Skin Permeation of Chemicals

Abstract

Chemical compounds which may contact the skin include chemicals such as pesticides used against cockroaches and mosquitoes, repellents put into drawers, and topically applied therapeutic drugs and cosmetics. Plasticizers contained in building materials, endocrine disrupting chemicals (environmental hormones), volcanic ash, and radioactive materials may also be exposed to skin. Table 2.1 summarizes these materials.

1 Chemical Compounds Contacting the Skin Surface

Chemical compounds which may come in contact with the skin include chemicals such as pesticides used against cockroaches and mosquitoes, repellents put into drawers, and topically applied therapeutic drugs and cosmetics. Plasticizers contained in building materials, endocrine disrupting chemicals (environmental hormones), volcanic ash, and radioactive materials may also be exposed to skin. Table 2.1 summarizes these materials.

Table 2.1 Classification of chemical compounds applied or exposed to skin

Are these chemicals permeating the skin? We expected to see permeation for intensively applied pharmaceuticals as the active drugs in topical formulations and transdermal drug delivery systems (TDDS) must penetrate and permeate into and through skin. Are functional cosmetics or cosmeceuticals with active ingredients that show whitening effects designed to penetrate into the skin? We intensively apply soaps and body shampoos on skin: Do we design for the active ingredients to penetrate? Are there any skin-exposed chemicals that are permeable to skin? Are they safe if they permeate through skin? Are active materials in pesticides targeting cockroaches and mosquitoes safe to human beings? How do we understand the safety of topically exposed chemicals?

Generally, chemicals applied or exposed on skin can be distinguished between leave-on and rinse-off materials. Topical drug formulations, TDDS, and cosmetics are usually left on the skin for hours. On the other hand, soaps and body soaps are generally rinsed off. Thus, intentionally applied materials on skin can be easily divided into leave-on or rinse-off materials. However, it is somewhat difficult to distinguish either category for non-intentionally exposed materials on skin. Although we must take care not to be exposed to environmental chemicals, such chemicals may be exposed to or left on our skin for long periods of time.

2 Process of Skin Permeation or Percutaneous Absorption

Chemical compounds which come into contact with the skin may be distributed to the skin surface and absorbed through the skin barrier. The possibility and rate of contact with leave-on chemicals is generally much higher than with rinse-off chemicals. The amount and rate of chemicals distributed on the skin and permeated and absorbed through the skin are dependent on the resident period of those chemicals.

Chemicals dissolved in medium can be permeated and absorbed through skin as for example with the gastrointestinal absorption of drugs. Gases like carbon dioxide are also permeated through biomembranes. Chemicals dissolved in vehicles on the skin may be partitioned to the skin, penetrated into deeper skin, and absorbed through the skin into the systemic circulation (Fig. 2.1). Some ingredients in cosmetics dissolved in other ingredients may also permeate through skin. Fatty acid anions in soaps may be absorbed through skin after dissolving in water which is used during washing. Fragrances used in aromatherapy may be absorbed through skin because they are applied on skin using aroma oil carriers such as jojoba oil. In addition, pesticides dissolved in kerosene may be permeated through skin, because atomized oil may be dropped on the floor or adhered to a wall and our skin might be exposed to that oil. Even chemicals and radioactive compounds have the possibility of being exposed to skin, dissolved in our sweat or sebum, and permeated through skin.

Fig. 2.1

Percutaneous absorption process of chemical compounds

The trigger for the percutaneous absorption of chemicals is “distribution” or “partition” of the chemicals dissolved in vehicles and media. The partition phenomena take place between the vehicles (media) and the skin tissue. These are the same phenomena found between two solvents that are immiscible with each other. The partition is an equilibrium phenomenon which is observed in a very short period. We have to pay attention to three phenomena at the partition as follows: (i) the solvent itself may be rapidly absorbed through the skin, (ii) some solvents may change the barrier function (or stratum corneum) in skin to markedly increase the skin permeation of chemicals dissolved in the solvent, and (iii) chemicals may be adsorbed on skin. Adsorption to skin was found in our skin permeation experiments using catechins [1]. The chemicals adsorbed to the skin are not permeated through the skin. Thus, “adsorption” is another phenomenon in addition to partition or distribution.

Partition is followed by diffusion of chemicals through skin. Chemicals distributed to the skin surface are diffused through the skin matrix. Chemicals are migrated by their concentration gradients from the skin surface to deeper skin. Skin diffusion is a kinetic process. The surface layer of the stratum corneum is desquamated daily, so that slowly penetrating chemicals cannot be diffused into deeper skin tissue and are eliminated from the skin [2]. In contrast, most chemicals that migrate into the deeper skin may be taken up by cutaneous capillaries and then absorbed into the systemic circulation [3, 4]. Figure 2.1 also shows these processes.

3 Partition and Diffusion of Chemical Compounds to and Through Skin

When an aqueous solution containing a chemical compound is applied to vegetable oil and thoroughly mixed and then left to stand for a few hours, vegetable oil is layered on the water layer. The partition coefficient of the chemical, K _ow , can be determined as follows:

$$ {K}_{ow}=\frac{C_o}{C_w} $$

(2.1)

Partition between the vehicle and skin must be a similar phenomenon to these two solvents. In contrast, diffusion is thought to be a dynamic process like a random walk [5]. As shown on the left side in Fig. 2.2, 100 drunkards are in room A, in the center of 5 × 5 rooms. Where do they move a few hours later, since they can move randomly and unconsciously? The answer is shown on the right side in Fig. 2.2. The high-density room for the drunkards must be room A, followed by B, and then C. Thus, existent probability becomes a concentric circle. This shows a two-dimensional diffusion.

Fig. 2.2

Movement of drunks and diffusion (two-dimensional diffusion). Drunks in room A concentrically diffuse outward

Next, one-dimensional diffusion is explained using the same example of 100 drunkards (Fig. 2.3). The 100 drunkards start to move from the center room in one line of 25 rooms. They move over a few hours as shown in the upper and right side of Fig. 2.3. The existent probability is highest at A, followed by B, C, etc. After diffusion, the probability shows normal distribution as shown in the right and lower part of Fig. 2.3.

Fig. 2.3

Movement of drunks and diffusion (one-dimensional diffusion). Drunks in room A move to B and then C… The left and lower figure shows the Dirac’s delta pulse and the right and lower figure shows normal distribution

If there are infinite numbers of rooms in a line, a rectangle shown in the left and lower part of Fig. 2.3 becomes a vertical line. This is called the Dirac delta distribution in mathematics or the impulse function in control engineering. Integrating the delta distribution or impulse function becomes unity (100%).

This one-dimensional distribution (Fig. 2.3) can be modified as a model for skin diffusion of chemicals as follows. The chemical compound is only in the left-most column (vehicle-skin surface) and diffused from the left to the right (skin-blood surface) under an infinite condition where the concentration of the chemical in the left column (at the vehicle-skin surface) is not decreased as shown in Fig. 2.4. This results in constant concentration and sink conditions where the concentration of the chemical in the right-most column (at the skin-blood surface) is zero independent of time after starting the chemical diffusion. In this case the concentration (density) gradient of the chemical compound (drunkards) in any place must be constant.

Fig. 2.4

One-dimensional diffusion showing concentration-distance profile in skin. Under the assumption of infinite dose condition and sink condition, the concentration gradient against the depth of skin barrier must be the same independent of the depth

Figure 2.5 shows a typical concentration-distance profile of a chemical compound across skin. C _v is the chemical concentration in the vehicle, K is the partition coefficient (skin/vehicle) of chemical, so that KC _v is the chemical concentration at the skin surface. In addition, chemical concentration at the other surface of skin at the boundary between skin and blood vessels is 0. Hence, a constant concentration gradient is found across the skin at steady-state condition.

Fig. 2.5

Concentration-distance profile of chemical compound in the skin barrier

4 Diffusion Equation

Chemical diffusion in skin was theoretically analyzed by “Fick’s law of diffusion,” as reported by Germany physiologist and physicist, Adolf Eugen Fick in 1855. Fick’s first and Fick’s second law of diffusion were presented.

