A spatially-distributed computational model to quantify behaviour of contrast agents in MR perfusion imaging

A lattice Boltzmann approach to mathematical modeling of myocardial perfusion

A mathematical model of myocardial perfusion based on the lattice Boltzmann method (LBM) is proposed and its applicability is investigated in both healthy and diseased cases. The myocardium is conceptualized as a porous material in which the transport and mass transfer of a contrast agent in blood flow is studied. The results of myocardial perfusion obtained using LBM in 1D and 2D are confronted with previously reported results in the literature and the results obtained using the mixed-hybrid finite element method. Since LBM is not suitable for simulating flow in heterogeneous porous media, a simplified and computationally efficient 1D-analog approach to 2D diseased case is proposed and its applicability discussed.

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.

Quantitative transport mapping (QTM) for differentiating benign and malignant breast lesion: Comparison with traditional kinetics modeling and semi-quantitative enhancement curve characteristics

PURPOSE

To test the feasibility of using quantitative transport mapping (QTM) method, which is based on the inversion of transport equation using spatial deconvolution without any arterial input function, for automatically postprocessing dynamic contrast enhanced MRI (DCE-MRI) to differentiate malignant and benign breast tumors.

MATERIALS AND METHODS

Breast DCE-MRI data with biopsy confirmed malignant (n = 13) and benign tumors (n = 13) was used to assess QTM velocity (|u|) and diffusion coefficient (D), volume transfer constant (K^trans), volume fraction of extravascular extracellular space (V_e) from kinetics method, and traditional enhancement curve characteristics (ECC: amplitude A, wash-in rate α, wash-out rate β). A Mann-Whitney U test and receiver operating characteristic curve (ROC) analysis were performed to assess the diagnostic performance of these parameters for distinguishing between benign and malignant tumors.

RESULTS

Between malignant and benign tumors, there was a significant difference in |u| and K^trans, (p = 0.0066, 0.0274, respectively), but not in D, V_e, A, α and β (p = 0.1119, 0.2382, 0.4418,0.2592 and 0.9591, respectively). ROC area-under-the-curve was 0.82, 0.75 (95% confidence level 0.60-0.95, 0.51-0.90) for |u| and K^trans, respectively.

CONCLUSION

QTM postprocesses DCE-MRI automatically through deconvolution in space and time to solve the inverse problem of the transport equation. Comparing with traditional kinetics method and ECC, QTM method showed better diagnostic accuracy in differentiating benign from malignant breast tumors in this study.

Perfusion Imaging: An Advection Diffusion Approach

Perfusion imaging is of great clinical importance and is used to assess a wide range of diseases including strokes and brain tumors. Commonly used approaches for the quantitative analysis of perfusion images are based on measuring the effect of a contrast agent moving through blood vessels and into tissue. Contrast-agent free approaches, for example, based on intravoxel incoherent motion and arterial spin labeling, also exist, but are so far not routinely used clinically. Existing contrast-agent-dependent methods typically rely on the estimation of the arterial input function (AIF) to approximately model tissue perfusion. These approaches neglect spatial dependencies. Further, as reliably estimating the AIF is non-trivial, different AIF estimates may lead to different perfusion measures. In this work we therefore propose PIANO, an approach that provides additional insights into the perfusion process. PIANO estimates the velocity and diffusion fields of an advection-diffusion model best explaining the contrast dynamics without using an AIF. PIANO accounts for spatial dependencies and neither requires estimating the AIF nor relies on a particular contrast agent bolus shape. Specifically, we propose a convenient parameterization of the estimation problem, a numerical estimation approach, and extensively evaluate PIANO. Simulation experiments show the robustness and effectiveness of PIANO, along with its ability to distinguish between advection and diffusion. We further apply PIANO on a public brain magnetic resonance (MR) perfusion dataset of acute stroke patients, and demonstrate that PIANO can successfully resolve velocity and diffusion field ambiguities and results in sensitive measures for the assessment of stroke, comparing favorably to conventional measures of perfusion.

A poroelastic immersed finite element framework for modelling cardiac perfusion and fluid-structure interaction

Modern approaches to modelling cardiac perfusion now commonly describe the myocardium using the framework of poroelasticity. Cardiac tissue can be described as a saturated porous medium composed of the pore fluid (blood) and the skeleton (myocytes and collagen scaffold). In previous studies fluid-structure interaction in the heart has been treated in a variety of ways, but in most cases, the myocardium is assumed to be a hyperelastic fibre-reinforced material. Conversely, models that treat the myocardium as a poroelastic material typically neglect interactions between the myocardium and intracardiac blood flow. This work presents a poroelastic immersed finite element framework to model left ventricular dynamics in a three-phase poroelastic system composed of the pore blood fluid, the skeleton, and the chamber fluid. We benchmark our approach by examining a pair of prototypical poroelastic formations using a simple cubic geometry considered in the prior work by Chapelle et al (2010). This cubic model also enables us to compare the differences between system behaviour when using isotropic and anisotropic material models for the skeleton. With this framework, we also simulate the poroelastic dynamics of a three-dimensional left ventricle, in which the myocardium is described by the Holzapfel-Ogden law. Results obtained using the poroelastic model are compared to those of a corresponding hyperelastic model studied previously. We find that the poroelastic LV behaves differently from the hyperelastic LV model. For example, accounting for perfusion results in a smaller diastolic chamber volume, agreeing well with the well-known wall-stiffening effect under perfusion reported previously. Meanwhile differences in systolic function, such as fibre strain in the basal and middle ventricle, are found to be comparatively minor.

