Originally published In Press as doi:10.1194/jlr.M600302-JLR200 on September 11, 2006
Originally published In Press as doi:10.1194/jlr.M600302-JLR200 on September 1, 2006
Journal of Lipid Research, Vol. 47, 2738-2753, December 2006
Copyright © 2006 by American Society for Biochemistry and Molecular Biology
Studying apolipoprotein turnover with stable isotope tracers: correct analysis is by modeling enrichments
Rajasekhar Ramakrishnan1
Department of Pediatrics, Columbia University College of Physicians and Surgeons, New York, NY 10032
Published, JLR Papers in Press, September 11, 2006.
1 To whom correspondence should be addressed. e-mail: rr6{at}columbia.edu
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ABSTRACT
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Lipoprotein kinetic parameters are determined from mass spectrometry data after administering mass isotopes of amino acids, which label proteins endogenously. The standard procedure is to model the isotopic content of the labeled precursor amino acid and of proteins of interest as tracer-to-tracee ratio (TTR). It is shown here that even though the administered tracer alters amino acid mass and turnover, apolipoprotein synthesis is unaltered and hence the apolipoprotein system is in a steady state, with the total (labeled plus unlabeled) masses and fluxes remaining constant. The correct model formulation for apolipoprotein kinetics is shown to be in terms of tracer enrichment, not of TTR. The needed mathematical equations are derived. A theoretical error analysis is carried out to calculate the magnitude of error in published results using TTR modeling. It is shown that TTR modeling leads to a consistent underestimation of the fractional synthetic rate. In constant-infusion studies, the bias error percent is shown to equal approximately the plateau enrichment, generally <10%. It is shown that, in bolus studies, the underestimation error can be larger. Thus, for mass isotope studies with endogenous tracers, apolipoproteins are in a steady state and the data should be fitted by modeling enrichments.
Supplementary key words amino acid metabolism • fractional synthetic rate • mass isotopes • tracer kinetics • tracer-to-tracee ratio • precursor-product relationship
Abbreviations: apoB, apolipoprotein B; E, enrichment; FCR, fractional catabolic rate; FSR, fractional synthetic rate; M, total (tracer and tracee) mass; m, tracer mass; P, amount or plateau level of precursor; R, total (tracer and tracee) flux; S, product synthetic rate; t, time; TG, triglyceride; TTR, tracer-to-tracee ratio; U, unlabeled or tracee mass
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INTRODUCTION
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Beginning in the 1970s, apolipoprotein kinetics were routinely studied with exogenous tracers, for instance by isolating VLDL or LDL from a subject, radioiodinating it, and injecting it back into the subject (1–4). Endogenous labeling, with a labeled precursor of the metabolite of interest, has the virtue of labeling the synthetic pathways and not altering tracer metabolic properties, as might happen with exogenous labeling. Whole body cholesterol metabolism was studied with tritiated water (5), and tritiated leucine has been used to study lipoprotein kinetics (6–8). Highly sensitive gas chromatography-mass spectrometry and refinements thereof, and the availability of synthetic amino acids and other molecules that are multiply labeled with mass isotopes, have altered the field to the point that endogenous labeling with mass isotopes is now the norm in human turnover studies (9).
An important aspect of mass isotopes is that the amount of tracer introduced is not negligible in relation to the amount in plasma of the tracee. Cobelli, Toffolo, and Foster (10) and Foster et al. (11) considered this problem and advocated the use of tracer-to-tracee ratio (TTR) in place of the previously standard use of tracer enrichment in atoms percent excess or moles percent excess (12–15). Since then, nearly all investigators have used TTR in analyzing mass isotope data to calculate lipoprotein turnover parameters. In what follows, we revisit this issue and derive the mathematical relationships needed for the analysis of tracer data from endogenous labeling. In particular, we show that the apolipoprotein system is in a steady state and that the correct formulation is in terms of tracer enrichment or concentration, not TTR. We show that compartmental models and the usual fractional synthetic rate (FSR) equations are valid provided that they are written for tracer enrichments but not for TTRs. The error in using TTR is shown for constant-infusion studies to be in the range of the plateau tracer enrichment, usually 5–10%; the error is shown to be higher for bolus studies.
