Bone metastasis treatment modeling via optimal control

Mathematical modeling of brain metastases growth and response to therapies: A review

Brain metastases (BMs) are the most common intracranial tumor type and a significant health concern, affecting approximately 10% to 30% of all oncological patients. Although significant progress is being made, many aspects of the metastatic process to the brain and the growth of the resulting lesions are still not well understood. There is a need for an improved understanding of the growth dynamics and the response to treatment of these tumors. Mathematical models have been proven valuable for drawing inferences and making predictions in different fields of cancer research, but few mathematical works have considered BMs. This comprehensive review aims to establish a unified platform and contribute to fostering emerging efforts dedicated to enhancing our mathematical understanding of this intricate and challenging disease. We focus on the progress made in the initial stages of mathematical modeling research regarding BMs and the significant insights gained from such studies. We also explore the vital role of mathematical modeling in predicting treatment outcomes and enhancing the quality of clinical decision-making for patients facing BMs.

Development of a precision tumor bone metastasis model by a magnetic micro-living-motor system

An ideal bone metastasis animal model is critical and fundamental for mechanistic research and following development of new drug and treatment. Caudal artery (CA) injection allows bone metastasis in the hindlimb, while in-depth targeted and quantitative studies of bone metastasis require a new model to overcome its limitations. Here, we developed a targeted, quantitative, and highly consistent method for the modeling of bone metastasis with cell-based magnetic micro-living-motor (MLM) system created by effectively combining Fe3O4-PDA-Au with biosafety. The MLM system can achieve efficient migration, target site colonization and control tumorigenesis in bone precisely with the application of a magnetic field. In vivo, day 3 post cell injection, tumor bone metastasis signals were observed locally in the injected femur among 82.76% mice of the MLM group as compared to the 56.82% in the CA group, and the signal intensity was 45.1 and 95.9 times stronger than that in the left and right lower limbs of the CA group, respectively. Post-injection day 28, metastasis in vital organs was reduced by approximately 90% in the MLM group compared to the CA group. Our innovative use of the MLM system in the field of tumor modeling opens a new avenue for exploring the mechanisms of tumor bone metastasis, recurrence and drug resistance.

Reclassified the phenotypes of cancer types and construct a nomogram for predicting bone metastasis risk: A pan-cancer analysis

BACKGROUND

Numerous of models have been developed to predict the bone metastasis (BM) risk; however, due to the variety of cancer types, it is difficult for clinicians to use these models efficiently. We aimed to perform the pan-cancer analysis to create the cancer classification system for BM, and construct the nomogram for predicting the BM risk.

METHODS

Cancer patients diagnosed between 2010 and 2018 in the Surveillance, Epidemiology, and End Results (SEER) database were included. Unsupervised hierarchical clustering analysis was performed to create the BM prevalence-based cancer classification system (BM-CCS). Multivariable logistic regression was applied to investigate the possible associated factors for BM and construct a nomogram for BM risk prediction. The patients diagnosed between 2017 and 2018 were selected for validating the performance of the BM-CCS and the nomogram, respectively.

RESULTS

A total of 50 cancer types with 2,438,680 patients were included in the construction model. Unsupervised hierarchical clustering analysis classified the 50 cancer types into three main phenotypes, namely, categories A, B, and C. The pooled BM prevalence in category A (17.7%; 95% CI: 17.5%-17.8%) was significantly higher than that in category B (5.0%; 95% CI: 4.5%-5.6%), and category C (1.2%; 95% CI: 1.1%-1.4%) (p < 0.001). Advanced age, male gender, race, poorly differentiated grade, higher T, N stage, and brain, lung, liver metastasis were significantly associated with BM risk, but the results were not consistent across all cancers. Based on these factors and BM-CCS, we constructed a nomogram for predicting the BM risk. The nomogram showed good calibration and discrimination ability (AUC in validation cohort = 88%,95% CI: 87.4%-88.5%; AUC in construction cohort = 86.9%,95% CI: 86.8%-87.1%). The decision curve analysis also demonstrated the clinical usefulness.

CONCLUSION

The classification system and prediction nomogram may guide the cancer management and individualized BM screening, thus allocating the medical resources to cancer patients. Moreover, it may also have important implications for studying the etiology of BM.

