In a significant academic achievement, the Department of Mathematics is proud to announce that Dr Koyel Chakravarty, an Assistant Professor, along with her diligent PhD Scholar, Ms Amrutha Sreekumar, has made a remarkable contribution to the field of biological mathematics. Their paper, titled “Exploring the Impact of PTH Therapy on Bone Remodeling: A Mathematical Investigation,” has been officially accepted for publication in the prestigious Journal of Biological Systems by World Scientific.

This paper presents a comprehensive mathematical model that investigates the effects of Parathyroid Hormone (PTH) therapy on bone remodelling processes. The research provides valuable insights that could potentially lead to more effective treatments for bone-related diseases and conditions.

The university community extends its heartfelt congratulations to Dr Chakravarty and Ms Sreekumar for their dedication and hard work.

Abstract

The regulatory role of parathyroid hormone (PTH) on bone, a crucial calcium reservoir, is influenced by sex steroids, notably estrogen. A mathematical model elucidates PTH-mediated bone remodeling mechanisms, examining plasma PTH effects and external dosages. Daily PTH injections, with their dual anabolic or catabolic action, offer a notable treatment for severe osteoporosis. This study predicts osteogenic responses to PTH, integrating factors like TGF-β, RANKL, and bisphosphonates in osteoblast-osteoclast signaling, alongside PTH’s effects on glands and regulatory molecules like Runx2, pCREB, and Bcl2. Using various methods, including simulations and sensitivity analysis, it aims to understand PTH therapy’s impact on bone volume, enhancing its clinical relevance.

Explanation of the Research in Layperson’s Terms

The role of parathyroid hormone (PTH) in calcium storage within bones is significant, with its regulation influenced by other hormones, notably estrogen. A mathematical model has been developed to comprehend the effects of PTH on bones, examining both endogenous levels in the blood and exogenous intake requirements. It has been observed that daily PTH injections can be beneficial in treating severe bone conditions like osteoporosis, as these injections can induce either bone formation or resorption, depending on their administration method. Additionally, the response of bones to PTH is under scrutiny in this study, considering various substances such as TGF-β, RANKL, and bisphosphonates, and their interplay in maintaining bone health. Through the utilization of different methodologies including computer simulations, efforts are being made to precisely understand how PTH injections influence bone health, thereby enhancing their efficacy in addressing bone-related ailments.

Title of the Research paper in the Citation Format

Exploring the Impact of PTH Therapy on Bone Remodeling: A Mathematical Investigation

Practical Implementation or the Social Implications Associated
Personalized Treatment Plans:
Mathematical modeling of bone remodeling can facilitate the development of personalized treatment plans for individuals with bone disorders such as osteoporosis. By considering factors like hormone levels, genetic predispositions, and lifestyle factors, healthcare providers can tailor treatment strategies to optimize bone health outcomes.
1. Drug Development:
Insights gained from mathematical models can aid in the development of new drugs for bone disorders. By simulating the effects of potential therapeutic agents on bone remodeling processes, researchers can identify promising candidates for further investigation, potentially accelerating the drug discovery process.
2. Improved Clinical Decision Making:
Healthcare professionals can use mathematical models to make more informed clinical decisions regarding the management of bone disorders. By integrating patient-specific data into predictive models, clinicians can better predict treatment outcomes and adjust therapeutic interventions accordingly.
3. Enhanced Surgical Planning:
Mathematical modeling can also be valuable in surgical planning for procedures such as bone grafting or joint replacement. By simulating the effects of surgical interventions on bone remodeling, surgeons can optimize surgical techniques to promote more effective healing and long-term outcomes for patients.
4. Public Health Interventions:
Understanding the factors influencing bone remodeling at a population level can inform public health interventions aimed at reducing the burden of bone disorders. By identifying modifiable risk factors and developing targeted prevention strategies, policymakers can promote bone health and reduce the incidence of conditions like osteoporosis on a larger scale.
5. Educational Tools:
Mathematical models of bone remodeling can serve as educational tools for healthcare professionals, students, and patients. By visualizing complex biological processes in a simplified manner, these models can enhance understanding of bone physiology and the mechanisms underlying bone disorders, ultimately improving patient care and outcomes.