The word "enrichment" (E) is used here solely to denote tracer concentration, defined as the amount of tracer divided by the sum of the amounts of tracer and tracee (16). The word is used sometimes to denote TTR, but not here.
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CONSTANT TRACEE FLUX IMPLIES THAT PROTEIN AMOUNT CHANGES WITH TIME
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That mass isotope tracers have nonnegligible mass has been recognized from early on. Matthews et al. (16) adjusted for it by assuming that the total flux is increased by a constant tracer infusion, so that a tracer balance can be written as:
 | (Eq. 1) |
where Q is the tracee flux, i is the constant infusion rate of the tracer, Ep is the plasma enrichment of the traced molecule (amino acid in Ref. 16), and Ei is the infusion enrichment. With a radiotracer, because of its negligible mass, there would be no i on the left side of the equation, and the right side would be the infusion rate of radioactivity.
The question relevant to lipoprotein turnover is whether every component of the flux changes similarly. Foster et al. (11) assume that incorporation into specific proteins follows the same pattern. This approach is shown by them to lead to simple equations for protein kinetics. The tracee masses and fluxes are constant, and linear differential equations can be written for TTR, the equations identical in form to those written for radiotracers.
Figure 1
shows the essential part of their model (Fig. 1 in Ref. 11) for leucine incorporation into VLDL apolipoprotein B (apoB), leaving out other pathways. Before tracer infusion, there is Uleu of unlabeled leucine, being incorporated into VLDL apoB with a rate constant of kleu; UB is the tracee mass of VLDL apoB. During the tracer study, by the assumption of tracee steady state, Uleu, kleu, and UB do not change; at any time t, there is mleu(t) of tracer, being incorporated into labeled VLDL apoB with the same rate constant as the tracee, kleu. [Since an apoB molecule has multiple leucine molecules, a labeled leucine combines with unlabeled leucine in the same apoB molecule, so this model is not precise, but it is easily rectified by combining the two apoB pools into one. None of the results below are affected, and the figure is drawn to be close to Fig. 1 in Foster et al. (11).] Thus, the constant tracee flux assumption predicts that, compared with the steady state before the study, more leucine should be incorporated into VLDL apoB during the tracer study and the total amount of VLDL apoB should increase correspondingly. If the TTR of leucine in VLDL apoB approaches 5%, which is typical, then, under the constant tracee flux assumption, VLDL apoB mass should be higher by 5% at the end of the constant infusion.
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Fig. 1. Adapted from Ref. 11 to show the essential part of their Fig. 1. Leucine pools and synthesis of a protein of interest [e.g., VLDL apolipoprotein B (apoB)]. The larger pool is for the tracee, and the smaller pool is for the tracer. Under the assumption of tracee steady state, tracee incorporation into VLDL apoB is unaltered during the study, but tracer mass and incorporation change with time, indicated by (t); the tracer incorporation rate constant is the same as for the tracee. Other pathways for leucine and for VLDL apoB are not shown. The dashed rectangles denote total leucine and VLDL apoB.
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THE CONSTANT TRACEE FLUX ASSUMPTION LEADS TO A CONTRADICTION
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There are two reasons why the assumption of constant tracee masses and fluxes may be invalid. One is that the assumption of constant tracee flux, applied simultaneously to multiple amino acids, leads to a contradiction. The other is that there is evidence that apolipoprotein synthesis is unaltered by tracer infusion. These reasons are elaborated on below.
