1. Introduction

The systemic administration of chemotherapeutic medicaments highly depends on the pharmacokinetic profile and blood perfusion rate. Despite this administration method can pose some disadvantages due to the potential occurrence of side effects on healthy tissues, it is still the most common method to deliver drug to cancerous tissues since it facilitates drug distribution through the tumor microvasculature [1]. Electroporation has arisen as a suitable technique that enables to reach the target concentration of chemotherapeutic agents within specific regions of the tissue, thereby localizing the cytotoxic effects of the treatment. The appropriate choice of parameters such as the voltage level, pulse form, spacing, length, and frequency, as well as number and duration of electroporation treatments, and electrodes location is crucial for achieving the primary objective of reversible electroporation (RE), namely, to augment the cell membrane permeability and facilitate the drug translocation towards the intracellular space while maintaining the cell viability [2–4]. Additionally, the utilization of high-voltage, short-duration electric pulses can induce modification of vessel diameter and endothelial permeability, which in turns leads to the alteration of convective drug transport within the vessel, being this particularly critical in fenestrated and/or distal tumor vessels [2, 5, 6]. As previously studied in Vélez et al. [7], the type of pharmacokinetic profile and the bloodstream velocity are very influential factors in the spatio-temporal evolution of the extracellular, free intracellular and bound intracellular concentrations of drug in electroporated and vasoconstricted tissues. This implies that the reaction and transport mechanisms (RTMs) involved in the drug passage from the circulatory system to the cell can be also modified by these parameters. These mechanisms are represented in Fig 1, where they can be identified: transport along the blood vessels (TBV), transvascular transport (TVT), extracellular transport (ECT), transmembrane transport (TMT), and intracellular transport (ICT).

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Fig 1. Reaction and transport mechanisms of electro-chemotherapeutic treatments.

https://doi.org/10.1371/journal.pone.0315194.g001

The TBV mechanism is modelled here using an advection/convection equation, which is based on an established model for fluid flow through porous circular tubes and incorporates a previously published model [7] to address the vessel vasoconstriction and subsequent recovery. The transvascular transport is denoted as TVT+ or EV when it occurs from the vessel to the interstitial space (extravasation), and as TVT− or LD when it occurs in the opposite direction (lymphatic drainage). In the present work, it is employed a Fickian diffusion model that incorporates an effective vessel wall permeability and is valid when the transvascular transport is mainly driven by the species concentration gradients rather than by the pressure gradient across the vessel wall [8–10].

On the other hand, the term ECT+ is used to denote a positive net extracellular transport in the Representative Unitary Cell (RUC), indicating an increase in the extracellular concentration (C₁), while ECT− refers to a negative net extracellular transport, signifying a decrease in C₁. In the original tumor cord tumor model, it is assumed that the dominant mode of drug transport in the extracellular space is diffusion [8–10]. A modified tumor cord model that incorporates a convective term to account for the volume change of the tissue when vasoconstriction occurs was used in Vélez et al. [7] and is taken up here again.

Besides, given that cellular uptake and efflux of drug can occur simultaneously inside the same RUC, the net rate of transmembrane transport (TMT) is considered here, in such a way that the terms TMT+ or INT denote a net internalization of drug, whereas TMT− or EX indicate a net externalization. In the tumor cord approach used here the transmembrane transport is incorporated by source/sink terms into the species transport equations for extracellular concentration (C₁) and free intracellular concentration (C₂), taking into account the effective permeability of the cell membrane (K₁) modified by a function known as the degree of reversible electroporation (DOE_R) proposed by Boyd and Becker [11].

Regarding the reaction mechanisms in the intracellular environment, the processes of drug association (AS) and dissociation (DIS) with proteins play a significant role in determining the free intracellular (C₂) and bound intracellular concentrations (C₃). More details about the RTMs are presented in the Supplementary material. In the tumor cord model employed in this study, the aforementioned reaction mechanisms are incorporated into the species transport equations as source/sink factors, depending on the drug association (K₂) and dissociation rates (K₋₂), which in turns are considered unaltered with the electric pulse application.

As observed, the transport and reaction processes associated with the drug passage from blood vessels to tumor cells exhibit a high level of complexity, which in turns can be increased by the electric pulse application. As a result, there has been an increasing demand of computational tools to better comprehend the intricate interaction between these mechanisms, and how this interaction influences the spatial and temporal changes of drug concentration within the tissue domain [12–18]. For instance, Mohammad et. al. [18] developed a multi-scale numerical model of drug delivery into solid tumors that included transport along blood vessel, transvascular and extravascular transport. This model was used to analyze how the microvascular network and vascular normalization from anti-angiogenic therapy affect the convective and diffusive transport of drugs. In the present study, the influence of two parameters of the systemic drug administration on the transport and reaction mechanisms in electroporated and vasoconstricted tissues is investigated, namely, the inlet bloodstream velocity (λ_inl) and pharmacokinetic profile (PK), considering several electric field magnitudes (E). Firstly, the spatio-temporal evolution of the ratios C₂/C₁, C₃/C₁, and C₃/C₂ is obtained using a formerly developed, calibrated and validated in-silico tool that employs the Global Method of Approximate Particular Solutions (GMAPS) to solve the governing Eq [7]. This code incorporates the effects of applying electric pulses on some parameters, including vessel diameter, vessel wall permeability, cell membrane permeability, and interstitial porosity. Then, logical inference rules are applied here to determine relationships between the previously mentioned mechanisms (TBV, TVT+ or EV, TVT− or LD, ECT+, ECT−, TMT+ or IN, TMT− or EX, AS, DIS). The simulation campaign conducted in the present study comprises three different magnitudes of electric field: E = 0 kV/m (no electroporation), E = 46 kV/m (reversible threshold), and E = 70 kV/m (irreversible threshold). Additionally, two different pharmacokinetic profiles, PK (one-short tri-exponential and mono-exponential), and three inlet blood velocities, λ_inl (1x10⁻⁴m/s, 1x10⁻³m/s and 1x10⁻²m/s), are deemed.

The pharmacokinetic profiles stand for the time evolution of drug concentration (C_v) along the bloodstream considering processes of absorption, distribution, metabolism and elimination with little physiological details [19]. These profiles depend on the drug type, loading and maintenance doses, infusion time, perfusion rate, tissue affinity, drug lipophilicity or hydrophilicity, plasma protein binding, among other factors [19, 20]. Both mono-exponential (MPK) and tri-exponential profiles (TPK) has been considered for intravenous (IV) injection of doxorubicin in previous works [9, 10, 21–26]. The MPK is a one-compartment, bolus IV injection profile, where total drug dose is rapidly injected (order of seconds) and the time decay rate of C_v is constant since the drug distribution and elimination rates are assumed to be uniform for all tissue groups [19, 20]. On the other hand, the TPK is a multi-compartment, one-short IV injection profile, where the total drug injection does not occur immediately, but in short period of time (order of minutes), to then decrease tri-exponentially due to the consideration of different distribution and elimination rates according to the tissue group [19, 20]. In general, a mono-exponential decay is typically evidenced when the drug is prone to be distributed to only richly perfused tissues and organs, such as liver, kidney and brain [27]. The parameters of both pharmacokinetic profiles (MPK and TPK) for doxorubicin were obtained from Groh et al. [9], where the total drug exposure, defined as the Area Under the Curve (AUC) of C_v vs t, namely, , which is a more useful target parameter at clinical practice, is the same for both profiles (AUC≈2.78 μMh).

The bloodstream velocity (λ_inl), which is the other parameter considered here, is directly related with the blood perfusion rate, which in turns plays a major role for the systemic administration of drugs. At clinical level, the low blood perfusion in some tumors due to the vessels compression and leakiness can lead to hypoperfusion and hypoxia. While hypoperfusion reduces the medication delivery, hypoxia can foster the tumor growth through immunosuppression, increase of metastatic potential and drug resistance, and generation of disordered angiogenesis [28]. Therefore, several therapeutic strategies have been developed to improve the blood perfusion and therapy outcomes, such as [28, 29]: vascular normalization with anti-angiogenic agents, stroma normalization with mechano-therapeutics, tumor microenvironment (TME) normalization with metronomic chemotherapy or nanomedicine, ultrasound sono-permeabilization, among others. These strategies are focused on the regularization of the bloodstream velocity in the vessels by reducing the endothelial gaps associated to the leakage flow rate, and/or alleviating the mechanical stresses exerted by the tumor microenvironment (TME) surrounding these vessels. Accordingly, the influence of the bloodstream velocity (λ_inl) on the reaction and transport mechanisms (RTMs) of chemotherapeutic drugs can have clinical importance.

The principal purpose of the present research is to investigate the influence of the combined input parameters (E, λ_inl, PK) on the interaction between the aforementioned reaction and transport mechanisms in the context of electro-chemotherapeutic treatments, aiming to propose potential clinical implications resulting from this combination. As detailed in the Supplementary Material, the transport and reaction phenomena involved in the drug passage from the circulatory system to the intracellular space have been studied at both macroscopic and microscopic scales by experiments and numerical simulations. The cited in-silico studies have used mainly meshed techniques (Finite Element Methods, Finite Volume Method, Finite Difference Method, etc.). Particularly, this research group have developed several in-silico tools using meshless and mesh-boundary techniques to predict the extracellular (C₁), free intracellular (C₂) and bound intracellular concentration (C₃) of drug in electroporated and vasoconstricted tissues at both microscopic [7, 17, 30] and macroscopic scale [31, 32]. To the best of our knowledge, the evolution of the reaction and transport mechanisms (RTMs) considering the spatio-temporal behavior of these concentration fields (C₁, C₂ and C₃) has been tackled using merely descriptive analyses, but not a rigorous logical inference analysis. Accordingly, one of the main contributions of this work lies on the development of a Boolean model to accurately establish the presence, interaction and rates of the RTMs in electroporated and vasoconstricted tissues applying logical inference laws for all possible combinations of the ratios C₂/C₃, C₃/C₁ and C₃/C₂, which in turns are retrieved from numerical simulations using a GMAPS code previously developed, calibrated and validated [7].

