Osteoporosis induced by cellular senescence: A mathematical model.

Osteoporosis is a disease characterized by loss of bone mass, where bones become fragile and more likely to fracture. Bone density begins to decrease at age 50, and a state of osteoporosis is defined by loss of more than 25%. Cellular senescence is a permanent arrest of normal cell cycle, while maintaining cell viability. The number of senescent cells increase with age. Since osteoporosis is an aging disease, it is natural to consider the question to what extend senescent cells induce bone density loss and osteoporosis. In this paper we use a mathematical model to address this question. We determine the percent of bone loss for men and women during age 50 to 100 years, and the results depend on the rate η of net formation of senescent cell, with η = 1 being the average rate. In the case η = 1, the model simulations are in agreement with empirical data. We also consider senolytic drugs, like fisetin and quercetin, that selectively eliminate senescent cells, and assess their efficacy in terms of reducing bone loss. For example, at η = 1, with estrogen hormonal therapy and early treatment with fisetin, bone density loss for women by age 75 is 23.4% (below osteoporosis), while with no treatment with fisetin it is 25.8% (osteoporosis); without even a treatment with estrogen hormonal therapy, bone loss of 25.3% occurs already at age 65.

Generating PET scan patterns in Alzheimer's by a mathematical model.

Alzheimer disease (AD) is the most common form of dementia. The cause of the disease is unknown, and it has no cure. Symptoms include cognitive decline, memory loss, and impairment of daily functioning. The pathological hallmarks of the disease are aggregation of plaques of amyloid-ß (Aß) and neurofibrillary tangles of tau proteins (τ), which can be detected in PET scans of the brain. The disease can remain asymptomatic for decades, while the densities of Aß and τ continue to grow. Inflammation is considered an early event that drives the disease. In this paper, we develop a mathematical model that can produce simulated patterns of (Aß,τ) seen in PET scans of AD patients. The model is based on the assumption that early inflammations, R and [Formula: see text], drive the growth of Aß and τ, respectively. Recently approved drugs can slow the progression of AD in patients, provided treatment begins early, before significant damage to the brain has occurred. In line with current longitudinal studies, we used the model to demonstrate how to assess the efficacy of such drugs when given years before the disease becomes symptomatic.

A mathematical model of obesity-induced type 2 diabetes and efficacy of anti-diabetic weight reducing drug.

The dominant paradigm for modeling the obesity-induced T2DM (type 2 diabetes mellitus) today focuses on glucose and insulin regulatory systems, diabetes pathways, and diagnostic test evaluations. The problem with this approach is that it is not possible to explicitly account for the glucose transport mechanism from the blood to the liver, where the glucose is stored, and from the liver to the blood. This makes it inaccurate, if not incorrect, to properly model the concentration of glucose in the blood in comparison to actual glycated hemoglobin (A1C) test results. In this paper, we develop a mathematical model of glucose dynamics by a system of ODEs. The model includes the mechanism of glucose transport from the blood to the liver, and from the liver to the blood, and explains how obesity is likely to lead to T2DM. We use the model to evaluate the efficacy of an anti-T2DM drug that also reduces weight.

IL-27 in combination with anti-PD-1 can be anti-cancer or pro-cancer.

Interleukin-27 (IL-27) is known to play opposing roles in immunology. The present paper considers, specifically, the role IL-27 plays in cancer immunotherapy when combined with immune checkpoint inhibitor anti-PD-1. We first develop a mathematical model for this combination therapy, by a system of Partial Differential Equations, and show agreement with experimental results in mice injected with melanoma cells. We then proceed to simulate tumor volume with IL-27 injection at a variable dose F and anti-PD-1 at a variable dose g. We show that in some range of "small" values of g, as f increases tumor volume decreases as long as fFc(g), where Fc(g) is a monotone increasing function of g. This demonstrates that IL-27 can be both anti-cancer and pro-cancer, depending on the ranges of both anti-PD-1 and IL-27.

Treatment of leishmaniasis with chemotherapy and vaccine: a mathematical model.