• (a) Fick’s First Law of Diffusion

Fick’s first law of diffusion can be applied at steady state when the skin concentration profile is independent of time after chemical application on skin. The percutaneous absorption (skin permeation) rate of chemical compound, J, is expressed by Fick’s first law of diffusion as follows:

$$ J=-D\frac{dC}{dx} $$

(2.2)

where J is defined as flux, the amount of chemical compound permeated across a unit area of skin per predetermined time. It is shown in mg/cm²/h for example. D is the diffusion coefficient shown in cm²/h for example. C is the concentration of chemical compound in the skin and shown in mg/cm³, and so on. x indicates the depth of skin shown in cm, for example.

Although chemical diffusion in skin can be obtained even for the directions through the y-axis and z-axis, these diffusions can be ignored compared to that in x-axis.

• (b) Fick’s second law of diffusion

Fick’s second law of diffusion is usually applied at the nonsteady-state diffusion when chemical concentration in skin is changeable against time. Most of the diffusion in skin is analyzed by Fick’s second law of diffusion. The equation is shown as follows:

$$ \frac{\partial C}{\partial t}=D\frac{\partial^2C}{\partial {x}^2} $$

(2.3)

Thus, chemical concentration in skin is expressed as function of depth of skin, x, and time after chemical application, t.

5 Skin Permeation (Percutaneous Absorption) Rate

Generally, biomembrane permeation is obtained by active transport, facilitated diffusion, and phagocytosis or pinocytosis as well as passive diffusion, which is explained by Fick’s law. However, in cases of skin permeation, the biggest barrier is diffusion through the dead cell layer on top of the skin, the stratum corneum. No active transport takes place in the stratum corneum, suggesting that the skin permeation profile can be analyzed by Fick’s law.

When a suspended solution containing a chemical compound is applied to the stratum corneum side (donor compartment) of excised skin in the in vitro skin permeation experiment, dissolved chemical concentration is constant throughout the experiment (i.e., infinite condition). In addition, a skin condition is assumed at the receiver compartment. Initial condition (I.C.) and boundary condition (B.C.) are shown as follows:

$$ {\displaystyle \begin{array}{ccc}\left[\mathrm{I}.\mathrm{C}.\right]& C=0& 0<x<L\\ {}\left[\mathrm{B}.\mathrm{C}.\right]& C=K{C}_v& x=0\\ {}& C=0& x=L\end{array}} $$

(2.4)

where L is thickness of skin (cm) (skin is assumed to be a homogenous membrane in this case), K is the partition coefficient from vehicle to skin, and C _v is the chemical concentration in vehicle (mg/cm³) as explained in Fig. 2.5. The following equation can be derived by Fick’s second law of diffusion using I.C. and B.C. (Eq. (2.4)) [6].

$$ C=K{C}_v\left[\left(1-\frac{x}{L}\right)-\frac{2}{\pi}\sum \limits_{n=1}^{\infty}\frac{1}{n}\sin \left(\frac{n\pi x}{L}\right)\exp \left(-\frac{D}{L^2}{n}^2{\pi}^2t\right)\right] $$

(2.5)

On the other hand, the skin permeation rate, J, is expressed by Eq. (2.2). Equation (2.2) was differentiated by x to obtain dC/dx, and the resultant dC/dx was introduced to Eq. (2.2) to obtain J under conditions of Eq. (2.4) as follows:

$$ J=\frac{K{C}_vD}{L}\left[1+2\sum \limits_{n=1}^{\infty }{\left(-1\right)}^n\exp \left(-\frac{D}{L^2}{n}^2{\pi}^2t\right)\right] $$

(2.6)

The cumulative amount of chemical compound permeated through skin, Q (mg/cm²/h), is

$$ Q= KL{C}_v\left[\frac{D}{L^2}t-\frac{1}{6}-\frac{2}{\pi^2}\sum \limits_{n=1}^{\infty}\frac{{\left(-1\right)}^n}{n^2}\exp \left(-\frac{D}{L^2}{n}^2{\pi}^2t\right)\right] $$

(2.7)

Figure 2.6 illustrates the profiles shown in Eqs. (2.6) and (2.7) as a function of time, t. J and Q at steady state (t→∞) are shown in Eqs. (2.8) and (2.9), respectively.

$$ J=\frac{\mathrm{d}Q}{\mathrm{d}t}=\frac{K{C}_vD}{L}=P{C}_v $$

(2.8)

$$ Q=\frac{K{C}_vD}{L}\left(t-\frac{L^2}{6D}\right) $$

(2.9)

where P is the permeability coefficient (P = KD/L, cm/s). Product of P and effective permeation area become clearance which is used as in hepatic clearance, total clearance and so on.

Fig. 2.6

Typical permeation rate, J, and cumulative amount, Q, of chemical compounds through skin. J approaches to a constant value, whereas Q becomes proportional to time after a certain period of time. (a) Infinite dose system, (b) Finite dose system

In addition, Q in Eq. (2.9) is extrapolated to the time-axis to obtain lag time, LT, as follows:

$$ LT=\frac{L^2}{6D} $$

(2.10)

LT is also shown in Fig. 2.6.

6 Analysis of Fick’s 2nd Law of Diffusion

6.1 Analysis of Ordinary Differential Equation Using Laplace Transformed Method

Fick’s second law of diffusion as shown in Fig. 2.3 is a partial differential equation having two variables. Before solving this partial differential equation, an analytical method for an ordinary differential equation is explained. The first-ordered chemical reaction equation is a derivative with one variable as follows:

$$ -\frac{dC}{dt}= kC $$

(2.11)

This equation shows that the decreasing rate of chemical, −dC/dt, is proportional to the chemical concentration, C, and the proportional rate constant is k. The equation can be changed to:

$$ \frac{dC}{C}=- kdt $$

where I.C. is C = C ₀. Then, Eq. (2.11) becomes

$$ {\int}_{C_0}^C\frac{dC}{C}=-k{\int}_0^t dt $$

When integrating dC/C and dt to ln C and t, respectively, the equation then becomes

$$ {\left[\ln C\right]}_{C_0}^C=-k{\left[t\right]}_0^t $$

And finally, the following equation

$$ \ln C=\ln {C}_0- kt $$

(2.12)

is obtained.

The Laplace transformation technique is used to solve complex ordinary differential equations. Laplace transformation causes a differential equation with a variable, time t, f(t), to become F(s) [7]. Where s is a complex variable as follows:

$$ F(s)={\int}_0^{\infty }f(t){e}^{- st} dt $$

(2.13)

This is a Laplace integration, where Laplace transformation (L: transformation hereinafter) and inverse Laplace transformation (L ⁻¹) mean a conversion from f(t) to F(s) and vice versa, respectively.

$$ F(s)=L\left[f(t)\right] $$

where f (t) and F(s) is a primitive function and image function, respectively. Reference 1 shows the principle of Laplace transformation.

Ref. 1

Equation (2.11) is solved by the Laplace transformation. First, left- and right-hand sites are Laplace transformed and equaled to obtain the following equation.

$$ -\left(s\overline{C}-{C}_0\right)=k\overline{C} $$

where $\overline{C}$ is the Laplace form of C. This equation becomes

$$ \overline{C}=\frac{C_0}{s+k} $$

(2.14)

Then, the equation is reverse transformed (L ⁻¹) to

$$ C={C}_0{e}^{- kt} $$

(2.15)

(see Table 2.2). It is apparent that Eq. (2.15) is the same as Eq. (2.12). These techniques are shown in books written by Wagner [7, 8].

Table 2.2 A typical Laplace transformation

Laplace transformation is very useful to solve the blood (or plasma) concentration after intravenous injection of drugs where the elimination kinetics obey a 2-compartment model. Then, the equation corresponding to Eq. (2.14) is as follows:

$$ \overline{C}=\frac{X_0\left(s+{k}_{21}\right)}{V_1\left(s+\alpha \right)\left(s+\beta \right)} $$

(2.16)

where X ₀ is dose, V ₁ is the volume of distribution of the central compartment, and α, β, and k ₂₁ are first-rate constants. Generally, the Heaviside expansion theorem and Benet’s method are usually applied to solve drug diffusion in skin using the Laplace transform.

According to Benet’s method at a reverse Laplace transformation, the reverse Laplace transformation of a polynomial expression, P(s)/Q(s) can be derived as follows:

$$ {L}^{-1}\left[\frac{P(s)}{Q(s)}\right]=\sum \limits_{i=1}^n\left(s-{\lambda}_i\right)\frac{P_i\left({\lambda}_i\right)}{Q_i\left({\lambda}_i\right)}{e}^{\lambda_it} $$

(2.17)

where λ _i are roots of the polynomial expression, and Q(λ _i ) are polynomial expression by substitution of λ _i to s in Q(s). In addition, Q(s) must be a higher degree equation than P(s) as a function of s, and no exponential terms such as (s + λ _i )² and (s + λ _i )³ are included in Q(s) when using Benet’s method in the reverse Laplace transformation.

