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“pH phoresis”: A new concept that can be used for improving drug delivery to tumor cells
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Journal of Controlled Release 170 (2013) 396–400
Contents lists available at SciVerse ScienceDirect
Journal of Controlled Release journal homepage: www.elsevier.com/locate/jconrel
“pH phoresis”: A new concept that can be used for improving drug delivery to tumor cells You-Yeon Won a,b,c,⁎, Hoyoung Lee a a b c
School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, USA Bindley Bioscience Center, Purdue University, West Lafayette, IN 47907, USA Center for Theragnosis, Biomedical Research Institute, Korea Institute of Science and Technology (KIST), Seoul 136-791, Republic of Korea
a r t i c l e
i n f o
Article history: Received 11 March 2013 Accepted 8 June 2013 Available online 19 June 2013 Keywords: Drug delivery Tumor Cancer pH Nanoparticle Polyamine
a b s t r a c t We propose a new concept describing how nanoparticles composed of weak polybases (such as polyamines) would behave when they are exposed to a pH gradient; weak polybase-containing particles will tend to accumulate preferentially in low pH regions under a pH gradient environment. This phenomenon, which we term “pH phoresis”, may provide a useful mechanism for improving the delivery of drugs to cancer cells in solid tumor tissues. © 2013 Elsevier B.V. All rights reserved.
1. Explanation of the concept We wish to propose a new concept describing how nanoparticles composed of weak polybases (such as polyamines) would behave when they are exposed to a pH gradient; weak polybase-containing particles will tend to accumulate preferentially in low pH regions under a pH gradient environment. This phenomenon, which we term “pH phoresis”, may provide a useful mechanism for improving the delivery of drugs to cancer cells in solid tumor tissues. This concept applies to any type of polybase-containing particles, such as hydrated molecular coils or small cross-linked hydrogels of weak polybases, or micellar aggregates formed from polybase-based amphiphilic block copolymers, and thus when we speak of “(polybase) particles” in this article, we intend to mean any such discrete objects composed of polybase molecules dispersed in water. In order to explain why/how weak polybase particles concentrate in low pH areas, let us consider a simplified situation as described in Fig. 1. In this situation, we consider a suspension of Brownian particles in a container. The container is divided into two sub-compartments, A and B. These compartments are separated by a membrane through which the particles can pass freely in both directions; so there is no size exclusion of the particles by the dividing membrane. Assume that (by a certain mechanism) the particles become swollen when they are located in ⁎ Corresponding author at: School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, USA. E-mail address: [email protected] (Y.-Y. Won). 0168-3659/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jconrel.2013.06.016
compartment B with their hydrodynamic radius increasing by a factor of α (>1) relative to when they are placed in compartment A (i.e., RB = αRA). Then it can be deduced that the speed of the particles diffusing from A to B is greater by the same factor (α) than that of the particles diffusing in the opposite direction; that is, vA → B = αvB → A. This is because for particles in a viscous medium the diffusion speed under a constant driving force (f) is inversely proportional to the friction coefficient (ξ), i.e., v = f/ξ, and this friction coefficient is related by a linear equation (called Stokes' formula) to the hydrodynamic radius of the particles (R), i.e., ξ = 6πηR where η is the viscosity of the medium [1]. When the system is in equilibrium, the net particle flux across the dividing membrane is zero, i.e., |JA → B| = |JB → A|. Since |J| = cv (c is the concentration of the particles) [2], the above condition creates an interesting situation where the particles preferentially concentrate in compartment B, forming a concentration gradient across the compartment boundary (cB (=αcA) > cA). Now one might question how the above-assumed situation connects to the problem of drug delivery to tumors. The connection becomes apparent when one considers the following two points: (1) solid tumors have a significantly lower extracellular pH (≈ 6.5– 6.9) compared to normal tissue (which has an average pH of 7.4) [3,4], and (2) weak polybases/polyamines swell at lower pH (i.e., their average hydrodynamic size increases with decreasing pH value) due to the increased protonation of the amine nitrogens and thus the increased repulsion between the monomers along the chain in that condition [5,6]. Therefore, the two compartments, A and B, discussed with reference to Fig. 1 can be taken, respectively, as diagrammatic
Fig. 1. A suspension of Brownian particles is contained in a container composed of two sub-compartments, A and B. The particles increase in size when located in B relative to when they are located in A. This size difference causes a buildup of an equilibrium concentration gradient of the particles across the compartment boundary.
representations of the normal and tumor tissue compartments in the body, and the particles with variable size described in Fig. 1 can be considered as representing drug-containing nanoparticles constructed using weak polybases; detailed design considerations for a “pH-phoretic” drug carrier will be discussed later in this paper. It is previously well known that small molecular weak bases (such as chloroquine) accumulate in acidic intracellular compartments; this phenomenon occurs because as a weak base becomes protonated under acidic environment, the resulting positive charge retards the movement of this compound across the lipid membrane [7]. However, it should be noted that the pH phoresis concept we propose in this paper is fundamentally different from this previously documented phenomenon in that in our situation a concentration gradient of weak polybase particles will naturally form even in the absence of compartmentalization of space into regions of different pH separated by semipermeable membranes; the pH-phoretic migration of weak polybase particles will occur even when they are subjected to a continuously varying pH field. In order to analyze this phenomenon quantitatively, let us consider the situation depicted in Fig. 2, where weak polybase particles are dispersed in a pH gradient solution. The solution is contained in a horizontally-positioned tube. At one end of the tube (x =0), the pH value is 7.4, and it monotonically decreases along the tube axis (x direction) to a value of 6.5 at the other end of the tube (x = L). For the reason discussed above, the average size of the weak polybase particles will vary as a function of pH and thus as a function
Fig. 2. Qualitative plots demonstrating how the hydrodynamic radius (R) and concentration (c) of weak polybase molecules dispersed in a pH gradient solution vary as functions of pH and thus as functions of position (x).
