5.2 Glucose Tolerance Studies
6.1.1 Single-Compartment Model
The single-compartment model is a very simplified model where the plasma is assumed to consist of a single-compartment with first-order absorption and elimination and is illustrated in Figure 6.1.
Central Compartment
Ic
Vd
ke ka
Isc
Rin
D
Figure 6.1: Single-compartment model. The symbols in the model are explained in the text.
6.1 PK Models 57 The two states in the model areIsc andIc. Isc describes the amount of insulin remaining to be absorbed from the SC tissue andIc rep-resents the amount of insulin in the central compartment.
The parameterska and ke are the rate constants for the irreversible absorption to and elimination from the central compartment, re-spectively. The parameterVd is the volume of the central compart-ment. Vd should not be mistaken with the plasma volume but can be thought of as theapparent volume of distribution in the body.
The two inputs to the system are D and Rin. D describes the SC injected insulin dose of either type A or B whileRinis the IV infusion of regular human insulin given throughout the study to suppress the secretion of insulin from the pancreas.
Assumptions
The model assumptions are:
A1: The insulin is mixed instantaneously in the plasma. The actual time taken for mixing is approx. a few minutes and is therefore insignificant compared with the sampling time.
A2: The insulin absorption and elimination is assumed to fol-low first-order kinetics meaning that the rate of change of insulin concentration is directly proportional to the re-maining concentration of insulin. This assumption leads to a linear model.
A3: The amount of insulin removed from the SC tissue is equal to the amount absorbed in the central compartment. This assumption is made because the break down of SC insulin is not modelled.
A4: No insulin is secreted from the pancreas because the IV in-fusion of regular human insulin suppresses the production.
The small amount of insulin that actually is secreted in the pancreas is corrected by using C-peptide measurements. It
is therefore reasonable not to include any feedback mecha-nisms in the model since they have been disrupted.
Model equations
The difference equation of finite differences ∆t for the amount of insulin is shown in Table 6.1 where t is the time from SC injection of insulin A/B.
Table 6.1: Integral mol balance for insulin.
Accumulated = In − Out
Isc(t+ ∆t)−Isc(t) = D − kaIsc∆t Ic(t+ ∆t)−Ic(t) = (kaF Isc+Rin)∆t − keIc∆t
The parameter F is the bioavailability factor which is included to describe the fraction of the injected doseDwhich is available in the SC depot.
The differential equation is obtained by dividing with ∆tand letting the time step tend to zero in the difference equation, i.e.:
dIsc
dt = D·δ(t)−kaIsc (6.1a) dIc
dt = kaF Isc+Rin−keIc (6.1b) whereδ(t) is a Dirac delta function.
The specified model in (6.1) is structural unidentifiable (see Ap-pendix A.2) but can be made identifiable by setting the bioavail-ability factor F equal 1, thereby assuming that all the SC injected insulin is available.
6.1 PK Models 59 Analytical solution
The amount of insulin remaining to be absorbed from the SC depot can be found by solving (6.1a), i.e.:
Isc=D·e−kat (6.2)
with the initial conditionIsc =D fort= 0.
By substituting (6.2) into (6.1b), the change in the amount of insulin in the central compartment can be written as:
dIc
dt =kaF De−kat+Rin−keIc (6.3) The deterministic equation (6.3) is split into two domains:
dIc,1
dt = Rin−keIc,1 −90 < t < 0 (6.4) dIc,2
dt = kaF De−kat+Rin−keIc,2 0 < t (6.5) since IV infusion of regular human insulin is given throughout the study (t∈[−90,600]) while the SC injection of insulin A/B is given att= 0.
The solution to (6.4) is:
Ic,1= Rin ke
³1−e−keτ´
+Ic,0e−keτ (6.6) with the initial conditionIc =Ic,0 forτ =t+ 90 = 0.
The solution to (6.5) can be found using the Panzer equation, i.e.:
Ic,2 =e−kethZ eket³
kaF De−kat+Rin´
dt+Ci
=F D ka ka−ke
³e−ket−e−kat´ +Ic,1
(6.7)
Thereby, the analytical solution for the insulin concentrationCc= VIc The analytical solution (6.8) and (6.9) can be expressed as a linear combination of exponential terms and a constant termK, i.e.:
Cc = (φ1+φ2)·e−φ3τ+φ4·e−φ3t+φ5·e−φ6t+K (6.10) whereke =φ3>0, and ka=φ6 >0.
The sum of the parametersφ1 and φ2 can be identified from (6.10) but not the individual parameters since they cannot be distinguished from one another. The biexponential termsφ4·e−φ3t and φ5·e−φ6t for SC insulin are not identifiable in the sense of having a unique vector of parameters associated with a given set of predictions since the parameters φ3 and φ6 may be exchanged without changing the predictions. Identifiability of the biexponential terms is ensured by requiring thatφ3> φ6 so that the first exponential term determines the initial absorption phase of insulin while the terminal elimination phase primarily is determined by the second exponential term.
Fundamental PK Parameters
Three fundamental pharmacokinetic parameters, that frequently are used to characterize the insulin profile in the plasma, are the time to maximum insulin concentration (tmax), the maximum insulin concen-tration (Cmax), and the area under the insulin concentration profile (AUC). These parameters are derived from the single-compartment model presented above.
6.1 PK Models 61 Time to Maximum Insulin Concentration: The maximum in-sulin concentration Cmax, occurs at time tmax. At this time, the insulin profile is at its peak and the slope is zero. tmax can be found by setting the first derivative of (6.9) equal to zero and solving for tmax [58, p. 171]:
Maximum Insulin Concentration: The maximum insulin con-centrationCmaxcan be found by inserting the timetmaxwhich solves (6.11) into (6.9):
Area Under Curve: The area under the insulin concentration profile (AUC) describes how much of the insulin is absorbed. AUCT0 is found by integrating (6.9) fromt= 0 to T:
The expression in (6.13) for AUC depends on a good estimation of the rate constants and of a possible lag-time [44, p. 30]. Therefore,
the Trapezoidal-method is used to determine the AUC, i.e.:
AU C0T =
N
X
i=0
Cc,i+1+Cc,i
2
¡ti+1−ti¢
(6.14) whereN is the number of measurements.