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

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Abstract

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.

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Abbreviations

F :

flux

f :

vector of rate constants for entry from outside

FCR:

fractional catabolic rate

FSR:

fractional synthetic rate

g :

vector of rate constants for exit to outside

K,R:

system matrix of rate constants

k,r:

rate constant

P :

diagonal matrix of pool masses

Q :

total mass or activity of tracer+tracee

q :

mass or activity of tracer

q-model:

model for tracer activities or amounts

S :

synthetic flux

s :

synthetic rate constant

t :

time

u :

a vector whose every element is 1

w :

precursor enrichment

y :

tracer enrichment

y-model:

model for tracer enrichments or specific activities

z :

observed tracer enrichment

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Authors and Affiliations

  1. Department of Pediatrics, Columbia University College of Physicians and Surgeons, New York, NY, 10032, USA

    Rajasekhar Ramakrishnan

  2. Department of Mathematics, Institut Camille Jordan, Université Lyon 1, 69622, Villeurbanne cedex, France

    Janak D. Ramakrishnan

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  1. Rajasekhar Ramakrishnan
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  2. Janak D. Ramakrishnan
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Correspondence to Rajasekhar Ramakrishnan.

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Ramakrishnan, R., Ramakrishnan, J.D. A State Space Transformation Can Yield Identifiable Models for Tracer Kinetic Studies with Enrichment Data. Bull. Math. Biol. 72, 2019–2046 (2010). https://doi.org/10.1007/s11538-010-9522-7

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