Benjamin R. Ecclestone, Zohreh Hosseinaee, Nima Abbasi, Kevan Bell, Deepak Dinakaran, John Mackey, Parsin Haji Reza
Histological images are critical in the diagnosis and treatment of cancers. Unfortunately, the current method for capturing these microscopy images require resource intensive tissue preparation that delays diagnosis for many days to a few weeks. To streamline this process, clinicians are limited to assessing small macroscopically representative subsets of tissues. Here, we present a combined photoacoustic remote sensing (PARS) microscope and swept source optical coherence tomography (SS-OCT) system designed to circumvent these diagnostic limitations. The proposed multimodal microscope provides label-free three-dimensional depth resolved virtual histology visualizations, capturing nuclear and extranuclear tissue morphology directly on thick unprocessed specimens. The capabilities of the proposed method are demonstrated directly in unprocessed formalin fixed resected tissues. Here, we present the first images of nuclear contrast in resected human tissues, and the first 3-dimensional visualization of subsurface nuclear morphology in resected Rattus tissues, captured with a non-contact photoacoustic system. Moreover, we present the first co-registered OCT and PARS images enabling direct histological assessment of unprocessed tissues. This work represents a vital step towards the development of a real-time histological imaging modality to circumvent the limitations of current histopathology techniques.
David Neiman, John Mackey, Marijn Heule
Tournaments are orientations of the complete graph, and the directed Ramsey number $R(k)$ is the minimum number of vertices a tournament must have to be guaranteed to contain a transitive subtournament of size $k$, which we denote by $TT_k$. We include a computer-assisted proof of a conjecture by Sanchez-Flores that all $TT_6$-free tournaments on 24 and 25 vertices are subtournaments of $ST_{27}$, the unique largest TT_6-free tournament. We also classify all $TT_6$-free tournaments on 23 vertices. We use these results, combined with assistance from SAT technology, to obtain the following improved bounds: $34 \leq R(7) \leq 47$.
Joshua Brakensiek, Marijn Heule, John Mackey, David Narváez
We consider three graphs, $G_{7,3}$, $G_{7,4}$, and $G_{7,6}$, related to Keller's conjecture in dimension 7. The conjecture is false for this dimension if and only if at least one of the graphs contains a clique of size $2^7 = 128$. We present an automated method to solve this conjecture by encoding the existence of such a clique as a propositional formula. We apply satisfiability solving combined with symmetry-breaking techniques to determine that no such clique exists. This result implies that every unit cube tiling of $\mathbb{R}^7$ contains a facesharing pair of cubes. Since a faceshare-free unit cube tiling of $\mathbb{R}^8$ exists (which we also verify), this completely resolves Keller's conjecture.
John Mackey, Bernardo Subercaseaux
Erdős and Guy initiated a line of research studying $μ_k(n)$, the minimum number of convex $k$-gons one can obtain by placing $n$ points in the plane without any three of them being collinear. Asymptotically, the limits $c_k := \lim_{n\to \infty} μ_k(n)/\binom{n}{k}$ exist for all $k$, and are strictly positive due to the Erdős-Szekeres theorem. This article focuses on the case $k=5$, where $c_5$ was known to be between $0.0608516$ and $0.0625$ (Goaoc et al., 2018; Subercaseaux et al., 2023). The lower bound was obtained through the Flag Algebra method of Razborov using semi-definite programming. In this article we prove a more modest lower bound of $\frac{5\sqrt{5}-11}{4} \approx 0.04508$ without any computation; we exploit``planar-point equations'' that count, in different ways, the number of convex pentagons (or other geometric objects) in a point placement. To derive our lower bound we combine such equations by viewing them from a statistical perspective, which we believe can be fruitful for other related problems.
Natasha Komarov, John Mackey
We find an exact formula for the number of directed 5-cycles in a tournament in terms of its edge score sequence. We use this formula to find both upper and lower bounds on the number of 5-cycles in any $n$-tournament. In particular, we show that the maximum number of 5-cycles is asymptotically equal to $\frac{3}{4}{n \choose 5}$, the expected number 5-cycles in a random tournament ($p=\frac{1}{2}$), with equality (up to order of magnitude) for almost all tournaments. Note that this means that almost all $n$-tournaments contain the maximum number of $5$-cycles.
