesize rapidly enough (Fig. 4) to maintainATP homeostasis at exercise onset (Fig. 5). Similarly,to Vicini and Kushmerick,62 we described cellularenergy balance in skeletal muscle without including theglycolytic pathway as a source of ATP synthesis. Undercertain conditions, such as maximal or supramaximal

Model of Respiration during Exercise 963

exercise, as well as at the onset of high intensity exer-cise, the glycolytic contribution to ATP synthesis maybe significant. The results of our simulation at the onsetof heavy and very heavy intensity exercise would mostlikely be different, were the glycolytic contribution betaken into account. Specifically, we speculate that therate of phosphocreatine breakdown would be reducedin proportion to the net rate of lactate formation fromglycolysis throughout the exercise period.

The simulated dynamic response of muscle oxygenutilization (indicated by /OxPhos) was always signifi-cantly faster than the response of pulmonary oxygenuptake at all intensities (Fig. 3).

The dynamic response sUO2mof simulated UO2m or

/OxPhos was always faster than sjVO2pof measured

VO2p at all exercise intensities ðsjVO2p, M: 65 ± 7 s;

100 ± 24 s; 82 ± 31 s).42 Furthermore, sUO2mbased

on model simulation in which /OxPhos depends onoxygen concentration Eq. (7) was 2–3 s larger thansUO2m

obtained using an empirically derived exponen-tial function for UO2m.

42 This difference vanishedwhen /OxPhos did not include oxygen dependence. Thedependence of /OxPhos on oxygen concentration maybecome evident at low oxygen concentration values,especially at high-intensity exercise.

It should be noted that without the oxygen depen-dence, the mean Vmax

V estimated is reduced from 44 to37 mmol l)1 min)1. Therefore, the original expressionfor /OxPhos proposed by Vicini and Kushmerick62 wasmodified in the present model by including an oxygendependency, which is important at higher exerciseintensities where oxygen could be very low and affectmetabolism.67–69 During exercise, oxygen dependencyaffects the dynamics of muscle oxygen utilization, butdoes not affect muscle oxygen uptake sjVO2m

because thevenous oxygen concentration dynamics is not greatlyaffected. It is worth noticing that, in our model, weassume the rate of ATP production associated withoxidative phosphorylation to be six times the rate ofoxygen consumption (i.e., b = 6). Although the P/O2

ratio may be 5 or 6, as proposed in some authors,45,56 arecent review suggests values between 3 and 6.32 Nev-ertheless, when different values of b were used in thesimulations, the estimated dynamics of oxidativephosphorylation remain the same at all exercise inten-sities, even though it affected the value of kATPase

j andthe estimation of Vmax

j .The kinetic parameters of cellular metabolism

Eq. (9–10) were previously estimated for nerve stimu-lation of finger and wrist flexors.62 The same parametervalues were used in this study even though musclesengaged in bicycle exercise were voluntarily activated.Nevertheless, model simulations of the dynamics ofoxidative phosphorylation and PCr, characterized bythe mean response times sUO2m

and sPCr, are in

agreement with previously published studies performedin human subjects under similar conditions.53

The model equations account for cellular ATPase,oxidative phosphorylation and CK fluxes variations inskeletal muscle during exercise, where the cellular res-piration is regulated by feedback control with depen-dence from ADP and oxygen concentrations.13–15,67–69

It is worth to note that, in the literature, several feed-back and feed forward control models have beenproposed to describe the regulation of cellular respi-ration in vivo, such as (1) feedback control using aMichaelis-Menten relationship between oxidativephosphorylation and [ADP], (2) higher-order feedbackcontrol using an expression in which the Hill coeffi-cient is greater than 1, or (3) a more fundamentalexpression of relating oxidative phosphorylation to thefree energy of ATP hydrolysis.34,41,63 Furthermore,some other scientists have proposed feed forwardmechanisms to control oxidative phosphorylation.2,40

However, the experimental data available in the pres-ent study are not sufficient to address this issue;therefore, we adopted the approach successfully ap-plied previously to in vivo data obtained by NMRspectroscopy.62