Equation (2.16) is solved using the Benet method [9]. First, Eqs. (2.16) and (2.17) are combined to:

$$ \overline{C}=\frac{P(s)}{Q(s)}=\frac{X_0\left(s+{k}_{21}\right)}{V_1\left(s+\alpha \right)\left(s+\beta \right)} $$

The roots for Q(s) = 0 are −α and −β, then

$$ {L}^{-1}\left[\frac{P(s)}{Q(s)}\right]=C=\frac{X_0\left({k}_{21}-\alpha \right)}{V_1\left(\beta -\alpha \right)}{e}^{-\alpha t}+\frac{X_0\left({k}_{21}-\beta \right)}{V_1\left(\alpha -\beta \right)}{e}^{-\beta t} $$

is obtained (see Table 2.1). The following equation is found in general pharmacokinetics textbooks:

$$ C=A{e}^{-\alpha t}+B{e}^{-\beta t} $$

6.2 Analysis of a Partial Differential Equation, Fick’s Second Law of Diffusion Using Laplace Transformed Method

When the Fick’s second law of diffusion is Laplace transformed, the following equation is obtained:

$$ s\overline{C}-C(0)=D\frac{d^2\overline{C}}{d{x}^2} $$

where $\overline{C}$ is Laplace transform of C as mentioned above, and C(0) is the chemical concentration in skin at t = 0. Chemical is applied or exposed on skin at t = 0, then C (0) = 0. Thus,

$$ s\overline{C}=D\frac{d^2\overline{C}}{d{x}^2} $$

(2.18)

Equation (2.18) means that $\overline{C}$ is a function of only the skin depth, x. The equation becomes an ordinary differential equation. Thus, Fick’s second law with two variables, t and x, becomes an ordinary differential equation having one variable of x. Equation (2.18) is an inhomogeneous linear differential equation. Its general analytical method is shown in Ref. 2. The characteristic equation becomes under $\overline{C}={e}^{\lambda x}$,

$$ D{\lambda}^2-s=0 $$

Then, two roots are obtained as follows:

$$ \lambda =\pm \sqrt{s/D} $$

When displaced $\sqrt{s/D}$ to q, finally a general answer for $\overline{C}$ becomes,

$$ \overline{C}={C}_1{e}^{qx}+{C}_2{e}^{- qx} $$

Laplace transform of Eq. (2.4) is,

$$ {\displaystyle \begin{array}{ccc}\left[\mathrm{B}.\mathrm{C}.\right]& \overline{C}=K{C}_v/s& x=0\\ {}& \overline{C}=0& x=L\end{array}} $$

(2.19)

$\overline{C}$ can be obtained by substitution of x = 0 and x = L in Eq. (2.13) to obtain C ₁ and C ₂. However, in this case, the following equation is solved by referring B.C. at C = 0.

$$ \overline{C}=A\sinh \kern0.50em q\left(x-L\right) $$

where a hyperbolic function, sinh z is defined as follows:

$$ \sinh z=\frac{e^z-{e}^{-z}}{2} $$

Table 2.3 summarizes the hyperbolic functions and related equations.

Table 2.3 Physicochemical properties of chemical compounds used in the skin permeation study

A is determined by substitution of B.C. at x = 0, and then the following equation is derived by introducing the obtained A.

$$ \overline{C}=-K{C}_v\frac{\sinh \kern0.50em q\left(x-L\right)}{s\sinh \kern0.50em qL}=\frac{P(s)}{sQ(s)} $$

(2.20)

Then, the reverse Laplace transformation is started. According to Benet’s method, the denominator on the left side of Eq. (2.20) is substituted by zero; i.e., sQ(s) = 0. The obtained roots become exponential coefficients. There are two kinds of roots for s: s = 0 and s = − Dη _n ² (n = 1,2,3,…,∞). From the latter case, q = iη _n . It can then be changed to sinhiη _n L = 0 and sinη _n L = 0. Thus, η _n L = nπ (n = 1, 2, 3 . …).

When s = 0,

$$ {\left[\frac{P(s)}{Q(s)}\right]}_{s\to 0}=-K{C}_v{\left[\frac{\sinh \kern0.50em q\left(x-L\right)}{\sinh \kern0.50em qL}\right]}_{s\to 0} $$

If zero is substituted to s in the denominator and numerator, both become zero. Then, both the denominator and numerator are differentiated and zero is substituted to s, and it becomes (cosh 0 = 1):

$$ {\left[\frac{P(s)}{Q(s)}\right]}_{s\to 0}=-K{C}_v{\left[\frac{\left(x-L\right)\cosh \kern0.50em q\left(x-L\right)}{L\cosh \kern0.50em qL}\right]}_{s\to 0}=K{C}_v\left(1-\frac{x}{L}\right) $$

(2.21)

When s = − Dη _n ², we need some idea to solve. The reader is referred to the Heaviside expansion theorem for the rational function with a single root in the denominator as shown in Ref. 4. Then,

$$ {\displaystyle \begin{array}{l}{\left[\frac{P(s)}{s{Q}^{\prime }(s)}\right]}_{s\to -D{\eta_n}^2}=-K{C}_v{\left[\frac{\sinh \kern0.50em q\left(x-L\right)}{s\frac{L}{2\sqrt{Ds}}\cosh \kern0.50em qL}\right]}_{s\to -D{\eta_n}^2}=-K{C}_v{\left[\frac{2\sinh q\left(x-L\right)}{qL\cosh\ qL}\right]}_{s\to -D{\eta_n}^2}\\ {}=-2K{C}_v\left[\frac{\sinh \left[-i{\eta}_n\left(x-L\right)\right]}{-i{\eta}_nL\cosh \left[-i{\eta}_nL\right]}\right]=-2K{C}_v\left[\frac{i\sin {\eta}_n\left(x-L\right)}{i{\eta}_nL\cos {\eta}_nL}\right]\\ {}=-2K{C}_v\left[\frac{\sin {\eta}_nx\cos {\eta}_nL-\cos {\eta}_nx\sin {\eta}_nL\Big)}{\eta_nL\cos {\eta}_nL}\right]=-2K{C}_v\left[\frac{\sin {\eta}_nx\cos n\pi -\cos {\eta}_nx\sin n\pi \Big)}{n\pi \cos n\pi}\right]\\ {}=-2K{C}_v\left[\frac{\sin \frac{n\pi x}{L}}{n\pi}\right]\end{array}} $$

(2.22)

In this calculation, definition of the hyperbolic function together with sin nπ = 0 and cos nπ = 1, as shown in Ref. 3. Finally, Eq. (2.5) can be derived by a combination of Eqs. (2.21) and (2.22). Then, Eq. (2.5) is differentiated by x to,

$$ \frac{dC}{dx}=-\frac{K{C}_v}{L}\left[1+2\sum \limits_{n=1}^{\infty}\cos \left(\frac{n\pi x}{L}\right)\exp \left(-\frac{D}{L^2}{n}^2{\pi}^2t\right)\right] $$

According to Fick’s first law of diffusion, flux, J, is obtained as follows:

$$ {\displaystyle \begin{array}{l}J=-D{\left(\frac{dC}{dx}\right)}_{x=L}=\frac{DK{C}_v}{L}\left[1+2\sum \limits_{n=1}^{\infty}\cos \left( n\pi \right)\exp \left(-\frac{D}{L^2}{n}^2{\pi}^2t\right)\right]\\ {}=\frac{K{C}_vD}{L}\left[1+2\sum \limits_{n=1}^{\infty }{\left(-1\right)}^n\exp \left(-\frac{D}{L^2}{n}^2{\pi}^2t\right)\right]\end{array}} $$

(2.6)

In addition, the cumulative amount of chemical that permeated through skin, Q (mg/cm²/h) as shown in Eq. (2.7) is obtained by integrating Eq. (2.6) with time, t. Please refer to Crank [10] for solving the process for Eqs. (2.5)–(2.7).

