A Hybrid Model for Cardiac Perfusion: Coupling a Discrete Coronary Arterial Tree Model with a Continuous Porous-Media Flow Model of the Myocardium.

This paper presents a novel hybrid approach for the computational modeling of cardiac perfusion, combining a discrete model of the coronary arterial tree with a continuous porous-media flow model of the myocardium. The constructive constrained optimization (CCO) algorithm captures the detailed topology and geometry of the coronary arterial tree network, while Poiseuille's law governs blood flow within this network. Contrast agent dynamics, crucial for cardiac MRI perfusion assessment, are modeled using reaction-advection-diffusion equations within the porous-media framework. The model incorporates fibrosis-contrast agent interactions and considers contrast agent recirculation to simulate myocardial infarction and Gadolinium-based late-enhancement MRI findings. Numerical experiments simulate various scenarios, including normal perfusion, endocardial ischemia resulting from stenosis, and myocardial infarction. The results demonstrate the model's efficacy in establishing the relationship between blood flow and stenosis in the coronary arterial tree and contrast agent dynamics and perfusion in the myocardial tissue. The hybrid model enables the integration of information from two different exams: computational fractional flow reserve (cFFR) measurements of the heart coronaries obtained from CT scans and heart perfusion and anatomy derived from MRI scans. The cFFR data can be integrated with the discrete arterial tree, while cardiac perfusion MRI data can be incorporated into the continuum part of the model. This integration enhances clinical understanding and treatment strategies for managing cardiovascular disease.

Enhanced optimization-based method for the generation of patient-specific models of Purkinje networks.

Cardiac Purkinje networks are a fundamental part of the conduction system and are known to initiate a variety of cardiac arrhythmias. However, patient-specific modeling of Purkinje networks remains a challenge due to their high morphological complexity. This work presents a novel method based on optimization principles for the generation of Purkinje networks that combines geometric and activation accuracy in branch size, bifurcation angles, and Purkinje-ventricular-junction activation times. Three biventricular meshes with increasing levels of complexity are used to evaluate the performance of our approach. Purkinje-tissue coupled monodomain simulations are executed to evaluate the generated networks in a realistic scenario using the most recent Purkinje/ventricular human cellular models and physiological values for the Purkinje-ventricular-junction characteristic delay. The results demonstrate that the new method can generate patient-specific Purkinje networks with controlled morphological metrics and specified local activation times at the Purkinje-ventricular junctions.

Modelling the response of urban lichens to broad-scale changes in air pollution and climate.

To create more resilient cities, it is important that we understand the effects of the global change drivers in cities. Biodiversity-based ecological indicators (EIs) can be used for this, as biodiversity is the basis of ecosystem structure, composition, and function. In previous studies, lichens have been used as EIs to monitor the effects of global change drivers in an urban context, but only in single-city studies. Thus, we currently do not understand how lichens are affected by drivers that work on a broader scale. Therefore, our aim was to quantify the variance in lichen biodiversity-based metrics (taxonomic and trait-based) that can be explained by environmental drivers working on a broad spatial scale, in an urban context where local drivers are superimposed. To this end, we performed an unprecedented effort to sample epiphytic lichens in 219 green spaces across a continental gradient from Portugal to Estonia. Twenty-six broad-scale drivers were retrieved, including air pollution and bio-climatic variables, and their dimensionality reduced by means of a principal component analysis (PCA). Thirty-eight lichen metrics were then modelled against the scores of the first two axes of each PCA, and their variance partitioned into pollution and climate components. For the first time, we determined that 15% of the metric variance was explained by broad-scale drivers, with broad-scale air pollution showing more importance than climate across the majority of metrics. Taxonomic metrics were better explained by air pollution, as expected, while climate did not surpass air pollution in any of the trait-based metric groups. Consequently, 85% of the metric variance was shown to occur at the local scale. This suggests that further work is necessary to decipher the effects of climate change. Furthermore, although drivers working within cities are prevailing, both spatial scales must be considered simultaneously if we are to use lichens as EIs in cities at continental to global scales.

A Poroelastic Approach for Modelling Myocardial Oedema in Acute Myocarditis.

