Absorption Kinetics of Insulin Mixtures after ...

Chapter 15

Absorption Kinetics of Insulin Mixtures after Subcutaneous Administration Christian Hove Rasmussen, Tue Søeborg, Erik Mosekilde, and Morten Colding-Jørgensen

Insulin is provided in a number of different variants with specific absorption kinetics. A detailed understanding of the processes determining this kinetics is important both to optimize the treatment of the individual patients and to reduce the risks associated with fluctuations in the absorption rate.

15.1 Introduction Day after day, millions of diabetic patients throughout the World must take injections of insulin in order to keep their blood glucose concentrations within an acceptable range [70]. To achieve the desired control it is important that the insulin concentration in the blood plasma, while coordinated with the supply of glucose, is kept within specific limits. Moreover, to ensure a consistent and predictable drug effect, the plasma insulin profiles (appearance curves) must be similar from injection to injection. Despite the fact that insulin has been used in the treatment of diabetes since the 1920s [2] and although the biological action of insulin is well described in

C.H. Rasmussen () M. Colding-Jørgensen Novo Nordisk A/S, Novo Allé, DK-2880 Bagsværd, Denmark e-mail: [email protected]; [email protected] T. Søeborg Copenhagen University Hospital, Blegdamsvej 9, DK-2100 Copenhagen, Denmark e-mail: [email protected] E. Mosekilde Department of Physics, Technical University of Denmark, Fysikvej 1, DK-2800 Lyngby, Denmark e-mail: [email protected] E. Mosekilde et al. (eds.), Biosimulation in Biomedical Research, Health Care and Drug Development, DOI 10.1007/978-3-7091-0418-7 15, © Springer-Verlag/Wien 2012

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the literature, the absorption process remains poorly understood. Large inter- and intra-patient variability is observed in most clinical data, and the processes that contribute to this variability have never been described in detail. Moreover, only a fraction of the injected dose becomes useful to the patient. Both the high variability and the reduced bioavailability are likely to be related to events that take place at the injection site during the absorption process. As described in Chap. 2, absorption of subcutaneously injected soluble insulin is a rather complicated process that combines polymerization of the insulin molecules with diffusion and binding in the subcutaneous (s.c.) tissue and depends on factors such as the administered dose, the applied insulin concentration, and the binding capacity of the tissue [49]. Other factors such as variations in local blood flow and skin temperature are also known to contribute to the observed variability [31], and a recent publication by the present authors [63] describes several additional mechanisms that become significant for biphasic insulin mixtures, i.e. for mixtures of soluble and crystalline insulin. Mixtures of this type are used to attain a slower and more constant supply of insulin to the blood. The rate of degradation in the s.c. tissue determines the bioavailability of the insulin variants. Soluble forms of insulin appear to be degraded by enzymatic processes at the injection site. Crystalline insulin, on the other hand, forms dense heaps around the center of the injection site and their degradation has been found to also involve the cells (macrophages) of the immune system. This explains why the bioavailability is lower for NPH (Neutral Protamin Hagedorn) insulin than for soluble insulin, and it also accounts for a major contribution to the absorption rate variability for crystalline insulin: The crystal heaps will vary in size and shape from injection to injection, and the surface area from which insulin is dissolved therefore also varies in a manner that is difficult to control. The first mechanism-based model describing the absorption kinetics of soluble insulin was developed by Mosekilde et al. [49], and several subsequent models have been based on the same ideas, e.g. Trajanoski et al. [66] and Li and Math [44]. To our knowledge, the first attempt to describe the absorption kinetics for mixtures of soluble and crystalline NPH insulin was made by Clausen et al. [8], but these authors did not apply a mechanism-oriented approach that allows predictions to be made outside of the investigated range of experimental conditions. The purpose of the present chapter is to describe the absorption kinetics for biphasic mixtures of soluble insulin and suspensions of NPH insulin by the means of a detailed model of the processes assumed to govern the rate of absorption. Our model is based on previous work by the present authors [62, 63], but the focus will now be on a quantification of the bioavailability for different mixtures and a characterization of their inter- and intra-patient variability. In this way the simple absorption model described in Chap. 2 is transformed into a useful tool in the development of new insulin variants. At the same time, the discussion will emphasize the approach to develop a consistent mechanistic description of the interplay between the involved processes, the background investigations drawn upon to establish a proper understanding of the conditions in the s.c. tissue, and the factors that affect the absorption rate. Even though our examples will relate to soluble

15 Absorption Kinetics of Insulin Mixtures after Subcutaneous Administration

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human insulin (HI), insulin aspart (IA), NPH insulin suspensions, and mixtures of these insulins, most of the described mechanisms apply with little or no modification to other types of insulin as well.

15.2 Mechanisms of Subcutaneous Absorption In this section we present the mechanisms that govern the absorption of s.c. injected insulin. The focus will be on identifying the mechanisms and describing them mathematically.

15.2.1 The Injection Depot Let us start the analysis by considering the conditions in the s.c. depot immediately after an insulin injection. Although many publications exist regarding tissue structure, few provide information about what happens to the injection fluid and insulin after administration. Studies done by groups at Novo Nordisk A/S (E. Hasselager (2009) and M. Poulsen and co-workers (2011), personal communication) illustrate both the layout of the s.c. depot and the placement of the injected insulin. Figure 15.1 shows a histological cross section excised immediately after a s.c. injection of soluble IA mixed with contrast fluid in a pig. The insulin (colored red) is visible between the fat cells, capillaries, and connective tissue below the injection

Fig. 15.1 Histological cross section of subcutis immediately after an injection of 0:1 mL soluble IA (colored red) mixed with contrast fluid in the s.c. tissue of a pig. The length of the black bar is 2 mm. Courtesy of M. Poulsen and co-workers (2011), Novo Nordisk A/S

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Fig. 15.2 CT scanning of the injection in Fig. 15.1 (same excision) showing the location of the insulin/contrast fluid. The image is not a cross section, but a projection of several layers on top of each other. The scale is approximately the same as in Fig. 15.1. Courtesy of M. Poulsen and co-workers (2011), Novo Nordisk A/S

canal. Figure 15.2 shows a CT scan of the insulin/contrast fluid in the injection (same excision). The figures suggest that the injection fluid containing insulin flows in between the cells of the s.c. tissue and replaces the interstitial fluid. Limited or no bursting of the tissue seems to have taken place. For NPH/crystalline insulins, the case is different. In Fig. 15.3 we have shown a histological cross section of subcutis excised 1 h after the injection of 100% NPH insulin in a live pig. The insulin (dark purple) has piled up near the center of the injection and some bursting of the tissue seems to have taken place. Figure 15.4 shows a close up of a similar cross section (right pane) and a view of the insulin crystals in the vial (left pane). Here, it is evident that the NPH crystals have been concentrated substantially compared to the suspension in the vial. A study by Hewitt [29] shows a similar picture. A histological cross section of 0:2 mL subcutaneously injected soluble and suspended dye was excised from a mouse (Fig. 15.5). Closer inspection of the figure shows that the dark region corresponding to the soluble dye (left side) is approximately 2:0 mL or 10 times the injected volume. On the other hand, the dark region stemming from the suspended dye (right side) is only about 0:02 mL or 10 times less than the injected volume. A weak trace of the solute from the suspended dye constitutes a volume similar to the colored region from the soluble dye injection.


