SIAM Nuggets

Overcoming structural uncertainty in computer models

What is good enough to aid health economics decision making?

Philadelphia, PA—A computer model is a representation of the functional relationship between one set of parameters, which forms the model input, and a corresponding set of target parameters, which forms the model output. A true model for a particular problem can rarely be defined with certainty. The most we can do to mitigate error is to quantify the uncertainty in the model.

In a recent paper published in the SIAM/ASA Journal on Uncertainty Quantification, authors Mark Strong and Jeremy Oakley offer a method to incorporate judgments into a model about structural uncertainty that results from building an “incorrect” model.

“Given that ‘all models are wrong,’ it is important that we develop methods for quantifying our uncertainty in model structure such that we can know when our model is ‘good enough’,” author Mark Strong says. “Better models mean better decisions.” Read the rest of this entry »

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Applying math to cancer, climate, crime and cameras

  Some SIAM journal papers you should know about:

Improving radiation therapies for cancer mathematically

In a paper published in December in the SIAM Journal on Scientific Computing, authors Li-Tien Cheng, Bin Dong, Chunhua Men, Xun Jia, and Steve Jiang propose a method to optimize radiation therapy treatments in cancer patients.

Radiation therapy is one of the primary methods used for cancer treatment, along with chemotherapy and surgery. While doses of radiation are delivered to eliminate cancerous tissue, care is taken to keep radiation within acceptable levels so as not to affect neighboring tissues and organs. The most common type of therapy delivers high-energy radiation via a medical linear accelerator mounted on a rotating apparatus to adjust the direction, and a collimator to shape the beam of radiation. In the recently developed volumetric modulated arc therapy (VMAT), beams continuously deliver doses as the delivery device rotates around the patient.  Enhancement of radiotherapy treatment is challenged by complexities of shape optimization, due to the mechanics of the equipment involved as well as the apertures of devices delivering the beams of radiation. Read the rest of this entry »

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Statistics research could build consensus around climate predictions

 Philadelphia, PA—Vast amounts of data related to climate change are being compiled by research groups all over the world. Data from these many and varied sources results in different climate projections; hence, the need arises to combine information across data sets to arrive at a consensus regarding future climate estimates.

In a paper published last December in the SIAM Journal on Uncertainty Quantification, authors Matthew Heaton, Tamara Greasby, and Stephan Sain propose a statistical hierarchical Bayesian model that consolidates climate change information from observation-based data sets and climate models.

“The vast array of climate data—from reconstructions of historic temperatures and modern observational temperature measurements to climate model projections of future climate—seems to agree that global temperatures are changing,” says author Matthew Heaton. “Where these data sources disagree, however, is by how much temperatures have changed and are expected to change in the future.  Our research seeks to combine many different sources of climate data, in a statistically rigorous way, to determine a consensus on how much temperatures are changing.” Read the rest of this entry »

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Algorithms enhance imaging techniques for breast cancer diagnosis

 Philadelphia, PA—Over 200,000 women are diagnosed with breast cancer every year. The five-year survival rate for afflicted women is 97% if the cancer is localized and discovered before it spreads to other parts of the body.

“Currently, mammography is the technique used most often for breast cancer screening, but since it gives only two-dimensional (2D) projection information of a three-dimensional (3D) anatomical structure, inaccuracies in screening often occur,” says James Nagy, co-author of a paper on breast tomosynthesis image reconstruction published last fall in the SIAM Journal on Scientific Computing (along with Veronica Mejia Bustamante, Steve Feng, and Ioannis Sechopoulos).

While conventional x-ray mammography produces 2D projection images of 3D objects, digital technologies such as tomosynthesis can produce 3D image information of an object by using slightly modified conventional digital x-ray systems. Conventional mammography is limited by superposition of breast tissue, which can sometimes mimic or obscure malignant pathology. Read the rest of this entry »

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Math models analyze the evolution of epidemics during air travel

 Philadelphia, PA—Air travel connects distant worldwide territories like never before. The benefits of high-connectedness are numerous, but epidemic outbreaks in recent decades highlight one pitfall of the global air travel network. Commercial flights remain a leading factor in the spread of commonly-known infectious diseases like tuberculosis, measles, and seasonal influenza.