Ref. 4

7 Infinite Dose System and Finite Dose System

In the infinite dose system, the amount of drug applied on skin is large enough so as to ignore the drug decrease in the vehicle. In a finite dose system, on the other hand, the drug concentration in vehicle is gradually decreased with the passage of time after application on skin. When a small amount of a drug is applied on skin, the drug in the vehicle is mostly absorbed into the skin and systemic circulation and only several tens of a percent are sometimes left in the vehicle. First, the finite dose system is considered.

7.1 Equation for Skin Permeation Rate at Finite Dose System

Although the B.C. becomes Eq. (2.4) in the infinite dose system, B.C. for the drug concentration at x = 0 in skin coming into contact with the vehicle is K·C _v , the B.C. at the finite dose system becomes as follows:

$$ {\displaystyle \begin{array}{cccc}\left[\mathrm{B}.\mathrm{C}.\right]& C=K{C}_v(t),& V\frac{d{C}_v(t)}{dt}= AD\frac{dC}{dx}& x=0\\ {}& C=0& & x=L\end{array}} $$

(2.23)

where V is the volume of vehicle (cm³), C _v (t) is the drug concentration in the vehicle at time, t, and A is application area (cm²) of the vehicle.

Figure 2.7 compares the concentration-distance profile between the infinite dose system and the finite dose system. At the infinite dose system, the drug concentration in vehicle (donor) is a constant value, C _v , as shown in Fig. 2.7a, so that the drug concentration gradient through skin becomes constant. Thus, the skin concentration is shown as a function of the skin position of x, as follows:

$$ C=K{C}_v\left(1-\frac{x}{L}\right) $$

Fig. 2.7

Effect of infinite (a) or finite condition (b) on the concentration-distance profile of a chemical compound in the skin barrier

In addition, skin concentration increases with the passage of time from t = t ₁ to t ₂ and so on to become a steady-state value at t = ∞, as shown in Fig. 2.7a. In the case of the finite dose system, on the other hand, drug concentration in vehicle decreases with time from t = 0 to t = t ₁−t ₂, so that the concentration gradient across skin is also decreased with time, as shown in Fig. 2.7b.

Laplace transformation of B.C. shown in eq. (2.23) becomes:

$$ {\displaystyle \begin{array}{l}\left[\mathrm{B}.\mathrm{C}.\right]\ \overline{C}=K{\overline{C}}_v,V\left(s{\overline{C}}_v-{C}_0\right)= AD{\overline{C}}^{\hbox{'}}\kern0.50em x=0\\ {}\kern1.50em \overline{C}=0\kern7.50em x=L\end{array}} $$

(2.24)

where $\overline{C}$, ${\overline{C}}_v$, and ${\overline{C}}^{\prime }$ show a Laplace form of skin concentration, vehicle concentration, and concentration gradient across skin ($d\overline{C}/ dx$).

When Eq. (2.3) is Laplace-transformed using I.C. (C = 0 at t = 0) and B.C. at x = 0, it becomes:

$$ \overline{C}={C}_1\sinh \kern0.50em q\left(x-L\right) $$

where C ₁ is a coefficient. The differential form of $\overline{C}$ becomes:

$$ d\overline{C}/ dx={C}_1q\cosh \kern0.50em q\left(x-L\right) $$

When these equations are introduced to Eq. (2.24), they become:

By combining these equations, C ₁ can be determined as follows:

$$ {C}_1=-\frac{VK{C}_0}{Vs\sinh \kern0.50em qL+ AKDq\cosh \kern0.50em qL} $$

Then, the Laplace-transformed equation showing skin concentration becomes:

$$ \overline{C}=-\frac{VK{C}_0\sinh \kern0.50em \ q\left(x-L\right)}{Vs\sinh \kern0.50em \ qL+ AK Dq\cosh \kern0.50em \ \ qL}=-\frac{qVK{C}_0\sinh \kern0.50em \ q\left(x-L\right)}{s\left( Vq\ \ \sinh \kern0.50em \ qL+ AK\cosh \kern0.50em qL\right)} $$

(2.25)

Permeation flux through skin, J, can be obtained by Fick’s first law of diffusion as follows:

$$ J=-D{\left(\frac{dC}{dx}\right)}_{x=L} $$

The Laplace transformation of this equation is:

$$ \overline{J}=-D{\left(\frac{d\overline{C}}{dx}\right)}_{x=L} $$

Then, Laplace transformation showing the skin permeation rate is:

$$ \overline{J}=\frac{VK{C}_0}{Vq\sinh \kern0.50em \ qL+ AK\kern0em \cosh \kern0.50em \ \ qL} $$

(2.26)

Laplace transformation of the cumulative amount of drug permeated through skin, Q, can be easily obtained using Eq. (2.25) as follows:

$$ \overline{Q}=\frac{VK{C}_0}{s\left( Vq\sinh \kern0.50em qL+ AK\cosh \kern0.50em qL\right)} $$

(2.27)

It is not easy to reverse transform Eqs. (2.25)–(2.27). Thus, let us directly treat these Laplace transformed equations. For example, we can estimate K and D from Q−t data from these equations using the nonlinear least-squares method in the Laplace dimension [11].

7.2 Analysis of Fick’s Second Law of Diffusion Using the Difference Equation Method

Fick’s second law of diffusion can be solved using the difference equation method [12, 13]. The left and right sides of Eq. (2.3) are transformed to Eqs. (2.28) and (2.29).

$$ \frac{d{C}_{i,j}}{dt}=\frac{1}{\varDelta t}\left({C}_{i,j+1}-{C}_{i,j}\right) $$

(2.28)

$$ \frac{d^2{C}_{i,j}}{d{x}^2}=\frac{1}{\varDelta {x}^2}\left({C}_{i-1,j}-2{C}_{i,j}+{C}_{i+1,j}\right) $$

(2.29)

where C _i,j indicates the drug concentration in skin at a depth of i and a time of j (where i and j are natural numbers), and ∆x and ∆t show x _i+1 − x _i and t _j+1 − t _j , respectively. Substituting Eqs. (2.28) and (2.29) with Eq. (2.3) yields:

$$ {C}_{i,j+1}= rD{C}_{i-1,j}+\left(1-2 rD\right){C}_{i,j}+ rD{C}_{i+1,j} $$

(2.30)

where r means ∆t/∆x ². The skin permeation rate, J, and the cumulative amount of skin permeation, Q, are expressed using the difference equation as follows:

$$ {J}_j=-D\frac{C_{n+1,j}-{C}_{n,j}}{\varDelta x} $$

(2.31)

$$ {Q}_j={Q}_{j-1}+{J}_j\times \varDelta t $$

(2.32)

where n is the segmentation number of the skin.

7.3 Analysis of the Difference Equation Method Using a Spreadsheet

J _j as shown in Eq. (2.31) can be analyzed using a spreadsheet, Microsoft^® Excel, as follows. Figures 2.8 and 2.9 illustrate examples for the analyses of Fick’s second law of diffusion. Figure 2.8 shows an example of an infinite dose system (B.C. as shown in Eq. (2.4)) and Figure 2.9 is an example of a finite dose system (B.C. in Eq. (2.23)).

Fig. 2.8

Typical analyzing sheet using difference equation for infinite dose system

Fig. 2.9

Typical analyzing sheet using difference equation for finite dose system

7.3.1 Infinite Dose System

The analytical method of the difference equation for an infinite dose system is explained using Fig. 2.8. The figure is shown in the cells in a Microsoft^® Excel spreadsheet. The position shows parameters used in this calculation ①. Cells drawn in yellow are parameters that we have to input before calculation. The thickness of the skin barrier, L (cm), and application concentration, C _v (μg/mL), are the input parameters. In an example in Fig. 2.8, L and C _v are 0.05 cm and 5000 μg/ml, respectively. Cells drawn in pink show calculated values using the nonlinear least-squares method (those are diffusion coefficients of the chemical compound in the skin barrier, D, and the partition coefficient from vehicle to the skin barrier, K).

In this analysis, the skin barrier was divided into ten segments (n = 10), so that Δx is L/10 and 5 × 10⁻³ cm. It is very important to set D·Δt/Δx ² at less than 0.5. Otherwise, the answer cannot be obtained due to divergence. Then, Δt was set to be 0.0033 h in this analysis. D·Δt/Δx ² and K·C _v were determined by D and K which were determined by the non-linear least-squares method ①. A parameter, α, relates to the weighting for the calculation.