Quantitative transport mapping (QTM) of the kidney with an approximate microvascular network

Purpose

Proof‐of‐concept study of mapping renal blood flow vector field according to the inverse solution to a mass transport model of time resolved tracer‐labeled MRI data.

Theory and Methods

To determine tissue perfusion according to the underlying physics of spatiotemporal tracer concentration variation, the mass transport equation is integrated over a voxel with an approximate microvascular network for fitting time‐resolved tracer imaging data. The inverse solution to the voxelized transport equation provides the blood flow vector field, which is referred to as quantitative transport mapping (QTM). A numerical microvascular network modeling the kidney with computational fluid dynamics reference was used to verify the accuracy of QTM and the current Kety’s method that uses a global arterial input function. Multiple post‐label delay arterial spin labeling (ASL) of the kidney on seven subjects was used to assess QTM in vivo feasibility.

Results

Against the ground truth in the numerical model, the error in flow estimated by QTM (18.6%) was smaller than that in Kety’s method (45.7%, 2.5‐fold reduction). The in vivo kidney perfusion quantification by QTM (cortex: 443 ± 58 mL/100 g/min and medulla: 190 ± 90 mL/100 g/min) was in the range of that by Kety’s method (482 ± 51 mL/100 g/min in the cortex and 242 ± 73 mL/100 g/min in the medulla), and QTM provided better flow homogeneity in the cortex region.

Conclusions

QTM flow velocity mapping is feasible from multi‐delay ASL MRI data based on inverting the transport equation. In a numerical simulation, QTM with deconvolution in space and time provided more accurate perfusion quantification than Kety’s method with deconvolution in time only.

Time-Lapsing Perfusion: Proof of Concept of a Novel Method to Study Drug Delivery in Whole Organs

Perfusion is one of the most important processes maintaining organ health. From a computational perspective, however, perfusion is among the least-studied physiological processes of the heart. The recent development of novel nanoparticle-based targeted cardiac therapy calls for novel simulation methods that can provide insights into the distribution patterns of therapeutic agents within the heart tissue. Additionally, resolving the distribution patterns of perfusion is crucial for gaining a full understanding of the long-term impacts of cardiovascular diseases that can lead to adverse remodeling such as myocardial ischemia and heart failure. In this study, we have developed and used a, to our knowledge, novel particle-tracking-based method to simulate the perfusion-mediated distribution of nanoparticles or other solutes. To model blood flow through perfused tissue, we follow the approach of others and treat the tissue as a porous medium in a continuum model. Classically, solutes are modeled using reaction-advection-diffusion kinetics. However, because of the discrepancy of scales between advection and diffusion in blood vessels, this method becomes practically numerically unstable. Instead, we track a bolus of solutes or nanoparticles using particle tracking based purely on advection in arteries. In capillaries, we employ diffusion kinetics, using an effective diffusion coefficient to mimic capillary blood flow. We first demonstrate the numerical validity and computational efficiency of this method on a two-dimensional benchmark problem. Finally, we demonstrate how the method is used to visualize perfusion patterns of a healthy and ischemic human left ventricle geometry. The efficiency of the method allows for nanoparticle tracking over multiple cardiac cycles using a conventional laptop, providing a framework for the simulation of experimentally relevant timeframes to advance preclinical research.

Extended quantitative dynamic contrast-enhanced cardiac perfusion imaging in mice using accelerated data acquisition and spatially distributed, two-compartment exchange modeling

The objective of the present work was to improve data acquisition and quantification of dynamic contrast-enhanced perfusion imaging in the in vivo murine heart. Four-fold undersampled data were acquired in 14 mice and reconstructed using k-t SPARSE. A two-compartment exchange model was employed to provide additional characterization of myocardial tissue based on compartment volumes and the permeability surface area product. The feasibility of the proposed method was tested using compartment-based analysis of contrast-enhanced perfusion data acquired with intravascular and extracellular contrast agents. A significantly different permeability surface area product was measured for the intravascular versus extracellular contrast agent (0.13-0.15 ml/g/min vs 0.86-0.88 ml/g/min). The reduced extravasation also resulted in significantly smaller interstitial volumes of the intravascular versus extracellular agent (9.8-11% vs 45-47%). No difference was found for myocardial blood flow (6.5-7.2 ml/g/min vs 6.0-7.0 ml/g/min). The results presented here show that two-compartment exchange modeling in the in vivo murine heart is feasible and gives access to tissue parameters beyond myocardial blood flow.

