Theory of effective drug release from medical implants based on the Higuchi model and physico-chemical hydrodynamics
Combining the approach of colloid transport with the generalized Higuchi theory of drug release and with the concept of minimum inhibitory concentration (MIC) known in microbiology, the theory of effective drug release from implants has been developed. Effective release of an antibiotic at a concentration above MIC is a necessary condition to achieve protection against infection from implants such as central catheters. The Higuchi theory in its present form is not predictive of the therapeutic effect from medical implants. The theory of effective release presented in this paper specifies two release modes, namely: one with therapeutic usefulness (effective release) and another without therapeutic effect. Therapeutic usefulness may be achieved when the antibiotic concentration, Cti, on the implant surface kills the organisms of interest and prevents the formation and propagation of biofilm when Cti exceeds the corresponding MIC of the released antibiotic compound. Currently, neither the Higuchi theory nor any other theory can provide such prediction. The present approach requires quantification of the antibiotic transport from the drug\xe2\x80\x93polymer blend implant surface into the tissue and accounts for its coupling with drug diffusion inside the blend, a task that has not been developed in existing theories. Our solution to this task resulted in the derivation of an equation for the time of duration of effective release, Te, which depends on MIC, the Higuchi invariant and the characteristics of convective diffusion within the tissue. The latter characteristics include: diffusivity Dti and diffusion layer thickness \xce\xb4 which is controlled by the velocity of the interstitial fluid in tissue. A smaller Dti is favorable because transport from the catheter surface is weaker, while a thinner diffusion layer is harmful because this transport is stronger. The influence of the tangential component of interstitial velocity in the tissue is especially harmful because the diffusion within the incision exit site (IES) will be extremely enhanced such that it may decrease Cti to zero. The incorporation of convective diffusion into the theory of antibacterial protection by means of antibiotic release has revealed that physicochemical mechanisms predict the effectiveness of antibiotic-loaded catheters and defines the conditions necessary to achieve better protection by means of combining the level of catheter loading with antibiotics and the use of wound (IES) dressing.
1. Introduction<!>2. Controlled drug release and effective drug release<!>3. The Higuchi theory<!>4. Theory of effective release for antimicrobial protection from an implant based on the Higuchi generalized model<!>5. Importance of convective diffusion in providing effective release of antibiotics<!>6. Discussion<!>7. Conclusions
<p>Drug release kinetics and related pharmaceutical applications [1,2] are largely based on the principles of surface and colloid science. While drug delivery usually focuses on therapeutic applications to treat disease [2], drug release from implants to surrounding tissues has not so far received sufficient consideration. For example, antimicrobial protection of surgical sites after catheter placement is needed to prevent serious catheter-related infections [3,4]. In the cases of central catheters, once bacteria grows on the external surface of the catheter, biofilm develops and propagates until it reaches the bloodstream, which ultimately leads to serious catheter-related bloodstream infections (CRBSIs). Although antibiotic-impregnated catheters [3,4] are used in some instances, their effectiveness in preventing CRBSIs is still debatable. Effective release of antibiotics from the implants at levels above a critical concentration is necessary to prevent such infections. However, predictive models that define the above criteria are still lacking in the literature.</p><p>We have studied the "effective release" of antibiotics from a new class of tympanostomy tubes (ear tubes) used to treat chronic otitis media (OM), which is an infection of the middle ear. We have developed polymeric implant materials made by incorporating a special combination of antibiotics, and have demonstrated their long-term effectiveness, both in vitro and in vivo, in inhibiting bacterial attachment and biofilm growth [5]. Achieving effective release above the minimum inhibitory concentration (MIC) from the ear tubes was necessary to provide long-term protection against infection from organisms implicated in OM in the chinchilla animal model.</p><p>In order to prevent CRBSIs, long-term "effective release" of antimicrobial agents is required for up to about 8 weeks in some cases. In order to achieve these results, other conditions need to be satisfied beyond the optimization of the polymer–drug blend. During long-term release from implants, release rates may become rather low over time, and this leads to an important question: what is the minimal release rate that may be sufficient to effect robust antimicrobial protection against infection in the case of central venous catheters?