2. Mathematical modeling and numerical methods

The current study uses an axisymmetric, continuum, tumor cord model for the species transport (see Fig 2). Usually, this model has been used considering stationary domains [8, 9]. In a recent work [7], the governing equation for the interstitial space was modified by incorporating a divergence term that accounts for the advective transport resulting from the tissue expansion and contraction due to the vessel vasoconstriction. A summary of the deduction of the modified model is presented in the Appendix section of Vélez et al. [7]. The modified tumor cord equations read: (1) (2) (3) where the field variables consist of the extracellular drug concentration (C1), free intracellular drug concentration (C₂) and bound intracellular drug concentration (C₃). The geometric-dependent properties δ₁, δ₂ and α stand for the extracellular volume content, intracellular volume content, and ratio of total surface area of cell membrane to the cord domain volume, respectively. The transport properties of the tissue are the interstitial diffusion coefficient (D) and cell membrane permeability (k₁), which are dependent on the magnitude of the electric field and time. On the other hand, the drug association rate (k₂) and dissociation rate (k₋₂) are the reaction properties. It is crucial to emphasize that the effective diffusivity, D, is dependent on the porosity of the extracellular space, δ₁, and can be updated as D = D₀. (δ₁/δ_1,0) when the time change of porosity is small, with D₀ and δ_1,0 as the diffusion coefficient and porosity corresponding to the original cord domain (without electroporation). The velocity field of the tissue induced by the vasoconstriction and subsequent recovery of vessel radius is assumed to have only radial components, namely, .

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Fig 2. Boundary conditions of the physical domain.

https://doi.org/10.1371/journal.pone.0315194.g002

The boundary conditions depicted in Fig 2 include a Robin-type boundary condition at the tissue-vessel wall interface, as shown in the following equation: (4) where r_v is the vessel radius that is considered changeable due to vasoconstriction, k_v(E,t) is the vessel wall permeability, C_v(t,z) is the time and space-dependent drug concentration along the bloodstream, C₁(r_v,z,t) is the extracellular drug concentration at the tissue-vessel wall interface, and ∂C₁/∂n is the normal gradient of C₁. At the remaining boundaries, a non-flux condition is considered.

In a previous study [7], it was provided a detailed description about how the application of electric pulses affects the vessel wall permeability (k_v), vessel radius (r_v), cell membrane permeability (k₁), interstitial diffusion coefficient (D), and geometric-dependent properties of the tissue (δ₁, δ₂ and α). In summary, the following was implemented:

• For the electro-permeabilization of the vessel wall and cell membrane, a function for the transient degree of electroporation (DOE_R) is considered [11, 33]:

(5)

With the electric field strength-dependent functions DOE(E) and DIE(E) as given by: (6) (7) where E, E_irrev, t_p and τ_i stand for electric field magnitude, irreversible threshold of the field magnitude, time elapsed from last pulse application, and decay time coefficient for the permeability, whereas σ(E), σ_min, σ_max and σ_irrev are the effective, minimum, maximum and irreversible electrical conductivity, respectively. For a given value of E, the effective electrical conductivity can be computed using a sigmoidal-type function [34]: (8) where the fitting parameters a* and b* are computed in terms of the reversible threshold (E_rev) and irreversible threshold (E_irrev) of the electric field magnitude: (9) (10)

The permeabilization state of the cell membrane and vessel wall is given by: (11) where k_i,max and k_i,min are the maximum and minimum values of the permeability, with i = [1, v] in such way that subindex “1” stands for the cell membrane, whereas subindex "v" represents the vessel wall.

• For the vessel vasoconstriction, the mathematical model proposed in [7] is also considered, where the normalized vessel radius, , as defined by the quotient between the current and the original vessel radius, i.e., , is estimated using a double-exponential function:

(12) where , , m_r and τ_r stand for the normalized ratio, minimum normalized ratio, coefficient of vasoconstriction, and decay time coefficient of vasoconstriction, respectively.

• Assuming that the motion of the tissue-vessel wall interface occurs in the radial direction and is uniform, that the opposite boundary remains fixed, and the function of the interface velocity does not depend on the space, the radial velocity (u_r) at any point along the tissue can be calculated following the approach used by [35, 36] for cell migration in changing tissue domains, obtaining the following expression:

(13) where r, r_v and R_cord stand for the radial coordinate, current vessel radius and cord radius, respectively, whereas dr_v/dt is the interface velocity, which in turns comes from the time derivative of Eq (12).

The iterative GMAPS scheme developed in [7] is implemented here. The principal characteristics of this scheme are:

• A fully implicit formulation is used for the time discretization.
• Scale factors S_L, S_T and S_C are used for the spatial domain, time and concentration to improve the conditioning of the final system.
• A Gauss-Seidel iterative scheme is used for solving the concentration fields (C₁, C₂ and C₃) at each time instant considering a relative error of 1x10⁻⁶.
• Under-relaxation is applied to update the concentration fields during the iterative process.
• An advection/convection model is used to calculate the concentration profile of drug along the bloodstream, C_v, which in turns depends on the pharmacokinetic profile of drug (PK), flow velocity in vessel (λ), inlet blood pressure (p_inl), plasma viscosity (μ), permeability coefficient of vessel wall (k_vc), and radius of blood vessel (r_v). The equation is given by:

(14)

Non-dimensionalization, time implicit scheme, and backward scheme for the spatial gradients can be applied in Eq (14) to get a recurrence relationship for the drug concentration profile, C_v. This is shown in [7].

• To reduce the computational cost, vasoconstriction effect is not considered when the relative reduction of the vessel radius is above 90%. Otherwise, this effect is deemed by recalculating the mesh of the physical domain, the geometry-dependent matrices of the GMAPS scheme, and the geometry-dependent properties of the tissue (δ₁, δ₂ and α).
• The flow absorption through the vessel wall, which in turns leads to the change of the blood velocity along the vessel (λ), is captured using a model for flow through porous circular tubes [37], which is given in terms of the plasma viscosity (μ), permeability coefficient of vessel wall (k_vc), inlet blood velocity (λ_inl), inlet flow rate (Q_inl), inlet blood pressure (p_inl) and permeate pressure (p_p). Accordingly, the blood velocity, λ(z), can be computed from the following formulae:

(15) where λ_inl is the inlet blood velocity, and Q_leak(z) is the leakage flow rate through the vessel wall as computed by: (16)

With coefficients ϑ* and A* given by the following equations: (17) (18)

• It is important to emphasize that an accuracy and convergence analysis of the GMAPS code employed here was carried out in [7]. The number of collocation points (N_points), time step size (Δt) and under-relaxation factor (ω) selected there are used here as well.

Fig 3 depicts a simplified flowchart of the GMAPS scheme employed in the present study. A more detailed explanation of this flowchart can be found in [7].

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Fig 3. Simplified flowchart of the GMAPS scheme.

https://doi.org/10.1371/journal.pone.0315194.g003

It is worth mentioning that when simulations are conducted at the macroscopic scale (tissue domain of order of millimeters), the electric field magnitude, E, shall be considered as variable and its distribution highly depends on the electrode positioning [11, 31–33]. On the other hand, for simulations at the microscopic scale (Representative Unitary Cell), as the ones considered in the present work, bearing in mind that the characteristic length of the tumor cord domain is usually from one to three orders of magnitude smaller than such of the tissue domain, a constant electric field magnitude, E, can be supposed, and electrode positioning is not relevant in the simulations [7, 9, 10, 17, 30].

3.1 Physical significance of ratios C₂/C₁ and C₃/C₁ at a given time instant

In the context of diffusion-dominated transmembrane transport (TMT), the drug transport direction across the cellular membrane is principally determined by the concentration gradient between C₁ and C₂, as shown in Eqs 1 and 2. It is worth noticing that the bound drug cannot escape into the extracellular space. Considering this, the net rate of TMT at a given time instant and RUC of the tissue domain is related with the ratio C₂/C₁, in such a way that when C₂/C₁<1 the drug concentration gradient is inward the cell, leading to the drug internalization, whereas C₂/C₁>1 implies a reverse concentration gradient, occurring drug externalization. In the particular case of C₂/C₁ = 1, partial species equilibrium is reached. Regarding the drug association in the intracellular space, it is worth-mentioning that this is constrained by the binding site concentration, C₀, in such a way that when C₃ reaches the value of C₀, further drug association is not possible. The logical propositions and implications for the ratios C₂/C₁ and C₃/C₁, as well as their physical meaning, are shown in Table 1.

