Mixed uncertainty analysis on pumping by peristaltic hearts using Dempster-Shafer theory.

In this paper, we introduce the numerical strategy for mixed uncertainty propagation based on probability and Dempster-Shafer theories, and apply it to the computational model of peristalsis in a heart-pumping system. Specifically, the stochastic uncertainty in the system is represented with random variables while epistemic uncertainty is represented using non-probabilistic uncertain variables with belief functions. The mixed uncertainty is propagated through the system, resulting in the uncertainty in the chosen quantities of interest (QoI, such as flow volume, cost of transport and work). With the introduced numerical method, the uncertainty in the statistics of QoIs will be represented using belief functions. With three representative probability distributions consistent with the belief structure, global sensitivity analysis has also been implemented to identify important uncertain factors and the results have been compared between different peristalsis models. To reduce the computational cost, physics constrained generalized polynomial chaos method is adopted to construct cheaper surrogates as approximations for the full simulation.

The role of asymmetry and volume of thrombotic masses in the formation of local deformation of the aneurysmal-altered vascular wall: An in vivo study and mathematical modeling.

Currently, the primary factor indicating the necessity of an operation for an abdominal aortic aneurysm (AAA) is the diameter at its widest part. However, in practice, a large number of aneurysm ruptures occur before reaching a critical size. This means that the mechanics of aneurysm growth and remodeling have not been fully elucidated. This study presents a novel method for assessing the elastic properties of an aneurysm using an ultrasound technique based on tracking the oscillations of the vascular wall as well as the inner border of the thrombus. Twenty nine patients with AAA and eighteen healthy volunteers were considered. The study presents the stratification of a group of patients according to the elastic properties of the aneurysm, depending on the relative volume of intraluminal thrombus masses. Additionally, the neural network analysis of CT angiography images of these patients shows direct (r = 0.664271) correlation with thrombus volume according to ultrasound data, the reliability of the Spearman correlation is p = 0.000215. The use of finite element numerical analysis made it possible to reveal the mechanism of the negative impact on the AAA integrity of an asymmetrically located intraluminal thrombus. The aneurysm itself is considered as a complex structure consisting of a wall, intraluminal thrombus masses, and areas of calcification. When the thrombus occupies > 70% of the lumen of the aneurysm, the deformations of the outer and inner surfaces of the thrombus have different rates, leading to tensile stresses in the thrombus. This poses a risk of its detachment and subsequent thromboembolism or the rupture of the aneurysm wall. This study is the first to provide a mechanistic explanation for the effects of an asymmetrical intraluminal thrombus in an abdominal aortic aneurysm. The obtained results will help develop more accurate risk criteria for AAA rupture using non-invasive conventional diagnostic methods.

Digital twinning of cardiac electrophysiology for congenital heart disease.

In recent years, blending mechanistic knowledge with machine learning has had a major impact in digital healthcare. In this work, we introduce a computational pipeline to build certified digital replicas of cardiac electrophysiology in paediatric patients with congenital heart disease. We construct the patient-specific geometry by means of semi-automatic segmentation and meshing tools. We generate a dataset of electrophysiology simulations covering cell-to-organ level model parameters and using rigorous mathematical models based on differential equations. We previously proposed Branched Latent Neural Maps (BLNMs) as an accurate and efficient means to recapitulate complex physical processes in a neural network. Here, we employ BLNMs to encode the parametrized temporal dynamics of in silico 12-lead electrocardiograms (ECGs). BLNMs act as a geometry-specific surrogate model of cardiac function for fast and robust parameter estimation to match clinical ECGs in paediatric patients. Identifiability and trustworthiness of calibrated model parameters are assessed by sensitivity analysis and uncertainty quantification.

Hemodynamics with mechanical circulatory support devices using a cardiogenic shock model.

Mechanical circulatory support (MCS) devices, including veno-arterial extracorporeal membrane oxygenation (VA-ECMO) and Impella, have been widely used for patients with cardiogenic shock (CS). However, hemodynamics with each device and combination therapy is not thoroughly understood. We aimed to elucidate the hemodynamics with MCS using a pulsatile flow model. Hemodynamics with Impella CP, VA-ECMO, and a combination of Impella CP and VA-ECMO were assessed based on the pressure and flow under support with each device and the pressure-volume loop of the ventricle model. The Impella CP device with CS status resulted in an increase in aortic pressure and a decrease in end-diastolic volume and end-diastolic pressure (EDP). VA-ECMO support resulted in increased afterload, leading to a significant increase in aortic pressure with an increase in end-systolic volume and EDP and decreasing venous reservoir pressure. The combination of Impella CP and VA-ECMO led to left ventricular unloading, regardless of increase in afterload. Hemodynamic support with Impella and VA-ECMO should be a promising combination for patients with severe CS.