Myocarditis is a general set of mechanisms that manifest themselves into the inflammation of the heart muscle. In 2017, more than 3 million people were affected by this disease worldwide, causing about 47,000 deaths. Many aspects of the origin of this disease are well known, but several important questions regarding the disease remain open. One of them is why some patients develop a significantly localised inflammation while others develop a much more diffuse inflammation, reaching across large portions of the heart. Furthermore, the specific role of the pathogenic agent that causes inflammation as well as the interaction with the immune system in the progression of the disease are still under discussion. Providing answers to these crucial questions can have an important impact on patient treatment. In this scenario, computational methods can aid specialists to understand better the relationships between pathogens and the immune system and elucidate why some patients develop diffuse myocarditis. This paper alters a recently developed model to study the myocardial oedema formation in acute infectious myocarditis. The model describes the finite deformation regime using partial differential equations to represent tissue displacement, fluid pressure, fluid phase, and the concentrations of pathogens and leukocytes. A sensitivity analysis was performed to understand better the influence of the most relevant model parameters on the disease dynamics. The results showed that the poroelastic model could reproduce local and diffuse myocarditis dynamics in simplified and complex geometrical domains.

Timing the race of vaccination, new variants, and relaxing restrictions during COVID-19 pandemic.

Late in 2019, China identified a new type of coronavirus, SARS-CoV-2, and due to its fast spread, the World Health Organisation (WHO) declared a pandemic named COVID-19. Some variants of this virus were detected, including the Delta, which caused new waves of infections. This work uses an extended version of a SIRD model that includes vaccination effects to measure the impact of the Delta variant in three countries: Germany, Israel and Brazil. The calibrated models were able to reproduce the dynamics of the above countries. In addition, hypothetical scenarios were simulated to quantify the impact of vaccination and mitigation policies during the Delta wave. The results showed that the model could reproduce the complex dynamics observed in the different countries. The estimated increase of transmission rate due to the Delta variant was highest in Israel (7.9), followed by Germany (2.7) and Brazil (1.5). These values may support the hypothesis that people immunised against COVID-19 may lose their defensive antibodies with time since Israel, Germany, and Brazil fully vaccinated half of the population in March, July, and October. The scenario to study the impact of vaccination revealed relative reductions in the total number of deaths between 30% and 250%; an absolute reduction of 300 thousand deaths in Brazil due to vaccination during the Delta wave. The second hypothetical scenario revealed that mitigation policies saved up to 300 thousand Brazilians; relative reductions in the total number of deaths between 24% and 120% in the three analysed countries. Therefore, the results suggest that both vaccination and mitigation policies were crucial in decreasing the spread and the number of deaths during the Delta wave.

Design and Comparison of Image Hashing Methods: A Case Study on Cork Stopper Unique Identification.

Cork stoppers were shown to have unique characteristics that allow their use for authentication purposes in an anti-counterfeiting effort. This authentication process relies on the comparison between a user's cork image and all registered cork images in the database of genuine items. With the growth of the database, this one-to-many comparison method becomes lengthier and therefore usefulness decreases. To tackle this problem, the present work designs and compares hashing-assisted image matching methods that can be used in cork stopper authentication. The analyzed approaches are the discrete cosine transform, wavelet transform, Radon transform, and other methods such as difference hash and average hash. The most successful approach uses a 1024-bit hash length and difference hash method providing a 98% accuracy rate. By transforming the image matching into a hash matching problem, the approach presented becomes almost 40 times faster when compared to the literature.

The Quixotic Task of Forecasting Peaks of COVID-19: Rather Focus on Forward and Backward Projections.

Over the last months, mathematical models have been extensively used to help control the COVID-19 pandemic worldwide. Although extremely useful in many tasks, most models have performed poorly in forecasting the pandemic peaks. We investigate this common pitfall by forecasting four countries' pandemic peak: Austria, Germany, Italy, and South Korea. Far from the peaks, our models can forecast the pandemic dynamics 20 days ahead. Nevertheless, when calibrating our models close to the day of the pandemic peak, all forecasts fail. Uncertainty quantification and sensitivity analysis revealed the main obstacle: the misestimation of the transmission rate. Inverse uncertainty quantification has shown that significant changes in transmission rate commonly precede a peak. These changes are a key factor in forecasting the pandemic peak. Long forecasts of the pandemic peak are therefore undermined by the lack of models that can forecast changes in the transmission rate, i.e., how a particular society behaves, changes of mitigation policies, or how society chooses to respond to them. In addition, our studies revealed that even short forecasts of the pandemic peak are challenging. Backward projections have shown us that the correct estimation of any temporal change in the transmission rate is only possible many days ahead. Our results suggest that the distance between a change in the transmission rate and its correct identification in the curve of active infected cases can be as long as 15 days. This is intrinsic to the phenomenon and how it affects epidemic data: a new case is usually only reported after an incubation period followed by a delay associated with the test. In summary, our results suggest the phenomenon itself challenges the task of forecasting the peak of the COVID-19 pandemic when only epidemic data is available. Nevertheless, we show that exciting results can be obtained when using the same models to project different scenarios of reduced transmission rates. Therefore, our results highlight that mathematical modeling can help control COVID-19 pandemic by backward projections that characterize the phenomena' essential features and forward projections when different scenarios and strategies can be tested and used for decision-making.

