A mathematical model of cancer chemotherapy with an optimal selection of parameters

An optimal parameter selection model of cancer chemotherapy is presented which describes the treatment of a tumor over a fixed period of time by the repeated administration of a single drug. The drug is delivered at evenly spaced intervals over the treatment period at doses to be selected by the model. The model constructs a regimen that both minimizes the tumor population at the end of the treatment and satisfies constraints on the drug toxicity and intermediate tumor size. Numerical solutions show that an optimal regimen withholds the bulk of the doses until the end of the treatment period. When a drug used is of either moderate or low effectiveness, an optimal regimen is superior to a schedule that delivers all of the drug at the beginning of the treatment. This study questions whether the current method for the administration of chemotherapy is optimal and suggests that alternative regimens should be considered.     

Targeting apoptosis by hydroxymethylacylfulvene in combination with gamma radiation in prostate tumor cells

Hydroxymethylacylfulvene (HMAF) is a novel agent with alkylating activity and is a potent inducer of apoptosis that is currently undergoing Phase II clinical trials for prostate cancer. This study explored the pro-apoptosis and anti-proliferative potential of HMAF in combination with gamma radiation in human prostate tumor cell lines. Apoptosis was assessed based on the generation of fragmented DNA, a terminal transferase flow cytometry assay, and cell morphology. In each of the tumor cell lines examined, radiation alone induced a marginal level of apoptosis, even after a prolonged 48-h incubation after exposure. In contrast, HMAF alone was a potent inducer of apoptosis in prostate tumor cells but not in normal cells. Marked levels of apoptosis in tumor cells were also observed for the combination of HMAF with gamma radiation. When drug treatment preceded irradiation, at least additive levels of apoptosis were observed in both androgen-responsive and androgen-independent cells. The combined treatment with ionizing radiation and HMAF reduced the radiation dose needed for the same level of clonogenic survival up to 2.5-fold. The potentiation of apoptosis and reduction in the clonogenic survival of tumor cells occurred at HMAF concentrations lower than that which reduced survival to 10% and at doses up to 6 Gy. No potentiation of apoptosis or clonogenic inhibition was noted in normal cells. These results suggest that the combination of HMAF with gamma radiation may have clinical utility for treatments of prostate cancer.     

T and B lymphocyte populations of tumor-bearing mice treated with 1,3-bis(chloroethyl)-1-nitrosourea (BCNU) or pyran

Summary The immunostimulator pyran copolymer and the antitumor agent 1,3-bis(2-chloroethyl)-1-nitrosourea (BCNU) have been studied to determine their effect on T and B lymphocytes from BALB/c mice bearing a syngeneic tumor, Madison lung 109 carcinoma. Indirect immunofluorescent and mitogenic assays revealed that the tumor-bearing controls usually had T and B cell populations lower than those observed for the normal controls. The tumor-bearing group treated with BCNU generally had T and B cell levels lower than the tumor controls. The administration of pyran resulted in an increase of T cells and a temporary increase in the B cell population. Although this increase in the B cell population of tumor-bearing mice treated with pyran is higher than that seen for the untreated tumor controls it was never higher than in the normal control mice. The experimental results indicate that pyran appears to reconstitute the depressed T cell population and has a transitory effect on the B cell population resulting from tumor burden.     

Design of optimal cancer chemotherapy using a continuous-time state model of cell kinetics

A continuous bilinear model in state space is used to describe the cell kinetics of a tumor-cell population under the effects of chemotherapy. Firstly, the time-course behavior of a Chinese-hamster-ovary (CHO) cell population is simulated to demonstrate the utility of the model. Then, an optimal strategy for cancer treatment is derived, based on the need to balance the effects on both cancerous and normal tissues. The performance index minimized is the sum of the weighted tumor population and the weighted total drug dosage. The optimization problem has resulted in a two-point boundary-value problem (TPBVP) with a bang-bang control policy, which is solved by the switching-time variation method (STVM). Computer simulation of CHO cells is also carried out as a numerical example of determining optimal cancer chemotherapy.     

