Spatial Description Research Articles

Dynamics of living income for coffee farming households in agricultural area development policy: Socio-spatial integration

A decent income is an important part of overcoming economic disparities in agricultural development, especially in developing countries where most of the population are small farmers. As a developing country, Indonesia has also established a decent standard of living by setting a minimum wage as a reference for a decent income at the national and regional levels. However, this benchmark is not relevant to be applied uniformly at all levels of workers. This research determines the national coffee development area as the study center. We developed the Anker living wage methodology as a simple concept for determining living income for certain worker communities, especially for small farmers in rural areas who dominate the type of work in Indonesia. a socio-spatial approach is used to visualize the distribution of the dynamics of a decent life in various conditions of farming households. We found that 96.6% of coffee farming households in the national coffee development area had an inadequate living income, and only 3.4% were at an adequate level. We conclude that the current state of agricultural land management does not guarantee a decent income, even though efforts have been made to maximize agricultural crop productivity. The spatial description also shows that this condition is evenly distributed throughout residential areas. It is hoped that this approach can become an essential reference in implementing agricultural development programs that focus on welfare and equitable development as benchmarks for sustainable development goals in the future.

Geostatistical Simulation of Daily Rainfall Fields—Performance Assessment for Extremes in West Africa

Abstract The spatial description of high-resolution extreme daily rainfall fields is challenging because of the high spatial and temporal variability of rainfall, particularly in tropical regions due to the stochastic nature of convective rainfall. Geostatistical simulations offer a solution to this problem. In this study, a stochastic geostatistical simulation technique based on the spectral turning bands method is presented for modeling daily rainfall extremes in the data-scarce tropical Ouémé River basin (Benin). This technique uses meta-Gaussian frameworks built on Gaussian random fields, which are transformed into realistic rainfall fields using statistical transfer functions. The simulation framework can be conditioned on point observations and is computationally efficient in generating multiple ensembles of extreme rainfall fields. The results of tests and evaluations for multiple extremes demonstrate the effectiveness of the simulation framework in modeling more realistic rainfall fields and capturing their variability. It successfully reproduces the empirical cumulative distribution function of the observation samples and outperforms classical interpolation techniques like ordinary kriging in terms of spatial continuity and rainfall variability. The study also addresses the challenge of dealing with uncertainty in data-poor areas and proposes a novel approach for determining the spatial correlation structure even with low station density, resulting in a performance boost of 9.5% compared to traditional techniques. Additionally, we present a low-skill reference simulation method to facilitate a comprehensive comparison of the geostatistical simulation approaches. The simulations generated have the potential to provide valuable inputs for hydrological modeling.

Coherent point drift with Skewed Distribution for accurate point cloud registration

Point cloud registration methods based on Gaussian Mixture Models (GMMs) exhibit high robustness. However, GMM cannot precisely depict point clouds, because the Gaussian distribution is spatially symmetric and local surfaces of point clouds are typically non-symmetric. In this paper, we propose a novel method for rigid point cloud registration, termed coherent point drift with Skewed Distribution (Skewed CPD). Our method employs an asymmetric distribution constructed from the local surface normals and curvature radii. Compared to the Gaussian distribution, this skewed distribution provides a more accurate spatial description of points on local surfaces. Additionally, we integrate an adaptive multiplier to the covariance, which reallocates the weight of the covariance for different components in the probabilistic mixture model. We employ the EM algorithm to address this maximum likelihood estimation (MLE) issue and leverage GPU acceleration. In the M-step, we adopt an unconstrained optimization technique rooted in a Lie group and Lie algebra to attain the optimal transformation. Experimental results indicate that our method outperforms state-of-the-art methods in both accuracy and robustness. Remarkably, even without loop closure detection, the cumulative error of our approach remains minimal.

