Computational Model of Drug-Coated Balloon Delivery in a Patient-Specific Arterial Vessel with Heterogeneous Tissue Composition

Abstract

Balloon angioplasty followed by local delivery of antiproliferative drugs to target tissue is increasingly being considered for the treatment of obstructive arterial disease, and yet there is much to appreciate regarding pharmacokinetics in arteries of non-uniform disease. We developed a computational model capable of simulating drug-coated balloon delivery to arteries of heterogeneous tissue composition comprising healthy tissue, as well as regions of fibrous, fibro-fatty, calcified and necrotic core lesions. Image processing using an unsupervised clustering technique was used to reconstruct an arterial geometry from a single, patient-specific color image obtained from intravascular ultrasound-derived virtual histology. Transport of free drug was modeled using a time-dependent reaction-diffusion model and the bound, immobilized drug using the time-dependent reaction equation. The governing equations representing the transport of free as well as bound drug along with a set of initial settings and boundary conditions were solved numerically using an explicit finite difference scheme that satisfied the Courant-Friedrichs-Lewy stability criterion. Our results support previous findings related to the transport and binding of drug in arteries where tissue retention is strongly dependent on local pharmacologic properties. Additionally, modeling results indicate that non-uniform disease composition leads to heterogeneous arterial drug distribution patterns, although further validation using animal studies is required to fully appreciate pharmacokinetics in disease-laden arteries.

INTRODUCTION

Drug-coated balloon (DCB) therapy has recently emerged as a viable therapeutic strategy for treating obstructive arterial disease.^{38,8,22,3,31,7} With this technology, short-term transfer of anti-proliferative drugs to the arterial wall can be achieved without the requirement of permanent indwelling delivery systems such as metallic stents,^{39,33,11,12,1,16,14,28,23,20,17} and with potentially reduced risk of chronic inflammation and incomplete healing. Quantifying arterial pharmacokinetics is central to appreciating balloon-coated technologies, and emerging studies have defined the mechanisms by which tissue uptake and retention of hydrophobic drugs such as Zotarolimus can be achieved.¹⁵ For a specific formulation of Zotarolimus with an excipient coated on a balloon catheter, this study showed that a large bolus of balloon-released drug and its constituents transfer during inflation; some of the drug coating pervades the tissue while a fraction of the coating adheres to the tissue-lumen (or mural) interface. Using a computational model, they demonstrated that diffusion mediates transport into the tissue while reversible binding to tissue ultrastructural elements determines drug retention in an arterial-bed dependent manner. While this study provides the basis to appreciate tissue uptake and retention after DCB delivery, a natural question arises to understand variations in arterial pharmacokinetics in vessels with heterogeneous tissue composition. Indeed, arterial vessels with disease are the target sites for endovascular intervention, and quantifying drug distribution patterns for these scenarios is needed towards full appreciation of DCB and like technologies.^37,9,24

We developed a computational model by leveraging a finite difference scheme for a staggered grid setting that provided us with the flexibility to simulate the transport of a model drug and its retention in an arterial vessel with heterogeneous tissue composition comprising healthy tissue, and regions of fibrous, fibro-fatty, calcified and necrotic core lesions. A reaction-diffusion model that describes saturable reversible kinetics was developed to quantify arterial pharmacokinetics.^15,36 The computational domain was constructed using an RGB image (.jpg format) of a single, patient-specific arterial vessel obtained from the literature.^18,19 In this case, intravascular ultrasound (IVUS) imaging was performed to obtain a longitudinal cross-section of the vessel followed by virtual histology on the derived image to identify plaque composition. An image segmentation technique based on an unsupervised clustering algorithm was leveraged to automatically identify different sub-domains of the RGB image and construct the computational domain.¹³ Simulations performed by solving the transport equations under various drug release kinetics and tissue boundary conditions revealed strong dependence of arterial drug distribution on the pharmacologic property of local tissue composition characterized by the drug diffusion coefficent.

