Modelling oral adrenal cortisol support
David J. Smith, Alessandro Prete, Angela E. Taylor, Niki Karavitaki, Wiebke Arlt
MModelling oral adrenal cortisol support
David J. Smith ∗ , Alessandro Prete , Angela E. Taylor ,Niki Karavitaki and Wiebke Arlt School of Mathematics & Institute for Metabolism and Systems Research,University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK, Centre for Endocrinology, Diabetes and Metabolism, Birmingham Health Partners,Birmingham, B15 2GW, UK.
Abstract
A simplified mathematical model of oral hydrocortisone delivery in adrenal insufficiency isdescribed; the model is based on three components (gastric hydrocortisone, free serum cortisoland bound serum cortisol) and is formulated in terms of linear kinetics, taking into accountthe dynamics of glucocorticoid–protein binding. Motivated by the need to optimise cortisolreplacement in the situations of COVID-19 infection, the model is fitted to recently-publisheddata on 50 mg dosing and earlier data on 10 mg dosing. The fitted model is used to predicttypical responses to standard dosing regimes, which involve a larger dose in the morning and 1or 2 smaller doses later in the day, and the same regimes with doses doubled. In all cases thereis a circadian-like response, with early morning nadir. The model is also used to consider analternative dosing strategy based on four equal and equally-spaced doses of 10, 20 or 30 mgper 24 h, resulting in a more even response resembling a response to sustained inflammatorystress.
Background
The Prevention of Adrenal Crisis in Stress (PACS) study [1] aimed to identify the parenteral hy-drocortisone dose and administration mode most suitable for glucocorticoid stress dose cover inpatients with adrenal insufficiency exposed to major stress, such as trauma, surgery or sepsis. Thisincluded experimental data on serum cortisol levels measured by mass spectrometry after adminis-tering 200mg hydrocortisone in four different administration modes (50 mg qds orally or via bolusintramuscular or intravenous injections, and continuous intravenous infusion of 200 mg/24 h). Thatpaper included a model of intravenous hydrocortisone administration and clearance, from whichpredictions could be made, identifying an initial bolus of 50mg or 100 mg hydrocortisone followedby continuous intravenous infusion of 200 mg hydrocortisone per 24 hours as the most appropriateintervention.Patients with adrenal insufficiency are required to increase their usual oral hydrocortisone dosewhen experiencing intermittent illness with fever, which usually involves doubling of their regularglucocorticoid replacement dose, with twice the regular dose taken at the same timepoints as usual.Further deterioration then requires switching to parenteral hydrocortisone replacement for major ∗ [email protected] a r X i v : . [ q - b i o . T O ] M a y tress dose cover as described above. During the time of COVID-19, however, patients with adrenalinsufficiency might require higher oral stress doses already initially, as the viral illness caused bySAYS-cov2 often comes with significant fever, sweating and malaise early on. In addition, the highand frequently continuous fever requires a more sustained delivery of hydrocortisone that adjusts indose and timing to the permanent inflammatory stress. Thus, we used the experimental data onoral hydrocortisone administration [1] together with building on a previously developed approach [2]to adapt the model for oral hydrocortisone administration, enabling us to select the most suitabledose to recommend. Model of oral administration with binding kinetics