Functional Traits in Lichen Ecology: A Review of Challenge and Opportunity.

Community ecology has experienced a major transition, from a focus on patterns in taxonomic composition, to revealing the processes underlying community assembly through the analysis of species functional traits. The power of the functional trait approach is its generality, predictive capacity such as with respect to environmental change, and, through linkage of response and effect traits, the synthesis of community assembly with ecosystem function and services. Lichens are a potentially rich source of information about how traits govern community structure and function, thereby creating opportunity to better integrate lichens into 'mainstream' ecological studies, while lichen ecology and conservation can also benefit from using the trait approach as an investigative tool. This paper brings together a range of author perspectives to review the use of traits in lichenology, particularly with respect to European ecosystems from the Mediterranean to the Arctic-Alpine. It emphasizes the types of traits that lichenologists have used in their studies, both response and effect, the bundling of traits towards the evolution of life-history strategies, and the critical importance of scale (both spatial and temporal) in functional trait ecology.

Characterization of the COVID-19 pandemic and the impact of uncertainties, mitigation strategies, and underreporting of cases in South Korea, Italy, and Brazil.

By April 7th, 2020, the Coronavirus disease 2019 (COVID-19) has infected one and a half million people worldwide, accounting for over 80 thousand of deaths in 209 countries and territories around the world. The new and fast dynamics of the pandemic are challenging the health systems of different countries. In the absence of vaccines or effective treatments, mitigation policies, such as social isolation and lock-down of cities, have been adopted, but the results vary among different countries. Some countries were able to control the disease at the moment, as is the case of South Korea. Others, like Italy, are now experiencing the peak of the pandemic. Finally, countries with emerging economies and social issues, like Brazil, are in the initial phase of the pandemic. In this work, we use mathematical models with time-dependent coefficients, techniques of inverse and forward uncertainty quantification, and sensitivity analysis to characterize essential aspects of the COVID-19 in the three countries mentioned above. The model parameters estimated for South Korea revealed effective social distancing and isolation policies, border control, and a high number in the percentage of reported cases. In contrast, underreporting of cases was estimated to be very high in Brazil and Italy. In addition, the model estimated a poor isolation policy at the moment in Brazil, with a reduction of contact around 40%, whereas Italy and South Korea estimated numbers for contact reduction are at 75% and 90%, respectively. This characterization of the COVID-19, in these different countries under different scenarios and phases of the pandemic, supports the importance of mitigation policies, such as social distancing. In addition, it raises serious concerns for socially and economically fragile countries, where underreporting poses additional challenges to the management of the COVID-19 pandemic by significantly increasing the uncertainties regarding its dynamics.

Stabilized hybrid discontinuous Galerkin finite element method for the cardiac monodomain equation.

Numerical methods for solving the cardiac electrophysiology model, which describes the electrical activity in the heart, are proposed. The model problem consists of a nonlinear reaction-diffusion partial differential equation coupled to systems of ordinary differential equations that describes electrochemical reactions in cardiac cells. The proposed methods combine an operator splitting technique for the reaction-diffusion equation with primal hybrid methods for spatial discretization considering continuous or discontinuous approximations for the Lagrange multiplier. A static condensation is adopted to form a reduced global system in terms of the multiplier only. Convergence studies exhibit optimal rates of convergence and numerical experiments show that the proposed schemes can be more efficient than standard numerical techniques commonly used in this context when preconditioned iterative methods are used for the solution of linear systems.

A personalized computational model of edema formation in myocarditis based on long-axis biventricular MRI images.

