Finding the right math for medical problems

The linked article is for SIAM News, the magazine for members of the Society for Industrial and Applied Mathematics (SIAM). However, even though the main audience for this magazine is professional mathematicians, I wrote it to be understandable even if you gloss over the math. And it involves the word “tortuosity”, which is just fun to say.

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A Nonparametric Swiss Army Knife for Medicine

For SIAM News:

The complexity of living things is frequently humbling for mathematicians. Even a single cell contains a plethora of processes and complicated interactions that tractable mathematical models cannot easily describe. Researchers have applied nonlinear dynamics, mechanical analogs, and numerous other techniques to understand biological systems, but the tradeoffs of modeling often err on the side of reductionism.

For this reason, Heather Harrington of the University of Oxford and her collaborators are turning to global mathematical methods and drawing on experimental data to identify the best techniques. Harrington described several of these methods during her invited talk at the 2021 SIAM Conference on Applications of Dynamical Systems, which took place virtually earlier this year.

“The way that we look at dynamical systems is usually in a small region of the parameter space,” Harrington said. This approach is helpful if one knows a lot about the model and its parameters, but it can be hard to extract detailed predictions from the model if the parameters in question range over large values. “In biology, we often don’t know if the system is very close to a value in parameter space because the variables or parameters are difficult to measure or the data is too messy,” she added.

[read the rest at SIAM News]

The danger of climate change may be its rate

As with many of my other contributions to SIAM News, the article “It’s Not the Heat, It’s the Rate: Rate-Inducted Tipping’s Relation to Climate Change” includes some mathematical equations, but I’ve tried to write the piece so you can understand it even if you gloss over that part. And this article in particular has some important concepts relating to the biggest issue facing humanity today: climate change.

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It’s Not the Heat, It’s the Rate

Rate-Inducted Tipping’s Relation to Climate Change

For SIAM News:

For many years, scientists have warned that the Atlantic meridional overturning circulation (AMOC)—the thermal cycle that drives currents in the Atlantic Ocean—is getting weaker [1]. Among other effects, the AMOC carries warm water to Ireland and the U.K. and returns cooler water from the north to southern regions. Instability in this circulation cycle could result in its complete collapse and cause widespread disruptions in temperature, changes in rain and snowfall patterns, and other natural disasters.

The potential loss of the AMOC represents a possible tipping point due to human-driven climate change. Global increases in temperature lead to warmer ocean water and melting polar ice, both of which decrease water density (see Figure 1). The subsequent lower-density water does not sink as much as it cools, thus disrupting the thermal cycle. When the AMOC collapsed in the prehistoric past, it jolted Earth’s climate and affected every ecosystem.

[Read the rest at SIAM News]

Bicycles, networks, and biological homeostasis

The linked article is for SIAM News, the magazine for members of the Society for Industrial and Applied Mathematics (SIAM). The audience for this magazine, in other words, is professional mathematicians and related researchers working in a wide variety of fields. While this article contains equations, I wrote it to be understandable even if you gloss over the math.

[ This blog is dedicated to tracking my most recent publications. Subscribe to the feed to keep up with all the science stories I write! ]

Balancing Homeostasis and Health

For SIAM News:

Human beings are not bicycles. However, mechanistic metaphors for the human body abound. For instance, we compare athletes to finely-tuned machines and look for equations that are derived from mechanics to describe biological processes — even when the relationship is no better than an analogy.

However, the concept of homeostasis clearly exemplifies the breakdown of mechanistic models when one applies them to the human body. Homeostasis is the process by which an organism maintains a stable output regardless of input (within reasonable limits). The most familiar example is human body temperature, which stays within a remarkably small range of values regardless of whether one is sitting in a cold room or walking outside on a hot day.

“In a bicycle, you know what each part is for,” Michael Reed, a mathematician at Duke University, said. “We are not machines with fixed parts; we are a large pile of cooperating cells. The question is, how does this pile of cooperating cells accomplish various tasks?”

[ Read the rest at SIAM News ]

Ecological stability far from equilibrium

toxic algae on Lake Erie, as seen by the Landsat 8 satellite

The linked article is for SIAM News, the magazine for members of the Society for Industrial and Applied Mathematics (SIAM). The audience for this magazine, in other words, is professional mathematicians and related researchers working in a wide variety of fields. While this article contains equations, I wrote it to be understandable even if you gloss over the math.

[ This blog is dedicated to tracking my most recent publications. Subscribe to the feed to keep up with all the science stories I write! ]

Ecological Transients and the Ghost of Equilibrium Past

For SIAM News:

The sight and smell of eutrophication—in the form of a layer of stinking green algae on a lake or pond—is likely familiar to many readers. The result is detrimental, even toxic, to other species that rely on the water, ranging from tiny animals to birds and even humans. For example, eutrophication on Lake Erie affects millions (see Figure 1). But the real culprit is actually the substance that feeds the algae: excess phosphorous that is produced by human activities like fertilizer runoff and leaky septic systems.