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Table 1. Logical propositions and implications for C₂/C₁ and C₃/C₁ at each time instant and for any point along the tissue domain.

https://doi.org/10.1371/journal.pone.0315194.t001

3.2 Influence of transport and reaction mechanisms on the evolution of ratios C₂/C₁, C₃/C₁ and C₃/C₂

The transport and reaction mechanisms discussed above have varying effects on the spatio-temporal changes of C₁, C₂ and C₃. Accordingly, the extracellular concentration (C₁) rises at the tissue-vessel wall interface when positive transvascular transport (TVT+) or extravasation (EV) occurs. Additionally, C₁ increases at any location of the interstitial space as a result of negative transmembrane transport (TMT−) or cellular externalization (EX), and/or of positive extracellular transport (ECT+). As reasonable, C₁ exhibits a reduction in response to opposing mechanisms, namely, lymphatic drainage (LD), cellular internalization (IN) and negative extracellular transport (ECT−). On the other hand, the process of cellular internalization (IN or TMT+) and drug dissociation (DIS) results in an increment of the free intracellular concentration (C₂). Conversely, the cellular externalization (EX orTMT−) and drug association (AS) brings about an opposing influence on C₂. With respect to the bound intracellular concentration (C₃), it increases solely through drug association (AS) and decreases through drug dissociation (DIS). Table 2 presents a summary of the logical propositions and implications that describe the impact of the transport and reaction mechanisms on the concentrations changes (C₁, C₂ and C₃) and their corresponding ratios (C₂/C₁, C₃/C₁ and C₃/C₂). In general, an increase in C₂ and a decrease in C₁ are sufficient but not necessary conditions for an increase in the C₂/C₁ ratio. Similar premises can be applied to the ratios C₃/C₁ and C₃/C₂. Some logical propositions of Table 2 involve the comparison between C₁, C₂, C₃, and their respective ratios at two times instants identified by t_i and t_i+1. It is important to emphasize that in a given Representative Unitary Cell (RUC), extravasation and drainage, positive and negative extracellular transport, cellular internalization and externalization, and drug association and dissociation, can be present simultaneously, but the predominant mechanisms inside each RUC are the ones considered in the present work. Accordingly, from a Boolean viewpoint, EV (extravasation) and LD (lymphatic drainage), ECT+ (positive extracellular transport) and ECT− (negative extracellular transport), IN (internalization) and EX (externalization), and AS (association) and DIS (dissociation) are considered mutually exclusive each other in the RUC.

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Table 2. Logical propositions and implications for the influence of the transport and reaction mechanisms on the concentration fields.

https://doi.org/10.1371/journal.pone.0315194.t002

3.3 Boolean model for the interaction of transport and reaction mechanisms

In this subsection, the logical inference rules are applied to deduce how the time behavior of C₂/C₁, C₃/C₁ and C₃/C₂ allows revealing the interaction of the reaction and transport mechanisms (RTMs) involved in the systemic drug delivery. For instance, let us consider the logical statement P₇⋀¬P₅→P₁₅ presented in Table 2, which means that the increment of C₃ and the decrease of C₁ between time instants t_i and t_i+1 is sufficient but not necessary condition for the increase of the ratio C₃/C₁. Applying inference rules, this statement can be transformed as follows: (19) (20) by transposition law (21) by De Morgan and double-negation laws (22) by second law of constructive dilemma, biconditional and transposition law, considering the statements P₁₂↔P₇ and P₁₀ΔP₁₁↔P₅ from Table 2.

Similar expressions can be deduced for the time behavior of C₂/C₁ and C₃/C₂. For instance, for the time increase of C₂/C₁ and time decrease of C₃/C₂, the same inference analysis for the logical statements ¬P₆⋀P₅→¬P₁₄ and P₇⋀¬P₆→P₁₆ (see Table 2) results in the following expressions: (23) (24)

The simultaneous occurrence of P₁₄, ¬P₁₅ and ¬P₁₆, which corresponds to the increment of the ratio C₂/C₁, and decrease of the ratios C₃/C₁ and C₃/C₂, between the time instants t_i and t_i+1, can be represented as: (25)

Using inference rules, expression (25) can be developed as follows: (26) by Modus Ponendo Ponens considering the statements (22) to (24). (27) by distributivity law.

By replacing the logical propositions P₈ to P₁₂ defined in Table 2 on the expression (27), they are obtained all possible combinations of the transport and reaction mechanisms considered here that bring about the increase of the ratio C₂/C₁, and the decrease of the ratios C₃/C₁ and C₃/C₂, between two time instants (t_i and t_i+1). However, it is possible to considerably simplify the expression (27) based on the time behavior of the intracellular bound concentration (C₃) and extracellular concentration (C₁), as well as on the value of the ratio C₂/C₁ at each time instant. In this point, it is important to remind that IN and EX, AS and DIS, ECT+ and ECT− are considered here as mutually exclusive mechanisms. Accordingly, net drug association (AS) and dissociation (DIS) cannot take place at the same time at a given point of the tissue domain in the framework of the present analysis. Considering this and the fact that in the present model the drug association (AS) is the only mechanism that brings about a growth of the bound intracellular concentration (C₃) between two time instants (t_i and t_i+1), when C₃(t_i)<C₃(t_i+1) it can be inferred that association (AS) imposes over dissociation (DIS) between t_i and t_i+1, signifying that P₁₂ = 1 (true), and P₈ΔP₉ = P₈ by definition of exclusive disjunction and setting P₉ = 0 (false) since net dissociation does not exist. Under such condition, expressions (27) can be written as: (28) (29) by disjunctive identity (30) by association law (31) by distributivity law (32) by disjunction and conjunction definition.

Now, the statement (32) can still be simplified considering the ratio C₂/C₁. As formerly shown in Table 1, the direction of the net transmembrane transport (TMT) at a given time instant is related with the ratio C₂/C₁, in such a way that when C₂/C₁<1, net internalization occurs (∃IN, ∄EX), leading to P₁₀ = 0 (false). By definition of exclusive disjunction, expression (32) can then be written as: (33)

By replacing the logical propositions P₈ and P₁₁ defined in Table 2 in the conditional statement (33), the following is obtained: (34)

Assuming a non-equilibrium condition (IN≠AS, IN≠ECT+), this means that when net internalization (IN) and association (AS) occurs between two time instants (t_i and t_i+1), the increment of the ratio C₂/C₁, and decrease of C₃/C₁ and C₃/C₂, indicates that the internalization rate (IN) imposes over the association rate (AS) but it is smaller than the extracellular transport rate (ECT+). Following the procedure exposed above, the logical statements for the remaining scenarios of the time behavior of ratios C₂/C₁, C₃/C₁ and C₃/C₂ can be obtained. The statements for all possible combinations of C₂/C₁, C₃/C₁ and C₃/C₂ are deduced and summarized in Table 3, where “I” stands for the time increase of the concentration ratio and “D” for the time decrease. The Boolean scheme that summarizes the results obtained from the Boolean analysis is shown in Figs 4 and 5. In both Table 3 and Figs 4 and 5, “Tautology” means that the scenario resulting from the time change of C₂/C₁, C₃/C₁ and C₃/C₂, and the presence of drug association/dissociation and internalization/externalization is always true, and do not allow concluding anything about the interaction between the transport and reaction mechanisms considered here. For instance, when C₂/C₁ increases, C₃/C₁ decreases, C₃/C₂ decreases, and drug dissociation and internalization take place, the existing reaction and transport mechanisms (RTMs) can interact in any way. On the other hand, “Contradiction” means that the corresponding scenario is not physically consistent. For example, when the time behavior of the ratios C₂/C₁, C₃/C₁ and C₃/C₂ remains as in the last example, but drug association takes place, the cellular externalization is not physically possible.

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Fig 4. Boolean schemes for the reaction and transport mechanisms (I).

https://doi.org/10.1371/journal.pone.0315194.g004

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Fig 5. Boolean schemes for the reaction and transport mechanisms (II).

https://doi.org/10.1371/journal.pone.0315194.g005

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Table 3. Equivalent logical statements for possible scenarios of C₂/C₁, C₃/C₁ and C₃/C₂.

https://doi.org/10.1371/journal.pone.0315194.t003

4. Data and simulation campaign

As formerly mentioned, this work is aimed to study the influence of the electric field magnitude (E), type of pharmacokinetic profile (PK) and inlet blood velocity (λ_inl) on the interaction between the RTMs during the electro-chemotherapeutic treatment using the Boolean model previously presented. Firstly, the concentration fields (C₁, C₂, C₃) are obtained from the solution of the tumor cord model defined by Eqs (1)–(3) under the boundary conditions shown in Fig 2. The influence of the pulses’ characteristics on the vessel wall permeability (k_v), vessel radius (r_v), cell membrane permeability (k₁), interstitial diffusion coefficient (D), and geometric-dependent properties of the tissue (δ₁, δ₂ and α) are considered from Eqs (5)–(13). On the other hand, the spatio-temporal evolution of the drug concentration along the bloodstream (C_v) is obtained by solving Eq (14), whereas the change of the blood velocity (λ_inl) along the vessel is achieved from Eq (15), where the leakage flow rate through the vessel wall (Q_leak) is computed from Eq (16). The GMAPS iterative scheme represented in Fig 3 is used to solve these coupled equations. More details about the numerical development, calibration and validation of this numerical scheme can be found in [7]. Once the concentration fields are obtained at several points and time instants, ratios C₂/C₁, C₃/C₁ and C₃/C₂ can be computed. The Boolean model presented in Section 3 allows determining the presence and interaction of the different reaction and transport mechanisms (RTMs) considered here by defining logical propositions and implications related with the spatio-temporal behavior of the concentration ratios (C₂/C₁, C₃/C₁, C₃/C₂), Tables 1 and 2, and applying Boolean algebra to deduct equivalent logical statements for all possible combinations of these ratios, Table 3. Software Matlab^TM was used for the algorithm development and plots generation.