BACKGROUND: Myocarditis is defined as the inflammation of the myocardium, i.e. the cardiac muscle. Among the reasons that lead to this disease, we may include infections caused by a virus, bacteria, protozoa, fungus, and others. One of the signs of the inflammation is the formation of edema, which may be a consequence of the interaction between interstitial fluid dynamics and immune response. This complex physiological process was mathematically modeled using a nonlinear system of partial differential equations (PDE) based on porous media approach. By combing a model based on Biot's poroelasticity theory with a model for the immune response we developed a new hydro-mechanical model for inflammatory edema. To verify this new computational model, T2 parametric mapping obtained by Magnetic Resonance (MR) imaging was used to identify the region of edema in a patient diagnosed with unspecific myocarditis. RESULTS: A patient-specific geometrical model was created using MRI images from the patient with myocarditis. With this model, edema formation was simulated using the proposed hydro-mechanical mathematical model in a two-dimensional domain. The computer simulations allowed us to correlate spatiotemporal dynamics of representative cells of the immune systems, such as leucocytes and the pathogen, with fluid accumulation and cardiac tissue deformation. CONCLUSIONS: This study demonstrates that the proposed mathematical model is a very promising tool to better understand edema formation in myocarditis. Simulations obtained from a patient-specific model reproduced important aspects related to the formation of cardiac edema, its area, position, and shape, and how these features are related to immune response.

Effects of left ventricle wall thickness uncertainties on cardiac mechanics.

Computational models of the heart have reached a level of maturity that enables sophisticated patient-specific simulations and hold potential for important applications in diagnosis and therapy planning. However, such clinical use puts strict demands on the reliability and accuracy of the models and requires the sensitivity of the model predictions due to errors and uncertainty in the model inputs to be quantified. The models typically contain a large number of parameters, which are difficult to measure and therefore associated with considerable uncertainty. Additionally, patient-specific geometries are usually constructed by semi-manual processing of medical images and must be assumed to be a potential source of model uncertainty. In this paper, we assess the model accuracy by considering the impact of geometrical uncertainties, which typically occur in image-based computational geometries. An approach based on 17 AHA segments diagram is used to consider uncertainties in wall thickness and also in the material properties and fiber orientation, and we perform a comprehensive uncertainty quantification and sensitivity analysis based on polynomial chaos expansions. The quantities considered include stress, strain and global deformation parameters of the left ventricle. The results indicate that important quantities of interest may be more affected by wall thickness, and highlight the need for accurate geometry reconstructions in patient-specific cardiac mechanics models.

Modeling the provision of air-quality regulation ecosystem service provided by urban green spaces using lichens as ecological indicators.

The UN Sustainable Development Goals states that urban air pollution must be tackled to create more inclusive, safe, resilient and sustainable cities. Urban green infrastructures can mitigate air pollution, but a crucial step to use this knowledge into urban management is to quantify how much air-quality regulation can green spaces provide and to understand how the provision of this ecosystem service is affected by other environmental factors. Considering the insufficient number of air quality monitoring stations in cities to monitor the wide range of natural and anthropic sources of pollution with high spatial resolution, ecological indicators of air quality are an alternative cost-effective tool. The aim of this work was to model the supply of air-quality regulation based on urban green spaces characteristics and other environmental factors. For that, we sampled lichen diversity in the centroids of 42 urban green spaces in Lisbon, Portugal. Species richness was the best biodiversity metric responding to air pollution, considering its simplicity and its significative response to the air pollutants concentration data measured in the existent air quality monitoring stations. Using that metric, we then created a model to estimate the supply of air quality regulation provided by green spaces in all green spaces of Lisbon based on the response to the following environmental drivers: the urban green spaces size and its vegetation density. We also used the unexplained variance of this model to map the background air pollution. Overall, we suggest that management should target the smallest urban green spaces by increasing green space size or tree density. The use of ecological indicators, very flexible in space, allow the understanding and the modeling of the provision of air-quality regulation by urban green spaces, and how urban green spaces can be managed to improve air quality and thus improve human well-being and cities resilience.

Ectopic beats arise from micro-reentries near infarct regions in simulations of a patient-specific heart model.