METHODS

Geometry Reconstruction

We obtained a color IVUS-derived virtual histology (VH-IVUS) image (.jpg format) of an arterial vessel of a single patient from the literature (Fig. 1a).^18,19 IVUS is a tomographic imaging tool that allows visualization of atherosclerosis, elucidating plaque area, plaque distribution, lesion length and coronary remodeling.^19,27,26 VH-IVUS uses advanced radiofrequency analysis of ultrasound signals and provides a more detailed analysis of plague morphology.²⁵ The motivation for selecting this image was simply due to its availability in the public domain. Moreover, this image represented a longitudinal view of the arterial vessel with plaque components already identified. This representation allowed us to straightforwardly define the computational domain, apply boundary conditions such as balloon transfer of drug to the mural surface, and examine the variability in drug distribution patterns in the arterial vessel with non-uniform disease.

FIGURE 1.

VH-IVUS derived patient-specific geometry. (a) Longitudinal color image of the arterial vessel obtained in .jpg format. (b) An image processing technique based on k-means clustering was used to identify different sub-domains of the arterial vessel: fibrous (dark green), fibrofatty (light green), dense calcium (white), necrotic core (red) and gray (healthy tissue). (c) Outline of the final geometry derived for computational modeling.

Image segmentation was performed on the patient-specific arterial vessel using an unsupervised clustering algorithm followed by construction of the computational domain. At first, we divided the image into two sections, where one comprised of the intimal region with five compositions identified by VH-IVUS (fibrous cap, fibrofatty, necrotic core, calcified lesion and healthy tissue) while the remaining part of the vessel including the medial and perivascular regions as a separate section (Fig. S1). Segmentation of lesion-specific components was performed only on the former section (Fig. 1b). The process of clustering based on color information is described in two parts: (1) the generation of initial centroids using color histograms and (2) color-based image segmentation using the k-means algorithm. Note that the k-means clustering technique follows a simple procedure to classify a given data set (RGB color space) through a certain number of clusters that are determined a priori. The following steps describe the rest of the procedure performed on the RGB image obtained in a jpeg format and the computational sequences for the generation of initial centroids.

1. – Derive the intensity distributions (histograms) for each color channel of the longitudinal RGB image of a patient-specific arterial vessel.

   – Partition each histogram linearly into R sections where R > K (K = 5 for this study indicating healthy tissue, and regions of fibrous, fibro-fatty, calcified and necrotic core lesions).

   – For each section of the histogram, mark the bin that has the highest number of pixels.

   – Rank the peaks in each histogram in descending order.

   – Generate initial centroid for the highest peak p_i. If p_i ∈ R-channel, mark the pixels in red channel and calculate the mean values of marked pixels from green and blue channels or similarly for the other two cases.

   – Construct the initial centroid and discard p_i from the list.

   – Repeat the previous two steps until the desired number of centroids have been obtained.

2. – Generate clusters by assigning each data point to its nearest centroids.

   – Recalculate the centroids for each cluster.

   – Repeat the previous two steps until no pixels change clusters.

After few iterations, all the pixels are presumed to be distributed into 5 clusters, such that X ∈ S_j(n) if ∥X − c_j (n)∥ < ∥X − c_j (n)∥, ∀i = 1, 2, 3, 4, 5; i ≠ j where n is the iteration number, S_j (n) denotes the j^th cluster whose centroid is c_j(n)and ∥.∥ is the Euclidean metric between the data point and the cluster centroid. When new members (or pixels) are added, the new cluster centroids are calculated as $c_j(n+1)=\frac{1}{N_j}{\sum }_{X_i\in S_j(n)}{X}_i;j=1,2,3,4,5$. In order to reduce processing time, the algorithm ceases when ∥ c_j (n + 1) − c_j (n)∥ < ϵ where ϵ is a positive number that is small enough not to affect the overall outcome.

Computational Model

The computational domain comprised of an arterial segment derived via image processing (Fig. 1c). The volume-averaged molar concentration of free drug is denoted as c_f and the volume-averaged molar concentration of bound drug, completely immobilized in the tissue, is denoted as c_b. The reversible kinetics of drug between the unbound plasma phase and bound phase is controlled by the following equations:

$$\frac{\partial c_f}{\partial t}=D_T^i\left(\frac{{\partial }^2{c}_f}{\partial x^2}+\frac{{\partial }^2{c}_f}{\partial y^2}\right)-\frac{\partial c_b}{\partial t},$$ (1)

$$\frac{\partial c_b}{\partial t}=k_a^i{c}_f\left(B_M-c_b\right)-k_d^i{c}_b,$$ (2)

