Bulletin of Mathematical Biology

, Volume 72, Issue 8, pp 2019–2046 | Cite as

A State Space Transformation Can Yield Identifiable Models for Tracer Kinetic Studies with Enrichment Data

  • Rajasekhar RamakrishnanEmail author
  • Janak D. Ramakrishnan
Original Article


Tracer studies are analyzed almost universally by multicompartmental models where the state variables are tracer amounts or activities in the different pools. The model parameters are rate constants, defined naturally by expressing fluxes as fractions of the source pools. We consider an alternative state space with tracer enrichments or specific activities as the state variables, with the rate constants redefined by expressing fluxes as fractions of the destination pools. Although the redefinition may seem unphysiological, the commonly computed fractional synthetic rate actually expresses synthetic flux as a fraction of the product mass (destination pool). We show that, for a variety of structures, provided the structure is linear and stationary, the model in the enrichment state space has fewer parameters than that in the activities state space, and is hence better both to study identifiability and to estimate parameters. The superiority of enrichment modeling is shown for structures where activity model unidentifiability is caused by multiple exit pathways; on the other hand, with a single exit pathway but with multiple untraced entry pathways, activity modeling is shown to be superior. With the present-day emphasis on mass isotopes, the tracer in human studies is often of a precursor, labeling most or all entry pathways. It is shown that for these tracer studies, models in the activities state space are always unidentifiable when there are multiple exit pathways, even if the enrichment in every pool is observed; on the other hand, the corresponding models in the enrichment state space have fewer parameters and are more often identifiable. Our results suggest that studies with labeled precursors are modeled best with enrichments.