Leishmaniasis, an infectious disease, manifests itself mostly in two forms, cutaneous leishmaniasis (CL) and, a more severe and potentially deadly form, visceral leishmaniasis (VL). The current control strategy for leishmaniasis relies on chemotherapy drugs such as sodium antimony gluconate (SAG) and meglumine antimoniate (MA). However, all these chemotherapy compounds have poor efficacy, and they are associated with toxicity and other adverse effects, as well as drug resistance. While research in vaccine development for leishmaniasis is continuously progressing, no vaccine is currently available. However, some experimental vaccines such as LEISH-F1+MPL-SE (V) have demonstrated some efficacy when used as drugs for CL patients. In this paper we use a mathematical model to address the following question: To what extent vaccine shots can enhance the efficacy of standard chemotherapy treatment of leishmaniasis? Starting with standard MA treatment of leishmaniasis and combining it with three injections of V , we find, by Day 84, that efficacy increased from 29% to 65-91% depending on the amount of the vaccine. With two or just one injection of V , efficacy is still very high, but there is a definite resurgence of the disease by end-time.

Breast Cancer Exosomal microRNAs Facilitate Pre-Metastatic Niche Formation in the Bone: A Mathematical Model.

Pre-metastatic niche is a location where cancer cells, separating from a primary tumor, find "fertile soil" for growth and proliferation, ensuring successful metastasis. Exosomal miRNAs of breast cancer are known to enter the bone and degrade it, which facilitates cancer cells invasion into the bone interior and ensures its successful colonization. In this paper, we use a mathematical model to first describe, in health, the continuous remodeling of the bone by bone-forming osteoblasts, bone-resorbing osteoclasts and the RANKL-OPG-RANK signaling system, which keeps the balance between bone formation and bone resorption. We next demonstrate how breast cancer exosomal miRNAs disrupt this balance, either by increasing or by decreasing the ratio of osteoclasts/osteoblasts, which results in abnormal high bone resorption or abnormal high bone forming, respectively, and in bone weakening in both cases. Finally we consider the case of abnormally high resorption and evaluate the effect of drugs, which may increase bone density to normal level, thus protecting the bone from invasion by cancer cells.

Cancer therapy with immune checkpoint inhibitor and CSF-1 blockade: A mathematical model.

Immune checkpoint inhibitors (ICIs) introduced in recent years have revolutionized the treatment of many metastatic cancers. However, data suggest that treatment has benefits only in a limited percentage of patients, and that this is due to immune suppression of the tumor microenvironment (TME). Anti-tumor inflammatory macrophages (M1), which are attracted to the TME, are converted by tumor secreted cytokines, such as CSF-1, to pro-tumor anti-inflammatory macrophages (M2), or tumor associated macrophages (TAMs), which block the anti-tumor T cells. In the present paper we develop a mathematical model that represents the interactions among the immune cells and cancer in terms of differential equations. The model can be used to assess treatments of combination therapy of anti-PD-1 with anti-CSF-1. Examples are given in comparing the efficacy among different strategies for anti-CSF-1 dosing in a setup of clinical trials.

Optimal timing of steroid initiation in response to CTLA-4 antibody in metastatic cancer: A mathematical model.

Immune checkpoint inhibitors, introduced in recent years, have revolutionized the treatment of many cancers. However, the toxicity associated with this therapy may cause severe adverse events. In the case of advanced lung cancer or metastatic melanoma, a significant number (10%) of patients treated with CTLA-4 inhibitor incur damage to the pituitary gland. In order to reduce the risk of hypophysitis and other severe adverse events, steroids may be combined with CTLA-4 inhibitor; they reduce toxicity, but they also diminish the anti-cancer effect of the immunotherapy. This trade-off between tumor reduction and the risk of severe adverse events poses the following question: What is the optimal time to initiate treatment with steroid. We address this question with a mathematical model from which we can also evaluate the comparative benefits of each schedule of steroid administration. In particular, we conclude that treatment with steroid should not begin too early, but also not very late, after immunotherapy began; more precisely, it should start as soon as tumor volume, under the effect of CTLA-4 inhibitor alone, begins to decrease. We can also compare the benefits of short term treatment of steroid at high doses to a longer term treatment with lower doses.

A cancer model with nonlocal free boundary dynamics.

