Abstract
A model for brain bilirubin uptake (BBU) predicts that BBU in jaundiced newborns typically depends on the plasma total bilirubin concentration (TBC) and the bilirubinalbumin dissociation rate constant (k_{1}) rather than the unbound bilirubin (B_{f}). The model's validity was tested by 1) evaluating its requirement that k_{3} >>> k_{2}, where k_{3} and k_{2} are the rate constants for BBU and B_{f}albumin association, respectively, and 2) determining whether the calculated BBU is ≤5% of the bilirubin production rate, the approximate BBU expected if brain bilirubin levels are <1% of the miscible bilirubin pool as reported in the literature. The model was investigated using peroxidase test measurements of TBC, B_{f}, k_{1}, and k_{2} from 185 jaundiced newborns. Mean k_{2} was compared with the reported k_{3} value of 0.08/s. BBU calculated from TBC and k_{1} was expected to be ≤0.005 μg/kg/s given the reported bilirubin production rate of 0.1 μg/kg/s. BBU calculated using B_{f} was also compared with the bilirubin production rate. The mean k_{2} of 8.9 L/μmol/s was greater than k_{3}, and the mean BBU of 0.72 μg/kg/s exceeded the expected range of ≤0.005 μg/kg/s. However, mean BBU using B_{f} (0.00073 μg/kg/s) was within the expected range. A mathematical model calculating BBU as a function of TBC and k_{1} could not be validated. BBU calculated from B_{f} is consistent with the observation that <1% of the miscible bilirubin pool is distributed in the brain.
Main
The rate of BBU in jaundiced newborns is generally thought to be a function of the plasma nonalbumin bound or “free” bilirubin (B_{f}), and current thinking is that this tiny fraction of the TBC is an important indicator of the risk of bilirubin toxicity (1–3). Recent publications by the American Academy of Pediatrics and the National Institutes of Health aimed at improving the clinical management of jaundiced newborns recommend, among other things, more research into the relationship between B_{f} and bilirubin toxicity (4–6).
B_{f} is a function of the TBC, the fraction (α) of TBC bound [α = (TBC − B_{f})/TBC ≈ 1 at clinically relevant TBC], the albumin concentration (A), and the bilirubinalbumin dissociation and association rate constants, k_{1} and k_{2}, respectively:
equation 1
Because B_{f} is < 0.1% of plasma bilirubin but only about 90% of plasma bilirubin may be bound to albumin in adults (7), A in equation 1 may be considered the “average” concentration of all the plasma species reversibly binding bilirubin (e.g., HDL) and k_{1} and k_{2} the “average” constants without fundamentally changing the arguments that follow. Several recent studies suggest that B_{f} as measured by the peroxidase test is indeed a more sensitive and specific indicator of bilirubin toxicity than the TBC alone (3,8–10).
Despite these considerations, the clinical management of jaundiced newborns is conventionally based on the less reliable but readily obtainable TBC (4). Although this approach has been questioned (11), there is an oftenoverlooked but important paper that supports, in theory, the use of the TBC in most clinical situations. About 18 y ago, Robinson and Rapoport applied a general mathematical model they had developed for brain uptake of protein bound drugs to BBU (12,13). Their model predicts that at TBC levels below about 454 μM (27 mg/dL), the plasma factors determining BBU are TBC and k_{1}. B_{f} becomes important only when the TBC is above 27 mg/dL or unusual circumstances are present (e.g., drugs that interfere with plasma bilirubin binding). Because k_{1} is a constant, BBU will therefore be directly proportional to TBC.
According to their model, as blood enters the cerebral capillaries, B_{f} is instantaneously taken up at a rate given by −k_{3} · B_{f}, where k_{3} is the net first order rate constant for brain uptake of bilirubin (they estimate k_{3} ≈0.08/s) (13). The instantaneous change in B_{f} caused by −k_{3} · B_{f} perturbs the equilibrium B_{f} (B_{feq}) at which dB_{feq}/dt is zero per equation 2 below, and B_{feq} rapidly falls to the steady state level B_{fss}, where dB_{fss}/dt remains close to zero (equation 3) for the remainder of the trip through the capillaries.
