Substantial effort is being invested in the creation of a virtual human—a model which will improve our understanding of human physiology and diseases and assist clinicians in the design of personalised medical treatments. A central challenge of achieving blood flow simulations at full-human scale is the development of an efficient and accurate approach to imposing boundary conditions on many outlets. A previous study proposed an efficient method for implementing the two-element Windkessel model to control the flow rate ratios at outlets. Here we clarify the general role of the resistance and capacitance in this approach and conduct a parametric sweep to examine how to choose their values for complex geometries. We show that the error of the flow rate ratios decreases exponentially as the resistance increases. The errors fall below 4% in a simple five-outlets model and 7% in a human artery model comprising ten outlets. Moreover, the flow rate ratios converge faster and suffer from weaker fluctuations as the capacitance decreases. Our findings also establish constraints on the parameters controlling the numerical stability of the simulations. The findings from this work are directly applicable to larger and more complex vascular domains encountered at full-human scale.

Substantial effort is being invested in the creation of a virtual human — a model which will improve our understanding of human physiology and diseases and assist clinicians in the design of personalised medical treatments. A central challenge of achieving blood flow simulations at full-human scale is the development of an efficient and accurate approach to imposing boundary conditions on many outlets. A previous study proposed an efficient method for implementing the two-element Windkessel model to control the flow rate ratios at outlets. However, no study to date has examined the conditions for this approach to hold in complex geometries. Here we clarify the general role of the resistance and capacitance in this approach. We show that the error of the flow rate ratios decreases exponentially as the resistance increases. The errors fall below 4% in a simple five-outlets model and 7% in a human artery model comprising 10 outlets. Moreover, the flow rate ratios converge faster and suffer from weaker fluctuations as the capacitance decreases. Our findings also establish constraints on the parameters controlling the numerical stability of the simulations. The findings from this work are directly applicable to larger and more complex vascular domains encountered at full-human scale.

Solution verification is crucial for establishing the reliability of simulations. A central challenge is to estimate the discretization error accurately and reliably. Many approaches to this estimation are based on the observed order of accuracy; however, it may fail when the numerical solutions lie outside the asymptotic range. Here we propose a grid refinement method which adopts constant orders given by the user, called the Prescribed Orders Expansion Method (POEM). Through an iterative procedure, the user is guaranteed to obtain the dominant orders of the discretization error. The user can also compare the corresponding terms to quantify the degree of asymptotic convergence of the numerical solutions. These features ensure that the estimation of the discretization error is accurate and reliable. Moreover, the implementation of POEM is the same for any dimensions and refinement paths. We demonstrate these capabilities using some advection and diffusion problems and standard refinement paths. The computational cost of using POEM is lower if the refinement ratio is larger; however, the number of shared grid points where POEM applies also decreases, causing greater uncertainty in the global estimates of the discretization error. We find that the proportion of shared grid points is maximized when the refinement ratios are in a certain form of fractions. Furthermore, we develop the Method of Interpolating Differences between Approximate Solutions (MIDAS) for creating shared grid points in the domain. These approaches allow users of POEM to obtain a global estimate of the discretization error of lower uncertainty at a reduced computational cost.

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