The computational experiments consider 3 levels for E (0 kV/m, 46 kV/m, 70 kV/m), 2 levels for PK (One-short Tri-exponential, Mono-exponential) and 3 levels for λ_inl (1x10⁻⁴m/s, 1x10⁻³m/s, 1x10⁻²m/s). The other parameters are fixed. Both variable and fixed parameters are shown in Table 4. In this table, T_chemo represents the simulation time of electro-chemotherapy.

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Table 4. Variable and fixed parameters of the numerical simulations.

https://doi.org/10.1371/journal.pone.0315194.t004

The time evolution of drug concentration in the bloodstream (C_v) depends on the pharmacokinetic profile considered for the drug supply. For the profiles considered here, the Equations are [9, 22, 58]:

One-short, tri-exponential injection (TPK) (35) where D_o = 1.1983x10² μmol, τ′ = 180 s, A′ = 7.46x10⁻² l⁻¹, A′′ = 2.69x10⁻³ s⁻¹, B′ = 2.49x10⁻³ l⁻¹, B′′ = 2.83x10⁻⁴ s⁻¹, C′ = 5.52x10⁻⁴ l⁻¹ and C′′ = 1.18x10⁻⁵ s⁻¹.

Mono-exponential injection (MPK). (36) where A′ = 50 μM and A′′ = 0.005 s⁻¹.

The time behavior of TPK and MPK profiles until t = 1h is shown in Fig 6. Semilog-y plots are reported for both profiles to identify the change of the exponential rates. As can be observed, the bolus IV injection of MPK leads to the higher drug concentration,C_v, at the beginning of the treatment, with a subsequent exponential decay at a constant rate. ForTPK, a slower IV infusion with duration of τ′ = 180 s is considered, leading to a significant exponential increase of C_v at the first three minutes of the treatment, after which an exponential decay, with varying rates much smaller than such of MPK, is obtained. It is important to emphasize that both profiles entail the same total drug exposure or Area Under the Curve, i.e, .

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Fig 6. Time behavior of TPK and MPK profiles until t = 1h.

https://doi.org/10.1371/journal.pone.0315194.g006

For sake of clarity, a simple workflow summarizing the present numerical study is represented in Fig 7.

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Fig 7. Workflow of the present numerical study.

https://doi.org/10.1371/journal.pone.0315194.g007

5. Results and discussion

In the present section, the influence of the electric field magnitude (E), type of pharmacokinetic profile (PK) and inlet blood velocity (λ_inl) on the contour plots of RTMs, C₂/C₁ and C₃ is analyzed. For sake of simplicity, boundaries of the tissues are named as shown in Fig 8, namely, TVWI represents the tissue-vessel wall interface where drug extravasation takes place, FBV stands for the farthest boundary from the vessel where a non-flux condition is assumed, FIB represents the front inlet boundary, and ROB is the rear outlet boundary. To limit the manuscript length, some of the contour plots corresponding to the results of the following subsections are reported as supplementary material.

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Fig 8. Boundaries of the tumor cord domain: TVWI is the tissue–vessel wall interface, FBV is the farthest boundary from the vessel, FIB is the front inlet boundary, and ROB is the rear outlet boundary.

https://doi.org/10.1371/journal.pone.0315194.g008

The contour plots of the RTMs considers both equilibrium (equal rates of RTMs) and non-equilibrium (different rates of RTMs) conditions, but it is worth mentioning that the last condition type is most common during the whole treatment since drug concentrations usually vary between two times instants, t_i and t_i+1, according to numerical simulations. Additionally, it is important to take into account that the presence of black markers in any zone of the RTMs contour plot means that the extracellular concentrations, C₁, decreases between two time instants, t_i and t_i+1.

5.1 Analysis of non-electroporated tissue

5.1.1. One-short tri-exponential profile (TPK).

Interaction between reaction and transport mechanisms (RTMs). In Figs 9 and 10, and S1 Fig of supplementary material the spatio-temporal change of the interaction between the different reaction and transport mechanisms (RTMs) is represented for the non-electroporated tissue and TPK. It is important to emphasize that both equilibrium (equal rates of RTMs) and non-equilibrium (different rates of RTMs) conditions can occur along the tissue, with the last condition type as the predominant since the drug concentration usually changes between two time instants, t_i and t_i+1, as shown later. For λ_inl = 0.0001m/s (Fig 9) at the first time interval (0h to 0.5h), internalization rate imposes over or is equal to association rate (IN≥AS). For the next time interval (0.5h to 1h), IN can be lower or equal than AS and/or than the positive extracellular transport rate (ECT+). Afterwards, it occurs that (IN≥AS)∨(IN≥ECT+) up to the end of treatment, and the absence of black markers at these time instants means that the extracellular concentration (C₁) increases, indicating that it is not possible that IN≥ECT+, signifying that (IN≥AS) as for the first-time interval. For λ_inl = 0.001m/s (Fig 9), at the first-time interval, it is observed that IN≥AS, like in the previous case (λ_inl = 0.0001m/s). In the next interval (0.5h to 1h), IN can be lower or equal than AS and/or than ECT+, and the presence of black markers indicates a decrease in C₁ in all points of the tissue, concluding that IN≤ECT+ is false and IN≤AS is true. Next, until the end of the treatment, it can only be inferred that there exist dissociation and internalization (∃DIS∧∃IN), and that the extracellular concentration (C₁) is continuously decreasing. For the greater velocity, λ_inl = 0.01m/s (S1 Fig of supplementary material), the behavior of the RTMs is similar to such observed with λ_inl = 0.0001m/s, except for the presence of black markers from 0.5h until the end of treatment (C₁ continuously decreases from this time instant onwards), indicating that IN≤AS between t = 0.5h and t = 1h, and that (IN≥AS)∨(IN≥ECT+) from t = 1h until the end of the treatment.

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Fig 9. Spatio-temporal evolution of the presence and interaction of reaction and transport mechanisms– E = 0kV/m, λ_inl = 0.0001m/s and TPK.

https://doi.org/10.1371/journal.pone.0315194.g009

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Fig 10. Spatio-temporal evolution of the presence and interaction of reaction and transport mechanisms– E = 0kV/m, λ_inl = 0.001m/s and TPK.

https://doi.org/10.1371/journal.pone.0315194.g010

Space and time behavior of internalization and/or externalization rate. According to Fig 11, for λ_inl = 0.0001m/s, the values of C₂/C₁ are lower than 1 at all time instants, indicating the existence of the internalization mechanism. It is also noticed that the highest internalization rates (IN) occur at the first hour of evaluation (t = 1h). Results shows that after one hour of treatment, internalization rate (IN) continuously decreases until the end of treatment (C₂/C₁ ratios increase after this time instant). Discriminating by zone of the tissue, it can be noticed that for t = 0.5h and t = 1h, IN is higher in TVWI (where C₂/C₁ ratios are smaller), and after t = 2h, this rate is greater in the opposed boundary (FBV). Moreover, the difference between (C₂/C₁)_max and (C₂/C₁)_min allows quantifying the radial uniformity of IN. For the present velocity (λ_inl = 0.0001m/s) this difference is increasing over time, which means a reduction in the radial uniformity. For instance, for t = 0.5h such difference is 0.0016, for t = 12h it is 0.0037, and for t = 24h it is 0.013. For the intermediate velocity, λ_inl = 0.001m/s (S2 Fig of supplementary material), in a similar fashion as for the lower velocity (λ_inl = 0.0001m/s), the values of C₂/C₁ are less than 1 at all time instants evaluated, i.e., internalization is also present, with the corresponding rates (IN) increasing until the first hour and then continuously decreasing during the whole treatment. For this velocity (λ_inl = 0.001m/s), IN is also higher in TVWI at t = 0.5h and t = 1h, and then higher in FBV. According to the difference between (C₂/C₁)_max and (C₂/C₁)_min, the radial distribution of IN is more uniform at t = 1h, time from which this uniformity reduces until the end of the treatment. For the larger velocity (λ_inl = 0.01m/s), according to S3 Fig of supplementary material, the values and behavior of the ratio C₂/C₁ are very similar to the previous case with λ_inl = 0.001m/s, which indicates that the increase in the blood speed has an irrelevant influence on the internalization rate (IN) for these two cases.

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Fig 11.

Spatio-temporal evolution of C₂/C₁ ratios that account for internalization and externalization rates—E = 0kV/m, λ_inl = 0.0001m/s and TPK.

https://doi.org/10.1371/journal.pone.0315194.g011

On the other hand, the straightness of the contour lines of C₂/C₁ accounts for the axial uniformity of drug internalization and/or externalization. In this regard, it is observed that for λ_inl = 0.0001m/s the internalization rate (IN) is not uniform at all time instants, unlike the other blood speeds (λ_inl = 0.001m/s and λ_inl = 0.01m/s) where this rate is evidently more uniform, indicating that the increase of λ_inl favors this uniformity.