A His bundle pacing protocol for suppressing ventricular arrhythmia maintenance and improving defibrillation efficacy.

BACKGROUND: The excitable gap (EG), defined as the excitable tissue between two subsequent wavefronts of depolarization, is critical for maintaining reentry that underlies deadly ventricular arrhythmias. EG in the His-Purkinje Network (HPN) plays an important role in the maintenance of electrical wave reentry that underlies these arrhythmias. OBJECTIVE: To determine if rapid His bundle pacing (HBP) during reentry reduces the amount of EG in the HPN and ventricular myocardium to suppress reentry maintenance and/or improve defibrillation efficacy. METHODS: In a virtual human biventricular model, reentry was initiated with rapid line pacing followed by HBP delivered for 3, 6, or 9 s at pacing cycle lengths (PCLs) ranging from 10 to 300 ms (n=30). EG was calculated independently for the HPN and myocardium over each PCL. Defibrillation efficacy was assessed for each PCL by stimulating myocardial surface EG with delays ranging from 0.25 to 9 s (increments of 0.25 s, n=36) after the start of HBP. Defibrillation was successful if reentry terminated within 1 s after EG stimulation. This defibrillation protocol was repeated without HBP. To test the approach under different pathological conditions, all protocols were repeated in the model with right (RBBB) or left (LBBB) bundle branch block. RESULTS: Compared to without pacing, HBP for >3 seconds reduced average EG in the HPN and myocardium across a broad range of PCLs for the default, RBBB, and LBBB models. HBP >6 seconds terminated reentrant arrhythmia by converting HPN activation to a sinus rhythm behavior in the default (6/30 PCLs) and RBBB (7/30 PCLs) models. Myocardial EG stimulation during HBP increased the number of successful defibrillation attempts by 3%-19% for 30/30 PCLs in the default model, 3%-6% for 14/30 PCLs in the RBBB model, and 3%-11% for 27/30 PCLs in the LBBB model. CONCLUSION: HBP can reduce the amount of excitable gap and suppress reentry maintenance in the HPN and myocardium. HBP can also improve the efficacy of low-energy defibrillation approaches targeting excitable myocardium. HBP during reentrant arrhythmias is a promising anti-arrhythmic and defibrillation strategy.

Cell-specific models of hiPSC-CMs developed by the gradient-based parameter optimization method fitting two different action potential waveforms.

Parameter optimization (PO) methods to determine the ionic current composition of experimental cardiac action potential (AP) waveform have been developed using a computer model of cardiac membrane excitation. However, it was suggested that fitting a single AP record in the PO method was not always successful in providing a unique answer because of a shortage of information. We found that the PO method worked perfectly if the PO method was applied to a pair of a control AP and a model output AP in which a single ionic current out of six current species, such as IKr, ICaL, INa, IKs, IKur or IbNSC was partially blocked in silico. When the target was replaced by a pair of experimental control and IKr-blocked records of APs generated spontaneously in human induced pluripotent stem cell-derived cardiomyocytes (hiPSC-CMs), the simultaneous fitting of the two waveforms by the PO method was hampered to some extent by the irregular slow fluctuations in the Vm recording and/or sporadic alteration in AP configurations in the hiPSC-CMs. This technical problem was largely removed by selecting stable segments of the records for the PO method. Moreover, the PO method was made fail-proof by running iteratively in identifying the optimized parameter set to reconstruct both the control and the IKr-blocked AP waveforms. In the lead potential analysis, the quantitative ionic mechanisms deduced from the optimized parameter set were totally consistent with the qualitative view of ionic mechanisms of AP so far described in physiological literature.

Computational study of effect of hybrid nanoparticles on hemodynamics and thermal transfer in ruptured arteries with pathological dilation.