Since dB_{fss}/dt ≈ 0, equation 3 can be rearranged to give:
The general model (12) assumes that the perturbation of equilibrium (−k_{3} · B_{eq}) is of such magnitude that B_{fss} <<< B_{feq}, generating the inequalities k_{2} · [A − (α · TBC)] · B_{fss} <<< k_{1} · α · TBC and k_{2} · [A − (α · TBC)] <<< k_{3}. Under these circumstances, equation 4 becomes:
From equation 5, it follows directly that BBU = k_{3} · B_{fss} ≈ k_{1} · α · TBC, the basis of the Robinson and Rapoport model. By combining the capillary transit time, α (≈1), k_{1}, and k_{3} into the constant E (see equation 9 in “Methods”), which is the fraction of the TBC extracted during passage through the capillaries, they obtained equation 6, which gives BBU as a function of cerebral blood flow, the constant E, and the TBC (12,13):
The Robinson and Rapoport model for BBU has never been validated, but it exists in the literature as credible support (assuming little variability in the constants) for using only the TBC, at least when managing jaundiced but otherwise well term newborns whose TBC levels are below 27 mg/dL. In this study, we use measurements of TBC, B_{f}, k_{1}, and k_{2} from a diverse population of jaundiced newborns to investigate the validity of their model by testing 1) whether k_{2} · [A − (α · TBC)] <<< k_{3} is a valid assumption in the derivation of their model (see equations 4 and 5), and 2) whether BBU calculated by substituting actual TBC and k_{1} measurements into equation 6 yields BBU values that are about 5% of the reported rate of bilirubin production (14), a necessary condition if the BBU is to produce brain bilirubin levels that are <1% of the miscible bilirubin pool as reported in the literature (15,16).
METHODS
Population.
We reviewed the medical records of 256 newborns who received plasma TBC and B_{f} measurements as part of their evaluation for newborn jaundice to obtain a set of TBC values and their corresponding B_{f}, k_{1}, k_{2}, and albumin concentration for use in the study as outlined below. The review of medical records for this purpose was approved by the California Pacific Medical Center Institutional Review Board.
Hypotheses to Be Evaluated.
1) k_{2} <<< k_{3}. equation 5 depends on the inequality k_{2} · [A − (α · TBC)] <<< k_{3}, and Robinson and Rapoport estimate that k_{3} is about 0.08/s (13). Inasmuch as A − (α · TBC) will be numerically greater than 0.08 at all the TBC and albumin levels of the population, k_{2} · [A − (α · TBC)] <<< k_{3} will be true only if k_{2} <<< k_{3}. We therefore tested the validity of the inequality k_{2} · [A − (α · TBC)] <<< k_{3} by determining whether k_{2} was substantially less than 0.08. If the assumption is not valid and the hypothesis rejected, all the mass action factors (TBC, A, k_{1}, and k_{2}) are the relevant plasma factors determining B_{fss} per equation 4.
2) BBU calculated using TBC and k_{1} (equation 6) will be ≤0.005 μg/kg/s. The mean bilirubin production rate reported for well, term newborns is 8.5 mg/kg/24 h or 0.10 μg/kg/s (14). Since it is estimated that <1% of the miscible bilirubin pool distributes in the brain (15,16), the model, if valid, should provide rates of BBU somewhere in the neighborhood of 5% or less (≤0.005 μg/kg/s) of the bilirubin production rate. For example, assume BBU is 5% of the bilirubin production rate and that the bilirubin excretion rate is 60% of the production rate (14). If 10% of the cardiac output is directed to the brain and x mg of bilirubin are produced in a period of time, 0.005x will be taken up by the brain and 0.6x will be excreted. Of the 0.4x remaining in the miscible bilirubin pool, 1.25% (0.005/0.4) will reside in the brain.
Measurement of TBC, B_{f}, k_{1} and k_{2}.
