Local alliances and rivalries shape near-repeat terror activity of al-Qaeda, ISIS, and insurgents.

We study the spatiotemporal correlation of terrorist attacks by al-Qaeda, the Islamic State of Iraq and Syria (ISIS), and local insurgents, in six geographical areas identified via k-means clustering applied to the Global Terrorism Database. All surveyed organizations exhibit near-repeat activity whereby a prior attack increases the likelihood of a subsequent one by the same group within 20 km and on average 4 (al-Qaeda) to 10 (ISIS) weeks. Near-response activity, whereby an attack by a given organization elicits further attacks from a different one, is found to depend on the adversarial, neutral, or collaborative relationship between the two. When in conflict, local insurgents respond quickly to attacks by global terror groups while global terror groups delay their responses to local insurgents, leading to an asymmetric dynamic. When neutral or allied, attacks by one group enhance the response likelihood of the other, regardless of hierarchy. These trends arise consistently in all clusters for which data are available. Government intervention and spillover effects are also discussed; we find no evidence of outbidding. Understanding the regional dynamics of terrorism may be greatly beneficial in policy making and intervention design.

Mathematical modeling in cancer nanomedicine: a review.

Cancer continues to be among the leading healthcare problems worldwide, and efforts continue not just to find better drugs, but also better drug delivery methods. The need for delivering cytotoxic agents selectively to cancerous cells, for improved safety and efficacy, has triggered the application of nanotechnology in medicine. This effort has provided drug delivery systems that can potentially revolutionize cancer treatment. Nanocarriers, due to their capacity for targeted drug delivery, can shift the balance of cytotoxicity from healthy to cancerous cells. The field of cancer nanomedicine has made significant progress, but challenges remain that impede its clinical translation. Several biophysical barriers to the transport of nanocarriers to the tumor exist, and a much deeper understanding of nano-bio interactions is necessary to change the status quo. Mathematical modeling has been instrumental in improving our understanding of the physicochemical and physiological underpinnings of nanomaterial behavior in biological systems. Here, we present a comprehensive review of literature on mathematical modeling works that have been and are being employed towards a better understanding of nano-bio interactions for improved tumor delivery efficacy.

A Mathematical Model of the Effects of Aging on Naive T Cell Populations and Diversity.

The human adaptive immune response is known to weaken in advanced age, resulting in increased severity of pathogen-born illness, poor vaccine efficacy, and a higher prevalence of cancer in the elderly. Age-related erosion of the T cell compartment has been implicated as a likely cause, but the underlying mechanisms driving this immunosenescence have not been quantitatively modeled and systematically analyzed. T cell receptor diversity, or the extent of pathogen-derived antigen responsiveness of the T cell pool, is known to diminish with age, but inherent experimental difficulties preclude accurate analysis on the full organismal level. In this paper, we formulate a mechanistic mathematical model of T cell population dynamics on the immunoclonal subpopulation level, which provides quantitative estimates of diversity. We define different estimates for diversity that depend on the individual number of cells in a specific immunoclone. We show that diversity decreases with age primarily due to diminished thymic output of new T cells and the resulting overall loss of small immunoclones.

Theory and Experimental Validation of a Spatio-temporal Model of Chemotherapy Transport to Enhance Tumor Cell Kill.

AUTHOR SUMMARY: Cancer treatment efficacy can be significantly enhanced through the elution of drug from nano-carriers that can temporarily stay in the tumor vasculature. Here we present a relatively simple yet powerful mathematical model that accounts for both spatial and temporal heterogeneities of drug dosing to help explain, examine, and prove this concept. We find that the delivery of systemic chemotherapy through a certain form of nano-carriers would have enhanced tumor kill by a factor of 2 to 4 over the standard therapy that the patients actually received. We also find that targeting blood volume fraction (a parameter of the model) through vascular normalization can achieve more effective drug delivery and tumor kill. More importantly, this model only requires a limited number of parameters which can all be readily assessed from standard clinical diagnostic measurements (e.g., histopathology and CT). This addresses an important challenge in current translational research and justifies further development of the model towards clinical translation.

A hybrid agent-based model of the developing mammary terminal end bud.