The intended research aims to explore the convection phenomena of a hybrid nanofluid composed of gold and silver nanoparticles. This research is novel and significant because there is a lack of existing studies on the flow behavior of hybrid nanoparticles with important physical properties of blood base fluids, especially in the case of sidewall ruptured dilated arteries. The implementation of combined nanoparticles rather than unadulterated nanoparticles is one of the most crucial elements in boosting the thermal conduction of fluids. The research methodology encompasses the utilization of advanced bio-fluid dynamics software for simulating the flow of the nanofluid. The physical context elucidates the governing equations of momentum, mass, momentum, and energy in terms of partial differential equations. The results are displayed in both tabular and graphical forms to demonstrate the numerical and graphical solutions. The effect of physical parameters on velocity distribution is illustrated through graphs. Furthermore, the study's findings are unique and original, and these computational discoveries have not been published by any researcher before. The finding implies that utilizing hybrid nanoparticles as drug carriers holds great promise in mitigating the effects of blood flow, potentially enhancing drug delivery, and minimizing its impact on the body.

In Silico TRials guide optimal stratification of ATrIal FIbrillation patients to Catheter Ablation and pharmacological medicaTION: the i-STRATIFICATION study.

AIMS: Patients with persistent atrial fibrillation (AF) experience 50% recurrence despite pulmonary vein isolation (PVI), and no consensus is established for secondary treatments. The aim of our i-STRATIFICATION study is to provide evidence for stratifying patients with AF recurrence after PVI to optimal pharmacological and ablation therapies, through in silico trials. METHODS AND RESULTS: A cohort of 800 virtual patients, with variability in atrial anatomy, electrophysiology, and tissue structure (low-voltage areas, LVAs), was developed and validated against clinical data from ionic currents to electrocardiogram. Virtual patients presenting AF post-PVI underwent 12 secondary treatments. Sustained AF developed in 522 virtual patients after PVI. Second ablation procedures involving left atrial ablation alone showed 55% efficacy, only succeeding in the small right atria (<60 mL). When additional cavo-tricuspid isthmus ablation was considered, Marshall-PLAN sufficed (66% efficacy) for the small left atria (<90 mL). For the bigger left atria, a more aggressive ablation approach was required, such as anterior mitral line (75% efficacy) or posterior wall isolation plus mitral isthmus ablation (77% efficacy). Virtual patients with LVAs greatly benefited from LVA ablation in the left and right atria (100% efficacy). Conversely, in the absence of LVAs, synergistic ablation and pharmacotherapy could terminate AF. In the absence of ablation, the patient's ionic current substrate modulated the response to antiarrhythmic drugs, being the inward currents critical for optimal stratification to amiodarone or vernakalant. CONCLUSION: In silico trials identify optimal strategies for AF treatment based on virtual patient characteristics, evidencing the power of human modelling and simulation as a clinical assisting tool.

Mathematical and numerical analysis for PDE systems modeling intravascular drug release from arterial stents and transport in arterial tissue.

This paper is concerned with the PDE (partial differential equation) and numerical analysis of a modified one-dimensional intravascular stent model. It is proved that the modified model has a unique weak solution by using the Galerkin method combined with a compactness argument. A semi-discrete finite-element method and a fully discrete scheme using the Euler time-stepping have been formulated for the PDE model. Optimal order error estimates in the energy norm are proved for both schemes. Numerical results are presented, along with comparisons between different decoupling strategies and time-stepping schemes. Lastly, extensions of the model and its PDE and numerical analysis results to the two-dimensional case are also briefly discussed.

A lumped parameter model for evaluating coronary artery blood supply capacity.

The coronary artery constitutes a vital vascular system that sustains cardiac function, with its primary role being the conveyance of indispensable nutrients to the myocardial tissue. When coronary artery disease occurs, it will affect the blood supply of the heart and induce myocardial ischemia. Therefore, it is of great significance to numerically simulate the coronary artery and evaluate its blood supply capacity. In this article, the coronary artery lumped parameter model was derived based on the relationship between circuit system parameters and cardiovascular system parameters, and the blood supply capacity of the coronary artery in healthy and stenosis states was studied. The aortic root pressure calculated by the aortic valve fluid-structure interaction (AV FSI) simulator was employed as the inlet boundary condition. To emulate the physiological phenomenon of sudden pressure drops resulting from an abrupt reduction in blood vessel radius, a head loss model was connected at the coronary artery's entrance. For each coronary artery outlet, the symmetric structured tree model was appended to simulate the terminal impedance of the missing downstream coronary arteries. The particle swarm optimization (PSO) algorithm was used to optimize the blood flow viscous resistance, blood flow inertia, and vascular compliance of the coronary artery model. In the stenosis states, the relative flow and fractional flow reserve (FFR) calculated by numerical simulation corresponded to the published literature data. It was anticipated that the proposed model can be readily adapted for clinical application, serving as a valuable reference for diagnosing and treating patients.