‘The ‘Vector-Model’ wavefunction: spatial description and wavepacket formation of quantum-mechanical angular momenta’

We introduce the concept of an asymptotic spatial angular-momentum wavefunction, Xmjφ,θ,χ=eimφδθ−θm eij+12χ, which treats j in the jm state as a three-dimensional entity; φ, θ, χ are the Euler angles, and θm is the Vector-Model polar angle, given by cosθm=m/j. A wealth of geometric information about j can be deduced from the eigenvalues and spatial transformation properties of the Xmjφ,θ,χ. Specifically, the Xmjφ,θ,χ wavefunction gives a computationally simple description of the sizes of the particle and orbital-angular-momentum wavepackets (constructed from Gaussian distributions in j and m ), given by effective wavepacket angular uncertainty relations for ΔmΔφ, ΔjΔχ, and ΔφΔθ. The Xmjφ,θ,χ also predicts the position of the particle-wavepacket angular motion in the orbital plane, so that the particle-wavepacket rotation can be experimentally probed through continuous and non-destructive j -rotation measurements. Finally, we use the Xmj(φ,θ,χ) to determine geometrically well-known asymptotic expressions for Clebsch–Gordan coefficients, Wigner d-functions, the gyromagnetic ratio of elementary particles, g=2, and the m -state-correlation matrix elements. Interestingly, for low j, even down to j=1/2, these expressions are either exact (the last two) or excellent approximations (the first two), showing that the Xmj(φ,θ,χ) give a useful spatial description of quantum-mechanical angular momentum, and provide a smooth connection with classical angular momentum.

Research on spatial carving method of glutenite reservoir based on opacity voxel imaging

The glutenite reservoir in an exploration area in eastern China is well-developed and holds significant exploration potential as an important oil and gas alternative layer. However, due to the influence of sedimentary characteristics, the glutenite reservoir exhibits strong lateral heterogeneity, significant vertical thickness variations, and low accuracy in reservoir space characterization, which affects the reasonable and effective deployment of development wells. Seismic data contains the three-dimensional spatial characteristics of geological bodies, but how to design a suitable transfer function to extract the nonlinear relationship between seismic data and reservoirs is crucial. At present, the transfer functions are concentrated in low-dimensional or high-dimensional fixed mathematical models, which cannot accurately describe the nonlinear relationship between seismic data and complex reservoirs, resulting in low spatial description accuracy of complex reservoirs. In this regard, this paper first utilizes a fusion method based on probability kernel to fuse seismic attributes such as wave impedance, effective bandwidth, and composite envelope difference. This provide a more intuitive reflection of the distribution characteristics of glutenite reservoirs. Moreover, a hybrid nonlinear transfer function is established to transform the fused attribute cube into an opaque attribute cube. Finally, the illumination model and ray casting method are used to perform voxel imaging of the glutenite reservoirs, brighten the detailed characteristics of reservoir space, and then form a set of methods for ' brightening reservoirs and darkening non-reservoirs ', which improves the spatial engraving accuracy of glutenite reservoirs.

SpaNCMG: improving spatial domains identification of spatial transcriptomics using neighborhood-complementary mixed-view graph convolutional network.

The advancement of spatial transcriptomics (ST) technology contributes to a more profound comprehension of the spatial properties of gene expression within tissues. However, due to challenges of high dimensionality, pronounced noise and dynamic limitations in ST data, the integration of gene expression and spatial information to accurately identify spatial domains remains challenging. This paper proposes a SpaNCMG algorithm for the purpose of achieving precise spatial domain description and localization based on a neighborhood-complementary mixed-view graph convolutional network. The algorithm enables better adaptation to ST data at different resolutions by integrating the local information from KNN and the global structure from r-radius into a complementary neighborhood graph. It also introduces an attention mechanism to achieve adaptive fusion of different reconstructed expressions, and utilizes KPCA method for dimensionality reduction. The application of SpaNCMG on five datasets from four sequencing platforms demonstrates superior performance to eight existing advanced methods. Specifically, the algorithm achieved highest ARI accuracies of 0.63 and 0.52 on the datasets of the human dorsolateral prefrontal cortex and mouse somatosensory cortex, respectively. It accurately identified the spatial locations of marker genes in the mouse olfactory bulb tissue and inferred the biological functions of different regions. When handling larger datasets such as mouse embryos, the SpaNCMG not only identified the main tissue structures but also explored unlabeled domains. Overall, the good generalization ability and scalability of SpaNCMG make it an outstanding tool for understanding tissue structure and disease mechanisms. Our codes are available at https://github.com/ZhihaoSi/SpaNCMG.