Enrichments Estimation Identifiability Modeling Rate constants 





vector of rate constants for entry from outside


fractional catabolic rate


fractional synthetic rate


vector of rate constants for exit to outside


system matrix of rate constants


rate constant


diagonal matrix of pool masses


total mass or activity of tracer+tracee


mass or activity of tracer


model for tracer activities or amounts


synthetic flux


synthetic rate constant




a vector whose every element is 1


precursor enrichment


tracer enrichment


model for tracer enrichments or specific activities


observed tracer enrichment


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  1. Anderson, D.H., 1983. Compartmental Modeling and Tracer Kinetics. Springer, Berlin. zbMATHGoogle Scholar
  2. Arad, Y., Ramakrishnan, R., Ginsberg, H.N., 1990. Lovastatin therapy reduces low density lipoprotein apoB levels in subjects with combined hyperlipidemia by reducing the production of apoB-containing lipoproteins: implications for the pathophysiology of apoB production. J. Lipid Res. 31, 567–582. Google Scholar
  3. Barrett, P.H.R., Chan, D.C., Watts, G.F., 2006. Design and analysis of lipoprotein tracer kinetics studies in humans. J. Lipid Res. 47, 1607–1619. CrossRefGoogle Scholar
  4. Basu, R., Di Camillo, B., Toffolo, G., Basu, A., Shah, P., Vella, A., Rizza, R., Cobelli, C., 2003. Use of a novel triple-tracer approach to assess postprandial glucose metabolism. Am. J. Physiol.-Endocrinol. Metab. 284, E55–E69. Google Scholar
  5. Bellman, R., Åström, K.J., 1970. On structural identifiability. Math. Biosci. 7, 329–339. CrossRefGoogle Scholar
  6. Berglund, L., Witztum, J.L., Galeano, N.F., Khouw, A.S., Ginsberg, H.N., Ramakrishnan, R., 1998. Three-fold effect of lovastatin treatment on low density lipoprotein metabolism in subjects with hyperlipidemia: increase in receptor activity, decrease in apoB production, and decrease in particle affinity for the receptor. Results from a novel triple-tracer approach. J. Lipid Res. 39, 913–924. Google Scholar
  7. Berman, M., Schoenfeld, R., 1956. Invariants in experimental data on linear kinetics and the formulation of models. J. Appl. Phys. 27, 1361–1370. CrossRefGoogle Scholar
  8. Berman, M., Weiss, M.F., Shahn, E., 1962. Some formal approaches to the analysis of kinetic data in terms of linear compartmental systems. Biophys. J. 2, 289–316. CrossRefGoogle Scholar
  9. Bright, P.B., 1973. Volumes of some compartment systems with sampling and loss from one compartment. Bull. Math. Biol. 35, 69–79. Google Scholar
  10. Brown, R.F., Godfrey, K.R., 1978. Problems of determinacy in compartmental modeling with application to bilirubin kinetics. Math. Biosci. 40, 205–224. CrossRefzbMATHGoogle Scholar
  11. Chapman, M.J., Godfrey, K.R., 1985. Some extensions to the exhaustive modelling approach to structural identifiability. Math. Biosci. 77, 305–323. CrossRefMathSciNetzbMATHGoogle Scholar
  12. Chau, N.P., 1985. Parameter identification in n-compartment mammillary models. Math. Biosci. 74, 199–218. CrossRefMathSciNetzbMATHGoogle Scholar
  13. Chen, B.C., Landaw, E.M., DiStefano, J.J. 3rd, 1985. Algorithms for the identifiable parameter combinations and parameter bounds of unidentifiable catenary compartmental models. Math. Biosci. 76, 59–68. CrossRefMathSciNetzbMATHGoogle Scholar
  14. Cobelli, C., DiStefano, J.J. 3rd, 1980. Parameter and structural identifiability concepts and ambiguities: a critical review and analysis. Am. J. Physiol.-Endocrinol. Metab. 239, R7–R24. Google Scholar
  15. Cobelli, C., Toffolo, G., 1984. Identifiability from parameter bounds, structural and numerical aspects. Math. Biosci. 71, 237–243. CrossRefMathSciNetzbMATHGoogle Scholar
  16. Cobelli, C., Toffolo, G., 1987. Theoretical aspects and practical strategies for the identification of unidentifiable compartmental systems. In: Walter, E. (Ed.), Identifiability of Parametric Models, pp. 85–91. Pergamon, Oxford. Google Scholar
  17. Cobelli, C., Lepschy, A., Jacur, G.R., 1979. Identifiability results on some constrained compartmental systems. Math. Biosci. 47, 173–195. CrossRefMathSciNetzbMATHGoogle Scholar