The site of the Excel table shows the skin concentration, C, in contact with vehicle (x = 0) ②. C becomes K·C _v independent of time after application in the case of an infinite dose system (C = 12,500 μg/ml in Fig. 2.8). The site is the main part of the table ③, and was determined by Eq. (2.30). The site shows the skin concentration in relation to the receiver (x = L) ④. C must be 0 under a sink condition. The skin permeation rate, J, shown in the site can be determined by Eq. (2.31), and the cumulative amount of skin permeation, Q, can be determined by Eq. (2.32) ⑤.

Next, data for the cumulative amount of chemical permeated through skin were input into the site ⑦. The residual sum of squares, RSS, between calculated and observed values in the site can be obtained by the following equation ⑧.

$$ RSS=\frac{{\left({Q}_{cal}-{Q}_{obs}\right)}^2}{{\left({Q}_{obs}\right)}^{\alpha }} $$

(2.33)

The weighting parameter, α, is 0 in cases of high weighting to large values, 2 in cases weighting to small data values, or 1 in cases of even weighting. The site shows RSS ⑪. RSS is shown as follows:

$$ RS S=\sum RS $$

Although a certain number have already been written in whole cells in Fig. 2.9, each equation must be introduced to cells of ②, ⑥, ⑧, and ⑫ during calculation. When input parameters into the site, predetermined values are introduced ①.

7.3.2 Finite Dose System

Next, calculation method for finite dose system is explained in Fig. 2.9. Input parameters are shown in the site written in yellow as the same in the infinite system (Fig. 2.8) ①. The thickness of skin barrier, L, application concentration, C _v , effective permeation area, A, and donor volume are 0.05 cm, 5000 μg/mL, 0.95 cm², and 1.0 mL, respectively, for the finite system. The other parameters are the same as in the infinite dose system. Although the site in Fig. 2.9 is similar to that in Fig. 2.8, the skin concentration facing the vehicle decreases with a passage of time ②. Thus, C _1,j+1 in the site becomes as follows according to Eq. (2.3) ③.

$$ {C}_{1,j+1}=K\left({C}_v-\frac{A{Q}_{in,j}}{V}\right) $$

(2.34)

where Q _in,j shows the cumulative amount of chemical compound decreased in the donor cell. The decreasing rate of chemical compound in the donor compartment, J _in,j is determined by the following equation as shown in the site ④.

$$ {J}_{in,j}=-D\frac{C_{2,j}-{C}_{1,j}}{\varDelta x} $$

(2.35)

Equation (2.35) can be determined by the following Fick’s first law of diffusion using the difference equation method:

$$ {J}_{in}=-\frac{V}{A}\frac{d{C}_v(t)}{dt}=-D{\left(\frac{dC}{dx}\right)}_{x=0} $$

In addition, Q _in is expressed as follows:

$$ {Q}_{in}=-D{\int}_0^t{\left(\frac{dC}{dx}\right)}_{x=0} dt $$

(2.36)

Using the difference equation method, Q _in,j in the site can be expressed using J _in,j as follows ⑤:

$$ {Q}_{in,j}={Q}_{in,j-1}+{J}_{in,j}\cdot \varDelta t $$

(2.37)

By substituting Q _in,j which is obtained by Eq. (2.37) to Eq. (2.34), C _1,j+1 can be obtained as shown in the site ③. It is not so difficult when the author tries using PC. The sites ⑥ to ⑪ in Fig. 2.9 for the finite dose system are the same as in the sites ③ to ⑫ in Fig. 2.8 for the infinite dose system. In addition, the sites ②, ⑫, ⑩, and ⑪ are input on the right side of the equation.

7.4 Curve-Fitting Using the Least-Squares Method in a Spreadsheet

The determination method of permeation parameters is shown by curve-fitting to the cumulative amount of chemical compound permeated though skin, Q, using the least-squares method in spreadsheet. First, observed data for Q are input in the sheet, and equations showing RS are also input into corresponding cells having observed data, so that RSS can be determined.

Then, tool bars are opened in Microsoft^® Excel to use solver. When sober is not installed in the PC, solver is checked to install into the Microsoft^® Excel spreadsheet.

The cell for RSS is adjusted to objective cell in the solver, goal must be “the minimum,” and changeable cells are set to those for D and K. Conditions of the solver are 100 times for repeated cycles, accuracy is 0.000001, crossing is 5%, convergence is 0.001, and algorithm is set to pseudo-Newtonian (these can be changed by the authors). Then, “run” is clicked to produce a convergent solution. When a convergent solution is not obtained, “run” can be clicked again. After obtaining a solution, click “write solution” to show the determined D and K in the pink cells and draw a figure showing observed data and the fitting curve.

The fitting curves for Q are also shown in Figs. 2.8 and 2.9. Figure 2.10 compares these fitting curves in both the infinite and finite dose conditions. In this presentation, data under finite dose conditions were selected. Thus, fitting curves in Fig. 2.8 and left of Fig. 2.10 are not fitted to the observed ones, whereas those in Fig. 2.9 and right of Fig. 2.10 are well fitted.

Fig. 2.10

Comparison of observed and calculated cumulative amount of chemical compound permeated through skin. Curve-fitting was done by difference equation for Fick’s second law of diffusion

8 Modeling of Skin Barrier to Evaluate Skin Permeation Rate

8.1 Parallel Permeation Pathway Model

As explained in the previous chapter, the stratum corneum pathway as well as the appendage pathway are found in the skin barrier. Skin permeation rates of many chemical compounds having different hydrophilic or lipophilic properties may be predicted by considering those properties through these different pathways. Figure 2.11 shows a typical concentration-distance profile of a chemical compound at steady state using a parallel permeation model.

Fig. 2.11

Typical steady-state concentration-distance profile of chemical compound using Parallel permeation pathway model

Under steady state, the skin permeation rate through a lipophilic domain, J _lip is

$$ {J}_{lip}=\left(1-a\right)\frac{K{C}_v{D}_{lip}}{L}={P}_{lip}{C}_v $$

(2.38)

where K is partition coefficient from vehicle to lipophilic domain of skin, D _lip is diffusion coefficient through a lipophilic domain (cm²/h for example), and α is a ratio of the hydrophilic domain in the skin barrier (0 < α < 1). In addition, P _lip is the permeability coefficient through the lipophilic domain (cm/s for example). On the other hand, the permeation rate through a hydrophilic domain, J _aq is

$$ {J}_{aq}=a\frac{C_v{D}_{aq}}{L}={P}_{aq}{C}_v $$

(2.39)

where D _aq and P _aq are the diffusion coefficient and permeability coefficient, respectively, of a chemical compound through a hydrophilic domain. When water or an aqueous solution is used in a vehicle, the partition coefficient from vehicle to hydrophilic domain should be in unity.

The whole skin permeation rate, J _tot′ must be the sum of J _lip and J _aq, then

$$ {J}_{tot}^{\prime }={J}_{lip}+{J}_{aq}=\left({P}_{lip}+{P}_{aq}\right){C}_n $$

(2.40)

where one prime must be different from two primes and no prime. Whole skin permeability coefficient, P _tot′ becomes

$$ {P}_{tot}^{\prime }={P}_{lip}+{P}_{aq} $$

(2.41)

Molecular weight or volume affects D _lip and D _aq. We selected several model chemicals having a molecular weight of about 300, and measured their permeations through hairless rat and human skin [14]. When using similar molecular weight compounds, K as shown in Eq. (2.38) is the most important parameter for determining J _tot′ or P _tot′. Thereafter we measured the relation between the n-octanol water partition coefficient, which is closely related to K and P _tot (P _tot′).

8.2 Can We Predict Skin Permeation Rate Only From n-octanol Water Partition Coefficient?

The permeation rate of a chemical compound through a lipophilic domain, J _lip is proportional to K, as shown in Eq. (2.38). Generally, a relation between K and K _ow is expressed as follows [14]:

$$ K={a}^{\prime }{K}_{ow}{}{}^b $$

(2.42)

where a′ and b are coefficients. Then, J _tot′, which is the sum of Eqs. (2.38) and (2.39), is shown as a function of K _ow as follows:

$$ {P_{tot}}^{\hbox{'}}=a{K}_{ow}{}{}^b + c $$

(2.43)

where a and c are also coefficients. These aK _ow ^b and c in Eq. (2.43) correspond to the lipophilic and hydrophilic domains, respectively. The skin permeation rate of lipophilic compounds is generally much higher than that of hydrophilic compounds, thus aK _ow ^b is also much higher than c.