Mammary gland ductal elongation is spearheaded by terminal end buds (TEBs), where populations of highly proliferative cells are maintained throughout post-pubertal organogenesis in virgin mice until the mammary fat pad is filled by a mature ductal tree. We have developed a hybrid multiscale agent-based model to study how cellular differentiation pathways, cellular proliferation capacity, and endocrine and paracrine signaling play a role during development of the mammary gland. A simplified cellular phenotypic hierarchy that includes stem, progenitor, and fully differentiated cells within the TEB was implemented. Model analysis finds that mammary gland development was highly sensitive to proliferation events within the TEB, with progenitors likely undergoing 2-3 proliferation cycles before transitioning to a non-proliferative phenotype, and this result is in agreement with our previous experimental work. Endocrine and paracrine signaling were found to provide reliable ductal elongation rate regulation, while variations in the probability a new daughter cell will be of a proliferative phenotype were seen to have minimal effects on ductal elongation rates. Moreover, the distribution of cellular phenotypes within the TEB was highly heterogeneous, demonstrating significant allowable plasticity in possible phenotypic distributions while maintaining biologically relevant growth behavior. Finally, simulation results indicate ductal elongation rates due to cellular proliferation within the TEB may have a greater sensitivity to upstream endocrine signaling than endothelial to stromal paracrine signaling within the TEB. This model provides a useful tool to gain quantitative insights into cellular population dynamics and the effects of endocrine and paracrine signaling within the pubertal terminal end bud.

An integrated computational/experimental model of lymphoma growth.

Non-Hodgkin's lymphoma is a disseminated, highly malignant cancer, with resistance to drug treatment based on molecular- and tissue-scale characteristics that are intricately linked. A critical element of molecular resistance has been traced to the loss of functionality in proteins such as the tumor suppressor p53. We investigate the tissue-scale physiologic effects of this loss by integrating in vivo and immunohistological data with computational modeling to study the spatiotemporal physical dynamics of lymphoma growth. We compare between drug-sensitive Eµ-myc Arf-/- and drug-resistant Eµ-myc p53-/- lymphoma cell tumors grown in live mice. Initial values for the model parameters are obtained in part by extracting values from the cellular-scale from whole-tumor histological staining of the tumor-infiltrated inguinal lymph node in vivo. We compare model-predicted tumor growth with that observed from intravital microscopy and macroscopic imaging in vivo, finding that the model is able to accurately predict lymphoma growth. A critical physical mechanism underlying drug-resistant phenotypes may be that the Eµ-myc p53-/- cells seem to pack more closely within the tumor than the Eµ-myc Arf-/- cells, thus possibly exacerbating diffusion gradients of oxygen, leading to cell quiescence and hence resistance to cell-cycle specific drugs. Tighter cell packing could also maintain steeper gradients of drug and lead to insufficient toxicity. The transport phenomena within the lymphoma may thus contribute in nontrivial, complex ways to the difference in drug sensitivity between Eµ-myc Arf-/- and Eµ-myc p53-/- tumors, beyond what might be solely expected from loss of functionality at the molecular scale. We conclude that computational modeling tightly integrated with experimental data gives insight into the dynamics of Non-Hodgkin's lymphoma and provides a platform to generate confirmable predictions of tumor growth.

Three-dimensional multispecies nonlinear tumor growth-II: Tumor invasion and angiogenesis.

We extend the diffuse interface model developed in Wise et al. (2008) to study nonlinear tumor growth in 3-D. Extensions include the tracking of multiple viable cell species populations through a continuum diffuse-interface method, onset and aging of discrete tumor vessels through angiogenesis, and incorporation of individual cell movement using a hybrid continuum-discrete approach. We investigate disease progression as a function of cellular-scale parameters such as proliferation and oxygen/nutrient uptake rates. We find that heterogeneity in the physiologically complex tumor microenvironment, caused by non-uniform distribution of oxygen, cell nutrients, and metabolites, as well as phenotypic changes affecting cellular-scale parameters, can be quantitatively linked to the tumor macro-scale as a mechanism that promotes morphological instability. This instability leads to invasion through tumor infiltration of surrounding healthy tissue. Models that employ a biologically founded, multiscale approach, as illustrated in this work, could help to quantitatively link the critical effect of heterogeneity in the tumor microenvironment with clinically observed tumor growth and invasion. Using patient tumor-specific parameter values, this may provide a predictive tool to characterize the complex in vivo tumor physiological characteristics and clinical response, and thus lead to improved treatment modalities and prognosis.

A mathematical model to predict nanomedicine pharmacokinetics and tumor delivery.