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of position (x). For the sake of demonstration, let us assume that the hydrodynamic radius of the particle increases by a factor of α as it moves from x = 0 (pH 7.4) to x = L (pH 6.5); i.e., R(L) = αR(0) (α > 1). According to the Stokes–Einstein relation (D = kT/6πηR where k is the Boltzmann constant), this means that at x = L the diffusion coefficient is one α-th of its value at x = 0; i.e., D(L) = D(0)/α. For simplicity, we further assume that the function describing the dependence of the diffusion coefficient on the position along the x axis takes the linear form, D(x) = D(0)[1 + ((1 − α)/α)(x/L)]. In this exemplary situation, it can be shown that the pH gradient, in effect, forces the polybase particles to move towards the area with lower pH, leading to the formation of a corresponding concentration gradient. It needs to be pointed out that for this calculation, the original Fick's first law equation, J(x,y,z) = −D∇c(x,y,z) (or J(x) = −D(dc(x)/dx) in our case), cannot be used, because this equation was derived under the assumption that the diffusion coefficient (D) is a position-independent parameter; in the above equation, a bold typeface was used for the flux function because it is a vector quantity. As explained in the Supplementary material, it can be shown that for a situation where the diffusion coefficient is a spatially varying quantity, the Fick's first law equation should be modified to the following form: Jðx; y; zÞ ¼ ∇ðDðx; y; zÞcðx; y; zÞÞ ¼ −Dðx; y; zÞ∇−cðx; y; zÞ∇Dðx; y; zÞ; ð1Þ that is, J(x) = −D(x)(dc(x)/dx) − c(x)(dD(x)/dx) in our case. At equilibrium, the net particle flux is zero everywhere (J(x) = 0 at all x). Therefore, to obtain the concentration profile c(x), one simply needs to solve the first-order differential equation, − D(x)(dc(x)/ dx) − c(x)(dD(x)/dx) = 0, with an integral boundary condition, ∫ L0c(x)dx = N/A where N is the total number of the weak polybase particles in the system, and A is the cross-sectional area of the tube container. The solution is c(x) = N(α − 1)/{Aln(α)[αL + (1 − α)x]}. As can be seen from the c vs. x graph shown in Fig. 2, the polybase particles accumulate preferentially in the low pH areas. Further, the c(x) equation suggests that the factor of α difference in particle size (i.e., R(L) = αR(0)) produces the exact same level of difference in polymer concentration between the pH 7.4 and pH 6.5 regions (c(L) = αc(0)). This result has useful implications for drug carrier design as will be further discussed next. 2. Examination of the feasibility This concept that weak polybase particles will effectively flow from a high to low pH region has useful implications, we believe, for improving systemic drug delivery to solid tumors. Specifically, we propose the following idea: Design a drug carrier such that it expands its size to a larger structure when exposed to the lower tumor interstitial pH. Such pH-responsive characteristic can be achieved, for instance, with a micelle-type drug carrier constructed using a polyamine-containing block copolymer. However, as suggested by the result shown in Fig. 2, whether the incorporation of a polyamine component in a drug carrier will indeed produce a sufficient amount of pHphoretic effect for practical use will depend on how much increase there will be in the overall size of the carrier particle as the pH of the surrounding medium is changed from about 7.4 to about 6.5. In order to examine the feasibility of achieving a high degree of low pHtriggered size expansion of polyamine-based block copolymer micelles, we performed a theoretical study. For this theoretical analysis, we used a sophisticated, so-called self-consistent field (SCF) model which has been shown to be able to predict a correct description of an experimentally observed behavior of weak polyelectrolyte brushes [6]; an explanation of this theoretical model is presented in the Supplementary material. We used parameter values for the model that represent realistic situations. Specifically, in the analysis presented in the upper panel of Fig. 3, we considered a block copolymer micelle consisting of a hydrophobic core of about 10 nm radius surrounded by polyamine brush
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Y.-Y. Won, H. Lee / Journal of Controlled Release 170 (2013) 396–400
chains with a degree of polymerization (DP) of 200 at a dimensionless grafting density of σ · πRg2 = 3.0 (where Rg denotes the radius of gyration of the polyamine chain in the random-walk configuration); the micelle contains about 100 polyamine chains in the micelle's corona. Considering that most amine-based polymers are inherently hydrophobic, the effective Flory–Huggins interaction parameter of the polyamine brush chains was assumed to be significantly greater than that corresponding to the θ condition in Flory–Huggins solutions (i.e., χ ≈ χθ + 0.33). The aqueous solvent was assumed to have an ionic strength of 150 mM NaCl equivalent to mimic the physiological environment. As shown in the upper left panel of Fig. 3, the overall size of the polyamine-containing micelle is confirmed to grow with decreasing pH due to the increased stretching of the polyamine chains; on the other hand, at high pH where the fraction of protonated amine groups is too low to produce sufficient monomer–monomer repulsion necessary to overcome the effect of the hydrophobicity of the polyamine, the chains are predicted to collapse into a dense state (see Figs. S2 and S3 of the Supplementary Material). Also, as can be seen from the figure, the micelle size typically exhibits a rapid increase with decreasing pH over a relatively narrow range of pH, and the exact range of pH over which such increase in micelle size occurs depends on the pKa characteristic (and thus the detailed chemistry) of the polyamine material. From the data presented in the upper left of Fig. 3, the values of the expansion ratio, defined as the ratio of the micelle sizes between the two tumor-relevant pH limits α ≡ R(pH 6.5)/R(pH 7.4), were estimated for the various values of pKa°, and the results are shown in the upper right panel; here, pKa° denotes the intrinsic value of pKa for the polyamine in its monomeric (i.e., DP → 1) limit. As shown in the figure, the highest degree of micelle
