Approximate Cell-State Aggregation with Preserved Outputs

Why Approximate Cell-State Aggregation?

A mechanistic model may distinguish several cell states whose transition behavior is similar. Keeping every state can make a model harder to interpret and simulate, but merging states changes the dynamics unless the states satisfy exact aggregation conditions. Approximate aggregation makes this trade-off explicit: it replaces detailed states with a smaller set of aggregate counts, then assesses how much the aggregate trajectories and selected outputs differ from the original model.

The aim is not to claim that merged states are biologically identical. Rather, the reduction is useful when the approximation error is acceptably small for the quantities and conditions of interest. A reduction that preserves total cell count may still distort a biomarker, and a reduction that agrees at the initial time may drift later as the composition of a merged group changes.

Aggregation and Approximate Closure

For the approximation to be meaningful, the original and reduced models must use the same inputs and parameter values, and the initial reduced state should agree with the aggregation of the original initial state: \(\hat z(0)=Px(0)\). The original model should have well-defined solutions over the time interval being assessed. If counts represent populations, the original transition rates and the reduced rates should also be biologically admissible, typically nonnegative, and the reduction should not create negative counts.

In exact state aggregation, the aggregate derivative depends only on the aggregate state, so one can choose \(g\) such that \(P f(x,u,\theta)=g(Px,u,\theta)\) for every admissible \(x\). Approximate aggregation relaxes that equality. A common construction uses transition rates averaged over the initial within-group composition. This matches the initial aggregate slope, but generally does not guarantee a match later, when the relative abundance of the merged states changes.

Constructing a Reduced Transition Rate

Consider \(r\) cell states in one group. Suppose state \(i\) leaves the group at rate \(k_i\) per unit time and has count \(x_i\). The exact aggregate count \(z=\sum_{i=1}^{r}x_i\) has derivative \(-\sum_i k_i x_i\), apart from any inflows. If the within-group fractions are approximately stable at weights \(w_i\), with \(w_i\geq 0\) and \(\sum_i w_i=1\), a natural reduced exit rate is \(\bar k=\sum_i w_i k_i\). The reduced outflow is then \(\bar k z\).

When the weights are taken from the initial composition, \(w_i=x_i(0)/z(0)\), the initial slopes agree:

This matching is local in time. If states with faster exit rates become depleted more quickly, the actual composition shifts toward slower-exiting states, and the fixed value \(\bar k\) may no longer represent the aggregate rate.

Justifying and Bounding the Aggregate Error

A useful way to assess a reduction is to separate the error caused by imperfect closure from the error that accumulates while the reduced trajectory evolves. Define the closure residual at the exact state as \(\rho(t)=P f(x(t),u,\theta)-g(Px(t),u,\theta)\), and let \(e(t)=\hat z(t)-z(t)\). The residual is the mismatch between the exact aggregate derivative and the derivative prescribed by the reduced model at the exact aggregate state.

Suppose \(g\) is Lipschitz in its state argument with constant \(L\) on the region considered, meaning \(\|g(a)-g(b)\|\leq L\|a-b\|\), and suppose \(\|\rho(t)\|\leq\varepsilon\) over \(0\leq t\leq T\). With matching initial conditions, \(e(0)=0\). Since \(\hat z'=g(\hat z)\) and \(z'=g(z)+\rho\), subtracting gives \(e'=g(\hat z)-g(z)-\rho\). The triangle inequality then yields \(\|e'\|\leq L\|e\|+\varepsilon\). Integrating this differential inequality gives the bound below.

This is a conservative bound, not a prediction that the actual error will reach its upper limit. It shows why even a small closure mismatch can matter over a long interval or in a reduced system with strongly amplifying dynamics.

Preserving Selected Biomarker Outputs

A cell-count aggregation does not automatically preserve a biomarker. Suppose a selected output is a linear production rate \(y=c^\mathsf{T}x\), where \(c_i\) is the per-cell output rate for state \(i\). If all states in an aggregate group have the same coefficient \(c_i=q\), their contribution is exactly \(qz\). The output can then be calculated from the aggregate count without knowing the within-group composition. If the coefficients differ, the group total alone does not generally determine the output.

More generally, an output-preserving aggregate representation requires a reduced output map \(\hat y=q^\mathsf{T}\hat z\) that matches the original output over the relevant states, exactly or within a stated tolerance. For any chosen map, the output error can be separated into a within-group output mismatch and a trajectory mismatch:

The first term measures whether the output can be represented from the exact aggregate counts; the second reflects error in the reduced aggregate trajectory. Making only the second term small is insufficient when distinct cell states produce different amounts of the biomarker.

Composition Drift, Diagnostics, and Alternatives

Composition drift is a central failure mode. The initial weighted rate matches the initial slope, but cells with faster transitions leave their group more rapidly. The within-group weights therefore change, and a fixed average rate may cease to describe the group. The following hypothetical calculation shows that the fitted rate depends on the starting composition.

Diagnostics should be tied to the intended use of the reduced model. Compare exact and reduced aggregate trajectories over the time interval of interest, inspect the closure residual across plausible within-group compositions, and calculate errors in each selected output with its own units and tolerance. Check that cell counts remain nonnegative and that conserved totals or required transition balances are maintained. A small error at one time point does not establish a small error throughout a trajectory.

Approximate aggregation is distinct from exact aggregation through lumpability: exact aggregation satisfies the closure equality for every admissible within-group composition, whereas approximate aggregation accepts a residual and evaluates its consequences. Retaining the original states avoids this particular aggregation error but keeps the model larger. Splitting a problematic group or tracking an additional composition variable can reduce error, at the cost of a less compact model. These choices should be based on which outputs and conditions the model must represent.