where B_M is the net tissue binding capacity, and $k_a^i$ and $k_d^i(i=1,2,3,4,5)$ are the association and dissociation rate constants, respectively, which were computed as $k_a^i=\frac{D_T^{i}Da}{B_M{W}^2}$ and $k_d^i=k_a^i{K}_D$. Here Da and W are the Damköhler number and mean wall thickness respectively, and i = 1, 2, 3, 4, 5 correspond to fibrous cap, fibrofatty, calcified, necrotic core lesions and healthy tissue, respectively. Note that the arterial wall thickness was not uniform in the patient-specific image (Fig. 1a). Therefore, we computed the mean wall thicknesses for both the upper and lower wall widths respectively at pixel-level resolution along the vessel length (Table 1). $D_T^i(i=1,2,3,4,5)$ are the coefficients of free drug diffusivity for each lesion type where

$$D_T^i=\left(1+\frac{B_M}{K_D}\right)D_{eff}^i,$$ (3)

where K_D is the equilibrium dissociation constant whose value was taken from porcine model¹⁵ and $D_{eff}^i$, the effective diffusivity of free drug. In vessels with atherosclerotic disease, the fibrous cap consists of macrophages and smooth muscle cells, and the necrotic core is rich in fatty deposits. Such differences natually point to differential transport properties. Therefore, effective diffusivity in fibrous cap was derived using the Stokes–Einstein equation,¹⁰ whereas the effective diffusivity in the necrotic core was computed from its diffusivity in blood and then scaled by octanol-water partition coefficient for the drug.⁶ A more rigorous model should include differential local binding properties due to presence of heterogeneous plaque composition - an aspect that has been overlooked in the present model due to lack of experimental data. Also, we assumed similar effective diffusivities of fibrous cap and fibrofatty lesions, as well as for the necrotic core and the calcified lesions, respectively. We intend to revisit these assumptions once relevant experimental data becomes available. For the purpose of numerical computation of the desired quantities, the following parameters were selected (Table 1).

TABLE 1.

Parameters used in the computational model

Description	Parameter	Value	Reference
Empirical constant (1/s)	k 1	0.009221	Kolachalama et al.15
Empirical constant (gm/L)	A 1	23.95	Kolachalama et al.15
Molecular weight (gm/mol)	Z MW	966.21	Zotarolimus40
Tissue-binding capacity (mM)	B M	1.3	Kolachalama et al.15
Equilibrium dissociation constant (mM)	K D	0.136	Kolachalama et al.15
Damköhler number	Da	2700	Tzafriri et al.36
Radius (cm)	r 0	0.2126	König and. Klauss18
Mean wall thickness (cm)	W
Upper wall		0.1621	König and. Klauss18
Lower wall		0.1446	König and. Klauss8
Interstitial velocity (cm/s)	v y	5.8e-6	Tedgui and Lever35
Effective diffusivity (cm2/s)
Fibrous cap	${\text{D}}_{eff}^1$	6.23e–8	Hossain et al.10
Fibrofatty	$D_{eff}^2$	6.23e–8	Our study
Necrotic core	$D_{eff}^3$	1.45e–13	Hossain et al.10
Calcified lesion	$D_{eff}^4$	1.45e–13	Our study
Healthy tissue	$D_{eff}^5$	3.65e–8	Kolachalama et al.16,5

Symmetry boundary conditions for both the free and bound drugs were applied at the proximal (${\Gamma }_p^u$, ${\Gamma }_p^l$) and distal (${\Gamma }_d^u$, ${\Gamma }_d^l$) walls for both the upper and lower regions (Fig. 1c).² In line with previous, related work,^15,6 convective drug transport was assumed to be negligible in our study. Impermeable boundary condition for the bound drug was assumed at the perivascular wall (Γ_u, Γ_l) and the lumen-tissue interfaces. Zero concentration condition was applied at the perivascular end for the free drug. Once the balloon has been deflated and retracted from the arterial lumen, bulk transferred drug at the lumen-tissue interface becomes exposed to the streaming blood. Since a proper boundary condition at lumen-tissue interface is not readily apparent, the following interface condition was used:

$$\lambda J_f+(1-\lambda )v_y{c}_f=0,$$ (4)

