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Downloaded by guest on September 30, 2021 www.pnas.org/cgi/doi/10.1073/pnas.1620636114 a Blume-Peytavi Ulrike Schulz Robert profiles of free-energy interplay and the diffusivity to function barrier relates human penetration drug of modeling Data-based rlmdl o h emaino rg hog knexist, skin through Sev- drugs intercellu- bilayers. of lipid the permeation stacked is the of for mortar composed models is the eral which whereas , corneocytes, lar the are flat- gradually are toward migrating () when SC. cells corneocytes the the into skin transformed VE, and which and (SG), the tened corneocytes, granulosum of the stratum part the cells, stra- is In skin the (VE). con- into layer dead outermost viable divided dried-out thick 20-µm further of mechani- to is 10- sisting from the epidermis body (SC), thick, corneum The the tum mm (4). 2 protects stress typically and microorgan- is cal vessels, harmful which , of 100 the entrance contains about and the irritants, of or and thickness isms loss a water with prevents epidermis the 1A): medicine, in biology. interest and central for development, of required drug also in- only but The not delivery, is (1–3). drug function controlled delivery barrier drug of for consequently, understanding challenge depth highly molecules; severe the desired a allow present certain to they cases of to the some transfer designed in skin, and generally regulated out the are material barriers and foreign These keep mucosa; organ. respiratory human largest and mouth, intestinal, M equation Smoluchowski diffusion barrier biological of timescales entire deteri- samples the vivo span functioning. ex thus where and permeation, orate, long-time drug as feasible, short-time well not as predict are measurements from can experimental diffusivity where we penetration, disentangle Thereby of to effects. kinds us all free-energetic allows to diffusors, applicable and generally barriers underneath, is Our dermis which limit. long-time the approach, the modeling to in distribution epidermis barrier drug the free-energy the determines pronounced which from layer), a transition epidermal and the diffu- outermost times, at drug (the short reduced corneum at substantially stratum dominant to a the shown of in is combination sivity times function the further barrier consecutive on skin three without rely dexamethasone, at depth pro- For microscopy concentration skin used. X-ray drug are of from input, function derived only a the files As as pro- assumptions. free-energy drug the model equa- the and diffusion diffusivity of the the files invert obtain numerically thereby and we tion this, diffusion and diffusivity For general the energy. in 1D variations free spatial the for the accounts model by that equation we penetration skin, drug human time-dependent 2016) in 15, antiin- dexamethasone December the review drug of for profiles (received flammatory depth 2017 17, concentration February experimental approved on and Based PA, Philadelphia, University, Temple Klein, L. 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Direct PNAS a is article This interest. of conflict and no experiments. declare performed R.S. authors and data; The designed H E.R. analyzed S. and R.F., A.K., M.S.-K., K.Y., R.R.N. and Hedtrich, model; and S. the U.B.-P., designed A.V., R.S. F.R., R.R.N. and research; R.S. paper; performed the wrote R.R.N. R.R.N. and E.R., M.S.-K., free- position-dependent the on depends profile and profile energy concentration time 1D and a space of evolution the describes 16). with (15, slabs nm skin 100 thin below in resolution DXM a kinds X-ray unlabeled all absolute soft of 2D for profiles because generate concentration used to chosen us be is allows can drug spectromicroscopy and This general barriers. very permeation ex of is skin, in human (DXM) vivo input lipophilic dexamethasone three the only for glucocorticoid at test the we skin antiinflammatory which as the approach, Our in and times. profiles consecutive assumptions concentration-depth model drug any requires 13 make refs. not see does outline overview historical detailed a a and 14. For and models 12). experi- skin-diffusion (11, reproduce of certain studies inferred to profiles tape-stripping a as concentration-depth from such models assumed, or adjusted, all are permeabilities are mental parameters SC In posteriori transport 3D 12). a various the which (11, and models for structure detailed skin accounted partition more is is and in structure diffusivities structure and skin different (6–10), stratified with layers the coefficients 1D models com- by simplest of reflected levels the different In on structure plexity. skin the incorporate which uhrcnrbtos ...... H S. R.F., A.K., K.Y., R.S., contributions: Author owo orsodnesol eadesd mi:[email protected]. Email: addressed. be should correspondence whom To eieysrtge hog knms eetteeproperties. these reflect must skin through der- drug strategies Targeted a the delivery permeation. to whereas long-time epidermis reduces skin, underneath the mis the from step stratum into free-energy outer penetration pronounced the fast in hinders epi- diffusion of corneum slow mechanism function: basic barrier the dermal reveals highly which are profiles inhomogeneous, diffusivity modeling and example free-energy the both specific that dexamethasone, shows the drug For antiinflammatory the times. only of penetration the distinct as measured three profiles that at concentration introduced model which experimental is requires A skin out, input within challenging. materials diffusion harmful delivery drug for keep drug to pre- transcutaneous to and designed makes loss is and water layers distinct vent of consists skin Human Significance c esatwt h eea Ddfuineuto,which equation, diffusion 1D general the with start We which i.e., approach, modeling data-based a here describe We catR Eckart , b tfnH Stefan , uhl ¨ F onzke ¨ PNAS (z b n oadR Netz R. Roland and , ) n h ifsvt profile diffusivity the and | c irnaRancan Fiorenza , tBri,115Bri,Germany; Berlin, 14195 Berlin, at ¨ pi ,2017 4, April . lnc 01 eln Germany Berlin, 10117 Clinic, e ´ | nk,FR,AV,UB-. .Hedtrich, S. U.B.-P., A.V., F.R., onzke, ¨ o.114 vol. www.pnas.org/lookup/suppl/doi:10. a,1 d niaVogt Annika , | D o 14 no. (z ), c nttt of Institute | c 3631–3636 (z d , onzke, , t ¨ ) in

