Langmuir 1995,11, 3356-3368
3356
A Rigorous Theory of Remote Loading of Drugs into Liposomes B. Ceht and D.D.la sic*^^^^ Chemistry Department, University of Ljubljana, AskerEeva 5, Ljubljana, Slovenia, and Liposome Technology, Inc., 1050 Hamilton Court, Menlo Park, California 94025 Received January 20, 1995. In Final Form: June 6, 1995@
Various gradients have been used to actively load molecules into preformed colloidalparticles separated from a medium by a semipermeablemembrane. Therapeutic applications of liposomeshave predominantly utilized pH and ammonium sulfate gradient methods to load drug molecules, which are weak bases or acids, into preformed liposomes. Except for the influence of a few simple parameters in the case of the pH gradient loading, the influences of many variables, involving different mechanisms and processes affectingthe loading, are not well understood. Here we present a rigorous theoretical treatment to explain these phenomena and simulate loading processes as a function of any given variable. Firstly, taking into account acidhase equilibria, we develop an Ansatz to calculate the pH of any solution as well as concentration of any species under given conditions. We use this concept to explain pH gradient loading and exchange gradient loadingin the cases of simple or coupled redistribution of speciesacross a semipermeablemembrane in one or two directions. Finally we add the interaction of permeated species with the bilayer or with the liposome preencapsulated material and evaluate both concepts of loading for the case of drug binding to the membrane and formation of precipitate in the liposome. As a result, some examples of possible drug loading systems are simulated, discussed, and represented in the form of families of curves, showing the most significantparameters in the equilibrated liposome systemsaf't.er loading. In general, the mathematical methodology presented in this article can be used to simulate physical and chemical processes which occur in any two compartment system separated by a semipermeable membrane.
I. Introduction Liposomeshave become one of the most studied colloidal drug delivery systems. Their advantages are altered pharmacokinetics and biodistribution of the encapsulated drug molecules (giving rise to improved efficacy orland reduced toxicity of the chemotherapy), while one of the major disadvantages can be the low efficiency of the encapsulation and retention of given hydrophilic drug molecules. Especially weakly hydrophobic molecules tend to leak quickly after entrapment into 1iposomes.l Several methods exist to improve loading of such molecules. Mostly, they are simply based on the passive entrapment taking into account special geometrical considerations. At high lipid concentrations liposomes encapsulate up to 70% of the sample volume and upon dilution large fractions of drug can be encapsulated. These samples are, however, difficult to size down, andifpossible, an active loading method is preferablea2 Active loading methods consist of forced entrapment of drug molecules upon establishing certain concentration gradients between liposome interior and exterior. Molecules which can respond to these gradients must be weak bases or weak acids coexisting in aqueous solutions in neutral and charged form. Mostly the driving forces are pH gradients and electrostatic and transmembrane potential differences across the vesicle membrane. The molecules (andpossibly ions in the presence ofionophores and uncoupling agents) redistribute between the interior ~ - ~ gradients, and exterior according to the g ~ a d i e n t . These +
University of Ljubljana.
* Liposome Technology, Inc.