Radiation necrosis after radiation therapy treatment of brain metastases: A computational approach

Metastasis is the process through which cancer cells break away from a primary tumor, travel through the blood or lymph system, and form new tumors in distant tissues. One of the preferred sites for metastatic dissemination is the brain, affecting more than 20% of all cancer patients. This figure is increasing steadily due to improvements in treatments of primary tumors. Stereotactic radiosurgery (SRS) is one of the main treatment options for patients with a small or moderate number of brain metastases (BMs). A frequent adverse event of SRS is radiation necrosis (RN), an inflammatory condition caused by late normal tissue cell death. A major diagnostic problem is that RNs are difficult to distinguish from BM recurrences, due to their similarities on standard magnetic resonance images (MRIs). However, this distinction is key to choosing the best therapeutic approach since RNs resolve often without further interventions, while relapsing BMs may require open brain surgery. Recent research has shown that RNs have a faster growth dynamics than recurrent BMs, providing a way to differentiate the two entities, but no mechanistic explanation has been provided for those observations. In this study, computational frameworks were developed based on mathematical models of increasing complexity, providing mechanistic explanations for the differential growth dynamics of BMs relapse versus RN events and explaining the observed clinical phenomenology. Simulated tumor relapses were found to have growth exponents substantially smaller than the group in which there was inflammation due to damage induced by SRS to normal brain tissue adjacent to the BMs, thus leading to RN. ROC curves with the synthetic data had an optimal threshold that maximized the sensitivity and specificity values for a growth exponent β* = 1.05, very close to that observed in patient datasets.

Prevalence and factors associated with the occurrence of pathological fractures and their impact on the overall survival of patients with bone metastases under palliative care

OBJECTIVES

To assess the prevalence and factors associated with the occurrence of pathological fractures and their impact on the overall survival of patients with bone metastases under palliative care.

METHODS

An observational retrospective cohort assessment concerning both male and female patients with cancer presenting with bone metastases referred to a palliative care unit. Sociodemographic and clinical data were obtained before and after care unit referral. Patients were followed up until death or the last follow-up (4 years after referral). Logistic regression models and survival curves employing a log-rank test were applied.

RESULTS

A total of 348 patients were included in the study. Most were <65 years (65%) and female (62%), and the most frequent primary tumour site was the breast (40%). The prevalence of pathological fractures was 28%, more frequent in the axial skeleton (49%), with no association with overall patient survival (p=0.348). Patients with breast cancer exhibited a 2.96-fold higher chance (95% CI: 1.80 to 4.86) of developing a fracture compared with other tumours, and not receiving previous radiotherapy increased the chances of fracture occurrence by 5.60-fold (95% CI: 2.46 to 12.77).

CONCLUSION

A high prevalence of pathological fractures was observed. Presenting with breast cancer and not having undergone previous radiotherapy increase the chances of fracture occurrence, although this is not associated with overall survival in patients under palliative care.

Optimal treatment strategy of cancers with intratumor heterogeneity

Intratumor heterogeneity hinders the success of anti-cancer treatment due to the interaction between different types of cells. To recapitulate the communication of different types of cells, we developed a mathematical model to study the dynamic interaction between normal, drug-sensitive and drug-resistant cells in response to cancer treatment. Based on the proposed model, we first study the analytical conclusions, namely the nonnegativity and boundedness of solutions, and the existence and stability of steady states. Furthermore, to investigate the optimal treatment that minimizes both the cancer cells count and the total dose of drugs, we apply the Pontryagin's maximum(or minimum) principle (PMP) to explore the combination therapy strategy with either quadratic control or linear control functionals. We establish the existence and uniqueness of the quadratic control problem, and apply the forward-backward sweep method (FBSM) to solve the optimal control problems and obtain the optimal therapy scheme.

Antigen receptor therapy in bone metastasis via optimal control for different human life stages

In this work we propose a bone metastasis model using power law growth functions in order to describe the biochemical interactions between bone cells and cancer cells. Experimental studies indicate that bone remodeling cycles are different for human life stages: childhood, young adulthood, and adulthood. In order to include such differences in our study, we estimate the model parameter values for each human life stage via bifurcation analysis. Results reveal an intrinsic relationship between the active period of remodeling cycles and the proliferation of cancer cells. Subsequently, using optimal control theory we analyze a possible antigen receptor therapy as a new treatment for bone metastasis. Theoretical results such as existence of optimal solutions are proved. Numerical simulations for late stages of bone metastasis are presented and a discussion of our results is carried out.

Dynamic modeling of bone remodeling, osteolytic metastasis and PK/PD therapy: introducing variable order derivatives as a simplification technique

Bone is constantly being renewed: in the adult skeleton, bone resorption and formation are in a tightly coupled balance, allowing for a constant bone density to be maintained. Yet this micro-environment provides the necessary conditions for the growth and proliferation of tumor cells, and thus bone is a common site for the development of metastases, mainly from primary breast and prostate cancer. Mathematical and computational models with differential equations can replicate this bone remodeling process. These models have been extended to include the effects of disruptive tumor pathologies in the bone dynamics, as metastases contribute to the decoupling between bone resorption and formation and to the self-perpetuating tumor growth cycle. Such models may also contemplate the counteraction effects of currently used therapies, and, in the case of treatments with drugs, their pharmocokinetics and pharmacodynamics. We present a thorough overview of biochemical models for bone remodeling, in the presence of a tumour together with anti-cancer and anti-resorptive therapy, formulated as systems of first-order differential equations, or simplified using variable order derivatives. The latter models, of which some are new to this paper, result in equations with fewer parameters, and allow accounting for anomalous diffusion processes. In this way, more compact and parsimonious models, that promptly highlight tumorous bone interactions, are achieved, providing an effective framework to counteract the loss of bone integrity on the affected areas.