Consider a study with a primed constant infusion of a tracer of leucine and a simultaneous bolus injection of a tracer of glycine, as in Parhofer et al. (17). Figure 2A
shows three amino acids that go into the synthesis of apoB. The assumption that the tracee remains in a steady state can be applied to each of the three amino acids. The incorporation from each tracee pool is shown as constant, whereas the incorporation from each tracer pool varies with time, denoted by (t). Figure 2B shows the hypothetical total rate of incorporation of each precursor (tracer plus tracee) resulting from the assumption of tracee steady state. The values before tracer infusion are at the mol%s of the three amino acids in apoB. The total incorporation rate of leucine increases to a higher steady state, that of glycine increases sharply and declines with the clearance of the glycine tracer, and that of alanine remains unchanged. Thus, if the tracee fluxes are constant, the relative amounts of leucine, glycine, and alanine in newly synthesized apoB begin at the known values for apoB, but soon after the tracer study begins, newly synthesized apoB has 14% leucine and 9% glycine, the percentages changing every moment, a stoichiometric impossibility as this would mean a changing amino acid composition of the protein.
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Fig. 2. Schemata of three amino acid precursors, and their rates of incorporation into a specific apolipoprotein in a hypothetical study with a primed constant infusion of a tracer of leucine and a simultaneous bolus injection of a tracer of glycine. Under the assumption of tracee steady state, A shows the precursor pools, with the small pools representing tracers. If the tracee (unlabeled) masses Uleu, Ugly, and Uala are constant, the rates of incorporation vary with time, as given by the formulas under the arrows and as shown in B. The numbers along the y axis are in arbitrary units. The curves begin at the leucine, glycine, and alanine contents of apolipoprotein B before tracer infusion. Soon after 0 h, leucine is at 14 instead of 12.5, and glycine is at 9 instead of 4.5, resulting in a changing stoichiometry of the apoB product, an impossibility. C shows the tracer-to-tracee ratio (TTR) data from Fig. 4 of Parhofer et al. (17), presented here, under the assumption of tracee steady state, as the change in total mass of VLDL apoB. If tracee apoB were constant, total apoB increased by nearly 4% to a new steady level, according to the leucine TTR data, whereas the glycine TTR data would suggest that total apoB increased quickly by nearly 4% and then declined. The dashed horizontal line indicates that an untraced amino acid should be interpreted as no change in VLDL apoB. Thus, the assumption of a tracee steady state is contradicted by the data of Parhofer et al. (17). AA, amino acid.
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Fig. 4. Schematic bar graphs showing the changes with time in precursor amino acid (P), incorporation rate into apolipoprotein (S), and amino acid (AA) in apolipoprotein (M) for two types of tracer studies. The full height of each bar represents the total of tracer and tracee, whereas the hatched portion is for the tracer. The subscript T stands for total (tracer plus tracee), L stands for tracer (label), C stands for constant infusion, B stands for bolus, and 1, 8, and 15 indicate 1, 8, and 15 h. The left panels are for a primed constant infusion. The upper left panel shows that the tracer in the precursor increases and stays at a constant fraction from 1 to 15 h. The middle left panel shows that apolipoprotein incorporation is unchanged, with the tracer contributing that same fraction from 1 to 15 h. The bottom left panel shows that the apolipoprotein mass remains the same while the amount of label increases from 0 to 1 to 8 h, approaching a plateau at 15 h. The right panels are for a bolus study. The upper right panel shows that the tracer in the precursor increases and then declines from 1 to 15 h. The middle right panel shows that the apolipoprotein incorporation is unchanged, with the tracer contributing a fraction equal to its fraction in the precursor. The bottom right panel shows that the apolipoprotein mass remains the same while the amount of label increases from 0 to 1 h and then decreases to 15 h.
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This can also been seen in the composition of the product, VLDL apoB. Under the assumption of constant tracee mass, the TTR in VLDL apoB leucine or glycine is equivalent to the change in total VLDL apoB. If leucine TTR is 4% at some time, then VLDL apoB at that moment is 4% higher than before the study. Likewise, if glycine TTR in VLDL apoB is 2%, then VLDL apoB is up by 2% at that moment. For an amino acid such as alanine, there is no tracer and so total VLDL apoB should be unchanging. Figure 2C shows the VLDL apoB TTR data from Parhofer et al. (17), with the TTR values presented here, under the tracee constancy assumption, as changes in total VLDL apoB mass. The horizontal line indicates that an untraced amino acid such as alanine would imply no change in VLDL apoB. The three curves in Fig. 2C show very different behaviors for total VLDL apoB. Thus, the data of Parhofer et al. (17) are inconsistent with the tracee steady state assumption. The data of Demant et al. (18), with simultaneous leucine bolus and phenylalanine constant infusion, would also support this conclusion. Indeed, any double-tracer study in which the TTRs of the two tracers in VLDL apoB are not identical contradicts the assumption of tracee steady state.