TBC and the B_{f} were measured using an FDAapproved peroxidase method (Arrows UB Analyzer, Arrows Co., Ltd., Osaka, Japan) (17). HRP (EC 1.11.17) catalyzed oxidation of B_{f} by peroxide was measured at HRP concentrations of 23.8 μg/mL (0.54 μM) and 11.9 μg/mL to obtain B_{f1} and B_{f2}, respectively. Two HRP concentrations are used to avoid underestimating B_{feq} because, just as −k_{3} · B_{feq} perturbs B_{feq} during BBU (see equations 2 and 3), the onset of B_{f} oxidation, which is given by −k_{p} · [HRP] · B_{feq}, where k_{p} is the bilirubin oxidation rate constant per unit HRP concentration (18) and [HRP] is the HRP concentration, also perturbs the equilibrium. Substituting k_{p} · [HRP] for k_{3} in equation 4 gives (19):
In contrast to the requirement that k_{2} · [A − (α · TBC)] <<< k_{3} for the Robinson and Rapoport model (12), the peroxidase test requires kp · [HRP] <<< k_{2} · [A − (α · TBC)] if B_{fss} is to approximate B_{feq} (19). When B_{f2} > 1.05 · B_{f1} (5% is the error of the method) the required condition is not met and B_{eq} is underestimated at both peroxidase concentrations by at least 5%. Fortunately, the reciprocal of equation 7 provides a linear relationship between 1/B_{fss} and [HRP] with the reciprocal of the y intercept providing B_{fss} when [HRP] is zero, i.e., B_{feq}:
In this circumstance (and assuming α = 1), k_{1} can be obtained from the slope of equation 8 as k_{1} = k_{p}/(TBC · slope), following which k_{2} can be obtained from the intercept as k_{2} = intercept · k_{1} · TBC/(A −TBC) (19). The mean k_{p} for the stock peroxidase was 18/s per μmol/L of HRP (SD 1.1, n = 16).
An important caveat is that sample dilution alters bilirubinalbumin binding in plasma as well as purified albumin solutions (20,21). B_{feq} at the 42fold sample dilution used in this study is likely to be much smaller than B_{feq} in the undiluted sample, but it has been shown that the levels do correlate linearly (22). Dilution appears to primarily affect k_{2} rather than k_{1} (23) and because B_{f} is usually 2to 5fold lower at a 42fold sample dilution (20,21), the impact of an order of magnitude increasein B_{feq} or decrease ink_{2} on the results was taken into consideration when analyzing the data.
Calculation of BBU by equation 6 (the Robinson and Rapoport model).
BBU was calculated for each sample using equation 6. E was calculated from equation 9, which is derived from the general equation for E just above it using α = 1, the measured k_{1} for the sample, and the values provided by Robinson and Rapoport of 0.08/s for k_{3} and 1.75 s for the mean brain capillary transit time (τ) (13):
Inserting E for the sample along with the measured TBC and a cerebral blood flow of 0.5 mL/s/kg obtained from the reported blood flow rate 0.5 mL/s/100 g of brain (24) and a brain weight of 100 g/kg (25) into equation 6 provided BBU according to the Robinson and Rapoport model as:
Calculation of BBU using B_{f}.
If BBU is determined primarily by B_{f}, the instantaneous decrease in B_{feq} as blood enters the brain capillaries is given by:
Integrating equation 11 between the time that blood enters the capillary (t = 0) and the time it exits the capillary (τ) during which B_{feq} falls to B_{fss}, one obtains: equation 12
The fraction of B_{feq} removed on passing through the brain capillaries, comparable to E in equation 10, is:
Fraction of B_{f} extracted
If k_{3}=0.08/s and τ=1.75s, BBU is given as: equation 14
RESULTS
A total of 293 B_{f} measurements were made in a population of 256 jaundiced newborns. Of these, 198 samples from 185 newborns had B_{f2} levels that exceeded B_{f1} by 5% or more (mean 17%, SD 11%, range 6–69%). The mean birth weight and gestational age were 2401 g (SD = 1103, range 406–4483, n = 185) and 34 wk (SD 5, range 24–42 wk, n = 185), respectively. The mean TBC and B_{f} were 15.3 mg/dL (SD 7.0, range 2.8–34.2) and 1.12 μg/dL (SD 0.89, range 0.11–7.63), respectively. The mean B_{f1} and B_{f2} were 0.77 μg/dL (SD 0.55, range 0.09–3.83, n = 198) and 0.90 μg/dL, SD 0.66, range 0.10–5.1), respectively. Albumin was measured in 181 samples (91%), and the mean was 3.4 g/dL (SD 0.7, range 1.0–4.7). The various constants are given according to birth weight in Table 1.
Is k_{2} <<< k_{3}?
