Tercera Etapa

The primary aim of the insulin model was to establish smooth trajectories that would accurately describe the insulin-time courses under various provocations of NiAc, rather than explain all of the mechanistic aspects of insulin dynamics. To this end, the model structure chosen was as simple as possible. The insulin model could subsequently be used to provide

a) b)

Figure 5.2: Illustration of the biophase models used in this case study. Fig. (a) represents a zero-order input into, and first-order elimination from, the biophase—used for modelling intravenous drug administration. Fig. (b) represents a first-order absorption into, and a first-order elimination from, the biophase—used for modelling subcutaneous administration.

input to the FFA model, enabling a quantitative analysis of the antilipolytic effects of insulin. Given this premise, we applied a phenomenologically based modelling approach. Under the assumption that NiAc perturbs insulin, the characteristics seen in the data were used to establish an insulin model with NiAc as input. The characteristic behaviour of the data for acute and long-term NiAc provocations in lean and obese rats is illustrated in Fig. 5.3. Attributes seen include indirect action, tolerance, rebound, and complete adaptation. Data with similar properties to those observed in the acute experiments (Figs. 5.3a and 5.3c) have been modelled using turnover equations with moderator feedback control (Ahlstr¨om et al., 2013a; Gabrielsson and Peletier, 2007). Furthermore, to capture the different long-term adaptive behaviours with (Fig. 5.3b), and without (Fig. 5.3d) rebound, a ’NiAc action compartment’ was included, as well as an integral feedback control (a schematic illustation of the model is shown in Fig. 5.4).

The insulin dynamics are given by dI(t) dt =kinI·RI(t)·HNI(Ab(t))· M0I M1I(t) −koutI· M2I(t) M0I ·I(t), (5.4) dM1I(t) dt =ktolI· I(t)−M1I(t) , (5.5) dM2I(t)

dt =ktolI· M1I(t)−M2I(t)

, (5.6)

with initial conditions

I(0) =I0, (5.7)

and

M1I(0) =M2I(0) =M0I=I0, (5.8)

whereI(t) denotes the observed insulin level (expressed in nM), Ab(t) the biophase amount, andM1I(t) and M2I(t) the first and second moderator compartments, respectively (both

Acute Chronic

Lean

a) _b)

Obese

c) d)

Figure 5.3: Example of insulin-time course data during acute NiAc dosing (a) and (c), and chronic NiAc dosing (continuous infusion) (b) and (d) for lean and obese rats, respectively. The blue lines represent the NiAc treated animals and the red lines the vehicle control group.

expressed in nM). The parameters kinI (nM min−1) and koutI (min−1) are the turnover rate and fractional turnover rate of insulin, respectively, andktolI (min−1) is the fractional turnover rate of the moderators. The regulatorRI(t) compartment is given by

dRI(t)

dt =kinRI−kRI·I(t) with RI(0) = 1, (5.9) wherekinRI (min−1) is the turnover rate, koutRI (min−1nM−1) the fractional turnover rate, andI(t) the insulin concentration. The regulator is initially at steady-state with

dRI(0) dt =kinRI−koutRI·I0= 0 (5.10) or I0= kinRI koutRI . (5.11)

By integrating Eq. (5.9), the dynamics ofRI(t) can be expressed as

RI(t) = 1 +

Z t

0

kinRI−koutRI·I(τ) dτ. (5.12)

Hence, by construction,RI(t) represents the output of an insulin-driven integral feedback controller (Glad and Ljung, 2000) withI0as the set-point andkoutRI as the integral gain

parameter (koutRI will from hereon be referred to as the integral gain parameter). The integral feedback controller ensures that insulin levels return to the baseline I0, despite persistent external effects on insulin turnover and fractional turnover. The inhibitory NiAc function for insulin is given by