</p><p>The MIC of antibiotic/antimicrobial compounds, the break point concentration at which microbial culture does not grow, is recognized as an important key parameter in microbiology. It is obvious that expressing the released drug concentration in the surrounding tissue from an implant in terms of MIC units would be helpful in addressing the above question. Surprisingly, this indispensable notion has been overlooked in the published theories of drug release. Accordingly, the current state of drug release theory does not permit optimization of polymer–drug blend formulations nor does it provide guidance regarding drug delivery from implanted medical devices. This optimization should allow achieving therapeutically effective long-term drug release despite the decrease in the release rate with time. Also, optimization of release kinetics will avoid risk from antibiotic overdose, which may cause irritation or inflammation of the surrounding tissues. The latter is especially important in long-term drug release situations around the implant site.</p><p>The objectives of this paper are: (1) to introduce MIC conditions into the theory of drug release, (2) to consider the variables that may affect effective therapeutic concentration from implants, and (3) to create the scientific basis for the optimization of the polymer–drug blend used as implants. Although special focus will be placed on central catheter-related infections, the developed theory should be applicable to other classes of drug therapies from implants, including anticancer, anti-inflammatory, etc. Therefore, the results of this work can be extended to cover other implant devices, other drug therapies and other surgical sites. Although the biological objectives of this study are clear, our approach will make use of the theories of colloid transport [6–9] and of physico-chemical hydrodynamics to develop the theory of effective release [14].</p><!><p>The notion of controlled drug release [2] is well known and has been covered in numerous papers. Here, we introduce the relevant notion of "effective release" for the first time to address the special topic of therapies from implanted medical devices. We selected the example of antibiotic release from central venous catheters to elaborate the theory of effective release in this paper.</p><p>It is now accepted that organisms in the form of biofilm develop on implant surfaces (e.g., catheters) [10] and then propagate along the surface of the device where they ultimately cause infections. Thus, antibiotic concentration at and near the surface of a device during implantation is important to ensure that no bacterial growth can occur. Drug release at the surface of an implant results in the distribution of the antimicrobial agent concentration in the adjacent tissue, Cti(x, t), where x is the distance to the implant surface and t is time. In particular, the drug concentration in the tissue at the implant surface is Cti(0, t), where the biofilm formation to be prevented must satisfy the following condition,
(1)Cti(0,t)>MIC.</p><p>The true dependence of cumulative release M(t) and Cti(0, t) is schematically illustrated in Fig. 1, while their quantification will be described later. It is obvious that the decrease in the release rate with time causes Cti(0, t) to decrease. In general, as Cti(0, t) decreases, the entire release process may be divided into two portions where the requirement according to Eq. (1) is satisfied during the first portion of release, and is violated during the second. The "effective release" is defined as the portion of the release process where the condition according to Eq. (1) is satisfied. Accordingly, effective time Te is determined by the following conditions:
(2)Cti(0,t)>MIC(t<Te)
(3)Cti(0,t)<MIC(t>Te)
Te is the time duration of the effective release. This newly introduced notion has many practical applications. In fact, Te is an important characteristic of any polymer–drug blend and other forms of drug delivery vehicles used as implant devices. Such devices are useful only when the in-service use time is smaller than Te. When t > Te, the drug delivery device may become harmful in cases where excessive release of antibiotics/drugs may cause irritation or inflammation of the surrounding tissues. This leads to the need to define the optimal thickness of the drug delivery "slab" for the drug-polymer blend to be considered. Larger thickness may be harmful because a thickness increase above h0m does not prolong the effective release. Depending solely on the experimental approach to determine Te is not reliable or even possible because methods to measure the distribution of antibiotic concentration in tissue are unproven. Consequently, developing a theory for "effective release" is deemed necessary. This theory will provide equations for predicting an important set of practical characteristics: Te, h0m, etc. When the slab thickness is h0m or less, the release ceases spontaneously as soon as Cti becomes lower than the MIC. Since optimization of the materials is especially important in achieving long-term drug release, we restricted our focus to nonporous monolithic polymer implant systems [2]. The Higuchi theory [11,12] for drug release from monolithic dispersion has received wide recognition in drug delivery science and has become the accepted standard in assessing drug release. Further development of the "theory of effective release" will be useful in the design and in the safe use of drug-releasing implants.</p><!><p>The Higuchi theory [11] may be generalized to derive an equation for
Cti′(t), i.e., Cti(0, t).