APPLIED PHYSICAL BIOPHYSICS AND SCIENCES COMPUTATIONAL BIOLOGY ∂ ∂  ∂  c(z, t) = D(z)e−βF(z) c(z, t)eβF(z) , [1] als and Methods). We demonstrate our method using experi- ∂t ∂z ∂z mental 1D concentration-depth profiles of DXM, a medium-size drug molecule that in water is only poorly soluble (33). Starting where β = 1/(kB T ) is the inverse thermal energy. The free- the drug penetration by placing a 1.5% DXM formulation in a energy profile F (z) reflects the local affinity and determines how hydroxethyl cellulose (HEC) on top of excised at the substance, in our specific case DXM, partitions in equilib- −βF(z) time 0, 2D concentration profiles of DXM in the epidermal skin rium, namely ceq(z) ∝ e . The diffusivity profile D(z) is layer are obtained after penetration times of 10 min, 100 min, a local measure of the velocity at which the substance diffuses and 1,000 min, using soft X-ray absorption spectromicroscopy in the absence of external forces. The diffusion equation not (15, 16) (Materials and Methods). only describes passive and active particle transport in structured To determine both the free-energy profile F (z) and the dif- media, it has in the past also been used for modeling marketing fusivity profile D(z) in the epidermal skin layer as well as the strategies (17), decision making (18), and epigenetic phenotype diffusive DXM properties in the drug-containing HEC gel and fluctuations (19). The importance of a spatially varying diffusivity in the dermis, we need three experimental concentration pro- profile D(z) has been recognized for the relative diffusion of two files recorded at different times as input. We demonstrate that particles (20), transmembrane transport (21), particle diffusion both the diffusivity profile D(z), which dominates drug pene- at interfaces (22, 23), folding (24–27), and multidimen- tration for short times, and the free-energy profile F (z), which sional diffusion (28). The 1D diffusion equation can be solved dominates long-time drug concentration profiles, are needed to analytically only in simple limits; for general F (z) and D(z) pro- describe drug penetration quantitatively. files the c(z, t) must be numerically calculated. Our analysis reveals that skin barrier function results from The inverse problem, i.e., extracting F (z) and D(z) from sim- an intricate interplay of different skin layer properties: Whereas ulation or experimental data, is much more demanding and has the entrance of lipophilic DXM into the SC is free-energetically recently attracted ample theoretical attention. With one notable favored, the drug diffusivity in the SC is about 1,000 times lower exception (29), most inverse approaches need single-particle than in the VE and thus slows down the passage of DXM through stochastic trajectories and are not suitable to extract informa- the SC. In addition, a pronounced free-energy barrier from the tion from concentration profiles (20–22, 30–32). For the typi- epidermis to the dermis prevents DXM from penetrating into cal experimental scenario, where a few concentration profiles at the lower dermal layers. In essence, the epidermis has a high different times are available, we here present a robust method affinity for the lipophilic drug DXM, which together with the low that yields the F (z) and D(z) profiles with minimal numeri- diffusivity in the SC efficiently prevents DXM penetration into cal effort and for general open boundary conditions (Materi- the dermis. Results and Discussion epidermis Experimental Concentration Profiles. Fig. 1B shows 2D absorption A profiles recorded at photon energy 530.1 eV for penetration viable epidermis (VE) times of 10 min, 100 min, and 1,000 min. This photon energy SC SG selectively excites DXM (15). The skin-depth coordinate z is shifted such that the outer skin surface is aligned to z = 0. The color codes the transmitted intensity, with blue indicating low gel dermis and yellow high transmission, and allows insights into the skin structure. Note that the three profiles originate from different skin samples from the same donor, because the analysis of the identical skin sample for different penetration times is not possi- B 10 min ble (16). Accordingly, these three samples have been chosen for maximal similarity of the SC and VE thicknesses. The region below z < 0 µm is the Epon resin used for skin SC VE dermis V e embedding. For all three samples, the layered structure from