Present address of D. D. Lasic: MegaBios, Corp., Lipid and Liposome Department, 863A Mitten Rd., Burlingame, CA 94010. Abstract published inAdvance ACSAbstracts, August 15,1995. (1)Lasic, D.D. Liposomes: From Physics to Applications; Elsevier: Amsterdam, 1993. 8
*
however, can dissipate and the encapsulated molecules can leak out. To bypass this disadvantage, a method which uses a specific salt, i.e., chemical potential, gradient was recently i n t r ~ d u c e d . ~ This method consists of encapsulating ammonium sulfate into liposomes, and after dialysis into isotonic saline or sucrose solution, a weak base, such as the anticancer agent doxorubicin (adriamycin), is added into the external phase and the nonencapsulated drug is removed aRer incubation. Loading efficiencies approaching 100%were achievedl~~ and it was shown that, contrary to the pH gradient loading, drug molecules do not leak from the vesicle^.^,^ Here we present a rigorous treatment of loading/ unloading of permeable neutral forms (including the pH as well as ammonium salt gradient methods as special cases) of weak bases and acids into liposomes. The same mathematical formalism can be applied to any system separated with a membrane permeable to neutral species and impermeable to charged species or, in a more general case, with differential permeability. The model can also be also scaled up to many-compartment systems. ~
(2)Deamer, W. D.; Prince, R. C.; Crofts,A. R. Biochim. Biophys. Acta 1972,274,323. (3)Lee, H. C.; Forte, J. G. Biochim. Biophys. Acta 1978,508, 339. (4) Hanigan, P.R.; Wong, K. F.; Redelmeier, T. E.; Wheler, J. J.; Cullis, P.R. Biochim. Biophys. Acta 1993,149,329. (5)Haran, G.;Cohen, R.; Bar, L. K ; Barenholz, Y. Biochim. Biophys. Acta 1993., 1151.201. (6) Lasic, D. D.yFrederik, P. M.; Stewart, M. C. A.; Barenholz, Y.; McIntosh, T. J. FEBS Lett. 1992,312,255. (7) Nozaki, Y.; Tanford, C. Proc. Natl. Acad. Sci. U.S.A. 1981, 78, ~~~
~
~~~~
~~~~
4324. ~.~ (8) Bangham, A. D. Chem. Phys. Lipids 1993,64, 275. ~
(9)Cafiso, D. S.;Hubbell, W. L. Biochemistry 1978,17,3871
0743-7463/95/24I1-3356$09.OO/O0 1995 American Chemical Society
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11. Theory The interior of the liposomes is separated from the surrounding medium by the boundary consisting of one or more amphiphatic lipid bilayers. When liposomes, filled with a given (inner) aqueous solution, are placed in an (outer) aqueous solution of different composition, interaction between these two systems results in the equalization of concentrations of different species on both sides of membraneb) andor build-up of membrane potential andor osmotic pressure. Different species permeate the membrane with very different rates. Because of the amphiphatic nature of lipid membranes, the permeability of many neutral species is known to be many orders of magnitude faster than that of the charged specie^.^^^^^ Permeation of species occurs, in general, in both directions and is due to the gradient of chemical potential which represents the driving force for “loading“(i.e., accumulation of active molecules in the liposome interior) or “leakage” (release of the same) of mainly neutral species. After molecules cross the membrane, the following processes are possible. Firstly, there is always complete rearrangement of all acidhase equilibria in the outer as well in the inner solution. Secondly, additional events may occur: charged forms of permeated molecules may bind onto the membranesg or form complexeslprecipitates with appropriate ions, if they are present inside the liposomes.6 Therefore, the extent ofuptake of molecules into the liposomes and their leakage can be affected to a considerable extent by appearance of such additional processes. We also shall postulate that only neutral species can cross the bilayer and that activity coefficients of all the species present are unity. A. Stationary System. Under the influence of polar solvent molecules, the solutes either dissociate into ions or dissolve as neutral molecules. Because they represent sources of a series of chemically related molecules and ions, we shall call them “starting point species”. Dealing with aqueous solutions the starting-point species may be divided into two groups. In the first one there are all those species which, with the exception of hydration, do not undergo any chemical reactions with water molecules-the “inert” species. In the second one there are the members, which, in accordancewith Bronsted-Lowry theory, react with water molecules giving in general a series of “sister-species”differing in charge and acidity. Each of these sister-species can act as an acid or a base, depending on the conditions in a given aqueous solution; the extent and kind of acidhase reactivity of each of them depends on its nature, on the nature of other species present, on concentrations, and also on the amount and nature of species added or eliminated from the system. Before starting with the calculation of pH in any “stationary system” (i.e., a closed system where matter cannot cross the boundary), we shall assume that the species present do not form precipitates or complex ions. Suppose that each starting-point species