Eason Chen, Sophia Judicke, Kayla Beigh, Xinyi Tang, Isabel Wang, Nina Yuan, Zimo Xiao, Chuangji Li, Shizhuo Li, Reed Luttmer, Shreya Singh, Maria Yampolsky, Naman Parikh, Yvonne Zhao, Meiyi Chen, Scarlett Huang, Anishka Mohanty, Gregory Johnson, John Mackey, Jionghao Lin, Ken Koedinger
We evaluate GPTutor, an LLM-powered tutoring system for an undergraduate discrete mathematics course. It integrates two LLM-supported tools: a structured proof-review tool that provides embedded feedback on students' written proof attempts, and a chatbot for math questions. In a staggered-access study with 148 students, earlier access was associated with higher homework performance during the interval when only the experimental group could use the system, while we did not observe this performance increase transfer to exam scores. Usage logs show that students with lower self-efficacy and prior exam performance used both components more frequently. Session-level behavioral labels, produced by human coding and scaled using an automated classifier, characterize how students engaged with the chatbot (e.g., answer-seeking or help-seeking). In models controlling for prior performance and self-efficacy, higher chatbot usage and answer-seeking behavior were negatively associated with subsequent midterm performance, whereas proof-review usage showed no detectable independent association. Together, the findings suggest that chatbot-based support alone may not reliably support transfer to independent assessment of math proof-learning outcomes, whereas work-anchored, structured feedback appears less associated with reduced learning.
Danny Crytser, Natasha Komarov, John Mackey
We consider "Containment": a variation of the graph pursuit game of Cops and Robber in which cops move from edge to adjacent edge, the robber moves from vertex to adjacent vertex (but cannot move along an edge occupied by a cop), and the cops win by "containing" the robber---that is, by occupying all $°(v)$ of the edges incident with a vertex $v$ while the robber is at $v$. We develop bounds that relate the minimal number of cops, $ξ(G)$, required to contain a robber to the well-known "cop-number" $c(G)$ in the original game: in particular, $c(G) {\le} ξ(G) {\le} γ(G) Δ(G)$. We note that $ξ(G) {\geq} δ(G)$ for all graphs $G$, and analyze several families of graphs in which equality holds, as well as several in which the inequality is strict. We also give examples of graphs which require an unbounded number of cops in order to contain a robber, and note that there exist cubic graphs with $ξ(G) \geq Ω(n^{1/6})$.
Mikhail Lavrov, Mitchell Lee, John Mackey
In [5] Graham and Rothschild consider a geometric Ramsey problem: finding the least n such that if all edges of the complete graph on the points {+1,-1}^n are 2-colored, there exist 4 coplanar points such that the 6 edges between them are monochromatic. They give an explicit upper bound: F(F(F(F(F(F(F(12))))))), where F(m) = 2^^(m)^^3, an extremely fast-growing function. By reducing the problem to a variant of the Hales-Jewett problem, we find an upper bound which is between F(4) and F(5).
Bernardo Subercaseaux, John Mackey, Marijn J. H. Heule, Ruben Martins
We present a comprehensive demonstration of how automated reasoning can assist mathematical research, both in the discovery of conjectures and in their verification. Our focus is a discrete geometry problem: What is $μ_{5}(n)$, the minimum number of convex pentagons induced by $n$ points in the plane? In the first stage toward tackling this problem, automated reasoning tools guide discovery and conjectures: we use SAT-based tools to find abstract configurations of points that would induce few pentagons. Afterward, we use Operations Research tools to find and visualize realizations of these configurations in the plane, if they exist. Mathematical thought and intuition are still vital parts of the process for turning the obtained visualizations into general constructions. A surprisingly simple upper bound follows from our constructions: $μ_{5}(n) \leq \binom{\lfloor n/2 \rfloor}{5} + \binom{\lceil n/2 \rceil}{5}$, and we conjecture it is optimal. In the second stage, we turn our focus to verifying this conjecture. Using MaxSAT, we confirm that $μ_5(n)$ matches the conjectured values for $n \leq 16$, thereby improving both the existing lower and upper bounds for $n \in [12, 16]$. Our MaxSAT results rely on two mathematical theorems with pen-and-paper proofs, highlighting once again the rich interplay between automated and traditional mathematics.