Ectopic beats are known to be involved in the initiation of a variety of cardiac arrhythmias. Although their location may vary, ectopic excitations have been found to originate from infarct areas, regions of micro-fibrosis and other heterogeneous tissues. However, the underlying mechanisms that link ectopic foci to heterogeneous tissues have yet to be fully understood. In this work, we investigate the mechanism of micro-reentry that leads to the generation of ectopic beats near infarct areas using a patient-specific heart model. The patient-specific geometrical model of the heart, including scar and peri-infarct zones, is obtained through magnetic resonance imaging (MRI). The infarct region is composed of ischemic myocytes and non-conducting cells (fibrosis, for instance). Electrophysiology is captured using an established cardiac myocyte model of the human ventricle modified to describe ischemia. The simulation results clearly reveal that ectopic beats emerge from micro-reentries that are sustained by the heterogeneous structure of the infarct regions. Because microscopic information about the heterogeneous structure of the infarct regions is not available, Monte-Carlo simulations are used to identify the probabilities of an infarct region to behave as an ectopic focus for different levels of ischemia and different percentages of non-conducting cells. From the proposed model, it is observed that ectopic beats are generated when a percentage of non-conducting cells is near a topological metric known as the percolation threshold. Although the mechanism for micro-reentries was proposed half a century ago to be a source of ectopic beats or premature ventricular contractions during myocardial infarction, the present study is the first to reproduce this mechanism in-silico using patient-specific data.

An Optimization-Based Algorithm for the Construction of Cardiac Purkinje Network Models.

OBJECTIVE: This work presents a new algorithm for the construction of a model for the Purkinje network (PN) of the heart. METHODS: The algorithm is based on a method called constructive constrained optimization (CCO), which was reformulated for the specific case of automatic PN generation. The proposed optimization-based algorithm is referred to as constructive optimization (CO). The CO method iteratively constructs the PN by minimizing the total length of the generated PN tree. In addition, it can take into account some important topological information of the PN, such as the location of the Purkinje-muscle junctions and the average bifurcation angle found in the literature. RESULTS: To validate the model, the new method was compared with the classical L-system method for generating PN models and to a recently proposed image-based technique. CONCLUSION: The results show that the CO is able to construct PNs with geometric features and activation times that are in good agreement with those reported in the literature and to those obtained by the other aforementioned alternatives.

Preconditioned augmented Lagrangian formulation for nearly incompressible cardiac mechanics.

Computational modeling of the heart is a subject of substantial medical and scientific interest, which may contribute to increase the understanding of several phenomena associated with cardiac physiological and pathological states. Modeling the mechanics of the heart have led to considerable insights, but it still represents a complex and a demanding computational problem, especially in a strongly coupled electromechanical setting. Passive cardiac tissue is commonly modeled as hyperelastic and is characterized by quasi-incompressible, orthotropic, and nonlinear material behavior. These factors are known to be very challenging for the numerical solution of the model. The near-incompressibility is known to cause numerical issues such as the well-known locking phenomenon and ill-conditioning of the stiffness matrix. In this work, the augmented Lagrangian method is used to handle the nearly incompressible condition. This approach can potentially improve computational performance by reducing the condition number of the stiffness matrix and thereby improving the convergence of iterative solvers. We also improve the performance of iterative solvers by the use of an algebraic multigrid preconditioner. Numerical results of the augmented Lagrangian method combined with a preconditioned iterative solver for a cardiac mechanics benchmark suite are presented to show its improved performance.

Performance evaluation of GPU parallelization, space-time adaptive algorithms, and their combination for simulating cardiac electrophysiology.

The use of computer models as a tool for the study and understanding of the complex phenomena of cardiac electrophysiology has attained increased importance nowadays. At the same time, the increased complexity of the biophysical processes translates into complex computational and mathematical models. To speed up cardiac simulations and to allow more precise and realistic uses, 2 different techniques have been traditionally exploited: parallel computing and sophisticated numerical methods. In this work, we combine a modern parallel computing technique based on multicore and graphics processing units (GPUs) and a sophisticated numerical method based on a new space-time adaptive algorithm. We evaluate each technique alone and in different combinations: multicore and GPU, multicore and GPU and space adaptivity, multicore and GPU and space adaptivity and time adaptivity. All the techniques and combinations were evaluated under different scenarios: 3D simulations on slabs, 3D simulations on a ventricular mouse mesh, ie, complex geometry, sinus-rhythm, and arrhythmic conditions. Our results suggest that multicore and GPU accelerate the simulations by an approximate factor of 33×, whereas the speedups attained by the space-time adaptive algorithms were approximately 48. Nevertheless, by combining all the techniques, we obtained speedups that ranged between 165 and 498. The tested methods were able to reduce the execution time of a simulation by more than 498× for a complex cellular model in a slab geometry and by 165× in a realistic heart geometry simulating spiral waves. The proposed methods will allow faster and more realistic simulations in a feasible time with no significant loss of accuracy.