where J_f represents the interfacial flux for the free drug and v_y is the transmural filtration velocity, and its value was estimated using filtration studies on the rabbit thoracic aorta.³⁵ Here, λ is an adjustable parameter that provided us with the flexibility to choose a non-trivial boundary condition (0 ≤ λ ≤ 1). When λ = 0, we have one extreme that blood is extremely efficient at washing out mural-adhered drug, modeled as a zero concentration boundary condition whereas λ = 1 gives another extreme indicating that mural-adhered drug is insensitive to flowing blood, modeled as zero flux boundary condition. For all other settings (0 < λ < 1), we obtain a condition that exemplifies a combination of zero flux and zero concentration settings, noted here as a hybrid boundary condition. At the mural surface, the flux from expanded DCB is governed by $J_b(t)=k_1{A}_1{e}^{-k_{1}t}/Z_{MW}$, t ≤ t₀ where k₁ and A₁ are empirical constants that were estimated previously using bench-top experiments;¹⁵ Z_MW, the molecular weight of the drug and t₀ is the balloon inflation time (Fig. 2).

FIGURE 2.

Drug release kinetics during balloon inflation time of 180 s.

The governing equations were solved using a finite difference method by imposing the release kinetics (simulating two inflation times 30 s and 180 s) and boundary conditions defined above. The image-derived computational geometry comprised of 26,111 pixels, encompassing the arterial wall where each pixel was considered as a control volume (cell). For this type of grid alignment, the concentrations for free and bound drug were calculated at the cell centers. The discretization of the time derivative term was based on the first order accurate two-level forward time-differencing formula, while the diffusive term was discretized by second order accurate three-point central difference formula. In order to have second order spatial accuracy of the boundary conditions, some fictitious grid points outside the physical domain were considered. We defined x_i = iδx; y_j = jδy and tⁿ = nδt in which n refers to time direction, δt, the time increment and δx, δy are the dimensions of the pixels. The finite difference approximation of Eq. (1) at the (i, j) cell becomes

$$\frac{{\left(c_f\right)}_{i,j}^{n+1}-{\left(c_f\right)}_{i,j}^n}{\delta t}=D_T^i{\left[{\left(c_f\right)}_{xx}+{\left(c_f\right)}_{yy}\right]}_{i,j}^n+{[R]}_{i,j}^n,$$ (5)

Where ${\left(c_f\right)}_{xxi,j}^n$ and ${\left(c_f\right)}_{y{y}_{i,j}}^n$ are the finite difference discretizations of $\frac{{\partial }^2{c}_f}{\partial x^2}$ and $\frac{{\partial }^2{c}_f}{\partial y^2}$ at the n^th iteration at (i, j) cell respectively and ${[R]}_{i,j}^n$ is the discretized version of the reaction term.

Likewise, the discretized version of Eq. (2) is written as

$$\frac{{\left(c_b\right)}_{i,j}^{n+1}-{\left(c_b\right)}_{i,j}^n}{\delta t}={[R]}_{i,j}^n.$$ (6)

For the numerical simulation, the well known Courant–Friedrichs–Lewy (CFL) stability criterion for two-dimensional diffusion equation requires³⁴

$$\delta t\left(\frac{1}{\delta x^2}+\frac{1}{\delta y^2}\right)max\left(D_T^i\right)<\frac{1}{2}.$$

Simulations for each case were performed until there was a 10⁻⁶ reduction in the drug transport residual.

RESULTS

Arterial drug distribution patterns were modulated by lesion-specific transport properties of the drug (exemplified as the diffusion coefficient) and the set of boundary conditions (i.e. zero flux, zero concentration and hybrid settings) defined at the lumen-tissue and the perivascular interfaces. The impact of lumen-tissue interface condition on the temporal variation of averaged (mean) free drug concentration on the upper and lower arterial walls is depicted in Figs. 3a(i) and 3a(ii), respectively. These results indicate biphasic kinetics with an initial first-order decline phase followed by slower efflux of free drug. Simulations predicted that when the flowing blood is extremely efficient at clearing mural-adhered drug (zero concentration, λ = 0), free drug taken up by the arterial wall from inflated balloon was cleared rapidly for 180 s balloon inflation time. Despite perivascular washout, clearance of free drug was delayed for the hybrid condition and even more prolonged for zero flux interface condition. Specifically, for the case of zero concentration interface boundary condition, concentrations of free drug at 25 hours reduced to ~0.21% and ~0.14% of its values at 180 s for upper and lower walls, respectively. Similarly, when the hybrid boundary condition was used, only ~1.1% and ~0.6% of free drug concentration remained on the upper and lower walls, respectively, and ~1.9% and ~1% of the concentration of free drug remained for the case of zero flux interface conditions. On the other hand, peak concentration of bound drug was higher for the zero flux interface condition (λ = 1) than the zero concentration (λ = 0) condition. Further, averaged concentration of bound drug for zero flux interface condition were higher at all times due to higher residence time of free drug within the arterial tissue. Specifically, ~41% of bound drug remained after 25 hours on the upper wall when zero flux condition was used, and ~30% bound drug remained when hybrid condition was used and only ~10% remained for zero concentration condition (See Table S1 for more details). These results indicate that temporal kinetics of transport are modulated by the wall boundary conditions with the highest tissue free and bound drug concentration occurring when mural-adhered drug is insensitive to flowing blood (λ = 1).