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APOLIPOPROTEIN SYNTHESIS UNAFFECTED BY AMINO ACID INFUSION
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The other reason for the likely invalidity of a constant tracee flux to apolipoproteins is that there is no evidence that infusion of a single amino acid affects apolipoprotein synthesis. Apolipoprotein synthesis is regulated by many factors affecting transcription, mRNA stability, translation, and posttranslational degradation, but a single amino acid availability is not known to be such a factor.
Cohn et al. (14) and Lichtenstein et al. (19) measured VLDL apoB a number of times during a 15 h constant-infusion study. No time trend in VLDL apoB mass was seen, whereas VLDL apoB TTR increased to 
6%. This constancy has been replicated in a number of studies by that group (20–23), in which multiple VLDL apoB fractions were obtained during the study. Other studies (24–31) have also found no change in apolipoprotein concentrations during constant-infusion studies. There appear to be no reports of an increase in VLDL apoB mass during a constant-infusion study.
Davis and coworkers (32) have shown in pigs that, once the neonatal phase is over, amino acid infusion or protein intake increases muscle protein synthesis, but the effect on liver protein synthesis is quite modest, as was found earlier in rats (33, 34). Of particular relevance to apolipoproteins, Motil et al. (35) studied two different protein intakes in five women and found that leucine oxidation increased with protein intake but lysine incorporation into apoB did not. These animal and human studies are consistent with cell culture studies showing that a significant fraction of newly synthesized apoB is degraded, with secretion determined largely by lipid availability (36). Intracellular apoB degradation has been estimated by compartmental modeling to be 90% in HepG2 cells (37, 38) and >30% in primary hepatocytes (39).
Thus, there is no reason to expect that a tracer amino acid infusion alters apolipoprotein synthesis. Assuming that apolipoprotein synthesis is unaltered by tracer infusion avoids the contradiction implied by the constant tracee flux assumption and is consistent with the published data cited above.
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PRECURSOR ENRICHMENT IS FORCING FUNCTION FOR APOLIPOPROTEIN KINETICS
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We next show that the introduction of tracer alters tracee incorporation (as opposed to the total of tracer and tracee) into apolipoproteins. Consider Fig. 3
, which shows, for amino acid 1, the tracer with mass m1(t) and tracee with mass U1(t) at the site of synthesis of the apolipoprotein of interest. The rate of incorporation into the protein remains at f1S before and after tracer is introduced, where S is the protein synthesis and f1 is the fraction from amino acid 1. Since the tracer and tracee are indistinguishable, the tracer incorporation rate is obtained simply by multiplying f1S by the tracer fraction of the pool, 
, which is the enrichment Ep(t) of the precursor pool. The tracee incorporation rate, therefore, is (1 – Ep(t))f1S, compared with f1S in the absence of tracer.
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Fig. 3. The tracer and tracee precursor pools for an amino acid whose tracer is introduced. Both pool sizes can vary with time, indicated by (t), but the total (tracer plus tracee) incorporation rate into an apolipoprotein is unchanged. The separate incorporation rates of tracer and tracee are proportional to their respective masses, as given by the formulas. The fluxes on other pathways, such as oxidation, clearance, or other storage pools, may bear different relationships to the masses, as indicated by question marks to mean "unknown."
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Thus, for any apolipoprotein, after tracer infusion: 1) the apolipoprotein synthesis and the rate of incorporation from each amino acid remain unaltered; 2) tracee flux to the apolipoprotein is altered; and 3) tracer flux to the apolipoprotein equals precursor pool enrichment multiplied by the incorporation rate of that amino acid.