1

.

0 z ≈ 0 µm to z ≈ 10 µm is the SC, whereas the VE extends

3

5 from a depth of about z ≈ 10 µm to a variable depth ranging

t

a from z ≈ 50 µm to z ≈ 80 µm. Within the VE oval-shaped ker- 100 min I

y

t atinocyte nuclei are discerned, which move gradually toward the

i

s

n SC where they differentiate and flatten into corneocytes. The

e t dermis, separated from the VE by the basal layer and the basal

n

i

d membrane, is clearly distinguished from the VE by the differ-

e

t

t

i ent transmission intensity. Note that the variation of the VE

x lateral position [ m] m thickness across different samples is an inherent property of skin

s

1000 min n

a and must be kept in mind in the analysis. From the ratio of the

r

T 2D transmission profiles at photon energies of 530.1 eV and 528.0 eV the 2D concentration profiles of DXM are determined (15, 16) (see Materials and Methods, DXM Concentration Profiles from X-Ray Microscopy, and Figs. S1 and S2 for details). Our aim is not to describe DXM diffusion at the cellular z skin depth [ m] level, for which 3D concentration profiles at different penetra- tion times of the same sample would be needed; our goal rather Fig. 1. (A) Schematic of epidermal skin structure. (B) Normalized 2D- is to model the 1D diffusion of DXM from the HEC gel on the transmission intensity profiles from X-ray scanning microscopy at photon energy 530.1 eV after 10 min, 100 min, and 1,000 min penetration time. The skin surface through the epidermis into the deeper skin layers. profiles allow us to distinguish different skin layers, schematically indicated For this, we laterally average the experimental 2D concentra- by white dividing lines, and demonstrate the skin sample variation. Whereas tion profiles; the resulting 1D concentration data are shown in the SC has a rather uniform thickness, the VE thickness varies considerably Fig. 2A (black circles). We base our modeling on cubic between the three samples. smoothing spline fits (black lines) in the range 0 < z < 80 µm,

3632 | www.pnas.org/cgi/doi/10.1073/pnas.1620636114 Schulz et al. Downloaded by guest on September 30, 2021 Downloaded by guest on September 30, 2021 cuze al. et Schulz 2A experi- Fig. the in in profiles concentration DXM mental decreasing the reflects which oteV,tefe nryi ahrcntn.I h range the In constant. rather is energy free the 50 VE, (34). the sustains range SC the that to depth across SC the gradient the potential In across chemical water change steep by structural the smoothly formu- significant rather gel a increases HEC ing energy the free prefers with the which compared ≈ SC, DXM, SC the of lipid-rich In character the lation. lipophilic in the be with to line in is ∆ tn ntegladstto set and gel the in stant the details). pass more that for estimates the of SD threshold the error reflect bars error The approximate stant. we data, experimental derive the 2 we for equation, diffusion estimates the best for the solver range inverse our depth into time the in profiles 0 concentration DXM Profiles. experimental Diffusivity and Free-Energy has Extracting SC small. the rather be the in to and seen concentration 100-min is the the profiles between 1,000-min difference whereas The VE, much. DXM changed the penetra- penetration not min min in 100 10 After found after SC. is already the that entered has seen DXM is tion It available. is profile ela h range the as well irgrigrsda X nteEo ei for resin Epon the in DXM residual disregarding line. dashed horizontal a by indicated value stationary predicted the to relaxes it before epidermis the ls otefe-ouinvle h o ifsvt nteS ntedphrne10 range depth the in SG the in diffusivity beyond low and The gel value. based free-solution are the and to layers skin close (B indicate SC (B lines the 1B. dividing in Vertical Fig. increase lines). in (red profiles equation diffusion intensity profiles. the concentration transmission on 2D based predictions the theoretical on with compared are lines) (black 2. Fig. A < B F 1.5

