Optimal control for selected cancer chemotherapy ODE models: a view on the potential of optimal schedules and choice of objective function

In this article, four different mathematical models of chemotherapy from the literature are investigated with respect to optimal control of drug treatment schedules. The various models are based on two different sets of ordinary differential equations and contain either chemotherapy, immunotherapy, anti-angiogenic therapy or combinations of these. Optimal control problem formulations based on these models are proposed, discussed and compared. For different parameter sets, scenarios, and objective functions optimal control problems are solved numerically with Bock’s direct multiple shooting method.In particular, we show that an optimally controlled therapy can be the reason for the difference between a growing and a totally vanishing tumor in comparison to standard treatment schemes and untreated or wrongly treated tumors. Furthermore, we compare different objective functions. Eventually, we propose an optimization-driven indicator for the potential gain of optimal controls. Based on this indicator, we show that there is a high potential for optimization of chemotherapy schedules, although the currently available models are not yet appropriate for transferring the optimal therapies into medical practice due to patient-, cancer-, and therapy-specific components.     

The optimal dose of CAF regimen in adjuvant chemotherapy for breast cancer patients at stage IIB

This paper deals with the idea of balancing drug effects on tumor and normal cell populations based on a variety of criteria, which is evaluated by the oncologist for breast cancer patients at stage IIB. In this paper, the optimal controller represents the optimal drug dosage of CAF (Cyclophosphamide, Adriamycin and Fluorouracil) regimen in adjuvant chemotherapy after surgery for these patients. We determined the doses of CAF regimen by minimizing a cost function with some constraints. The cost function includes the cancer cell and the normal cell growth dynamics with prescribed weighting coefficients for each patient. The physician determines these weighting coefficients based on some individual parameters. The optimal treatment schedules are computed based on a trade-off between the cancer cell reduction and the normal cell preserving. Numerical simulations are given to illustrate the accuracy of the optimal controller.     

Microvascular endothelial cell heterogeneity: general concepts and pharmacological consequences for anti-angiogenic therapy of cancer

Microvascular endothelial cells display a large degree of heterogeneity in function depending on their location in the vascular tree. The existence of organ-specific, microvascular-bed-specific, and even intravascular variations in endothelial cell gene expression emphasizes their high cell-to-cell variability, which is furthermore extremely adaptable to altering conditions. The ability of microvascular endothelial cells to respond dynamically to pathology-related microenvironmental changes is particularly apparent in tumor-growth-associated angiogenesis. An understanding of how they behave, how their behavior varies between and within tumors, and how their behavior is related to responsiveness to drugs is critical for the development of effective anti-angiogenic treatment strategies. In this review, we introduce some general issues concerning organ-imprinted microvascular heterogeneity and highlight the importance of studying microvascular endothelial cell behavior in an in vivo context. This is followed by an overview of state-of-the-art knowledge regarding the nature of the variation in microenvironmental conditions in pre-clinical and clinical tumors and their consequences for tumor endothelial behavior. We provide recent insights into the in vivo molecular activation status of the endothelium and, finally, outline our current understanding of the way that anti-angiogenic drugs affect tumor endothelial cells in relation to their anti-tumor effects.     

Theoretical analysis of conjugate localization in two-step cancer chemotherapy

 Considerable research has been aimed at improving the efficacy of chemotherapeutic agents for cancer therapy. A promising two-step approach that is designed to minimize systemic drug toxicity while maximizing activity in tumors employs monoclonal antibody-enzyme conjugates for the activation of anti-cancer prodrugs. A mathematical model based on the biology of human 3677 melanoma xenografts in nude mice is presented, analyzed, and numerically simulated to study the biodistribution, pharmacokinetics, and intratumoral localization properties of L49-β-lactamase fusion proteins in solid tumor masses. The model predictions were compared with published experimental data and an excellent correlation was found to exist. Analytic expressions for the total concentration of conjugate in the tumor, the time at which the concentration is maximal, and the half life of conjugate in the tissue were derived. From these results, key parameters were isolated; and the effects of the tumor vasculature, binding kinetics, and administration schedule were investigated. The antibody-antigen dissociation ratio, the conjugate permeability, and the inter-capillary half distance within the tumor mass were found to strongly influence localization and retention in the tumor. The model was used to examine various dosing strategies in an attempt to determine which regimen would provide the best biodistribution results. The results of administering a uniform dose of conjugate via bolus injection, multiple injections, and continuous infusion were compared. The model predicts that when saturation of binding sites does not occur, dosing strategy has little effect on the amount of conjugate that localizes in the tumor. Received: 20 April 1998 / Revised version: 12 October 1998     