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Fig. 15.3 Histological cross section of subcutis 1 h after the injection of 100% NPH HI in a live pig. The NPH crystals are visible as the dark purple area near the bottom center of the image. Courtesy of E. Hasselager (2009), Novo Nordisk A/S

Fig. 15.4 Close up of a histological cross section of subcutis 1 h after the injection of 100% NPH HI in a live pig (right pane) and the NPH crystal suspension in the vial (left pane). The scale to the left is 10 m and applies to both panes. The right pane clearly shows the condensed crystal heaps (bright red) between the connective tissue (yellow). Courtesy of E. Hasselager (2009), Novo Nordisk A/S

The total volume covered by the injection fluid Vsc will therefore exceed the injected volume V0 , since Vsc also accounts for the volume of fat cells and connective tissue. By how much will depend on the amount of suspended particles in the injection fluid. For a solution (with no particles), Vsc depends only on the fraction " of extracellular space:

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Fig. 15.5 Histological cross section of the s.c. tissue of a mouse following the injection of 0:2 mL injected soluble dye (left) and suspended dye (right). Reprint, with permission, of a figure shown by Hewitt [29]

Vsc D

V0 : "

(15.1)

The value of " is reported to be approximately 10 % [11], a value that corresponds well with Fig. 15.5. For an injected suspension the relation is different. Figures 15.3–15.5 show that the crystals are retained near the center of the injection site and have caused some bursting of the tissue. The explanation for this could be that the fluid pressure itself has burst the tissue, resulting in one or more local fluid cavities. The contracting tissue would then filter the crystals and cause heap formation near the injection center. On the other hand, the particles may simply clog the tissue and burst it, while the fluid diffuses out in the tissue and replaces the interstitial fluid. Either case would give the result shown in the figures. Further away from the center, the particles will only partially fill up the intercellular space. Consequently, (15.1) still applies, but the value of " now varies locally where the tissue has burst. The degree of bursting and, thus, the local value of " will depend on the suspension concentration as well as the amount (dose) and size of the particles.

15.2.2 Soluble Insulin Equilibria In both the vial and the injection depot, most types of soluble insulin consist of a mixture of various oligomers in a chemical equilibrium [25]. Higher molecular weight species are favored at higher insulin concentrations, as Brange et al. [3]


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and Hansen [21] found that the fraction of hexamer insulin (6 insulin monomers) for zinc-free soluble HI at total concentrations of U331 , U167, and U250 were 50%, 75%, and 100%, respectively. At concentrations below U5, the absorption of soluble HI occurs at the same rate as monomeric insulin [4]. The most general assumption is that soluble insulin exists in an equilibrium between insulin monomers (M), dimers (D), and hexamers (H): H • 3D

(15.2a)

D • 2 M:

(15.2b)

Using the chemical rate laws, this translates into the following net reaction kinetics: @cH D PDH .KDH cD 3 cH / @t @cD D PMD .KMD cM 2 cD / @t

(15.3a) (15.3b)

where cH , cD , and cM are the s.c. concentrations of insulin monomers, dimers, and hexamers, respectively. KDH and KMD are the chemical equilibrium constants for the dimer-hexamer and monomer-dimer transitions, respectively. The parameters PDH and PMD are transformation rate constants. According to Mosekilde et al. [49] there is no indication that the transformations rate constants should be limiting factors for the reactions, so they will most likely be very high (>0:5 min1 ). The equilibrium constants depend on the presence of several auxiliary substances, including zinc and phenolic substances. In the presence of both, the hexamers that are observed in the zinc-free T6 state may form the much more stable R6 state, in which each hexamer contains two zinc atoms and 6 m-cresol [1]. This state is desired for long term storage purposes. Values for KDH and KMD without auxiliary substances have been reviewed by Søeborg et al. [62] and they tend to vary greatly in size depending on the type of experiment. KDH is of the order 108 M2 , while KMD is of the order 1010 M1 . KDH has been reported to be about 400 times higher in the presence of auxiliary substances in concentrations typically used [37]. The removal of the auxiliary substances is most likely proportional to the concentrations of substances, so an approximate description of the time evolution of the hexamer-dimer equilibrium constant KDH would be: KDH D KDH1 C .KDH0 KDH1 / exp.DH t/

(15.4)

This implies that, following the injection, KDH changes its value from KDH0 to KDH1 with a half time of ln.2/=DH .

1 U or IU (International Unit) is defined as 6 nmol. U33 is a concentration of 33 U mL1 , while 33 U is a dose of 33 times 6 nmol.

1

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Many rapid-acting soluble insulin analogues, such as IA, insulin lispro, and insulin glulisine, are characterised by their reduced tendency to form dimers relative to regular HI. For IA, the monomer-dimer equilibrium constant, KMD , has been found to be reduced by a factor of 200–300 relative to regular HI [5]. The molecular reformulation in IA only alters the monomer-monomer interaction, leaving the reaction surface of the IA dimer similar to that of HI [68]. Therefore, the value of KDH is the same for IA and HI.

15.2.3 Crystal Dissolution Suspended insulins tend to prolong the effect of a given basal dose when compared to soluble insulins. The use of suspended insulin in formulations was first suggested by Hagedorn et al. [20] in a form called NPH (Neutral Protamine Hagedorn). It is an intermediate acting insulin and one of the most widely used insulin suspensions. NPH can be based on both HI and IA, and structural studies have shown the neutrally charged NPH crystal to consist of R6 hexamers and the salmon protein protamine [13]. At pH 7.3, these two components are present in a molar ratio of 5 R6 hexamers to 6 protamine [60]. NPH crystals contain about 50% water and have a density of approximately 1:22 g mL1 [1]. NPH crystals and particles in general behave very differently from soluble substances when injected subcutaneously. Figure 15.3 indicates that the NPH crystals are retained near the center of the injection site due to the tissue acting as a sieve as suggested by Hagedorn et al. [19]. The individual NPH crystals are rod-shaped and measure about 20 5 5 m [51], but the heaps of NPH are as large as 200 m in diameter [47]. It is not known whether the heaps (gel-like according to P. Balschmidt (2009), Novo Nordisk A/S, personal communication) are permeable to water, but in either case the dissolution of the heaps will depend on the (effective) surface area. Following injection, the auxiliary substances are released from the NPH crystals within a short period of time. As suggested by Hagedorn [18], this may be caused by a protamine-splitting enzyme present in the s.c. tissue. When this happens, the kinetics of NPH crystals can be regarded as the net result of a dissolution process proportional to the total heap surface area and a recrystallization process which depends (approximately equally) on both the surface area of the heaps and the concentrations of protamine and hexamer insulin. The result is a modified NoeyesWhitney equation [15]: 1 @MNPH D s ONPH .1 ˛ cH cP /; @t