Efficient electrocardiogram generation based on cardiac electric vector simulation model.

This study introduces a novel Cardiac Electric Vector Simulation Model (CEVSM) to address the computational inefficiencies and low fidelity of traditional electrophysiological models in generating electrocardiograms (ECGs). Our approach leverages CEVSM to efficiently produce reliable ECG samples, facilitating data augmentation essential for the computer-aided diagnosis of myocardial infarction (MI). Significantly, experimental results show that our model dramatically reduces computation time compared to conventional models, with the self-adapting regression transformation matrix method (SRTM) providing clear advantages. SRTM not only achieves high fidelity in ECG simulations but also ensures exceptional consistency with the gold standard method, greatly enhancing MI localization accuracy by data augmentation. These advancements highlight the potential of our model to generate dependable ECG training samples, making it highly suitable for data augmentation and significantly advancing the development and validation of intelligent MI diagnostic systems. Furthermore, this study demonstrates the feasibility of applying life system simulations in the training of medical big models.

A multi-objective optimisation approach for the linear modelling of cerebral autoregulation system.

OBJECTIVE: Dynamic cerebral autoregulation (dCA) has been addressed through different approaches for discriminating between normal and impaired conditions based on spontaneous fluctuations in arterial blood pressure (ABP) and cerebral blood flow (CF). This work presents a novel multi-objective optimisation (MO) approach for finding good configurations of a cerebrovascular resistance-compliance model. METHODS: Data from twenty-nine subjects under normo and hypercapnic (5% CO2 in air) conditions was used. Cerebrovascular resistance and vessel compliance models with ABP as input and CF velocity as output were fitted using a MO approach, considering fitting Pearson's correlation and error. RESULTS: MO approach finds better model configurations than the single-objective (SO) approach, especially for hypercapnic conditions. In addition, the Pareto-optimal front from the multi-objective approach enables new information on dCA, reflecting a higher contribution of myogenic mechanism for explaining dCA impairment.

A study on the correlation between lung injury severity and cardiac function through a closed-loop model.

OBJECTIVE: Numerous clinical and pathological studies have confirmed that lung injury can cause cardiovascular disease, but there is no explanation for the mechanism by which the degree of lung injury affects cardiac function. We attempt to reveal this mechanism of influence by simulating a cyclic model. METHOD: This study established a closed-loop cardiovascular model with a series of electrical parameters. Including the heart, lungs, arteries, veins, etc., each part of the cardiovascular system is modeled using centralized parameters. Adjusting these lung resistances to alter the degree of lung injury is aimed at reflecting the impact of different degrees of lung injury on cardiac function. Finally, analyze and compare the changes in blood pressure, aortic flow, atrioventricular volume, and atrioventricular pressure among different lung injuries to obtain the changes in cardiac function. RESULTS: In this model, the peak aortic flow decreased, the earlier the trough appeared, and the total aortic flow decreased. Left atrial blood pressure decreased from 6.5 mmHg to around 5.5 mmHg, left ventricular blood pressure decreased from 100 mmHg to around 50 mmHg, and aortic blood pressure also decreased from 100 mmHg to around 50 mmHg. The blood pressure in the pulmonary artery, right atrium, and right ventricle increases. The right ventricular blood pressure decreased from 20 mmHg to around 40 mmHg, while the right atrial blood pressure slightly increased. It can be seen that the increase in impedance has a greater impact on ventricular blood pressure than on atrium. Pulmonary arterial pressure significantly increases, rising from 20 mmHg to around 50 mmHg, forming pulmonary hypertension. The left ventricular end-systolic potential energy, filling energy, stroke work, stroke output, left ventricular filling period, maximum blood pressure during ventricular ejection period, and stroke energy efficiency decrease. CONCLUSION: We established a closed-loop cardiovascular model that reveals that the more severe lung injury, the higher blood pressure in the pulmonary artery, right atrium, and right ventricle, while the lower blood pressure in the left atrium, left ventricle, and aorta. The increase in pulmonary impedance leads to abnormalities in myocardial contraction, diastolic function, and cardiac reserve capacity, leading to a decrease in cardiac function. This closed-loop model provides a method for pre assessment of cardiovascular disease after lung injury.