swelling in tumors (α ≈ 1.8) is achieved when pKa° ≈ 7.0 ± 0.5. Further, as shown in the lower panel of Fig. 3, we tested whether the pH sensitivity of the micelle size at pKa° = 7.0 can be further increased by adjusting the various micelle parameters, i.e., polyamine degree of polymerization (DP), chain grafting density (σ · πRg2), micelle core radius (Rc), salt concentration (Cs), and the effective Flory–Huggins interaction parameter (χ). The results shown in the figure suggest that, for instance, an increase of DP or χ, a decrease of σ or Rc, or a combination of such changes from their original values used to generate the data shown in the upper panel would cause a further increase in the tumor pH responsiveness of the micelle size; the value of α can be increased to about two. 3. Carrier design The results demonstrated in the previous subsection suggest that in order to maximize the pH-responsiveness of the polymer micelle carrier systems to the intratumoral pH, it is most critical to choose an appropriate polyamine material for use as the size-controlling element; the polyamine should undergo a significant change in its conformation as the pH is changed between the pH values of human tumors and adjacent normal tissues (i.e., between approximately 6.5 and 7.4). Based on the data presented in Fig. 3, for instance, polyethylenimine (PEI for short) appears to be inappropriate for this use, because the PEI polymer has a pKa at about 4.2 [5] which is too far away from the optimal pKa° value of about 7.0. Another common polyamine called poly(2-(dimethylamino)ethyl methacrylate) (PDMAEMA) is also expected not to be an optimal material for constructing tumor pH-sensitive drug carriers, because its hydrophobicity is only marginal
Fig. 3. (Upper left) Radius of a polyamine-based block copolymer micelle (R) estimated by the SCF theory as a function of solution pH at various values of polyamine's intrinsic pKa°. It is assumed that the hydrophobic micelle core of 10 nm radius (Rc) is surrounded by polyamine brush chains of degree of polymerization DP = 200 with Kuhn monomer length b = 6 Å, monomer volume v = 179 Å3 and Flory-Huggins interaction parameter in water significantly greater than that corresponding to the θ condition (i.e., χ ≈ χθ + 0.33) at a dimensionless grafting density σ · πRg2 = 3.0 (where Rg denotes the radius of gyration of the polyamine chain in the random-walk configuration). The micelle is dispersed in an aqueous solvent with an ionic strength (Cs) of 150 mM NaCl equivalent. The water molecular volume is assumed to be v = 30 Å3. (Upper right) The values of the ratio of the micelle sizes between the pH values of human tumors and normal tissues α ≡ R(pH 6.5)/R(pH 7.4) estimated from the data presented in the upper left for the various values of pKa°. (Lower) Plots demonstrating how the tumor pH responsiveness of the micelle size (α) at a fixed pKa° of 7.0 changes in response to variations of other parameter, such as polyamine degree of polymerization (DP), chain grafting density (σ · πRg2), micelle core radius (Rc), salt concentration (Cs), or the effective Flory–Huggins interaction parameter of the polyamine-water pair (χ), from their original values used in the upper panel; the original parameters produce an α value of about 1.8 as denoted with a red dotted line. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
(even at high pH this polymer does not normally become completely water-insoluble [8]). The best candidate material appears to be poly(L-histidine) (PHis), which has an apparent pKa of about 6.5 [9]; this polymer has been shown to exhibit a dramatic conformational change as the pH of the environment is varied through the intratumoral pH values [9]. In real applications of polyamine nanoparticles in in vivo drug delivery, the amine moieties of the polyamine chains need to be shielded (e.g., by “PEGylation”) to enhance the stability of the particles against self-association and also to minimize toxicity-inducing nonspecific electrostatic interactions with serum proteins and tissue components. As schematically demonstrated in Fig. 4, there are ways of incorporating the size-controlling, polyamine component into the coronal structure of the polymer micelle without compromising the benefits of the PEGylation method commonly adopted for achieving prolonged blood circulation times. In fact, a PHis-based polymer micelle (similar to the one shown in Fig. 4(bottom)) has previously been proposed for enhanced delivery of drugs to tumors [9–11]; in these previous studies, in vitro tests confirmed that the pH-responsiveness of PHis can be used for achieving low pH-triggered surface expression of cell-interacting ligands. It will be interesting to conduct an in vivo pharmacokinetic study to determine whether/how much the incorporation of PHis to a polymer micelle system will