h reenergy free The 2

µm c [ g/(cm m)] gel/SC z 10 10 10 15 and 0 5 0 5 0 5 < Jmloe ag of range a over kJ/mol n-iesoa xeietlDMcnetainpolsa he ifrn eerto ie baksldcrls n ui mohn splines smoothing cubic and circles) solid (black times penetration different three at profiles concentration DXM experimental One-dimensional (A) 80 < 1000min 100min 10min − ≈ ntegladi h ems u oteasneof absence the to due dermis, the in and gel the in C; -20 20406080 0 z fe 0mn 0 i,ad100mnpenetration min 1,000 and min, 100 min, 10 after µm < ei C SC resin C 8. epi z 80 6 = < σ Jmla h onaybtengladS.This SC. and gel between boundary the at kJ/mol rmeprmns(lc oi ice)adter rdln) hwn aiu tpntaintime penetration at maximum a showing line), (red theory and circles) solid (black experiments from 80 , h reeeg lgtyicessagain, increases slightly energy free the µm z Inset F idctdb etcldse ie the line) dashed vertical a by (indicated µm 0.6 > F (z 10 (z 80 slwi h nieeiems(CadV)cmae oteHCgladt h emsadehbt ml u infiatgradual significant but small a exhibits and dermis the to and gel HEC the to compared VE) and (SC epidermis entire the in low is ) ,adteeiemstu xiislppii affinity. lipophilic exhibits thus epidermis the and ), µg/(cm ) z skindepth[ µm hr nyasnl concentration single a only where µm SG prxmtdt econ- be to approximated 2B, Fig. in F o 0 i n ,0 i.Compar- min. 1,000 and min 100 for (z F < viable epidermis(VE) ) gel 2 10 )(see ·µm) and z 0 = < Fg 2B, (Fig. µm 50 D xiisajm onby down jump a exhibits , F (z (z m] ,wihcorresponds which µm, ) ) aeil n Methods and Materials rfie hw nFig. in shown profiles and edn h three the Feeding D and theory exp. fitted experiment (z ,reflect- Inset), C ) z reeeg profile Free-energy ) < ob con- be to dermis 0 as µm F (z and ) D C B

D(z 2 2

D(z C [ g/cm ] D [ m /s] tn fdxmtaoei uewater, pure in con- dexamethasone diffusion of estimated the stant than smaller somewhat and gel HEC 400 of depth For a to up value D D previous with line in epi- affinity, lipophilic the (36). high conclusions whereas with sink water-like, a essentially is dermis as dermis the identifies to ing gel HEC the of of is range jump variations depth sample free-energy broad skin The a to over due out which the smeared 1B dermis, from transition the Fig. the to by in caused epidermis profiles is increase intensity free-energy this transmission that the with ison h G hc skont elctdrgtblwteS,is SC, the barrier SG that below The Note right 1. SG. Fig. located in the be profiles with transmission distinct to the VE, is known in the structurally visible is not but to which SC belongs SG, the and the in SC as the low from as is layer diffusivity the the associate tentatively We in DXM maximal the water, from derived difference dermis, energy the and gel HEC ∆F the Dermis between difference and free-energy in Epidermis the explained com- Between constitutes is Height dermis value this as rier the scattering bound, to data lower to affinity Due a DXM VE. the different with the pared der- to the into related penetration DXM mis for barrier pronounced a reflects epi 1000 nertdaon fDMin DXM of amount Integrated (D) constant. as approximated are profiles ) 200 400