  18. Cobelli, C., Toffolo, G., Ferrannini, E., 1984. A model of glucose kinetics and their control by insulin, compartmental and noncompartmental approaches. Math. Biosci. 72, 291–315. CrossRefzbMATHGoogle Scholar
  19. Cobelli, C., Toffolo, G., Bier, D.M., Nosadini, R., 1987. Models to interpret kinetic data in stable isotope tracer studies. Am. J. Physiol.-Endocrinol. Metab. 253, E551–E564. Google Scholar
  20. Cobelli, C., Foster, D., Toffolo, G., 2000. Tracer kinetics in biomedical research: from data to model. Kluwer Academic, New York. Google Scholar
  21. Demant, T., Packard, C.J., Demmelmair, H., Stewart, P., Bedynek, A., Bedford, D., Seidel, D., Shepherd, J., 1996. Sensitive methods to study human apolipoprotein B metabolism using stable isotope-labeled amino acids Am. J. Physiol.-Endocrinol. Metab. 270(6), E1022–E1036. Google Scholar
  22. DiStefano, J.J. 3rd, 1983. Complete parameter bounds and quasiidentifiability conditions for a class of unidentifiable linear systems. Math. Biosci. 65, 51–68. CrossRefzbMATHGoogle Scholar
  23. DiStefano, J.J. 3rd, Chen, B.C., Landaw, E.M., 1988. Pool size and mass flux bounds and quasiidentifiability relations for catenary models. Math. Biosci. 88, 1–14. CrossRefMathSciNetzbMATHGoogle Scholar
  24. Eisenfeld, J., 1996. Partial identification of underdetermined compartmental models: a method based on positive linear Lyapunov functions. Math. Biosci. 132, 111–140. CrossRefMathSciNetzbMATHGoogle Scholar
  25. Evans, N.D., Erlington, R.J., Shelley, M., Feeney, G.P., Chapman, M.J., Godfrey, K.R., Smith, P.J., Chappell, M.J., 2004. A mathematical model for the in vitro kinetics of the anti-cancer agent topotecan. Math. Biosci. 189, 185–217. CrossRefMathSciNetzbMATHGoogle Scholar
  26. Ferrannini, E., Smith, J.D., Cobelli, C., Toffolo, G., Pilo, A., DeFronzo, R.A., 1985. Effect of insulin on the distribution and disposition of glucose in man. J. Clin. Investig. 76, 357–364. CrossRefGoogle Scholar
  27. García-Meseguer, M.J., Vidal de Labra, J.A., García-Moreno, M., García-Cánovas, F., Havsteen, B.H., Varón, R., 2003. Mean residence times in linear compartmental systems. Symbolic formulae for their direct evaluation. Bull. Math. Biol. 65, 279–308. CrossRefGoogle Scholar
  28. Garlick, P.J., Mcnurlan, M.A., Essen, P., Wernerman, J., 1994. Measurement of tissue protein-synthesis rates in-vivo—a critical analysis of contrasting methods. Am. J. Physiol.-Endocrinol. Metab. 266(3), E287–E297. Google Scholar
  29. Gastaldelli, A., Schwarz, J.M., Caveggion, E., Traber, I.D., Traber, D.L., Rosenblatt, J., Toffolo, G., Cobelli, C., Wolfe, R.R., 1997. Glucose kinetics in interstitial fluid can be predicted by compartmental modeling. Am. J. Physiol.-Endocrinol. Metab. 272, E494–E505. Google Scholar
  30. Hart, H.E., 1955. Analysis of tracer experiments in non-conservative steady-state systems. Bull. Math. Biophys. 17, 87–94. CrossRefGoogle Scholar
  31. Hart, H.E., 1965. Determination of equilibrium constants and maximum binding capacities in complex in vitro systems: I. The mammillary system. Bull. Math. Biophys. 27, 87–98. CrossRefGoogle Scholar
  32. Hearon, J.Z., 1963. Theorems on linear systems. Ann. N. Y. Acad. Sci. 108, 36–68. CrossRefMathSciNetzbMATHGoogle Scholar
  33. Hearon, J.Z., 1974. A note on open linear systems. Bull. Math. Biol. 36, 97–99. MathSciNetzbMATHGoogle Scholar
  34. Jacquez, J.A., 1985a. Richard Bellman. Math. Biosci. 77, 1–4. CrossRefMathSciNetzbMATHGoogle Scholar
  35. Jacquez, J.A., 1985b. Compartmental Analysis in Biology and Medicine. University of Michigan, Ann Arbor. Google Scholar
  36. Jacquez, J.A., Simon, C.P., 1993. Qualitative theory of compartmental systems. SIAM Rev. 35, 43–79. CrossRefMathSciNetzbMATHGoogle Scholar
  37. Landaw, E.M., Chen, B.C., DiStefano, J.J. 3rd, 1984. An algorithm for the identifiable parameter combinations of the general mammillary compartmental model. Math. Biosci. 72, 199–212. CrossRefzbMATHGoogle Scholar
  38. Lindell, R., DiStefano, J.J. 3rd, Landaw, E.M., 1988. Statistical variability of parameter bounds for n-pool unidentifiable mammillary and catenary models. Math. Biosci. 91, 175–199. CrossRefzbMATHGoogle Scholar