Towards clinical translation of cancer nanomedicine, it is important to systematically investigate the various parameters related to nanoparticle (NP) physicochemical properties, tumor characteristics, and inter-individual variability that affect the tumor delivery efficiency of therapeutic nanomaterials. Comprehensive investigation of these parameters using traditional experimental approaches is impractical due to the vast parameter space; mathematical models provide a more tractable approach to navigate through such a multidimensional space. To this end, we have developed a predictive mathematical model of whole-body NP pharmacokinetics and their tumor delivery in vivo, and have conducted local and global sensitivity analyses to identify the factors that result in low tumor delivery efficiency and high off-target accumulation of NPs. Our analyses reveal that NP degradation rate, tumor blood viscosity, NP size, tumor vascular fraction, and tumor vascular porosity are the key parameters in governing NP kinetics in the tumor interstitium. The impact of these parameters on tumor delivery efficiency of NPs is discussed, and optimal values for maximizing NP delivery are presented.

Dynamics of T cell receptor distributions following acute thymic atrophy and resumption.

Naive human T cells are produced and developed in the thymus, which atrophies abruptly and severely in response to physical or psychological stress. To understand how an instance of stress affects the size and "diversity" of the peripheral naive T cell pool, we derive a mean-field autonomous ODE model of T cell replenishment that allows us to track the clone abundance distribution (the mean number of different TCRs each represented by a specific number of cells). We identify equilibrium solutions that arise at different rates of T cell production, and derive analytic approximations to the dominant eigenvalues and eigenvectors of the mathematical model linearized about these equilibria. From the forms of the eigenvalues and eigenvectors, we estimate rates at which counts of clones of different sizes converge to and depart from equilibrium values-that is, how the number of clones of different sizes "adjusts" to the changing rate of T cell production. Under most physiological realizations of our model, the dominant eigenvalue (representing the slowest dynamics of the clone abundance distribution) scales as a power law in the thymic output for low output levels, but saturates at higher T cell production rates. Our analysis provides a framework for quantitatively understanding how the clone abundance distribution evolves under small changes in the overall T cell production rate.

Development of a Physiologically-Based Mathematical Model for Quantifying Nanoparticle Distribution in Tumors.

Nanomedicine holds promise for the treatment of cancer, as it enables tumor-targeted drug delivery. However, reports on translation of most nanomedicine strategies to the clinic so far have been less than satisfactory, in part due to insufficient understanding of the effects of nanoparticle (NP) physiochemical properties and physiological variables on their pharmacological behavior. In this paper, we present a multiscale mathematical model to examine the efficacy of NP delivery to solid tumors; as a case example, we apply the model to a clinically detectable primary pancreatic ductal adenocarcinoma (PDAC) to assess tissue-scale spatiotemporal distribution profiles of NPs. We integrate NP systemic disposition kinetics with NP-cell interactions in PDAC abstractly described as a two-dimensional structure, which is then parameterized with human physiological data obtained from published literature. Through model analysis of delivery efficiency, we verify the multiscale approach by showing that NP concentration kinetics of interest in various compartments predicted by the whole-body scale model were in agreement with those obtained from the tissue-scale model. We also found that more NPs were trapped in the outer well-perfused tumor region than the inner semi-necrotic domain. Further development of the model may provide a useful tool for optimal NP design and physiological interventions.

Swarming in viscous fluids: Three-dimensional patterns in swimmer- and force-induced flows.

We derive a three-dimensional theory of self-propelled particle swarming in a viscous fluid environment. Our model predicts emergent collective behavior that depends critically on fluid opacity, mechanism of self-propulsion, and type of particle-particle interaction. In "clear fluids" swimmers have full knowledge of their surroundings and can adjust their velocities with respect to the lab frame, while in "opaque fluids" they control their velocities only in relation to the local fluid flow. We also show that "social" interactions that affect only a particle's propensity to swim towards or away from neighbors induces a flow field that is qualitatively different from the long-ranged flow fields generated by direct "physical" interactions. The latter can be short-ranged but lead to much longer-ranged fluid-mediated hydrodynamic forces, effectively amplifying the range over which particles interact. These different fluid flows conspire to profoundly affect swarm morphology, kinetically stabilizing or destabilizing swarm configurations that would arise in the absence of fluid. Depending upon the overall interaction potential, the mechanism of swimming ( e.g., pushers or pullers), and the degree of fluid opaqueness, we discover a number of new collective three-dimensional patterns including flocks with prolate or oblate shapes, recirculating pelotonlike structures, and jetlike fluid flows that entrain particles mediating their escape from the center of mill-like structures. Our results reveal how the interplay among general physical elements influence fluid-mediated interactions and the self-organization, mobility, and stability of new three-dimensional swarms and suggest how they might be used to kinetically control their collective behavior.

Development of a diffusion-based mathematical model for predicting chemotherapy effects.