Collaborators:
• Prof. D.C. Dalal, Professor, Department of Mathematics, IIT Guwahati
• Dr. L.N. Guin, Associate Professor, Department of Mathematics, Visva-Bharati

Future research plans.
a) Incorporating Multi-Scale Modeling: Future research could focus on integrating multi-scale modeling approaches to capture the intricate interactions occurring at different levels of bone structure, from the molecular to the tissue level. By incorporating information on cellular processes, tissue mechanics, and systemic factors, these models could provide a more comprehensive understanding of bone remodeling dynamics.
b) Accounting for Heterogeneity: There is a need to develop mathematical models that account for the heterogeneity observed in bone remodeling processes across individuals and within different anatomical sites. By considering factors such as age, sex, genetics, and bone quality, researchers can create more personalized and accurate models to predict individual responses to treatment and disease progression.
c) Integration of Advanced Imaging Techniques: Advances in imaging technologies such as micro-computed tomography (micro-CT) and magnetic resonance imaging (MRI) provide detailed insights into bone structure and function. Future research could focus on integrating data from these advanced imaging techniques into mathematical models to enhance their predictive capabilities and enable non-invasive monitoring of bone remodeling in clinical settings.
d) Exploring Therapeutic Interventions: Researchers could use mathematical modeling to explore the efficacy of novel therapeutic interventions for bone disorders, such as drug treatments, exercise regimens, and dietary interventions. By simulating the effects of these interventions on bone remodeling processes, researchers can identify optimal treatment strategies and accelerate the development of new therapies.

Pictures Related to the Research

Link to the article

DOI: https://doi.org/10.1142/S021833902450027X

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The Department of Mathematics had organised an International Conference on Women in Pure and Applied Mathematics (WPAM). The five-day-long conference featured luminaries in the field of Mathematics and was funded by three prominent Indian government research bodies: SERB, NBHM, and CSIR.

The event saw Prof. Raman Parimala, a renowned Indian Mathematician, acclaimed for her contributions to the field of Algebra. The Conference was held with the purpose of providing an empathetic platform for women mathematicians to present their cutting-edge research work and to share their concerns about the gender gap in mathematical science.

Prof. Sanoli Gun (President of Asian Oceanian Women in Mathematics), Prof. Vijaylaxmi Trivedi (Chairperson, Indian Women in Mathematics), and Prof. Anisa Chorwadwala (Member, Indian Women in Mathematics) motivated the students and provided the students with valuable inputs on how to pursue their career further in Mathematics. The luminaries also discussed activities conducted by their organisations to encourage established women researchers, women PhD scholars and advanced undergraduate-level women students in Mathematics.

During the conference, activities such as poster presentations were held to facilitate mathematical interaction between students. The conference ended with the hope that there will be more opportunities to organise similar events.

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The combat against HIV, the virus responsible for AIDS has witnessed consistent advancements and studies put forward by researchers in various fields. To identify a more intelligent and effective approach to combat HIV-1 and enhance the understanding of the workings of its treatments, Dr Koyel Chakravarty, Assistant Professor, Department of Mathematics has published a paper titled “Mathematical modelling of HIV-1 transcription inhibition: a comparative study between optimal control and impulsive approach” in the Q2 journal, Journal of Computational and Applied Mathematics.

Abstract

By adopting a proactive strategy, this study facilitates the interaction with human immunodeficiency virus type I (HIV-1), successfully navigating its sequential fusion stages. This approach enables efficient infiltration of the virus into a target CD4+T helper cell within the host organism, initiating the virus’s replication cycle. As a retrovirus, HIV-1 orchestrates the conversion of its single-stranded viral RNA genome into a more stable double-stranded DNA structure. The newly formed DNA integrates with the host cell’s genetic material, and the pro-viral DNA transforms into functional messenger RNA (mRNA) with the assistance of the host enzyme RNA polymerase II (Pol II).

The ongoing research focuses on constructing a meticulous mathematical framework using a system of nonlinear differential equations. The investigation aims to assess the impact of a Tat inhibitor on suppressing the transcriptional activity of HIV-1, treating it as an optimal control problem. The study also evaluates the Tat inhibitor’s efficacy as a potential therapeutic intervention for HIV-1 infection. Employing a one-dimensional impulsive differential equation model to determine the mathematically derived maximum concentration of the elongating complex (P2), the research considers the crucial aspect of optimal timing between successive dosages. A comparative analysis contrasts the effects of continuous dosing with impulse dosing of the Tat inhibitor, using numerical analysis to evaluate outcomes. The findings underscore the superior effectiveness of impulsive dosing over continuous dosing in inhibiting HIV-1 transcription. Visual representations of the model’s parameter sensitivities enhance understanding of the intricate physiological and biochemical processes within the system.

