BIBECHANA Vol. 21, No. 3, December 2024, 233-240 ISSN 2091-0762 (Print), 2382-5340 (Online) Journal homepage: http://nepjol.info/index.php/BIBECHANA Publisher: Dept. of Phys., Mahendra Morang A. M. Campus (Tribhuvan University) Biratnagar Computational analysis of clot formation risk in diabetes: A mathematical modeling approach Shabab Akbar, Rohit Kumar Sharma, Mo Sadique, Kshiteendra Mohan Jaiswal, Purnima Chaturvedi, Vivek Kumar, Sapna Ratan Shah∗ School of Computational & Integrative Sciences, Jawaharlal Nehru University, New Delhi 110067, India ∗Corresponding author: Email: sapnarshah@mail.jnu.ac.in Abstract Diabetes (Type 1 or Type 2) is a serious condition that can make blood clot more easily. This can cause heart attacks and strokes, which are very dangerous. Our approach integrates physiological data, hemodynamic principles, and mathematical equations to simulate blood flow dynamics and clot formation processes within the vasculature of diabetic individuals. By incorporating key factors such as altered blood viscosity, resistance to flow, endothelial dys- function, and platelet aggregation, we obtained insights into the complex interplay between diabetes related factors and clotting propensity. Changes in blood composition, such as in- creased levels of fibrinogen and other clotting factors, can make blood thicker and more prone to clotting and the reason for increased resistance to flow and viscosity. As blood clots enlarge in blood vessels, they obstruct blood flow, increasing resistance. This makes blood movement harder. Clot size also affects nearby blood viscosity. Accumulating cells and clotting fac- tors thicken blood, worsening circulation. Larger clots heighten flow resistance and viscosity, potentially causing issues like tissue damage. Thus, larger clots worsen blood flow and cardio- vascular health. Through computational simulations, we explored various scenarios to assess the impact of different parameters on clot formation risk, thereby offering valuable insights for the development of preventive strategies and targeted interventions for diabetic patients. Keywords Clot formation, diabetes mellitus, Mathematical modeling, cardiovascular complications, Platelet aggregation. Article information Manuscript received: April 19, 2024; Revised: April 23, 2024; Accepted: June 6, 2024 DOI https://doi.org/10.3126/bibechana.v21i3.64973 This work is licensed under the Creative Commons CC BY-NC License. https://creativecommons. org/licenses/by-nc/4.0/ 233 http://nepjol.info/index.php/BIBECHANA sapnarshah@mail.jnu.ac.in https://doi.org/10.3126/bibechana.v21i3.64973 https://creativecommons.org/licenses/by-nc/4.0/ https://creativecommons.org/licenses/by-nc/4.0/ Shabab Akbar et al./ BIBECHANA 21 (2024) 233-240 234 1 Introduction In diabetes, blood tends to clot more easily, which can lead to serious problems like heart attacks and strokes. It is important to manage diabetes care- fully to reduce this risk. Recent data from the In- ternational Diabetes Federation (IDF) underscores the alarming scale of the epidemic, with 537 mil- lion adults aged 21–80 living with diabetes in 2021, a number projected to soar to 784 million by 2045. The disease exacts a heavy toll, causing 6.9 mil- lion demise in 2021 alone, translating to one in every two seconds. Moreover, Diabetes ranks as the third most prevalent comorbidity in COVID-19 cases, leading to increased disease severity, and ad- verse outcomes, including admission in ICU and at last death. It is considered a silent epidemic, with its prevalence steadily increasing globally and pos- ing significant public health challenges. All these statistics underscore the urgent need for effective interventions and policies to curb diabetes epidemic and its associated complications [1–4]. Diabetes is a chronic disorder characterized by high blood glu- cose levels and is associated with a myriad of com- plications, including cardiovascular diseases Fig- ure.1. One of the major cardiovascular complica- tions of diabetes is the increased risk of thrombotic events, such as myocardial infarction and stroke, attributed to the formation of blood clots within the vasculature [5–8]. Despite advances in under- standing the pathophysiology of diabetic vascular complications, the precise mechanisms underlying the heightened clot formation risk in diabetic in- dividuals remain incompletely understood [9–13]. Mathematical modeling and computational simu- lations offer a powerful approach to elucidate the complex hemodynamic and biochemical processes involved in clot formation, providing valuable in- sights into the interplay between diabetes related factors and thrombotic propensity [14–18]. Dia- betes is commonly categorized into two primary types: Type 1 and Type 2. In Type 1 diabetes, pan- creas doesn’t produce enough insulin, while Type 2 diabetes, more prevalent form, arises when the body becomes resistant to insulin or doesn’t pro- duce sufficient