We now describe an idealised model of oral hydrocortisone treatment. Comparing with the 1-component linear kinetics model of Prete et al. [1] for intravenous delivery, it is necessary to takeinto account the presence of a gastric compartment from which hydrocortisone must be absorbed,then transported to the blood. A model consisting of two compartments only (gastric and serumcortisol) was initially attempted, however it failed to replicate the dose response characteristics asan oral dose is increased from 10 mg to 50 mg. We therefore expanded the model to take intoaccount a limited bound component, modelling the effect of binding protein, which slows excretionat lower doses.In detail the components of the model are: gastric hydrocortisone dose ( S ( t ) mg, which isincreased by Q j each time a dose is taken), free serum cortisol F f ( t ) nmol/L and bound cortisol F b ( t ) nmol/L. The concentration of binding protein is accounted for via B ( t ) . Reactions will bemodelled as linear in all cases, however the finite quantity of binding protein available will result ina nonlinear response.The reactions in the model are: • Uptake from stomach to blood, at rate k abs and with dilution factor α . • Excretion of free cortisol at rate k ex . Bound cortisol is assumed not to undergo significantexcretion [3]. • Binding of free cortisol to binding protein at rate k b and • release of bound cortisol and protein at rate k r .The system takes the form, dSdt = − k abs S + q ( t ) , (1) dF f dt = αk abs S − k ex F − k b F f B + k r F b , (2) dF b dt = k b F f B − k r F b , (3) dBdt = − k b F f B + k r F b , (4)where q ( t ) is a function modelling the oral dosing. The initial conditions for an adrenal insufficientpatient will be approximated as S (0) = 0 , F f (0) = 0 , F b = 0 and B (0) = B where B is thephysiological level of binding protein (a value quoted in the literature is 650 nmol/L [4], althoughthis will vary between individuals). 2he case of oral administration of a N doses Q = ( Q , Q , . . . , Q N ) at times t = ( t , t , . . . , t N ) respectively can be represented by a sum of Dirac delta functions, q ( t ) = N (cid:88) n =1 Q d δ ( t − t n ) . (5)The differential equaiton for S ( t ) can be integrated, yielding, S ( t ) = exp( − k abs t ) (cid:90) tt q ( t (cid:48) ) exp( k abs t (cid:48) ) dt (cid:48) , (6)where starting time t precedes the first dose. Denoting the heaviside function by H , equation 6can be evaluated as, S ( t ) = N (cid:88) n =1 Q n H ( t − t n ) exp( k abs ( t n − t )) . (7)Adding equations (3) and (4) shows that F b ( t ) + B ( t ) is constant and hence equal to its initialvalue of B . Therefore the variable F b ( t ) = B − B ( t ) may be eliminated from the model, leadingto the two-variable system, dF f dt = α N (cid:88) n =1 Q n H ( t − t n ) exp( k abs ( t n − t )) − k ex F f − k b F f B + k r ( B − B ( t )) , (8) dBdt = − k b F f B + k r ( B − B ) . (9)The fraction of free to total cortisol F f / ( F f + F b ) is typically 5% [5]. Working on the assumptionthat the binding/unbinding processes occur faster than absorption and excretion leads to the quasi-steady approximation ≈ − k b F f B + k r ( B − B ) , (10)which can be justified formally via dimensional analysis. Then F f ≈ (cid:18) k r k b B (cid:19) ( B − B ) B B . (11)The first dimensionless parameter grouping will be denoted φ := k r / ( k b B ) ; this grouping quantifiesthe relative size of free to bound cortisol and will be found through fitting to time-course data. Thereason for working with this parameter grouping is that the full model is insensitive to the value of k b , meaning that it is challenging to fit directly. We then have the approximation F f ≈ φ ( B − B ) B B , (12)which enables the system to be reduced to a single variable.Noting that the total cortisol is given by F f + F b = F f + ( B − B ) , we may write d ( F f + F b ) dt = d ( F f − B ) dt , (13)then subtracting equation (9) from (8) yields, d ( F f − B ) dt = αk abs S − k ex F f . (14)3 Time (h) C o r t i s o l ( n m o l / L ) Figure 1: Oral dose data for cortisol (black: 50 mg every 6 hours [1]; red: 10 mg [6]) duringthe – h interval and associated 3-compartment model simultaneous fit to the data. Time axesare aligned so that the dose commences at the time designated t = 6 . Parameter estimates are α = 69 (nmol/L)/mg k ex = 1 . h − , k abs = 0 . h − , φ = 0 . and B = 1026 nmol/L.Applying equation (12) then leads to ddt (cid:18) φ ( B − B ) B B − B (cid:19) = αk abs S − k ex φ ( B − B ) B B , (15)hence dBdt = (cid:18) φ B B (cid:19) − (cid:18) − αk abs S + φk ex B (cid:18) B B − (cid:19)(cid:19) . (16)This model can then be solved numerically for B ( t ) , from which the total cortisol concentration F ( t ) = F f ( t ) + F b ( t ) is then given by equation (12). There are three remaining parameters toestimate by fitting to time series data for F ( t ) : the dilution factor α , absorption rate k abs andexcretion rate k ex . Parameter estimation
