In total, 20 neonates who received allopurinol and 17 neonates who received mannitol were included, and 97 allopurinol, 97 oxypurinol, 165 hypoxanthine, 159 xanthine, and 171 uric acid observations were analyzed, respectively. The observed concentration–time data of hypoxanthine, xanthine, and uric acid are shown in Fig. 1. Overall, 26 neonates were classified as moderate-to-severe HIE and underwent TH treatment. Details on the dataset and patient characteristics are summarized in Table 1.

Observed concentration–time profiles

The model described by Eqs. 1–5 was applied to the data but overpredicted the endogenous baseline of hypoxanthine and xanthine. As can be observed in Fig. 1, steady-state concentrations of hypoxanthine and xanthine were not reached within the observation period, while the model predicted the steady state being reached. Hypoxanthine and xanthine are products of purine nucleotides and nucleosides, which are highly dynamic under a hypoxic status owing to a substantial AMP breakdown [27]. In hypoxic animals, a dramatic increase in purine precursors (including purine nucleotides such as inosine monophosphate, guanosine monophosphate, and purine nucleosides such as inosine and guanosine) following a rapid decline has been identified [28, 29]. Therefore, the influence of dynamic purine precursors on the synthesis of hypoxanthine and xanthine was investigated.

An empirical compartment was added to the model, representing pooled precursors in purine metabolism. As no measurement was available for the precursors, the initial amount in compartment was assumed to be 1, and all relevant rate process were therefore, relative to the rate at time 0. At equilibrium, the turnover of precursors compartment composed of a zero-order synthesis rate constant (kin3) and a first-order elimination rate constant (k30) as described in Eq. 6.

$$\frac}} \right)}} = \frac} = k_}}} - k_ *A\left( 3 \right).$$

(6)

This equation could be further simplified given that the initial amount of precursors is dominated by the hypoxic state, and is extensively higher than the physiological steady-state (\(\frac3}\)) [30]. Therefore, kin3 was omitted resulting in a net first-order decline in total amount over time during the observation period as shown in Eq. 7:

$$\frac}} \right)}} = \frac} = - k_ *A\left( 3 \right).$$

(7)

To describe the impact of upstream precursor exhaustion on downstream hypoxanthine and xanthine synthesis, kin4 and kin5 were further regulated by the amount in the pooled precursors compartment [A(3)] as described in Eqs. 8–9:

$$\frac}} \right)}} = \frac} = k_ *A\left( 3 \right) - k_ *A\left( 4 \right)*} - k_ *A\left( 4 \right),$$

(8)

$$\frac}} \right)}} = \frac} = k_ *A\left( 3 \right) + k_ *A\left( 4 \right)*} - k_ *A\left( 5 \right)*}.$$

(9)

Incorporating the precursors effect in the model improved the description of hypoxanthine and xanthine endogenous turnover. Estimated k30 was 0.0043 1/h, implying that the production of hypoxanthine and xanthine was reduced by 0.43% per hour because of precursor exhaustion after hypoxia.

The final structure of the model is depicted in Fig. 2. All PK and PD parameters were estimated simultaneously, and the final estimates with corresponding 95% confidence intervals are shown in Table 2. In the final model, PK parameters were comparable to our previous estimations. Allopurinol clearance and volume of distribution was 0.74 L/h and 2.8 L, respectively. Oxypurinol clearance and volume of distribution relative to a formation fraction were 0.3 L/h and 10.39 L, respectively. The auto-inhibition of allopurinol-oxypurinol metabolism achieved a half-maximal effect under an oxypurinol concentration of 2.96 mg/L.

Final model structure. Gray shades, pharmacokinetic model of allopurinol and oxypurinol with an autoinhibition effect on allopurinol metabolism by oxypurinol. Red broken line, xanthine oxidase inhibition by allopurinol and oxypurinol (EFF). Blue broken line, the impact of purine nucleotides and nucleoside level on the generation of hypoxanthine and xanthine. EBASE4 hypoxanthine baseline value, EBASE5 xanthine baseline value, EBASE6 uric acid baseline value, Init4D disease-related hypoxanthine initial value, Init5D disease-related xanthine initial value, Init6D disease-related uric acid initial value, k30 precursors clearance, k40 hypoxanthine salvage clearance, k45 hypoxanthine to xanthine clearance, k56 xanthine to uric acid clearance, k60 uric acid clearance, kin4 hypoxanthine synthesis rate, kin5 xanthine synthesis rate, HXA hypoxanthine, XA xanthine, UA uric acid, XOR xanthine oxidoreductase.

In the final PD model, the estimated baseline production of hypoxanthine (kin4) and xanthine (kin5) was 0.49 mg/L*h and 0.66 mg/L*h, respectively. Hypoxanthine was readily salvaged or degraded to xanthine with rate constants of 0.5 1/h and 0.2 1/h, respectively. In the allopurinol and the mannitol group, the salvage pathway accounted for 75% and 68% of total hypoxanthine clearance, respectively. High disease-related initial level of hypoxanthine (Init4D), xanthine (Init5D), and uric acid (Init6D) was estimated (0.45 mg/L, 0.59 mg/L and 37.98 mg/L, respectively). The effect of drug treatment on XOR inhibition was described, where the half-maximal XOR inhibition (IC50) was achieved with a combined allopurinol and oxypurinol concentration of 0.68 mg/L. This IC50 was below all observed concentrations within 72 hours post-dose.

High IIVs in the disease-related initial level of hypoxanthine (Init4D), xanthine (Init5D), and uric acid (Init6D) were found. A trend towards higher disease-related hypoxanthine, xanthine, and uric acid estimates with a more severe HIE status, indicated by T-score or TH, was seen in the graphical analysis (Fig. 1 of the ESM). Furthermore, patients with very high LDH (> 2500 U/L) and ALT (> 100 U/L) values, suggesting more serious organ damage, demonstrated higher disease-related hypoxanthine and xanthine values (Fig. 2 of the ESM). The T-score and TH were therefore tested as binary covariates on Init4D, Init5D, and Init6D.

Adding TH as a binary covariate on Init4D and Init5D significantly improved the model goodness of fit with a OFV drop of 5.05 units, and reduced IIV in Init4D and Init5D by more than 35%. Neonates with moderate-to-severe HIE (thus treated with TH) had 2.9-fold and 1.2-fold higher Init4D and Init5D than neonates with mild HIE, respectively. Init6D was not significantly different between the TH and non-TH group. A correlation of 1 between IIV in Init4D and Init5D was estimated, which resulted in a 20-unit drop in OFV. To reduce model complexity, the same ETA distribution was assigned for IIV in Init4D and in Init5D with a scaling factor of 0.66 estimated for the latter. The correlation between either IIV Init4D or IIV Init5D and IIV Init6D was below 0.35; therefore, it was not included in the final model. As hypoxanthine and xanthine are derived from purine nucleotides inosine monophosphate and guanosine monophosphate, respectively, the strong correlation between IIV in Init4D and Init5D might indicate an impact of the hypoxic event.

The final model provided an adequate description for the data from both the allopurinol and mannitol groups. No substantial trend or bias was observed in the observed versus population-predicted and individual-predicted concentration plots as well as the conditional weighted residual versus population-predicted concentration and time plots (Fig. 3). In the visual predictive check, no considerable misspecification was observed (Fig. 4).

Goodness of fit. hr hours

Visual predictive checks. HIE hypoxic-ischemic encephalopathy, hrs hours