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.

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.

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.

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.

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 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.

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.

Modeling combination therapy for breast cancer with BET and immune checkpoint inhibitors.

CTLA-4 is an immune checkpoint expressed on active anticancer T cells. When it combines with its ligand B7 on dendritic cells, it inhibits the activity of the T cells. The Bromo- and Extra-Terminal (BET) protein family includes proteins that regulate the expression of key oncogenes and antiapoptotic proteins. BET inhibitor (BETi) has been shown to reduce the expression of MYC by suppressing its transcription factors and to down-regulate the hypoxic transcriptome response to VEGF-A. This paper develops a mathematical model of the treatment of cancer by combination therapy of BETi and CTLA-4 inhibitor. The model shows that the two drugs are positively correlated in the sense that the tumor volume decreases as the dose of each of the drugs is increased. The model also considers the effect of the combined therapy on levels of myeloid-derived suppressor cells (MDSCs) and the overexpression of TNF-α, which may predict gastrointestinal side effects of the combination.

Complex role of NK cells in regulation of oncolytic virus-bortezomib therapy.

In the present work, we investigated the role of natural killer (NK) cells in combination therapy with oncolytic virus (OV) and bortezomib, a proteasome inhibitor. NK cells display rapid and potent immunity to metastatic and hematological cancers, and they overcome immunosuppressive effects of tumor microenvironment. We developed a mathematical model to address the question of how the density of NK cells affects the growth of the tumor. We found that the antitumor efficacy increases when the endogenous NKs are depleted and also when exogenous NK cells are injected into the tumor. These predictions were validated by our in vivo and in vitro experiments.

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.

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.

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.

Mathematical model of chronic pancreatitis.

Chronic pancreatitis (CP) is a progressive inflammatory disease of the pancreas, leading to its fibrotic destruction. There are currently no drugs that can stop or slow the progression of the disease. The etiology of the disease is multifactorial, whereas recurrent attacks of acute pancreatitis are thought to precede the development of CP. A better understanding of the pathology of CP is needed to facilitate improved diagnosis and treatment strategies for this disease. The present paper develops a mathematical model of CP based on a dynamic network that includes macrophages, pancreatic stellate cells, and prominent cytokines that are present at high levels in the CP microenvironment. The model is represented by a system of partial differential equations. The model is used to explore in silico potential drugs that could slow the progression of the disease, for example infliximab (anti-TNF-[Formula: see text]) and tocilizumab or siltuximab (anti-IL-6/IL-6R).

Rheumatoid arthritis - a mathematical model.

Rheumatoid arthritis (RA) is a common autoimmune disease that mainly affects the joints. It is characterized by synovial inflammation, which may result in cartilage and bone destruction. The present paper develops a mathematical model of chronic RA. The model is represented by a system of partial differential equations (PDEs) in the synovial fluid, the synovial membrane, and the cartilage. The model characterizes the progression of the disease in terms of the degradation of the cartilage. More precisely, we assume a simplified geometry in which the synovial membrane and the cartilage are planar layers adjacent to each other. We then quantify the state of the disease by how much the cartilage layer has decreased, or, equivalently, how much the synovial layer has increased. The model is used to evaluate treatments of RA by currently used drugs, as well as by experimental drugs.

Mathematical modeling in scheduling cancer treatment with combination of VEGF inhibitor and chemotherapy drugs.

The present paper considers a treatment of cancer with a combination of anti-VEGF (bevacizumab) and a chemotherapy drug (docetaxel). Since anti-VEGF reduces the perfusion of chemotherapy drugs, the question arises whether it is more effective to administer the two drugs at the same time, or non-overlapping, in order to reduce tumor volume more effectively. To address this question we develop a mathematical model and use it to simulate different schedules. We find that the treatment of cancer would be far more effective if the two drugs are given non-overlappingly, with the chemotherapy drug at day 0 and anti-VEGF at day 7 in cycles of 21 days.

The Role of Exosomes in Pancreatic Cancer Microenvironment.

Exosomes are nanovesicles shed by cells as a means of communication with other cells. Exosomes contain mRNAs, microRNAs (miRs) and functional proteins. In the present paper, we develop a mathematical model of tumor-immune interaction by means of exosomes shed by pancreatic cancer cells and dendritic cells. Cancer cells' exosomes contain miRs that promote their proliferation and that inhibit immune response by dendritic cells, and by CD4+ and CD8+ T cells. Dendritic cells release exosomes with proteins that induce apoptosis of cancer cells and that block regulatory T cells. Simulations of the model show how the size of the pancreatic cancer can be determined by measurement of specific miRs (miR-21 and miR-203 in the case of pancreatic cancer), suggesting these miRs as biomarkers for cancer.

Exosomal microRNA concentrations in colorectal cancer: A mathematical model.

Colorectal cancer (CRC) is the second leading cause of cancer related deaths in the United States. Early detection increases survival very significantly. Indeed, five year survival for people diagnosed at stage I-II is 90%, while for those diagnosed at stage IV it is only 13%. The gold standard for early detection is colonoscopy, but this procedure is limited due to its invasive nature and its high cost. Hence there is a need to identify non-invasive biomarkers for CRC. Exosomal miRs secreted by cancer cells and overexpressed in the blood have been suggested as biomarkers for cancer. In particular, exosomal miRs 21, 23a, 92a and 1246 are overexpressed in CRC, and thus have the potential to be used as serum biomarkers for early detection of the disease. The present paper develops for the first time a mathematical model for early stage of CRC which includes the effect of these miRs on the growth of the cancer. The model is represented by a system of partial differential equations. Simulations of the model show a relationship between the growth of the tumor diameter and the total mass of these miRs under some of the common mutations which occur in CRC, namely, KRAS, PI3K, APC, p53 and SMAD mutations. The model may serve as a step toward establishing miRs 21, 23a, 92a and 1246 as reliable blood biomarkers for CRC as more experimental results and clinical data become available.

Granuloma formation in leishmaniasis: A mathematical model.

Leishmaniasis is a disease caused by the Leishmania parasites. The two common forms of leishmaniasis are cutaneous leishmaniasis (CL) and visceral leishmaniasis (VL). VL is the more severe of the two and, if untreated, may become fatal. The hallmark of VL is the formation of granuloma in the liver or the spleen. In this paper, we develop a mathematical model of the evolution of granuloma in the liver. The model is represented by a system of partial differential equations and it includes migration of cells from the adaptive immune system into the granuloma; the rate of the influx is determined by the strength of the immune response of the infected individual. It is shown that parasite load decreases as the strength of the immune system increases. Furthermore, the efficacy of a commonly used drug, which increases T cells proliferation, increases in an individual with stronger immune response. The model also provides an explanation why, in contrast to humans, mice recover naturally from VL in the liver.

Mathematical model of renal interstitial fibrosis.

Lupus nephritis (LN) is an autoimmune disease that occurs when autoantibodies complex with self-antigen and form immune complexes that accumulate in the glomeruli. These immune complexes initiate an inflammatory response resulting in glomerular injury. LN often concomitantly affects the tubulointerstitial compartment of the kidney, leading first to interstitial inflammation and subsequently to interstitial fibrosis and atrophy of the renal tubules if not appropriately treated. Presently the only way to assess interstitial inflammation and fibrosis is through kidney biopsy, which is invasive and cannot be repeated frequently. Hence, monitoring of disease progression and response to therapy is suboptimal. In this paper we describe a mathematical model of the progress from tubulointerstitial inflammation to fibrosis. We demonstrate how the model can be used to monitor treatments for interstitial fibrosis in LN with drugs currently being developed or used for nonrenal fibrosis.