1.      Introduction 

Over the years, much research effort has been put to clarify insulin pharmacokinetics (the plasma levels of insulin over time) and pharmacodynamics (the resulting effects on glucose over time). It is reported [1] that insulin pharmacokinetics is governed by a multitude of factors of importance, such as absorption process, distribution that includes binding to circulating antibodies (if present) and to insulin receptors, and its ultimate degradation and excretion. The duration of absorption naturally determines the time for equilibration between the amounts injected and absorbed. Therefore, special attention has been devoted to insulin absorption [2]. Radiotracer measurements have shown that absorption rate of iodine (I¹²⁵) labeled insulin, remaining at the injection site, depends on many factors, such as insulin form (dissolved, amorphous or crystalline) and concentration, injection volume, blood flow, and presence or absence of degradation at the injection site. For instance, if the injection is given in the abdominal area it results in the most rapid insulin absorption after subcutaneous administration, and the femoral region in the slowest resorption. The measurements were carried out with dissolved, amorphous and crystalline (Lente, Ultralente) insulins from pig and beef. It was found that external monitoring of the radioactivity remaining at the site of injection provides a reliable measure of the amount of insulin remaining in the tissue [2].

To suppress “Peaks and valleys” in blood concentration some insulin formulations contain micro-crystals, usually of Zn-insulin. Dissolving more slowly than the amorphous drugs, such crystals ensure a basal insulin replacement, and a prolonged therapeutic effect [3-5]; in some cases, up to one day. Preferred must be crystals of narrow size distributions because such crystals dissolve in a nearly parallel way. Otherwise, if differently sized, the smaller crystals dissolve sooner than the larger species, and with their disappearance the drug starts to deplete. 

To reduce crystal polydispersity internal seeding of equally-sized crystals was suggested recently [6], its advantage being avoidance of crystal grinding, sieving and any introduction of impurities. Nucleating all crystals quasi-instantaneously (for a sufficiently short time) and then abruptly depriving the system of its capability to further produce nuclei impels all seed crystals to grow from an initial nearly equal size, which is the critical nucleus size (or slightly larger). Then, during a slow uniform process, these crystals overgrow further to form the nearly monodisperse crystalline matter desired.

The long-acting basal insulins are combined with rapid acting insulin analogues, mainly intended for prandial doses. Varied patients and lifestyles necessitate a variety of basal and bolus doses. Therefore, selection of a proper basal and bolus balance must always address variations in an individual’s age, weight, gender, activity, live stile, and general physical condition. One of the four processes in personalized medicine strategy is to tailor individual insulin dose for each patient [7]. In view of this requirement, equilibration between insulin supply from dissolving crystals and its disappearance from the subcutaneous depot is considered here. In particular, we study the duration of the balance between insulin supply resulting from crystal dissolution and its consumption. The balance duration is estimated assuming insulin elimination obeying first order chemical reaction law[8].So, basal insulin doses are calculated from insulin half-life time in human body. 

2.                  Model Considerations 

For establishing the balance between insulin supply (resulting from dissolution of crystals in the anti-diabetic formulations) and its consumption in human body,we consider the dissolution driving force,the so-calledundersaturation, (c_e - c), where cis the actual concentration of the substance in bulk solution and c_e is solubility. The concentration, cincreases systematically during crystal dissolution, reaching finally c_e, when the dissolution driving force disappears.Then the dissolution process stops, so that the (dissolving) crystals cannot render concentration higher than c_e. Noyes-Whitney equation [9] describes the diffusion-controlled crystal dissolution flux,which is the amount of massdissolved per unit time and per unit dissolving area: 

dM/dt = –S_cryst(D/δ_cryst)(c_e - c)(1) 

where Mis mass of solid remaining at a specific timetand S_crystis the total crystal surface;D is the diffusioncoefficient ofthe solute and δ_crystis the diffusion layer thickness; the latter depends on temperature, solvent viscosity, dissolving surface geometry, and stirring solvent hydrodynamics.

Insulin disappears gradually from plasma in man, the liver and the kidneys being major sites of insulin degradation, and about 10% are appearing in the urine. Proteolytic degradation of insulin occurs both at cell surfaces and in the lysosomes. Therefore, equalization should be maintained in the body [8] between insulin supply (due to crystal dissolution) and its disappearance from the subcutaneous depot. The equilibration has another important impact, because it is known [10] that the receptors vary in number inversely with the insulinconcentration to which they are exposed. Therefore, keeping constant insulin concentration should ensure a constant number of receptors.

Diffusion-control of insulin elimination is assumed here. Thus, the equalization between the crystal dissolution flux and the insulin elimination flux yields:

S_cryst(D/δ_cryst)(c_e–C) = S_recept(D/δ_recept)C                                                                              (2)

where C is the equilibration concentration and c_recept= 0. Thus:

C =c_e/[1+(S_receptδ_cryst/S_crystδ_recept)]                                                                                        (3)

This result shows why the size of the crystals should be adjusted to the insulin receptors’ capacity. Therefore, personalmedical data will be needed to tailor individual insulindose for diabetes treatment. The ideal case,S_cryst = S_recep, and δ_cryst = δ_recept,is shown inFigure 1, where the two concentration gradients (the one from the crystals and the second one towards the receptors) are equal.(In such a special case,C =c_e/2.) 

Indeed, the blood stream diminishesthe thickness of the diffusion layer; but it never vanishes.Insulin diffusion supply to the receptors can be augmented, but so may insulin supply from dissolving crystals be increased. So, the crystals are dissolved only to the extent, which corresponds to the insulin demand. It is known thatin disease situation blood rheology is altered, and plasma and whole blood viscosity are markedly higher in diabetes cases.