improve its tumorspecific accumulation. Purely from the diffusion standpoint, a factor of two increase in the micelle size in response to the pH of the tumor microenvironment will, for instance, cause the same two-fold enhancement in the efficiency of delivery to tumor cells; such effect can be a “game changer”, especially when the absorbed dose of the drug within the tumor volume is close to the therapeutic threshold of the drug. Further, the overall delivery performance of the “pH-phoretic” micelles is expected to be even greater than what one would anticipate on the basis of the diffusion rates, because there can be other advantageous effects of using a polyamine-containing, “pH-phoretic” delivery system (such as an increased electrostatic interaction with cancer cells under low pH environments). A properly designed “pH phoretic” drug carrier
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having appropriate size (≪100 nm in diameter [12]) and surface (anti-fouling against collagen [13]) characteristics would even be able to traverse toward the deeper parts of the tumor due to the intratumoral pH gradient [14]. This “pH phoresis” concept can also be combined readily within other established drug delivery methodologies, such as the utilization of the enhanced permeability and retention (EPR) phenomenon [15] and tumor pH-specific dissociatable polymer micelles [16]. Many current anticancer drug formulations generally suffer low tumor “targeting” efficiencies [17]. We speculate that this “pH phoresis” concept will contribute to achieving an improvement in systemic drug delivery to solid tumors. Lastly, we would like to note that it is non-trivial to experimentally demonstrate the isolated effect of the pH-phoretic diffusion of polyamine nanoparticles on their distribution in a pH gradient, because the currently available methods for creating a stable pH gradient either in an electrophoretic gel or in a microfluidic device are inappropriate for testing the pH phoresis concept using polyamine particles in their most desirable form (i.e., in the form of polyamine-containing block copolymer micelles); commercial immobilized pH gradient (IPG) electrophoresis gels have too small pore sizes (much less than a few tens of nanometers), and in a microfluidic flow environment, the lifetime of a pH gradient created by a concentration gradient of acid or base molecules is much less than the diffusion time of the polyamine particles. Currently, efforts are under way to develop an appropriate experimental technique for this type of measurement. Acknowledgment The authors would like to thank the U.S. National Science Foundation (CBET-0828574 and DMR-0906567) and the Purdue University School of Chemical Engineering for providing financial support of this research. Y.Y.W. gratefully acknowledges the Global RNAi Carrier Initiative Program Fellowship from the Biomedical Research Institute of the Korea Institute of Science and Technology (KIST) and also the Research Fellowship from the Bindley Bioscience Center at Purdue University. Appendix A. Supplementary data Supplementary data to this article can be found online at http:// dx.doi.org/10.1016/j.jconrel.2013.06.016. References
Fig. 4. Cartoons describing exemplary designs of “pH-phoretic” polymer micelle systems. The red represents polyamine chains, the blue represents poly(ethylene glycol) (PEG) (or any such moiety that can confer stabilization of the micelle, e.g., serum albumin), and the purple represents the hydrophobic core domain of the micelle (where drugs can be loaded). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
[1] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, 2nd ed. Elsevier Butterworth-Heinemann, Oxford, U.K., 1987 [2] K.A. Dill, S. Bromberg, Molecular Driving Forces: Statistical Thermodynamics in Chemistry and Biology, Garland Science, New York, NY, 2003. [3] G. Helmlinger, F. Yuan, M. Dellian, R.K. Jain, Interstitial pH and pO(2) gradients in solid tumors in vivo: high-resolution measurements reveal a lack of correlation, Nat. Med. 3 (1997) 177–182. [4] J.L. Wikehooley, A.P. Vandenberg, J. Vanderzee, H.S. Reinhold, Human-tumor pH and its variation, Eur. J. Cancer Clin. Oncol. 21 (1985) 785–787, 789-791. [5] H. Lee, S.H. Son, R. Sharma, Y.Y. Won, A discussion of the pH-dependent protonation behaviors of poly(2-(dimethylamino)ethyl methacrylate) (PDMAEMA) and poly(ethylenimine-ran-2-ethyl-2-oxazoline) (P(EI-r-EOz)), J. Phys. Chem. B 115 (2011) 844–860. [6] K.N. Witte, S. Kim, Y.Y. Won, Self-consistent field theory study of the effect of grafting density on the height of a weak polyelectrolyte brush, J. Phys. Chem. B 113 (2009) 11076–11084. [7] R.G.W. Anderson, L. Orci, A view of acidic intracellular compartments, J. Cell Biol. 106 (1988) 539–543. [8] F.A. Plamper, A. Schmalz, M. Ballauff, A.H.E. Muller, Tuning the thermoresponsiveness of weak polyelectrolytes by pH and light: lower and upper critical-solution temperature of poly(N,N-dimethylaminoethyl methacrylate), J. Am. Chem. Soc. 129 (2007) 14538–14539. [9] E.S. Lee, K. Na, Y.H. Bae, Super pH-sensitive multifunctional polymeric micelle, Nano Lett. 5 (2005) 325–329. [10] E.S. Lee, Z.G. Gao, D. Kim, K. Park, I.C. Kwon, Y.H. Bae, Super pH-sensitive multifunctional polymeric micelle for tumor pH(e) specific TAT exposure and multidrug resistance, J. Control. Release 129 (2008) 228–236. [11] E.S. Lee, K.T. Oh, D. Kim, Y.S. Youn, Y.H. Bae, Tumor pH-responsive flower-like micelles of poly(L-lactic acid)-b-poly (ethylene glycol)-b-poly(L-histidine), J. Control. Release 123 (2007) 19–26.