Bone metastases: a comprehensive review of the literature

The last report of the World Health Organization (WHO) stated that approximately four million people experience bone pain due to malignant diseases. Among them, metastatic bone pain is one of the most important sources of complaint. The estimated median survival in the presence of bone metastases ranks from 10 to 12 weeks. Bone represents a potential target of distant metastases for the majority of malignant tumours. However, the exact incidence of bone metastases is unknown. Bone metastases have an important socio-economic impact, and due to the enhancement of the overall survivorship, their incidence is increasing. Malignant neoplasms such as lung, thyroid, renal cancer, multiple myeloma, and melanoma often metastasize to the bone. Bone metastases commonly localize to the spinal column, pelvis, shoulder, and distal femur. The proper treatment for painful skeletal metastases is still unknown. Hence, the purpose of this review of the literature was to update current evidence concerning the aetiogenesis, biological behaviour, and treatment algorithms for painful skeletal metastases.

Designing combination therapies using multiple optimal controls

Strategic management of populations of interacting biological species routinely requires interventions combining multiple treatments or therapies. This is important in key research areas such as ecology, epidemiology, wound healing and oncology. Despite the well developed theory and techniques for determining single optimal controls, there is limited practical guidance supporting implementation of combination therapies. In this work we use optimal control theory to calculate optimal strategies for applying combination therapies to a model of acute myeloid leukaemia. We present a versatile framework to systematically explore the trade-offs that arise in designing combination therapy protocols using optimal control. We consider various combinations of continuous and bang-bang (discrete) controls, and we investigate how the control dynamics interact and respond to changes in the weighting and form of the pay-off characterising optimality. We demonstrate that the optimal controls respond non-linearly to treatment strength and control parameters, due to the interactions between species. We discuss challenges in appropriately characterising optimality in a multiple control setting and provide practical guidance for applying multiple optimal controls. Code used in this work to implement multiple optimal controls is available on GitHub.

Mathematical modelling of the role of Endo180 network in the development of metastatic bone disease in prostate cancer

Metastatic bone disease (MBD) is a common complication of advanced cancer and recent research suggests that Endo180 expression is dysregulated through the TGFβ-TGFβR-SMAD2/3 signalling pathway during the invasion of tumour cells in the development of MBD. We here provide a model for the dysregulation of the Endo180 network to demonstrate its vital contribution to bone destruction as well as tumour cell growth. The model consisted of a set of ordinary differential equations and reconstructed variations in the bone cells, resultant bone volume, and biochemical factors involved in the TGFβ-TGFβR-SMAD2/3 signalling pathway over time. The model also investigated the underlying mechanism in which the change of TGFβ affects the TGFβ-TGFβR-SMAD2/3 signalling pathway and the resultant Endo180 expression in osteoblastic and tumour cells. The model links the appearance of tumour cells with the inhibition of TGFβ binding to its receptors on osteoblastic cells, to affect TGFβ-TGFβR-SMAD2/3 signalling and Endo180 expression. Temporal variation in bone cells, bone volume, and the biochemical factors involved in the TGFβ-TGFβR-SMAD2/3 pathway as demonstrated in the model simulations agree with published experimental data. The model can be refined based on further discoveries but allows the influence of Endo180 network dysregulation on bone remodelling in MBD to be established. This model could aid in the development of Endo180 targeted therapies for MBD in the future.

Modeling Public Health Campaigns for Sexually Transmitted Infections via Optimal and Feedback Control

Control of sexually transmitted infections (STIs) poses important challenges to public health authorities. Obstacles for STIs' control include low priority in public health programs and disease transmission mechanisms. This work uses a compartmental pair model to explore different public health strategies on the evolution of STIs. Optimal control and feedback control are used to model realistic strategies for reducing the prevalence of these infections. Feedback control is proposed to model the reaction of public health authorities relative to an alert level. Optimal control is used to model the optimization of available resources for implementing strategies. Numerical simulations are performed using trichomoniasis, gonorrhea, chlamydia and human papillomavirus (HPV) as study cases. HPV is non-curable, and it is analyzed only under transmission control such as condom promotion campaigns. Trichomoniasis, gonorrhea and chlamydia are curable STIs that are modeled here additionally under treatment control. Increased cost-effectiveness ratio is employed as a criterion to measure control strategies performance. The features and drawbacks of control strategies under the pair formation process are discussed.