FIGURE 3.

Temporal variation of averaged drug concentration for zero concentration (λ = 0), hybrid (λ = 0:5) and zero flux (λ = 1) lumen-tissue interface condition for balloon inflation time of 180 s. a(i) and a(ii) denote free drug concentration for the upper and lower walls, respectively. b(i) and b(ii) denote bound drug concentration for the upper and lower walls, respectively.

Figures 3b(i) and 3b(ii) show how the lumen-tissue interface conditions affect the temporal variation of averaged concentration of bound drug for the upper and lower walls, respectively. Simulations also predicted biphasic kinetics for bound drug where concentration increased in the first hour after balloon expansion and thereafter diminished with time. Predicted bound-to-free drug after exposure to 180 s balloon inflation for both the upper and lower walls are shown in Figs. 4a and 4b, respectively. Results show that initially this ratio increased exponentially and reached a steady state within few hours for zero concentration interface condition, but for zero flux and hybrid interface conditions, the time for reaching steady state was longer. These results are consistent with previous findings when healthy artery models were used to examine balloon-coated drug delivery.¹⁵ Uniform uptake kinetics of drug resulted in arterial distribution patterns that were dependent on drug transport parameters and interface boundary conditions (Figs. 5 and 6). Distribution patterns were significantly heterogeneous when variable diffusion coefficients associated with lesion composition were used to model drug transport (Figs. 5a, 5c, 5e, 6a, 6c, and 6e). Drug mostly accumulated in and around the intimal region but within the pockets of calcified and necrotic cores, lower drug concentration was observed in comparison to the regions of healthy tissue. This is because low drug diffusivity within the diseased sites results in slow transport of drug into these regions that eventually diminishes the rate of forward binding of drug.

FIGURE 4.

Ratio of bound to free drug over time for zero concentration (λ = 0), hybrid (λ = 0:5) and zero flux (λ = 1) lumen-tissue interface conditions for balloon inflation time of 180 s. (a) Upper wall, (b) Lower wall.

FIGURE 5.

Spatial variation of free drug concentration at t = 30 mins for balloon inflation time of 180 s. Arterial drug distribution maps for (a) variable coefficient of diffusivity, zero concentration (λ = 0), (b) constant coefficient of diffusivity, zero concentration (λ = 0), (c) variable coefficient of diffusivity, hybrid condition (λ = 0:5), (d) constant coefficient of diffusivity, hybrid condition (λ = 0:5), (e) variable coefficient of diffusivity, zero flux (λ = 1), (f) constant coefficient of diffusivity, zero flux (λ = 1), are shown.

FIGURE 6.

Spatial variation of bound drug concentration at t = 30 min for balloon inflation time of 180 s. Arterial drug distribution maps for (a) variable coefficient of diffusivity, zero concentration (λ = 0), (b) constant coefficient of diffusivity, zero concentration (λ = 0), (c) variable coefficient of diffusivity, hybrid condition (λ = 0:5), (d) constant coefficient of diffusivity, hybrid condition (λ = 0:5), (e) variable coefficient of diffusivity, zero flux (λ = 1), (f) constant coefficient of diffusivity, zero flux (λ = 1), are shown.