Vascular tumor growth and treatment: Consequences of polyclonality,competition and dynamic vascular support

 A mathematical model is presented to describe the evolution of a vascular tumor in response to traditional chemotherapeutic treatment. Particular attention is paid to the effects of a dynamic vascular support system in a tumor comprised of competing cell populations that differ in proliferation rates and drug susceptibility. The model consists of a system of partial differential equations governing intratumoral drug concentration, cancer cell density, and blood vessel density. The balance between cell proliferation and death along with vessel production and destruction within the tumor generates a velocity field which drives the expansion or regression of the neoplasm. Radially symmetric solutions are obtained for the case when only one cell type is present and when the proportion of the tumor occupied by blood vessels remains constant. The stability of these solutions to asymmetric perturbations and to a small semi-drug resistant cell population is then investigated. The analysis shows that drug concentrations which are sufficient to insure eradication of a spherical tumor may be inadequate for the successful treatment of non-spherical tumors. When the drug is continuously infused, linear analysis predicts that whether or not a cure is possible is crucially dependent on the proliferation rate of the semi-resistant cells and on the competitive effect of the sensitive cells on the resistant population. When the blood vessel density is allowed to change dynamically, the model predicts a dramatic increase in the tumors growth and decrease in its response to therapy. Received: 4 August 2000 / Revised version: 13 July 2001 / Published online: 21 February 2002     

Cancer stem cells and escape from drug-induced premature senescence in human lung tumor cells: Implications for drug resistance and in vitro drug screening models

In this study, using an in vitro human tumor model, we show that non-small lung adenocarcinoma A549 cells after treatment with DNA damaging antitumor drugs become permanently growth-arrested as a result of so-called drug-induced premature senescence (pseudo-senescence). However, a small fraction of drug-treated cells escapes pseudo-senescence that leads to re-growth of tumor cell population after drug treatment. We show that this re-growth is associated with the presence of cancer stem cells (CSCs) in lung tumor cell population. We also document that re-growth of CSCs can be greatly delayed if lung tumor cells are treated with drug/caffeine combination that leads to the inhibition of the ATM/ATR pathway and decreased phosphorylation of PKB/Akt at Ser473. We show that in non-treated A549 cells caffeine by itself induces a reversible growth arrest that is associated with increased fraction of so-called side population cells, containing CSCs. These results point to the existence of an unknown, caffeine-sensitive mechanism that controls the number of CSCs in lung tumor cell population. Full characterization of this mechanism may lead to the development of innovative cancer therapies which are based on small molecular weight inhibitors of CSC differentiation and self-renewal, that mimic caffeine action. Our results have also important implications for drug screening tumor models in vitro.     

Drug scheduling of cancer chemotherapy based on natural actor-critic approach

Recently, reinforcement learning methods have drawn significant interests in the area of artificial intelligence, and have been successfully applied to various decision-making problems. In this paper, we study the applicability of the NAC (natural actor-critic) approach, a state-of-the-art reinforcement learning method, to the drug scheduling of cancer chemotherapy for an ODE (ordinary differential equation)-based tumor growth model. ODE-based cancer dynamics modeling is an active research area, and many different mathematical models have been proposed. Among these, we use the model proposed by de Pillis and Radunskaya (2003), which considers the growth of tumor cells and their interaction with normal cells and immune cells. The NAC approach is applied to this ODE model with the goal of minimizing the tumor cell population and the drug amount while maintaining the adequate population levels of normal cells and immune cells. In the framework of the NAC approach, the drug dose is regarded as the control input, and the reward signal is defined as a function of the control input and the cell populations of tumor cells, normal cells, and immune cells. According to the control policy found by the NAC approach, effective drug scheduling in cancer chemotherapy for the considered scenarios has turned out to be close to the strategy of continuing drug injection from the beginning until an appropriate time. Also, simulation results showed that the NAC approach can yield better performance than conventional pulsed chemotherapy.     