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[12] F. Yuan, M. Leunig, S.K. Huang, D.A. Berk, D. Papahadjopoulos, R.K. Jain, Microvascular permeability and interstitial penetration of sterically stabilized (stealth) liposomes in a human tumor xenograft, Cancer Res. 54 (1994) 3352–3356. [13] R.G. Thorne, A. Lakkaraju, E. Rodriguez-Boulan, C. Nicholson, In vivo diffusion of lactoferrin in brain extracellular space is regulated by interactions with heparan sulfate, Proc. Natl. Acad. Sci. U. S. A. 105 (2008) 8416–8421.
[14] R.K. Jain, T. Stylianopoulos, Delivering nanomedicine to solid tumors, Nat. Rev. Clin. Oncol. 7 (2010) 653–664. [15] H. Maeda, Tumor-selective delivery of macromolecular drugs via the EPR effect: background and future prospects, Bioconjugate Chem. 21 (2010) 797–802. [16] E.S. Lee, Z.G. Gao, Y.H. Bae, Recent progress in tumor pH targeting nanotechnology, J. Control. Release 132 (2008) 164–170. [17] I.K. Kwon, S.C. Lee, B. Han, K. Park, Analysis of the current status of targeted drug delivery to tumors, J. Control. Release 164 (2012) 108–114.
Journal of Controlled Release 170 (2013) 396–400
Contents lists available at SciVerse ScienceDirect
Journal of Controlled Release journal homepage: www.elsevier.com/locate/jconrel
“pH phoresis”: A new concept that can be used for improving drug delivery to tumor cells You-Yeon Won a,b,c,⁎, Hoyoung Lee a a b c
School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, USA Bindley Bioscience Center, Purdue University, West Lafayette, IN 47907, USA Center for Theragnosis, Biomedical Research Institute, Korea Institute of Science and Technology (KIST), Seoul 136-791, Republic of Korea
a r t i c l e
i n f o
Article history: Received 11 March 2013 Accepted 8 June 2013 Available online 19 June 2013 Keywords: Drug delivery Tumor Cancer pH Nanoparticle Polyamine
a b s t r a c t We propose a new concept describing how nanoparticles composed of weak polybases (such as polyamines) would behave when they are exposed to a pH gradient; weak polybase-containing particles will tend to accumulate preferentially in low pH regions under a pH gradient environment. This phenomenon, which we term “pH phoresis”, may provide a useful mechanism for improving the delivery of drugs to cancer cells in solid tumor tissues. © 2013 Elsevier B.V. All rights reserved.
1. Explanation of the concept We wish to propose a new concept describing how nanoparticles composed of weak polybases (such as polyamines) would behave when they are exposed to a pH gradient; weak polybase-containing particles will tend to accumulate preferentially in low pH regions under a pH gradient environment. This phenomenon, which we term “pH phoresis”, may provide a useful mechanism for improving the delivery of drugs to cancer cells in solid tumor tissues. This concept applies to any type of polybase-containing particles, such as hydrated molecular coils or small cross-linked hydrogels of weak polybases, or micellar aggregates formed from polybase-based amphiphilic block copolymers, and thus when we speak of “(polybase) particles” in this article, we intend to mean any such discrete objects composed of polybase molecules dispersed in water. In order to explain why/how weak polybase particles concentrate in low pH areas, let us consider a simplified situation as described in Fig. 1. In this situation, we consider a suspension of Brownian particles in a container. The container is divided into two sub-compartments, A and B. These compartments are separated by a membrane through which the particles can pass freely in both directions; so there is no size exclusion of the particles by the dividing membrane. Assume that (by a certain mechanism) the particles become swollen when they are located in ⁎ Corresponding author at: School of Chemical Engineering, Purdue University, West Lafayette, IN 47907, USA. E-mail address: [email protected] (Y.-Y. Won). 0168-3659/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jconrel.2013.06.016
compartment B with their hydrodynamic radius increasing by a factor of α (>1) relative to when they are placed in compartment A (i.e., RB = αRA). Then it can be deduced that the speed of the particles diffusing from A to B is greater by the same factor (α) than that of the particles diffusing in the opposite direction; that is, vA → B = αvB → A. This is because for particles in a viscous medium the diffusion speed under a constant driving force (f) is inversely proportional to the friction coefficient (ξ), i.e., v = f/ξ, and this friction coefficient is related by a linear equation (called Stokes' formula) to the hydrodynamic radius of the particles (R), i.e., ξ = 6πηR where η is the viscosity of the medium [1]. When the system is in equilibrium, the net particle flux across the dividing membrane is zero, i.e., |JA → B| = |JB → A|. Since |J| = cv (c is the concentration of the particles) [2], the above condition creates an interesting situation where the particles preferentially concentrate in compartment B, forming a concentration gradient across the compartment boundary (cB (=αcA) > cA). Now one might question how the above-assumed situation connects to the problem of drug delivery to tumors. The connection becomes apparent when one considers the following two points: (1) solid tumors have a significantly lower extracellular pH (≈ 6.5– 6.9) compared to normal tissue (which has an average pH of 7.4) [3,4], and (2) weak polybases/polyamines swell at lower pH (i.e., their average hydrodynamic size increases with decreasing pH value) due to the increased protonation of the amine nitrogens and thus the increased repulsion between the monomers along the chain in that condition [5,6]. Therefore, the two compartments, A and B, discussed with reference to Fig. 1 can be taken, respectively, as diagrammatic
Fig. 1. A suspension of Brownian particles is contained in a container composed of two sub-compartments, A and B. The particles increase in size when located in B relative to when they are located in A. This size difference causes a buildup of an equilibrium concentration gradient of the particles across the compartment boundary.