These points are illustrated by the bar graphs in Fig. 4
; tracer content is shown by hatched areas. Two experimental designs are considered. The left panels are for a primed constant infusion. The top left panel shows the mass of free amino acid in the precursor pool at four different times. There is no tracer at time zero; with the introduction of tracer, the total mass increases and remains at a constant level from 1 to 15 h, as does the mass of tracer. The middle left panel shows the rate of apolipoprotein synthesis at the four times. The total synthesis, S, remains constant, whereas a fraction of the synthesis is labeled after time zero. By the principle of isotopic indistinguishability (40), the tracer fraction of the synthesis equals the tracer fraction in the precursor pool:
 | (Eq. 2) |
where S stands for synthetic rate (mass/time), P stands for the amount of precursor (free amino acid), the subscript L refers to the label, T refers to the total mass, C refers to the constant-infusion design, and 1, 8, and 15 refer to the time. The bottom left panel shows the mass of amino acid in the apolipoprotein of interest at the four times. The total amount, M, remains constant, whereas an increasing fraction of the mass is labeled over 15 h, approaching the fraction in the precursor pool (and the synthetic pathway) given in Eq. 2.
The right side of Fig. 4 is for a bolus injection. The top right panel shows the mass of free amino acid in the precursor pool at the four times. The difference from constant infusion is that there is a much greater perturbation at 1 h, with a high tracer content that decreases with time. The middle right panel shows the rate of apolipoprotein synthesis at the four times. As with constant infusion, total synthesis, S, remains constant, whereas the labeled fraction increases and decreases with the precursor pool:
 | (Eq. 3) |
where the subscript B refers to the bolus injection design. This is analogous to Eq. 2. The bottom right panel shows the mass of amino acid in the apolipoprotein of interest at the four times. As with constant infusion, the total amount, M, remains constant, whereas the labeled fraction changes over 15 h.
Figure 4 illustrates how, with a constant-infusion or bolus study, the precursor amino acid pool can be in an unsteady state while apolipoprotein synthesis and mass remain constant, with the rate of tracer incorporation proportional to tracer enrichment in the precursor.
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UNSTEADY-STATE PRECURSOR/STEADY-STATE APOLIPOPROTEIN
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These ideas are summarized in the schematic model in Fig. 5
for the kinetics of an amino acid and an apolipoprotein of interest. The model is similar to that used by many researchers for apoB (41–44). The details, regarding the number of pools and their connectivity, may vary with the apolipoprotein being studied and with the mode of tracer administration. The key element here is the dashed horizontal line that separates the free amino acid model from the apolipoprotein model. Before the tracer study, the study subject is in a steady state: that is, the masses and fluxes of the free amino acid and of the protein are constant with time. When the mass isotope tracer is introduced, it clearly increases the total amount of the free amino acid in plasma. This may or may not lead to changes in the rate constants and alter the masses and fluxes of the unlabeled free amino acid as well as of the total (labeled plus unlabeled) amino acid. Modeling amino acid turnover has been described by Cobelli et al. (40). Clearly, the free amino acid part of the system is in an unsteady state.
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Fig. 5. Scheme of the amino acid and apolipoprotein systems. When a tracer is introduced, that amino acid is in an unsteady state, but the apolipoprotein remains in a steady state. The dashed line separates the unsteady amino acid system from the steady apolipoprotein system. The number of pools is for illustrative purposes. The unaltered synthesis paths are shown by double arrows. Total masses are denoted by M, and fluxes are denoted by R. Tracer masses are denoted by m; labeled fluxes, which equal the corresponding total fluxes multiplied by the source pool enrichments, are given in terms of total fluxes and tracer-to-total mass ratios or enrichments. For instance, the total flux from protein pool 1 to pool 2 is R21, and the corresponding tracer flux is R21m1(t)/M1, or R21 multiplied by the tracer enrichment in pool 1. Tracer quantities are shown in italics, below or to the right of the corresponding total quantities.
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We next consider the double arrows crossing the horizontal line, indicating synthesis or incorporation of the amino acid into the apolipoprotein of interest (two arrows are given to allow for the possibility of synthesis into multiple lipoprotein fractions). This flux (mass per unit of time) is not limited by the amount of a single amino acid in the precursor pool. Whether the mass of that pool increases by 5% or even by 100%, it is not rate-limiting for apolipoprotein synthesis; hence, the total synthetic flux (labeled plus unlabeled) is not affected.