Suppression of antitumor immunity by macrophages in spleens of mice bearing a large MOPC-315 tumor

We had shown previously that progression of MOPC-315 plasmacytoma growth is associated with an increase in the percentage of macrophages in the spleen as well as a decrease in the ability of tumor-bearer spleen cells to mount an antitumor cytotoxic response upon in vitro immunization. Here we provide evidence that macrophages in the MOPC-315 tumor-bearer spleen are responsible at least in part for the suppression of the generation of antitumor cytotoxicity. Accordingly, removal of most macrophages by depletion of phagocytic cells or Sephadex G-10-adherent cells from spleens of mice bearing a large tumor resulted in augmented antitumor immune potential. Also, Sephadex G-10-adherent spleen cells from tumor-bearing (but not normal) mice drastically suppressed the in vitro generation of antitumor cytotoxicity by normal spleen cells. The suppressive activity of these adherent cells did not reside in contaminating suppressor T cells, since it was not reduced by treatment with monoclonal anti-Thy 1.2 antibody plus complement. The Sephadex G-10-adherent cell population from the tumor-bearer spleen suppressed the in vitro generation of antitumor cytotoxicity against autochthonous tumor cells but not against allogeneic EL4 tumor cells, and hence the suppression was apparently specific. The suppressive activity of the Sephadex G-10-adherent cell population from tumor-bearer spleens was overcome by treatment of the tumor-bearing mice with a low curative dose of cyclophosphamide. This immunomodulatory effect of a low dose of the drug in overcoming the suppression mediated by the Sephadex G-10-adherent cell population enables the effector arm of the immune system of tumor-bearing mice to cooperate effectively with the drug's tumoricidal activity in tumor eradication.     

The dose-dependent cytogenetic and growth-inhibiting effects of a new antitumor preparation from the nitrosourea group

Kinetics of growth-inhibiting and cytogenetic effect of a new carbon-substituted nitrosourea derivative (ADEKO) has been studied in a wide dose range. A linear dose-effect dependence was observed. The level of damaged cells in a population is connected with a number of chromosomal aberrations per cells with a semilogarithmic dependence. The drug has a pronounced clastogenic effect that reveals itself in total damaging of chromosomal structure of tumor cells. It also causes cell polyploidization with the increase in does and duration of action. Chromosomal aberrations induced by the drug are observed in the tumor long after the action of the drug and their level correlates positively with antitumor activity of ADEKO. ADEKO damages preferentially tumor cells as compared to bone marrow.     

Isocorydine Targets the Drug-Resistant Cellular Side Population through PDCD4-Related Apoptosis in Hepatocellular Carcinoma

Isocorydine (ICD), an anticancer agent under current evaluation, decreased the percentage of side population (SP) cells significantly in hepatocellular carcinoma (HCC) cell lines. ICD treatment sensitized cancer cells to doxorubicin (DXR), a conventional clinical chemotherapeutic drug for HCC. We found that ICD decreased the percentage of SP cells in HCC cell lines by preferentially killing SP cells. In the early stage of treatment, ICD inhibited SP cell growth by arresting cells in G2/M; later, it induced apoptosis. Our xenograft model confirmed that ICD selectively reduced the size and weight of SP-induced tumor masses in vivo. Furthermore, it was found that programmed cell death 4 (PDCD4), a tumor suppressor gene, was relatively low when expressed in SP cells compared with non-SP cells, and its expression level was remarkably elevated when cells were treated with ICD. Taken together, these data suggest that ICD is a drug that may target the SP cells of HCC.     

Conflicting objectives in chemotherapy with drug resistance

A system of differential equations for the control of tumor cells growth in a cycle nonspecific chemotherapy is presented. Spontaneously acquired drug resistance is accounted for, as well as the evolution in time of normal cells. In addition, optimization of conflicting objectives forms the aim of the chemotherapeutic treatment. For general cell growth, some results are given, whereas for the special case of Malthusian (exponential) growth of tumor cells and rather general growth rate for normal cells, the optimal strategy is worked out. The latter, from the clinical standpoint, corresponds to maximum drug concentration throughout the treatment.     