Cti′(t) will exceed MIC at t < Te. It has now been recognized that the initial theory of Higuchi [11] is not sufficient to derive Eq. (1) because the so-called sink boundary condition for drug concentration in water near the polymer slab surface (Fig. 2):
(4)Cs′(x=0)=0 was used [11], where x = 0 corresponds to the slab surface. In the meantime, the antibiotic concentrations
Cs′(t) and
Ca′(t) (see Fig. 3) are interconnected by the condition of local equilibrium, i.e.,
(5)C′s(t)Ca′(t)=CsCa=K where Ca is the antibiotic solubility in water.</p><p>Cs′(t)=Cs(t,x=0), where Cs(t, x) is the non-steady antibiotic distribution within the polymer zone 0 < x < h where drug particles have already been dissolved. Both
Cs′(t) and
Ca′(t) decrease in time during antibiotic release, but their ratio is invariant and is the same, as in the experiment, when the antibiotic solubility in water Ca is measured after the equilibrium between the slab of the polymer–antibiotic blend and the water in the system as a whole is established (Fig. 3). The essential limitation of the Higuchi model follows from comparison of conditions (4) and (5), which leads to
(6)Ca(x=0,t)=0 instead of the necessary condition defined in Eq. (1).</p><p>If the solid drug is finely divided, uniformly suspended and rapidly dissolved, then a two-zone system will develop, as illustrated in Fig. 2. In the region closest to the receiving chamber [0 < x < h(t)], all the solid drug will have dissolved, and the drug will be present in a concentration below the solubility limit, Cs. Further away from the slab surface, the drug will still be present as a solid suspension at the original starting concentration in the slab, C0.</p><p>About 50 years ago, Higuchi [11,12] assumed a linear concentration in the dissolved zone, as illustrated in Fig. 3, which is a reasonable assumption when C0 ≫ Cs. He then required that the dissolution boundary h(t) be located so that the mass of drug left in the formulation would be equal to the amount of drug which diffused across the boundary at x = 0 into the receiving chamber. In Higuchi development
(7)h(t)=2D(CsC0)t if Cs ≫ C0.</p><p>Interestingly, elimination of the sink condition from the Higuchi theory was made by Higuchi himself [13]. He clarified [13] that this approximation assumed that "the rate limiting step was the diffusion of drug from the matrix (matrix controlled)," while "the rate of diffusion from the surface of the matrix to the surrounding bulk solution will make a significant contribution to the total diffusion process" [13]. The notion of diffusion layer thickness δ [14], which accounts for possible spatial non-uniformity, was ignored in the initial Higuchi theory (Fig. 2). This enables one to describe the diffusion flux within the initial diffusion layer and at its boundary with the matrix as
(8a)dMdt=−DaC′a(t)δ where Da is the drug diffusivity in water and M is the amount of released drug normalized per unit area of S of the slab.</p><p>A similar expression of the Higuchi approximation is valid within the quasi-steady diffusion layer of matrices
(8b)dMdt=−D(Cs−C′s(t))h(t)</p><p>An equation for h(t) which is valid during both release stages was derived by Higuchi [13] and in [5] with the use of Eqs. (8a) and (8b)
(9a)h(t)=p2[−1+1+(2p)2qt] where p = 2δDK/Da and q = 2DCs/(C0 − (Cs/2)).</p><p>It is clear that
(9b)M(t)=h(t)[C0−(Cs2)]</p><!><p>Higuchi incorporated external convective diffusion into his model to interpret the kinetics of drug release under the conditions of the standard release experimental procedure, i.e., when a slab of polymer–drug blend is placed in a vial and the solution is stirred around it to provide a steady thickness of the diffusion layer, δ (Fig. 3). On the other hand, when drug release occurs within the tissue from an implant, the arising external diffusion transport of antibiotic is affected by convection as well; however, in this case, the transport will depend on the well-known movement of interstitial fluid in the surrounding tissue [15,16]. Interstitial fluid is the liquid which surrounds the cells in the body. Since the fluid velocity in the interstitium (interstitium is the space between cells in a tissue) is many orders of magnitude slower compared to the stirred experimental system used by Higuchi, δ is expected to be much thicker within the tissue adjacent to the implant. Nevertheless, δ may be much thinner than the catheter radius, a geometry which allows one to consider the local release from a curved surface of the catheter as a flat plate. This consideration permits us to apply the Higuchi model of one-dimensional diffusion.</p><p>The right-hand side in Eqs. (8a) and (8b) are equal, and interconnects three unknown functions of time
Cs′(t),
Ca′(t) and h(t) (Fig. 3).