Space and time behavior of association and/or dissociation rate. As can be seen in the Fig 12, for λ_inl = 0.0001m/s, the bound intracellular concentration (C₃) always increases in time, indicating the existence of association during the whole treatment. At all time instants, C₃ is maximum in the TVWI border and minimum in the FBV. The difference between C₃)_max and C₃)_min allows quantifying the radial uniformity of drug association and/or dissociation rate. This difference increases over time, in such a way that for t = 0.5h this difference is 1.7176x10⁻³ μM, for t = 1h it is 0.00355 μM; for t = 12h it is 0.0421 μM, and for t = 24h it is 0.0803 μM. This shows that the association rate (AS) is higher in the TVWI border than in the FBV border during the whole treatment. For λ_inl = 0.001m/s (S4 Fig of supplementary material), as in the previous case (λ_inl = 0.0001m/s), C₃ is maximum in TVWI and minimum in FBV, but in this case there is dissociation after t = 2h. The behavior of the difference between (C₃)_max and (C₃)_min is not monotonous in time, i.e., for t = 0.5h this difference is 0.2374 μM, for t = 1h it is 0.2505 μM, for t = 12h it is 0.1988 μM, and for t = 24h it is 0.2101 μM. This allows inferring that there is not predominance of the association or dissociation rate at any of the boundaries (TVWI and FBV) at all time instants. For λ_inl = 0.01m/s (S5 Fig of supplementary material), like in the first case (λ_inl = 0.0001m/s), there is association during the whole treatment, C₃ is maximum in TVWI and minimum in FBV, and the association rate (AS) is higher in TVWI than in FBV. After 24 hours of electrochemotherapy, the highest intracellular bound concentration (C₃) is obtained for the intermediate blood velocity, λ_inl = 0.001m/s.

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Fig 12. Spatio-temporal evolution of bound intracellular concentration C₃ that account for association and dissociation −E = 0kV/m, λ_inl = 0.0001m/s and TPK.

https://doi.org/10.1371/journal.pone.0315194.g012

On the other hand, the straightness of the contour lines of C₃ accounts for the axial uniformity of drug association and/or dissociation. The axial distribution of C₃ is not uniform for the three blood speeds, indicating that the association/dissociation rate is not the same for a set of points equidistant from the TVWI border.

5.1.2 Mono-exponential profile (MPK).

Interaction between reaction and transport mechanisms (RTMs). For the MPK profile and the three inlet blood velocities (λ_inl = 0.0001m/s, λ_inl = 0.001m/s and λ_inl = 0.01m/s), S13 Fig, as well as S6 and S7 Figs of supplementary material, represent the evolution of the interaction between the RTMs for the non-electroporated tissue. For λ_inl = 0.0001m/s (Fig 13) at the first-time interval (0h to 0.5h), IN≥AS is initially observed in all points in the tissue, however, from a certain time instant within this interval, externalization takes place in the lower part of TVWI. For the following two-time intervals (0.5h to 1h and 1h to 2h), externalization rate can be inferior or equal to negative extracellular transport rate (EX≤ECT−) and C₁ increases, but from 2h until the end of treatment only the existence of association and externalization (∃AS∧∃EX) can be inferred in most of the tissue; only in some points of the FBV boundary, it can be asserted that EX≤ECT− remains. For the entire treatment C₁ increases continuously. For λ_inl = 0.001m/s (S6 Fig of supplementary material), from t = 0h to t = 0.5h, the behavior is the same as in the previous case. For the next time interval (0.5h to 1h), EX≤ECT− and C₁ decreases, whereas from 1h to 2h, it occurs that (EX≤ECT−)∧(EX≥DIS), but from 2h onwards, it appears a zone in the tissue where (EX≥DIS)∨(EX≥ECT−) and C₁ continuously decreases, inferring that EX≥DIS is true. Therefore, for all points in the tissue, it can be concluded that externalization is present from t = 1h until the end of the treatment, with rates imposing over or being equal to dissociation ones (EX≥DIS). For λ_inl = 0.01m/s (S7 Fig of supplementary material), the behavior of the RTMs is very similar to such of the lower velocity, λ_inl = 0.0001m/s, with the difference that C₁ decreases from t = 0.5h until the end of the treatment.

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Fig 13. Spatio-temporal evolution of the presence and interaction of reaction and transport mechanisms −E = 0kV/m, λ_inl = 0.0001m/s and MPK.

https://doi.org/10.1371/journal.pone.0315194.g013

Space and time behavior of internalization and/or externalization rate. In Fig 14, for λ_inl = 0.0001m/s, it is observed that, contrary to the results of the TPK with the same velocity (Fig 11), the values of C₂/C₁ are greater than 1 at all time instants above t = 0.5h, which confirms the presence of the externalization mechanism for the MPK profile. It is also worth-noticing that externalization rates (EX) considerably increase until the first hour, where the values of C₂/C₁ significantly augments. After t = 1h, EX is almost stable until the end of the treatment. Discriminating by tissue areas, it can be concluded that EX is always higher in the TVWI. With regards to the radial uniformity of EX, the difference between (C₂/C₁)_max and (C₂/C₁)_min is 2.0897 for t = 0.5h, 9.0139 for t = 1h, 8.9682 for t = 12h and 8.9645 for t = 24h, pointing out a reduction of the radial uniformity of EX until t = 1h, but then a stabilization of this rate until the end of treatment. For λ_inl = 0.001m/s, S8 Fig of supplementary material, the values of C₂/C₁ are greater than 1 at all time instants above t = 0.5h (externalization is present too), and the externalization rate (EX) behaves the same in time and space as for λ_inl = 0.0001m/s, although in this case the values of EX are comparatively higher. The difference between (C₂/C₁)_max and (C₂/C₁)_min for t = 0.5h is2.1017, for t = 1h is 10.1706, for t = 12h is 10.1266 and for t = 24h is 10.1234, which illustrates that the time behavior of the radial uniformity of EX is the same as for λ_inl = 0.0001m/s, although in this case this uniformity is lower as shown by the greater differences between (C₂/C₁)_max and (C₂/C₁)_min. For λ_inl = 0.01m/s, S9 Fig of supplementary material, the contours of C₂/C₁ are very similar to the previous case of λ_inl = 0.001m/s.

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Fig 14. Spatio-temporal evolution of C₂/C₁ ratios that account for internalization and externalization rates −E = 0kV/m, λ_inl = 0.0001m/s and MPK.

https://doi.org/10.1371/journal.pone.0315194.g014

Regarding the axial distribution of the drug’s internalization, it can be noticed that this behaves similarly to the TPK profile, i.e., for λ_inl = 0.0001m/s the internalization rate (IN) is not uniform in the axial direction, but for the other two velocities (λ_inl = 0.001m/s and λ_inl = 0.01m/s) an axial homogeneity is obtained, confirming that the increase in the blood velocity benefits this homogeneity.

Space and time behavior of association and/or dissociation rate. As can be observed in Fig 15 for λ_inl = 0.0001m/s, in a similar fashion as for TPK, C₃ increases at all time instants (representing the existence of association) and C₃ is maximum in the TVWI border. The difference between (C₃)_max and (C₃)_min increases over time, in such a way that for t = 0.5h it is 1.4502x10⁻³μM, for t = 1h it is 0.00308 μM, for t = 12h it is 0.0354 μM, and for t = 24 h it is 0.0977 μM, showing that the association rate (AS) is higher in the TVWI. For λ_inl = 0.001m/s (S10 Fig of supplementary material), differently to the previous case (λ_inl = 0.0001m/s), C₃ increases only until t = 1h and then decreases due to the presence of dissociation during most of the treatment. Similar to the previous case, C₃ is maximum in the TVWI and minimum in the FBV. The difference between (C₃)_max and (C₃)_min does not behave monotonously over time, indicating that AS or DIS might be higher or lower in the TVWI versus the FBV. For λ_inl = 0.01m/s (S11 Fig of supplementary material), likewise the case of λ_inl = 0.0001m/s, there is association during the treatment and C₃ is maximum in the TVWI. Moreover, the difference between (C₃)_max and (C₃)_min is increasing over time, showing a larger AS in the TVWI. As for the TPK profile, the highest intracellular bound concentration (C₃) is obtained for the intermediate blood velocity, λ_inl = 0.001 m/s.

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Fig 15. Spatio-temporal evolution of bound intracellular concentration C₃ that account for association and dissociation −E = 0kV/m, λ_inl = 0.0001m/s and MPK.

https://doi.org/10.1371/journal.pone.0315194.g015

The straightness of the contour lines shows that the axial distribution of C₃ is uniform for the three blood velocities considered here, indicating that AS or DIS is almost the same for certain points equidistant from the TVWI.

5.1.3 Comparison with recent literature of non-electroporated tissues.

The geometric, transport, reaction and pharmacokinetic properties used in this paper are very similar to those used in Groh et al [9], where both TPK and MPK were considered for drug administration in non-electroporated tissues (E = 0 kV/m). Even though, contrary to the present work, the axial change of drug concentration in the vessel (C_v) was not considered there [9], some results are consistent with the present findings. For instance, in the present work, for all inlet blood velocities, λ_inl = [1x10⁻⁴ m/s, 1x10⁻³m/s, 1x10⁻² m/s], the cellular internalization is present during the whole treatment for TPK (Figs 9 and 10 of manuscript, and S1 Fig of supplementary material), whereas this mechanism is only present at the first instants for MPK, for which, in turns, the cellular externalization takes place during almost the entire treatment (Fig 13 of manuscript, and S6 and S7 Figs of supplementary material). Groh et al [9] found a significant increase of the extracellular concentration (C₁) at the first instants of treatment for both profiles (TPK and MPK), which was more pronounced as the points get closer to the vessel (TVWI boundary in this case). Because the initial intracellular concentration was zero, this sudden increment of C₁ generated cellular internalization at these first time instants, as founded here from t = 0 h to t = 0.5 h. Due to the nature of both profiles, Groh et al [9] predicted that, once it reaches the maximum value, C₁ decreases exponentially until t = 5h approximately for both profiles, but this reduction is more abrupt in MPK, boosting the drug externalization for this profile. In TPK, this reduction is smoother as the points get closer to the vessel, which could favor the internalization. Afterwards, Groh et al [9] predicted that the decrease of C₁ is kept along the whole tissue domain until t = 24 h for MPK, favoring the externalization, whereas C₁ increases at several points for TPK, which benefits the internalization.