Fig. 2.12 shows a relation between logP _tot′ and logK _ow of 16 different chemical compounds as shown in Table 2.3. From these values, the following correlation equations were obtained for hairless rat and human skin permeations [15] using a non-linear least squares method for Eq. (2.43) [15].

$$ \mathrm{Hairless}\ rat\ \mathrm{skin}\kern0.50em \ \kern0.50em {P}_{tot}^{\prime}\left( cm/\mathrm{s}\right)=4.78\times {10}^{-7}{K}_{ow}+8.33\times {10}^{-8} $$

(2.44)

$$ \mathrm{Human}\ \mathrm{skin}\kern0.50em \ \kern0.50em {P}_{tot}^{\prime}\left( cm/\mathrm{s}\right)=1.17\times {10}^{-7}{K}_{ow}+2.73\times {10}^{-8} $$

(2.45)

Fig. 2.12

Relationship between permeability coefficient of several compounds through human skin and hairless rat skin and n-octanol–water partition coefficient. Open circle: human skin, filled circle: hairless rat skin. Solid lines: Eqs. (2.7) and (2.8). Abbreviations for chemical compounds are shown in Table 2.3.

Solid lines in Fig. 2.12 were obtained by Eqs. (2.44) and (2.45). These solid lines are much closer to the observed data, suggesting that P _tot′ can be estimated by the K _ow of chemical compounds. The figure also illustrates a similar profile of P _tot′ for lipophilic compounds but a little lower P _tot′ through human skin than that through hairless rat skin for hydrophilic compounds. We then determined lipid content and water uptake in and to the stratum corneum. The obtained results are shown in Table 2.4. The lipid content in human skin is a little higher than in hairless rats, whereas water uptake in human skin is lower than hairless rats, suggesting that the ratio between lipophilic and hydrophilic pathways must be different between the two species.

Table 2.4 Lipid content and water uptake in and into human and hairless rat stratum corneum

8.3 Two-Layered Model

Although the biggest barrier to skin permeation is the stratum corneum, the underlying viable epidermis and dermis also have a role as a skin barrier. Thus, it is sometimes very important to determine the permeability coefficient through the viable epidermis and dermis, P _ved, which can be measured by the stratum corneum-stripped skin.

A relation between whole skin permeability coefficient, P _tot″, and P _tot″ and P _ved becomes

$$ \frac{1}{P_{tot}^{{\prime\prime} }}=\frac{1}{P_{sc}}+\frac{1}{P_{ved}} $$

(2.46)

where P _sc is permeability coefficient through stratum corneum.

8.4 Combination of the Parallel Permeation Pathway Model Together with the Two-Layered Model

Since skin consists of the stratum corneum and viable epidermis/dermis and the stratum corneum has lipophilic and aqueous domains, finally P _tot can be expressed by a combination of the parallel permeation pathway model and the two-layered model as follows. (This P _tot is different from P _tot′ and P _tot″ as shown above.)

$$ \frac{1}{P_{tot}}=\frac{1}{P_{lip}+{P}_{aq}}+\frac{1}{P_{\mathrm{ved}}} $$

(2.47)

This equation can be simplified depending upon the chemical compound tested. For example, the two-layered model is very useful for lipophilic drugs, whereas the one-layered model is sometimes enough for hydrophilic drugs due to a much lower barrier in the viable epidermis and dermis than in the stratum corneum for hydrophilic drugs. On the other hand, the parallel permeation pathway model is useful for analyzing skin permeation of hydrophilic drugs, whereas a one-permeation pathway model is enough for lipophilic drugs.

9 Effect of Molecular Size on Skin Permeation

9.1 Effect of Molecular Size

As mention above, the size of chemical compounds is one of the most important parameters for their skin permeation. From the 1980s when TDDS were introduced to the market, pharmaceutical researchers have studied a relation between physicochemical properties, i.e., polarity (n-octanol water partition coefficient) and molecular size, and skin permeation rate of chemical compounds. The following equation reported by Potts and Guy [17] has been frequently cited in this regard over the past two or three decades:

$$ \log p=-6.3+0.71\cdot \log {K}_{ow}-0.0061\cdot MW $$

(2.48)

where MW is molecular weight (in Daltons). Permeability coefficient, P, means moving the length (in cm) of chemical compounds through the skin barrier per unit time (in seconds), which can be used as an index for the skin permeation rate. Potts and Guy [17] introduced Eq. (2.48) by analyzing skin permeation data by Flynn [18]. Unfortunately, Eq. (2.48) cannot be extrapolated to compounds having a MW of more than 500 Da.

In addition, we have to realize the lower limit for P for the percutaneous absorption of chemical compounds. The stratum corneum consists of about 20 layers of corneocytes, although the exact number of layers is dependent on the body site. The thickness of the stratum corneum is about 20 μm, although the thickness is also dependent upon the site. The uppermost layer of corneocytes is peeled off (desquamated) from the stratum corneum in a day (24 h). Thus, the desquamation rate is 1.0 μm/24 h or 1.0 × 10⁻⁹ cm/s. This suggests that P less than this number indicates no skin penetration, because compounds having a low P are peeled off together with the desquamated layer.

Bos and Meinardi [19] empirically reported the 500-Da rule using a lot of data with regard to the fact that the most skin irritating compounds have a MW of less than 500 Da [1]. The MW of important drugs entrapped in TDDS and topical formulations is also less than 500 Da [2, 3]. Table 2.5 shows the P of compounds having different MW and log K _ow . It is clear that P increases with an increase in log K _ow . It is important to understand the effect of MW. An increase in MW decreases P. A zone drawn in pink means a lower P than the desquamation rate. A chemical compound in this zone cannot permeate skin.

Table 2.5 Effect of molecular weight and lipophilicity on the permeability coefficient P

9.2 Percutaneous Absorption of Macromolecules and Nanomaterials

Many researchers have reported skin permeation of polymers and nanomaterials. Nanomaterials may have a higher thermodynamic activity compared to original low MW materials. However, it is still very difficult for us to understand the skin permeation of high molecular weight compounds and nanomaterials by considering the 500-Da rule. These studies showing skin permeation of nanomaterials must be investigated, especially with regard to their absorption mechanism. Baroli et al. [20] reported that nanomaterials penetrate into the skin surface, although they do not permeate. It is reasonable for nanomaterials to penetrate into cracks and appendages in the stratum corneum. We have obtained similar data using TiO₂ nanomaterials [21].

We have also reported that higher molecular weight compounds permeated more through the shunt route or skin appendage routes like hair follicles and sweat ducts than through the stratum corneum [7]. Figure 2.13 compares fluorescent images of hairless rat skin with hair follicles, and a three-dimensional cultured human skin model (LSE-high, Toyobo, Osaka, Japan) without hair follicles after the application of fluorescein Na (FL) with a MW of 376, and FITC-dextran (FD-4) with a MW of 4000 [22]. A very small amount of FL was observed in the epidermis of LSE-high, but FD-4 was found only in the stratum corneum of LSE-high. In contrast, both FL and FD-4 penetrated into hair follicles in hairless rat skin. Results from in vitro skin permeation studies suggest that no skin permeation was observed for high molecular weight compounds, but they could penetrate into the hair follicles.

Fig. 2.13

Microfluoroimages of hairless rat skin (a, b) and LSE-high (c, d) after application of fluorescein Na (FL) (a, c) and FITC-dextran 4000 (FD-4) (b, d). SC stratum corneum, HF hair follicle. White bars shows 100 μm

A similar study was carried out using Fluoresbrite^® (diameter 500 nm, Polyscience, Inc., Warrington, PA, USA) [22]. Figure 2.14a shows a light micrograph of a cross section of porcine skin, and Fig. 2.14b and c show its fluorescence micrographs. Fluoresbrite^® particles were found in the hair follicles in porcine skin. These fluorescent particles were found at a depth of 80–100 μm, but not at 200–220 μm.