Cancer cells at the tumor boundary move in the direction of the oxygen gradient, while cancer cells far within the tumor are in a necrotic state. This paper introduces a simple mathematical model that accounts for these facts. The model consists of cancer cells, cytotoxic T cells, and oxygen satisfying a system of partial differential equations. Some of the model parameters represent the effect of anti-cancer drugs. The tumor boundary is a free boundary whose dynamics is determined by the movement of cancer cells at the boundary. The model is simulated for radially symmetric and axially symmetric tumors, and it is shown that the tumor may increase or decrease in size, depending on the "strength" of the drugs. Existence theorems are proved, global in-time in the radially symmetric case, and local in-time for any shape of tumor. In the radially symmetric case, it is proved, under different conditions, that the tumor may shrink monotonically, or expand monotonically.

A mathematical model of immunomodulatory treatment in myocardial infarction.

A heart attack, or acute myocardial infarction (MI) is caused by the acute occlusion of a coronary artery. MI is associated with 30% mortality; approximately half of the deaths occur prior to arrival at the hospital. Reperfusion therapy in the hospital is a medical treatment to restore blood flow through the blocked artery; treatment includes drugs and surgery. However, the damage to the heart muscles through the infarct area is permanent and there is additional damage around the infarct area due to inflammation or insufficient oxygen supply. Approximately half of the patients who survive MI are hospitalized again within one year after reperfusion treatment. In this paper we develop a mathematical model of MI and use it to assess the efficacy of drugs used, post reperfusion, to reduce the damage caused by inflammation in a region of the left ventricular wall surrounding the infarct area. The mathematical model, represented by a system of partial differential equations. The model variables include myocytes, endothelial cells, neutrophils, macrophages, fibroblasts and cytokines that play a role in the interactions among these cells. The drugs used to in the model include IL-1, TNF-α and TGF-ß inhibitors, and the delivery of VEGF. The model is based on mice data. In particular, we find that immunomodulatory treatment with TNF-α and IL-1 inhibitors can significantly increase the low density of myocytes bordering the infarct area by 50-60% and decrease the abnormally high density of ECM in a region surrounding the infarct area.

Combination therapy for mCRPC with immune checkpoint inhibitors, ADT and vaccine: A mathematical model.

Metastatic castration resistant prostate cancer (mCRPC) is commonly treated by androgen deprivation therapy (ADT) in combination with chemotherapy. Immune therapy by checkpoint inhibitors, has become a powerful new tool in the treatment of melanoma and lung cancer, and it is currently being used in clinical trials in other cancers, including mCRPC. However, so far, clinical trials with PD-1 and CTLA-4 inhibitors have been disappointing. In the present paper we develop a mathematical model to assess the efficacy of any combination of ADT with cancer vaccine, PD-1 inhibitor, and CTLA-4 inhibitor. The model is represented by a system of partial differential equations (PDEs) for cells, cytokines and drugs whose density/concentration evolves in time within the tumor. Efficacy of treatment is determined by the reduction in tumor volume at the endpoint of treatment. In mice experiments with ADT and various combinations of PD-1 and CTLA-4 inhibitors, tumor volume at day 30 was always larger than the initial tumor. Our model, however, shows that we can decrease tumor volume with large enough dose; for example, with 10 fold increase in the dose of anti-PD-1, initial tumor volume will decrease by 60%. Although the treatment with ADT in combination with PD-1 inhibitor or CTLA-4 inhibitor has been disappointing in clinical trials, our simulations suggest that, disregarding negative effects, combinations of ADT with checkpoint inhibitors can be effective in reducing tumor volume if larger doses are used. This points to the need for determining the optimal combination and amounts of dose for individual patients.

Analysis of a mathematical model of immune response to fungal infection.