Drug distribution patterns were less differential when same value for coefficient of diffusivity was assumed for all lesion types (Figs. 5b, 5d, 5f, 6b, 6d, and 6f). Moreover, model results indicate that when the delivered coating is assumed to remain adherent and impede luminal wash out (zero flux condition), concentration of bound drug increases due to longer residence time of free drug within the arterial tissue. Heterogeneity in drug distribution patterns can be better appreciated by looking at the transmural distribution profiles (Figs. 7, S2, S3 and S4). For example, transmural drug concentration was generally higher for the settings when same value for diffusivity was used for all lesion types (Figs. S2 and S4) in comparison to the cases when variable diffusivity values were used for each lesion type (Figs. 7 and S3). At both t = 5 & t = 30 mins, the transmural profile indicating the magnitude of free drug concentration was consistently lower than bound drug concentration magnitude when the zero concentration boundary condition was used (Figs. 7a, 7b, S2a, S2b, S3a, S3b, S4a, and S4b). However, for the other cases, this trend was not observed. Particularly, when variable diffusivity values were used, bound drug concentration was higher for hybrid and zero flux conditions (Figs. 7c–f, S3c, and S3e). For other cases, there was a crossover of the free and bound drug concentration profiles at different transmural depths (Figs. S2c–f, S3d, S3f, and S4c–f). In sum, our computational findings demonstrate the need for accouting the lesion-specific transport properties while modeling drug transport of diseased arterial vessels that exemplify real world scenarios.

FIGURE 7.

Transmural drug concentration (total, free and bound) at a longitudinal position of x = 0.27 cm for different lumen-tissue interface conditions at t = 30 mins for variable coefficient of diffusivity (balloon inflation time = 180 s) (a) Upper wall, zero concentration (λ = 0), (b) Lower wall, zero concentration (λ = 0), (c) Upper wall, hybrid condition (λ = 0:5), (d) Lower wall, hybrid condition (λ = 0:5), (e) Upper wall, zero flux (λ = 1), (f) Lower wall, zero flux (λ = 1).

Sensitivity analysis of the drug diffusivity coefficients in the fibrous cap, fibrofatty, necrotic core and the calcified lesions was performed while maintaining the ratio of the Damköhler number (Da) and the Peclet number ($P{e}^i=v_{y}W/D_T^i$) constant. Although not ideal, such assumptions about the Damköhler and Peclet numbers allowed us to perform sensitivity analysis and understand the effect of varying diffusion coefficients across lesion types on overall arterial drug distribution. Varying the diffusivity of fibrous cap and fibrofatty areas over several orders in magnitude (O(10⁻¹⁰) – O(10⁻⁶)) affected the peak averaged concentration for both upper and lower walls (Table S2, Figs. S5 and S6). Peak averaged concentration of free drug decreases with decreasing diffusivity of fibrous cap/fibrofatty for both upper and lower walls. More specifically, varying the diffusivity of fibrous cap over 4 orders of magnitude from 6.23e − 06 cm²/s to 6.23e − 10 cm²/s resulted in a decrease in peak averaged concentration from 0.947 mM to 0.26 mM for the upper wall and 1.194 mM to 0.287 mM for the lower wall (Table S2). A similar trend is observed for varying diffusivity of necrotic core as well. Figure S6 shows the temporal variation of peak averaged bound drug concentration for various diffusion coefficients of fibrous cap/fibrofatty and necrotic core/calcified lesion. Averaged concentration of bound drug attains its maximum (7.7e − 02 mM) for higher diffusivity of fibrous cap at 5e − 02 hours. When the coefficient of diffusivity of fibrous cap (6.23e − 06 cm²/s) is greater than the baseline value (6.23e − 08 cm²/s), the averaged concentration of bound drug attains its maximum at the end of balloon inflation (180 s). However, for other diffusivities considered, the averaged concentrations attain its maxima well after the balloon inflation time. On the contrary, variation of diffusivity of necrotic core from its threshold value (1.45e − 13 cm²/s) resulted in peak averaged concentration of bound drug always after balloon inflation time (Table S2). Figure S6 presents the fact that despite perivascular washout, averaged concentration of bound drug is higher for larger time in case of higher diffusivity of fibrous cap (6.23e − 06 cm²/s) and necrotic core (1.45e − 11 cm²/s), implying that short-term evolution is affected differentially than the long-term.