100 F [kJ/mol] SC gel -10 0.1 slwi h Cadicessb atro ,0 nteV,weei is it where VE, the in 1,000 of factor a by increases and SC the in low is ) h osatdfuiiyi h E e un u obe to out turns gel HEC the in diffusivity constant The 10 20 10 gel/derm F 0 4001 03 05 07 20,000 70 60 50 40 30 20 10 0 -400 µm ≈ ≈ 0 1 z 10 µm (z ( ) k ≥ 16 0.2 c e SC gel 0 B < H 2 n ifsvt profile diffusivity and B) T ,wihi infiatylre hntevlei the in value the than larger significantly is which /s, 2 15 102030405060700 µm z ≈ O µm < ln(c stationary 89 = 16.2 µm 2 15 2 c n rp nteS yafco fruhy8 to 80 roughly of factor a by SC the in drops and /s .Tedfuiiymitissc low a such maintains diffusivity The 2C). (Fig. /s gel gel ol niaetepeec ftgtjntos nthe In junctions. tight of presence the indicate could µm SG D gLat mg/L Jml=6.4 = kJ/mol 15 = /c 10 (z penetration timet+1[s] H ) PNAS 2 2 O xiisarte osatvleof value constant rather a exhibits /,vatepriincefiin accord- coefficient partition the via g/L, 5. = ) z -9 -8 -7 z skindepth[ viable epidermis(VE) 15 = 25 ∆F 0123456789 | ◦ pi ,2017 4, April 1k VE/derm 3) n h X ouiiyin solubility DXM the and (35), C D(z k B ,weei butyincreases. abruptly it where µm, 10 B T (C ) T 440 460 480 4 h reeeg rfiethus profile free-energy The . siaeo reEeg Bar- Free-Energy of Estimate sqiecoet h free- the to close quite is , 10 eie rmteexperimental the from derived ) t ≈ 10 ≈ 20.5 µm 6 m] D 4 | × H o.114 vol. 2 10 50 and < O Jmlat kJ/mol 10 4 10 ≈ z µm s < 680 = 6 nfact, In S3. Fig. 5 ,0 i (Inset min 1,000 15 | < 80 o 14 no. µm z ,where µm, 10 z < 2 80 = 6 (33). /s 80 reveals D | 10 VE µm. 3633 µm 8 ≈ )

APPLIED PHYSICAL BIOPHYSICS AND SCIENCES COMPUTATIONAL BIOLOGY function has been shown to be due to tight junctions (37–40), 2 10 SC 0s in line with the low local diffusivity in this region displayed in 1s 1min Fig. 2C. 10min m)] 1000min Summarizing, four distinct features are revealed by our anal- 0 10 7d ysis: (i) a low diffusivity in the SC, (ii) a low diffusivity in a thin 2 stat. layer just below the SC that we associate with the SG, (iii) a sud- gel g/(cm den drop in free energy from the gel to the SC and a slight but 10 -2