The mean k_{2} (Table 1) was 8.9 L/μmol/s, which is numerically about 100 times greater than k_{3} (0.08). Even if undiluted plasma has a mean k_{2} two orders of magnitude less (0.089 L/μmol/s), the assumption required by the Robinson and Rapoport model that k_{2} · [A − (α · TBC)] <<< k_{3} will not be valid. Review of the literature provided no evidence that k_{3} might be substantially larger than 0.08/s (26), and assuming A underestimates the total concentration of bilirubin binding sites only reinforces the conclusion that the assumption is not valid.
Is BBU ≤0.005 μg/kg/s?
The mean k_{1} (Table 1) was 0.093/s. E calculated by equation 9 ranged from 0.001 to 0.036 (mean 0.010, SD 0.006), indicating that as little as 0.1% to as much as 3.6% of the TBC according to the model would be extracted as blood passed through the cerebral capillaries.
The mean BBU calculated using equation 10 (Robinson and Rapoport model) was 0.72 μg/kg/s (SD 0.52, range 0.057–2.7), with the calculated BBU exceeding the reported bilirubin production rate of 0.1 μg/kg/s in 193 of 198 samples (97%). Assuming BBU equals the bilirubin production rate (it could not proceed any faster), about 25% of the miscible bilirubin pool rather than the reported <1% should be found in the brain.
The mean BBU as a function of B_{f} (equation 15) was 0.00073 μg/kg/s (SD 0.00058, range 0.000073–0.0050), which is 0.73% of the bilirubin production rate and much more in line with the anticipated BBU of ≤0.005 μg/kg/s. Interestingly, if B_{f} is actually an order of magnitude higher in undiluted plasma (20–22), BBU would be about 7% of the bilirubin production rate and about 1.8% of the miscible bilirubin pool would be expected to reside in the brain. This is still very close to the value expected, and BBU by the total bilirubin model would either be the same or increase depending on the degree to which k_{1} is affected by dilution (k_{2}/k_{1} decreases as B_{f} increases requiring some combination of a decrease in k_{2} and/or increase in k_{1}). Even if k_{2} is an order of magnitude smaller, it still is 10fold higher than k_{3}.
DISCUSSION
Our results show that the Robinson and Rapoport mathematical model predicting that BBU will be largely a function of the TBC at TBC levels below 27 mg/dL is invalid when measured values for TBC and the bilirubinalbumin dissociation rate constant (k_{1}) along with constants from the literature for brain bilirubin blood flow, capillary transit time (τ), and brain bilirubin uptake (k_{3}) are applied to the model. B_{feq} when k_{3} = 0.08/s and τ = 1.75 s will not fall sufficiently in the cerebral capillaries to meet the assumptions required for equation 5 to be operative. For the Robinson and Rapoport model to be operative, the expression ${\text{e}}^{{\text{k}}_{3}\xb7\text{\tau}}$ in equation 13 must be nearly zero, indicating that virtually 100% of the B_{f} is taken up within a few milliseconds of the blood entering the capillary. This requires −k_{3} · τ to be much greater than previously reported. Since τ is unlikely significantly greater, k_{3} in humans would need to be orders of magnitude larger than the values reported in animals. Even assuming k_{3} = 80/s (1000fold higher than the value in animals), at the mean k_{2} of our study (8.9 L/μmol/s), a TBC of 10 mg/dL (171 μM), and the mean study albumin of 3.4 g/dL (514 μM), k_{2} · [A − (α · TBC)] in equation 4 = 3052.7/s versus a k_{3} = 80/s, which clearly invalidates the assumption in equation 4 required by the Robinson and Rapoport model that k_{2} · [A − (α · TBC)] is so small relative to k_{3} that it can be ignored giving equation 5. Thus, all the plasma mass action variables determining BBU must be operative (TBC, A, k_{1}, and k_{2}), and because the fraction of bilirubin extracted during brain capillary transit is constant (equation 15), the overall amount of bilirubin taken up by the brain during capillary transit will be a function of B_{feq}.
