Organ-Resolved Glucose Mass Balances

Why Resolve Glucose by Organ?

A plasma glucose concentration is the result of several simultaneous flows. Glucose can enter plasma from endogenous production and, after a meal, from the gut; it can leave plasma through hepatic uptake and peripheral-tissue uptake. The same concentration change can therefore arise from different combinations of organ fluxes. A mass balance makes these contributions explicit without, by itself, specifying the regulatory mechanisms that control them.

The central bookkeeping distinction is between production within an organ, transport between compartments, and utilization or storage within a compartment. For example, endogenous glucose formed in the liver is not automatically identical to glucose released by the liver into plasma: some may be used or retained within the liver. Keeping these terms separate makes the balance auditable.

Three Compartments and Their Fluxes

Let \(M_P\), \(M_L\), and \(M_T\) denote glucose amounts in the plasma, liver, and peripheral-tissue compartments, respectively. These are amounts, not concentrations. If a compartment has fixed volume \(V_i\), its concentration is \(C_i=M_i/V_i\); the volume must be specified before translating a change in amount into a concentration change.

Use the following fluxes, all expressed per kg body weight per minute:

• \(E_H\): endogenous glucose formation within the liver compartment, entering its exchangeable glucose pool.
• \(J_{LP}\): glucose transported from liver to plasma; \(J_{PL}\): glucose transported from plasma to liver.
• \(A_G\): glucose appearance from the gut into plasma. It is zero in the fasting examples below and may be positive in a fed state.
• \(J_{PT}\): glucose uptake from plasma into peripheral tissues.
• \(S_H\): hepatic glucose utilization or retention out of the exchangeable hepatic pool.
• \(R_T\): peripheral glucose utilization out of the modeled tissue glucose pool.

The uptake terms \(J_{PL}\) and \(J_{PT}\) describe movement into organs, not the subsequent fate of glucose after entry. \(S_H\) and \(R_T\) account for modeled removal from the exchangeable liver and tissue pools. These balances do not require a particular kinetic form for any flux: rates may be constant over a short interval or depend on time and system state.

Each term appears with a sign determined by the compartment being balanced. Liver-to-plasma transport is a loss for the liver and a gain for plasma; plasma-to-liver transport has the opposite signs. Peripheral uptake is a plasma loss and a tissue gain. Endogenous formation and gut appearance are external inputs to this three-compartment representation, while hepatic and tissue utilization are exits from the represented exchangeable pools.

Steady State, Transients, and a Whole-System Check

A compartment is at steady state when its amount does not change, so its net inflow equals its net outflow. This is a condition on the balance at a particular state, not an assertion that every individual flux is zero. In a transient, the difference between inflows and outflows determines the direction and rate of change.

Adding the three equations gives a useful consistency check. Each internal transport term appears once as an inflow and once as an outflow, so the terms cancel:

This result follows directly by addition: \(+J_{PL}\) cancels \(-J_{PL}\), \(+J_{LP}\) cancels \(-J_{LP}\), and \(+J_{PT}\) cancels \(-J_{PT}\). Internal transport can redistribute glucose and change individual compartment amounts, but cannot by itself change the total amount across these three compartments. The total changes only through inputs and exits at the boundary of the modeled system.

Fed-State Inputs and Transient Changes

In a fed-state balance, gut appearance \(A_G\) can be positive. This adds glucose to plasma, but it does not imply that plasma glucose must rise: the amount rises only if total inflow exceeds total outflow at that time. Similarly, positive hepatic production does not force hepatic release to equal production, because hepatic uptake and removal also affect the liver pool.

Short Derivation: Why Internal Fluxes Cancel

The cancellation in the whole-system equation is a consequence of defining compartments consistently. To see this, sum the organ equations without changing any signs:

Collecting like terms leaves \(E_H+A_G-S_H-R_T\), because the three internal transfers have zero net contribution to the sum. If an internal flux fails to cancel, either its direction or its sign has been assigned inconsistently, or the compartment receiving or supplying it has been omitted. This is a practical way to detect bookkeeping errors before interpreting model behavior.

Diagnostics, Alternatives, and Limitations

Several checks help identify a balance that is going wrong:

• Check units and scaling. Every term in a compartment equation must have the same units, such as mg/kg/min. Do not add a concentration directly to a mass flux.
• Check directions and signs. A flux from plasma to liver must be negative in the plasma equation and positive in the liver equation.
• Check the system boundary. Gut appearance is external to the three compartments, while liver–plasma transport is internal. Treating an internal transfer as a system input double-counts glucose.
• Check amount and concentration assumptions. A positive \(dM_P/dt\) indicates increasing plasma amount. It implies increasing concentration only if plasma volume is fixed.
• Check the whole-system sum. Its derivative must agree with external inputs minus exits. A mismatch points to a missing, duplicated, or mis-signed flux.

A lumped one-compartment model can be useful when only plasma glucose is needed and organ fluxes are not the question; it summarizes net appearance and disappearance without locating them. The organ-resolved model instead distinguishes where formation, uptake, and utilization occur, but requires more information to estimate its separate fluxes. A more elaborate model could subdivide organs or add storage pools, but that changes the system boundary and requires additional balance equations and measurements.

These equations are bookkeeping identities conditional on the chosen compartments and flux definitions. They do not determine the flux values or explain how they respond to hormones, meals, or disease. In particular, a single plasma concentration measurement cannot, by itself, identify all the simultaneous organ fluxes. More measurements or justified model structure are needed to distinguish flux combinations that produce the same net plasma balance.