representations of the normal and tumor tissue compartments in the body, and the particles with variable size described in Fig. 1 can be considered as representing drug-containing nanoparticles constructed using weak polybases; detailed design considerations for a “pH-phoretic” drug carrier will be discussed later in this paper. It is previously well known that small molecular weak bases (such as chloroquine) accumulate in acidic intracellular compartments; this phenomenon occurs because as a weak base becomes protonated under acidic environment, the resulting positive charge retards the movement of this compound across the lipid membrane [7]. However, it should be noted that the pH phoresis concept we propose in this paper is fundamentally different from this previously documented phenomenon in that in our situation a concentration gradient of weak polybase particles will naturally form even in the absence of compartmentalization of space into regions of different pH separated by semipermeable membranes; the pH-phoretic migration of weak polybase particles will occur even when they are subjected to a continuously varying pH field. In order to analyze this phenomenon quantitatively, let us consider the situation depicted in Fig. 2, where weak polybase particles are dispersed in a pH gradient solution. The solution is contained in a horizontally-positioned tube. At one end of the tube (x =0), the pH value is 7.4, and it monotonically decreases along the tube axis (x direction) to a value of 6.5 at the other end of the tube (x = L). For the reason discussed above, the average size of the weak polybase particles will vary as a function of pH and thus as a function
Fig. 2. Qualitative plots demonstrating how the hydrodynamic radius (R) and concentration (c) of weak polybase molecules dispersed in a pH gradient solution vary as functions of pH and thus as functions of position (x).
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of position (x). For the sake of demonstration, let us assume that the hydrodynamic radius of the particle increases by a factor of α as it moves from x = 0 (pH 7.4) to x = L (pH 6.5); i.e., R(L) = αR(0) (α > 1). According to the Stokes–Einstein relation (D = kT/6πηR where k is the Boltzmann constant), this means that at x = L the diffusion coefficient is one α-th of its value at x = 0; i.e., D(L) = D(0)/α. For simplicity, we further assume that the function describing the dependence of the diffusion coefficient on the position along the x axis takes the linear form, D(x) = D(0)[1 + ((1 − α)/α)(x/L)]. In this exemplary situation, it can be shown that the pH gradient, in effect, forces the polybase particles to move towards the area with lower pH, leading to the formation of a corresponding concentration gradient. It needs to be pointed out that for this calculation, the original Fick's first law equation, J(x,y,z) = −D∇c(x,y,z) (or J(x) = −D(dc(x)/dx) in our case), cannot be used, because this equation was derived under the assumption that the diffusion coefficient (D) is a position-independent parameter; in the above equation, a bold typeface was used for the flux function because it is a vector quantity. As explained in the Supplementary material, it can be shown that for a situation where the diffusion coefficient is a spatially varying quantity, the Fick's first law equation should be modified to the following form: Jðx; y; zÞ ¼ ∇ðDðx; y; zÞcðx; y; zÞÞ ¼ −Dðx; y; zÞ∇−cðx; y; zÞ∇Dðx; y; zÞ; ð1Þ that is, J(x) = −D(x)(dc(x)/dx) − c(x)(dD(x)/dx) in our case. At equilibrium, the net particle flux is zero everywhere (J(x) = 0 at all x). Therefore, to obtain the concentration profile c(x), one simply needs to solve the first-order differential equation, − D(x)(dc(x)/ dx) − c(x)(dD(x)/dx) = 0, with an integral boundary condition, ∫ L0c(x)dx = N/A where N is the total number of the weak polybase particles in the system, and A is the cross-sectional area of the tube container. The solution is c(x) = N(α − 1)/{Aln(α)[αL + (1 − α)x]}. As can be seen from the c vs. x graph shown in Fig. 2, the polybase particles accumulate preferentially in the low pH areas. Further, the c(x) equation suggests that the factor of α difference in particle size (i.e., R(L) = αR(0)) produces the exact same level of difference in polymer concentration between the pH 7.4 and pH 6.5 regions (c(L) = αc(0)). This result has useful implications for drug carrier design as will be further discussed next. 2. Examination of the feasibility This concept that weak polybase particles will effectively flow from a high to low pH region has useful implications, we believe, for improving systemic drug delivery to solid tumors. Specifically, we propose the following idea: Design a drug carrier such that it expands its size to a larger structure when exposed to the lower tumor interstitial pH. Such pH-responsive characteristic can be achieved, for instance, with a micelle-type drug carrier constructed using a polyamine-containing block copolymer. However, as suggested by the result shown in Fig. 2, whether the incorporation of a polyamine component in a drug carrier will indeed produce a sufficient amount of pHphoretic effect for practical use will depend on how much increase there will be in the overall size of the carrier particle as the pH of the surrounding medium is changed from about 7.4 to about 6.5. In order to examine the feasibility of achieving a high degree of low pHtriggered size expansion of polyamine-based block copolymer micelles, we performed a theoretical study. For this theoretical analysis, we used a sophisticated, so-called self-consistent field (SCF) model which has been shown to be able to predict a correct description of an experimentally observed behavior of weak polyelectrolyte brushes [6]; an explanation of this theoretical model is presented in the Supplementary material. We used parameter values for the model that represent realistic situations. Specifically, in the analysis presented in the upper panel of Fig. 3, we considered a block copolymer micelle consisting of a hydrophobic core of about 10 nm radius surrounded by polyamine brush
