- Email: [email protected]
Modeling Stable Isotope Tracer Data: The Effect of a Nonnegligible Mass of Tracer
Copyright C IFAC ModeJin& md Control in Biomedical
Systems. GalVestol1. Texas. USA. 1994
MODELING STABLE ISOTOPE TRACER DATA THE EFFECT OF A NONNEGLIGIBLE MASS OF TRACER G. Toffolo and C.Cobel/i Dapt. of Electronics and Informatics, University of Padova, Padova, Italy
The use of stable isotope tracers to study the kinetics of metabolites offers a number of advantages as compared to radioisotope tracers. However, at variance with weightless radioactive tracers, stable isotope tracers have a nonnegligible mass, and the input of these tracers into a system, by altering the total amount of the substrate (tracer+tracee), can induce a perturbation on the endogenous steady state. We have developed in [1] a kinetic formalism for the analysis of stable isotope tracer data, assuming that the tracer has a non negligible mass, but its administration does not perturb the endogenous steady state. This assumption requires that the production of tracee is not altered, and the system kinetics are linear in the range of values during the experiment. We discuss here the situation where these conditions are not met, so that the endogenous system is perturbed by the tracer experiment. Indistinguishability between tracer and tracee will be assumed, and for sake of simplicity the analysis will be performed on a single pool system.
dQ(t)/dt = U(t) - F(t)
The assumption that tracer and tracee are indistinguishable implies that the probability that a particle leaving the pool is a tracer particle is equal to the probability that a particle in the pool is tracer. This provides the link between F(t) and f(t) : f(t) F(t)+ f(t)
dQ/dt = U - F = 0 A tracer is introduced at time zero at a rate u(t), q(t) is tracer mass, f(t) the tracer disposal. The mass balance equation for the tracer is :
dq(t)/dt = u(t) - f(t)
q(O) = 0
=_q.:..:...(t....... ) _ O(t)+q(t)
Modeling analysis is now much more complex than in the ideal tracer case, since structural descriptions must be postulated for disposal and production. In general, they are function of the pool mass O(t)+q(t), other control signals e.g. hormones and constant unknown parameters. To clarify these points, assume that disposal F follows a Michaelis-Menten kinetics. Prior to the experiment: F=U= a ·O b+Q
MODEL EOUATIONS:EFFECT OF PERTURBATION Prior to the experiment, the tracee system is in a constant steady state, that is tracee production U is constant and equals tracee disposal F, hence tracee mass 0 is constant:
0(0) = 0
where a and b are constant coefficients. A weightless tracer introduced into the system doesn't perturb the endogenous steady state, and tracer kinetics is linear, time invariant with a rate constant equat to ko1=F/O=al(b+0). Conversely, a non negligible mass tracer perturbs the system: the total disposal flux F(t)+f(t) obeys the Michaelis-Menten kinetics: F(t) + f(t) = a[O(t) + q(t)] b+ O(t)+ q(t) and from tracer tracee indistinguishability, the tracee and tracer components are:
Suppose that the tracer mass q(t) is not negligible compared with the pre-test tracee mass 0, and that the system kinetics is not intrinsically linear, so that a modification of the amount of material in the system induces a perturbation on the tracee kinetics: tracee mass and fluxes become time varying functions, and the mass balance equation for the tracee in this situation is:
301
f(t) _
a . q(t) b+O(t)+q(t)
F(t) _
a · O(t) b+O(t)+q(t)
Tracer kinetics are nonlinear since f(t) is a nonlinear function of q(t). The tracee system is perturbed from steady state since F is no longer a constant function and in addition feedback inhibition mechanisms on the tracee production could be activated. In this case a description of the way U varies with time needs to be specified.
Suppose now that the perturbation is non negligible but small in magnitude compared to the steady state level Q, so that the nonlinear disposal flux can be approximated with the first order term of the Taylor series expansion: F(t) + f(t) = F + k *01 [~O(t)+ q(t)]
and
k *01 = o[F(t) + f(t)] o[O(t) + q(t)]O(I}+q(I)=o ~Q(t)
Y1 (t) = [0 + ~Q(t}+q(t}]N
= Q(t) - Q.