Eason Chen, Sophia Judicke, Kayla Beigh, Xinyi Tang, Zimo Xiao, Chuangji Li, Shizhuo Li, Reed Luttmer, Shreya Singh, Maria Yampolsky, Naman Parikh, Yi Zhao, Meiyi Chen, Scarlett Huang, Anishka Mohanty, Gregory Johnson, John Mackey, Jionghao Lin, Ken Koedinger
We evaluate the effectiveness of LLM-Tutor, a large language model (LLM)-powered tutoring system that combines an AI-based proof-review tutor for real-time feedback on proof-writing and a chatbot for mathematics-related queries. Our experiment, involving 148 students, demonstrated that the use of LLM-Tutor significantly improved homework performance compared to a control group without access to the system. However, its impact on exam performance and time spent on tasks was found to be insignificant. Mediation analysis revealed that students with lower self-efficacy tended to use the chatbot more frequently, which partially contributed to lower midterm scores. Furthermore, students with lower self-efficacy were more likely to engage frequently with the proof-review-AI-tutor, a usage pattern that positively contributed to higher final exam scores. Interviews with 19 students highlighted the accessibility of LLM-Tutor and its effectiveness in addressing learning needs, while also revealing limitations and concerns regarding potential over-reliance on the tool. Our results suggest that generative AI alone like chatbot may not suffice for comprehensive learning support, underscoring the need for iterative design improvements with learning sciences principles with generative AI educational tools like LLM-Tutor.
Benjamin Przybocki, John Mackey, Marijn J. H. Heule, Bernardo Subercaseaux
Ramsey-good graphs are graphs that contain neither a clique of size $s$ nor an independent set of size $t$. We study doubly saturated Ramsey-good graphs, defined as Ramsey-good graphs in which the addition or removal of any edge necessarily creates an $s$-clique or a $t$-independent set. We present a method combining SAT solving with bespoke LLM-generated code to discover infinite families of such graphs, answering a question of Grinstead and Roberts from 1982. In addition, we use LLMs to generate and formalize correctness proofs in Lean. This case study highlights the potential of integrating automated reasoning, large language models, and formal verification to accelerate mathematical discovery. We argue that such tool-driven workflows will play an increasingly central role in experimental mathematics.
Marian Boktor, Benjamin Ecclestone, Vlad Pekar, Deepak Dinakaran, John R. Mackey, Paul Fieguth, Parsin Haji Reza
Histopathological visualizations are a pillar of modern medicine and biological research. Surgical oncology relies exclusively on post-operative histology to determine definitive surgical success and guide adjuvant treatments. The current histology workflow is based on bright-field microscopic assessment of histochemical stained tissues and has some major limitations. For example, the preparation of stained specimens for brightfield assessment requires lengthy sample processing, delaying interventions for days or even weeks. Hence, there is a pressing need for improved histopathology methods. In this paper, we present a deep-learning-based approach for virtual label-free histochemical staining of total-absorption photoacoustic remote sensing (TA-PARS) images of unstained tissue. TA-PARS provides an array of directly measured label-free contrasts such as scattering and total absorption (radiative and non-radiative), ideal for developing H&E colorizations without the need to infer arbitrary tissue structures. We use a Pix2Pix generative adversarial network (GAN) to develop visualizations analogous to H&E staining from label-free TA-PARS images. Thin sections of human skin tissue were first virtually stained with the TA-PARS, then were chemically stained with H&E producing a one-to-one comparison between the virtual and chemical staining. The one-to-one matched virtually- and chemically- stained images exhibit high concordance validating the digital colorization of the TA-PARS images against the gold standard H&E. TA-PARS images were reviewed by four dermatologic pathologists who confirmed they are of diagnostic quality, and that resolution, contrast, and color permitted interpretation as if they were H&E. The presented approach paves the way for the development of TA-PARS slide-free histology, which promises to dramatically reduce the time from specimen resection to histological imaging.