Effects of ozone nano-bubble water on mucositis induced by cancer chemotherapy

No effective, reliable treatment for stomatitis associated with cancer therapy has been established. This study focused on the its effectiveness of ozone nano-bubble water (ONBW) for the treatment of chemotherapy-induced stomatitis. Oral mucositis was induced in 14-week-old male Sprague-Dawley rats (N = 21). The animals were randomly divided into 3 groups: 7 without treatment (control); 7 treated with physiological salt solution (saline); and 7 treated with ONBW. Animals were weighed on Days 7, 9, 11, and 16. Stomatitis grade evaluation and bacterial count measurements were performed before rinsing in all animals 3, 5, and 10 days after acetic acid irritation (Days 9, 11, and 17 respectively). Weight loss after stomatitis creation was observed in all groups, with significant differences between the control and ONBW groups and between the saline and ONBW groups on Day 16. The stomatitis grade did not worsen during the experimental period in any group, with the lowest grades in the ONBW group on Days 11 and 16. Significant differences were identified between the control and ONBW groups and between the saline and ONBW groups on Days 11 and 16. Oral bacterial counts tended to decrease over time in all three groups, with the greatest decrease in the ONBW group, followed by the saline group. The decrease in the bacterial count was steepest in the ONBW group. Rinsing out the oral cavity with ONBW decreased bacterial counts and encouraged the healing of oral chemotherapy-induced stomatitis. ONBW may be an effective treatment for chemotherapy-induced stomatitis.     

The role of neoadjuvant and adjuvant chemotherapy regimens consisting of different combinations of drugs in the treatment of advanced oral cancer

We performed a retrospective analysis on the effect of neoadjuvant chemotherapy with three cycles of methotrexate (100 mg/m² on day 1), cisplatin (90 mg/m² on day 1) and bleomycin (20 mg/m² on day 1–5) with 21 d gap between each cycle in 44 patients with advanced squamous cell carcinoma of the cheek, lip and tongue followed by surgery and adjuvant chemotherapy consisting of cisplatin (90 mg/m² on day 1), Mitomycin C (6 mg/m² on day 1) and 5-fluorouracil (1000 mg/m² 120 h continuous infusion from day 1) repeated every 3 weeks for three cycles. Following induction chemotherapy, complete response was observed in 11 out of 44 patients (25%), and a partial response in a further 28 patients (64%). The overall median survival of all patients was 29 months and those in stage III and stage IV were 30 and 15 months respectively (P<0.001). The median duration of the time to relapse in patients who responded to adjuvant chemotherapy was 28 months. The main toxic effect was vomiting followed by hematological toxicity. No treatment-related deaths occurred. The regimen showed a significant response, encouraging median survival and a good tolerability profile.     

Epithelial cell plasticity defines heterogeneity in lung cancer

Lung cancer is the leading cause of cancer death for both men and women and accounts for almost 18.4% of all deaths due to cancer worldwide, with the global incidence increasing by approximately 0.5% per year. Lung cancer is regarded as a devastating type of cancer owing to its high prevalence, reduction in the health-related quality of life, frequently delayed diagnosis, low response rate, high toxicity, and resistance to available therapeutic options. The highly heterogeneous nature of this cancer with a proximal-to-distal distribution throughout the respiratory tract dramatically affects its diagnostic and therapeutic management. The diverse composition and plasticity of lung epithelial cells across the respiratory tract are regarded as significant factors underlying lung cancer heterogeneity. Therefore, definitions of the cells of origin for different types of lung cancer are urgently needed to understand lung cancer biology and to achieve early diagnosis and develop cell-targeted therapies. In the present review, we will discuss the current understanding of the cellular and molecular alterations in distinct lung epithelial cells that result in each type of lung cancer.     