(10)DaδCa′(t)=Dh(t)[Cs−Cs′(t)]</p><p>An additional interconnection is given by Eq. (5), which may be used in Eq. (10)
(11)Ca′(t)=DδDah(t)[Cs−KC′a(t)]
Ca′(t) or
Cti′(t) may be found from this equation
(12)Câ²ti(t)=CsDÎ´/Dtih(t)1+KDÎ´/Dtih(t)
Da in Eq. (11) is replaced by Dti because the diffusivity within the tissue is noticeably lower [15,16] than that in water (see Appendix A).</p><p>When t = 0, the dissolved zone (Fig. 2) is absent, i.e., h(t) = 0 and the adjacent water-tissue is in equilibrium with
Cs′(t)=Cs. Hence, the concentration has to be the same as that in water
(13)Ca′(t=0)=Ca</p><p>This test for the derived Eq. (12) is satisfied because 1 in the denominator may be neglected in comparison with the very large second term in the denominator, when h(t) → 0. Afterwards, the right-hand side in Eq. (12) reduces to Cs/K which yields the qualitatively predicted result:
Ca′(t=0)=Cs/K=Ca.</p><p>This maximal value for
Cti′(t) preserves during times which satisfy the condition
(14)KDÎ´Dtih(t)>>1</p><p>At longer times, corresponding to the opposite condition
(15)KDÎ´Dtih(t)<<1 the denominator in Eq. (12) transforms into 1 and Eq. (12) takes the form:
(16)C′ti(t)=CsDδDtih(t) i.e., the decrease in
Cti′(t) starts at condition (15) and continues to decrease approximately as t−1/2.</p><p>The onset of the second stage is inevitable for long-term release and the determination of the thickness of the wall h0m to provide effective release, i.e., release satisfying condition (Eq. (1)), must explore the asymptotics (16), i.e., [MIC] has to be substituted in the left-hand side of Eq. (16), while h0m in the right-hand side yields an estimate for the thickness of the cylindrical wall
(17)h0m~Î´D/DtiCs/MIC</p><p>The increase in the thickness of the cylindrical wall above h0m is senseless because the duration of effective release will not be longer than that at h0m.</p><p>The maximal duration of the effective release follows from Higuchi Eq. (7), i.e., by means of the substitution h0m, which yields
(18)Te=DCsC02(MIC)2(Î´Dti)2
Te decreases with decreasing wall thickness h0
(19)Te(h0)=T1(h0m)(h0h0m)2</p><p>Measurements of the parameters which are present in Eq. (18) are necessary for the prediction of h0m and Tmax for a certain drug-blend.</p><p>Ca is known for many antibiotics or can be measured using standard experimental methods. Da is also known for these compounds. Measurement of the release rate at a sufficiently long period of time and its interpretation with the Higuchi Eq. (7) allow us to predict the product DCs.</p><!><p>Convective diffusion plays a role only during the first drug release stage when it controls transport as a whole. In contrast, convective diffusion plays a cardinal role during the second stage of effective release when transport as a whole is controlled by diffusion within the polymer blend. This manifests itself in Eq. (16). The larger Î´ is, the higher
Cti′(t) will be. Expanding the diffusion layer (i.e., δ increase) is favorable for the transition from the non-effective mode of release to an effective one. A primitive explanation of this is based on the interrelationship between the concentration gradient and δ
(20)|gradC|=Cti′(t)δ</p><p>At the same release rate (the same grad C), the increase in the denominator in Eq. (20) leads to an increase in the nominator, i.e.,
Cti′(t). This approach is useful with respect to the second stage of release, which is controlled by diffusion in the polymer matrix according to Eqs. (9a) and (9b), and correspondingly does not depend on convective diffusion. However, the rate of convective diffusion, expressed by Eq. (8a), is equal to dM/dt, expressed by Eq. (8b). Hence, combining Eqs. (8a) and (8b) yields
(21)C′ti(t)=(δDa)dMdt where dM/dt is determined by Eq. (8b), and Da has to be replaced by the diffusivity within the tissue, Dti, in our case.</p><p>Ca′(t) during release from the portion of a coated catheter surrounded by tissue is affected by the flow of interstitial fluid [15]. Its velocity is estimated to be in the range of 5 × 10−4 to 5 × 10−3 cm/s [16] and the direction is parallel to the skin surface because the flow normal to skin is absent within the sub-epithelial layer.</p><p>To determine if convection affects diffusion transport may be predicted by estimation of the Peclet number (Pe) [14], which is an estimate for the ratio of convective flux to diffusion flux
(22)Pe=LVD where L is a characteristic length. L may be specified as the catheter radius R with respect to the radial component of velocity. As the angle between the catheter axis and the skin plane is small, this affects Vr, namely Vr ~ 0.1 V, which yields for
(23)Per>>1</p><p>Peclet number in this range indicates that convection may be important.</p><p>Tissue swelling concomitant with inflammation leads to an increase in tissue diffusivity, Dti. In turn, this manifests itself as a decrease in