Fungi are cells found as commensal residents, on the skin, and on mucosal surfaces of the human body, including the digestive track and urogenital track, but some species are pathogenic. Fungal infection may spread into deep-seated organs causing life-threatening infection, especially in immune-compromised individuals. Effective defense against fungal infection requires a coordinated response by the innate and adaptive immune systems. In the present paper we introduce a simple mathematical model of immune response to fungal infection consisting of three partial differential equations, for the populations of fungi (F), neutrophils (N) and cytotoxic T cells (T), taking N and T to represent, respectively, the innate and adaptive immune cells. We denote by [Formula: see text] the aggressive proliferation rate of the fungi, by [Formula: see text] and [Formula: see text] the killing rates of fungi by neutrophils and T cells, and by [Formula: see text] and [Formula: see text] the immune strengths, respectively, of N and T of an infected individual. We take the expression [Formula: see text] to represent the coordinated defense of the immune system against fungal infection. We use mathematical analysis to prove the following: If [Formula: see text], then the infection is eventually stopped, and [Formula: see text] as [Formula: see text]; and (ii) if [Formula: see text] then the infection cannot be stopped and F converges to some positive constant as [Formula: see text]. Treatments of fungal infection include anti-fungal agents and immunotherapy drugs, and both cause the parameter I to increase.

TGF-ß inhibition can overcome cancer primary resistance to PD-1 blockade: A mathematical model.

Immune checkpoint inhibitors have demonstrated, over the recent years, impressive clinical response in cancer patients, but some patients do not respond at all to checkpoint blockade, exhibiting primary resistance. Primary resistance to PD-1 blockade is reported to occur under conditions of immunosuppressive tumor environment, a condition caused by myeloid derived suppressor cells (MDSCs), and by T cells exclusion, due to increased level of T regulatory cells (Tregs). Since TGF-ß activates Tregs, TGF-ß inhibitor may overcome primary resistance to anti-PD-1. Indeed, recent mice experiments show that combining anti-PD-1 with anti-TGF-ß yields significant therapeutic improvements compared to anti-TGF-ß alone. The present paper introduces two cancer-specific parameters and, correspondingly, develops a mathematical model which explains how primary resistance to PD-1 blockade occurs, in terms of the two cancer-specific parameters, and how, in combination with anti-TGF-ß, anti-PD-1 provides significant benefits. The model is represented by a system of partial differential equations and the simulations are in agreement with the recent mice experiments. In some cancer patients, treatment with anti-PD-1 results in rapid progression of the disease, known as hyperprogression disease (HPD). The mathematical model can also explain how this situation arises, and it predicts that HPD may be reversed by combining anti-TGF-ß to anti-PD-1. The model is used to demonstrate how the two cancer-specific parameters may serve as biomarkers in predicting the efficacy of combination therapy with PD-1 and TGF-ß inhibitors.

A mathematical model of the multiple sclerosis plaque.

Multiple sclerosis is an autoimmune disease that affects white matter in the central nervous system. It is one of the primary causes of neurological disability among young people. Its characteristic pathological lesion is called a plaque, a zone of inflammatory activity and tissue destruction that expands radially outward by destroying the myelin and oligodendrocytes of white matter. The present paper develops a mathematical model of the multiple sclerosis plaques. Although these plaques do not provide reliable information of the clinical disability in MS, they are nevertheless useful as a primary outcome measure of Phase II trials. The model consists of a system of partial differential equations in a simplified geometry of the lesion, consisting of three domains: perivascular space, demyelinated plaque, and white matter. The model describes the activity of various pro- and anti-inflammatory cells and cytokines in the plaque, and quantifies their effect on plaque growth. We show that volume growth of plaques are in qualitative agreement with reported clinical studies of several currently used drugs. We then use the model to explore treatments with combinations of such drugs, and with experimental drugs. We finally consider the benefits of early vs. delayed treatment.

Mathematical Model of Chronic Dermal Wounds in Diabetes and Obesity.

Chronic dermal-wound patients frequently suffer from diabetes type 2 and obesity; without treatment or early intervention, these patients are at risk of amputation. In this paper, we identified four factors that impair wound healing in these populations: excessive production of glycation, excessive production of leukotrient, decreased production of stromal derived factor (SDF-1), and insulin resistance. We developed a mathematical model of wound healing that includes these factors. The model consists of a system of partial differential equations, and it demonstrates how these four factors impair the closure of the wound, by reducing the oxygen flow into the wound area and by blocking the transition from pro-inflammatory macrophages to anti-inflammatory macrophages. The model is used to assess treatment by insulin injection and by oxygen infusion.