The computation of CADP at steady-state duringconstant work rate exercise (M: 2.74 Æ 10)3 mM; H:5.94 Æ 10)3 mM; V: 1.54 Æ 10)2 mM), based on theequilibrium assumption using the CK equilibriumconstant Keq = 177 (pH = 7, T = 38 �C)57 andCATP, CPCr, CCr obtained from simulations using fullkinetic expressions for CK (Eqs. 9–10), is consistentwith the CADP values simulated (M: 2.32 Æ 10)3 mM;H: 5.26 Æ 10)3 mM; V: 1.45 Æ 10)2 mM). Moreover,our approach allows for dynamic information on themetabolic fluxes, which is useful to investigate themechanisms regulating the dynamics of oxidativephosphorylation during exercise.

Model Transport Parameter

In this model, mass transport of oxygen betweencapillary blood and tissue depends on permeability-surface area coefficient PS, which varies with muscleblood flow (Eq. 13) that in turn depends on exerciseintensity.11 The dependency of PS on Qm

j allows themodel to simulate resting and exercise conditions. Al-though the expression used to describe the variation ofPS with work rate is phenomenological, it is consistentwith tissue oxygen concentration values at rest and inagreement with the hypothesis of a capillary recruit-ment during exercise.33 Regardless whether there iscapillary recruitment during exercise or just an increasein blood flow rate through the already recruited capil-laries, in our simulations, PS coefficient must be set to asufficiently high value to ensure enough oxygen supply

LAI et al.964

to contracting muscle fibers to match the energy de-mand at all intensities.42 The parameter QC that affectsthe PS)Qm

j relationship (Eq. 13), must be sufficientlysmall so that simulations of arterio-venous oxygen dif-ferences are consistent with the blood flow increase withexercise intensity. Also, from Eq. (12), the blood flowincreases between steady states, DQm

j were in agreementwith those found previously at all intensities.42

Limitations of Experimental Data

Model simulations indicate that a reduction of theamount of active muscle, Vm, from 40 to 9 l and thus areduction of capillary and tissue volume (Vcap, Vtis)engaged during exercise, affects the temporal profile ofthe metabolites, the maximum rate of oxidative phos-phorylation Vmax

j and the mean response time of cel-lular oxygen utilization sjUO2m

. In particular, the smallerthe Vm values are the faster the dynamics of oxygenconcentration in capillary blood and in tissue appears,for a fixed value of Vmax

j . The Vmaxj value estimated by

Vicini and Kushmerick62 (30 mM min)1) falls in therange of values determined in our simulations (45–25 mM min)1) when varying Vm between 40 and 9 l.The estimation of the Vmax

j parameter is limited by theexperimental uncertainty in determining active musclevolume and the microvascular volume distributionduring exercise. In the present study, the volume ofmuscle recruited by each subject during exercise wasassumed to be a percentage of the whole body volume;thus, a more accurate determination of the functionalrelationship between active muscle volume engaged andwork rate is needed. Experimental measurements of keymetabolites, such as PCr in large muscle groups, couldbe coupled to the model developed to estimate the ac-tive muscle volume unknown. Future model develop-ment will take into consideration motor unitrecruitment pattern accounting for an increase ofmuscle volume during exercise.

Another limitation related to the unknown musclevolume engaged during exercise is the lack of accuracyand specificity of NIRS measurements in contractingskeletal muscle. Specifically, the limitation of usingNIRS technique to evaluate StO2m in skeletal muscle isits lack of discriminatory power to distinguish amongthe relative contributions of skin, adipose tissue,43

capillaries, and small arterioles and venules to theNIRS signal obtained from an uncertain volume of theregion investigated. Therefore, the absolute values maybe misleading with respect to the rate of adjustment ofoxidative metabolism. Changes in capillary bloodvolume can affect the concentrations of oxygenatedand deoxygenated hemoglobin measured by NIRS andcause variability in the evaluation of oxygen con-sumption.10,18 Another potential source of interference

is the relative contribution of Hb and Mb44,60 to theNIRS signal. Most studies focus on Hb changes, sinceit has been reported that intracellular Mb accounts forless than 10% of the total NIRS signal.28 Currently,available NIRS instrumentation cannot accuratelydetermine the relative contribution of myoglobin (Mb)to the total NIRS signal.