Graph methods to infer spatial disturbances: Application to Huntington's Disease's speech

ObjectiveHuntington's Disease (HD) is an inherited neurodegenerative disease caused by the mutation of the Htt gene, impacting all aspects of living and functioning. Among cognitive disabilities, spatial capacities are impaired, but their monitoring remains scarce as limited by lengthy experts' assessments. Language offers an alternative medium to evaluate patients' performance in HD. Yet, its capacities to assess HD's spatial abilities are unknown. Here, we aimed to bring proof-of-concept that HD's spatial deficits can be assessed through speech. MethodsWe developed the Spatial Description Model to graphically represent spatial relations described during the Cookie Theft Picture (CTP) task. We increased the sensitivity of our model by using only sentences with spatial terms, unlike previous studies in Alzheimer's disease. 78 carriers of the mutant Htt, including 56 manifest and 22 premanifest individuals, as well as 25 healthy controls were included from the BIOHD & (NCT01412125) & Repair-HD (NCT03119246) cohorts. The convergence and divergence of the model were validated using the SelfCog battery. ResultsOur Spatial Description Model was the only one among the four assessed approaches, revealing that individuals with manifest HD expressed fewer spatial relations and engaged in less spatial exploration compared to healthy controls. Their graphs correlated with both visuospatial and language SelfCog performances, but not with motor, executive nor memory functions. ConclusionsWe provide the proof-of-concept using our Spatial Description Model that language can grasp HD patient's spatial disturbances. By adding spatial capabilities to the panel of functions tested by the language, it paves the way for eventual remote clinical application.

Bearing fault diagnosis based on transfer learning with dual-flow manifold ResNet and improved CapsNet

In intelligent fault diagnosis for rolling bearings, the variable operating conditions of rotating machinery lead to inconsistent distributions between training and testing data, inevitably decreasing fault diagnosis accuracy. To overcome this issue, we propose a bearing fault diagnosis method based on transfer learning with a dual-flow manifold residual network and improved capsule network (DMRCN). Firstly, we introduce a dual-flow manifold residual network structure consisting of continuous wide-kernel convolution, continuous narrow-kernel convolution, and manifold learning spatial information description blocks. This architecture is utilized to extract low-level features from the data. Then, we present a novel architecture of the capsule network with self-attention output. We incorporate a self-attention mechanism into the dynamic routing algorithm to transfer richer high-level features from the low-level capsule layer to the higher-level capsule layer. During training, we propose a loss function known as global-local distribution discrepancy loss to enhance the model’s generalization ability and accelerate model convergence. Finally, we validate the performance of DMRCN in cross-condition transfer fault diagnosis on both the Case Western Reserve University public dataset and a laboratory-built bearing experimental rig dataset. Experimental results demonstrate that DMRCN outperforms other commonly used deep transfer learning methods regarding fault diagnosis across different operating conditions.

Spatial Dispersion and Statistical Description of Organized Cumulus Cloud Ensembles in Radiative Convective Equilibrium

AbstractDescriptions of cloud ensembles in radiative convective equilibrium (RCE) rely mostly on the assumption that clouds are randomly distributed in space, a hypothesis that obviously fails in presence of strong organization. In this work, idealized RCE simulations at horizontal grid spacings ranging from 2 km to 125 m are analyzed, displaying a transition between complete randomness at coarse resolution to a high degree of cold pool driven organization at high resolution. The objective is then twofold: (a) characterizing the spatial cloud patterns focusing on correlation scales and association with covariates; (b) providing a statistical description of organized cloud ensembles to generalize existing theories based on complete randomness. It is demonstrated that organized cloud ensembles may be treated as heterogeneous point processes where individual clouds do not interact substantially with their neighbors. In other words, clouds may be considered as randomly distributed if we restrict the space to regions of high cloud density. In addition, it is shown that in highly organized situations local cloud counts may be modeled using negative binomial distributions, while cloud mass fluxes follow lognormal distributions. The total mass flux within regions of varying sizes can then be predicted by compounding the two aforementioned distributions. The total mass flux variability at scales ≲5 km is dominated by the heavy‐tails of the mass flux distributions, whereas at scales ≳5 km, it is entirely controlled by the spatial cloud count variability (the spatial cloud pattern). These results have important implications for the parameterization of organized convection in quasi‐equilibrium.