significant free-energy increase in the SC, and (iv) a pronounced c [ free energy barrier from the epidermis to the dermis. We stress viable epidermis (VE) that these features in the free-energy and diffusivity profiles are 10 -4 -400 -200 20 40 60 80 10,000 20,000 not put in by way of our analysis method, but rather directly fol- z skin depth [ m] low from the experimental concentration profiles. We note in passing that the steep increase of the diffusivity at the bound- Fig. 3. Comparison of theoretical DXM concentration profiles for a wide 2 range of different penetration times. At time 0 the drug is entirely in the ary from the putative SG (with DSG ≈ 1 µm /s) to the VE (with 2 gel. Already at t = 1 s DXM penetrates into the SC. At intermediate times DVE ≈ 400 µm /s) is nothing one could directly identify from the experimental concentration profiles shown in Fig. 2A. t = 1 min and t = 10 min the VE is gradually filled, whereas at longer times t = 1,000 min and t = 7 d the profile below the epidermis approaches the stationary profile (indicated by a dotted curve). Predicting Concentration Profiles. In Fig. 2A we demonstrate that the numerical of the diffusion equation (red lines), F (z) D(z) based on the free-energy and diffusivity profiles and which no experimental data are available. We see that for short B C in Fig. 2 and , reproduce the experimental concentration pro- penetration times the concentration profile in the HEC gel is files very well (black lines). Small deviations are observed for the inhomogeneous, so the simplifying assumption of a constant con- drug-concentration profile after 10 min penetration time, and the centration in the gel becomes invalid. Interestingly, already at agreement is almost perfect for the 100-min and 1,000-min pro- t = 1 s DXM enters the SC. The 1,000-min profile is indistin- F (z) files. This not only means that our method for extracting guishable from the stationary profile in the HEC gel and the epi- D(z) and from concentration profiles works; we also conclude dermis, whereas below the epidermis, extending from z = 80 µm 1 that the diffusion equation Eq. describes the concentration to z = 2 cm, even after 7 d the stationary (flat) concentration pro- time evolution in skin very well. file has not yet been reached (note the inhomogeneous depth We define the time-dependent integral DXM amount that has scale and the logarithmic concentration scale). Not surprisingly, penetrated into the epidermis over the distance range 0 < z < molecular diffusion over a macroscopic length scale of 2 cm takes 80 µm as a long time. Z 80 µm C = c(z)dz. [2] epi Checking Model Validity. We check the robustness of our diffu- 0 µm sion model by comparison with two simplified models. In the In Fig. 2D we compare the experimental data for Cepi (solid constant-F model we restrict the free energy in the epidermis to black circles), which are directly obtained by integrating over the be constant (with the diffusivity still being a variable function), experimental concentration profiles in Fig. 2A, with the theoret- whereas in the constant-D model we restrict the diffusivity in ical prediction based on the diffusion equation and the deter- the epidermis to be constant (with the free energy being a vari- mined F (z) and D(z) profiles (red line). Note the logarithmic able function). This means that the number of adjustable model 8 timescale that extends from t = 1 s to t = 10 s ≈3 y. According parameters drops from 162 in the full model down to 83; other- to the experimental protocol, at time t = 0 the entire amount of 2 wise we use the same methods for finding inverse solutions of the DXM, corresponding to a surface concentration of 600 µg/cm , diffusion equation as before (Materials and Methods). is located in the gel and thus Cepi is zero. With increasing time, The free-energy profiles of the constant-F (blue) and the the theoretically predicted Cepi increases gradually and reaches constant-D (green) models in Fig. 4A are rather similar and do 2 4 a maximum of Cepi = 471.9 µg/cm at t ≈ 6 × 10 s = 1,000 min, not differ much from the full model result (red); in particular, at which time only ≈1.2 µg/cm2 DXM has penetrated into the the free-energy jumps from the HEC gel to the SC and from the dermis [Calculating the DXM Penetration Amount from Estimated VE to the dermis come out roughly the same. The diffusivity pro- F(z) and D(z) Profiles]. In the long-time limit, which is reached file of the constant-F model in Fig. 4B is again similar to the full above t ≈ 107 s ≈ 115 d, as seen in Fig. 2D, Inset, theory predicts model, whereas the constant-D model obviously misses the dif- eq 2 the equilibrium value Cepi = 465.0 µg/cm , denoted by a horizon- fusivity jump from the SC to the VE region. tal dashed line. In this hypothetical limit, longer than the exfoli- When we look at the predicted DXM concentration profiles in ation time of skin, which is 30–40 d, theory predicts that ≈10.0 Fig. 4C, we clearly see the shortcomings of the restricted models: µg/cm2 DXM has penetrated into the dermis, whereas ≈125.0 The constant-F model (blue lines) correctly predicts the short- µg/cm2 DXM still resides in the gel [see Calculating the DXM time behavior including the 10-min profile, but fails severely for Penetration Amount from Estimated F(z) and D(z) Profiles for the the long penetration time of 1,000 min, which is close to the full calculation]. Note that in vivo, dermal blood perfusion is cru- stationary equilibrium limit. In contrast, the constant-D model cial and could be easily taken into account in a generalized diffu- (green line) produces a concentration profile for 1,000 min that sion model by an additional reaction term. is indistinguishable from the full model and thus describes the The theoretically predicted curve for Cepi in Fig. 2D agrees experimental profile very nicely, whereas it fails at the short pen- well with the experimental data, which is not surprising in light etration time of 10 min. The comparison of the full model to the of the good agreement of the concentration profiles in Fig. 2A. restricted models demonstrates that both the free-energy and the This indicates that also the short- and long-time DXM penetra- diffusivity profiles are needed to correctly describe the experi- tion amounts, which are difficult to extract experimentally, are mental concentration profiles over the entire penetration time straightforwardly obtained from our model. range from 10 min to 1,000 min. We also understand from this In Fig. 3 we show calculated DXM concentration depth pro- comparison that the diffusivity profile is important at penetration files for a wide range of times. We here also plot the predicted times up to 10 min, whereas the free-energy profile is required to concentration profiles in the gel and below the epidermis, for describe the long-time and the equilibrium behavior accurately.