Future Directions

The complexity of the model required in futuredevelopments depends on the experimental informa-tion available and hypotheses to be tested. The validityof model simulations depends on the accuracy andprecision associated with estimating unknown param-eters. In modeling the linkage between whole body,tissue, and cellular processes with mass balances, theanalysis of dynamic responses depends on theestimated volumes of the tissues.

In practice, if the observation scale is reduced to themicrovascular level, the volume of perfused tissue un-der consideration is uncertain. Furthermore, the spa-tial distribution and temporal variation of blood flowand oxygen concentration can have a significant effecton the interpretation of the measurements.48 Withappropriate experimental data, more general detailedmodels,5,6,17,22 can take into account (a) spatial dis-tribution of transport and reaction processes in thecapillaries and extra-vascular tissue, (b) intracellularcompartmentation,46 and (c) structural and metaboliccharacteristics of the tissue (e.g., muscle fiber types).Specifically, for human studies of skeletal musclemetabolism in response to exercise, improvements inthe analysis of pulmonary and muscle oxygendynamics could be made by experiments that simul-taneously combine invasive and non-invasive tech-niques.18,28,29,39,50,59 The variables to be measured arevenous oxygen content, muscle blood flow, and muscleoxygenation during exercise. Measurements of muscleoxygen utilization should be as close as possible to thesite of cellular respiration in order to assess thedynamics of oxidative phosphorylation in vivo duringexercise. Measurements of 31P and 17O by magneticresonance spectroscopy could provide key speciesconcentrations in oxidative phosphorylation.16

Nowadays measurements of pulmonary oxygenuptake, VO2p are used to make inferences on the bio-energetic processes involved during exercise. The VO2p

average measured in our population at the end ofconstant work rate exercise may be expressed as per-cent of the maximal pulmonary oxygen uptake.42:49 ± 11%, 68 ± 9% and 87 ± 6% at moderate,heavy and very heavy exercise intensity, respectively.

Presence of the slow component was discernablefrom our data during heavy and very heavy intensity

Model of Respiration during Exercise 965

exercise. To date, there is no unequivocal explanationfor such behavior because several factors can lead toan excess VO2p during high intensity constant workrate exercise, as already reported by severalauthors.4,20,21,52 The predominant portion of the slowcomponent (86%) may be associated with the con-tracting muscles,51 whereas fiber recruitment at higherlevel may affect the pattern of VO2p measured at themouth. Fiber recruitment may only contribute in partto the slow component. The model developed heredoes not incorporate components that may lead to aslow component for VO2; however, it can be enhancedto enable testing of putative mechanisms resulting inexcess oxygen consumption during high intensityexercise.

Additionally, if dynamic measurements of O2, CO2,and pH were obtained in the venous blood perfusingthe contracting muscle, then the effect of acid-baseregulation1,17 could be modeled to investigate thecontrol of respiration during exercise. Although directin vivo measurements of UO2m are not feasible duringexercise involving a large muscle mass, indirect evalu-ation of UO2m can be achieved with a multi-scalesystems biology approach. This would combine com-putational modeling and simulation with experimentaldata with many types of measurements at the wholebody, tissue, and cellular levels.