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chains with a degree of polymerization (DP) of 200 at a dimensionless grafting density of σ · πRg2 = 3.0 (where Rg denotes the radius of gyration of the polyamine chain in the random-walk configuration); the micelle contains about 100 polyamine chains in the micelle's corona. Considering that most amine-based polymers are inherently hydrophobic, the effective Flory–Huggins interaction parameter of the polyamine brush chains was assumed to be significantly greater than that corresponding to the θ condition in Flory–Huggins solutions (i.e., χ ≈ χθ + 0.33). The aqueous solvent was assumed to have an ionic strength of 150 mM NaCl equivalent to mimic the physiological environment. As shown in the upper left panel of Fig. 3, the overall size of the polyamine-containing micelle is confirmed to grow with decreasing pH due to the increased stretching of the polyamine chains; on the other hand, at high pH where the fraction of protonated amine groups is too low to produce sufficient monomer–monomer repulsion necessary to overcome the effect of the hydrophobicity of the polyamine, the chains are predicted to collapse into a dense state (see Figs. S2 and S3 of the Supplementary Material). Also, as can be seen from the figure, the micelle size typically exhibits a rapid increase with decreasing pH over a relatively narrow range of pH, and the exact range of pH over which such increase in micelle size occurs depends on the pKa characteristic (and thus the detailed chemistry) of the polyamine material. From the data presented in the upper left of Fig. 3, the values of the expansion ratio, defined as the ratio of the micelle sizes between the two tumor-relevant pH limits α ≡ R(pH 6.5)/R(pH 7.4), were estimated for the various values of pKa°, and the results are shown in the upper right panel; here, pKa° denotes the intrinsic value of pKa for the polyamine in its monomeric (i.e., DP → 1) limit. As shown in the figure, the highest degree of micelle
swelling in tumors (α ≈ 1.8) is achieved when pKa° ≈ 7.0 ± 0.5. Further, as shown in the lower panel of Fig. 3, we tested whether the pH sensitivity of the micelle size at pKa° = 7.0 can be further increased by adjusting the various micelle parameters, i.e., polyamine degree of polymerization (DP), chain grafting density (σ · πRg2), micelle core radius (Rc), salt concentration (Cs), and the effective Flory–Huggins interaction parameter (χ). The results shown in the figure suggest that, for instance, an increase of DP or χ, a decrease of σ or Rc, or a combination of such changes from their original values used to generate the data shown in the upper panel would cause a further increase in the tumor pH responsiveness of the micelle size; the value of α can be increased to about two. 3. Carrier design The results demonstrated in the previous subsection suggest that in order to maximize the pH-responsiveness of the polymer micelle carrier systems to the intratumoral pH, it is most critical to choose an appropriate polyamine material for use as the size-controlling element; the polyamine should undergo a significant change in its conformation as the pH is changed between the pH values of human tumors and adjacent normal tissues (i.e., between approximately 6.5 and 7.4). Based on the data presented in Fig. 3, for instance, polyethylenimine (PEI for short) appears to be inappropriate for this use, because the PEI polymer has a pKa at about 4.2 [5] which is too far away from the optimal pKa° value of about 7.0. Another common polyamine called poly(2-(dimethylamino)ethyl methacrylate) (PDMAEMA) is also expected not to be an optimal material for constructing tumor pH-sensitive drug carriers, because its hydrophobicity is only marginal
Fig. 3. (Upper left) Radius of a polyamine-based block copolymer micelle (R) estimated by the SCF theory as a function of solution pH at various values of polyamine's intrinsic pKa°. It is assumed that the hydrophobic micelle core of 10 nm radius (Rc) is surrounded by polyamine brush chains of degree of polymerization DP = 200 with Kuhn monomer length b = 6 Å, monomer volume v = 179 Å3 and Flory-Huggins interaction parameter in water significantly greater than that corresponding to the θ condition (i.e., χ ≈ χθ + 0.33) at a dimensionless grafting density σ · πRg2 = 3.0 (where Rg denotes the radius of gyration of the polyamine chain in the random-walk configuration). The micelle is dispersed in an aqueous solvent with an ionic strength (Cs) of 150 mM NaCl equivalent. The water molecular volume is assumed to be v = 30 Å3. (Upper right) The values of the ratio of the micelle sizes between the pH values of human tumors and normal tissues α ≡ R(pH 6.5)/R(pH 7.4) estimated from the data presented in the upper left for the various values of pKa°. (Lower) Plots demonstrating how the tumor pH responsiveness of the micelle size (α) at a fixed pKa° of 7.0 changes in response to variations of other parameter, such as polyamine degree of polymerization (DP), chain grafting density (σ · πRg2), micelle core radius (Rc), salt concentration (Cs), or the effective Flory–Huggins interaction parameter of the polyamine-water pair (χ), from their original values used in the upper panel; the original parameters produce an α value of about 1.8 as denoted with a red dotted line. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