(15.5)

where is the density of the NPH crystals, MNPH is the total mass of the crystals, s is the dissolution rate constant, and ONPH is the total surface of the heaps. ˛ is the NPH dissolution inhibition constant that slows the dissolution of the heaps if cH or cP becomes large and recrystallization therefore becomes substantial.


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The dissolution kinetics will depend on the relationship between the mass MNPH and the surface ONPH of the NPH crystals. Assuming that the heaps are small spheres distributed in their radii r this relationship will in turn depend on the actual distribution. Rasmussen [56] calculated that for triangular distributions there is approximate proportionality between the total volume and surface of such heaps, a result which may be generalized to (calculations not shown): ONPH

VNPH D MNPH ; L L

(15.6)

with a proportionality factor of 3:9. Proportionality between surface and volume has also been found for many other types of distributions (data not shown). For distributions with a large proportion of infinitely or very small heaps, the proportionality fails, but for most other distributions it holds. A special case is when the heaps all have the same radius rNPH0 . In this case, we obtain: 1

ONPH

3 VNPH0 3 2 D VNPH 3 ; rNPH0

(15.7)

where VNPH0 is the initial total volume of the heaps. This was used in a simpler model presented by Søeborg et al. [62]. Although it does not change the results significantly compared to using (15.7), the surface-volume proportionality in (15.6) will be used in the present chapter. The initial conditions for soluble insulin species are approximately constant concentration throughout the injection area. For NPH crystals, the initial conditions are somewhat different. In a U100 formulation of NPH insulin the crystals account for some 3.6% of the total volume. With complete separation due to tissue sifting, the crystals are retained in the innermost 3.6% of the volume (33% of the radius), while the solute will spread and reach the same space as when injecting pure solute, if the tissue is intact. With partial separation, the crystals will reach between 33% and 100% of the radius, typically in an irregular fashion. This corresponds to a variation in the local value of " in (15.1) depending on the degree of bursting. The value will be larger near the center of the injection site and decline towards the periphery. In the present chapter it is assumed that the initial distribution of NPH in the depot is such that in the center the NPH concentration is large and then it decreases linearly with radius until it reaches zero at the depot boundary r D r0 . This is equivalent to: r cT0 ; cNPH0 .r/ D 4 1 r0

r r0 ;

(15.8)

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where is the fraction of crystalline insulin in the vial and cT0 is the total concentration of insulin in the vial. Furthermore, it is assumed that the tissue does not burst following injection and that the volume of the NPH crystals is negligible.

15.2.4 Insulin Absorption In the vial, the formulation contains auxiliary substances to ensure that soluble insulin is almost entirely in its hexameric and most stable form. Following s.c. administration, the auxiliary substances disappear rapidly and the equilibrium shifts gradually towards dimers and monomers [37]. It has been shown that 65 Zn disappears faster than 125 I following s.c. administration of soluble 65 Zn-125 Iinsulin [57] which indicates that a considerable fraction of the hexamers dissociate into dimers before being absorbed. For smaller molecules, such as the insulin monomer (~6 kDa) and dimer (~12 kDa), it is likely that absorption from the s.c. depot into the blood stream takes place via the capillary wall. Although it is thought that the absorption mostly takes place via monomers and dimers [49], evidence of direct hexamer absorption was found by Kurtzhals and Ribel [41] using the very stable Co.III/-insulin hexamer. Molecules larger than 16–20 kDa are generally thought to be taken up via the lymphatic system [55], while proteins in the 30–40 kDa range are absorbed almost completely via the peripheral lymphatics in sheep as reviewed by McLennan et al. [48]. Therefore, it is likely that the observed insulin hexamer (~36 kDa) absorption takes place via the lymphatics. Due to the size of the heaps, NPH insulin cannot be absorbed via the capillary wall, nor can it be be absorbed via the lymph. Consequently, the absorption from the s.c. depot must be regarded as a combination of monomer, dimer, and hexamer absorption via both the capillaries and the lymphatic system. Evidence suggests that when one soluble insulin species dominates (e.g. the monomer), the disappearance of insulin from the depot will be a monoexponentially decreasing function of time. This suggests that the absorption of soluble species into plasma takes place at rates proportional to the respective concentrations (first order, non-saturable under physiological conditions). For the soluble species x the rate would be: @cx D Bx cx @t

(15.9)

The time constant for the absorption of a given species is equal to the first order absorption rate constant Bx . A value for this constant for monomeric/dimeric insulin has been estimated by Mosekilde et al. [49] to 9 103 min1 based on the terminal slope of a disappearance curve of porcine 125 I-insulin. Brange et al. [4] used the same approach to estimate BH , BD , and BM using the disappearance of 125 I-labelled HI, a dimer analogue (AspB9 , GluB27 ), and a monomer analogue (AspB10 , insulin


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aspart). The HI disappearance curve was found to consist of three separate slopes corresponding to the three rate constants, although the values of BH and BD are associated with some uncertainty, since the HI in subcutis is a mixture of all the species. BM was in all cases estimated to 1 102 min1 .