Fig. 2.14

Microphotographs of skin section 12 h after application of 500 nm Fluroresbrite^® on the porcine ear skin. (a) Micrograph of cross section of skin, (b and c) microfluorographs of the site b and c shown in a. White bars show 200 μm

10 Skin Permeation and Thermodynamic Activity of Chemicals

10.1 Thermodynamic Activity

Thermodynamic activity (symbol a or A) is a measure of the effective concentration of chemical compounds introduced by Gilbert Newton Lewis who wanted to modify a difference between an ideal system and a real system. In this chapter, a is used for thermodynamic activity, whereas x is molar fraction. In an ideal mixture, the chemical potential of a component i, μ _i , becomes

$$ {\mu}_i\left(P,T\right)={\mu_i}^{\circ }(T)+ RT\ln {x}_i $$

(2.49)

where μ _i ^° is a standard chemical potential of a component i, x _i is a molar fraction of a component i, and R is gas constant. Values in parenthesis are variables (P: pressure, T: absolute temperature). On the other hand, the equation is shown as follows in a real system:

$$ {\mu}_i\left(P,T\right)={\mu}_i(T)+ RT\ln {a}_i $$

(2.50)

Then, the difference in chemical potential μ _i , Δμ _i becomes

$$ \varDelta {\mu}_i= RT\ln {a}_i $$

(2.51)

Thermodynamic activity is shown as follows:

$$ {a}_i=\exp \left(\varDelta {\mu}_i/ RT\right) $$

(2.52)

Activity coefficient, γ is defined as follows:

$$ {a}_i={\gamma}_i{x}_i $$

(2.53)

The activity coefficient is an index to show difference from the ideal system.

10.2 Raoult’s Law and Henry’s Law

Raoult’s law and Henry’s law are important for understanding the concept of thermodynamic activity of chemical compounds. Raoult’s law was reported by François-Marie Raoult. It states that vapor pressure of each component in a mixed solution is determined by vapor pressure of neat component and molar fraction of the component in the mixed solution. Thus, the vapor pressure of a component i, P _i becomes

$$ {P}_i={P_i}^{\ast }{x}_i $$

(2.54)

where P _i ^* and x _i are the vapor pressure of neat solution and a molar fraction, respectively, of a component i. Total vapor pressure, P _tot, is a sum of vapor pressure of each component. Thus,

$$ {P}_{tot}=\sum \limits_i{P}_i=\sum \limits_i{P}_i{x}_i $$

(2.55)

A solution supposed to follow Raoult’s law at predetermined molar fractions is called an ideal solution. No molecular interaction is found in the ideal solution. In general, mixed solution consisting of resemble components is closed to ideal solution.

On the other hand, Henry’s law, reported by William Henry, is related to the physical properties of gas. It states that at a constant temperature, the concentration of a given gas that dissolves in a given type, and volume of liquid, C, is directly proportional to the partial pressure of that gas, p, in equilibrium with that liquid.

Raoult’s law can also be applied for solvents (much amount components) in many real solutions, whereas it is rarely applied to solutes (less amount components) in solutions. However, a relation is found between vapor pressure, P, and the molar fraction, x, of solute.

$$ P={K}_Hx $$

(2.56)

where K _H is a proportional constant.

Solutions where solutes obey Henry’s law can be called ideal dilute solutions. When the solutes are gas, Eq. (2.56) is the same as Henry’s law. Thus, Henry’s law can be applied to dilute solutions.

The ideal solution is often used when Raoult’s and Henry’s laws are explained. From a thermodynamic view point, molecules in a liquid are rather randomized compared with those in a solid. In the ideal solution, a solvent molecule interacts equally with other solvent molecules, so that these intermolecular bondages cannot be distinguished. Thus, the ideal solution involves a kind of thermodynamic concept. Raoult’s law was introduced from this theory. In other words, any solutions which obey Raoult’s law are ideal solutions. Since intermolecular interaction can be ignored in dilute solutions, these are similar to ideal solutions, so that Henry’s law as well as Raoult’s law can be applied to dilute solutions.

Colligative properties can be applied to ideal solutions and probably to dilute solutions. Decrease in vapor pressure, increase in boiling temperature, decrease in freezing temperature, and increase in osmotic pressure are observed by a decrease in the chemical potential of solutes when the solutes are diluted by the solvent. Since the level of chemical potential of solutes is dependent on their molar fraction, these phenomena (colligative properties) are directly related to the molar fraction (more accurate for thermodynamic activity) of solutes.

10.3 Relation Between Chemical Concentration and Its Activity or Osmotic Pressure

Osmotic pressure is selected to explain the concept of thermodynamic activity. Let us imagine a situation where Sumo wrestlers are practicing in a square room. Two of them are practicing with each other in some cases, whereas several wrestlers push the wall in a room. Pressure to the wall by wrestlers looks like the osmotic pressure by molecules in a solution. When the number of wrestlers in the room (molecules in solution) becomes doubled, the pressure outside of the room (osmotic pressure) is also doubled. Thus, colligative properties can be applied. However, if the number of wrestlers or molecules increases by ten times, the moving activity of wrestlers or molecules is limited, so that it is no longer a dilute solution and it cannot be approximated to the ideal solution. Wrestlers or molecules move and their activity is proportional to their number in a diluted room or solution.

Figure 2.15a shows a relation between concentration and activity of a certain chemical compound in different vehicles A, B, and C. Thus, increasing the concentration of chemical compound in solution increases its thermodynamic activity (or osmotic pressure). In the dilute solution, the activity (or osmotic pressure) is proportional to thermodynamic activity. For example, glucose activity in water is proportionally increased by its concentration in water. However, the activity of glucose becomes constant after saturation of glucose in water. In addition, glucose is rapidly saturated to be a constant activity when it solubilizes in oleic acid, for example.

Fig. 2.15

Relation between chemical concentration and thermodynamic activity. (a) Illustrates the relation between concentration and thermodynamic activity of a certain chemical compound in different vehicles A, B, and C. (b) Shows the relation between concentration and thermodynamic activity of different chemical compounds A, B, and C in a certain vehicle

It is very important and interesting to see the activity in the saturated water solution which is the same to that in the saturated oleic acid solution. The molar fraction of a chemical compound in its solid should be 1.0, since only the chemical compound is in the solid. Thus, the thermodynamic activity of a chemical compound in saturated or suspended solution which contain solids must be 1.0.

One more explanation will be shown using Fig. 2.16. Figure 2.16a shows six molecules of a chemical in a vehicle (this can be thought of as 6 nM or 6 mM). These molecules are dissolved in the vehicle. There are ten molecules in Fig. 2.16b (this also can be thought of as 10 nM or 10 mM), where the concentration is thought to be solubility. Figure 2.16c is a suspended vehicle, so that some of the chemical is in the solid. Thermodynamic activity in a molecular fraction of the vehicle shown in Fig. 2.16c must be 1.0. The number of soluble molecules in Fig. 2.16b is the same as that in Fig. 2.16c, so that the activity of the chemical in Fig. 2.16b is also 1.0. Since there are six molecules in Fig. 2.16a, the thermodynamic activity in molecular activity is 0.6. Figure 2.16d illustrates the relation between the concentration and activity of chemical in the vehicle.

Fig. 2.16

Chemical concentration and thermodynamic activity in dilute solution (a), saturated solution (b) and suspended solution (c)

Fig. 2.15b shows a relation between concentration and activity of chemical compounds A, B, and C in a certain vehicle. Even when the same vehicle is used, the maximum thermodynamic activity at the saturated or suspended vehicle is different depending on the chemical. This is because the activity is intrinsic to chemical species A, B, and C. In addition, the solubility is also dependent on the chemical, so that a bending point of the solubility varies among the vehicles. Figure 2.16d also shows a relation between the chemical concentration in vehicle and its thermodynamic activity, and Fig. 2.16a–c illustate three cases of the vehicle (solution, saturated solution, and suspension, respectively).

10.4 Thermodynamic Activity and Percutaneous Absorption of Chemical Compound

A relation between the thermodynamic activity and percutaneous absorption of chemical compounds was first explained by Takeru Higuchi [23] more than a half century ago.

$$ J=\frac{dQ}{dt}=\frac{a_vD}{\gamma_sL} $$

(2.57)

where a _v is thermodynamic activity of chemical in vehicle and γ _s is the activity coefficient in the skin barrier. Figure 2.17 shows a relation between the activity and percutaneous absorption (skin permeation) rate. This figure markedly resembles Fig. 2.15, although the y-axis in Fig. 2.17 is the percutaneous absorption (skin permeation) rate, suggesting that the percutaneous absorption (skin permeation) rate is proportional to the thermodynamic activity of chemical in vehicle.