Kevan Bell, Saad Abbasi, Deepak Dinakaran, Muba Taher, Gilbert Bigras, Frank K. H. van Landeghem, John R. Mackey, Parsin Haji Reza
Histological visualizations are critical to clinical disease management and are fundamental to biological understanding. However, current approaches that rely on bright-field microscopy require extensive tissue preparation prior to imaging. These processes are labor intensive and contribute to delays in clinical feedback that can extend to two to three weeks for standard paraffin-embedded tissue preparation and interpretation. Here, we present a label-free reflection-mode imaging modality that reveals cellular-scale morphology by detecting intrinsic endogenous contrast. We accomplish this with the novel photoacoustic remote sensing (PARS) detection system that permits non-contact optical absorption contrast to be extracted from thick and opaque biological targets with optical resolution. PARS was examined both as a rapid assessment tool that is capable of managing large samples (>1 cm2) in under 10 minutes, and as a high contrast imaging modality capable of extracting specific biological contrast to simulate conventional histological stains such as hematoxylin and eosin (H&E). The capabilities of the proposed method are demonstrated in a variety of human tissue preparations including formalin-fixed paraffin-embedded tissue blocks and unstained slides sectioned from these blocks, including normal and neoplastic human brain, and breast epithelium involved with breast cancer. Similarly, PARS images of human skin prepared by frozen section clearly demonstrated basal cell carcinoma and normal human skin tissue. Finally, we imaged unprocessed murine kidney and achieved histologically relevant subcellular morphology in fresh tissue. This represents a vital step towards an effective real-time clinical microscope that overcomes the limitations of standard histopathologic tissue preparations and enables real-time pathology assessment.
Utkarsh Chauhan, Kaiqiong Zhao, John Walker, John R. Mackey
The Kaplan-Meier estimator (KM) is widely used in medical research to estimate the survival function from lifetime data. KM is a powerful tool to evaluate clinical trials due to simple computational requirements, a logrank hypothesis test, and the ability to censor patients. However, KM has several constraints and fails to generalize to ordinal variables of interest such as toxicity and ECOG performance. We devised Weighted Trajectory Analysis (WTA) to combine the advantages of KM with the ability to compare treatment groups for ordinal variables and fluctuating outcomes. To assess statistical significance, we developed a new hypothesis test analogous to the logrank test. We demonstrate the functionality of WTA through 1000-fold clinical trial simulations of unique stochastic models of chemotherapy toxicity and schizophrenia progression. At several increments of sample size and hazard ratio, we compare the performance of WTA to both KM and Generalized Estimating Equations (GEE). WTA generally required half the sample size to achieve comparable power to KM; advantages over GEE include its robust non-parametric approach and summary plot. We also apply WTA to real clinical data: the toxicity outcomes of melanoma patients receiving immunotherapy and the disease progression of patients with metastatic breast cancer receiving ramucirumab. The application of WTA demonstrates that using traditional methods such as percent incidence and KM can lead to both Type I and II errors by failing to model illness trajectory. This article outlines a novel method for clinical outcome assessment that extends the advantages of Kaplan-Meier estimates to ordinal outcome variables.
Benjamin R. Ecclestone, James E. D. Tweel, Marie Abi Daoud, Hager Gaouda, Deepak Dinakaran, Michael P. Wallace, Ally Khan Somani, Gilbert Bigras, John R. Mackey, Parsin Haji Reza
Apr 25, 2025·q-bio.QM·PDF Photon Absorption Remote Sensing (PARS) enables label-free imaging of subcellular morphology by observing biomolecule specific absorption interactions. Coupled with deep-learning, PARS produces label-free virtual Hematoxylin and Eosin (H&E) stained images in unprocessed tissues. This study evaluates the diagnostic performance of PARS virtual H&E images in excisional skin biopsies, including Squamous (SCC), Basal (BCC) Cell Carcinoma, and normal skin. Sixteen unstained formalin-fixed paraffin-embedded skin excisions were PARS imaged, virtually H&E stained, then chemically stained and imaged at 40x. Seven fellowship trained dermatopathologists assessed all images. Example PARS and chemical H&E whole-slide images from this study are available at the BioImage Archive (https://doi.org/10.6019/S-BIAD2324). Concordance analysis indicates 95.5% agreement between primary diagnoses from PARS versus H&E images (Cohen's k=0.93). Inter-rater reliability was near-perfect for both image types (Fleiss' k=0.89 for PARS, k=0.80 for H&E). For subtype classification, agreement was near-perfect 91% (k=0.73) for SCC and was perfect for BCC. For malignancy confinement (e.g., cancer margins), agreement was 92% between PARS and H&E (k=0.718). During assessment dermatopathologists could not reliably distinguish image origin (PARS vs. H&E), and diagnostic confidence was equivalent. Inter-rater reliability for PARS virtual H&E was consistent with reported histologic evaluation benchmarks. These results indicate that PARS virtual histology may be diagnostically equivalent to chemical H&E staining in dermatopathology diagnostics, while enabling assessment directly from unlabeled slides. In turn, the label-free PARS virtual H&E imaging workflow may preserve tissue for downstream analysis while producing data well-suited for AI integration potentially accelerating and enhancing skin cancer diagnostics.
