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Interdisciplinary NIST Team Claims qPCR Improvements Using Unusual Math Tweaks


NEW YORK – Compelled by the COVID-19 pandemic to take a new approach, an interdisciplinary team of researchers at the National Institute of Standards and Technology has developed a method it claims will improve real-time PCR results. The team has used unusual mathematical techniques to essentially pull additional data from the noise in qPCR amplification curves and provide confidence in positive results.

Similar studies published recently are also bringing a fresh math and physics perspective to decades-old qPCR technology in the hopes of improving outcomes, although some experts appear skeptical that software tweaks can make much of a dent in noise caused by sample processing.

The new NIST method, described last month in Analytical and Bioanalytical Chemistry, uses mathematical approaches called uncertainty quantification and affine transformation to elucidate that true positive qPCR curves are essentially identical for a given target and can therefore be resized and reoriented using mathematical tricks to pull out additional positives that would otherwise be lost in noise.

Specifically, the team showed that it could detect true positive samples at fluorescence values somewhere between a factor of two and a factor of 10 lower than a typical thresholding technique. The technique was validated with dilutions of SARS-CoV-2 reference material generated at NIST.

The interdisciplinary team involved applied mathematicians and mathematical modelers Paul Patrone and Anthony Kearsley, as well as Peter Vallone, Erica Romsos, and Megan Cleveland, experts at NIST's applied genetics group who specialize in optimizing qPCR.

Patrone and Kearsley had recently applied affine transformation — a mathematical technique that scales and squeezes a dataset while preserving certain fixed relationships within it — to investigate DNA origami folding and melt curve analysis.

Vallone, Romsos, and Cleveland, meanwhile, are part of a team at NIST that recently developed SARS-CoV-2 qPCR reference material.

The team claims its mathematical analysis reveals that, essentially, qPCR amplification curves for a given target and chemistry are identical, and that some data that would normally be discarded can be scaled to map onto a master curve and counted as a true positive signal. Similarly, curves that do not conform to a master curve could be more readily considered true negatives.

So, the NIST method can use the shape of the curve, rather than a thresholding approach, to determine which curves are true positives and which are likely to be noise.

By way of analogy, Patrone said mapping curves to a master curve in this way is akin to if someone were given a jumble of triangles of different scale and orientation. With just the right rotation and scaling, all right triangles would map on top of each other, as would all isosceles triangles and all obtuse triangles.

"For a specific chemistry, so for example the N1 element of the SARS-CoV-2 assay, qPCR curves are identical," he said. "They may be different from the N2 and N3 segments, but all N1 [curves] for a given chemistry should be the same," Patrone said. By scaling them with an affine transformation, all N1 curves will fall on top of each other "to within the noise of the [qPCR] instrument," he said, adding, "It is really beautiful."

The NIST investigation also undertook uncertainty quantification, which is an integral part of the analysis but also offers an added degree of consistency check within the data analysis, Patrone said.

Overall, the team found the results surprising, Kearsley said.

"If you look at the [qPCR] data Pete and Erica gave us, it looks like spaghetti — it has all these curves going — and when we tried to shrink them and put them on top of each other, they were all landing right on top of each other ... It was fundamentally, mathematically the same curve," he said, noting that the differences fell within 1 percent. "That's when we knew that we were correct, and this was in fact the right way to view these curves," he said.

Being able to work with the people developing reference material for SARS-CoV-2 was also beneficial, Patrone said. "We got to be one of the first users of the material, so we had these really pristine SARS-CoV-2 RNA samples ... There was a lot of concern because we were moving so quickly, and this is so new, that we really hope this works for coronavirus, and lo and behold we are working with people who are developing the standard — that was really an integral part of this," he said. 

Aware of occasional IP squabbles in the qPCR space, the team also pored over patents to make sure its method is original, and discovered surprises there as well.

"We found five or six patents that did linear regression, and they all had different names for it; one said, 'fit a line,' one said 'fit the equation y=ax+b,' one said 'find the slope' — there were literally five different terms for the same thing, so this must be why [developers] are terrified, because everyone has patented fitting a line," Patrone said.

Nevertheless, the NIST team determined its algorithm is novel and it has now patented it and is offering it for a no-cost license, Kearsley said.

"People could put this in software now. It might take a good programmer a couple of days to a week," Patrone said.

Vallone pointed out that currently, if something looks off in a curve by eye, an expert qPCR user will instinctively know something is not right. But this new method is "more subtle, in terms of comparing curve morphologies, [so you] know the curve is coming from the type of target that you are expecting," he said.

Patrone also commented that while everyone presumes there is exponential growth in qPCR curves up to a point, the problem is, "'Up to a point' is where the dark magic is." The affine transformation can now detect whether a curve matches the master curve even if someone put too few reagents in, for example, and this holds for the entire curve, too, not just the exponential phase, he said.

"The idea of having some underlying math that might work in the software analysis, that is not costing you anything but giving you a little more information about the robustness for validation, that is a positive from a wet-lab standpoint," Vallone said.