Two datasets are used for fitting the model: the PACS [1] data on oral administration of 50 mg every6 h in primary adrenal insufficiency patients, and published data from ref. [6] on oral administrationof a single dose of 10 mg in HPA-suppressed healthy individuals. Data from the latter study wereextracted digitally from the electronic journal article. The parameters α , k abs , k ex , φ and B arefitted to the two datasets simultaneously.A high degree of fidelity is unlikely in this situation due to attempting to fit to two differentpatient groups, dosing formulations and assays simultaneously with a relatively simple model; figure 1shows reasonably good agreement, although appearing to under-predict the response at 10 mg andover-predict at 50 mg. Nevertheless, the model does appear to be capable of giving approximateinformation over a 5-fold change in dose.The model can then be used to predict typical responses to a range of dosing regimes, as detailedin table 1. Figure 2(a,b) show the predicted time courses for the standard (regime I: 15 mg at 7:00followed by 10 mg at 13:00; regime II: 10 mg at 7:00 followed by 5 mg at 13:00 and 5 mg at 17:00),4hort name Summary Detail Parameters15–10 standard regime I 15 mg at 7:00,10 mg at 13:00 t = (7 , , Q = (15 , t = (7 , , , Q = (10 , , t = (7 , , Q = (30 , t = (7 , , , Q = (20 , , t = (0 , , , , Q = (10 , , , t = (0 , , , , Q = (20 , , , t = (0 , , , , Q = (30 , , , Table 1: Dosing regimes used to produce the model results in figure 2.5nd doubled regimes, which replicate a circadian-like pattern with nadir just before the first dose at7:00, and maintained concentrations through most of the day from around 7:30 to midnight.Motivated by the aim to replicate the adrenal response to continuous physiological stress, fig-ure 2(c) shows the predicted time courses with 6-hourly equal doses of each of 10, 20 and 30 mg.In all cases the low cortisol nadir is avoided.
Model limitations and discussion
The model described here is somewhat idealised and does not attempt to replicate in detail multiplephysiological compartments, heterogeneity between patients nor characteristics such as age, sexand ethnicity: more detailed pharmacokinetic-pharmacodynamic models (such as ref. [7]) wouldgive more detail and information around variability in responses. The fitted model appears tounderpredict peak responses at 10 mg and over-predicts at 50 mg, in each case by around 25%.Oral dosing is also subject to first-pass liver metabolism, which metabolise steroids for downstreamexcretion, and indeed the PACS study showed higher excretion of hydrocortisone metabolites in urinethan equivalent parenteral doses ([1], suppl. fig. 2). It is also possible that physiological response toCOVID-19, in particular body temperature, may affect rates in the system such as binding globulinaffinity [8]. Results should be interpreted with these limitations in mind.Nevertheless, the model provides an approximation of the shape of the 24 hour time-courseswhich would result from the dosing strategies considered. Dosing every 6 h clearly indicates thatan early morning nadir in cortisol can be avoided, which may be valuable for patients suffering fromthe stress of viral infection.
Funding
Data were generated via the
Prevention of adrenal crisis in surgery study [1], supported by MedicalResearch Council UK (program grant G0900567), the Oxfordshire Health Services Research Com-mittee, and the National Institute for Health Research (NIHR) Birmingham Biomedical ResearchCentre at the University Hospitals Birmingham NHS Foundation Trust and the University of Birm-ingham (grant reference number BRC-1215-2009). AP is a Diabetes UK Sir George Alberti ResearchTraining Fellow (grant reference number 18/0005782).The views expressed are those of the authors and not necessarily those of the NIHR or theDepartment of Health and Social Care UK.
Acknowledgements
The authors thank Profs. Richard Ross and Simon Pearce for valuable discussions.
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