5.2 Analysis for electroporated tissue (E = 46 kV/m)

5.2.1 One-short tri-exponential profile (TPK).

Interaction between reaction and transport mechanisms (RTMs). In the S12–S14 Figs of supplementary material, they are shown the interactions between the RTMs arising in the tissue electroporated with an electrical field magnitude of 46 kV/m (reversible threshold), for the three different blood speeds (λ_inl = 0.0001m/s, λ_inl = 0.001m/s and λ_inl = 0.01m/s). For λ_inl = 0.0001m/s (S12 Fig of supplementary material) and λ_inl = 0.001m/s, (S13 Fig of supplementary material), the behavior of the RTMs is similar to such observed for the non-electroporated tissue (E = 0 kV/m), with the only difference that in the interval from t = 9h to t = 12h, C₁ decreases for all points of the domain. For λ_inl = 0.01m/s (S14 Fig of supplementary material), the behavior of the RTMs is the same as for the non-elecroporated tissue with this velocity.

Space and time behavior of internalization and/or externalization rate. According to the Fig 16 of manuscript, and S15 and S16 Figs of supplementary material, some aspects of the behavior of C₂/C₁ change with respect to the non-electroporated tissue when electrical pulses are applied, while others are not altered. For all blood velocities, it is possible to verify the presence of the internalization mechanism during the whole treatment. For the particular case of λ_inl = 0.0001m/s, S16 Fig of manuscript, from the first hour to the end of the treatment, the internalization rate (IN) decreases, coinciding with the non-electroporated tissue. It is also observed that the internalization rate (IN) is smaller in the lower part of the TVWI at all time instants. As for the non-electroporated tissue, the radial uniformity of IN decreases with the time according to the evolution of the difference between (C₂/C₁)_max and (C₂/C₁)_min. For λ_inl = 0.001m/s (S15 Fig of supplementary material) and λ_inl = 0.01m/s (S16 Fig of supplementary material), IN behaves the same as for λ_inl = 0.0001 m/s, namely, it decreases over the time after the first hour, being lower along the whole TVWI. For both blood velocities (λ_inl = 0.001m/s and λ_inl = 0.01m/s), the decrease in the internalization rate (IN) after the first hour coincides with the results of the non-electroporated tissue. Furthermore, the increment of the difference between (C₂/C₁)_max and (C₂/C₁)_min indicates that the radial uniformity of IN decreases after the first hour for these two velocities.

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Fig 16. Spatio-temporal evolution of C₂/C₁ ratios that account for internalization and externalization rates—E = 46kV/m, λ_inl = 0.0001m/s and TPK.

https://doi.org/10.1371/journal.pone.0315194.g016

On the other hand, contour lines of C₂/C₁ depict that for λ_inl = 0.0001m/s and λ_inl = 0.001m/s, internalization rate (IN) is not axially uniform at all times, as opposed to λ_inl = 0.01 m/s where this rate is homogeneous, confirming the positive influence of increasing λ_inl on the longitudinal uniformity of IN.

Space and time behavior of association and/or dissociation rate. As observed in Fig 17 of manuscript, for λ_inl = 0.0001m/s, C₃ increases during the treatment (there is association) and this concentration is maximum in TVWI and minimum in FBV, being this result similar to the one obtained for the non-electroporate tissue, although in this case (E = 46 kV/m) the maximum values of C₃ are concentrated in the inferior zone of the TVWI. The difference between (C₃)_max and (C₃)_min increases over time, indicating that AS remains higher in the TVWI, also coinciding with the non-electroporate tissue. For the velocity of λ_inl = 0.001m/s (S17 Fig of supplementary material), the changes of C₃ show that there is association at the first hour, and afterwards, dissociation occurs, which also agrees with the non-electroporated tissue. On the contrary, for λ_inl = 0.01m/s (S18 Fig of supplementary material), association occurs during the whole treatment, also coinciding with the case of E = 0 kV/m. For these two last speeds (λ_inl = 0.001m/s, λ_inl = 0.01m/s), C₃ is also maximum in the inferior part of the TVWI, but the difference between (C₃)_max and (C₃)_min is not monotonous, concluding that there is no predominance of the dissociation rate (DIS) for λ_inl = 0.001m/s or association rate (AS) for λ_inl = 0.01m/s in any of the radial boundaries (TVWI or FBV).

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Fig 17. Spatio-temporal evolution of bound intracellular concentration C₃ that account for association and dissociation—E = 46kV/m, λ_inl = 0.0001m/s and TPK.

https://doi.org/10.1371/journal.pone.0315194.g017

In a similar fashion to the case with E = 0 kV/m, the axial distribution of C₃ is not uniform for the three blood velocities, namely, AS/DIS differs for points equidistant to the TVWI, and the highest values of C₃ are also achieved for the intermediate velocity, λ_inl = 0.001m/s.

5.2.2 Mono-exponential profile (MPK).

Interaction between reaction and transport mechanisms (RTMs). For λ_inl = 0.0001m/s, S19 Fig of supplementary material, the behavior of the RTMs is similar to the one observed for the non-electroporated tissue and this profile (MPK), with the differences that in the interval from t = 9h to 12h, C₁ decreases for all points of the domain, and from t = 2h onwards, the phenomenon EX≤ECT− does not necessarily occur in some points close to the FBV, where only the existence of association and externalization without any particular interaction (∃AS∧∃EX) can be inferred at this time. For λ_inl = 0.001m/s, S20 Fig of supplementary material, at the first three-time intervals, the behavior of the RTMs is the same than for non-electroporated tissue. For the following three intervals (2h to 3h, 3h to 6h, 6h to 9h), it occurs that (EX≥DIS)∨(EX≥ECT−) in some parts of the tissue, with the last inequality as false since C₁ decreases, leading to EX≥DIS as true. In some parts of the tissue closed to the FBV, it happens that (EX≤ECT−)∧(EX≥DIS), namely, both conditions shall be complied. At the time interval from t = 9h to t = 12h, (EX≥DIS)∨(EX≥ECT−) and C₁ increases, whereas from t = 12h to t = 15h, EX≤ECT− and C₁ decreases. From t = 15h up to the end of the treatment, the behavior of RTMs is similar to such between 2h and 9h. For λ_inl = 0.01m/s (S21 Fig of supplementary material), the behavior of RTMs is similar to such observed for the non-electroporated tissue with the same blood velocity.

Space and time behavior of internalization and/or externalization rate. In the Fig 18 of manuscript, for λ_inl = 0.0001m/s, resembling the phenomenon observed for the non-electroporated tissue, externalization occurs, and an important increment of externalization rates (EX) takes place until the first hour, from which EX remains almost constant in most of tissue domain until the end of treatment. There is a small difference with the non-electroporated tissue, since EX is not higher along the whole TVWI for the present electric field magnitude (E = 46 kV/m), but only in the inferior part of this boundary. Regarding the radial uniformity of EX, which is accounted for by the difference between (C₂/C₁)_max and (C₂/C₁)_min, the behavior is the same as for the protocol without electroporation (E = 0 kV/m). For the other two velocities, λ_inl = 0.001 m/s (S22 Fig of supplementary material) and λ_inl = 0.01 m/s, (S23 Fig of supplementary material), the behavior of contours C₂/C₁ is similar to such observed in the non-electroporated tissues with this profile and these same velocities, hence, the aforementioned observations can be extended for these cases. Regarding the axial uniformity of EX, similar conclusions to the ones of the non-electroporated tissue with the same profile (MPK) can be addressed.

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Fig 18. Spatio-temporal evolution of C₂/C₁ ratios that account for internalization and externalization rates—E = 46kV/m, λ_inl = 0.0001m/s and MPK.

https://doi.org/10.1371/journal.pone.0315194.g018

Space and time behavior of association and/or dissociation rate. Observing the Fig 19 of manuscript, for λ_inl = 0.0001m/s, C₃ increases during the treatment (there is association), it is maximum in TVWI, and the incremental difference between (C₃)_max and (C₃)_min points out that the association rate (AS) is higher in TVWI during the whole treatment. This agrees with the corresponding case with E = 0 kV/m. For λ_inl = 0.001m/s (S24 Fig of supplementary material), C₃ is also maximum in the TVWI and minimum in the FBV. For this velocity, association is present in all points of the tissue until the second hour (t = 2h), from which dissociation arises. The behavior of the difference between (C₃)_max and (C₃)_min is not monotonous over time, indicating that there is no predominance of the rate of association or dissociation in any of the radial boundaries (TVWI and FBV) at all time instants. For λ_inl = 0.01m/s (S25 Fig of supplementary material), the behavior of C₃ is similar as for the lower velocity, λ_inl = 0.0001 m/s.