reacts with water molecules forming a series of (at least two) species where S p is the sister-speciesj in series i having charge zv and SplS$!!; are conjugated acidhase pairs, respectively. The starting-point species S:ik, introduced into solution by a given source, may be any of the sister-species designated j , i.e., 1 Ij INi. In the state of equilibrium, the concentrations of the first and each of other sister-species are connected by the product of the corresponding acidity constants:
Here, Kil, ..., Ku are the acidity constants, S:? is the concentration of the first, highest charged member in the sisterspecies series designated i, and [H+lis the proton concenteration. After equilibrium is reached, the concentration of any starting-point species i introduced into the system is equal to the sum of the concentrations of all sister-species originating from it. The mass balance equation can be written as
where ci is the concentration of the starting-point species k, placed into the system in the form of S:jk, Introducing the right-hand expressions of eq 2, for each j in eq 3, one gets
Here, fi[H+]will be called the “concentration function” and can be written in a shorter form:
The charge contribution of all sister-species from particular starting-point species i may be expressed as follows:
3358 Langmuir, Vol. 11, No. 9, 1995
Ceh and Lasic
Here, Qi is charge contribution of all sister-species members of the corresponding starting-point species i, and Qii is the charge of the particular sister-species having a charge zy. From the above equation and in accordance with eq 4, the charges of all species originating from the source i can be expressed as a function of the concentration of its first sister-species member, Siu,
and by analogy with eq 5 it follows that i-1
where function gi[H+l will be called "charge function", Taking into account all acidhase sister-species series, then all neutral species, and finally [H+l and [OH-](i.e.,proton and hydroxide concentrations, which are connected through water autoprotolysis constant, KW= [H+][OH-]),the charge balance equation expressing the electroneutrality of the equilibrated solution can be obtained:
Here, M represents the number of inert species (which do not react with water, e.g., Na+, K+, Cl-), and zl and CI are their charges and concentrations, respectively. In the above equation, only the first and third terms are [H+l(or pH) dependent. Finally, by introduction of eqs 4 and 7 into the above equation, and taking into account all sister-species series present in the solution, the function depending only on [H+l is obtained:
This expression, which we shall call the master equation, allows one to calculate [H+I and pH in any solution assuming the activity coefficients are equal to unity, (y& = 1. Combining eqs 2 and 4, the concentration of any sister-species designated 6 may be obtained from
B. Exchangeable System without Membrane Binding or Precipitate Formation. In the most general case such a system represents any two compartment system separated by a semipermeable membrane. We postulate that due to the concentration gradients the portions of particular starting-point species permeate the vesicle membrane in either direction in the form of their neutral sister-species reaching the interiorlexterior of the liposome. The fundamental presumptions of our derivations are, firstly, equality of the equilibria constants in both aqueous compartments, and, secondly, equality of the equilibrium concentrations of all those neutral species which can cross the liposome membrane. Symbols relating to the chemical species inside and outside liposomes are provided by the subscript I and 0. Suppose that in liposome internal and external solution the equilibria are reached before either loading or leakage of neutral species SfJ commences (zil= 0). In accordance with eq 10 the expressions for states in both systems are shown below. For the outer solution the master equation is
and for the inner one
Here, CkO and CmI are starting concentrations of the source-species k outside (0)and of source species m inside (I) of liposomes, gkO and gmI the corresponding charge functions and f k 0 and f m I are, similarly, concentration functions, respectively. Each of the terms uo([H+l$)and U ~ ( [ H + I ; represents ~) the sum of the second and third member in eq 10, respectively.