The effect of chemotherapy on Babesia bigemina in the tick vector Boophilus microplus

Percentages of feeding ticks in which Babesia bigemina could be detected (infection rates) were determined following treatment of bovine hosts with each of four babesicides. Infection rates were suppressed by imidocarb dipropionate, quinuronium sulphate and amicarbalide, reaching minimum levels 3–4 days after treatment, but imidocarb dihydrochloride had comparatively little effect. Total elimination of the parasite from ticks was not achieved. Treatment of tick infested hosts with imidocarb dipropionate or quinuronium sulphate failed to prevent transmission of B. bigemina by transovarian passage or by transfer of adult male ticks. These findings indicate that the use of babesicides for chemotherapy is unlikely to have a significant effect on the rate of transmission of B. bigemina.     

The role of monoclonal antibody A7 as a drug modifier in cancer therapy

An anticancer antibiotic, neocarzinostatin (NCS), was covalently conjugated to the murine monoclonal antibody A7 (mAb A7), which recognizes the glycoprotein on the cell surface of human colon cancer. The biological and pharmacological properties of the conjugate (A7-NCS) were examined and compared with those of unconjugated NCS. A7-NCS exhibited a strong binding and cytotoxicity to the cell and an antigen-specific tumor accumulation. Significant tumoricidal effects in vivo were observed in the antigen-positive tumor-bearing mice treated with A7-NCS, whereas NCS mixed with mAb A7 and NCS alone were relatively ineffective. In the antigennegative tumor, the tumoricidal effect of A7-NCS was lower than in the antigen-positive tumor. The NCS concentration in blood and tumor were significantly elevated by conjugation to mAb A7. The NCS localization in tumor was higher in the antigen-positive tumor than in the antigen-negative tumor. Death due to acute toxicity was observed at a dose of 20 units (U) NCS in mice treated with unconjugated NCS, whereas toxicity was seen with a much higher dose of NCS (100 U) if the drug was conjugated to the mAb. These findings show that mAb A7 confers more favorable pharmacological properties on an anticancer drug, making it potentially more useful for cancer chemotherapy.This work has been supported in part by a grant-in-aid for cancer research from the Ministry of Education and for the comprehensive 10-year strategy for cancer control from the Ministry of Health and Welfare, Japan     

Role of optimal control theory in cancer chemotherapy

This paper presents a review of the ways in which optimal control theory interacts with cancer chemotherapy. There are three broad areas of investigation. One involves miscellaneous growth kinetic models, the second involves cell cycle models, and the third is a classification of "other models." Both normal and tumor cell populations are included in a number of the models. The concepts of deterministic optimal control theory are applied to each model in such a way as to present a cohesive picture. There are applications to both experimental and clinical tumors. Suggestions for designing better chemotherapy strategies are presented.     

Cell sorting in cancer research—Diminishing degree of cell heterogeneity

Increasing evidence of intratumor heterogeneity and its augmentation due to selective pressure of microenvironment and recent achievements in cancer therapeutics lead to the need to investigate and track the tumor subclonal structure. Cell sorting of heterogeneous subpopulations of tumor and tumor-associated cells has been a long established strategy in cancer research. Advancement in lasers, computer technology and optics has led to a new generation of flow cytometers and cell sorters capable of high-speed processing of single cell suspensions. Over the last several years cell sorting was used in combination with molecular biological methods, imaging and proteomics to characterize primary and metastatic cancer cell populations, minimal residual disease and single tumor cells. It was the principal method for identification and characterization of cancer stem cells. Analysis of single cancer cells may improve early detection of tumors, monitoring of circulating tumor cells, evaluation of intratumor heterogeneity and chemotherapeutic treatments. The aim of this review is to provide an overview of major cell sorting applications and approaches with new prospective developments such as microfluidics and microchip technologies.     