Coverage, mapping and barriers to complete vaccination for age among children under 5 years in 2021: case of Adjara-Hounvè and Ahouicodji villages in southern Benin

Vaccination is a protective measure against infectious diseases and remains one of the best investments in public health. Some African countries are still struggling to reach the required child immunization coverage. Several factors are responsible for limiting immunization coverage. Most of the factors considered to limit immunization coverage are related to the health system. In addition, inaccessibility to care, especially during the critical period of the Covid-19 pandemic, greatly reduced vaccination coverage rates. In Benin, several vaccines are included in the Expanded Programme on Immunization or are administered as part of routine immunization. However, cases of non-compliance with the vaccine and persistent flaccid paralysis are still recorded in the commune of Ouidah in southern Benin. The aim of this study was to investigate the coverage and factors associated with full immunization for age in children aged 0-5 years. A cross-sectional survey was conducted from August to October 2021 in two villages (Adjara-Hounvè and Ahouicodji) in southern Benin. All the households were included. The survey regarded children under 5 for whom a vaccination record was presented. A couple child/mother was recruited after informed consent of the mother and her child. An univariate analysis followed by a multivariate analysis was performed by using a logistic regression model to identify the variables that influence vaccine completeness. Spatial description of vaccine completeness was performed using the kriging method using ArcGIS 10.8 mapping software. Results. Of the 414 mothers surveyed, 57.49% had an immunization card, from which information was collected. Of the 238 children recruited, 141 were in Adjara-Hounvè and 97 in Ahouicodji. Of the 238 children with an immunization card, 20.6% were fully immunized for their age. All children received Baccille Calmette Guérin vaccine at birth. Since poliomyelitis, pentavalent, pneumococcal conjugate, and rotavirus are three-dose vaccines, the percentage of children who received these vaccines decreased as the number of doses increased: 96.6%, 88.2%, 78.1% and 72.3% for the four doses of polio respectively. According to 53.4% of the respondents the reception at the vaccination site was poor, and according to 70.3% of them waiting time for vaccination sessions was long. Several reasons justified the absence of complete vaccination for the age of the children: vaccination site too far from the place of residence (59.54%), lack of financial means (29.78%) and the mother's ignorance (12.76%). Education level "primary" vs "none" (ORa = 3.32; CI95% 1.07-10.25), occupation "health staff" vs "housewife" (ORa = 21.18; CI95% 3.07-145.94), mothers' knowledge of Expanded Programme on Immunization diseases (ORa = 2, 20; CI95% 1.03-4.68) and children's age 0-2 months vs ≥ 16 months (ORa = 8.53; CI95% 2.52-28.85) and 9-15 months vs ≥ 16 months (ORa = 2.99; CI95% 1.24-7.23) increased complete immunization status for age. The homogeneity of behaviour related to age-complete immunization coverage in children under 5 years was evident at mapping. Age-complete immunization coverage in children under 5 years of age is very low, with a spatial homogeneity in community immunization uptake behaviour. Age-complete immunization coverage is an innovative indicator that can contribute to achieving age-specific immunization targets.

AA-RGTCN: reciprocal global temporal convolution network with adaptive alignment for video-based person re-identification.

Person re-identification(Re-ID) aims to retrieve pedestrians under different cameras. Compared with image-based Re-ID, video-based Re-ID extracts features from video sequences that contain both spatial features and temporal features. Existing methods usually focus on the most attractive image parts, and this will lead to redundant spatial description and insufficient temporal description. Other methods that take temporal clues into consideration usually ignore misalignment between frames and only focus on a fixed length of one given sequence. In this study, we proposed a Reciprocal Global Temporal Convolution Network with Adaptive Alignment(AA-RGTCN). The structure could address the drawback of misalignment between frames and model discriminative temporal representation. Specifically, the Adaptive Alignment block is designed to shift each frame adaptively to its best position for temporal modeling. Then, we proposed the Reciprocal Global Temporal Convolution Network to model robust temporal features across different time intervals along both normal and inverted time order. The experimental results show that our AA-RGTCN can achieve 85.9% mAP and 91.0% Rank-1 on MARS, 90.6% Rank-1 on iLIDS-VID, and 96.6% Rank-1 on PRID-2011, indicating we could gain better performance than other state-of-the-art approaches.

