Tissue transport affects how treatment scheduling increases the efficacy of chemotherapeutic drugs

Abstract

A method to predict the effect of tissue transport on the scheduling of chemotherapeutic treatment could increase efficacy. Many drugs with desirable pharmacokinetic properties fail in vivo due to poor transport through tissue. To predict the effect of treatment schedule on drug efficacy we developed an in silico method that integrates diffusion through tissue and cell binding into a pharmacokinetic model. The model was evaluated with an array of theoretical drugs that had different rates of diffusivity, binding, and clearance. The efficacy of each drug, quantified as the fraction of cells killed, was calculated for twenty dosage schedules. Simulations showed that efficacy strongly depended on tissue transport, with a range of 0.00 to 99.99%, despite each drug having equal plasma areas under the curve (AUC). For most drugs, schedules that increased exposure also increased efficacy. Drugs with fast clearance benefited the most from increasing the number of doses and this was most effective for those with intermediary binding. All drugs with slow diffusivity were ineffective. For a subset of drugs, increasing the number of doses decreased efficacy. This phenomenon was unexpected because, when considering uptake into tissue, sustained plasma levels from multiple doses are generally assumed to be more effective. This counterintuitive decrease in efficacy was caused by drug retention within tumor tissue. These results established a set of rules that suggests how transport parameters affect the efficacy of drugs at different schedules. The two most predominant rules are (1) multiple doses improve efficacy for drugs with fast clearance, fast diffusivity and low to intermediate cell binding; and (2) one dose is most effective for drugs with slow clearance, slow diffusivity or strong cell binding. Understanding the role of tissue transport when determining drug treatment schedules would improve the outcome of preclinical animal experiments and early clinical trials.

Keywords: Cell binding; Chemotherapy; Diffusion; Optimal treatment schedule; Pharmacokinetics; Retention; Tissue transport model.

Fig. 1.

Mechanisms that control chemotherapeutic drug efficacy in tumors. The computational model of chemotherapeutic (stars) efficacy contains four main mechanisms: diffusion (D), binding (R), cytotoxicity (α), and clearance (t_1/2). Drugs are delivered through the vasculature (dark red) and diffuse, with rate D, through tumor tissue (orange) made up of cells. After binding to cells with rate constant R, drugs cause cell death (red star) with maximum rate, α. Drugs diffuse out of tissue with rate D and are cleared from the blood at rate t_1/2. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 2.

Drug transport in tumor tissue after a single dose. (A) Transport of a theoretical drug (Drug A) with similar transport properties as doxorubicin (D = 10⁻¹⁰ m²/s; B = 10⁰; and t_1/2 = 0.5 h). Drug was introduced at a normalized concentration of one at time t = 0 hr. Initially the proximal concentration (C_P) was greater than the distal concentration (C_D). With time, retention caused C_D to be greater than C_P. (B) The average concentration in the tissue (C^ave) shows the combined effects of clearance and retention, which affected the proximal and distal concentrations in (A). In the presence of drug, killed cells (Fk) decreased until the drug was cleared from the system.

Fig. 3.

Effect of dosage schedule and clearance rate on effective plasma concentration. (A, B) Effective plasma concentration for a drug with half-life of 0.5 h and (A) 2-doses and (B) 10 doses. The amount of drug added per dose was ten times greater in A compared to B to keep the total drug administered constant. (C, D) Effective plasma concentration for a drug with half-life of 15 h and (C) 2-doses and (D) 10 doses.

Fig. 4.

Drug transport parameters affected how treatment schedule influenced efficacy. (A) For a drug (Drug B) with moderate diffusivity (10⁻⁸ m²/s) and binding (10⁻²), increasing the number of doses increased efficacy. (B) For a drug (Drug C) with lower diffusivity (10⁻¹² m²/s) and higher binding (10²), more doses decreased efficacy.

Fig. 5.