3634 | www.pnas.org/cgi/doi/10.1073/pnas.1620636114 Schulz et al. Downloaded by guest on September 30, 2021 Downloaded by guest on September 30, 2021 cuze al. et Schulz fHlik udlnsadtesmlswr bandatrtesge con- signed the after obtained Declaration were the samples with the accordance and in guidelines was Helsinki It of 2015). Charit July the updated of study EA/1/135/06, Committee The Ethics samples. the skin by abdominal approved 2-cm-thick human was vivo ex use and 16) (15, Profiles. Concentration Experimental or Methods small and Materials is barrier affinity epidermis–dermis absent. the consist charac- even that could lipophilic–hydrophilic such balanced remedy with ter The drugs pronounced drugs. modified have lipophilic of down to of slows severely show permeation dermis we the hydrophilic the which bar- to in free-energy affinity, epidermis, the diffusivity lipophilic the second, low And from overcome. the be rier First, to needs challenges: SC two the meets thus dermis dermis. the the into to transport leads DXM that slow combination and the low is exceptionally permeation it DXM but the epidermis, of the reduces low of Each through severely a epidermis. itself combination entire and by the SC the properties in the these solubility) on in high diffusivity (i.e., rely low energy a to free namely the shown properties, against key is two function DXM short-time barrier of the skin permeation reproducing Epidermal for penetration. needed drug is profile whereas to profiles, diffusivity skin essential concentration the long-time the is the describe profile describe to free-energy correctly important inhomogeneous diffusiv- are The both profiles barrier: that free-energy demonstrate to and results penetrat- our generalized DXM ity skin, of be human example also into specific can ing the and For available dimensions. are higher profiles times concentration differ- different resolved and at situations spatially is barrier whenever approach of The diffusors kinds presented. ent all is to skin human applicable in penetra- generally drugs different of from three times profiles at tion diffusivity profiles and concentration free-energy experimental deriving for method A Conclusions data limited which pre- at correctly times to provided. for able are even is profiles model the concentration that dict conclude we which in from shown is dataset reduced a short-time the predict to fails model constant-D the and C ( min, 1,000 and and diffusivity, min ( B) 100 free-energy, times min. (A) penetration 10 the for profile profiles concentration of concentration terms long-time in experimental the lines) (black profiles concentration 4. Fig. B A

h eino fcetdu eieymtostruhteepi- the through methods delivery drug efficient of design The from obtained results with comparison by check robustness A D [ m 2/s] 1000 F [kJ/mol] 100 -10 0.1 10 20 30 10 (A–C 4001 03 05 07 20,000 70 60 50 40 30 20 10 0 -400 0 1 oprsno h etitdcntn- bu ie)adtecntn- genlns oest h ulmdl(e ie)adexperimental and lines) (red model full the to models lines) (green constant-D the and lines) (blue constant-F restricted the of Comparison ) e CSG SC gel 102030405060700 z skindepth[ ottapn Analysis Bootstrapping viable epidermis(VE) u tde eedtie previously detailed were studies Our m] lncBri (approval Berlin Clinic e ´ and i.S4, Fig. 80 h atta nyfe-nrydfeecsaddfuiiysm fneighbor- of to sums due Eq. diffusivity 162 enter and to sites differences 164 ing free-energy from only reduced that is fact parameters con- the of experimental number the and total for surface The appropriate gel as ditions. boundary, upper subdermal the param- lower at free-fitting the as used at treated are are conditions and boundary constant Reflective be eters. to assumed are layer mal vial,wt 5stsadattltikeso m si h experiment. the in as cm, 2 is of F sites thickness discretization of total number a total are and The total data sites the concentration a 15 and experimental dermis and with no the available, sites also discretize We seven which experiment. for with the layer, gel in subcutaneous as HEC mm, the 0.4 concentration of discretize experimental thickness We no available. penetration, are the during data source DXM the as F 2 have thus We balance. detailed and (41) use we rates 80 transition of the depth with For skin µm. discretization a to equidistant up an epidermis use the in resolution available are micrometer data at concentration experimental Because schemes: cretization where (41) equation Master a of Equation. form Eq. Diffusion diffusion of the Inversion solution and Solution details. for Numerical concentra- 16 DXM and local 15 1s refs. absolute O see the law; the derive Beer–Lambert’s via using to eV by us 530.1 tion allows at This DXM (16). exclusively probe transition to Germany) tuned 100 (Berlin, was of II energy range BESSY depth scan facility a radiation in were synchrotron studies the microscopy at X-ray sil- thickness. performed on 100-nm placed and of thickness, membranes 350-nm nitride of icon sections into sliced and resin, epoxy buffer in the Na-cacodylate ded subse- at M were 0.1 chamber samples in humidified K glutaraldehyde gel, with a 2.5% HEC in in the K treated removing 305 quently gently of After temperature point. a saturation at concentration min a 1,000 with and DXM containing of formulation layer 0.4-mm-thick gel of a HEC to ethanol exposed was 70% surface a skin The patients. the of sent derm C (z c