Moving down to the apolipoprotein part of the model, since the synthetic rates are not affected, the total (labeled plus unlabeled) masses and fluxes of the apolipoprotein remain constant during the tracer study. Thus, there is a steady state for total apolipoprotein. As labeled amino acid is incorporated into protein, the tracer content of the protein masses will change, much as with exogenous tracers.
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KEY FEATURES OF ENDOGENOUS LABELING
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Thus, the key features of endogenous labeling, with radioactive or mass isotopes, are as follows.- By using a labeled precursor, not rate-limiting for the synthesis of the apolipoprotein of interest, even in nontracer amounts (whether as a bolus or as a primed constant infusion), the synthetic pathways are labeled without affecting the steady state of the apolipoprotein: the masses, fluxes, and rate constants of apolipoprotein turnover are not affected.
- In contrast, the precursor itself is in an unsteady state, with the introduction of the label altering the total masses and possibly fluxes and/or rate constants.
- The kinetics of the apolipoprotein differ in a subtle way from those of an exogenous radiolabel, in which the amount of tracer is so small that the tracee (unlabeled) masses, fluxes, and rate constants may be assumed to be constant. With endogenous labeling with a mass isotope, the total masses, fluxes, and rate constants do not change but the amounts of tracer and tracee do change in the course of the study.
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MATHEMATICAL MODELING OF APOLIPOPROTEIN KINETICS
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Since mass isotopes, unlike radiotracers, have significant mass, it is possible that conventional pool models may not apply. We first show that, since the apolipoprotein system is in a steady state, with constant total masses and fluxes, it is possible to use pool models for tracer enrichment. Consider Fig. 5, which shows two protein pools both receiving newly synthesized protein. Total (labeled plus unlabeled) masses are denoted by M, total fluxes by R, and tracer masses by m. As in Fig. 3, tracer fluxes equal total fluxes multiplied by the source pool enrichments. For instance, the total flux from protein pool 1 to pool 2 is R21, and the corresponding tracer flux is R21m1(t)/M1, or R21 multiplied by y1(t), the tracer enrichment in pool 1. (Tracer quantities are shown in italics, below or to the right of the corresponding total quantities.) The rate of change of m1(t), tracer mass in pool 1, is given by a differential equation:
 | (Eq. 4) |
where w(t) is the precursor enrichment, equal to m0(t)/M0(t). The equation can be rewritten in terms of tracer enrichment, since m1(t) equals M1y1(t), and M1 is constant:
 | (Eq. 5) |
where L values are rate constants, with Lij defined as the flux into pool i from pool j divided by the mass of pool i. Proceeding similarly, a differential equation for tracer enrichment in pool 2 is obtained:
 | (Eq. 6) |
It is seen that the differential equations 5 and 6 are linear and stationary with constant coefficients, identical to what would be obtained with exogenous tracers. Thus, although mass isotopes introduce a nonnegligible mass into the precursor pools, the apolipoprotein system is modeled as with exogenous tracers (45–49).
 | (Eq. 7) |
is the classic pool model in matrix-vector notation, where y(t) is the tracer enrichment vector in moles percent excess, A is the matrix of rate constants (Aii is the negative of the total flux out of pool i divided by the mass of that pool, and Aij is the flux into pool i from pool j divided by the mass of pool i), si is the direct synthetic flux into pool i divided by the mass of pool i, and w(t) is the precursor tracer enrichment function, which may not be describable by a pool model since the precursor system is in an unsteady state.
To clarify, consider a single pool for a protein, as is done in modeling apolipoprotein [a] (50) or apoC-III (51). The mass balance for the tracer is written as:
 | (Eq. 8) |
where S is the synthetic rate, equal to R, the flux out of the pool. Since the protein is in a steady state, M is constant, and so the equation can be rewritten as:
 | (Eq. 9) |
With a primed constant infusion, if the precursor enrichment w can be assumed constant, an analytical solution is available:
 | (Eq. 10) |
Both w and y are tracer enrichments.