Cti′ according to Eq. (16). An important conclusion follows that inflammation may cause an essential decrease in
Cti′(t). In this case, effective release may transition into a non-effective release during inflammation depending on the system under consideration.</p><!><p>The concepts and fundamentals of drug release and their applications in controlled drug delivery systems are presented in monograph [2]. The following statement is provided in [2]: "It is the drug delivery module that will deliver a specific amount of a drug at the right time to the right place. The delivery of drugs using drug-delivery modules is collectively known as controlled drug delivery." As to the determination of "specific amount of drug at the right time and right place," this is a task of pharmacokinetics and pharmacodynamics, which requires comprehensive empirical testing in animal models and humans. The most difficult stage of this empirical testing process is when volunteers are involved in the safety and efficacy clinical studies.</p><p>With respect to medical implants like catheters, an easier and less expensive strategy is possible because site localization of biofilm formation is known and because the in vitro methodology for the determination of the MIC of antibiotics is well developed and is also reliable for further prediction of the in vivo effectiveness.</p><p>As soon as
Cti′(t) is incorporated into the theory of drug release for the case of an implant in a tissue, the theory for therapeutically effective release can be developed. One may suggest that controlled drug release according to its cited definition may be sufficient to determine the therapeutically effective conditions. However, this can only be achieved by performing expensive pharmacokinetic testing. The latter type of testing is not eliminated completely when the theory of effective release is used because regulatory approval of a drug delivery device mandates further in vivo and clinical testing. However, a higher level of success and less cost will be achieved with a new strategy based on the application of the theory of effective release presented in this paper.</p><p>Minimization of side effects of a drug-delivery system is emphasized in [1]. However, this task may only be achieved by a different form of pharmacokinetic testing. With respect to implanted catheters, such side effects may be manifested as irritation or inflammation of the surgical site, especially during long-term placement of the implant. In this case, the condition for infection suppression without irritation of tissue may also be defined in the framework of the theory of effective release. First, condition (Eq. (1)) should not be replaced with one with too high
Cti′. Second, the slab thickness has to be optimized according to Eq. (17) in order to eliminate excessive drug release beyond what is needed to achieve the desired therapeutic effects.</p><p>Eq. (18) allows the prediction of the duration of effective release, Te, as well as its dependence on three factors. We specify these dependencies in Table 1 within accepted representative ranges for these factors. As to δ modeling, both the normal and tangential components of the interstitial velocity in adjacent tissue are important. This leads to two components of Peclet numbers, Pe∥ and Pe⊥. An estimate
(24)δ⊥≈RPe⊥0.5 is possible for δ controlled by the flow normal to the catheter axis according to Levich (Eqs. 72.12 and 72.13 in [14]). The interstitial velocity is parallel to the skin and almost parallel to the catheter surface because angle α (Fig. 4) is small. Hence
(25)u⊥=u∥sinα≈0.2u∥</p><p>The velocity 3 × 10−5 < u⊥ < 2 × 10−3 cm/s yields the range for Peclet number 6 < Pe⊥ < 200, when Dti ~ 10−6 cm2/s is accounted for. The substitution into Eq. (24) yields the range for δ ⊥ values
(26)0.007<δ⊥<0.04cm when R = 0.1 cm.</p><p>Since a rather high value of the Higuchi invariant (dM/dt)2t~3× 103 was achieved in our systematic experimental research of antibiotic release from a polymer blend, this value is acceptable. The Higuchi invariant value is calculated on the basis of the measurement of the rate of drug release for a period of time while it is time independent according to Higuchi Eq. (7). Te values calculated with Eq. (18) specified for Dti = 0.1 cm2/day for two values of δ⊥ and Higuchi invariant 2 × 103 µg2/day cm2 are presented in Table 1.</p><p>It can be seen that short-term effective release (Te < 4 days) is possible even at high interstitial velocity u∥ ~ 0.01 cm/s, if MIC is below 2. Long-range effective release is impossible even at MIC = 1 when the velocity is high. It is possible at low velocity u∥ ~ 3 µm/s, if MIC is about 3 or lower.</p><p>Unfortunately, the influence of u⊥ on diffusion layer formation predominates only at a large distance from the incision exit site (IES), while u∥ influence dominates at a small distance and causes much lower thickness δ∥, whose dependence on y is shown in Fig. 5 (Appendix B). The axial dependence of diffusion layer thickness causes the axial dependence of the antibiotic concentration on the catheter surface