Spatial Heterogeneity in Drug Concentrations Can Facilitate the Emergence of Resistance to Cancer Therapy

Acquired resistance is one of the major barriers to successful cancer therapy. The development of resistance is commonly attributed to genetic heterogeneity. However, heterogeneity of drug penetration of the tumor microenvironment both on the microscopic level within solid tumors as well as on the macroscopic level across metastases may also contribute to acquired drug resistance. Here we use mathematical models to investigate the effect of drug heterogeneity on the probability of escape from treatment and the time to resistance. Specifically we address scenarios with sufficiently potent therapies that suppress growth of all preexisting genetic variants in the compartment with the highest possible drug concentration. To study the joint effect of drug heterogeneity, growth rate, and evolution of resistance, we analyze a multi-type stochastic branching process describing growth of cancer cells in multiple compartments with different drug concentrations and limited migration between compartments. We show that resistance is likely to arise first in the sanctuary compartment with poor drug penetrations and from there populate non-sanctuary compartments with high drug concentrations. Moreover, we show that only below a threshold rate of cell migration does spatial heterogeneity accelerate resistance evolution, otherwise deterring drug resistance with excessively high migration rates. Our results provide new insights into understanding why cancers tend to quickly become resistant, and that cell migration and the presence of sanctuary sites with little drug exposure are essential to this end.     

Optimization of the exposure of normal and transformed cell populations to phase-specific agents

Dynamic behavior of stem cells population of the "critical" tissue (normal population) and tumor cell population under periodic treatment with a phase-specific cytotoxic agent was considered. The results were used for optimization of anticancer chemotherapy. The schedules of treatment were found which provide a maximum rate of tumor-cell elimination for any given rate of the normal population size decrease. If the mean generation times of normal and tumor populations differ (which was stated for many tumors), usage of the optimal period markedly increases the selectivity of therapy, while application of other periods can result in selective elimination of the normal population. Problems concerned with practical realization of the proposed regimes are discussed.     

Vitamin C Effect on Mitoxantrone-Induced Cytotoxicity in Human Breast Cancer Cell Lines

In recent years the use of natural dietary antioxidants to minimize the cytotoxicity and the damage induced in normal tissues by antitumor agents is gaining consideration. In literature, it is reported that vitamin C exhibits some degree of antineoplastic activity whereas Mitoxantrone (MTZ) is a synthetic anti-cancer drug with significant clinical effectiveness in the treatment of human malignancies but with severe side effects. Therefore, we have investigated the effect of vitamin C alone or combined with MTZ on MDA-MB231 and MCF7 human breast cancer cell lines to analyze their dose-effect on the tumor cellular growth, cellular death, cell cycle and cell signaling. Our results have evidenced that there is a dose-dependence on the inhibition of the breast carcinoma cell lines, MCF7 and MDA-MB231, treated with vitamin C and MTZ. Moreover, their combination induces: i) a cytotoxic effect by apoptotic death, ii) a mild G2/M elongation and iii) H2AX and mild PI3K activation. Hence, the formulation of vitamin C with MTZ induces a higher cytotoxicity level on tumor cells compared to a disjointed treatment. We have also found that the vitamin C enhances the MTZ effect allowing the utilization of lower chemotherapic concentrations in comparison to the single treatments.     

Suppression of antitumor immunity by macrophages in spleens of mice bearing a large MOPC-315 tumor

Summary We had shown previously that progression of MOPC-315 plasmacytoma growth is associated with an increase in the percentage of macrophages in the spleen as well as a decrease in the ability of tumor-bearer spleen cells to mount an antitumor cytotoxic response upon in vitro immunization. Here we provide evidence that macrophages in the MOPC-315 tumor-bearer spleen are responsible at least in part for the suppression of the generation of antitumor cytotoxicity. Accordingly, removal of most macrophages by depletion of phagocytic cells or Sephadex G-10-adherent cells from spleens of mice bearing a large tumor resulted in augmented antitumor immune potential. Also, Sephadex G-10-adherent spleen cells from tumor-bearing (but not normal) mice drastically suppressed the in vitro generation of antitumor cytotoxicity by normal spleen cells. The suppressive activity of these adherent cells did not reside in contaminating suppressor T cells, since it was not reduced by treatment with monoclonal anti-Thy 1.2 antibody plus complement. The Sephadex G-10-adherent cell population from the tumor-bearer spleen suppressed the in vitro generation of antitumor cytotoxicity against autochthonous tumor cells but not against allogeneic EL4 tumor cells, and hence the suppression was apparently specific. The suppressive activity of the Sephadex G-10-adherent cell population from tumor-bearer spleens was overcome by treatment of the tumor-bearing mice with a low curative dose of cyclophosphamide. This immunomodulatory effect of a low dose of the drug in overcoming the suppression mediated by the Sephadex G-10-adherent cell population enables the effector arm of the immune system of tumor-bearing mice to cooperate effectively with the drug's tumoricidal activity in tumor eradication.This paper was presented in part at the annual meeting of the American Association of Immunologists, Chicago, Illinois, 10–15 April 1983     