It is not possible to write a linear differential equation for TTR because tracee mass and flux are not constant. The equation for TTR analogous to equation 8 is the following:
 | (Eq. 11) |
 | (Eq. 12) |
This is a nonlinear differential equation, which has to be solved along with a separate differential equation for Mu(t), the subscript u denoting the unlabeled tracee. In general, there is no simple solution.
Thus, the linear differential equations with constant coefficients used in various modeling programs such as SAAM (49) and Poolfit (52) are valid only for tracer enrichments, not for TTR.
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MATHEMATICAL MODELING OF PRECURSOR KINETICS
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As seen in the top panels of Fig. 4, the precursor is not in a steady state. Total and tracer masses and fluxes are varying with time. Cobelli et al. (40) have discussed various possibilities. We have seen that tracee fluxes to proteins are affected, since the total apolipoprotein synthesis is not altered while part of it gets labeled, as seen in the middle panels of Fig. 4. The modeling of the precursor is quite complex, and data may not be available to do justice to the problem. As Barrett et al. (53) stated, "The kinetics of amino acids are complex, so a possible approach to incorporating plasma amino acid data into the development and fitting of a compartmental model to tracer data is to use a forcing function. In this way, the system can be decoupled and the plasma amino acid data can be used as the source of tracer." Following their reasoning, the precursor is merely a forcing function. So any model (or even no model) is adequate as long as the observed data for the precursor are fitted well. The data fitted can be TTR or enrichment.
As seen above, the model for apolipoprotein turnover requires precursor tracer enrichment. Therefore, regardless of how the precursor data are fitted, they have to be converted to enrichments for use as w(t) in equation 7 for any apolipoprotein model.
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ERROR IN USING TTR
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Since most investigators have calculated kinetic parameters from TTR data, and not enrichments, the question arises of the magnitude of error in reported results. Some theoretical analysis is provided here to give an idea of the magnitude of the error.
Since TTR equals E/(1 – E), it is always greater than E. We can also compare the shapes of the two curves by looking at the time derivative of TTR:
 | (Eq. 13) |
Since the denominator on the right side is always <1, the slope of TTR is always greater in magnitude than that of E: if E is increasing, TTR increases faster; if E is decreasing, TTR decreases faster. Figure 6A
shows an illustration of this for a single pool receiving a bolus injection (the rest of the figure will be described below).
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Fig. 6. In a bolus study, precursor TTR is higher and sharper than precursor tracer enrichment (E). A: Theoretical situation with a single precursor pool whose enrichment declines monoexponentially. B: TTR data taken from Parhofer et al. (17) and enrichments calculated from their TTR data, along with fitted curves. The area under the TTR curve is 39% higher in A and 35% higher in B than the corresponding area under the enrichment curve. This overestimation of the forcing function leads to an underestimate of fractional synthetic rates with TTR modeling.
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The error in using TTR instead of enrichment E is given by: error = TTR – E. The error is a positive bias, which can be expressed as a fraction:
 | (Eq. 14) |
Thus, the TTR curve is greater than the E curve by a fraction that equals TTR. If TTR is 10%, then E is 9.09%, and the error is 0.91%, which is 10% of the enrichment. The relative error increases from zero at zero enrichment to a maximum at the peak.
We consider three simple situations that are amenable to theoretical analysis. In what follows, FSRTTR is used to denote the FSR computed by TTR modeling, and FSRE denotes the correct FSR from modeling enrichment.