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We postulate that due to the concentration gradient the portion of a particular (designated *) starting-point species k*lm* (aplP,*) permeates the membrane in the form of its neutral sister-species designated Sil reaching the opposite compartment, i.e., interiorlexterior of the liposomes. Therefore, the sum of concentrations ol'the corresponding k*lm* sister-species, remaining in either compartment aRer permeation into the opposite one, decreases. Thus, the acidities in both compartments are affected by -*I&*. After influx of a portion of one species, k*, only, for the outer solution the following equation is valid:
/Silm
A similar expression can also be obtained for the inner solution
where [H+ld[H+l~ is the externallinternal proton concentration in both compartments after permeation of only one species in either direction. The amount of the species k*lm* which permeates the membrane can be expressed in terms of concentration, volume and ak*lB,*
Vock,oak*= vflk*I
(16)
and
Vsm*$m* = VOCm*O
(17)
Here, VO and VI are external and internal volume of the liposome system, respectively, and ck*IIcm*Ois the s u m of concentrations of all species formed insideloutside the liposomes after a portion of species k*lm*, i.e., a&,,*, is loaded released. After infldefflux of either species,the concentration of [H+land of all other species in the innerlouter solution is now affected by a portion of the correspondingloadedreleased neutral one. The situation in the innerlouter compartment is now described as
and
Assuming that there are many neutral species which can cross the liposome membrane in either direction and that equilibria for all these species are achieved, one can express the state in the outer liposome solution by the following master equation:
A second, similar equation can also be obtained describing the situation in the inner liposome compartment:
Here the %eq and p2 are the equilibrium percentages of loadedreleased species, and Ak and B, are either 1 or 0, depending on whether o r not the species k or m is allowed to permeate the membrane in corresponding directions. If there is no neutral species existing for particular starting-point species k or m , the Ak or B, have to be equal to zero. Now, the a? can be expressed as
czt
where nkI, nko, cgt, and are total aniounts and total concentrations of all sister-species originating from the corresponding starting-point species k in the equilibrated inner and outer solution, respectively. Similarly assigned quantities and relations are also valid for all the species leaking out of the liposomes:
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Henceforth, for the sake of simplicity, the [H'I?
Ceh and Lasic
and [H+13will be written x and y , and the ratio V&I will be called
K,,respectively. Further, in accordance with eqs 2 and 3, each of the terms ctFt, ciat, c.:& andf:c can be expressed by the concentration of the neutral species, existing in each of series k or m. So, for the influxed portion of the species k one gets
and a similar expression is valid also for the portion of the same species remaining in the outer solution after influx Ceq,t = [SzkZ eq kO k110 f k O b )
=
(25)
When the sister-species in a corresponding series are arranged by their decreasing charges, the index !!k is a position number of the neutral sister-species Zk in this numbered series. After the transmembrane equilibrium is reached, the concentrations of neutral species of each source-species k, [S&l;q,and [S&l2, inside and outside of the liposomes, should be the same:
By introducingeqs 24 and 25 in eq 22, one obtains anx and y dependent equation expressing the percentage of the species
k, loaded into the liposome interior:
By a similar procedure, an analogous expression for calculation of portion of the species m , efluxed from the liposomes, can also be derived:
However, taking into account the possibility of the influxiefflux of all starting-point species with appropriate neutral forms capable of permeating the membrane in either direction, i.e., introducing eqs 27 and 28 into expressions 20 and 21, two generalized equations with two unknowns are obtained. The first one represents the equilibrium state in the outer solution
and the second one describes the equilibrium state in the inner solution
Here, gk0, gkI, g,o, gmI, bo, f k I , fmo, and fmI are charge and concentration functions of the corresponding species k and m , respectively. Ak andB, are the factors which "allow" (=l),or not (=O), to load or to leak the neutral species designated k1 or ml. The above pair of equations completely describes the equilibrated state in both liposome compartments, i.e., after permeation of an optional number of species in both directions. C. Exchangeable Systems in Which One of the Charged Species Is Bound onto the Membrane Surface. We shall assume that one of the charged sister-species originating from starting-point species designated k and having charge zkj may bind to the inner and outer membrane surface. When the membrane is of symmetric composition, the external and internal membrane binding constants can be assumed t o be equals
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where c z and c? are concentrations of charged species bound to the inner and outer membrane surface layers and c2* and cF* are concentrations of the same species remaining in both liposome compartments. According to the CafisoHubbell model,gthe internal as well as external liposome volume has to be reduced for the volume of the membrane layer formed. To compare the situation where one species is membrane-bound to that in the previous section, the concentrations (see eq 31) both in solution and in the interior and exterior layers, have to be recalculated for the new volumes and respectively. Now, for the inner solution one obtains
6,
and analogously,
e can be calculated for the outer solution
Here, V d and Vm0 are volumes of the inner and outer membrane layer occupied by the charged species kj, and all concentrations designated are recalculated on reduced volumes. The relationship between K, and reduced is now
*
If the fact is taken into account that usually Vmo