Effect of transport properties on efficacy. (A) An array of theoretical drugs was evaluated over all dosage schedules. The maximum efficacy (MxE) across all schedules was highly dependent on diffusivity (D) and binding (R). Across the range of these properties and at t_1/2 = 0.5 h, maximum efficacy went from ineffective (MxE = 0) to completely effective (MxE = 1). The behavior of five representative drugs were investigated (a: D = 10⁻¹⁰ m²/s, R = 10⁰; b: D = 10⁻⁸ m²/s, R = 10⁻²; c: D = 10⁻¹² m²/s, R = 10²; d: D = 10⁻⁷ m²/s, R = 10²; and e: D = 10⁻⁷ m²/s, R = 10). The transport properties of doxorubicin (dox) are (D = 2 × 10⁻¹¹ m²/s, R = 4.8 × 10¹). (B) Multiple-dose improvement (MDI) is the difference in efficacy between 20 and 1 doses. It indicates how efficacy would improve with increased doses. The measured range for MDI over all investigated drugs was (−0.092, 0.641). (C) For each drug the optimum number of doses (OpD) was determined as the minimum dose that produced at least 98% of the maximum efficacy. (D) Drugs were arranged into five groups based on transport properties and the effect of doses on efficacy: Group I, low diffusivity; Group II, moderate diffusivity and high binding; Group III, strong binding; Group IV moderate binding; and Group V, weak binding.

Fig. 6.

Increased exposure throughout tissue increased efficacy. (A–D) For a theoretical drug (Drug D), with half-life of 0.5 h, diffusivity of 10⁻⁷ m²/s and binding constant of 10², increasing overall exposure increased efficacy. (A) With one dose, the drug concentration at the distal edge (C_D) was initially large but quickly dropped to zero. (B) With ten doses, C_D was smaller, but continually reappeared throughout treatment. (C) With one dose, cell death (Fk) stopped after drug was cleared from the tissue (Eff_{D, 1} = 0.485). (D) After ten doses, almost all cells in the tissue were killed (Eff_{D, 10} = 0.993). E–F) Normalized distal drug exposure (DDE’) for all theoretical drugs after 1 (E) and 10 (F) doses. G) The change in distal exposure ΔDDE’ matches the effect of increasing doses on efficacy (MDI; Fig. 5B).

Fig. 7.

Drugs retention and negative MDI. (A–D) A theoretical drug (Drug C; t_1/2 = 0.5 h, D = 10⁻¹² m²/s and R = 10²) in Group II was more effective after 1 dose than after 20 doses. (A) For Drug C the fraction of killed cells (Fk) was higher for 1 dose than for 20 doses throughout the treatment. (B) The average concentration across the length of the tissue (C^ave) was initially greater for 1 dose compared to 20 doses. The average concentration at T/2 (36 h; C^hf) was positive and non-zero indicating that the drug had not washed out of the tissue. (C) The concentration at the proximal edge (C_P) quickly dropped to zero after each dose, regardless of the number of doses. (D) The concentration at the distal edge (C_D) was greater after 1 dose for most of the treatment period. (E) Positive drug retention (Rt) indicates that the distal concentration was greater than the proximal concentration. (F) Drugs with negative MDI values (MDIn) had decreased performance with more doses.

Fig. 8.

Effect of half-life. (A–B) Maximum efficacy (MxE) over all treatment schedules for drugs with half-lives of 5 h (A) and 15 h (B). (C–D) Multiple dose improvement (MDI) for drugs with half-lives of 5 h (C) and 15 h (D). (E–F) Optimal dose (OpD) for drugs with half-lives of 5 h (E) and 15 h (F).

Fig. 9.

Half-live increased the concentration at the distal edge. (A) After two doses, Drug E (D = 10⁻⁴ m²/s; R = 10⁰), with t_1/2 = 0.5 h, appeared quickly at the distal edge and was cleared quickly. (B) After two doses of a similar drug with the same D and R, but slower clearance (t_1/2 = 5 h), clearance from the distal region was slower. (C) Each dose of the slow clearing (t_1/2 = 0.5 h) drug killed approximately 20% of the cells. (D) The first dose of the slower clearing drug (t_1/2 = 5 h) killed most of the cells, so the second dose was less effective.