Utkarsh Chauhan, Daylen Mackey, John R Mackey
Analyzing and effectively communicating the efficacy and toxicity of treatment is the basis of risk benefit analysis (RBA). More efficient and objective tools are needed. We apply Chauhan Weighted Trajectory Analysis (CWTA) to perform RBA with superior objectivity, power, and clarity. We used CWTA to perform 1000-fold simulations of RCTs using ordinal endpoints for both treatment efficacy and toxicity. RCTs were simulated with 1:1 allocation at defined sample sizes and hazard ratios. We studied the simplest case of 3 levels each of toxicity and efficacy and the general case of the advanced cancer trial, with efficacy graded by five RECIST 1.1 health statuses and toxicity by the six-point CTCAE scale (6 x 5 matrix). The latter model was applied to a real-world dose escalation phase I trial in advanced cancer. Simulations in both the 3 x 3 and the 6 x 5 advanced cancer matrix confirmed that drugs with both superior efficacy and toxicity profiles synergize for greater statistical power with CWTA-RBA. The CWTA-RBA 6 x 5 matrix reduced sample size requirements over CWTA efficacy-only analysis. Application to the dose finding phase I clinical trial provided objective, statistically significant validation for the selected dose. CWTA-RBA, by incorporating both drug efficacy and toxicity, provides a single test statistic and plot that analyzes and effectively communicates therapeutic risks and benefits. CWTA-RBA requires fewer patients than CWTA efficacy-only analysis when the experimental drug is both more effective and less toxic. CWTA-RBA facilitates the objective and efficient assessment of new therapies throughout the drug development pathway. Furthermore, several advantages over competing tests in communicating risk-benefit will assist regulatory review, clinical adoption, and understanding of therapeutic risks and benefits by clinicians and patients alike.
Benjamin R. Ecclestone, Kevan Bell, Saad Abbasi, Deepak Dinakaran, Muba Taher, John R. Mackey, Parsin Haji Reza
Mohs micrographic surgery (MMS) is a precise oncological technique where layers of tissue are resected and examined with intraoperative histopathology to minimize the removal of normal tissue while completely excising the cancer. To achieve intraoperative pathology, the tissue is frozen, sectioned and stained over a 20- to 60-minute period, then analyzed by the MMS surgeon. Surgery is continued one layer at a time until no cancerous cells remain, meaning MMS can take several hours to complete. Ideally, it would be desirable to circumvent or augment frozen sectioning methods and directly visualize subcellular morphology on the unprocessed excised tissues. Employing photoacoustic remote sensing (PARS) microscopy, we present a non-contact label-free reflection-mode method of performing such visualizations in frozen sections of human skin. PARS leverages endogenous optical absorption contrast within cell nuclei to provide visualizations reminiscent of histochemical staining techniques. Presented here, is the first true one to one comparison between PARS microscopy and standard histopathological imaging in human tissues. We demonstrate the ability of PARS microscopy to provide large grossing scans (>1 cm2, sufficient to visualize entire MMS sections) and regional scans with subcellular lateral resolution (~300 nm).