Uniformization method for solving cardiac electrophysiology models based on the Markov-chain formulation.

This paper compares different numerical methods for the solution of myocyte models of cardiac electrophysiology. In particular, it presents how the technique called uniformization method substantially increases the stability of simple first-order methods such as Euler explicit method and Rush-Larsen (RL) method, for the solution of modern electrophysiology models that are based on continuous-time Markov chains (MCs) for the description of subcellular structures, such as ion channels. The MCs are often associated with stiff ordinary differential equations that severely limit the time step used by these traditional methods. By using the uniformization method, we could significantly increase the time steps for the solution of different cardiac electrophysiology models and improve the computational performance up to 150 times compared to the performance of Euler's and RL's methods.

Prevalência e fatores associados ao consumo de queijo não pasteurizado entre pacientes com tuberculose de uma área urbana do Brasil / Prevalence and associated factors to unpasteurized cheese consumption inpatients with tuberculosis in an urban area of Brazil

No Brasil o consumo de leite cru e derivados lácteos elaborados a partir de leite cru é responsável por 30% da produção total. Um estudo de corte transversal foi realizado para estimar a prevalência e, adicionalmente, avaliar fatores associados ao consumo de queijo feito com leite cru em pacientes com tuberculose na região de Juiz de Fora, Minas Gerais. Foram construídos modelos explicativos de regressão linear univariado e multivariado. A taxa de consumo atual de queijo feito com leite cru encontrada foi de 30,7%. A mediana de consumo foi de 279 unidades de queijo feito com leite cru por indivíduo ao longo da vida. As variáveis sexo, idade e ocupação relacionada à pecuária ou agroindústria de alimentos de origem animal apresentaram-se associadas (p 0,05) ao consumo de queijo feito com leite cru. Os resultados reforçam a necessidade de medidas efetivas de controle do comércio de queijo feito com leite cru e de programas de conscientização dos consumidores sobre os riscos de consumo de lácteos elaborados a partir de leite cru, assegurados pelas autoridades de saúde e agricultura. Esse estudo objetivou estimar a prevalência atual ou passada de consumo, bem como as quantidades de ingestão ao longo da vida, de queijo feito com leite cru e os fatores de risco associados a esse consumo.

In Brazil the consumption of unpasteurized milk and dairy products accounted for 30% of total production. A cross-sectional studywas conducted to estimate the prevalence and additionally assess factors associated to unpasteurized cheese consumption inpatients with tuberculosis in the region of Juiz de Fora, Minas Gerais. Explanatory univariate and multivariate linear regressionmodels were constructed. The current consumption rate of unpasteurized cheese was 43.9%. The median of consumption was279 units of unpasteurized cheese per individual over the lifetime. Sex, age and occupation related to farming or agribusinessof foodstuffs of animal origin, were associated (p ≤ 0.05) to unpasteurized milk consumption. The results reinforce the need foreffective measures to control informal cheese trade, highlighting the need of health and agriculture authorities assure educationalprograms focusing health hazards of drinking raw milk. This study aimed to estimate the prevalence of current or past consumption,as well as the amounts of intake throughout life, cheese made with raw milk and the risk factors associated with consumption.

Automatic Parameterization Strategy for Cardiac Electrophysiology Simulations.

Driven by recent advances in medical imaging, image segmentation and numerical techniques, computer models of ventricular electrophysiology account for increasingly finer levels of anatomical and biophysical detail. However, considering the large number of model parameters involved parameterization poses a major challenge. A minimum requirement in combined experimental and modeling studies is to achieve good agreement in activation and repolarization sequences between model and experiment or patient data. In this study, we propose basic techniques which aid in determining bidomain parameters to match activation sequences. An iterative parameterization algorithm is implemented which determines appropriate bulk conductivities which yield prescribed velocities. In addition, a method is proposed for splitting the computed bulk conductivities into individual bidomain conductivities by prescribing anisotropy ratios.