Increase hemoglobin level in severe malarial anemia while controlling parasitemia: A mathematical model.

Macrophage migration inhibitory factor (MIF) is a pleiotropic cytokine produced by immune cells; it can play a protective or deleterious role in response to pathogens. The intracellular malaria parasite secretes a similar protein, PMIF. The present paper is concerned with severe malarial anemia (SMA), where MIF suppresses the recruitment of red blood cells (RBCs) from the spleen and the bone marrow. This suppression results in a decrease of the hemoglobin (Hb) in the blood to a dangerous level. Indeed, SMA is responsible for the majority of death-related malaria cases. Artesunate is the first line of treatment of SMA; it accelerates the death of infected RBCs (iRBCs), thereby decreasing parasitemia. However, artesunate does not increase the level of Hb, and, in some cases, post-artesunate hemolytic anemia requires blood transfusion. In order to avoid this situation, we explore combining artesunate with another drug so that the Hb level is increased to healthy levels while parasitemia is still controlled. In this paper we show, by a mathematical model, that increasing the Hb levels while controlling parasitemia in malarial anemia can be done with the experimental drug Epoxyazadiradione (Epoxy) in combination with artesunate. Epoxy acts as MIF inhibitor and thus has the potential to increase the Hb level. Simulations of the model show that the two drugs compliment each other: while artesunate is primarily responsible for decreasing parasitemia, Epoxy is primarily responsible for increasing the hemoglobin level.

TNF-α inhibitor reduces drug-resistance to anti-PD-1: A mathematical model.

Drug resistance is a primary obstacle in cancer treatment. In many patients who at first respond well to treatment, relapse occurs later on. Various mechanisms have been explored to explain drug resistance in specific cancers and for specific drugs. In this paper, we consider resistance to anti-PD-1, a drug that enhances the activity of anti-cancer T cells. Based on results in experimental melanoma, it is shown, by a mathematical model, that resistances to anti-PD-1 can be significantly reduced by combining it with anti-TNF-α. The model is used to simulate the efficacy of the combined therapy with different range of doses, different initial tumor volume, and different schedules. In particular, it is shown that under a course of treatment with 3-week cycles where each drug is injected in the first day of either week 1 or week 2, injecting anti-TNF-α one week after anti-PD-1 is the most effective schedule in reducing tumor volume.

Analysis of a mathematical model of rheumatoid arthritis.

Rheumatoid arthritis is an autoimmune disease characterized by inflammation in the synovial fluid within the synovial joint connecting two contiguous bony surfaces. The inflammation diffuses into the cartilage adjacent to each of the bony surfaces, resulting in their gradual destruction. The interface between the cartilage and the synovial fluid is an evolving free boundary. In this paper we consider a two-phase free boundary problem based on a simplified model of rheumatoid arthritis. We prove global existence and uniqueness of a solution, and derive properties of the free boundary. In particular it is proved that the free boundary increases in time, and the cartilage shrinks to zero as [Formula: see text], even under treatment by a drug. It is also shown in the reduced one-phased problem, with cartilage alone, that a larger prescribed inflammation function leads to a faster destruction of the cartilage.

Overcoming Drug Resistance to BRAF Inhibitor.

One of the most frequently found mutations in human melanomas is in the B-raf gene, making its protein BRAF a key target for therapy. However, in patients treated with BRAF inhibitor (BRAFi), although the response is very good at first, relapse occurs within 6 months, on the average. In order to overcome this drug resistance to BRAFi, various combinations of BRAFi with other drugs have been explored, and some are being applied clinically, such as a combination of BRAF and MEK inhibitors. Experimental data for melanoma in mice show that under continuous treatment with BRAFi, the pro-cancer MDSCs and chemokine CCL2 initially decrease but eventually increase to above their original level, while the anticancer T cells continuously decrease. In this paper, we develop a mathematical model that explains these experimental results. The model is used to explore the efficacy of combinations of BRAFi with anti-CCL2, anti-PD-1 and anti-CTLA-4, with the aim of eliminating or reducing drug resistance to BRAFi.