Practical implementation/social implications of the research

1. Treatment Optimization:

• Practical Implementation: Develop personalized treatment plans for individuals with HIV-1 based on the optimal control and impulsive approaches identified in the study.

• Social Implication: Improve the effectiveness of HIV-1 treatments, potentially leading to better health outcomes and a higher quality of life for individuals living with the virus.

2. Drug Administration Guidelines:

• Practical Implementation: Provide guidelines for healthcare professionals on the timing and dosage of Tat inhibitors using insights from the comparative study.

• Social Implication: Enhance the efficiency of drug administration, potentially reducing side effects and improving patient adherence to treatment regimens.

3. Public Health Planning:

• Practical Implementation: Incorporate the study’s findings into public health planning, considering the optimal and impulsive control strategies in broader HIV-1 prevention and treatment programs.

• Social Implication: Contribute to more effective and resource-efficient public health interventions, potentially reducing the overall burden of HIV-1 in communities.

4. Drug Development Strategies:

• Practical Implementation: Inform pharmaceutical companies and researchers about the comparative study results to guide the development of new HIV-1 inhibitors or improvements to existing drugs.

• Social Implication: Accelerate the development of more potent and targeted therapies, offering new options for managing HIV-1 infections.

5. Patient Education:

• Practical Implementation: Develop educational materials for individuals with HIV-1, explaining the importance of adherence to optimized treatment plans based on the study’s findings.

• Social Implication: Empower patients to actively participate in their treatment, potentially leading to better treatment outcomes and reduced transmission rates.

6. Policy Recommendations:

• Practical Implementation: Present policy recommendations to healthcare institutions and government agencies based on the study’s outcomes.

• Social Implication: Influence health policies to integrate the most effective strategies for HIV-1 transcription inhibition, potentially contributing to more efficient resource allocation and improved public health outcomes.

7. Global Health Impact:

• Practical Implementation: Collaborate with international health organizations to disseminate the study’s findings globally.

• Social Implication: Contribute to global efforts in controlling the HIV-1 pandemic, fostering collaboration and knowledge-sharing among nations.

8. Reduced Healthcare Costs:

• Practical Implementation: If the impulsive approach proves more cost-effective, healthcare systems can implement this strategy to potentially reduce the overall cost of HIV-1 treatment

• Social Implication: Alleviate financial burdens on both individuals and healthcare systems, making HIV-1 treatment more accessible

In summary, the practical implementation and social implications of this study extend from optimizing individual treatment plans to influencing global health policies, ultimately contributing to more effective HIV-1 management and improved public health outcomes.

Collaborations

• Prof. D C Dalal, Professor, Department of Mathematics, IIT Guwahati
• Prof. A K Sarkar, Professor, Centre for Mathematical Biology and Ecology, Department of Mathematics, Jadavpur University
• Dr L N Guin, Associate Professor, Department of Mathematics, Visva-Bharati

Future Prospects of the Research

• Mathematical Modelling of Cholesterol Dynamics.
• Mathematical Modelling of Muscle Regeneration.
• Mathematical Modelling of Bone Remodelling
• Mathematical Modelling of Glucose-Insulin Dynamics.
• Mathematical Modelling on HIV-1 Transcription.
• Mathematical Modelling of Population Dynamics for Patients suffering from Diabetes.

Link to the Article

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The Department of Mathematics, in collaboration with the National Centre for Mathematics, organised an AIS school on p-adic methods in Arithmetic from June 26 to July 15, 2023 at the campus. Over 58 students, research scholars, post-doc fellows and faculty from reputed universities such as IIT Hyderabad, IIT Guwahati, IISER, University of Hyderabad, Pondicherry University, Banaras Hindu University etc. participated in the three-week-long AIS school, which was conducted with the prime objective to study the p-adic numbers and the p-adic methods in arithmetic.

Prof. C S Rajan, Ashoka University; Dr Manish Kumar Pandey and Dr Sazzad Ali Biswas, SRM University-AP; Dr Amiya Kumar Mondal, IISER Berhampur; Dr Santosh Nadimpalli, IIT Kanpur; Dr Shaunak Deo, IISc, Bangalore; and Dr Mihir Sheth, Post-Doc, IISc, Bangalore were the speakers at the school. The sessions were delivered in collaboration with course associates /tutors. Dr Arindam Jana, Postdoc, TIFR Mumbai; Dr Ravitheja Vangola, Postdoc, IISC Bangalore; Mr Sagar Shrivastava, Mr Manodeep Raha, and Mr Niladri Sekhar Patra, PhD scholar, TIFR Mumbai were the tutors mentoring the participants.