To manage eutrophication, one must know whether the affected body of water resides in a eutrophic stable state, or if its state is a long transient. The second case mimics stability because it can last a long time but is sustained by another source of phosphorous in the lakebed sediments. According to Tessa Francis, an ecologist at the University of Washington Puget Sound Institute, the wrong management choice has major consequences in terms of costs and trade-offs.

“You’re investing all of this social, political, and economic capital into management, but you’re getting no results from it,” Francis said. “If you gave the system a bigger smack by adding an alternative management strategy to tackle the phosphorus pool at the bottom of the lake, that would be more likely to get your lake back to the state you want. This is just one consequence of long transients in terms of how they affect management decisions.”

[Read the rest at SIAM NEWS]

Fighting racial gerrymandering with math

The linked article is for SIAM News, the magazine for members of the Society for Industrial and Applied Mathematics (SIAM). The audience for this magazine, in other words, is professional mathematicians and related researchers working in a wide variety of fields. While this article contains equations, I wrote it to be understandable even if you gloss over the math.

[ This blog is dedicated to tracking my most recent publications. Subscribe to the feed to keep up with all the science stories I write! ]

The Mathematical Fight for Voting Rights

For SIAM News:

State and local governments will redraw voting districts based on new information following completion of the 2020 U.S. Census. Ideally, this process ensures fair representation. In practice, however, districting often involves gerrymandering: the deliberate planning of districts to dilute the voting power of certain groups in favor of others, which violates the law.

Racial gerrymandering—drawing districts to limit the power of voters of color to select candidates they favor—is a particularly pernicious problem. Section 2 of the Voting Rights Act (VRA) of 1965 specifically prohibits this practice, but that has not stopped authorities from doing it anyway. “A number of court decisions have purposefully asked mathematicians, political scientists, and statisticians to use specific methods to try and understand racial gerrymandering,” Matt Barreto, a professor of political science and Chicana/o studies at the University of California, Los Angeles, said.

Barreto and his colleagues employ powerful statistical methods and draw on census and other public data to identify gerrymandered districts. Utilizing these tools, mathematicians can test proposed district maps or draw their own, designing them from the ground up to prevent voter dilution.

[Read the rest at SIAM News…]

The science connecting extreme weather to climate change

The linked article is for SIAM News, the magazine for members of the Society for Industrial and Applied Mathematics (SIAM). The audience for this magazine, in other words, is professional mathematicians and related researchers working in a wide variety of fields. While this article contains (just a few simple) equations, I wrote it to be understandable even if you skip over the math.

[ This blog is dedicated to tracking my most recent publications. Subscribe to the feed to keep up with all the science stories I write! ]

Linking Extreme Weather to Climate Change

For SIAM News:

As the world’s climate changes, the warming atmosphere and oceans produce heavier rainfalls and more hurricanes, snowstorms, and other instances of extreme weather. Climate models predict the change in frequency of these events as a result of human-driven global warming. However, scientists and non-scientists alike are interested in whether climate change is responsible for specific weather events — such as Hurricane Maria, which devastated Puerto Rico in 2017.

“The kosher answer to this used to be that we can never say that climate change causes a specific event,” statistician Claudia Tebaldi of the University of Maryland’s Joint Global Change Research Institute said. “This has actually changed over time, because a few recent events were so extreme that the probability of observing them without climate change would have been practically zero.”

In other words, scientists and science communicators are growing increasingly confident about linking specific weather to global changes, a subfield of climate science and meteorology known as “event attribution.” Researchers calculate the probability of a particular event’s occurrence with or without climate change by considering a combination of factors, including human activity and variations that are independent of human contribution. Event attribution is a relatively recent discipline; scientists first used it to link climate change to the 2003 European heat wave [4], which killed thousands of people.

[Read the rest at SIAM News…]

Protecting privacy with mathematics

The linked article is for SIAM News, the magazine for members of the Society for Industrial and Applied Mathematics (SIAM). The audience for this magazine, in other words, is professional mathematicians and related researchers working in a wide variety of fields. While the article contains equations, I wrote it to be understandable even if you skip over the math.

[ This blog is dedicated to tracking my most recent publications. Subscribe to the feed to keep up with all the science stories I write! ]

Using Differential Privacy to Protect the United States Census

Census data must simultaneously be publicly available and protect the privacy of the people it describes. Differential privacy is a method that injects noise into the data to hide the presence of individual responses, while preserving the general statistical structure of the data. [Credit: moi, which is why I’m not a professional graphic artist]

For SIAM News:

In 2006, Netflix hosted a competition to improve its algorithm for providing movie recommendations to customers based on their past choices. The DVD rental and video streaming service shared anonymized rental records from real subscribers, assuming that their efforts to remove identifying information sufficiently protected user identities. This assumption was wrong; external researchers quickly proved that they could pinpoint personal details by correlating other public data with the Netflix database, potentially exposing private information.

This fatal flaw in the Netflix Prize challenge highlights multiple issues concerning privacy in the information age, including the simultaneous need to perform statistical analyses while protecting the identities of people in the dataset. Merely hiding personal data is not enough, so many statisticians are turning to differential privacy. This method allows researchers to extract useful aggregate information from data while preserving the privacy of individuals within the sample.