Approaching insulin disappears from plasma in man, we accept first order kinetics law (see [8]): 

c = c_o exp(– кt)                                                                                                      (4) 

where к [s^-1] is rate constant.It is known that in this case к is related to the half-life timeτ_1/2, τ_1/2 = ln2/к,and that both кand τ_1/2do not depend on the initial concentration. Indeed, biological half-life depends on the route of exposure. A typical half-life time, τ_1/2 ≈4 hours is observed after a subcutaneous injection; afteran intramuscular injectionτ_1/2is 2 hours[10]. 

3.                  Discussion

Although the site of administration is one place in the subcutaneous tissue,the insulin molecules from theinjection depotare absorbed via the capillaries into the blood stream. So, the latter conveys the physiologically active insulin monomers to the insulin receptors that are widely distributed throughout the whole human body. That is why, although highly simplified, our model canreflect adequately the pharmacokinetic and pharmacodynamic situation.The course of subcutaneous depot dissipation (i.e. insulindisappearance from it) is presented in Figure 2. The upper curve shows that about 7% of subcutaneous depot content is remaining after 15 hours (i.e. dissolved crystalvolume 93%).Correspondingly, about 99% of the crystal volume is dissolved during the same time after intramuscular injection (see the lower curve in Figure 2). The time course of insulin consumption presented inFigure 2 explains the direct (radiotracer) measurements of iodine (I¹²⁵) labeled residual amount of insulin at the site of injectionsee [1].The same exponential decay is measured in [1] but with half-life time τ_1/2 = 8 h. Keeping in mind the considerable injection regional differences [2] and the intrapatient variability in insulin absorption (which may reach 35%,see [1]),the results seem to correlate. Therefore, personalized tailoring of basal insulin dosescan be made based on the c vs t plots inFigure 2.

4.                  Materials and Methods

 Crystallization of rhombohedral insulin crystals (Figure 3) was studied experimentally using two different insulin sorts, from BioChemika (BioChemika, ≥ 85% (GE), ~24 IU/mg) and from SIGMA, Denmark, Lot # 080M1589V.Identical crystallization conditions were used withinsulin concentrations between 4.0 and 8.0 mg/ml. The crystallization solution included also 0.001M HCl, 0.05M citrate buffer at pH = 6.98, 0.005M ZnCl₂, 15% (v/v) acetone.The experimental procedure is simple. The weighted amount of insulin was dissolved in highly diluted HCl. Then consecutively citrate buffer (pH = 6.98), ZnCl₂and finally acetone was added. Bi-distilled water from quartz distiller was used. 

Insulin crystals are transparent and colorless, and to distinguish them in the water solution (from amorphous precipitates or from the precipitating agent crystals) we apply interference contrasting via differential image splitting. The application of this method required the use of cuvettes of optical quality. Therefore, custom made quasi-2D all-glasscells were used [11]. In such a cell, the crystallization proceeded in the thin solution layer confined in the gap between the two glass plates, the gaps being varied in series of cells, from 0.05 cm (volume 0.35 cm³) down to 0.002 cm.For cell production two circular glass plates of optical quality were welded in exactly parallel position (the letter being controlled interferometrically by using a Mach-Zehnder scheme). An advantage of the 2D-cell is that it has relatively small inside volume, and is characterized by low solute consumption. Besides, the cell allows perfect cleaning.The most important benefit of the cell is the rapid imposing of supersaturation shifts, via appropriate temperature changes. In view of temperature dependence of insulin solubility, supersaturation was altered using water baths having the desired temperature. Simply, the all-glasscell containing the protein solution was immersed in the water.

Indeed, systems with acetone are inappropriate for medical application; the crystallization technology used to produce therapeutic insulin crystals is quite different. This notwithstanding, the principle to steer the crystallization process aimed at reducing crystal polydispersity can find practical application.

And finally, dissolution of rhombohedral insulin crystals in human blood plasma was studied in-vitro. Sufficient details allowing replication of the experimental studies are provided in the original paper [12]. 

5.      Conclusion

Consideringthe balance betweeninsulin supply from dissolvingequally-sized crystals and insulinelimination, we explain theprolonged therapeutic effect achieved with low frequency of injection in parenteral controlled release.The calculation model presented here can help in designingpersonalized schemes for treatment of people with diabetes mellitus,by differentiatingbasal and bolus insulin doses. The insulin crystal mass in the drug formulation must be adjusted adequately. 

6.      Acknowledgements: The authors would like to acknowledge the contribution of the COST Action CM1402 Crystallize. The authors are grateful to the National Science Fond at Bulgarian Ministry of Science and Education for the financial support by NSF project No Т02/8/12.12.2014.

Figure 1: Schematic (not to scale) presentation of theequilibration between insulin crystal dissolution and insulin elimination. (Inthis ideal case:S_cryst = S_recep, andδ_cryst = δ_recept.).

Figure 3:Rhombohedral insulin crystals formed by {101} growth faces. (Crystal sizes are ~ 200 to 400 μm.)The crystals are visualized by interference contrasting via differential image splitting. The latter means displaying two object images at a distance shorter than the resolution limit of the microscope objective lens. The so-called Interphako method, involving a homogeneous microscopic field,is used with a microscope Peraval-Interphako, Carl Zeiss, Jena, Germany.The homogeneous microscopic field is obtained by fine-tuning the two fronts of the white light so as to situate them in a precise parallel position.Thus, the interference fringe of the selected coloris expanded to a degree allowing for the entire microscopic field to appear in a single color. In such an image, the protein crystals jutted-out at the single-color background of the optically homogeneous solution.

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