Multi-perspective analysis of sustainability metrics characterising the debris environment

To assess the effectiveness of mitigation measures, long-term simulations of the debris environment are often analysed and evaluated using simple metrics such as the total number of objects or the frequency of fragmentation events. In addition, more complex criticality metrics have been developed in the past to rank objects according to their environmental impact and evaluate the overall state of the debris environment. In this context, three different perspectives are particularly worth highlighting: object-specific metrics that characterise the criticality of individual objects and orbits, spatial metrics that can be used to identify particularly vulnerable orbital regions, and global metrics which describe the sustainability of the environment as a whole. Typically, the criticality metrics are specific to one of the respective perspectives, but unification across and convertibility between perspectives would allow for a more comprehensive assessment of sustainability. In this context, a methodology is developed to transform object-specific and spatial metrics into one another. The conversion concept enables a comparison of two structurally different metrics from their respective initial object-specific and spatial domain. Furthermore, the third perspective is taken into account by using different aggregation methods to retrieve a global metric from the initial object-specific and spatial description. The three perspectives then are compared by conducting environment evolution simulations for various baseline scenarios using a common space debris population in low Earth orbit. The comparative analysis identifies commonalities and differences of the chosen metrics in the three different perspectives and how the conversion between them affects their quantification of sustainability. The multi-perspective convertibility and aggregation of single-perspective metrics represents a contribution to the generalisation of sustainability metrics for different scopes of application.

Electrochemical detection of quinone reduced by Complex I Complex II and Complex III in full mitochondrial membranes

In the last decades enormous advances have been made in characterizing the atomic and molecular structure of respiratory chain supercomplexes [1]. However, it still remains a challenge to stitch this refined spatial atomistic description with functional information provided by biochemical studies of isolated protein material. Development of functional assays that detect respiratory chain complexes in their native membrane environment contribute to address the open questions related to the role played by their association and interactions. We present a characterization assay in which a functionalized gold electrode is modified with mitochondrial membrane fragments that allows monitoring electrochemically the activity of different respiratory chain complexes immersed in the mitochondrial membrane. We measure the intensity of the reducing current of the electron mediator CoQ1 at the electrode surface and its variation upon addition of the corresponding enzymatic substrates. The activities of Complex I, Complex II and Complex III were monitored by the way in which they reduce the current, reflecting the amount of quinone reduced by the complexes in the presence of their substrates. We detect that CoQ1H2 produced by Complex I remains partially trapped within the membrane and is more easily oxidized by Complex III or the electrode than the quinone reduced by Complex II.

A Mechanical Theory of Growth

A theory of growth is developed, utilizing the notion of a directional density function that captures the number and distribution of the material particles and their changes in time. A spatial (or Eulerian) description of kinematics is adopted, and the constitutive theory for a growing body is developed that relates the stress to the directional density function. The equation that governs the evolution of the directional density function is derived. An example of internal surface growth is presented.

Spatial relationship description model and algorithm of urban and rural planning in the smart city

Spatial relationship description model and algorithm of urban and rural planning in the smart city

Bivariate Spatial Pattern Based on The Characteristics of Women Smokers in Indonesia

The increasing prevalence of smoking among women globally has become a critical public health issue. World Health Organization (WHO) data shows 20 percent of women use tobacco. Women smokers face an increased risk of various diseases. Spatial studies need to be conducted as a basis for planning and providing targeted interventions to reduce the spread of diseases caused by smoking. The study aimed to determine the spatial description of smoking among women in Indonesia. This study analyzed the 2017 IDHS data and used an ecological study design with a descriptive approach and a spatial analysis model. The results are Papua was the province with the highest percentage of women smoking with no education and with the lowest de jure wealth quintile. Provinces using the Internet almost daily were Banten and East Java, and the province with women unemployed in the past 12 months before the survey was North Sulawesi. Based on the results, there are five provinces in Indonesia with a high percentage of smoking among women, assessed by education, internet access, wealth quintile, and non-working women. This study concludes that there is a need to implement smoke-free zones (SFZs), also conduct communication, information, and education (IEC) through mass media.

Peduncle collision-free grasping based on deep reinforcement learning for tomato harvesting robot