Benjamin R. Ecclestone, Kevan Bell, Saad Abbasi, Deepak Dinakaran, Frank K. H. van Landeghem, John R. Mackey, Paul Fieguth, Parsin Haji Reza
Malignant brain tumors are among the deadliest neoplasms with the lowest survival rates of any cancer type. In considering surgical tumor resection, suboptimal extent of resection is linked to poor clinical outcomes and lower overall survival rates. Currently available tools for intraoperative histopathological assessment require an average of 20 minutes processing and are of limited diagnostic quality for guiding surgeries. Consequently, there is an unaddressed need for a rapid imaging technique to guide maximal resection of brain tumors. Working towards this goal, presented here is an all optical non-contact label-free reflection mode photoacoustic remote sensing (PARS) microscope. By using a tunable excitation laser, PARS takes advantage of the endogenous optical absorption peaks of DNA and cytoplasm to achieve virtual contrast analogous to standard hematoxylin and eosin (H and E) staining. In conjunction, a fast 266 nm excitation is used to generate large grossing scans and rapidly assess small fields in real-time with hematoxylin-like contrast. Images obtained using this technique show comparable quality and contrast to the current standard for histopathological assessment of brain tissues. Using the proposed method, rapid, high-throughput, histological-like imaging was achieved in unstained brain tissues, indicating PARS utility for intraoperative guidance to improve extent of surgical resection.
Benjamin R. Ecclestone, Kevan Bell, Sarah Sparkes, Deepak Dinakaran, John R. Mackey, Parsin Haji Reza
In the past decades, absorption modalities have emerged as powerful tools for label-free functional and structural imaging of cells and tissues. Many biomolecules present unique absorption spectra providing chromophore-specific information on properties such as chemical bonding, and sample composition. As chromophores absorb photons the absorbed energy is emitted as photons (radiative relaxation) or converted to heat and under specific conditions pressure (non-radiative relaxation). Modalities like fluorescence microscopy may capture radiative relaxation to provide contrast, while modalities like photoacoustic microscopy may leverage non-radiative heat and pressures. Here we show an all-optical non-contact total-absorption photoacoustic remote sensing (TA-PARS) microscope, which can capture both radiative and non-radiative absorption effects in a single acquisition. The TA-PARS yields an absorption metric proposed as the quantum efficiency ratio (QER), which visualizes a biomolecules proportional radiative and non-radiative absorption response. The TA-PARS provides label-free visualization of a range of biomolecules enabling convincing analogues to traditional histochemical staining of tissues, effectively providing label-free Hematoxylin and Eosin (H&E)-like visualizations. These findings represent the establishment of an effective all-optical non-contact total-absorption microscope for label-free inspection of biological media.
Utkarsh Chauhan, Daylen Mackey, John R. Mackey
As Kaplan-Meier (KM) analysis is limited to single unidirectional endpoints, most advanced cancer randomized clinical trials (RCTs) are powered for either progression free survival (PFS) or overall survival (OS). This discards efficacy information carried by partial responses, complete responses, and stable disease that frequently precede progressive disease and death. Chauhan Weighted Trajectory Analysis (CWTA) is a generalization of KM that simultaneously assesses multiple rank-ordered endpoints. We hypothesized that CWTA could use this efficacy information to reduce sample size requirements and expedite efficacy signals in advanced cancer trials. We performed 100-fold and 1000-fold simulations of solid tumour systemic therapy RCTs with health statuses rank ordered from complete response (Stage 0) to death (Stage 4). At increments of sample size and hazard ratio, we compared KM PFS and OS with CWTA for (i) sample size requirements to achieve a power of 0.8 and (ii) time-to-first significant efficacy signal. CWTA consistently demonstrated greater power, and reduced sample size requirements by 18% to 35% compared to KM PFS and 14% to 20% compared to KM OS. CWTA also expedited time-to-efficacy signals 2- to 6-fold. CWTA, by incorporating all efficacy signals in the cancer treatment trajectory, provides clinically relevant reduction in required sample size and meaningfully expedites the efficacy signals of cancer treatments compared to KM PFS and KM OS. Using CWTA rather than KM as the primary trial outcome has the potential to meaningfully reduce the numbers of patients, trial duration, and costs to evaluate therapies in advanced cancer.