APPENDIX I

The oxygen mass balances of Eqs. (1) and (2) re-quire a relationship between total oxygen (Ccap

T , CtisT ) to

free oxygen (CcapF , Ctis

F ) in the capillaries and tissues.42

To relate the total oxygen concentration CxT to free

oxygen concentration CxF (x = art, cap, tis), we con-

sider oxygen in free and (hemoglobin) bound forms inarterial and capillary blood (Cart

F , CcapB ; Ccap

F , CartB ) and

in free and (myoglobin) bound forms in muscle tissue(Ctis

F , CtisB ). The total oxygen concentrations in arterial

and capillary blood and in muscle tissue are the sumsof the corresponding free and bound oxygen concen-tration as:

CTx ¼ CF

x þ CBx x 2 art, cap, tisf g ðA:1Þ

which are related by local chemical equilibrium. Inblood the relation is

CBcap¼4HctCrbc;HbScap;Hb¼4HctCrbc;Hb

KHb CFcap

� �n

1þKHb CFcap

� �n

ðA:2Þ

In tissue the relation is

CBtis ¼WmcCmc;MbStis;Mb ¼WmcCmc;Mb

KMbCFtis

1þ KMbCFtis

ðA:3Þ

These relations depend on Hb and Mb concentrationsin red blood cell and myocyte (Crbc,Hb, Cmc,Mb) andtheir respective volume fractions (Hct, Wmc) and oxy-gen saturations (Scap,Hb, Stis,Mb). From (A.1), we getthe differential relationships:

dCTx ¼ d CF

x þ CBx

� �¼ 1þ dCB

x =dCFx

� �dCF

x ¼ cxdCFx

ðA:4Þ

where

ccap ¼ 1þ4HctCrbc;HbKHbn CF

cap

� �n�1

1þ KHb CFcap

� �nh i2 ;

ctis ¼ 1þWmcCmc;MbKMb

1þ KMbCFtis

� �2

ðA:5Þ

Using the relationships above, Eq. (1) can be expressedas:

dCFcap

dt¼Qm CT

artðCFartÞ�CT

capðCFcapÞ

� ��PScap CF

cap�CFtis

� �

ccapVcap

ðA:6Þ

Similarly, we can express Eq. (2) as

dCFtis

dt¼

PS CFcap � CF

tis

� ��UO2m

ctisVtisðA:7Þ

It should be noted that this spatially lumped,two-compartment model of oxygen transport andmetabolism simulates the dynamic changes of oxygenconcentration in capillary blood and tissue cells withinthe skeletal muscle during exercise assuming perfectmixing. Thus, under this assumption, Ccap

F (t) is equiv-alent to the free oxygen concentration at the end of thecapillary, as well as to the venous oxygen concentration.

APPENDIX II

The values of the concentrations under steady-stateconditions at any exercise condition j must satisfy thefollowing equations:

0 ¼ Qjm CT

art � CT;jcapðCF;j

cap� �

� PSj CF;jcap � CF;j

tis

� �

ðA:8Þ

LAI et al.966

0 ¼ PSj CF;jcap � CF;j

tis

� ��UOj

2m ðA:9Þ

0 ¼ b/OxPhosðCjADP;C

F;jtis Þ � /ATPaseðCj

ADPÞ ðA:10Þ

0 ¼ /rCKðC

jADP;C

jATP;C

jCr;C

jPCrÞ

�/fCKðC

jADP;C

jATP;C

jCr;C

jPCrÞ

ðA:11Þ

CA;tot ¼ CjADP þ Cj

ATP ðA:12Þ

CCr;tot ¼ CjCr þ Cj

PCr ðA:13Þ

These 6 equations involve eight concentrations.Therefore, for any j we can specify two concentrationsand solve for the other six. At rest, j = R, we canspecify CATP

R and CCr,tot, then solve for the otherconcentrations including CA,tot. Under the warm-upconditions, j = W, we can specify CA,tot and CCr,tot

then solve for the other concentrations including CcapF,W,

CtisF,W, CATP

W and CPCrW .

ACKNOWLEDGMENTS

We thank Dr. Asit K. Saha for providing thoughtfuland constructive criticisms. This research was sup-ported by the grant GM-66309-01 from the NationalInstitute for General Medical Science (NIGMS) of theNational Institute of Health (NIH) for establishing theCenter for Modeling Integrated Metabolic Systems(MIMS) at Case Western Reserve University.

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