Cti′(y) (Fig. 6). The data obtained in Table 1 characterizes the release efficiency only for a portion of the catheter with a distance longer than 4 mm from the IES. The shorter the distance y is to the IES, the lower is
Cti′(y), which is calculated using Eq. (18). The data about δ(y) is presented in Fig. 5. The
Cti′(y) decrease with the approach to the IES causes a rapid decrease in Te, as shown in Fig. 7. Te(y) is calculated using Eq. (18) which is specified with the account for δ(y) dependence in Fig. 5.</p><p>At high velocity u∥ ~ 0.01 cm/s, even a short time release is not efficient at a small distance to the IES, while this difficulty disappears at low velocity. At a very small distance to the IES, the release becomes inefficient even at a low velocity.</p><p>Assuming the worst possibility of biofilm formation in the vicinity of the IES, the therapeutic consequence of this event may not be dangerous if the higher antibiotic concentration downstream of the IES prevents the extension of biofilm towards the bloodstream, as was established in our experiment [10].</p><p>Catheter insertion leads to further exuding from the incision through the IES on the skin surface. The direction of the exudate stream is opposite to that of the interstitial flow, which existed before insertion. Initially, the exuding velocity is high and causes the reversal of the tangential velocity. While the flow before exuding displaces the released drug from the IES into deeper into the tissue, this causes
Cti′ to decrease, as illustrated in Fig. 6. The opposite direction of the tangential flow during exuding will cause an increase of
Cti′ at the approach to the IES. The exuding is the flow of interstitial fluid from tissue through the wound in the skin.</p><p>The exuding flow along the catheter surface to the IES is enriched by the released antibiotic, which will cause an increase in its concentration within the IES. This is an important topic for further modeling.</p><p>After catheter placement, exuding gradually weakens with time and its favorable influence on the release efficiency within the IES may terminate. This indicates the necessity of preventing bacteria transport towards the IES along the skin. Wound dressings such as the Biopatch are known to be effective in this situation. However, wound dressing with release duration much longer than that of the Biopatch will be necessary.</p><p>Further development of the Higuchi model by many groups [17–29] confirms its recognition in spite of the reporting of another kind of release kinetics in some experimental works. These discrepancies are possible for cases when the properties of polymer–drug blends do not satisfy the idealistic conditions inherent in the Higuchi model, including, for example, that a solid drug has to be finely divided, uniformly suspended and rapidly dissolved in a polymer. Control of the above properties is not easy in practical systems. Accordingly, publications with comments regarding the relationship between measured release kinetics and the Higuchi theory usually do not predict the control of the properties of the polymer–drug blends used. As soon as such control is absent, the possibility of deviation from the kinetics predicted by Higuchi is clear. Consequently, such publications cannot contribute to the estimates of the Higuchi theory.</p><p>The gradient of drug concentration within tissue arising due to drug release induces capillary osmotic flow [35–39] within the tissue, which in turn affects the antibiotic concentration on the implant surface. However, the influence is negligible when antibiotic molecules are uncharged because capillary osmosis is weak for uncharged solutes especially at low concentrations. The antibiotic concentration is usually about 1 µg/ml, i.e., it is very low. Although the capillary osmosis may be stronger for charged solute, it is weak as well when the solute concentration is low and simultaneously when the concentration of background electrolyte is high. This occurs during drug release within tissue because the physiological concentration of salt is in the centinormal range.</p><p>With respect to the influence of convective diffusion on