amounts. Symptoms may include heightened thirst, hunger, fatigue, weight loss, fre- quent urination, infections, blurry vision, and slow wound healing [19–24]. Both type 1 and type 2 dia- betes can increase the risk of blood clotting, but the mechanisms may differ between the two types. In type 1 diabetes, where the body does not produce enough insulin, the risk of blood clotting may be re- lated to factors such as hyperglycemia (high blood sugar levels), which can lead to damage of blood vessels and impair blood flow. Additionally, indi- viduals with type 1 diabetes may have other risk factors such as increased levels of clotting factors in the blood. In type 2 diabetes, which is charac- terized by insulin resistance or reduced insulin pro- duction, the risk of blood clotting may be higher due to factors such as obesity, inflammation, and metabolic abnormalities associated with insulin re- sistance. These factors can contribute to changes in blood vessel function and increased clot formation. Both types of diabetes can predispose individuals to blood clotting, but the specific mechanisms and risk factors may vary [10,21]. Figure 1: Normal artery with diseased artery with a blood clot. Addressing these risk factors through lifestyle modifications and targeted interventions is crucial in combating the diabetes epidemic and reducing its burden on individuals and healthcare systems alike [25–30]. Diabetes can lead to severe compli- cations that affect various parts of the body, in- cluding heart, blood vessels, eyes, teeth, kidneys, nerves, and may ultimately result in demise. Dia- betic complications often necessitate amputations, causing permanent disability [31–36]. Diabetic pa- tients has to face more higher risk of blockage and heart strokes compared to the general population. Diabetic neuropathy, a type of nerve damage result- ing from high blood sugar levels, is another com- mon complication, particularly affecting the feet and increasing the risk of foot ulcers, infections, and subsequent limb amputations, especially when combined with poor blood flow [37–42]. Diabetic retinopathy, characterized by hurting to the tiny blood vessels in retina, is a leading cause of blind- ness worldwide, affecting nearly one million indi- viduals. Additionally, diabetic nephropathy, which damages the small blood vessels in the kidneys, leads to kidney disease and can eventually cause kidney failure, representing one of most prevalent causes for kidney failure [43–47]. In this study, we presented a computational analysis of clot forma- Shabab Akbar et al./ BIBECHANA 21 (2024) 233-240 235 tion risk in diabetes using a mathematical modeling approach. Our research aims to unravel the intri- cate mechanisms that contribute to the increased thrombotic risk in diabetic patients [48–51]. By leveraging mathematical models and computational simulations, we simulated various scenarios to ex- plore the impact of diabetes related factors, such as altered blood rheology, endothelial dysfunction, and platelet hyperactivity, on clot formation dynamics. Through comprehensive analyses of hemodynamic parameters, clot formation kinetics, and biochem- ical pathways, we aim to elucidate the underlying mechanisms driving clot formation in diabetic indi- viduals [52–56]. The significance of clot formation risk in diabetes lies in its potential to lead to severe and life-threatening complications. Diabetes is as- sociated with an increased propensity for blood clot formation, a condition known as hypercoagulability. Additionally, diabetic individuals are more suscep- tible to developing blood clots in smaller blood ves- sels, leading to conditions like deep vein thrombosis (DVT) and pulmonary embolism (PE). Therefore, understanding and mitigating the risk of clot for- mation in diabetes is crucial for preventing adverse cardiovascular outcomes and improving patient out- comes. This computational approach allows us to integrate diverse physiological and pathological fac- tors into a unified framework, providing a holis- tic understanding of the complex interplay between diabetes and thrombotic risk. By elucidating the key determinants of clot formation in diabetes, our study aims to identify potential therapeutic targets and strategies for mitigating the heightened throm- botic risk associated with this prevalent metabolic disorder [51,57–60] ersonalized interventions aimed at reducing the burden of cardiovascular complica- tions in diabetic patients. 