We constructed a stock-flow model in Insight Maker [12], which is redrawn in Fig. 1, to simulate two-compartment urea kinetics during dialysis. The model comprises four stocks and five flows.

The stocks are: intracellular volume, extracellular volume, intracellular urea, and extracellular urea. The flows are: urea generation rate, inter-compartmental urea transfer rate, urea elimination rate, inter-compartmental volume flow rate, and dialyzer ultrafiltration rate. The intracellular and extracellular urea concentrations are calculated dynamically during model simulation from the above stocks, as shown in Fig. 1.

Redrawn System Dynamics stock-flow diagram of the dual-compartment hemodialysis model created in Insight Maker

In keeping with the V–A model, we examined three variables for parameter estimation, namely: a urea compartmental mass transfer coefficient (\(\phi\)) expressed as a volumetric clearance in L/h, dialyzer clearance (K) in L/h, and urea generation rate (G) in mmol/h.

To compare the performance of our SD model with the V–A model, we used the same constants as Sano et al. [8] (expressed in alternative units), namely: intracellular volume 21.6 L, extracellular volume 14.4 L, inter-compartmental volume flow rate 0.72 L/ h, and dialyzer ultrafiltration rate 1.2 L/h. In our System Dynamics model, we did not include bulk flow of urea with the ultrafiltration.

A converter holds the measured clinical urea data, [\(U_\textrm\)] [8, 9] and a simple cost function to be minimized during optimization is defined as, \(\sum (]} - ]})^2\), where [\(U_\textrm\)] is our model output for extracellular urea concentration. Insight Maker performs a direct search optimization which is an adaptation of Powell’s method [12].

We digitized [13] the BUN data for the three patients [9] reported in Sano et al. [8], and converted these to urea concentrations in mmol/L. Approval for this secondary use of data was obtained from our institution’s Human Research Ethics Committee (Medical) (Ref. W-CBP-230315-01). We calibrated our model in Insight Maker by minimizing the cost function, to achieve a best-fit parameter estimation.

To remain consistent with the study by Sano et al. [8], we chose to optimize parameters G, \(\phi\), and K, which they term S, Ah, and K, respectively. The simulation was run for a 5 h time horizon with dialysis running for 4 h. Euler’s method with time increments of 0.001 h was used.

We ran a sensitivity analysis in Insight Maker for extracellular urea concentration for each patient by applying a uniform distribution of values for each of the three free parameters spanning the ± 50% range of the optimized values. The 50% and 95% confidence bounds for urea were plotted for each parameter as shown in the Supplementary Material.