(even at high pH this polymer does not normally become completely water-insoluble [8]). The best candidate material appears to be poly(L-histidine) (PHis), which has an apparent pKa of about 6.5 [9]; this polymer has been shown to exhibit a dramatic conformational change as the pH of the environment is varied through the intratumoral pH values [9]. In real applications of polyamine nanoparticles in in vivo drug delivery, the amine moieties of the polyamine chains need to be shielded (e.g., by “PEGylation”) to enhance the stability of the particles against self-association and also to minimize toxicity-inducing nonspecific electrostatic interactions with serum proteins and tissue components. As schematically demonstrated in Fig. 4, there are ways of incorporating the size-controlling, polyamine component into the coronal structure of the polymer micelle without compromising the benefits of the PEGylation method commonly adopted for achieving prolonged blood circulation times. In fact, a PHis-based polymer micelle (similar to the one shown in Fig. 4(bottom)) has previously been proposed for enhanced delivery of drugs to tumors [9–11]; in these previous studies, in vitro tests confirmed that the pH-responsiveness of PHis can be used for achieving low pH-triggered surface expression of cell-interacting ligands. It will be interesting to conduct an in vivo pharmacokinetic study to determine whether/how much the incorporation of PHis to a polymer micelle system will improve its tumorspecific accumulation. Purely from the diffusion standpoint, a factor of two increase in the micelle size in response to the pH of the tumor microenvironment will, for instance, cause the same two-fold enhancement in the efficiency of delivery to tumor cells; such effect can be a “game changer”, especially when the absorbed dose of the drug within the tumor volume is close to the therapeutic threshold of the drug. Further, the overall delivery performance of the “pH-phoretic” micelles is expected to be even greater than what one would anticipate on the basis of the diffusion rates, because there can be other advantageous effects of using a polyamine-containing, “pH-phoretic” delivery system (such as an increased electrostatic interaction with cancer cells under low pH environments). A properly designed “pH phoretic” drug carrier
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having appropriate size (≪100 nm in diameter [12]) and surface (anti-fouling against collagen [13]) characteristics would even be able to traverse toward the deeper parts of the tumor due to the intratumoral pH gradient [14]. This “pH phoresis” concept can also be combined readily within other established drug delivery methodologies, such as the utilization of the enhanced permeability and retention (EPR) phenomenon [15] and tumor pH-specific dissociatable polymer micelles [16]. Many current anticancer drug formulations generally suffer low tumor “targeting” efficiencies [17]. We speculate that this “pH phoresis” concept will contribute to achieving an improvement in systemic drug delivery to solid tumors. Lastly, we would like to note that it is non-trivial to experimentally demonstrate the isolated effect of the pH-phoretic diffusion of polyamine nanoparticles on their distribution in a pH gradient, because the currently available methods for creating a stable pH gradient either in an electrophoretic gel or in a microfluidic device are inappropriate for testing the pH phoresis concept using polyamine particles in their most desirable form (i.e., in the form of polyamine-containing block copolymer micelles); commercial immobilized pH gradient (IPG) electrophoresis gels have too small pore sizes (much less than a few tens of nanometers), and in a microfluidic flow environment, the lifetime of a pH gradient created by a concentration gradient of acid or base molecules is much less than the diffusion time of the polyamine particles. Currently, efforts are under way to develop an appropriate experimental technique for this type of measurement. Acknowledgment The authors would like to thank the U.S. National Science Foundation (CBET-0828574 and DMR-0906567) and the Purdue University School of Chemical Engineering for providing financial support of this research. Y.Y.W. gratefully acknowledges the Global RNAi Carrier Initiative Program Fellowship from the Biomedical Research Institute of the Korea Institute of Science and Technology (KIST) and also the Research Fellowship from the Bindley Bioscience Center at Purdue University. Appendix A. Supplementary data Supplementary data to this article can be found online at http:// dx.doi.org/10.1016/j.jconrel.2013.06.016. References
Fig. 4. Cartoons describing exemplary designs of “pH-phoretic” polymer micelle systems. The red represents polyamine chains, the blue represents poly(ethylene glycol) (PEG) (or any such moiety that can confer stabilization of the micelle, e.g., serum albumin), and the purple represents the hydrophobic core domain of the micelle (where drugs can be loaded). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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