DISCUSSION

There is much we need to understand regarding the mechanisms of balloon catheter delivery of antiproliferative drugs and their performance in real world settings. Questions remain as to when and why these devices achieve desirable efficacy levels and maintain them, and whether they could serve as better alternatives to other endovascular treatments such as bare metal or drug-eluting stents (DES) for specific disease scenarios. The literature suggests that local drug delivery can be impacted by lesion complexity and degree of atherosclerosis.³⁷ Moreover, for heavily calcified primary lesions of the femoropopliteal vasculature, balloon catherization using DCBs demonstrated clinical benefit after debulking the lesions prior to angioplasty.⁴ The underlying rationale for debulking is that efficient drug uptake can be achieved by removing the calcified lesions that presumably act as perfusion barriers. Furthermore, mechanical recanalization of the vessel allows for angioplasty to be performed at low balloon pressures, thereby reducing barotrauma to the vessel wall. Additional clinical data is however required to understand whether DCBs can provide significant benefit when introduced at the time of direct intervention on existing lesions of different complexity or when balloon catherization is performed following lesion debulking using atherectomy.³² Animal models and pre-clinical studies can guide these trials but majority of the preclinical efforts to date have utilized healthy arterial vessel models. We therefore sought to build a computational model of drug transport in diseased arterial vessels as a first step towards understanding the effect of non-uniform tissue composition on balloon catheter delivery and provide a mechanistic rationale that could pave way towards evaluating the impact of lesion debulking prior to catheterization.

As with the case of any computational modeling study, our findings are dependent on the assumptions we made regarding the governing equations, the parameters we selected (Table 1), and the boundary conditions that were assigned. Several parameters taken from the literature were estimated from a series of experiments performed using animal models of different species as well as bench-top experiments. Although not ideal, this way of obtaining parameters from several sources has allowed us to simulate DCB drug delivery and identify interesting aspects related to understanding the impact of physiologic forces on arterial pharmacokinetics. Derivation of all these parameters from human tissues may not be feasible and therefore, assignment of these derived parameter values in the computational model should be considered as an approximation towards quantifying arterial pharmacokinetics in humans.

Impact of Arterial Diffusivity

In modeling drug transport through biological tissues, the effective diffusivity of drug can be estimated using a number of techniques. For example, Levin et. al.,²⁰ experimentally estimated lumped effective diffusivities in the planar and transmural directions for hydrophobic drugs such as Paclitaxel and Rapamycin in calf carotid arteries. The dependence of effective diffusivity on the total drug concentration and binding capacity has been estimated by substituting approximations of the free fraction into the definition of effective diffusivity.³⁶ Some researchers estimated effective diffusion coefficient by absorbing porosity and tortuosity parameters into the free diffusion coefficient,³⁰ whereas the effective diffusivity of free and bound drug within the arterial tissue was estimated by considering the permeation depth and the total time of incubation.^5,15 In this case, effective diffusivity was presumed to be lower than the diffusivity in fluid because of the combined effects of steric hindrance and reversible binding to immobilized proteins. In another study, the effective diffusivity of drug was calculated from the averaging process of the diffusive term that accounts for reduced transport of drug on its tortuous path through the fibrous structure of the extracellular matrix.² They considered the effective diffusion coefficient to be a simple scaling of the free diffusivity by a factor that depends on the average radii of the drug molecule and the fibers of the extracellular matrix. In summary, these models point to the complex microscopic interactions that take place within the tissue and therefore, one has to recognize the strengths and limitations of each study and select these values appropriately.

Although drug transport properties through different plaque compositions are not available in the literature due to their wide range of variability, researchers have shed some light on the possible behavior of drugs based on their observations in healthy tissue. For example, the effective drug diffusivity of fibrous cap was derived using Stokes-Einstein equation by taking into account the permeability and partition coefficient of retinoic acid in collagen films.¹⁰ Levitt opined that properties of olive oil can be used to represent the transport behavior of drug through the lipid core.²¹ Applying the same rationale and considering hindrance due to binding and unbinding of drug, the effective diffusivity of Paclitaxel in olive oil was calculated to get the effective diffusivity in the lipid core.¹⁰ In our study, we used these estimations to define the effective diffusion coefficients of the fibrous cap and the lipid core (necrotic core).