Integrated modeling and simulation of recruitment of myocardial perfusion and oxygen delivery in exercise.

While exercise-mediated vasoregulation in the myocardium is understood to be governed by autonomic, myogenic, and metabolic-mediated mechanisms, we do not yet understand the spatial heterogeneity of vasodilation or its effects on microvascular flow patterns and oxygen delivery. This study uses a simulation and modeling approach to explore the mechanisms underlying the recruitment of myocardial perfusion and oxygen delivery in exercise. The simulation approach integrates model components representing: whole-body cardiovascular hemodynamics, cardiac mechanics and myocardial work; myocardial perfusion; and myocardial oxygen transport. Integrating these systems together, model simulations reveal: (1.) To match expected flow and transmural flow ratios at increasing levels of exercise, a greater degree of vasodilation must occur in the subendocardium compared to the subepicardium. (2.) Oxygen extraction and venous oxygenation are predicted to substantially decrease with increasing exercise level preferentially in the subendocardium, suggesting that an oxygen-dependent error signal driving metabolic mediated recruitment of flow would be operative only in the subendocardium. (3.) Under baseline physiological conditions approximately 4% of the oxygen delivered to the subendocardium may be supplied via retrograde flow from coronary veins.

The hybrid (physical-computational) cardiovascular simulator to study valvular diseases.

To better understand the impact of valvular heart disease (VHD) on the hemodynamics of the circulatory system, investigations can be carried out using a model of the cardiovascular system. In this study, a previously developed hybrid (hydro-numerical) simulator of the cardiovascular system (HCS) was adapted and used. In our HCS Björk-Shiley mechanical heart valves were used, playing the role of mitral and aortic ones. In order to simulate aortic stenosis (AS) and mitral regurgitation (MR), special mechanical devices have been developed and integrated with the HCS. The simulation results proved that the system works correctly. Namely, in the case of AS - the mean pulmonary arterial pressure was increased due to increased preload of the left ventricle and the decrease in right ventricular preload was caused by a decrease in systemic arterial pressure. The severity of AS was performed based on the transaortic pressure gradient as well as using the Gorlin and Aaslid equations. In the case of severe AS, when the mean gradient was above 40 mmHg, the aortic valve orifice area was 0.5 cm2, which is in line with ACC/AHA guidelines. For the case of MR - with increasing severity of MR, there was a decrease in the left ventricular pressure and an increase in left atrial pressure. Using mechanical heart valves to simulate VHD by the HCS can be a valuable tool for biomedical research, providing a safe and controlled environment to study and understand the pathophysiology of VHD.

HDL and plaque regression in a multiphase model of early atherosclerosis.

Atherosclerosis is a chronic disease of the arteries characterised by the accumulation of lipids and lipid-engorged cells in the artery wall. Early plaque growth is aggravated by the deposition of low density lipoproteins (LDL) in the wall and the subsequent immune response. High density lipoproteins (HDL) counterbalance the effects of LDL by accepting cholesterol from macrophages and removing it from the plaque. In this paper, we develop a free boundary multiphase model to investigate the effects of LDL and HDL on early plaque development. We examine how the rates of LDL and HDL deposition affect cholesterol accumulation in macrophages, and how this impacts cell death rates and emigration. We identify a region of LDL-HDL parameter space where plaque growth stabilises for low LDL and high HDL influxes, due to macrophage emigration and HDL clearance that counterbalances the influx of new cells and cholesterol. We explore how the efferocytic uptake of dead cells and the recruitment of new macrophages affect plaque development for a range of LDL and HDL influxes. Finally, we consider how changes in the LDL-HDL profile can change the course of plaque development. We show that changes towards lower LDL and higher HDL can slow plaque growth and even induce regression. We find that these changes have less effect on larger, more established plaques, and that temporary changes will only slow plaque growth in the short term.

Human induced pluripotent stem cell-derived atrial cardiomyocytes recapitulate contribution of the slowly activating delayed rectifier currents IKs to repolarization in the human atrium.