Our bilirubinalbumin binding data were from a diverse population of jaundiced newborns to provide a wide range of binding variables for the models, and because it was not possible to measure the bilirubin production rate, bilirubin pool distribution, and cerebral blood flow for each newborn, we applied our best estimate of each variable to the population. Variation in cerebral blood flow would apply to both models equally (equations 10 and 14) and not change the results. Because BBU by the Robinson and Rapoport model is 1000fold greater than that by the B_{f} model, the BBU as a percentage of bilirubin production rate would not change greatly with variation in bilirubin production (even at triple the bilirubin production rate as occurs in hemolysis (14) BBU by the Robinson and Rapoport model still exceeds the production rate and BBU by the B_{f} model remains below 1% of the production rate). The percentage of the miscible bilirubin pool residing in the brain could also be in error by an order of magnitude (10% versus <1%) and BBU by the Robinson and Rapoport model would still far exceed the BBU predicted.
Considering the above, it is not unexpected that the BBU calculated according to the model (equation 10) was far too high relative to the bilirubin production rate for the model to be considered an accurate representation of BBU. BBU rates calculated using B_{f} as the operative plasma species (equation 15), however, are within the range expected assuming the reports that < 1% of the miscible bilirubin pool distributes in the brain are reasonably accurate and applicable to human newborns (15,16). Although sample dilution may systematically alter the binding variables (20–23), even when order of magnitude errors in the variables are invoked, the Robinson and Rapoport model still fails.
It is ironic that were the model correct, the variability in k_{1} (Table 1) would still require routine measurement of bilirubinalbumin binding, but the desired result would usually be k_{1} (to calculate E) rather than B_{f}. This paradoxical “inconstancy” of constants (Table 1) is a subtle but important reminder that there is considerable variability in plasma bilirubin binding in jaundiced newborns. If not, B_{f} could be calculated from the TBC and albumin, assuming albumin binds most of the bilirubin in plasma (7) We found a 73% coefficient of variation in k_{1} and a 158% variation in the equilibrium association constant (k_{2}/k_{1}). Cashore (27), using the peroxidase test and a single peroxidase concentration, has reported a 40% coefficient of variation for k_{2}/k_{1} in well term newborns. The reasons for this variability are mostly unknown, but albumin dimerization and polymorphism (28) and small competing molecules generated during illness (29) may contribute. Variability in the nonplasma factors that can alter individual susceptibility to bilirubin toxicity, including interindividual variability in BBU, also deserve more study (30).
The Robinson and Rapoport model cannot be used as evidence supporting the use of the TBC alone when evaluating jaundiced newborns in any circumstances. Our data suggest that even if significant nonalbumin binding of bilirubin in plasma occurs, B_{f} as measured by the peroxidase test provides rates of BBU consistent with the reported bilirubin production rate and miscible bilirubin pool distribution. Continued investigation of the usefulness of B_{f} measured in minimally diluted plasma (22) as an adjunct in the evaluation of newborn jaundice seems warranted.
Abbreviations
 A:

plasma albumin concentration
 α:

fraction of the TBC bound = (TBC–B_{f})/TBC
 BBU:

brain bilirubin uptake
 B_{f}:

plasma free bilirubin concentration
 B_{feq}:

equilibrium concentration of free bilirubin
 B_{fss}:

steady state concentration of free bilirubin
 E:

TBC extraction fraction
 HRP:

horseradish peroxidase
 k_{1}:

bilirubinalbumin dissociation rate constant
 k_{2}:

bilirubinalbumin association rate constant
 k_{3}:

rate constant for the uptake of B_{f} by the brain
 k_{p}:

rate constant for the peroxidase catalyzed oxidation of bilirubin by peroxide
 τ:

brain capillary transit time
 TBC:

plasma total bilirubin concentration
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Ahlfors, C., Parker, A. Evaluation of a Model for Brain Bilirubin Uptake in Jaundiced Newborns. Pediatr Res 58, 1175–1179 (2005). https://doi.org/10.1203/01.pdr.0000185248.43044.cd
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