Prof. C S Rajan delivered 9 lectures – four lectures in the first week of the programme and five lectures at the end of the programme. The first part of his session covered topics such as Hensel’s Lemma and Local-Global Principle while the latter part focused on the Main Conjecture of Iwasawa Theory and Introduction to the work of Ribet (Converse to Herbrand) and subsequent work.

Dr Sazzad Ali Biswas offered his insights on topics like Construction of p-adic numbers, Various Properties (algebraic and Analytic) of p-adic numbers, Arithmetic operations of p-adic numbers and Local-Global Principle in five sessions. Dr Manish Kumar Pandey gave two lectures on the Special values of the Riemann zeta functions.

Local Fields, Global Fields and p-adic algebraic numbers were the focal points of the four lectures delivered by Dr Mihir Sheth. Dr Amiya Kumar Mondal took five lectures on Modular Forms and Dr Santosh Nadimpally gave five lectures on the Construction of p-adic zeta functions and the Various properties of p-adic Zeta function. Dr Shaunak Deo dealt with topics including Cyclotomic Fields, Iwasawa’s Construction of p-adic L-fucntions and p-adic Family of Modular Forms through six lectures.

The AIS School was organised by Prof. C S Rajan of Ashoka University and Assistant Professors Dr Manish Kumar Pandey and Dr Sazzad Ali Biswas of the Department of Mathematics at SRM University-AP. The programme successfully concluded with the active participation of students, scholars and faculty who assembled to exchange constructive discourse on p-adic methods in Arithmartics.

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The Department of Mathematics is glad to announce that the project of Dr Sandeep Kumar Verma, Assistant Professor, titled “Investigation and development of wavelet transform and its applications in the framework of fractional Dunkl transform”, has been selected to be funded by DST, Govt. of India under SERB-SURE Scheme, securing an approved grant of 20.13 Lacs. Dr Sandeep Kumar Verma is the Principal Investigator of the project.

Title of the Project: Investigation and development of wavelet transform and its applications in the framework of fractional Dunkl transform

Approved Grant: 20.13 Lacs (As per university calculation)

Funded by: DST, Govt. of India under SERB-SURE Scheme

PI: Dr Sandeep Kumar Verma, Assistant Professor, Department of Mathematics

Project Summary: Dunkl operators are differential-difference operators associated with finite reflection groups in a Euclidean space. Dunkl transform is attracting widespread interest and giving us new perspectives on familiar topics from Harmonic Analysis and Partial Differential Equations. The fractional Dunkl transform (FRDT), which is introduced with one extra parameter, is the generalization of the well-known signal processing operations, such as the Fourier transform (FT), the fractional Fourier transform (FRFT), the Fresnel transform, Hankel transforms, fractional Hankel transforms, and the Dunkl transform. The FRDT will also find applications in the solution of optical systems, filter design, time-frequency analysis, and many others. The wavelet transforms (WT), which has had a growing importance in optics and signal processing, has been shown to be a successful tool for time-frequency analysis and image processing. It has found many applications in time-dependent frequency analysis of short-transient signals, data compression, optical correlators, sound analysis, representation of fractal aggregates, and many others.

The fractional wavelet transforms (FRWT) based on the fractional Fourier transform, which generalizes the classical wavelet transform, has proven potentially useful for signal processing, data compression, pattern recognition, and computer graphics. Thus, fractional wavelet transform gives more flexibility for time-frequency analysis than the usual wavelet transforms. As a generalization of the WT, a novel fractional Dunkl wavelet transform (FRDWT) can combine the advantages of the WT and the FRDT, i.e., it will be a linear transformation without cross-term interference and will be capable of providing multiresolution analysis and representing signals in the fractional domain. Simultaneously, compared to the Dunkl transforms, FRDT is more flexible for its one extra degree of freedom and can be used frequently in time-frequency analysis and non-stationary signal processing. Inspired by the fractional wavelet transform (FRWT), we introduce the concept of the (FRDWT), combining the idea of FRDT and WT. The proposed transform will not only inherit the advantages of multiresolution analysis of the WT, but also will have the capability of signal representations in the fractional Dunkl transform domain. Compared with the existing WT, the fractional Dunkl wavelet transform can offer signal representations in the time-fractional-frequency plane in the FRDT domain.

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