  39. Nagashima, K., Lopez, C., Donovan, D., Ngai, C., Fontanez, N., Bensadoun, A., Fruchart-Najib, J., Holleran, S., Cohn, J.S., Ramakrishnan, R., Ginsberg, H.N., 2005. Effects of the PPARgamma agonist pioglitazone on lipoprotein metabolism in patients with type 2 diabetes mellitus. J. Clin. Investig. 115, 1323–1332. Google Scholar
  40. Packard, C.J., Demant, T., Stewart, J.P., Bedford, D., Caslake, M.J., Schwertfeger, G., Bedynek, A., Shepherd, J., Seidel, D., 2000. Apolipoprotein B metabolism and the distribution of VLDL and LDL subfractions. J. Lipid Res. 41(2), 305–317. Google Scholar
  41. Perl, W., Lassen, N.A., Effros, R.M., 1975. Matrix proof of flow, volume and mean transit time theorems for regional and compartmental systems. Bull. Math. Biol. 37, 573–588. zbMATHGoogle Scholar
  42. Pont, F., Duvillard, L., Verges, B., Gambert, P., 1998. Development of compartmental models in stable-isotope experiments—application to lipid metabolism Arterioscler. Thromb. Vasc. Biol. 18(6), 853–860. Google Scholar
  43. Ramakrishnan, R., 1984. An application of Berman’s work on pool-model invariants in analyzing indistinguishable models for whole-body cholesterol metabolism. Math. Biosci. 72, 373–385. CrossRefMathSciNetzbMATHGoogle Scholar
  44. Ramakrishnan, R., 2006. Studying apolipoprotein turnover with stable isotope tracers—correct analysis is by modeling enrichments. J. Lipid Res. 47, 2738–2753. CrossRefGoogle Scholar
  45. Ramakrishnan, R., Ramakrishnan, J.D., 2008. Utilizing mass measurements in tracer studies—a systematic approach to efficient modeling. Metab.-Clin. Exp. 57, 1078–1087. Google Scholar
  46. Ramakrishnan, R., Dell, R.B., Goodman, D.S., 1981. On determining the extent of side-pool synthesis in a three-pool model for whole body cholesterol kinetics. J. Lipid Res. 22, 1174–1180. Google Scholar
  47. Ramakrishnan, R., Leonard, E.F., Dell, R.B., 1984. A proof of the occupancy principle and the mean transit time theorem for compartmental models. Math. Biosci. 68, 121–136. CrossRefMathSciNetzbMATHGoogle Scholar
  48. Rescigno, A., Michels, L., 1973. Compartment modeling from tracer experiments. Bull. Math. Biol. 35, 245–257. Google Scholar
  49. Rescigno, A., Segre, G., 1964. On some topological properties of the systems of compartments. Bull. Math. Biophys. 26, 31–38. CrossRefMathSciNetzbMATHGoogle Scholar
  50. Rescigno, A., Segre, G., 1966. Drug and Tracer Kinetics. Blaisdell, Waltham. Google Scholar
  51. Rubinow, S.I., Winzer, A., 1971. Compartment analysis: an inverse problem. Math. Biosci. 11, 203–247. CrossRefMathSciNetzbMATHGoogle Scholar
  52. Shipley, R.A., Clark, R.E., 1972. Tracer Methods for Vivo Kinetics—Theory and Applications, Academic Press, New York. Google Scholar
  53. Tremblay, A.J., Lamarche, B., Cohn, J.S., Hogue, J.C., Couture, P., 2006. Effect of Ezetimibe on the in vivo kinetics of ApoB-48 and ApoB-100 in men with primary hypercholesterolemia. Arterioscler. Thromb. Vasc. Biol. 26(5), 1101–1106. CrossRefGoogle Scholar
  54. Vajda, S., 1984. Analysis of unique structural identifiability via submodels. Math. Biosci. 71, 125–146. CrossRefMathSciNetzbMATHGoogle Scholar
  55. Vajda, S., DiStefano, J.J. 3rd, Godfrey, K.R., Fagarasan, J., 1989. Parameter space boundaries for unidentifiable compartmental models. Math. Biosci. 97, 27–60. CrossRefMathSciNetzbMATHGoogle Scholar
  56. Vicini, P., Su, H., DiStefano, J.J. 3rd, 2000. Identifiability and interval identifiability of mammillary and catenary compartmental models with some known rate constants. Math. Biosci. 167, 145–161. CrossRefzbMATHGoogle Scholar
  57. Walter, E., 1987. Identifiability of Parametric Models. Pergamon, Oxford. zbMATHGoogle Scholar
  58. Zak, R., Martin, A.F., Blough, R., 1979. Assessment of protein turnover by use of radioisotopic tracers. Physiol. Rev. 59, 407–447. Google Scholar
  59. Zilversmit, D.B., 1960. The design and analysis of isotope experiments. Am. J. Med. 29, 832–848. CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2010

Authors and Affiliations

  • Rajasekhar Ramakrishnan
    • 1
    Email author
  • Janak D. Ramakrishnan
    • 2
  1. 1.Department of PediatricsColumbia University College of Physicians and SurgeonsNew YorkUSA
  2. 2.Department of Mathematics, Institut Camille JordanUniversité Lyon 1Villeurbanne cedexFrance

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