Yit}=~yv
If we denote by ~U(t) the deviation of the production from its steady state level, the deviation ~ 0 + q induced by the tracer experiment on the amount of material in the system obeys a first order linear differential equation: \ d[~Q(t) + q(t}] / dt = (1) = -k*ol [~Q(t)+q(t}]+~U(t)+u(t) ~Q(O} + q(O)
dq(t)/dt = ko1 q(t) + u(t) when the perturbation ~O(t}+q(t) is small compared to Q, so as it can be neglected. The linearized model consists of Eq.1 and 2, having as state variables the mass perturbation ~O(t)+q(t) and the tracer mass q(t}. Output equations need to be specified, relating the measurement variables to the state variables. Consider as an example the situation where U is not perturbed during the experiment, ~U(t}=O, so that the model is non linear but time invariant, having ko1' k*01 and Q as model parameters. They can be estimated by associating to the state equations two output equations:
THE L1NEARIZED MODEL
where
Eq. 2 shows that the tracer time course depends not only on the amount of tracer in the system, but also on the total amount O+~O(t)+q(t), and both the rate ko1 and the derivative term k* 01 appear as constant parameters. Note that the tracer equation simplifies into the equation for the ideal tracer:
Y1 expresses the total concentration measurement, Y2 is the tracer concentration to be derived from Y1 and the tracer to tracee ratio measurements [1]. V is the volume of distibution of the accessible pool, and it is related to parameter Q through the pre-test concentration c: Q=e V. CONCLUSION
=0
Administration of a nonweightless tracer into a system may induce a perturbation on the endogenous steady state. While the time course of the mass perturbation, Eq.1, reflects k* 01' the steady state value of the derivative of the disposal flux with respect to the mass in the system, the time course of the tracer mass, Eq.2, can provide information on the steady state rate constant ko1 . We have discussed here the single pool system, but similar considerations also apply for the general multicornpartmental case.
Eq. 1 is the well-known equation describing small signal perturbation: the kinetics is linear, time invariant as for the case of an ideal tracer equation, but the model parameter is now k*01' that is the derivative of the flux F with respect the amount of material Q in the pool, and not the steady state rate coefficient ko1= F/Q. In addition, a functional dependence needs to be specified for ~U(t}. The Taylor series expansion of the disposal flux F(t}+f(t} and the tracer tracee indistinguishability principle can be used to derive the following equation for the tracer mass q(t}:
This work was in part supported by a MURST 40% grant. REFERENCES
dq(t} / dt = -kolq(t}-(k * -k ) ~Q(t} + q(t) (t) + u(t) 01 01 Q+ ~Q(t}+ q(t) q
1. Cobelli C., G.,Toffolo, D.M.Foster : Tracer-to-tracee ratio for analysis of stable isotope tracer data: link with radioactive kinetic formalism . Am .J.Physiol. 262 E968-E975,1992.
(2)
q(O} = 0
302
Systems. GalVestol1. Texas. USA. 1994
MODELING STABLE ISOTOPE TRACER DATA THE EFFECT OF A NONNEGLIGIBLE MASS OF TRACER G. Toffolo and C.Cobel/i Dapt. of Electronics and Informatics, University of Padova, Padova, Italy
The use of stable isotope tracers to study the kinetics of metabolites offers a number of advantages as compared to radioisotope tracers. However, at variance with weightless radioactive tracers, stable isotope tracers have a nonnegligible mass, and the input of these tracers into a system, by altering the total amount of the substrate (tracer+tracee), can induce a perturbation on the endogenous steady state. We have developed in [1] a kinetic formalism for the analysis of stable isotope tracer data, assuming that the tracer has a non negligible mass, but its administration does not perturb the endogenous steady state. This assumption requires that the production of tracee is not altered, and the system kinetics are linear in the range of values during the experiment. We discuss here the situation where these conditions are not met, so that the endogenous system is perturbed by the tracer experiment. Indistinguishability between tracer and tracee will be assumed, and for sake of simplicity the analysis will be performed on a single pool system.