AIMS: Human induced pluripotent stem cell-derived atrial cardiomyocytes (hiPSC-aCM) could be a helpful tool to study the physiology and diseases of the human atrium. To fulfil this expectation, the electrophysiology of hiPSC-aCM should closely resemble the situation in the human atrium. Data on the contribution of the slowly activating delayed rectifier currents (IKs) to repolarization are lacking for both human atrium and hiPSC-aCM. METHODS AND RESULTS: Human atrial tissues were obtained from patients with sinus rhythm (SR) or atrial fibrillation (AF). Currents were measured in human atrial cardiomyocytes (aCM) and compared with hiPSC-aCM and used to model IKs contribution to action potential (AP) shape. Action potential was recorded by sharp microelectrodes. HMR-1556 (1 µM) was used to identify IKs and to estimate IKs contribution to repolarization. Less than 50% of hiPSC-aCM and aCM possessed IKs. Frequency of occurrence, current densities, activation/deactivation kinetics, and voltage dependency of IKs did not differ significantly between hiPSC-aCM and aCM, neither in SR nor AF. ß-Adrenoceptor stimulation with isoprenaline did not increase IKs neither in aCM nor in hiPSC-aCM. In tissue from SR, block of IKs with HMR-1556 did not lengthen the action potential duration, even when repolarization reserve was reduced by block of the ultra-rapid repolarizing current with 4-aminopyridine or the rapidly activating delayed rectifier potassium outward current with E-4031. CONCLUSION: I Ks exists in hiPSC-aCM with biophysics not different from aCM. As in adult human atrium (SR and AF), IKs does not appear to relevantly contribute to repolarization in hiPSC-aCM.

Eulerian- lagrangian dense discrete phase model (DDPM) of stenotic LAD coronary arteries in comparison with single phase modeling.

In computational fluid dynamic studies related to blood flow, investigating the behavior of blood particles is crucial, especially red blood cells as they constitute a significant proportion of blood particles. Additionally, studying red blood cell movements is necessary, especially in stenotic artery geometries. A new multiphase scheme was utilized to demonstrate the effect of red blood cells on hemodynamics in complex coronary arteries and investigate the consequence of their motion. To investigate the effect of red blood cell movement on flow, the dense discrete phase model (DDPM) was used. This simulation was performed in 3D coronary arteries with different degrees of stenosis, utilizing blood pressure as inlet and outlet boundary conditions while assuming the arterial wall to be rigid. The model prediction shows good agreement with experimental data. Velocity values were comparable in both single-phase and two-phase flow simulations, but the shear stress in two-phase modeling had higher values. In the two-phase DDPM modeling, the recirculation areas indicated a higher probability of atherosclerosis plaque re-formation in the pre-stenosis area compared to the stenosis and post-stenosis areas. The DDPM model was found to be more effective in obtaining shear stress values in the artery. Additionally, this model provides good results compared to the single-phase model in investigating the movement of particles along the artery as well as recirculation areas that lead to the deposition of particles.

Flexural pulse wave velocity in blood vessels.

Arteriosclerosis is a major risk factor for cardiovascular disease and results in arterial vessel stiffening. Velocity estimation of the pulse wave sent by the heart and propagating into the arteries is a widely accepted biomarker. This symmetrical pulse wave propagates at a speed which is related to the Young's modulus through the Moens Korteweg (MK) equation. Recently, an antisymmetric flexural wave has been observed in vivo. Unlike the symmetrical wave, it is highly dispersive. This property offers promising applications for monitoring arterial stiffness and early detection of atheromatous plaque. However, as far as it is known, no equivalent of the MK equation exists for flexural pulse waves. To bridge this gap, a beam based theory was developed, and approximate analytical solutions were reached. An experiment in soft polymer artery phantoms was built to observe the dispersion of flexural waves. A good agreement was found between the analytical expression derived from beam theory and experiments. Moreover, numerical simulations validated wave speed dependence on the elastic and geometric parameters at low frequencies. Clinical applications, such as arterial age estimation and arterial pressure measurement, are foreseen.

Analyzing solitary wave solutions of the nonlinear Murray equation for blood flow in vessels with non-uniform wall properties.

Solitary wave solutions are of great interest to bio-mathematicians and other scientists because they provide a basic description of nonlinear phenomena with many practical applications. They provide a strong foundation for the development of novel biological and medical models and therapies because of their remarkable behavior and persistence. They have the potential to improve our comprehension of intricate biological systems and help us create novel therapeutic approaches, which is something that researchers are actively investigating. In this study, solitary wave solutions of the nonlinear Murray equation will be discovered using a modified extended direct algebraic method. These solutions represent a uniform variation in blood vessel shape and diameter that can be used to stimulate blood flow in patients with cardiovascular disease. These solutions are newly in the literature, and give researchers an important tool for grasping complex biological systems. To see how the solitary wave solutions behave, graphs are displayed using Matlab.