“Even though researchers are just trying to learn facts about the world, their analyses might incidentally reveal sensitive information about particular people in their datasets,” Aaron Roth, a statistician at the University of Pennsylvania, said. “Differential privacy is a mathematical constraint you impose on an algorithm for performing
data analysis that provides a formal guarantee of privacy.”

[read the rest at SIAM News…]

Gaining time for brain cancer patients with mathematics

The linked article is for SIAM News, the magazine for members of the Society for Industrial and Applied Mathematics (SIAM). The audience for this magazine, in other words, is professional mathematicians and related researchers working in a wide variety of fields. While the article contains equations, I wrote it to be understandable even if you skip over the math.

[ This blog is dedicated to tracking my most recent publications. Subscribe to the feed to keep up with all the science stories I write! ]

Mathematical Modeling Gains Days for Brain Cancer Patients

For SIAM News:

Glioblastoma, or glioblastoma multiforme, is a particularly aggressive and almost invariably fatal type of brain cancer. It is infamous for causing the deaths of U.S. Senators John McCain and Ted Kennedy, as well as former U.S. Vice President Joe Biden’s son Beau. Though glioblastoma is the second-most common type of brain tumor—affecting roughly three out of every 100,000 people—medicine has struggled to find effective remedies; the U.S. Food and Drug Administration has approved only four drugs and one device to counter the condition in 30 years of research. The median survival rate is less than two years, and only about five percent of all patients survive five years beyond the initial diagnosis.

Given these terrible odds, medical researchers strive for anything that can extend the effectiveness of treatment. The nature of glioblastoma itself is responsible for many obstacles; brain tumors are difficult to monitor noninvasively, making it challenging for physicians to determine the adequacy of a particular course of therapy.

Figure 1. Magnetic resonance imaging scan of the brain. Public domain image.
Kristin Rae Swanson and her colleagues at the Mayo Clinic believe that mathematical models can help improve patient outcomes. Using magnetic resonance imaging (MRI) data for calibration, they constructed the proliferation-invasion (PI) model — a simple deterministic equation to estimate how cancer cells divide and spread throughout the brain. Rather than pinpoint every cell’s location, the model aims to categorize the general behavior of each patient’s cancer to guide individualized treatment.

[Read the rest at SIAM News]

The mathematics of knowledge networks in the brain

[ This blog is dedicated to tracking my most recent publications. Subscribe to the feed to keep up with all the science stories I write! ]

This article is for SIAM News, the magazine for members of the Society for Industrial and Applied Mathematics (SIAM). The audience for this magazine, in other words, is professional mathematicians and related researchers working in a wide variety of fields. While the article contains equations, I wrote it to be understandable even if you skip over the math.

Understanding Knowledge Networks in the Brain

For SIAM News:

One strength of the human mind is its ability to find patterns and draw connections between disparate concepts, a trait that often enables science, poetry, visual art, and a myriad of other human endeavors. In a more concrete sense, the brain assembles acquired knowledge and links pieces of information into a network. Knowledge networks also seem to have a physical aspect in the form of interconnected neuron pathways in the brain.

During her invited address at the 2018 SIAM Annual Meeting, held in Portland Ore., last July, Danielle Bassett of the University of Pennsylvania illustrated how brains construct knowledge networks. Citing early 20th century progressive educational reformer John Dewey, she explained that the goal of a talk—and learning in general—is to map concepts from the speaker/teacher’s mind to those of his or her listeners. When the presenter is successful, the audience gains new conceptual networks.

More generally, Bassett explored how humans acquire knowledge networks, whether that process can be modeled mathematically, and how such models may be tested experimentally. Fundamental research on brain networks can potentially facilitate the understanding and treatment of conditions as diverse as schizophrenia and Parkinson’s disease.

[Read the rest at SIAM News…]

The secret to good digital animation is physics

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This article is a little different from the fare you’re used to getting from me: it’s for SIAM News, which is the magazine for members of the Society for Industrial and Applied Mathematics (SIAM). The audience for this magazine, in other words, is professional mathematicians and related researchers working in a wide variety of fields. While the article contains equations, I wrote it to be understandable even if you skip over the math.

The Serious Mathematics of Digital Animation

For SIAM News:

While computer simulations have a wide range of uses, their goals are generally similar: find the simplest model that recreates the properties of the system under investigation. For scientific systems, this involves matching observed or experimental phenomena as precisely as necessary.

But what about movie simulations? Should they match the processes they replicate so closely? Computer-generated imagery (CGI) is a common feature in both animated and live-action films. For these CGI systems, creating visuals that look right is an important task. However, Joseph Teran of the University of California, Los Angeles believes that starting from physical models is still a good idea.

During his invited address at the 2018 SIAM Annual Meeting, held in Portland, Ore., this July, Teran pointed out that beginning with a mathematical system is often easier than drawing from real life. Many movies model a system’s various forces and internal structures with partial differential equations (PDEs) for this reason. While solving these equations to produce CGI is computationally expensive, such methods have become powerful tools for creating realistic visual cinematic effects.

[Read the rest at SIAM News]