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Fig 19. Spatio-temporal evolution of bound intracellular concentration C₃ that account for association and dissociation—E = 46kV/m, λ_inl = 0.0001m/s and MPK.

https://doi.org/10.1371/journal.pone.0315194.g019

The axial distribution of C₃ is slightly uneven for the three blood speeds (λ_inl = 0.0001m/s, λ_inl = 0.001m/s and λ_inl = 0.01m/s), indicating that AS/DIS can have minimum variations for certain points equidistant from the TVWI. As for the non-electroporated tissue, maximum values of C₃ are achieved for the intermediate velocity, λ_inl = 0.001 m/s.

5.2.3 Comparison with recent literature of tissues electroporated with the reversible electric field.

As it was shown in the present section, when tissue is electroporated with the reversible electric field magnitude (E = 46 kV/m), the C₂/C₁ ratio that accounts for the cellular internalization or externalization rates under the assumption of Fickian flow, decreases radially from the TVWI towards the FBV for both TPK (Fig of manuscript, and S15 and S16 Figs of supplementary material) and MPK (Fig 18 of manuscript, S22 and S23 Figs of supplementary material), being this more noticeable as λ_inl is greater. In a recent work [59], it was studied the influence of the pulse numbers (N_ep) on the extracellular and intracellular concentrations of a tissue electroporated with a reversible electric field magnitude of E = 28 kV/m, considering a uniform extracellular concentration of C₁ = 1M in one of the borders of the tumor domain (corresponding to the TVWI boundary in the present work). Despite the difference in the reversible thresholds (28 kV/m vs. 46 kV/m), it was found that at a given time instant, the ratio between intracellular and extracellular concentrations (C₂/C₁) is greater in the drug inlet boundary (TVWI in this case), decreasing towards the outermost boundary (FBV in this case), and this behavior remains unchanged by increasing the number of pulses (N_ep).

5.4 Transvascular transport at the first hour of treatment

Since both pharmacokinetic profiles (TPK and MPK) are exponential, the largest changes of the ratio C_v/C_1,TVWI, with C_v and C_1,TVWI as the drug concentration along the bloodstream and the extracellular concentration at the tissue-vessel wall interface, respectively, are expected at the first instants of the electro-chemotherapeutic treatment. The C_v/C_1,TVWI ratio is useful to determine the time instants where reverse diffusion takes place, namely, drug transport from the tissue towards the free plasma. This emulates a drug clearance back to the plasma by reverse concentration gradients, but it is not exactly a lymphatic drainage since lymphatic vessels are usually more permeable than the capillary vessel included in the tumor cord domain (Fig 2). The time evolution of the C_v/C_1,TVWI ratio calculated at the axial coordinate z = 0 until the first hour of treatment (t = 1h) is shown in Fig 20A and 20C for the three blood velocities (1x10⁻⁴m/s, 1x10⁻³m/s, 1x10⁻²m/s) and electric field magnitudes (0 kV/m, 46 kV/m, 70 kV/m) considered here. In these figures, the green line corresponds to C_v/C_{1,TVWI,z = 0} = 1, above which extravasation occurs and below which a reverse drug diffusion takes place. For visualization purposes, semilog-y plots are reported.

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Fig 20. Time behavior of the ratio of drug concentration at the bloodstream to the extracellular concentration at the tissue-vessel wall interface (TVWI) at z =0, C_v/C_{1,TVWI,z = 0} vs. t.

(a) λ_inl = 1x10⁻⁴ m/s, (b) λ_inl = 1x10⁻³ m/s, (c) λ_inl = 1x10⁻² m/s. The green line corresponds to C_v/C_{1,TVWI,z = 0} = 1, above which extravasation occurs and below which a reverse drug diffusion takes place (from tissue towards the bloodstream).

https://doi.org/10.1371/journal.pone.0315194.g020

For all values of λ_inl and E, the C_v/C_{1,TVWI,z = 0} ratios are larger in MPK by one order of magnitude at the beginning of the treatment, indicating greater rates of extravasation for this pharmacokinetic profile, which is associated to the IV bolus injection in a short period of time. However, the time decrease of C_v/C_{1,TVWI,z = 0} is more pronounced for MPK as well, suggesting a faster increase of the extracellular concentration at the TVWI regarding the plasma concentration, which leads to drug diffusion back towards the bloodstream at a time instant ranging between 0.31h to 0.48h depending on the combination between λ_inl and E. The exponential decay rate of C_v/C_{1,TVWI,z = 0} is maintained for MPK. On the other hand, for the TPK profile, C_v/C_{1,TVWI,z = 0} considerably decreases until the first half hour (t = 0.5h) approximately, from which the time change of C_v/C_{1,TVWI,z = 0} is significantly smaller, remaining almost constant for λ_inl = 1x10⁻⁴ m/s where it is predicted drug transport back to the vessel at very low rates when compared with MPK; for λ_inl = 1x10⁻³ m/s and λ_inl = 1x10⁻² m/s, C_v/C_{1,TVWI,z = 0} barely increases and decreases, respectively, and drug extravasation is kept for both velocities and all values of E.

6. Conclusions and clinical implications

In the present study, the influence of two parameters of the systemic administration of chemotherapeutic drugs on the presence, relative importance and rates of the transport and reaction mechanisms (RTMs) involved in electroporated and vasoconstricted tissues is investigated, namely, the bloodstream inlet velocity (λ_inl) and the pharmacokinetic profile (PK), considering several electric field magnitudes (E). Results were obtained by simulating several treatment scenarios with an in-silico tool previously developed, calibrated and validated, which is based on the Global Method of Approximate Particular Solutions (GMAPS) [7]. According to the numerical results, the RTMs for the One-short Tri-exponential (TPK) and Mono-exponential profile (MPK) can differ each other for a given combination of E and λ_inl. For TPK, the reversible threshold (E = 46 kV/m) does not affect the interaction of the RTMs with respect to the non-electroporated tissue (E = 0 kV/m) for the three blood speeds considered here, λ_inl = [1x10⁻⁴ m/s, 1x10⁻³ m/s, 1x10⁻² m/s], but this interaction is clearly influenced by λ_inl. Accordingly, in non-equilibrium conditions, the drug internalization rate (IN) imposes over the drug association rate (AS) at the first half hour, and then, from t = 0.5h to t = 1h a reverse behavior can occur for all values of λ_inl and both electric field magnitudes (E = 0 kV/m and E = 46 kV/m). However, from t = 1h to the end of the treatment, the existence and relative importance of RTMs depends on the blood velocity, in such a way that for λ_inl = 1x10⁻⁴ m/s, internalization rate (IN) is greater than association one (AS); for λ_inl = 1x10⁻³ m/s, there exist dissociation and internalization but nothing can be inferred about how they interact with each other; and for λ_inl = 1x10⁻² m/s, it can be asserted that internalization rate (IN) can impose over association rate (AS) and/or over positive extracellular transport rate (ECT+). On the other hand, the irreversible threshold (E = 70 kV/m) evidently affects the RTMs regarding the non-electroporated tissue (E = 0 kV/m), mainly after t = 1h, and this influence depends on λ_inl. Accordingly, in non-equilibrium conditions, for the lower velocity (λ_inl = 1x10⁻⁴ m/s), IN is greater than AS during most of the treatment after t = 1h and in the majority of points of the domain, but from t = 9h to t = 12h, IN imposes over AS and/or over ECT+, and externalization occurs in some points close to the TVWI. For the intermediate velocity (λ_inl = 1x10⁻³ m/s), internalization is present until t = 21h except in some points closed to TWVI where externalization occurs between t = 9h to t = 12h; additionally, periods of association and dissociation can be present from t = 1h to t = 21h, whereas positive extracellular transport is predominant from t = 3h to t = 21h. In the last three hours of the chemotherapeutic treatment (t = 21h to t = 24h), externalization, dissociation and negative extracellular transport take place, with the externalization rate (EX) lower than the negative extracellular transport rate (ECT−), but larger than the dissociation one (DIS). For the larger velocity considered here (λ_inl = 1x10⁻² m/s), internalization, association and positive extracellular transport are the predominant RTMs during the whole treatment, whereas externalization and negative extracellular transport only occurs in some points close to the TVWI at some time intervals. In most of the points of the domain, internalization rate (IN) surpasses the association rate (AS) but can be smaller or larger than the positive extracellular transport rate (ECT+) depending on the time interval.

For the MPK profile, the reversible threshold (E = 46 kV/m) and blood velocity (λ_inl) can affect the interaction between the RTMs. For both electric field magnitudes (E = 0 kV/m and E = 46 kV/m) and the three inlet velocities, λ_inl = [1x10⁻⁴ m/s, 1x10⁻³ m/s, 1x10⁻² m/s], in a different manner as for TPK, internalization is only present up to the first half hour, but then externalization imposes in the whole domain until the end of the treatment. Additionally, for both values of E and all values of λ_inl, in non-equilibrium conditions, externalization rate (EX) is inferior to the negative extracellular transport rate (ECT−) until t = 2h. The main difference appears after t = 2h, where association exists until the end of the treatment in the non-electroporated tissue (E = 0 kV/m) for λ_inl = 1x10⁻⁴ m/s and λ_inl = 1x10⁻² m/s, whereas dissociation is predominant for the intermediate velocity (λ_inl = 1x10⁻³ m/s). For E = 46 kV/m, association is also present until the end of treatment for λ_inl = 1x10⁻⁴ m/s and λ_inl = 1x10⁻² m/s, whereas both dissociation and negative extracellular transport can be present for λ_inl = 1x10⁻³ m/s, with the externalization rate (EX) imposing over dissociation rate (DIS), and being smaller or larger than the negative extracellular transport rate (ECT−) depending on the time instant. On the other hand, contrarily to the TPK profile, the application of the electric field magnitude of E = 70 kV/m does not considerably alter the interaction between the RTMs involved in the electro-chemotherapeutic treatment with respect to the reversible threshold (E = 46 kV/m).