Mathematical modeling of drug transport can complement current experimental and clinical investigations to understand drug resistance mechanisms, which eventually will help to develop patient-specific chemotherapy treatments. In this paper, we present a general time- and space-dependent mathematical model based on diffusion theory for predicting chemotherapy outcome. This model has two important parameters: the blood volume fraction and radius of blood vessels divided by drug diffusion penetration length. Model analysis finds that a larger ratio of the radius of blood vessel to diffusion penetration length resulted in to a larger fraction of tumor killed, thereby leading to a better treatment outcome. Clinical translation of the model can help quantify and predict the optimal dosage size and frequency of chemotherapy for individual patients.

Understanding Drug Resistance in Breast Cancer with Mathematical Oncology.

Chemotherapy is mainstay of treatment for the majority of patients with breast cancer, but results in only 26% of patients with distant metastasis living 5 years past treatment in the United States, largely due to drug resistance. The complexity of drug resistance calls for an integrated approach of mathematical modeling and experimental investigation to develop quantitative tools that reveal insights into drug resistance mechanisms, predict chemotherapy efficacy, and identify novel treatment approaches. This paper reviews recent modeling work for understanding cancer drug resistance through the use of computer simulations of molecular signaling networks and cancerous tissues, with a particular focus on breast cancer. These mathematical models are developed by drawing on current advances in molecular biology, physical characterization of tumors, and emerging drug delivery methods (e.g., nanotherapeutics). We focus our discussion on representative modeling works that have provided quantitative insight into chemotherapy resistance in breast cancer and how drug resistance can be overcome or minimized to optimize chemotherapy treatment. We also discuss future directions of mathematical modeling in understanding drug resistance.

A novel, patient-specific mathematical pathology approach for assessment of surgical volume: application to ductal carcinoma in situ of the breast.

We introduce a novel "mathematical pathology" approach, founded on a biophysical model, to identify robust patient-specific predictors of tumor growth useful in clinical practice to improve the accuracy of diagnosis/prognosis and intervention. In accordance with biological observations, our model simulates the diffusion-limited in situ tumors with a relatively short phase of fast initial growth, followed by a prolonged slow-growth phase where tumor size is constrained primarily by the relative weight of cell mitosis and death. The former phase may only last for a few months, so that at the time of diagnosis, we may assume that most tumors will have entered the phase where their size is changing slowly. Based on this prediction, we hypothesize that the volume of breast with ducts affected by in situ tumors at the time of diagnosis will be closely approximated by a model-derived mathematical function based on the ratio of tumor cell proliferation-to-apoptosis indices and on the extent of diffusion of cell nutrients (diffusion penetration length), which can be measured from immunohistochemical and morphometric analysis of patient histopathology specimens without the need for multiple-time measurements. We tested this idea in a retrospective study of 17 patients by staining breast tumor specimens containing ductal carcinoma in situ for mitosis with Ki-67 and for apoptosis with cleaved caspase-3 and counting cells positive for each marker. We also determined diffusion penetration by measuring the thickness of viable rims of tumor cells within ducts. Using the ensuing ratios, we applied the model to determine a predicted surgical volume or tumor size. We then corroborated our hypothesis by comparing the predicted size of each tumor based on our model with the actual size of the pathological specimen after tumor excision (R2 = 0.74-0.88). In addition, for the 17 cases studied, both histological grade and mammography were not found to correlate with tumor size (R2 = 0.08-0.47). We conclude that our mathematical pathology approach yields a high degree of accuracy in predicting the size of tumors based on the mitotic/apoptotic index and on diffusion penetration. By obtaining these ratios at the time of initial biopsy, pathologists can employ our model to predict the size of the tumor and thereby inform surgeons how much tissue to remove (surgical volume). We discuss how results from the model have implications concerning the current debate on recommendations for screening mammography, while the model itself may contribute to better planning of breast conservation surgery.

Multiparameter computational modeling of tumor invasion.

Clinical outcome prognostication in oncology is a guiding principle in therapeutic choice. A wealth of qualitative empirical evidence links disease progression with tumor morphology, histopathology, invasion, and associated molecular phenomena. However, the quantitative contribution of each of the known parameters in this progression remains elusive. Mathematical modeling can provide the capability to quantify the connection between variables governing growth, prognosis, and treatment outcome. By quantifying the link between the tumor boundary morphology and the invasive phenotype, this work provides a quantitative tool for the study of tumor progression and diagnostic/prognostic applications. This establishes a framework for monitoring system perturbation towards development of therapeutic strategies and correlation to clinical outcome for prognosis.