Interface Boundary Conditions

Model results qualitatively demonstrated strong dependence of arterial drug concentration on local transport properties, and the interface boundary conditions played a major role in determining the magnitude of arterial drug distribution patterns. Although the three interface boundary conditions (λ = 0, λ = 0.5 and λ = 1) used in this study allowed us to qualitatively demonstrate the relationship between arterial drug distribution and drug diffusivity, none of these conditions may truly depict the real-world drug transport process. In fact, the zero flux and zero concentration boundary conditions represent two opposing extremes of drug transport; application of them on our models heavily impacted our findings. This is particulary evident in Figs. 3b(i) and 3b(ii), where a change in interface boundary condition (λ = 0 to λ = 1) resulted in a significant difference in the averaged bound drug concentration. This discrepancy can be partially attributed as well to the boundary effects that cause the averaged values to shift in magnitude. Moreover, the use of the adjustable parameter such as λ is too simplified to account for the impact of blood flow on drug wash out. For example, areas where flow velocities are low (e.g., bifurcation regimes¹⁴) are likely to have much lower wash out rates than areas of high velocities, yet λ does not account for these differences. Nevertheless, trendlines of average bound drug concentration for all the cases with different boundary conditions were consistent in behavior and allowed us to generate similar inferences regarding pharmacokinetics in vessels with non-uniform disease. Indeed, boundary conditions are critical to developing model systems that provide us with a means by which to appreciate the mechanisms of action of DCB delivery. Careful validation using bench-top or animal studies may allow for better estimation of the interface boundary conditions and improved understanding of the underlying physicochemical phenomenon.

Image Processing for Sub-Domain Identification

The k-means algorithm implemented for segmentation is a distanced-based unsupervised, non-hierarchical clustering technique that follows a simple procedure to classify a given data set of RGB color values obtained from original image through certain clusters (here k = 5 denotes different types of tissue composition comprising healthy tissue, and regions of fibrous, fibro-fatty, calcified and necrotic core lesions) that are known. It updates clusters iteratively where the elements are exchanged between clusters based on a predefined metric (usually Euclidean distance between cluster centers and the RGB values). Each value is then assigned to a cluster whose distance is minimum. The algorithm is iterated until no elements are exchanged between clusters. A limitation of the k-means algorithm is its dependency on the clusters’ initialization of the centroids. Typically, random selection of centroids is performed as part of initialization and this strategy is not fully optimal. In this study, we assigned the initial centroids based on dominant color components that are extracted from the intensity distribution of the input image (i.e., from histogram values in the RGB color space).¹³ In summary, this clustering technique served as an efficient, automated approach for reconstructing arterial geometries using IVUS-derived, RGB color images.

Study Limitations and Future Directions

Image processing was performed on a longitudinal section of a single, patient-specific arterial image. It has to be noted that longitudinal view does not take into account the catheter rotation during image acquisition and this may have affected the overall geometry reconstruction. Future studies will consider improved image reconstruction techniques.²⁹ Numerically solving the governing equations on a domain with region-specific diffusivity values is particularly challenging to solve using commercial software. The finite difference model that we developed in house allowed us with the flexibility to assign variable diffusion coefficients across the domain, meet stability criterion and obtain solution convergence. We selected a single patient-specific arterial geometry derived from an IVUS image. For the inferences derived in this paper to be generalizable, simulations of this nature need to be performed on several other patient-specific geometries using a variety of boundary conditions followed by experimental validation.

Our model did not consider the impact of blood flow and the role of convection on governing arterial pharmacokinetics. Although flow-mediated transport is a much researched topic in the case of endovascular device-based drug therapies, DES and DCB have distinct mechanisms of action and the physiologic forces that determine pharmacokinetics in DES may not act similarly on DCB, i.e., flow-mediated drug transport was demonstrated to play an important role in the former case (DES) and not in the latter (DCB). Moreover, previous work has shown that effects of convection prevail in DES primarily due to the presence of permanently indwelling stent struts juxtaposed to each other that obstruct luminal flow and cause flow derangements that in turn act as secondary sources of drug deposition.^1,16,14,28 For the case of DCB, there is no likelihood that this feature could take into effect because there is no presence of a strut or any other permanent implant that can intervene with flow. For this reason, we believe neglecting convection for DCB is a rational simplification for the current study. Nevertheless, we intend to investigate this and other topics mentioned in this section as part of our future work.

CONCLUSIONS

Quantifying pharmacokinetics using computational models of arterial vessels with non-uniform disease composition as opposed to modeling healthy arterial vessels with uniform tissue-specific properties is appealing and provides a rational basis by which to understand real world scenarios. Drug distribution after balloon inflation within diseased arterial vessels tracks precisely with lesion-specific transport properties is an important consideration for local drug delivering technologies. Local tissue composition needs to be carefully characterized in the context of modeling endovascular drug delivery using balloon catheters.