2 Formulation of the problem Our mathematical model incorporates several key factors implicated in the pathogenesis of clot for- mation in diabetes, including altered blood viscos- ity, endothelial dysfunction, and platelet aggrega- tion. The model is based on fundamental principles of hemodynamics, fluid mechanics, and biochemi- cal kinetics, represented by a system of differential equations describing the dynamics of blood flow and clotting processes within the vascular network. Pa- rameters such as blood glucose levels, lipid profiles, and inflammatory markers are integrated into the model to capture the systemic effects of diabetes on vascular health [61, 62]. In this study, we con- sidered, a scenario where a narrowing in the artery, known as stenosis, forms asymmetrically along the artery’s length but maintains symmetry around its circumference. This narrowing depends on both the axial distance along the artery, denoted as z, and height of stenosis. In this situation, radius of artery, denoted as R(z), can be expressed as below: R(z) R0 = { 1− [L (m−1) 0 x− xm] d ≤ z ≤ d+ L0, 1 otherwise. (1) where x =z-d. In simpler terms, let’s break down the equation. We are talking about an artery with a blockage, which reduces its radius compared to its original size. The variable R(z) represents the current ra- dius of the artery with the blockage, while R0 repre- sents its original, unobstructed radius. The length of the blockage is denoted by L0 and d indicates where along the artery it is located. The parame- ter m, which must be greater than or equal to 2, describes the shape of the blockage. When m=2, it means the blockage is axially symmetric. Finally, there is another parameter, A, which is determined by the specific values of these variables and param- eters [10,31,36,50]. A = δ mm/(m−1) R0 Lm 0 (m− 1) This equation helps us find the maximum height of the blockage in the artery, indicated by the sym- bol . It is determined by factors such as the location of the blockage along the artery (z), its length (L0), and the shape parameter (m). The expression also involves a fractional calculation, with the result be- ing the maximum height of the blockage. 2.1 Conservation equation and boundary condition The equation describing the steady, and fully- developed flow of blood in an artery, under the con- ditions of laminar flow and incompressibility, sim- plifies as: 0 = −∂P ∂r + 1 r ∂(rτ) ∂z 0 = −∂P ∂r (2) The coordinates (z, r) represent the positional measurements, with z indicating the direction along the artery’s axis, and r denoting measurements per- pendicular to the artery’s axis. Coordinates (z, r) are used for pinpointing lo- cations within artery. The coordinate z represents positions along the artery’s length, while the co- ordinate r measures distances perpendicular to the artery’s axis. This system allows for precise spatial referencing within the artery, aiding in the analysis of various phenomena occurring within its struc- ture. The following conditions at the boundaries are applied to find the solution of the aforemen- tioned equations. Shabab Akbar et al./ BIBECHANA 21 (2024) 233-240 236  ∂u ∂r = 0 at r = 0 u− 0 at r = R(z) τ is finite at r = 0 P = P0 at z = 0 P = PL at z = L (3) 2.2 Casson’s fluid model Casson’s model is often expressed [16,27]:{ τ1/2 = τ 1/2 0 (µ)1/2(−du dr ) 1/2, if τ ≥ τ0 (dudr ), if τ < τ0 (4) with τ0 = −( dp dz ) Rc 2 where µ shows Casson’s viscosity coefficient, Rc represents radius of plug flow region, 0 indicates yield stress, and represents wall shear. The rate at which volume flows through a particular point in the system, as described by equation (16), is termed as: Q = π ∫ R 0 r(−du dr )dr (5) Upon integrating equation (17) with the assistance of equations (16) and (3), we obtain the following result: Q = πR4 8µ (−dp dz )[1− 16 7 ( Rc R )1/2+ 4 3 ( Rc R )− 1 21 ( Rc R )4] (6) Equation (18) can be rewritten as; Q = πR4 8µ (−dp dz )f(ȳ) with f(ȳ) = [1− 16 7 (ȳ)1/2 + 4 3 (ȳ)− 1 21 (ȳ)4] where ȳ = Rc R << 1 The pressure gradient, as derived from the equation above, can be expressed as below: −dp dz = 8µQ πR4f(ȳ) (7) By integrating equation (19) with the boundary conditions, It is obtained: ∆P = P − P0 = 8µQ πR4f(ȳ) ∫ L 0 dz (R(z)/R0)4f( ¯y(z)) (8) Resistance to flow, also known as resistive impedance, is represented by the symbol and is defined as below: λ = PL − P0 Q (9) Resistance to flow, obtained from above equations as a reference, can be expressed as follows: λ = 1− L0 L + f0 L ∫ d+L0 0 dz (R(z)/R0)4f( ¯y(z)) (10) where f0 = [1− 16 7 ( Rc R )1/2 + 4 3 ( Rc R )− 1 21 ( Rc R )4] Apparent viscosity (µapp) is defined as below: µapp = 1 (R(z)/R0)4f(ȳ) (11) Shear stress at wall may be obtained as below; τR = [τ 1/2 0 + (−µ du dr ) (12) 3 Results Our computational analysis unveils the intricate ways in which diabetic conditions, characterized by hyperglycemia, dyslipidemia, and chronic in- flammation, exert profound effects on blood rhe- ology and endothelial function, thereby predispos- ing diabetic individuals to heightened clot forma- tion. Through detailed simulations, we observe that the diabetic milieu significantly alters key hemo- dynamic and biochemical parameters, creating a prothrombotic environment within the vasculature [34,50,52]. Specifically, our model demonstrates that increased blood viscosity, attributed to ele- vated levels of circulating glucose and lipids, im- pedes blood