15.2.5 Subcutaneous Degradation The bioavailability of a drug is defined as the fraction of the administered dose that appears in plasma after the administration. The rest of the dose is in this case degraded or inactivated in subcutis. The bioavailability can be found by calculating the amount of substance which appears in plasma from time zero to infinity and dividing it by the dose. This is typically not possible so instead the bioavailability is estimated as the ratio between the s.c. and the intravenous area under the curve (AUC) from t D 0 to infinity (see e.g. Greenblatt and Koch-Weser [17]). Such a method is not always accurate for insulin, since the plasma elimination may be saturated following an intravenous injection, resulting in a lower overall elimination compared to a s.c. injection as discussed by Lauritzen [43]. Various other methods exist for determining the bioavailability, and an extensive review has been done by our group (see Table 1 in Søeborg et al. [63]). The table shows that measured values of the bioavailability vary substantially depending on the method and the insulin species. Moreover, the bioavailability has been found to decrease with increasing fraction of NPH crystals in insulin mixtures based on IA [64]. Decreasing bioavailability has also been seen for increasing NPL to insulin lispro ratios [28]. Studies have shown that many types of soluble insulin have the same bioavailability in man, e.g. human vs. porcine [53], human vs. aspart [33], human vs. lispro [36], and aspart vs. lispro [23]. Increasing the total insulin concentration has been found to decrease the bioavailability for HIand IA-based NPH insulin [38, 64], while it remained unchanged for soluble HI [38, 53]. For soluble insulin, this claim is supported by a study, in which patients shifting from U100 to U500 insulin formulations did not need to adjust their dose [50]. The exact mechanisms accounting for the bioavailability of s.c. injected insulin are not known. Some studies report that s.c. degradation of insulin is minimal [12], while other studies report it as substantial [54]. Co-administering various enzyme inhibitors such as “-cyclodextrin [65] and collagen [35] showed increased bioavailability of soluble insulin. Given this and the fact that the bioavailability of soluble insulin lies in most cases between 50% and 80% and even lower for NPH insulin [63], it is likely that s.c. insulin degradation is a governing factor in the bioavailability. In the present chapter we assume that the degradation of soluble insulin species in subcutis takes place at a rate proportional to the species concentration (first order, non-saturable under physiological conditions) as suggested by Hori et al. [34], or for the soluble species x:

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Fig. 15.6 Histological cross section of subcutis 48 h after the injection of NPH insulin in a live pig. The white cells are fat cells and the insulin heaps are colored bright red. The diameter of the largest heap is around 200 m. Courtesy of E. Hasselager (2009), Novo Nordisk A/S

@cx D Ax cx ; @t

(15.10)

where Ax is the degradation rate constant. In vitro studies have shown that the insulin dimers/monomers are more susceptible to degradation by ’-chymotrypsin than the insulin hexamers [45]. Therefore, Ax will be larger for insulin dimers and monomers when compared to hexamers. No values for the s.c. degradation rates of insulin have been found in the literature, but it is still possible to derive estimates, as is described in Sect. 15.3.2. A substantial part of injected NPH crystals are degraded by invading macrophages as observed by Markussen et al. [47] and E. Hasselager (2009), Novo Nordisk A/S, personal communication. The NPH heaps are too large to undergo phagocytosis [16], so the degradation takes place from the surfaces. This is illustrated in Figs. 15.6 and 15.7 which show a histological cross section of the s.c. tissue at different zoom levels 48 h after the s.c. injection of NPH insulin in a living pig. No macrophages can be seen inside the heaps, but they are instead seen degrading the heaps from the surface. Therefore, the rate of degradation due to macrophages is suggested to be proportional to the surface area of the heaps: 1 @MNPH D ONPH ; @t

(15.11)

where is a breakdown rate constant and ONPH obeys (15.6). Macrophages arrive at the injection site and reach their full number within a few hours after the injection (E. Hasselager (2009), Novo Nordisk A/S, personal communication).


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Fig. 15.7 Same histological cross section as in Fig. 15.6 at a higher zoom level. The macrophages (dark purple) are visible degrading the insulin crystals. Many of the macrophages have pieces of crystal heaps inside them. Courtesy of E. Hasselager (2009), Novo Nordisk A/S

15.2.6 Diffusion in Subcutis Injected soluble substances will diffuse in the winded, irregular extracellular space in the s.c. tissue. Although Fick’s law of diffusion pressumably still applies locally in the tissue, the overall diffusion will be restricted. This restriction can be approximated by dividing the diffusion constants for the soluble substances by a factor called the tortuosity. Thus, the corrected Fick’s law reads: Dx 2 @cx D r cx ; @t

(15.12)

where Dx is the diffusion constant of the soluble species x. Values for DH , DD , and DM have been found by Oliva et al. [52] to be 7:81107 cm2 s1 , 1:13106 cm2 s1 , and 1:60 106 cm2 s1 , respectively, while Lin and Larive [46] found DD to be 1:38 106 cm2 s1 . The value of the tortuosity for subcutis is typically 1.5–1.7 [59]. Diffusion causes a concentration gradient from high concentrations to low concentrations in the subcutis and it effectively dilutes the injected insulin. For soluble insulin, the concentration is initially almost constant in the injection area. At the boundary between the initial injection area and the surrounding tissue, there will be a gradual concentration gradient, since the injection fluid will mix with the interstitial fluid to a certain degree. The effect of diffusion will be largest for small volumes/doses, since the boundary is relative large compared to the volume.

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NPH insulin is too large to diffuse in the tissue [42] so the heaps will remain at their original position while they dissolve. The heaps are scattered in subcutis as described in Sect. 15.2.1 and they will act as local sources of insulin, making the overall absorption and diffusion process complex and anisotropic.

15.2.7 Local Subcutaneous Blood Flow The local s.c. blood flow (SBF) is a major source of the inter- and intra-subject variability observed in clinical trials [26]. Increased SBF increases the absorption rate of insulin formulations [32]. In turn, SBF has been known to increase with injection depth [30], body temperature [39], and hormonal activity [11]. Williams et al. [69] found that administered insulin could itself increase local SBF. No evidence for flow dependent degradation has been found. Under normal physiological conditions SBF can vary more than ˙50% from the normal flow which is around 4–6 mL min1 .100g/1 [30]. The consequent change in absorption rate from the s.c. depot can be quantitatively described as a change in the absorption constants Bx . Simultaneous measurements of the SBF and the terminal s.c. absorption rate factor (corresponding to a flow-dependent monomer absorption rate constant BM ) were done by Hildebrandt et al. [32]. The data suggests a saturable Michaelis–Menten relation as proposed by Claessen and Mortensen [7]: Bx D

Bxmax v ; kx C v

(15.13)

max D 0:018 min1 and the having values of the maximum absorption rate BM Michaelis constant kM D 4:3 mL min1 .100g/1 corresponding to monomeric soluble insulin. The reason for the dependence of the absorption rate on SBF is that the absorption rate depends on the concentration gradient over the capillary wall and not just the concentration in subcutis. If the SBF is very low, an equilibrium is established and the absorption is slowed down. At not too high constant flows, the absorption rate will depend mostly on the concentration of insulin in the s.c. depot and can thus be approximated as the first order process described in Sect. 15.2.4. At very high SBF, the main limiting factor for the absorption is the capillary permeability and, if the permeability is independent of the SBF, the absorption rate will therefore converge towards Bxmax .