In general, numerical results allow concluding that, for both pharmacokinetic profiles (TPK and MPK), the radial uniformity of internalization/externalization rates (IN/EX) for the non-electroporated tissues is negatively affected when the blood velocity increases, specially from λ_inl = 1x10⁻⁴ m/s to λ_inl = 1x10⁻³ m/s, whereas an opposite effect is reached for the axial homogeneity of IN/EX, i.e., the increase of λ_inl benefits this homogeneity. On the other hand, the effect of the blood velocity (λ_inl) on the radial and axial uniformity of association/dissociation rates (AS/DIS) is not necessarily monotonic for the conditions considered here. These conclusions can be extended for the electroporated tissues (E = 46 kV/m and E = 70 kV/m), but it is worth mentioning that, in general, the increase of E reduces the radial uniformity of IN/EX and AS/DIS regarding the non-electroporated tissue, whereas the axial uniformity is affected in a lower extent.

The Boolean model presented in this study could have practical applications in the development of electro-chemotherapy treatments since it could be useful for establishing appropriate drug dosages and pharmacokinetic profiles, as well as for determining the necessary electroporation parameters. Prior knowledge of the interaction between the reaction and transport mechanisms (RTMs) of the drug can aid in determining these parameters before the treatment implementation. For instance, when comparing two treatments with the same drug type and total drug exposure, one in which cellular internalization (IN) of drug occurs continuously over time in most of the tissue, and another in which internalization (IN) only happens at the beginning and is followed by externalization (EX), as TPK and MPK considered here, the former may be more advantageous because it leads to a higher possibility of increased the final cell death since drug resides inside the cells during more time. The above findings also suggest that the selection of the appropriate electric field magnitude (E) should consider the type and parameters of the PK profile, along with the expected blood perfusion rates in the target tissues, because the spatio-temporal behavior of the concentration fields and exposures of the drug, as well as the presence, interaction and rates of the RTMs can significantly change for a given value of E when the other parameters are modified. According to the numerical results, this can be particularly decisive as the electric field magnitude (E) is closer to the irreversible threshold (70 kV/m in the present case). Accordingly, the characterization, monitoring, control and/or correction of the PK parameters (specially when AUC-based dosing is employed) and the blood perfusion rate (vascular normalization, selective depletion of TME constituents, among others) could be relevant for the prescription of the electric field strength in electro-chemotherapeutic treatments.

As formerly mentioned, the total drug exposure or Area Under the Curve (AUC) is fixed for both profiles (MPK and TPK) in the present work, having a value of AUC = 2.78 μM. h. However, some parameters of the PK profiles can be fitted to reach a target AUC or target exposures to extracellular drug , free intracellular drug and/or bound intracellular drug after a given time (T′). For instance, by modifying the total drug dose (D₀) and time of infusion (τ’) in the Eq (35), which predicts the time evolution of C_v for TPK, different target values of AUC, C_1exp, C_2exp and C_3exp, which are more common parameters in the clinical practice related with the cell-kill capacity of the therapy, can be obtained using the present model. Despite the PK parameters of each profile are not modified here, it is worth mentioning that the present model could be used in future works for evaluating the effects of modifying these parameters on the drug concentration fields, drug exposures (which are more useful at clinical level), and the presence and interaction of the reaction and transport mechanisms (RTMs) of chemotherapeutic drugs.

The mechanisms of protein association (AS) or dissociation (DIS) are highly significant in clinical contexts. Accordingly, the high drug association in the blood vessels and the interstitial matrix slows down the rates of extravasation (EV) and cellular internalization (IN), respectively, which may imply higher drug doses or treatment duration to reach the target cell death. Nevertheless, association (AS) also delays the lymphatic drainage (LD), resulting in a longer drug residence time inside the tumor, which can favor the therapy efficiency. Within the intracellular environment, a high association (AS) can be advantageous for enhancing the therapy effectiveness by inhibiting the cellular externalization (EX). As reasonable, dissociation (DIS) produces reverse impacts. The protein content of cancer patients has a significant role in the side effects of the treatment. Patients with low protein levels, known as hypoproteinemia, are more susceptible to side effects and require lower drug doses. Additionally, these patients are less prone to drug association, with the implications that this entails for the treatment efficacy [62–65].

The extracellular transport (ECT) is also highly significant in a clinical context. Facilitating the interstitial transportation of drug (ECT+) to cancer cell receptors is advantageous for Antibody-Drug Conjugate (ADC) therapies, as the drug is only released upon internalization. On the other hand, the positive extracellular transport (ECT+) to healthy peripheral cells can promote the bystander effect. This occurs when the death of tumor cells is intensified due to the absence of feed and protection from the neighboring healthy cells that have been injured by the medicament. In clinical practice, the process of negative extracellular transport (ETC−) can be beneficial in some areas of the tissue (for example the tissue-vessel wall interface) since it could help prevent the formation of reverse concentration gradients or allows for the release of the interstitial pressure.

In general, any combination of parameters that promotes the radial and axial uniformity of the rates associated to the RTMs considered here brings about more homogenous cell-kill effects on the target tissue, which is very beneficial at clinical level, particularly in low-vascularized tumors.

It is worth-mentioning that the present model is not intended to replace the Heuristic methods commonly used for dosing of anticancer drugs (weight-based dosing, body surface area dosing, fixed-dose prescribing, AUC-based dosing, among others), but to complement these methods. Heuristic methods have been defined based on in vitro and in vivo studies, and clinical trials considering several aspects such as: type of drug, type and subtype of cancer, stage of cancer, previous health treatments, current patient’s medications and mainly, the patient’s characteristics (age, weight, height, health problems, among others). One of the main drawbacks of these methods is their poor relationship with the pharmacokinetic parameters of chemotherapeutic drugs (loading and maintenance dose, injection time, decay rates, among others), affecting the accuracy in the prediction of the drug exposure and drug clearance, and consequently, the individualization capacity of the therapy. Moreover, since some heuristic models are not able to correctly predict the drug clearance for some cases, the definition of the chemotherapy schedule can also carry some inaccuracies when defined by these methods. For instance, the Body Surface Area (BSA), which is still an extensively used variable for definition of the drug dose, has not shown an acceptable correlation with the AUC and clearance of doxorubicin [66] or cisplatin [67], which are common drugs employed in the cancer treatment. As a complement, the present model is able to predict the spatio-temporal evolution of drug concentration in the vessels, extracellular and intracellular space, as well as the transport and reaction mechanisms of the chemotherapeutic drug in electroporated and vasoconstricted tissues. Accordingly, the present model could assist the initial evaluation of the effects of the chemotherapy doses and schedule on the drug distribution, concentration and exposure along the vessel and tissues during the whole treatment prior to the application of the therapy.

Generally speaking, most anticancer agents have a very narrow therapeutic window, which means that a small change in dosage can lead to unexpected side effects in the patient. This is one of the main risks of some heuristic methods widely used in clinical practice, such as BSA, which, not being closely correlated with AUC, may generate different cytotoxic processes in patients with similar characteristics (weight, age, height, health problems, etc.), such as systemic DNA degeneration and inflammation of healthy tissues, in turns leading to less severe side effects (nausea, vomiting, dysgeusia, hair loss), life-threatening effects (myelosuppression, febrile neutropenia, severe oral mucositis, sepsis, among others) or late effects at cardiac, pulmonary, neurological, renal or hepatic level. The model for the concentration fields developed and validated in previous works [7, 17, 30], which was briefly presented in Section 2 of the manuscript and was used to generate the inputs of the Boolean model, allows estimating the exposure to the chemotherapeutic drug considering the parameters of the electroporation protocol, the reaction and transport properties of the drug in the tissue, the geometrical and electrical properties of the tissue, information of the chemotherapeutic treatment (type of drug, injected dose, pharmacokinetic profile, treatment time, among others), fitting parameters of the chemotherapeutic cell death models, and characteristics related to the blood perfusion rate (vasoconstriction, blood velocity and pressure, viscosity, etc.). Accordingly, under the premise of the high correlation between the drug exposure and side effects of the therapy, this model could be very useful to complement the design of chemotherapeutic protocols by heuristic methods, focused on reducing the side effects without considerably compromising the efficacy of the treatment. This could be addressed in detail in future works.

The scaling to macroscopic domains with complex vascular and lymphatic networks, the consideration of more specific reaction and transport mechanisms arising in several subdomains (like the endothelial wall), the inclusion of cell death models by electroporation and cytotoxic effects, among others, are challenges that represent opportunities of improvement of the present model. Additionally, it could be necessary to enhance the model by including the mechanisms of drug resistance of tumor cells, such as: (1) tumor heterogeneity due to mutations, deletions, chromosomal rearrangements, among others [68, 69], (2) inactivation of the medication by alteration of its molecular characteristics when binding to proteins [68, 69], (3) uncontrolled release of certain medications (Doxorubicin, Vinblastine, and Taxol) into the extracellular space due to certain substances like chloride transported by ATP-dependent carriers [68, 70], (4) changes in drug metabolism by enzymes [68, 71], (5) secondary mutations in cancer cells such as topoisomerase II [68, 72], (6) processes of DNA damage and repair, especially in platinum compounds and doxorubicin [68, 73].

Supporting information