and ) 2 gel

W c [ g/(cm m)] 10 10 10 15 n h diffusivities the and (t 4 i ,i 0 5 0 5 0 5 ∆t [(Fe(CN) c = = 020406080 i D(z (t 0) −W stetm iceiainse.W ac ifrn pta dis- spatial different match We step. discretization time the is ocnrto rfie.Tecntn- oe al opredict to fails model constant-F The profiles. concentration ) S ∆t + rfie nteeiemllyr nteHCgl hc serves which gel, HEC the In layer. epidermal the in profiles ) W = C i i ∆ mg/(cm 1.5 −1,i ,j 6 n OsO and ] = t ) 1000min 100min 10min − o oedtissee details more For 4. − D 2(∆z c i W i (t + i ) +1,i PNAS D ) 2 = D 4 2 j 1 m o eerto ie f1 i,10min, 100 min, 10 of times penetration for ·mm) hc aifiscnetaincnevto and conservation concentration satisfies which , gel o ii n X xto,dhdae,embed- dehydrated, fixation, DXM and lipid for exp W sdsrtzdi pc n ieadtksthe takes and time and space in discretized is ihase it f20n.Tephoton The nm. 200 of width step a with µm z skindepth[ i and ,i | −1  × pi ,2017 4, April − 80 c D v i −1 F i a derm 2k i = b − (t l B ∆z e 6 aaeesfo h discretized the from parameters 160 T ) F N e nteHCgladtesubepider- the and gel HEC the in + j p aibeDsrtzto fte1D the of Discretization Variable  = = i d W e 1 0.Tefe energies free The 102. r i ,i m | nterne0 range the in µm with +1 m] i s o.114 vol. c ( i V +1 E j ) (t = ) o h numerical the For + i | Full Theory D=const. F=const. Exp. ± W o 14 no. 1 i ,i c i (t dermis ), < ,we µm, F | gel z → < 3635 and π [3] [4] 80 ?

APPLIED PHYSICAL BIOPHYSICS AND SCIENCES COMPUTATIONAL BIOLOGY Diffusion Equation. The concentration profile ci(t) at time t = n∆t follows data points is necessarily higher than the number of 162 free-energy and dif- from Eq. 3 as fusivity parameters we need to determine. The profile ci(t = 0) corresponds

N N to the initial condition where DXM is homogeneously distributed in the gel X X ( ) = (1 + ∆ )n (0) = etW (0), only. For minimization of the error function Eq. 6 we use the trust-region ci t tW i,jcj i,j cj [5] j=1 j=1 iteration approach (42, 43); see Trust-Region Optimization for Constrained Nonlinear Problems for details. As an initial guess for the minimization we where W is the rate matrix defined in Eq. 4 and the continuous limit ∆t → 0 use a flat free-energy profile and choose random values for D(z) in the range has been taken to express the nth power by the matrix exponential. Eq. [10−1,..., 103] µm2/s. We perform 1,000 runs with different initial values 5 can be used to numerically solve the diffusion equation for any initial for D(z) with a maximal number of 250 iterations per run. For our results, we distribution cj(t = 0) and diffusivity and free-energy profiles D(z) and F(z). use only the best 1% of solutions with a residual error σ < 0.6 µg/(cm2·µm). To determine D(z) and F(z) based on experimental concentration profiles at A perfect solution with an error of σ = 0 is never observed, which reflects different times, we minimize the squared sum of deviations that our equation system is overdetermined and at the same time input data 2 concentration profiles come from different skin samples. In the constant-F Np N   k N 2 2 1 X 1 X exp X t and constant-D models we obtains errors σ < 1.6 µg/(cm ·µm) and σ < 0.9 σ = c (t ) − e kWc (0) , [6] data  i k i,j j  2 Np N µg/(cm ·µm) for the best 1% of the solutions, significantly larger than for k=1 k i=1 j=1 the full model. data where Nk is the number of experimental concentration data per profile and Np = 3 is the number of experimental concentration profiles. We use We thank M. Weigand, I. Bykova, and M. Bechtel data data data ACKNOWLEDGMENTS. N1 = 73 data points for the 10-min profile and N2 = N3 = 80 data from the Max-Planck Institute for Intelligent Systems for experimental data points for the 100-min and 1,000-min profiles, taken from the smoothed acquisition. This work was supported by the Deutsche Forschungsgemein- splines (black lines) in Fig. 2A. The total number of 233 = 73 + 80 + 80 input schaft within grants from Sonderforschungsbereich (SFB) 1112 and SFB 1114.

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3636 | www.pnas.org/cgi/doi/10.1073/pnas.1620636114 Schulz et al. Downloaded by guest on September 30, 2021