15.2.8 Plasma Insulin As described in Sect. 15.2.7, the absorption rate from the s.c. depot into plasma depends primarily on the depot concentration of the given insulin species for not too small flows. Once in plasma the insulin will rapidly distribute itself and


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be eliminated via various routes. It is a commonly accepted hypothesis that the clearance of insulin from plasma primarily takes place in the liver and the kidneys as reviewed by Duckworth et al. [14]. However, as found by Castillo et al. [6], insulin receptors act as an important buffer for plasma insulin and may dominate the insulin kinetics. This is supported by a report by Colding-Jørgensen [9], who concluded that most of the clearance of insulin once in plasma is receptor mediated (internalization). The elimination of insulin in plasma is therefore not only first order process, but rather a mixture between a saturable nonlinear component and a non-saturable linear component so the clearance changes substantially within physiological plasma concentrations [9]. It is, however, not within the scope of this chapter to describe this process, since the focus is on the processes of subcutis. Therefore, a simple one compartment model for plasma with first order elimination is assumed, in accordance with Søeborg et al. [62] and Mosekilde et al. [49]. This implies that the total flux of non-degraded insulin from the depot JT is equal to: @nT D JT D @t

Z .6 BH cH C 2 BD cD C BM cM / @Vsc ;

(15.14)

depot

where nT is the remaining amount of insulin in the s.c. depot measured in insulin monomers and @Vsc is a volume element of the injection depot. The plasma insulin equation therefore reads: dcpl 1 D JT Apl cpl ; dt Vpl

(15.15)

where cpl is the plasma concentration of monomer insulin, Vpl is the distribution volume, and Apl is the insulin elimination constant in plasma. The later is related to the metabolic insulin clearance rate by M CR D Apl Vpl . A value for the distribution volume, which is comprised of both the interstitial and the plasma volume, has be found to be 12 L in a study by Kraegen and Chisholm [40]. In accordance with Mosekilde et al. [49] and Søeborg et al. [62] the two volumes have been taken as one due to the fast onset of equilibrium between the two compartments. The value of Apl is approximately 0:1 min1 [22].

15.3 Constructing a Mechanistic Model In the previous section we have investigated and described the most important mechanisms governing the s.c. absorption of injected insulin. They will form the basis for the formulation of a mathematical model described in this section.

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15.3.1 Model Formulation Based on the mechanisms derived in the previous section, a set of equations can be formulated to describe the temporal development of depot and plasma insulin concentrations mathematically. The model is stated in the following coupled equations for the concentrations in the injection depot of NPH insulin, protamine (P), hexamer insulin (H), and dimer insulin (D) along with a compartment for plasma insulin (pl): @cNPH D ˇ cNPH .1 ˛ cH cP / ANPH cNPH @t

œ ™

(15.16a)

degraded

dissolution to H and P

DP 2 @cP D ˇ cNPH .1 ˛ cH cP / AP cP BP cP C r cP @t

œ ” ” ˜ P from dissolved NPH

degraded

absorbed

(15.16b)

diffusion

@cH D ˇ cNPH .1 ˛ cH cP / C PDH .KDH cD 3 cH / @t

œ H • D transition

H from dissolved NPH

DH

r c ” ” ˜

AH cH BH cH C degraded

absorbed

2

(15.16c)

H

diffusion

DD 2 @cD D PDH .KDH cD 3 cH / AD cD BD cD C r cD (15.16d) @t

” ” ˜ H • D transition

degraded

Z

@cpl 1 D @t Vpl

absorbed

•

.BH cH C BD cD / @Vdep Apl cpl

depot

diffusion

(15.16e)

removed

depot insulin absorbed into plasma

where KDH is time dependent according to (15.4). The reader will note that the terms in all the differential equations each correspond to a specific chemical or physiological mechanism related to the absorption process. To enable a simpler description, the following simplifications have been made: • Based on the results of Sect. 15.2.3 it is assumed that the total surface area of the heaps is proportional to total volume of the heaps, so ONPH / cNPH . • The crystal equations (15.5) and (15.11) describing dissolution and macrophage breakdown of the NPH heaps, respectively, have been redefined in terms of new parameters ˇ D s=L and ANPH D =L, since the values of s and have not been found in the literature.


345

• Despite that macrophages do not reach their full number at the injection site until a few hours after administration, ANPH is assumed to be independent of time. • Since insulin dimers and monomers have similar absorption and diffusion constants, they have been pooled in a single compartment, denoted dimers (D). • All concentrations concentrations are measured in units of monomers/dimers which implies the following variable and parameter transformations: cNPH ! 1 1 1 1 3 cNPH , cH ! 3 cH , cP ! 3 cP , 9 ˛ ! ˛, and 3 KDH ! KDH . Besides the plasma compartment, the model is a system of coupled partial differential equations. This means that the equation system describes the evolution of the various concentrations over both time and space. Based on the histological cross sections shown in this chapter, spherical symmetry of the spatial component of the equations is assumed.

15.3.2 Parameter Estimation In the previous section the equations of the mechanism-based model were established. This section will focus on determining the parameters of the model, of which some have already been stated in Sect. 15.2. Others could not be found in the literature and will be determined using a parameter estimation procedure based on clinical data. The purpose of the present model is not to describe a specific dataset. Consequently, population-based parameter estimation is not necessary, nor is it practically feasible. The model was developed to describe general tendencies attributed to s.c. injections and insulin mixtures which is why mean clinical plasma profiles will be a suitable input for a simple fitting procedure to estimate values of the parameters. Such data was found in the literature for different insulin types, concentrations, and doses for populations of healthy volunteers (Table 15.1). Data was selected based on availability and on whether or not the plasma profiles converged to zero concentration within the time frame of the sampling. Many plasma curves in the literature start at zero concentration, but converge to a value larger than zero for t ! 1, indicating that the baseline insulin concentration shifted during the experiment. This would make it difficult to determine whether the model

Table 15.1 Table of the datasets used in model validation and parameter estimation (modified from the given reference) Molecule HI HI IA HI IA

Soluble/NPH 100/0 100/0 100/0 30/70 30/70

Concentration 40 U mL1 100 U mL1 100 U mL1 100 U mL1 100 U mL1

Mean dose 15 U 15 U 16 U 23 U 23 U

Subjects 18 18 10 24 24

Reference Hübinger et al. [38] Hübinger et al. [38] Heinemann et al. [27] Weyer et al. [67] Weyer et al. [67]

346


Table 15.2 Table of model parameters for the soluble insulin system (15.16c)–(15.16e). The values have been determined either from the literature, using RMS error fitting X, or RMS error fitting restricted to near literature values (X). These parameter values are used in simulations unless otherwise stated Description Term Value Fitted Primary referencea Rate constant Eq. constant (s.c.) Eq. constant (vial) Eq. constant (s.c.) Eq. constant (vial) Time constant Abs. constant Abs. constant Deg. constant Deg. constant Diff. constant Diff. constant Interstitial fraction Tortuosity Sol. insulin bioavail. Elim. constant