Cti′(t,y), two portions of the coated catheter have to be considered separately, namely, that portion surrounded by the tissue and that portion submerged in the bloodstream. The blood velocity (about 20 cm/s) is orders of magnitude higher than the interstitial fluid velocity within the tissue. Correspondingly, diffusion layer thickness δle is approximately two orders of magnitude thinner in the portion submerged in the bloodstream. As δle is so small, it is doubtful that the condition similar to that of Eq. (1) may be satisfied for the bloodstream portion of the catheter, and consequently, the possibility of antimicrobial protection for this portion of the catheter with the use of a polymer antibiotic blend does not exist. On the other hand, positive results regarding the application of a coated catheter, when the placement time Tp is not long, is documented [3,4]. This contradiction will perhaps not exist if attention is paid to two factors. First, bacteria mainly penetrate through the IES from the skin. The conditions for biofilm growth are the most favorable near the IES. This is confirmed by our observations [10]. Second, bacteria in the bloodstream may be killed by immune cells which prevent biofilm formation on the catheter surface submerged in the bloodstream. Although there is a possibility of immune cells penetrating into the incision, their concentration under the skin is normally low. Taking this into account, we may conclude that bacteria suppression by means of antibiotic release is the most critical within the IES. Fortunately, conditions for protection by released antibiotics are more favorable within the incision where the harmful influence of convective diffusion is weaker. The positive results known for coated catheters (when the placement time is short) may be explained by antibiotic release into the tissue only. Whether antibiotic release from the portion of the catheter submerged in the bloodstream is useful remains uncertain.</p><!><p>Existing theories of drug release, including the generalized Higuchi theory, do not predict antibiotic concentration near an implant surface Cti which, in the case of catheters, has to exceed the MIC of the target organism to achieve antimicrobial protection. This shortcoming occurs because only the diffusion flux inside the polymer–drug blend towards the implant surface is considered in existing theories, while the flux from the surface into the tissue (or bloodstream), although critical, remains lacking. As the movement of interstitial liquid promotes drug transport, the convective diffusion within the tissue is a harmful factor which restricts the efficiency of released antibiotic. Methods of physico-chemical hydrodynamics are used for modeling the antibiotic transport from the catheter surface into the surrounding tissue.</p><p>Combining the Higuchi generalized theory with models of convective diffusion within tissue and by considering the microbiological approach known as MIC, we can derive the equation for antibiotic concentration near the catheter surface Cti as well as the equation for the duration of time Te of effective release. Derived Eq. (18) allows to predict the antimicrobial protection effectiveness of the catheter. In fact, this equation is the result of more complete modeling of the processes involved in catheter (implant) protection due to antibiotic release, although Eq. (18) and the Higuchi equation for drug release have to be analyzed in their combination. Eq. (18) may be called the equation of the efficiency of antimicrobial protection for catheters or implants. According to Eq. (18), Te depends on MIC, the Higuchi invariant, the characteristics of convective diffusion within tissue, diffusivity within tissue Dti and the diffusion layer thickness δ. The smaller Dti is favorable because the transport from the catheter surface is weaker. The thinner diffusion layer is harmful because the drug transport is stronger. The harmful enhancement of convective diffusion due to interstitial flow is quantified using methods of physico-chemical hydrodynamics taking into account the difference in the influence of the velocity components perpendicular and parallel to the catheter. The influence of the tangential component is especially harmful because the diffusion within the IES is extremely enhanced, which can decrease Cti to zero. This indicates the necessity of perfecting the suppression of bacteria transport along the skin towards the IES using proper wound dressing.</p><p>The release theory based on our models attains important new features. This theory is comprised of drug transport in the surrounding tissue, a process that is absent in the Higuchi theory. This predetermines the value of this new theory with respect to achieving therapeutic success from medical implants. Such progress becomes possible due to the application of colloid science principles, namely, colloid transport [6–9]. On the other hand, it must be recognized that rather crude estimations for Pe and δ were applied to the axial dependence for
Cti′(t) which occurs within the IES.</p><p>The incorporation of convective diffusion into the theory of antibacterial protection by means of antibiotic release has revealed physico-chemical mechanisms which can explain the limitations of antibiotic-loaded catheters, while at the same time define the conditions necessary to achieve better protection by means of combining catheter coating and wound dressing.</p>