Collision-free grasping of the thin, brief peduncles connecting cherry tomato clusters to the main stem was crucial for tomato harvesting robots. Recognizing that the optimal operating posture for each individual peduncle was various, this study proposed a novel peduncle grasping posture decision model using deep reinforcement learning (DRL) for tomato harvesting manipulators, to overcome the collision issue caused by fixed-posture grasping. This model could dynamically generated action sequences for the harvesting manipulator, ensuring that the end-effector approach to the peduncle along the collision-free path with the optimal grasping posture. Building upon prior research into the multi-task identification of tomato clusters, peduncles, and the main stem, a keypoint-based spatial pose description model for tomato bunches was devised. Through this, the optimal operating posture for the end-effector on the peduncle was established. An improved HER-SAC (Soft Actor Critic with Hindsight Experience Replay) algorithm was subsequently established to guide the end-effector in collision-free grasping motions. The reward function of this algorithm incorporated end-effector posture constraints obtained from the optimal posture plane. In the training phase, a heuristic strategy model, providing prior knowledge, was merged with a dynamic gain module to sidestep local optimal policies, collectively enhancing the learning efficiency. In the simulation, our method improved the success rate of the peduncle grasping by at least 14 %, compared with SAC, HER-DDPG and HER-TD3. For the identical scenarios, improved HER-SAC reached the desired posture with a minimum of 15.5 % fewer steps compared to other algorithms. In field experiments conducted in tomato greenhouses, the robot achieved a harvesting success rate of 85.5 %, which was an increase of 57.3 % and 43.0 % compared to traditional methods with fixed horizontal and parallel-to-main-stem postures, respectively. The average operation time, from identification to successful harvesting, was 11.42 s. Our findings offer a promising solution to enhancing the efficiency of tomato-harvesting robots.

On the need for multi-dimensional models for the safety analysis of (fast-spectrum) Molten Salt Reactors

This paper aims at characterizing the impact of adopting numerical models with different dimensionalities on the predicted behavior of fast-spectrum Molten Salt Reactors (MSRs). The study encompasses 1-D, 2-D, and 3-D representations of thermal-hydraulics and precursor transport/diffusion, along with spatial and point kinetics models for neutronics. We evaluate the accuracy of each model based on steady-state results and on the reactor response to 2 different transient initiators. The findings emphasize the significance of utilizing a 3-D representation with accurate thermal-hydraulics modeling, and with either spatial kinetics or carefully calibrated point kinetics incorporating a spatial description of precursors transport. 2-D and 1-D models can reproduce main trends and remain valuable tools for e.g. reactor design, control-oriented studies or uncertainty quantification. However, proper calibration of these models is needed and the user should be aware that alterations in flow patterns could jeopardize model calibration and hide first-order local effects.

IoUNet++: Spatial cross‐layer interaction‐based bounding box regression for visual tracking

AbstractAccurate target prediction, especially bounding box estimation, is a key problem in visual tracking. Many recently proposed trackers adopt the refinement module called IoU predictor by designing a high‐level modulation vector to achieve bounding box estimation. However, due to the lack of spatial information that is important for precise box estimation, this simple one‐dimensional modulation vector has limited refinement representation capability. In this study, a novel IoU predictor (IoUNet++) is designed to achieve more accurate bounding box estimation by investigating spatial matching with a spatial cross‐layer interaction model. Rather than using a one‐dimensional modulation vector to generate representations of the candidate bounding box for overlap prediction, this paper first extracts and fuses multi‐level features of the target to generate template kernel with spatial description capability. Then, when aggregating the features of the template and the search region, the depthwise separable convolution correlation is adopted to preserve the spatial matching between the target feature and candidate feature, which makes their IoUNet++ network have better template representation and better fusion than the original network. The proposed IoUNet++ method with a plug‐and‐play style is applied to a series of strengthened trackers including DiMP++, SuperDiMP++ and SuperDIMP_AR++, which achieve consistent performance gain. Finally, experiments conducted on six popular tracking benchmarks show that their trackers outperformed the state‐of‐the‐art trackers with significantly fewer training epochs.

Modeling of blood flow in the framework of micropolar theory

In this paper, we study the blood flow through blood vessels of various radii (including the case of variable cross section as well as modeling the blood flow through venae and arteries). Two approaches are discussed in order to mimic the dependence of blood viscosity on red blood cells aggregation, which changes with the shear rate and position inside the vessel: Two microstructural parameters together with empirical constitutive equations as a characteristic of aggregation are proposed, namely the microinertia as well as the volume fraction of blood particles (erythrocytes, platelets and leukocytes). Consequently, the Navier–Stokes system of equations for an incompressible fluid is supplemented by a constitutive equation for the moment of inertia in one case and for the volume fraction in another. The problems are solved numerically by the finite volume method for vessels of various geometries in spatial description. A comparison with experimental data for a narrow capillary shows the efficiency of the proposed constitutive equations for describing blood flow. Also, velocity profiles are obtained on the basis of compiled empirical formula for various sections of a blood vessel of variable radius. In addition, the flow through vessels of the human circulatory system, such as the inferior vena cava and the carotid artery, are studied.