PDH KDH1 (HI) KDH0 (HI) KDH1 (IA) KDH0 (IA) DH BH BD AH AD DH DD " fS Apl

> 0:5 min1 0:010 mL2 U2 4:0 mL2 U2 4:0 105 mL2 U2 0:016 mL2 U2 0:09 min1 1:7 103 min1 7:8 103 min1 9:2 104 min1 4:2 103 min1 4:7 105 cm2 min1 8:4 105 cm2 min1 0:1 1:6 0:65 0:11 min1

Mosekilde et al. [49] Søeborg et al. [62] Hvidt [37] Brems et al. [5] Estimated (X) Søeborg et al. [62] (X) Brange et al. [4] (X) Brange et al. [4] Neal [50] Neal [50] Oliva et al. [52] Lin and Larive [46] Crandall et al. [11] Sharkawy et al. [58] Søeborg et al. [63] (X) Hansen [22] Kraegen and Dist. volume Vpl 12 L (X) Chisholm [40] a Either a reference to a stated value or to the basis for a parameter fitting interval or calculation. (X)

is able to reproduce the observed AUC/bioavailability. Hence, all the selected data curves converge to zero. The fitting procedure used to estimate the model parameters was a simple root mean square (RMS) error method, in which a scan of unknown or uncertain parameters within specified regions of parameter space was performed. Based on parameter combinations with the lowest overall RMS error, a suitable set which reproduced the clinical data properly was selected. Since the model consists of the soluble insulin system in (15.16c)–(15.16e) and the NPH/soluble system in (15.16a)–(15.16e), the fitting procedure was performed separately for each system. The resulting parameters are shown in Table 15.2. Figure 15.8 shows the clinical data (plasma time curves) for the soluble insulin studies in Table 15.1 and Fig. 15.9 shows the corresponding simulations for the parameters shown in Table 15.2. Below are some comments concerning the parameters and the fitting procedure: • The system parameters are the same for the three simulated plasma curves. Only concentration, dose, and insulin type have been changed according to the experimental conditions. • Unless otherwise stated, the parameter values apply to both HI and IA.

15 Absorption Kinetics of Insulin Mixtures after Subcutaneous Administration 350 Plasma concentration [pM ]

Fig. 15.8 Clinical pharmacokinetic profiles described in Table 15.1 showing the plasma concentration of insulin (either HI or IA) as a function of time. Modified from Hübinger et al. [38] and Heinemann et al. [27]

Soluble HI U40 Soluble HI U100 Soluble IA U100

300 250 200 150 100 50 0

0

2

4 6 Time [h]

8

10

350 Plasma concentration [pM ]

Fig. 15.9 Simulations corresponding to the data curves in Fig. 15.8. The system parameters are the same for all three simulations. Only insulin type, concentration, and dose have been changed in accordance with the experimental conditions

347

Soluble HI U40 Soluble HI U100 Soluble IA U100

300 250 200 150 100 50 0

0

2

4 6 Time [h]

8

10

• PDH is assumed to be infinitely large which for all practical purposes means that it should be larger than 0.5 min1 . • Due to the large range of parameter values for KDH1 in the literature, it was included in the fitting procedure. • Pooling monomers and dimer implies that KDH1 for IA is reduced by a factor of 200–300 compared to HI (250 is used). KDH0 for IA is assumed to be 400 times larger than KDH1 for IA. • BH and BD have been allowed to vary slightly from the values in the literature. This is because the slope of the disappearance curves described in Sect. 15.2.4 most likely also holds a component of the disappearance of degraded insulin. This means that the slope, e.g. for the monomers/dimers, should be comparable to or slightly larger than AD C BD . The fitted parameter values reflect this. • The time constant DH was not found in the literature, but its lower bound was determined by assuming that the auxiliary substances disappear within the first 30–60 min.

348


• Soluble insulin bioavailability fS is assumed to be independent dose and concentration in accordance with the findings of Sect. 15.2.5. This means that the absorption and degradation constants are related according to: fS D

Bx Ax C Bx

(15.17)

• Although the data curves are from healthy volunteers, most parameters can be assumed to be the same for type I and II diabetic patients, since they relate directly to the s.c. depot. Once the parameters were estimated for the soluble system, the parameters of the NPH and protamine equations in the NPH/soluble system were estimated using the same fitting procedure, but with fixed parameter values for the soluble system (Table 15.2). The estimates are shown in Table 15.3. The clinical data for the NPH insulins of Table 15.1 is plotted in Fig. 15.10. The corresponding simulations using the parameters of Table 15.2 as well as the additional parameters of Table 15.3 are shown in Fig. 15.11. Below are some comments regarding the NPH parameters:

Table 15.3 Table of model parameters for the NPH insulin system (15.16a)–(15.16e). The values have been determined either from the literature, using RMS error fitting X, or RMS error fitting restricted to near literature values (X). These parameter values are used in simulations unless otherwise stated Description Term Value Fitted Primary referencea

Fig. 15.10 Clinical pharmacokinetic profiles described in Table 15.1 showing the plasma concentration of insulin (either HI or IA) as a function of time. Modified from Weyer et al. [67]

Plasma concentration [pM]

Inhib. constant ˛ 9 103 mL2 U2 X Søeborg et al. [63] Dissolut. constant ˇ 5 103 min1 X Søeborg et al. [63] Deg. constant ANPH 4 104 min1 X Søeborg et al. [63] Abs. constant BP 7:8 103 min1 Estimated Deg. constant AP 4:2 103 min1 Estimated Diff. constant DP 8:4 105 cm2 min1 Estimated a Either a reference to a stated value or to the basis for a parameter fitting interval or calculation.

200

70 % NPH HI 70 % NPH IA

150 100 50 0

0

4

8

12 Time [h]

16

20

24

Fig. 15.11 Simulations corresponding to the data curves in Fig. 15.10. The system parameters are the same for all three simulations. Only insulin type, NPH-to-soluble ratio, and dose have been changed according to the data



200

349

70 % NPH HI 70 % NPH IA

150 100 50 0

0

4

8

12 Time [h]

16

20

24

• System parameters are the same for the two simulated plasma curves. Only dose, insulin type, and NPH-to-soluble ratio have been changed in accordance with experimental conditions. • Parameters describing absorption, degradation, and diffusion of protamine were assumed to be identical to those for the insulin dimers, since the molecular weights are comparable. • The NPH/protamine parameters are applicable to both healthy volunteers and type I/II diabetics. Model parameters related to SBF dependence of absorption rates according to (15.13) are estimated to be kD D 4:3 mL min1 .100g/1 and BDmax D 0:018 min1 based on Hildebrandt et al. [32]. The value of kx is assumed to be the same for all soluble species, while Bxmax is scaled according to Bx (at normal SBF) for hexamer insulin and protamine. The simulations shown in Figs. 15.9 and 15.11 reproduce the corresponding clinical data in Figs. 15.8 and 15.10 quite well, including AUC and peak concentration (cmax ). Since the system parameters are required to be the same for all simulations (except KDH1 and KDH0 which differ for HI and IA) the number of degrees of freedom is reduced substantially and the model would therefore be less likely to reproduce all the clinical data, if it did not describe the mechanisms involved properly. However, if the mechanisms or parameters are different between the different populations in the data sets, the parameter estimation would be less accurate.