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ERROR IN FSRTTR OF A SINGLE RAPID POOL WITH A CONSTANT INFUSION
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Many investigators have used primed constant infusion and modeled the apolipoprotein of interest by a single pool (29, 30, 41, 50, 54–70). The FSR is calculated by fitting the TTR data by an increasing exponential:
 | (Eq. 15) |
where r is the FSR of interest. However, the modeling should be done for enrichment, which leads to:
 | (Eq. 16) |
where k is the true FSR, E is the enrichment, and P is the plateau enrichment of the precursor, E and P expressed as fractions. TTR is given by:
 | (Eq. 17) |
This is not a simple increasing exponential, but for the usual values of P (<0.1), it can be fitted by an increasing exponential quite adequately, which may be a reason that this error has not been reported. A general result for the error is not available; individual data sets have to be fitted both ways to calculate the errors. However, if we assume that the initial increase of the TTR data is well fitted by an increasing exponential, the initial slope of the fitted curve (equation 15) must equal that of the true TTR response (equation 17). By equating the time derivatives of the two equations, we get:
 | (Eq. 18) |
At very early times, this becomes, approximately
 | (Eq. 19) |
which leads to r = k(1 – P).
Thus, the FSR calculated by fitting the TTR data by an increasing exponential roughly equals the true FSR multiplied by (1 – P), an underestimate by a fraction roughly equal to the plateau enrichment.
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ERROR IN FSRTTR OF SLOWLY TURNING-OVER APOLIPOPROTEINS WITH A CONSTANT INFUSION
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Some studies have used primed constant infusions and looked at slowly turning-over apolipoproteins such as LDL apoB, apoC-III, and apolipoprotein [a], proteins whose enrichments show a linear increase with time during the course of the study; the FSR has been calculated in a "model-free" approach by fitting the early TTR data by a straight line and dividing the slope by the precursor TTR (19, 71–73):
 | (Eq. 20) |
The correct model-free method is to use the enrichment plateau and slope:
 | (Eq. 21) |
As shown above, the TTR denominator has a positive bias equal to plateau(TTR). The numerator, being a slope estimated by fitting a number of points, is more complicated to analyze. The ideal situation, in which protein enrichment data are available continuously from zero until the maximum on the linear portion, is solved in Appendix 1:
 | (Eq. 22) |
where g is the highest enrichment reached in the protein. Table 1
shows the error in a slow FSR calculated from TTR modeling. It is seen that the protein enrichment has little influence on the error, which is primarily determined by the error in the plateau. Thus, FSR from TTR data underestimates the true FSR of a slow pool by roughly the plateau enrichment (e.g., if the plateau enrichment is 10%, FSR from TTR underestimates the true FSR of a slow pool by 10%). The result is similar to that obtained for a fast apolipoprotein pool in equation 19.
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ERROR IN FSRTTR OF SLOWLY TURNING-OVER APOLIPOPROTEINS WITH A BOLUS INJECTION
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For a slowly turning-over apolipoprotein, we can ignore the clearance of tracer from that protein during the study so that the amount of tracer in the protein at the end of the study can be equated to the amount incorporated from the precursor over the duration of the study:
 | (Eq. 23) |
The equation can be solved for the true FSR:
 | (Eq. 24) |
(The equation simplifies to equation 21 when the precursor is at a constant level.) With TTR modeling, which assumes that the tracee mass and flux are constant, the equation above is derived with TTR in place of E:
 | (Eq. 25) |
It is shown in Appendix 2 that, for a single pool for the precursor, it is possible to get an estimate of the error in FSR from TTR data:
 | (Eq. 26) |
where c is the initial precursor enrichment. The error can be quite significant. As the magnitude of the bolus increases, the value of c increases and the integral ratio gets smaller, so the error is worse. For a small bolus, c may be 0.1, the integral ratio is 0.95, which means a small error from using TTR modeling. For a larger bolus, c may be 0.5 (TTR of 1), which is quite common (17, 18, 74), the integral ratio is 0.72, which means TTR modeling will underestimate the true FSR of a slowly turning-over apolipoprotein by >25%.
The underestimation error with TTR modeling appears to be larger in a bolus study than in a primed constant-infusion study. The reason is that the precursor TTR is generally <10% during a constant-infusion study, so the error in considering it as enrichment is <10% as well; on the other hand, in the early part of a bolus study, the precursor TTR can exceed 100%, which means an error of >100% if TTR is used as the forcing function.
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ERRORS IN RATE CONSTANTS IN A MULTICOMPARTMENTAL MODEL
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ABSTRACT
INTRODUCTION