Cancer immunotherapy, mathematical modeling and optimal control

Clinical immunologists, among other problems, routinely face a question: what is the best time and dose for a certain therapeutic agent to be administered to the patient in order to decrease/eradicate the pathological condition? In cancer immunotherapies the therapeutic agent is something able to elicit an immune response against cancer. The immune response has its own dynamics that depends on the immunogenicity of the therapeutic agent and on the duration of the immune response. The question then is "how can we decide when and how much of the drug to inject so to have a prolonged and effective immune response to the cancer?". This question can be addressed in mathematical terms in two stages: first one construct a mathematical model describing the cancer-immune interaction and secondly one applies the theory of optimal control to determine when and to which extent to stimulate the immune system by means of an immunotherapeutic agent administered in discrete variable doses within the therapeutic period. The solution of this mathematical problem is described and discussed in this article. We show that the method employed can be applied to find the optimal protocol in a variety of clinical problems where the kinetics of the drug or treatment and its influence on the physiologic/pathologic functions have been described by a system of ordinary differential equations.     

Multiscale heterogeneity in filamentous microbes

Microbial cells within clonal populations can display different morphologies or carry out different tasks. This heterogeneity is beneficial at the population level and allows microbes to spread risk or separate incompatible activities. Heterogeneity is also evident in filamentous bacteria and fungi, which form mycelial networks consisting of interconnected hyphae. Here, heterogeneity is observed between clonal mycelial particles, between different zones of colonies, between adjacent hyphae and even between adjacent compartments of individual hyphae. In this review, we compare this multiscale heterogeneity in filamentous bacteria and fungi and discuss the underlying mechanisms. These mechanisms might provide targets to improve the exploitability of these organisms as cell factories in the biotech sector.     

Influence of trial duration on the bias of the estimated treatment effect in clinical trials when individual heterogeneity is ignored

In randomized clinical trials where the times to event of two treatment groups are compared under a proportional hazards assumption, it has been established that omitting prognostic factors from the model entails an underestimation of the hazards ratio. Heterogeneity due to unobserved covariates in cancer patient populations is a concern since genomic investigations have revealed molecular and clinical heterogeneity in these populations. In HIV prevention trials, heterogeneity is unavoidable and has been shown to decrease the treatment effect over time. This article assesses the influence of trial duration on the bias of the estimated hazards ratio resulting from omitting covariates from the Cox analysis. The true model is defined by including an unobserved random frailty term in the individual hazard that reflects the omitted covariate. Three frailty distributions are investigated: gamma, log‐normal, and binary, and the asymptotic bias of the hazards ratio estimator is calculated. We show that the attenuation of the treatment effect resulting from unobserved heterogeneity strongly increases with trial duration, especially for continuous frailties that are likely to reflect omitted covariates, as they are often encountered in practice. The possibility of interpreting the long‐term decrease in treatment effects as a bias induced by heterogeneity and trial duration is illustrated by a trial in oncology where adjuvant chemotherapy in stage 1B NSCLC was investigated.     

Response of Potamogeton crispus root characteristics to sediment heterogeneity

Plant growth, biomass allocation, root distribution and plant nutrient content were investigated in the submerged macrophyte Potamogeton crispus growing in heterogeneous sediments. Three experimental sediments heterogeneous in nutrient content and phosphorus release capacity were used: sandy loam with low nutrient content (A), clay with intermediate nutrient content (B), and clay with high nutrient content (C). Biomass accumulation was significantly affected by the sediment type, and was highest in clay C (1.23 mg per plant dry weight) but lowest in sandy loam (0.69 mg per plant dry weight). The root:shoot ratios in treatments A, B and C were 0.30, 0.14 and 0.09, respectively. P. crispus allocated more biomass to roots in sandy loam compared with the other sediments. The average root numbers in sediments A, B and C were 16, 19 and 20, respectively, and the total root lengths in sediments A, B and C were 238.84, 200.36 and 187.21 cm, respectively. Almost 90% of the root biomass was distributed in the 0–15 cm depth in sediments B and C, compared with 64.53% in sediment A. The rank order of plant nitrogen and phosphorus concentrations in the sediment types was C > B > A. These results indicate that both sediment structure and nutrient availability influence the growth and distribution of the root system of P. crispus.