dQ(t)/dt = U(t) - F(t)
The assumption that tracer and tracee are indistinguishable implies that the probability that a particle leaving the pool is a tracer particle is equal to the probability that a particle in the pool is tracer. This provides the link between F(t) and f(t) : f(t) F(t)+ f(t)
dQ/dt = U - F = 0 A tracer is introduced at time zero at a rate u(t), q(t) is tracer mass, f(t) the tracer disposal. The mass balance equation for the tracer is :
dq(t)/dt = u(t) - f(t)
q(O) = 0
=_q.:..:...(t....... ) _ O(t)+q(t)
Modeling analysis is now much more complex than in the ideal tracer case, since structural descriptions must be postulated for disposal and production. In general, they are function of the pool mass O(t)+q(t), other control signals e.g. hormones and constant unknown parameters. To clarify these points, assume that disposal F follows a Michaelis-Menten kinetics. Prior to the experiment: F=U= a ·O b+Q
MODEL EOUATIONS:EFFECT OF PERTURBATION Prior to the experiment, the tracee system is in a constant steady state, that is tracee production U is constant and equals tracee disposal F, hence tracee mass 0 is constant:
0(0) = 0
where a and b are constant coefficients. A weightless tracer introduced into the system doesn't perturb the endogenous steady state, and tracer kinetics is linear, time invariant with a rate constant equat to ko1=F/O=al(b+0). Conversely, a non negligible mass tracer perturbs the system: the total disposal flux F(t)+f(t) obeys the Michaelis-Menten kinetics: F(t) + f(t) = a[O(t) + q(t)] b+ O(t)+ q(t) and from tracer tracee indistinguishability, the tracee and tracer components are:
Suppose that the tracer mass q(t) is not negligible compared with the pre-test tracee mass 0, and that the system kinetics is not intrinsically linear, so that a modification of the amount of material in the system induces a perturbation on the tracee kinetics: tracee mass and fluxes become time varying functions, and the mass balance equation for the tracee in this situation is:
301
f(t) _
a . q(t) b+O(t)+q(t)
F(t) _
a · O(t) b+O(t)+q(t)
Tracer kinetics are nonlinear since f(t) is a nonlinear function of q(t). The tracee system is perturbed from steady state since F is no longer a constant function and in addition feedback inhibition mechanisms on the tracee production could be activated. In this case a description of the way U varies with time needs to be specified.
Suppose now that the perturbation is non negligible but small in magnitude compared to the steady state level Q, so that the nonlinear disposal flux can be approximated with the first order term of the Taylor series expansion: F(t) + f(t) = F + k *01 [~O(t)+ q(t)]
and
k *01 = o[F(t) + f(t)] o[O(t) + q(t)]O(I}+q(I)=o ~Q(t)
Y1 (t) = [0 + ~Q(t}+q(t}]N
= Q(t) - Q.
Yit}=~yv
If we denote by ~U(t) the deviation of the production from its steady state level, the deviation ~ 0 + q induced by the tracer experiment on the amount of material in the system obeys a first order linear differential equation: \ d[~Q(t) + q(t}] / dt = (1) = -k*ol [~Q(t)+q(t}]+~U(t)+u(t) ~Q(O} + q(O)
dq(t)/dt = ko1 q(t) + u(t) when the perturbation ~O(t}+q(t) is small compared to Q, so as it can be neglected. The linearized model consists of Eq.1 and 2, having as state variables the mass perturbation ~O(t)+q(t) and the tracer mass q(t}. Output equations need to be specified, relating the measurement variables to the state variables. Consider as an example the situation where U is not perturbed during the experiment, ~U(t}=O, so that the model is non linear but time invariant, having ko1' k*01 and Q as model parameters. They can be estimated by associating to the state equations two output equations:
THE L1NEARIZED MODEL
where
Eq. 2 shows that the tracer time course depends not only on the amount of tracer in the system, but also on the total amount O+~O(t)+q(t), and both the rate ko1 and the derivative term k* 01 appear as constant parameters. Note that the tracer equation simplifies into the equation for the ideal tracer:
Y1 expresses the total concentration measurement, Y2 is the tracer concentration to be derived from Y1 and the tracer to tracee ratio measurements [1]. V is the volume of distibution of the accessible pool, and it is related to parameter Q through the pre-test concentration c: Q=e V. CONCLUSION
=0
Administration of a nonweightless tracer into a system may induce a perturbation on the endogenous steady state. While the time course of the mass perturbation, Eq.1, reflects k* 01' the steady state value of the derivative of the disposal flux with respect to the mass in the system, the time course of the tracer mass, Eq.2, can provide information on the steady state rate constant ko1 . We have discussed here the single pool system, but similar considerations also apply for the general multicornpartmental case.
Eq. 1 is the well-known equation describing small signal perturbation: the kinetics is linear, time invariant as for the case of an ideal tracer equation, but the model parameter is now k*01' that is the derivative of the flux F with respect the amount of material Q in the pool, and not the steady state rate coefficient ko1= F/Q. In addition, a functional dependence needs to be specified for ~U(t}. The Taylor series expansion of the disposal flux F(t}+f(t} and the tracer tracee indistinguishability principle can be used to derive the following equation for the tracer mass q(t}:
This work was in part supported by a MURST 40% grant. REFERENCES
dq(t} / dt = -kolq(t}-(k * -k ) ~Q(t} + q(t) (t) + u(t) 01 01 Q+ ~Q(t}+ q(t) q
1. Cobelli C., G.,Toffolo, D.M.Foster : Tracer-to-tracee ratio for analysis of stable isotope tracer data: link with radioactive kinetic formalism . Am .J.Physiol. 262 E968-E975,1992.
(2)
q(O} = 0
302