flow and promotes stasis, facilitating clot formation. Moreover, impaired endothelial ni- tric oxide production, a hallmark of diabetic en- dothelial dysfunction, disrupts the delicate balance between prothrombotic and antithrombotic factors, further exacerbating thrombotic propensity. Figure 2: Resistance to flow with stenosis shape. Shabab Akbar et al./ BIBECHANA 21 (2024) 233-240 237 Figure 3: Viscosity with stenosis size. Interestingly, our simulations elucidate a non- linear relationship between blood glucose levels and clot formation risk, with acute hyperglycemic spikes exerting particularly pronounced effects on platelet reactivity and activation of the clotting cascade. These findings underscore the impor- tance of glycemic control in mitigating acute throm- botic events in diabetic patients. Additionally, our sensitivity analyses highlight the potential ef- ficacy of multifaceted interventions targeting vari- ous pathophysiological pathways implicated in clot formation. By integrating blood glucose lowering strategies, lipid-lowering therapies, and antiplatelet agents, our model suggests synergistic effects in reducing clot formation risk and attenuating the burden of cardiovascular complications in diabetes [6,16,45,51]. Figure.2. show that as blood clots expand in size within the blood vessels, they cre- ate obstructions that impede the flow of blood. This obstruction results in an increase in resis- tance to blood flow, making it more difficult for the blood to move past the clot. The larger the clot, the greater the resistance it presents to the flow of blood. This increased resistance can lead to higher pressure within the blood vessel upstream of the clot and decreased pressure downstream, al- tering blood flow patterns and potentially causing complications such as ischemia or tissue damage. Therefore, as clot size increases, so does the re- sistance to blood flow, which can have significant implications for overall cardiovascular health and function. Figure.3. show that as blood clots grow larger within the vasculature, they exert significant effects on blood viscosity, the thickness or sticki- ness of blood. This increase in viscosity arises from several factors. First, the clot traps various blood components, including red blood cells and platelets, within its structure, leading to a concentration of blood constituents in the vicinity of the clot. Addi- tionally, the formation of fibrin, a protein essential for clot structure, results in the creation of a dense meshwork that impedes blood flow [4,15,23,50]. As more platelets aggregate to the clot site, they fur- ther contribute to the viscosity of the surrounding blood. The obstruction caused by a larger clot al- ters the flow dynamics within the blood vessel, in- fluencing shear forces and pressure gradients, which in turn affect viscosity. Ultimately, the growth of blood clots leads to an increase in local blood viscos- ity, impacting blood flow dynamics and potentially exacerbating thrombotic events. Overall, our com- putational approach offers valuable insights into the complex interplay between diabetes-related factors and thrombotic propensity, providing a framework for identifying novel therapeutic targets and opti- mizing treatment strategies for diabetic individuals at increased risk of thrombotic events [24,32,52]. By elucidating the underlying mechanisms driving clot formation in diabetes, our research aims to pave the way for personalized interventions aimed at mitigat- ing the heightened thrombotic risk associated with this prevalent metabolic disorder. 4 Conclusion Our study underscores the utility of mathematical modeling and computational analysis in elucidating the complex interplay between diabetes and throm- bosis. By integrating physiological data and com- putational simulations, we provided mechanistic in- sights into the pathophysiology of clot formation in diabetes and identify potential therapeutic targets for mitigating thrombotic complications in diabetic individuals. The computational framework devel- oped in this study offers a valuable tool for assessing clot formation risk, optimizing treatment strategies, and guiding clinical decision-making in diabetic pa- tients. As blood clots grow larger within the blood vessels, they create obstructions that impede the flow of blood. This obstruction increases the resis- tance to blood flow, making it more challenging for blood to move past the clot. At the same time, the clot’s size can also affect the viscosity of the blood in the vicinity of the clot. As more blood cells and clotting factors accumulate around the clot, the vis- cosity of the blood in that area can increase. 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International Journal of Biomedi- cal Engineering and Technology, 6(3):286–294, 2011. Introduction Formulation of the problem Conservation equation and boundary condition Casson’s fluid model Results Conclusion