Using mass measurements in tracer studies--a systematic approach to efficient modeling

Abstract

Tracer enrichment data are fitted by multicompartmental models to estimate rate constants and fluxes or transport rates. In apolipoprotein turnover studies, mass measurements are also available, for example, apolipoprotein B levels in very low-density lipoprotein, intermediate-density lipoprotein, and low-density lipoprotein, and are often essential to calculate some of the rate constants. The usual method to use mass measurements is to estimate pool masses along with rate constants. A systematic alternative approach is developed to use flux balances around pools to express some rate constants in terms of the other rate constants and the measured masses. The resulting reduction in the number of parameters to be estimated makes the modeling more efficient. In models that would be unidentifiable without mass measurements, the usual approach and the proposed approach yield identical results. In a simple two-pool model, the number of unknown parameters is reduced from 4 to 2. In a published five-pool model for apolipoprotein B kinetics with three mass measurements, the number of parameters is reduced from 12 to 9. With m mass measurements, the number of responses to be fitted and the number of parameters to be estimated are each reduced by m, a simplification by 1/4 to 1/3 in a typical pool model. Besides a proportionate reduction in computational effort, there is a further benefit because the dimensionality of the problem is also decreased significantly, which means ease of convergence and a smaller likelihood of suboptimal solutions. Although our approach is conceptually straightforward, the dependencies get considerably more complex with increasing model size. To generate dependency definitions automatically, a Web-accessible program is available at http://biomath.info/poolfit/constraints.

Figures

Figure 1

An extremely simple model for apoB turnover in VLDL and IDL with a labeled amino acid precursor as tracer. The square P denotes the precursor system, consisting of one or more pools. Pool 1 is for VLDL apoB and pool 2 for IDL apoB. The k symbols denote rate constants for the pathways, defined as fluxes divided by the source pool mass, with the first subscript indicating the destination and the second the source of the pathway. Thus, k02 is IDL apoB FCR, the rate constant into 0 from 2, equal to the mass flux to the outside from 2 (IDL) divided by Q2, the mass of the source pool 2 (IDL). The extension to this definition is that, when describing synthesis or entry from the outside, the second subscript is zero or P, and the division is by the destination pool mass. Thus, the FSR of VLDL apoB is k1P, the flux into 1 (VLDL) from P (the precursor) divided by Q1, the mass of the destination pool. A: The model has a direct removal pathway from VLDL (k01), making the relationship between the two masses an identifying constraint. B: The model has no direct removal pathway from VLDL (k01=0), making the relationship between the two masses a nonidentifying constraint.

Figure 2

A five-pool model for apoB turnover, used in Nagashima et al. [28]. Pools 1, 2 and 3 constitute VLDL apoB, pool 4 is IDL apoB, and pool 5 is LDL apoB. The k symbols are as defined in Figure 1.