15.4 Understanding and Predicting Clinical Results Having established a mechanistic model describing the s.c. absorption of insulin mixtures this section aims to use the model to analyze some aspects of the pharmacokinetic plasma profiles: Variability, bioavailability, and cmax .

350


15.4.1 Variability For intermediate and long acting insulins, a flat insulin plasma concentration profile is desirable in order to avoid e.g. night time hypoglycaemic events. In clinical studies, however, the absorption kinetics of s.c. injected insulin is associated with substantial variability. For 100% NPH HI the variability has been reported to be 10–15 % higher than that of soluble HI [24]. An example of the variability in plasma curves can be seen in Fig. 15.12 which is a plot showing the individual plasma profiles following the administration of 100% NPH insulin in healthy volunteers. Inspection of the figure reveals that there is variability both between subjects and over time for a given subject. One subject experiences cmax after 1 h, while another experiences it after 24 h. Some of this may be explained by taking into account the local SBF. The SBF in a local capillary next to an insulin heap may be virtually zero and the insulin concentration will thus be high and in equilibrium across the capillary. If the SBF in that capillary is increased due to e.g. heat or hormonal activity, a burst of insulin will reach the systemic circulation and result in a peak in the insulin concentration profile. Consequently, cmax and the time (tmax ) to cmax are poorly defined for the population, but also for the individual subject. Although the mean concentration curve looks flat, the large variability is simply leveled out when calculating the mean. Furthermore, the values of the individual cmax occurs at different times (tmax ) so the mean cmax is of reduced scientific value. By using the established model with the SBF and absorption rate constants Bx varying according to (15.13), a typical 24 h variation of SBF can be simulated during a basal injection of insulin. Sindrup et al. [61] studied SBF during five phases of sleep in healthy volunteers and found it to increase 140 % for approximately 100 min during the so-called hyperaemic phase. Data from this study was used to simulate the corresponding effects on the insulin concentration in plasma following s.c. injection of NPH insulin 4 h prior to the first phase of sleep. The simulation is shown in Fig. 15.13.

Fig. 15.12 Individual and mean plasma profiles following s.c. administration of 100% NPH HI in healthy volunteers


200 Individual Mean

150

100

50

0

0

4

8

12 Time [h]

16

20

24

15 Absorption Kinetics of Insulin Mixtures after Subcutaneous Administration 80

400

60

300

40

200 20

0

100 0

5

10 15 Time [h]

20

0

110 Relative bioavailability [%]

Fig. 15.14 The relative bioavailability of soluble and 100% NPH HI corresponding to injections of 20 U of U100 insulin at different time invariant values of the SBF. All values of the bioavailability are relative to U100

Normal SBF 500 Relative SBF [%]


Fig. 15.13 A simulation showing the plasma profile of a U100 injection of 20 U of 100% NPH HI (blue) with SBF varying according to the SBF curve (orange) modified from Sindrup et al. [61]. The injection is simulated to take place 4 h prior to the first phase of sleep

351

100

90 Soluble HI/IA 100 % NPH HI 80 50

75

100

125

150

Relative SBF [%]

It is evident from the figure that natural variations in SBF can cause large excursions in plasma insulin and thus provide an explanation for at least some of the time dependent variability observed in Fig. 15.12. The hyperaemic phase may also provide an explanation for night-time hypoglycaemic events for diabetic patients. Changes in local SBF can also affect the bioavailability. This is illustrated in Fig. 15.14 which shows the bioavailability as a function of the (time invariant) SBF for injections of 20 U of U100 soluble and 100% NPH HI. The simulation show that increased SBF yields an increased bioavailability, while decreasing SBF causes a notable drop in the bioavailability. The later is due to a decreased absorption rate which in turn increases the time available for s.c. insulin degradation, vice versa for increased SBF. It is noted that the bioavailability of NPH insulin is slightly more sensitive to changes in SBF compared to regular HI. In the case of increased SBF it is because the faster absorption of soluble species has the additional effect of increasing the dissolution of NPH heaps (due to lower hexamer presence), resulting in less time for NPH degradation. Another type of variability described in the previous sections is associated with the formation and dissolution of NPH heaps. Depending on where in the s.c. tissue

Fig. 15.15 The peak concentration (cmax ) of 100% NPH HI for varying values of the dissolution constant ˇ relative to the standard value shown in Table 15.3 for injections of 20 U of U100 insulin. All cmax values are relative to U100


Relative peak concentration [%]

352 120 100 80 60 40 10

40

70 100 130 Relative β [%]

160

190

the injection center is situated and the thickness and specific layout of the tissue, heap size distributions may vary substantially between injections, causing both inter- and intra-subject variability. This is included in the combined dissolution constant ˇ. Heap distributions with different mean values may then be simulated by changing ˇ from the standard value shown in Table 15.3. This has been done in Fig. 15.15. It is seen that especially a decrease in ˇ (corresponding to a smaller heap surface/volume ratio) yields a substantial decrease in cmax . This phenomenon also offers a plausible explanation for some of the variability observed for NPH mixtures.