Evolution of Pre-Existing versus Acquired Resistance to Platinum Drugs and PARP Inhibitors in BRCA-Associated Cancers

Platinum drugs and PARP inhibitors (“PARPis”) are considered to be effective in BRCA-associated cancers with impaired DNA repair. These agents cause stalled and collapsed replication forks and create double-strand breaks effectively in the absence of repair mechanisms, resulting in arrest of the cell cycle and induction of cell death. However, recent studies have shown failure of these chemotherapeutic agents due to emerging drug resistance. In this study, we developed a stochastic model of BRCA-associated cancer progression in which there are four cancer populations: those with (i) functional BRCA, (ii) dysfunctional BRCA, (iii) functional BRCA and a growth advantage, and (iv) dysfunctional BRCA and a growth advantage. These four cancer populations expand from one cancer cell with normal repair function until the total cell number reaches a detectable amount. We derived formulas for the probability and expected numbers of each population at the time of detection. Furthermore, we extended the model to consider the tumor dynamics during treatment. Results from the model were validated and showed good agreement with clinical and experimental evidence in BRCA-associated cancers. Based on the model, we investigated conditions in which drug resistance during the treatment course originated from either a pre-existing drug-resistant population or a de novo population, due to secondary mutations. Finally, we found that platinum drugs and PARPis were effective if (i) BRCA inactivation is present, (ii) the cancer was diagnosed early, and (iii) tumor growth is rapid. Our results indicate that different types of cancers have a preferential way of acquiring resistance to platinum drugs and PARPis according to their growth and mutational characteristics.     

When cytoprotective autophagy isn’t… and even when it is

Multiple papers have been published that have identified and/or characterized the cytoprotective function of autophagy, primarily in tumor cells exposed to chemotherapy or radiation. These studies have relied on pharmacological and/or genetic interference with autophagy to establish its protective function, often primarily by demonstrating that cells in which autophagy has been suppressed undergo increased apoptosis. The purpose of this Editor’s Corner is to emphasize that these approaches, while absolutely necessary, are of themselves insufficient to support the conclusion that autophagy is cytoprotective in a given experimental tumor line exposed to a particular agent; complementary studies are required that demonstrate that autophagy inhibition sensitizes the tumor cell to the autophagy-inducing treatment. Otherwise, autophagy may be responsible for the growth arrest and/or cell death that is observed with the drug or radiation treatment alone, and autophagy inhibition may simply be converting one form of growth inhibition/cell death to an alternative pathway that achieves the same end result in terms of sensitivity to the treatment.     

Proliferative and anti-proliferative effects of thymosin α1 on cells are associated with manipulation of cellular ROS levels

Reactive oxygen species (ROS) are constantly generated and eliminated in the biological system and play important roles in a variety of physiological and pathological processes. Previous studies indicate that modulation of cellular ROS affects cell proliferation. Thymosin alpha 1 (Tα1) is a naturally occurring thymic peptide and has previously been shown to be a potential therapy for some immunodeficiencies, malignancies, and infections. However, few reports have focused on manipulation of cellular ROS level effects of Tα1. In this study, the Tα1-treated leukomonocytes, which were isolated from mice spleens, exhibited a higher ROS level and a lower reduced glutathione (GSH) level; however, HepG2 cells treated with Tα1 exhibited lower ROS level and higher GSH level. In addition, after treatment with Tα1, the population of leukomonocytes in the G₂ phase increased, resulting in a slight increase in viability. However, in Tα1-treated HepG2 cells, the cell cycle was delayed in the G₁ phase, thereby inhibiting tumor cell proliferation; in addition, dephosphorylation of the serine/threonine kinase Akt was detected. In conclusion, we show that Tα1 has potent anti-proliferative activity against malignant human hepatoma cells and proliferative activity against leukomonocytes associated with manipulation of oxidative stress levels which indicates the potential of Tα1 as an antitumor drug.