Fig. 2.17

Relation between concentration in vehicle and skin permeation rate of chemical compound. (a) Illustrates the relation between concentration and skin permeation rate of a certain chemical compound in different vehicles A, B, and C, and (b) shows the relation between concentration and skin permeation rate of different chemical compounds A, B and C in a certain vehicle, as similar to Fig. 2.15.

10.5 An Example Showing the Relation Between the Skin Permeation Rate and Activity of Chemical

An example showing the relation between the skin permeation rate and activity of a chemical is shown here [24]. We determined the in vitro skin permeation rate of isosorbide dinitrate (ISDN) from different kinds of pressure sensitive adhesive (PSA) tapes. Figure 2.18 shows a relation between the ISDN concentration in PSA (mg/g) and its skin permeation rate (μg/cm²/h). The broken line shows the skin permeation rate of ISDN from suspended solution in water. When using the acrylic type of PSA, the skin permeation rate of ISDN was low from low concentrations of ISDN but increased to be constant with the maximum flux at its higher concentration. The maximum flux from the acrylic type PSA is almost the same as those from ISDN-suspended rubber and silicone type PSAs. In addition, these flux values are also similar to those from an aqueous suspension. These findings suggest that skin permeation of a drug is the same independent of the vehicle when the drug is suspended in the vehicle.

Fig. 2.18

Effect of isosorbide dinitrate (ISDN) concentration on its steady-state skin permeation rate from PSA tapes

11 Chemical and Physical Means to Increase Skin Permeation of Drugs

The simplest means for increasing the skin permeation rate of chemical compounds is to use chemical enhancers. Addition of chemical enhancer(s) to the vehicle increases the permeation parameters like K and D in Eqs. (2.6)–(2.9). These parameters can be changed to γ _s and D from a thermodynamic point of view as shown in Eq. (2.57). Thus, enhancers decrease γ _s and/or increase D. Typical enhancers are alcohols (including polyols), fatty acids and their esters, surfactants, terpenes, terpenoids, Azone and its derivatives, urea and its derivatives, pyrrolidones, sulfoxides, alkyl-N,N-2 substituted amino acetic acids, cyclodextrins, and so on. Please refer in detail to the other relevant chapters in this book.

Physical means with external energies are also evaluated as well as chemical enhancers [25]. Generally, these physical means enhance the skin permeation of drugs more than chemical enhancers. Iontophoresis, phonophoresis (sonophoresis), electroporation, microneedle arrays, and non-needle syringes are examples of the physical means. When using these physical means, external medical machines are sometimes needed, so that the system becomes “formulation with medical machine”. This may cause other problems. Other chapters in this book deal with this subject.

12 Conclusion

Theory and understanding are very important for fabricating and developing topical and transdermal drug delivery systems. This chapter covers mathematical and physical points for understanding the kinetics of skin permeation or percutaneous absorption. Hopefully this chapter will help in the understanding of skin permeation profiles of chemical compounds.

Ref. 1 Principle of Laplace Transform

Laplace transform of a function f(t), defined for all real numbers t ≥ 0, which is a unilateral transform defined by:

$$ F(s)={\int}_0^{\infty }f(t){e}^{- st}\mathrm{d}t $$

The parameter s is a real or complex number. Reverse Laplace transform function F(s) can be obtained by reverse operation of f(t) as follows:

$$ f(t)=\underset{p\to \infty }{\lim}\frac{1}{2\pi i}{\int}_{c- ip}^{c+ ip}F(s){e}^{st} ds $$

The relation of Laplace transformation (L) and reverse original Laplace transformation (L ⁻¹) becomes,

$$ L{L}^{-1}={L}^{-1}L=\mathrm{identity}. $$

In the Laplace transformation, the following equation can be applied for certain functions f(t) and (t).

$$ L\left[ af(t)+ bg(t)\right]= aF(s)+ bG(s) $$

Where constants, a and b are independent of time, t. In the reverse Laplace transformation, the equation can be applied.

$$ {L}^{-1}\left[ aF(s)+ bG(s)\right]= af(t)+ bg(t) $$

Laplace transform of the derivative with respect to time t becomes as a difference in the polynomial equation. Laplace transformation of the first derivative appears f(0). In case of second derivative, f′(0) as well as f(0) are found.

$$ L\left[\frac{df(t)}{dt}\right]= sF(s)-f(0) $$

$$ L\left[\frac{d^2f(t)}{d{t}^2}\right]={s}^2F(s)- sf(0)-{f}^{\prime }(0) $$

Finally, Laplace transform of the n ^th derivative becomes,

$$ {\displaystyle \begin{array}{l}\kern1em L\left[\frac{d^nf(t)}{d{t}^n}\right]={s}^nF(s)-\sum \limits_{k=0}^{n-1}{s}^{n-k-1}{f}^{(k)}(0)\\ {}={s}^nF(s)-{s}^{n-1}f(0)-{s}^{n-2}{f}^{(1)}(0)-{s}^{n-3}{f}^{(2)}(0)-\cdots -{f}^{\left(n-1\right)}(0).\end{array}} $$

On the other hand, Laplace transform of integral equations become,

$$ L\left[{\int}_0^tf(u) du\right]=\frac{1}{s}F(s). $$

Laplace transform of convolution equation used also in the pharmacokinetics can be shown as follows:

$$ L\left[f(t)\ast g(t)\right]=F(s)G(s). $$

Ref. 2 General Solution of Inhomogeneous Differential Equation

Solution of the following differential equation,

$$ {y}^{{\prime\prime} }+a{y}^{\prime }+ by=0\kern1.25em \left(a,b:\mathrm{real}\ \mathrm{number}\right) $$

can be solved using correspondence characteristic equation,

$$ {\lambda}^2+ a\gamma +b=0 $$

where roots are λ ₁ and λ ₂ (λ ₁≠λ ₂). Then, the following solutions are obtained,

$$ {y}_1={e}^{\lambda_1x}\kern0.50em \ {y}_2={e}^{\lambda_2x} $$

General solution becomes, when λ ₁≠λ ₂,

$$ y={C}_1{e}^{\lambda_1x}+{C}_2{e}^{\lambda_2x} $$

Or, using the definition of hyperbolic function (Ref. 3), it becomes,

$$ y=A\sinh {\lambda}_1x+B\cosh {\lambda}_2 $$

Ref. 3 Hyperbolic Functions Having Complex Variables

Hyperbolic functions having complex variables are defined as follows:

$$ \begin{array}{l}\sinh z=\frac{e^z-{e}^{-z}}{2}\kern0.50em \ \kern0.50em \ \tanh z=\frac{\sinh z}{\cosh z}\kern0.50em \ \kern0.78em sechz=\frac{1}{\cosh z}\\ {}\cosh z=\frac{e^z+{e}^{-z}}{2}\kern0.50em \ \kern0.50em \coth z=\frac{\cosh z}{\sinh z}\kern0.50em \ \kern0.50em cosechz=\frac{1}{\sinh z}\end{array} $$

Addition theorem for hyperbolic functions are,

$$ \begin{array}{l}\sinh iz=i\sin z\kern1em \tanh iz=i\tan z\kern1.28em \sec \mathrm{h}\ iz=\sec z\\ {}\cosh iz=\cos z\kern1.28em \coth iz=i\cot z\kern1.28em \cos ech\ iz= icosecz\end{array} $$

Thus, the following equations can be derived,

$$ \begin{array}{l}\sinh \left(x+ iy\right)=\pm i\sin \left(y\pm ix\right)=\sinh x\cos y\pm i\cosh x\sin y\\ {}\cosh \left(x+ iy\right)=\cos \left(y\pm ix\right)=\cosh x\cos y\pm i\sinh x\sin y\\ {}\tanh \left(x+ iy\right)=\frac{\sinh 2x\pm i\sin 2y}{\cosh 2x+\cos 2y}\end{array} $$

Ref. 4 Heaviside Expansion Theorem for Rational Function with the Only Single Root in the Denominator

Image function, F(s), of rational function where the denominator has only single root is

$$ F(s)\frac{P(s)}{Q(s)}=\frac{P(s)}{\left(s-{a}_1\right)\cdots \left(s-{a}_r\right)} $$

Reverse Laplace transform of this F(s) is,

$$ {L}^{-1}\left[F(s)\right]=\sum \limits_{i=1}^r\frac{P\left({a}_i\right)}{Q^{\prime}\left({a}_i\right)}\exp \left({a}_it\right) $$