Outsiders and outside perspectives

The upshot of asking very basic questions is that it can potentially lead to insights, Patrone said. "When you have a pair of fresh eyes on something, they bring you perspectives and can see things that the community hasn't always paid attention to," he said.

Real-time PCR is indeed a very standard lab technique. "Those curves have been around forever, as has the software to analyze them," Vallone said. But, he emphasized, the pandemic has invigorated people to think differently and bring new teams together, and the process has made his wet-lab team think more critically about the fundamentals of things they do many times a day.

In fact, the NIST team was so diverse, Kearsely said they had to first learn each other's jargon. "It took us a while to hone in on the right words," he said.

Interestingly, there doesn't seem to be a consensus name for using mathematical and data analytical approaches to qPCR. It can be math, physical chemistry, chemical physics, or chemometrics, for example. Kearsley said chemometrics is a phrase used to describe "well-known mathematical techniques in not well-known applications," but Patrone said he struggles with classifying the work as chemometrics because it "almost makes it sound too much like data analysis," when it is actually a different type of work.

For example, "We had a ton of conversations with Pete and Erica. We went to the lab. We have also taken instruments apart, literally using screwdrivers and breaking machines," Patrone said. "I think the thing that is hard to capture is how interdisciplinary this is," he said.

In the end, Kearsley said the diverse team was able to use "sophisticated mathematics, with sophisticated physics intuition, together with expert experimental insight and intuition."

Other researchers and companies are using qPCR information in new ways of late, such as by using data analytics to enhance multiplexing of existing qPCR instruments.

One recent more math-heavy example is a new approach to qPCR data analysis that aspires to increase the precision of the calibration curve assay with a strategy called pairwise analysis.

And, researchers in Italy recently created a tool in the web-based workbench Galaxy called PIPE-T that can be used for "parsing, filtering, normalizing, imputing, and analyzing RT-qPCR expression data." The tool was validated on two extant datasets, which some researchers suggest is an important next step in theoretical exploits.

Joel Tellinghuisen, a professor emeritus at Vanderbilt University who specializes in chemometrics, said the new NIST method looks promising. But, "the proof is in the pudding," he said. Comparison testing against existing methods and using large replicate datasets is important, he said in an email.

Tellinghuisen published a method last year on qPCR data analysis which teased, "Better results through iconoclasm." With his colleague, Andrej-Nikolai Spiess at University Hospital Hamburg-Eppendorf, Tellinghuisen fit qPCR growth profiles to a nonlinear algorithm then tested out the method using six qPCR data sets. The pair found some unexpected results that violate tenets of qPCR, such as that amplification efficiency appears to depend on the initial number of target molecules.

Spiess observed in an email that the NIST study does not apply the affine transformation approach to very late qPCR curves, above 35 cycles. "These late curves may be encountered by COVID-19 PCR in the presence of very low viral load," he said, noting also that "very late curves are not sigmoidal and may not be amenable to affine transformation."

Still, using affine transformations is an interesting approach, according to John Brunstein, president of molecular diagnostics consulting service PathoID who has written about qPCR curve fundamentals and was not associated with the NIST work.

"The underlying issue this seeks to address ­— 'Is my late rising real-time PCR trace real or not?' — is a real one, and commonly encountered," Brunstein said.

He observed that for late rising signals, "visual subjective comparison of the curve shape compared to in-run positive controls" might be something lab staff regularly do in trying to assess true versus spurious noise signals.

"What this paper does is provide an objective mathematical basis for this, which could be coded into instrument software algorithms and potentially allow for more accurate calling of weak samples," Brunstein said. "The concept is thus not necessarily novel, but the rigor and uniformity with which it could be employed is much improved."

While the method may be less generally applicable due to issues like variable inhibition and target sequence variation, "I would not be surprised to see some instrument manufacturers embedding this approach in particular assays in future, with improvements to accuracy of calling of weak, low-titer samples," Brunstein said.

Jan Ruijter, a qPCR expert recently retired from the Academic Medical Centre in Amsterdam, the Netherlands, concurred with the NIST team's underlying assumptions about qPCR curves.

"So-called late reactions, that would not even reach a high quantification threshold, can still be identified as generating the correct product and should be considered positive reactions, [w]hile the standard software of the qPCR machine might probably reject them because the plateau and threshold are not reached," he said.

On the other hand, Ruijter expressed concern that the method may not be able to distinguish correct amplification events from artefacts, like primer dimers.

And, standardizing sample collection and sample prep, particularly during the COVID-19 pandemic, is a major hurdle and causes differences in amplification efficiency, Ruijter said. A strictly applied rule that an amplification curve should fit the collapsed curve to be considered positive might miss positive samples because deviating curves with different slopes might be excluded, he suggested.

Jo Mailleux, who heads up the assay digitization team at Belguim-based UgenTec is also an expert in qPCR analysis. Result interpretation is complex and the default software that comes with instruments tends to impart significant challenges, he commented in an email.

"The affine amplification technique is innovative," he said. But, he noted that to be maximally useful the new scaling method might itself need to be carefully scaled in order to match increasingly complex laboratory workflows with multiplex panels and high sample volumes.