In a paper published this fall in the SIAM Journal on Applied Dynamical Systems, Diána H. Knipl, Gergely Röst, and Jianhong Wu formulate a model to describe the evolution of an epidemic in regions connected by international flights. Using the 2009 influenza A(H1N1) epidemic in Mexico and Canada as an example,  the model describes the spread of disease within and between two regions. Read the rest of this entry »

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A mathematical perspective of seasonal variations in Lyme disease transmission

 Philadelphia, PA—Lyme disease is a common tick-borne illness caused by a bacterium, which is transmitted to humans through the bite of infected ticks. The transmission dynamics of Lyme disease is dependent on a variety of factors, including the length of the tick’s life cycle, availability of hosts, climatic conditions and seasonal influences, which are important to understand for control strategies.

In a paper published last month in the SIAM Journal on Applied Mathematics, authors Yuxiang Zhang and Xiao-Qiang Zhao propose a reaction-diffusion model to study transmission dynamics of Lyme disease while taking into account seasonality.

Ticks live for roughly 2 years, and their life cycle includes three stages: larva, nymph and adult. Ticks climb on to host animals who brush against vegetation from the tips of grasses and shrubs. Once they attach themselves, they feed on blood by inserting their mouthparts into the skin of a host, thus transmitting the disease. After obtaining a blood meal—which can take anywhere between 3 and 5 days—ticks drop off their hosts and prepare for the next stage of the life cycle. Read the rest of this entry »

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Math models enhance current therapies for coronary heart disease

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Equations help explain key parameters of stents that combat artherosclerosis

Philadelphia, PA—Coronary heart disease accounts for 18% of deaths in the United States every year. The disease results from a blockage of one or more arteries that supply blood to the heart muscle. This occurs as a result of a complex inflammatory condition called artherosclerosis, which leads to progressive buildup of fatty plaque near the surface of the arterial wall.

In a paper published last month in the SIAM Journal on Applied Mathematics, authors Sean McGinty, Sean McKee, Roger Wadsworth, and Christopher McCormick devise a mathematical model to improve currently-employed treatments of coronary heart disease (CHD).

“CHD remains the leading global cause of death, and mathematical modeling has a crucial role to play in the development of practical and effective treatments for this disease,” says lead author Sean McGinty. “The use of mathematics allows often highly complex biological processes and treatment responses to be simplified and written in terms of equations which describe the key parameters of the system. The solution of these equations invariably provides invaluable insight and understanding that will be crucial to the development of better treatments for patients in the future.” Read the rest of this entry »

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An intersection of math and biology: Clams and snails inspire robotic diggers and crawlers

 Philadelphia, PA—Engineering has always taken cues from biology. Natural organisms and systems have done well at evolving to perform tasks and achieve objectives within the limits set by nature and physics.

That is one of the reasons Anette Hosoi, professor of mechanical engineering at the Massachusetts Institute of Technology, studies snails. Snails can move in any direction—horizontally, vertically, and upside down—on various surfaces, be it sand, shells, tree barks or slick walls and smooth glass. One of the reasons for this is the sticky substance on their underbellies, which acts as a powerful lubricant and reduces friction during movement. Read the rest of this entry »

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Are you ready to retire?

 Mathematical models estimate the value of pension plans

Philadelphia, PA—There comes a time in each of our lives when we consider starting a pension plan –either on the advice of a friend, a relative, or of our own volition. The plan of choice may depend on various factors, such as the age and salary of the individual, number of years of expected employment, as well as options to retire early or late.

One possible plan is a defined pension plan, where the benefit amount is typically based on the employee’s number of years of service at the time of retirement and the salary and/or average salary over an employment period. For instance, the employee may receive a fraction of the average salary during a certain number of years.

In a paper published last month in the SIAM Journal on Applied Mathematics, authors Carmen Calvo-Garrido, Andrea Pascucci, and Carlos Vázquez present a partial differential equation (PDE) model governing the value of a defined pension plan including the option for early retirement. Read the rest of this entry »

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Seamless photography: using mathematical models for image stitching

 Philadelphia, PA—A photo captures only as much as the camera in use will allow, and is therefore limited by the field of view of the camera’s lens. In the case of smartphones and many advanced cameras, the view from the lens is much smaller than the view from your own eyes.

Panoramic photographs were invented to capture large objects or scenes that could not otherwise fit within the constraints of a single photo. Panoramic photography is achieved through image stitching, a process that combines two or more photographs, seamlessly blending input images with overlapping regions into one picture. A paper published by Wei Wang and Michael Ng in the SIAM Journal on Imaging Sciences this summer aims to develop an algorithm for image stitching.

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