HNI(Ab(t)) = 1−ENI(NI(t))· An b(t) IDn_50NI+An b(t) , (5.13)

whereID50NI (µmol kg−1) is the potency of NiAc on insulin andnthe Hill coefficient of the inhibitory function. The termENI(NI(t)) represents the drug efficacy, which is fixed for lean rats and dependent on the concentration in a hypothetical NiAc action compartment,NI(t) (µmol kg−1), for obese rats, according to

ENI(NI(t)) =      ImaxNI lean ImaxNI 1− SNI·NIγ(t) N_50Iγ +N_Iγ(t) obese, (5.14)

whereImaxNI is the initial efficacy of NiAc for insulin,N50I (µmol kg−1) the potency of the NiAc action compartment,SNI the long-term NiAc efficacy loss, andγ the corresponding Hill coefficient of the efficacy relation. The dynamics ofNI are in turn given by

dNI(t)

dt =kNI·(Ab(t)−NI(t)), (5.15) withNI(0) =Ab(0). HerekNI(min−1) is the turnover rate of the NiAc action concentration. The NiAc action compartment is initially at steady-state with the biophase NiAc amount Ab. As infusions are initiated, and the biophase amount increases,NI(t) increases until it reaches the steady-state biophase NiAc amountNss(t) =Abss. With increasing levels in the NiAc action compartment,E(NI(t)) decreases to a minimum of 1−SNI and, consequently, the efficacy of NiAc as an insulin inhibitor is down-regulated. In other words, the system has developed tolerance to the drug. The turnover ratekNI determines the rate at which tolerance develops.

Assuming that the system in Eq. (5.4) is at an initial steady-state with d_dI_t = 0 and knowing that the initial biophase drug amount is zero, i.e.,Ab(0) = 0, the inhibitory NiAc function for insulin isHNI(Ab(0)) = 1 and we obtain

dI dt _t=0 =kinI−koutI·I0= 0, (5.16) or kinI=koutI·I0. (5.17)

Thus, we can eliminatekinI from the estimation procedure and obtain an estimate for this parameter from Eq. 5.17.

Figure 5.4: Mechanisms of insulin dynamics. The parameterskinI andkoutI represent the turnover rate and fractional turnover rate, respectively. The turnover of insulin is inhibited by the NiAc action functionHNI(Ab). Tolerance and rebound are captured by the moderator compartmentsM1I andM2I, which act on the turnover rate and fractional turnover rate of insulin, respectively. The integral feedback controller acts on the turnover rate of insulin, in that it strives to maintain the insulin baseline,I0, despite persistent external effects on the turnover.

5.3.0.1 Between-subject, inter-study, and residual variability

The insulin data illustrate individual variations (as can be seen in Fig. 5.3), which were described by incorporating random effects in the model and allow the parametersI0, ktolI, andID50NI to vary in the population. These parameters were assumed to be log-normally distributed. Thus, for example, the insulin baseline levelI0i of subjectiis given by

I0i =I0·expηi, (5.18)

whereI0 is the median population insulin baseline level andηi is the corresponding random

effect of subjecti. The covariance matrix of the random effects was assumed to be diagonal in order to simplify the estimations. Hence, the vector of random effectsηi of subjectiis

normally distributedηi∼ N(0,Ω) where

Ω=     ω11 0 0 0 ω22 0 0 0 ω33     . (5.19)

Here,ω11, ω22, and ω33 are the standard deviations of the random effects. The choice of these parameters was guided by ana priori sensitivity analysis (see Sec. 2.4 or Saltelli et al. (2008); Saltelli (2002) for more details). Moreover, the parametersI0and ktolI varied over

study groups according to fixed-study effects on both the mean and individual parameter distributions (Laporte-Simitidis et al., 2000). In other words, forS the number of groups,

the parameterI0 for an individualj was modelled as

I0j = (I01·Study1+. . .+I0S·StudyS)· (5.20)

exp(η1·Study1+. . .+ηS·StudyS), (5.21)

where Studyk= 1 if individualj is in groupkand 0 otherwise. The residual variability was

modelled using an additive model (with normally distributed errors).