15.4.2 Bioavailability and Peak The absorption kinetics of s.c. administered insulins depends on the concentration of the injected drug as can be seen in Fig. 15.8. For instance, there are indications that the bioavailability is negatively correlated to the drug concentration for NPH insulins as described in Sect. 15.2.5. However, the clinical data to support this claim is sparse, but the absorption model described in this chapter can be used to simulate the kinetic effects of increasing the drug concentration (for the same dose). This has been done in Fig. 15.16 which shows the bioavailability of various soluble and NPH insulin mixtures corresponding to injections of 20 U of U100 insulin. The bioavailabilities are relative to the respective U100 formulation of the given insulin. The simulations predict a notable drop in bioavailability for NPH insulin mixtures with concentration which gets larger with increasing fraction of NPH. For a HI formulation of 100% NPH this would make it difficult to achieve U100 bioequivalence2 for higher concentrations. The decreasing bioavailability is due to the fact that during dissolution of the NPH heaps, the solute with protamine and

For AUC and cmax the 90 % confidence interval for the ratio of the test and reference products should be contained within the acceptance interval of 80.00–125.00 % [10]

2

Fig. 15.16 The simulated relative bioavailability of different insulin types as a function of the administered concentration (dose 20 U). All values of the bioavailabilities are relative to the respective U100 value

Relative bioavailability [%]

15 Absorption Kinetics of Insulin Mixtures after Subcutaneous Administration 110 100 90 Soluble HI/IA 70 % NPH HI 70 % NPH IA 100 % NPH HI

80 70 0

100

200 300 400 Concentration [U mL −1 ]

110 Relative bioavailability [%]

Fig. 15.17 The simulated relative bioavailability of soluble HI and IA, respectively, for varying values of the hexamer degradation constant relative to the standard value shown in Table 15.2. All values of the bioavailability are relative to U100

353

500

Soluble HI Soluble IA

105 100 95 90 10

40

70 100 130 Relative AH [%]

160

190

insulin surrounding the heaps will have an increasingly higher concentration of hexamer insulin. This, in turn, slows down the net dissolution of the heaps due to recrystallization, increasing the time available for degradation by macrophages. The effect is less pronounced for IA due to the reduced tendency for such insulin to form hexamers. In the model, the bioavailability of soluble insulins is independent of the drug concentration. This is because the degradation constants are balanced, so the fraction of hexamers and dimers degraded during the entire absorption process is the same (enforced in (15.17)). Although no evidence has been found for the dependence of soluble insulin bioavailability on concentration, data is sparse. It may well be that the change is small enough to elude detection. To investigate the effect of unbalanced soluble (hexamer) insulin degradation, Fig. 15.17 shows the simulated relative bioavailability for soluble HI and IA when changing the relative value of the hexamer degradation constant AH . Inspection of the figure reveals that the relative bioavailability changes almost linearly with AH , and that the effect for HI only becomes substantial for a large change in the parameter. The effect is limited for IA which is again due to the low

Fig. 15.18 The simulated relative peak concentration (cmax ) of different insulin types as a function of the administered concentration (dose 20 U). All cmax values are relative to the respective U100 value


Relative peak concentration [%]

354 150

Soluble HI Soluble IA 70 % NPH HI 70 % NPH IA 100 % NPH HI

120

90

60

30

0

100 200 300 400 Concentration [U mL −1 ]

500

concentration of hexamers. A similar curve for the dimer degradation constant AD would have a larger effect for IA. It is well known that the cmax of a plasma profile changes with the dose of the injected insulin. In a simple linear system with only one species of insulin (monomer), cmax would not depend on the concentration of the formulation. However, since multiple species of insulin interact through constant chemical transitions, cmax will depend on the concentration, a phenomenon which has been simulated in Fig. 15.18. As is seen from the figure, the variation in cmax with concentration is substantial. For most of the insulin shown types this phenomenon alone would make it difficult to obtain bioequivalence between a U100 and higher concentrations. The effect is most pronounced for HI based insulins, especially NPH mixtures. This is again due to the increased concentration of insulin hexamers at higher total concentrations which cause a longer absorption time for NPH insulin in the depot. Since IA based insulins have a much lower concentration of hexamers they will be less sensitive to the concentration effect.

15.5 Conclusion In this chapter, the absorption kinetics of subcutaneously injected insulin mixtures has been described and the most important mechanisms have been determined via an extensive study of the literature. The injection depot itself is an important aspect of the absorption process as the depot varies in shape from injection to injection. Furthermore, the depot is about ten times larger than the injection volume due the presence of fat cells and connective tissue. NPH crystals in injected suspensions are filtered in the tissue and retained near the center of the injection site in large concentrated heaps. The dissolution of the heaps will take place from the surface and the kinetics will depend on the surface-to-volume ratio.


355

Soluble insulin is injected as oligomers in chemical equilibrium which shifts towards lower molecular weight species when the auxiliary substances in the injection fluid are absorbed. In subcutis, soluble insulin species are degraded enzymatically, while NPH heaps are degraded by invading macrophages. The absorption of insulin can take place via the capillary wall or the lymphatics for smaller and larger molecules, respectively, while the NPH heaps can only be absorbed by dissolving into soluble insulin which is subsequently absorbed. The absorption rate is increased with the subcutaneous blood flow. To obtain a quantitative description of the absorption process, the individual mechanisms were combined in a mathematical model to account for their complex interactions. The parameters of the model were determined from the literature when possible, otherwise by a simple fitting procedure based on multiple clinical data sets. With the determined parameters, the model provided a satisfactory description of all the clinical data. Several important aspects of insulin absorption were then investigated using the established model. The variability associated with clinical plasma profiles was explained by simulating the effect of changed subcutaneous blood flow as well as differences in the size distribution for the NPH heaps. The effects of injected insulin concentration on both the bioavailability and cmax were simulated for several mixtures of NPH and soluble insulin. The bioavailability and cmax were predicted to decrease with increasing concentration for NPH mixtures (more with increasing NPH fraction), while the bioavailability for soluble insulins was found to be independent of concentration. The overall effect was lower for aspart based insulins. The concentration effect could be explained by increased concentrations of insulin hexamers and subsequent recrystallization which slows down the absorption and leaves more time for macrophage degradation of the NPH heaps. The model proved useful for simulating mean plasma profiles of subcutaneous injections of several different types of insulin. The individual variations in plasma profiles were not simulated, since the aim of the model was to describe general effects rather than to describe a specific dataset. These effects were captured properly by the model and it was able to predict the effect of increasing the drug concentration, something which otherwise would be unpredictable due to the complex nature of the absorption process. Although the model was constructed to describe insulin mixtures of NPH and soluble human or aspart insulin, most of the elements in model relate to the injection process rather than the specific drug. By substituting the elements relating to the chemical transitions in subcutis with those for other subcutaneously injected drugs, a quantitative description of other drugs could also be obtained. Acknowledgements Mette Poulsen, Dan Nørtoft Sørensen, Bente Eyving, and Maria Thomsen from Materials and Device Characterisation as well as Susanne Primdahl, Maibritt C. Pedersen, and Jonas Kildegaard from Histology & Delivery at Novo Nordisk A/S are acknowledged for providing and commenting on Figs. 15.1 and 15.2. Erik Hasselager from Novo Nordisk A/S is acknowledged for providing and commenting on the rest of the histological samples presented

356


in this article. Prof. David R. Katz, editor of International Journal of Experimental Pathology, is acknowledged for providing us with the opportunity to reprint Fig. 15.5. This work was supported by the European Union through the Network of Excellence BioSim, Contract No. LSBH-CT-2004005137.

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