A new framework for assessing subject-specific whole brain circulation and perfusion using MRI-based measurements and a multi-scale continuous flow model

A large variety of severe medical conditions involve alterations in microvascular circulation. Hence, measurements or simulation of circulation and perfusion has considerable clinical value and can be used for diagnostics, evaluation of treatment efficacy, and for surgical planning. However, the accuracy of traditional tracer kinetic one-compartment models is limited due to scale dependency. As a remedy, we propose a scale invariant mathematical framework for simulating whole brain perfusion. The suggested framework is based on a segmentation of anatomical geometry down to imaging voxel resolution. Large vessels in the arterial and venous network are identified from time-of-flight (ToF) and quantitative susceptibility mapping (QSM). Macro-scale flow in the large-vessel-network is accurately modelled using the Hagen-Poiseuille equation, whereas capillary flow is treated as two-compartment porous media flow. Macro-scale flow is coupled with micro-scale flow by a spatially distributing support function in the terminal endings. Perfusion is defined as the transition of fluid from the arterial to the venous compartment. We demonstrate a whole brain simulation of tracer propagation on a realistic geometric model of the human brain, where the model comprises distinct areas of grey and white matter, as well as large vessels in the arterial and venous vascular network. Our proposed framework is an accurate and viable alternative to traditional compartment models, with high relevance for simulation of brain perfusion and also for restoration of field parameters in clinical brain perfusion applications.

Porous medium 3D flow simulation of contrast media washout in cardiac MRI reflects myocardial injury

PURPOSE

Myocardial blood-flow simulation based on laws of fluid mechanics is a valuable tool for understanding tissue behavior. Our aim is to evaluate the ability of a porous-media flow model approach to reflect disturbed washout of contrast media (CM) from the myocardium as observed by cardiovascular MR.

METHODS

A coupled advection-diffusion model is used to describe the CM flow in the vascular and extravascular space as separate compartments. Their exchange of CM is controlled by the exchange rate ExR , which in turn determines the washout behavior. We fitted simulations to CM concentration measurements, derived from T₁ maps of the midventricular slice. The CM concentration was extracted from 18 patients with myocarditis in the acute phase and during follow-up after 6 months. The results were compared with 18 sex- and age-matched controls. For each subject, the measurements were acquired before and during the first 10 minutes at 5 time points after CM administration, representing CM washout. Image registration was applied to compensate for motion between different time points.

RESULTS

Eight matched data sets had to be excluded due to low registration quality. Processing was successful in n = 10 matched data sets of acute and healed myocarditis as well as controls. Significant differences in ExR were observed when comparing patients with acute myocarditis to controls (P < .001), to their follow-up (P < .05), and the follow-up to controls (P < .05).

CONCLUSION

Our study suggests the feasibility of using the proposed porous-medium flow framework for the simulation of pathologic myocardial tissue.

Simulation of the Perfusion of Contrast Agent Used in Cardiac Magnetic Resonance: A Step Toward Non-invasive Cardiac Perfusion Quantification

This work presents a new mathematical model to describe cardiac perfusion in the myocardium as acquired by cardiac magnetic resonance (CMR) perfusion exams. The combination of first pass (or contrast-enhanced CMR) and late enhancement CMR is a widely used non-invasive exam that can identify abnormal perfused regions of the heart via the use of a contrast agent (CA). The exam provides important information to the diagnosis, management, and prognosis of ischemia and infarct: perfusion on different regions, the status of microvascular structures, the presence of fibrosis, and the relative volume of extracellular space. This information is obtained by inferring the spatiotemporal dynamics of the contrast in the myocardial tissue from the acquired images. The evaluation of these physiological parameters plays an important role in the assessment of myocardial viability. However, the nature of cardiac physiology poses great challenges in the estimation of these parameters. Briefly, these are currently estimated qualitatively via visual inspection of images and comparison of relative brightness between different regions of the heart. Therefore, there is a great urge for techniques that can help to quantify cardiac perfusion. In this work, we propose a new mathematical model based on multidomain flow in porous media. The model is based on a system of partial differential equations. Darcy's law is used to obtain the pressure and velocity distribution. CA dynamics is described by reaction-diffusion-advection equations in the intravascular space and in the interstitial space. The interaction of fibrosis and the CA is also considered. The new model treats the domains as anisotropic media and imposes a closed loop of intravascular flow, which is necessary to reproduce the recirculation of the CA. The model parameters were adjusted to reproduce clinical data. In addition, the model was used to simulate different scenarios: normal perfusion; endocardial ischemia due to stenosis in a coronary artery in the epicardium; and myocardial infarct. Therefore, the computational model was able to correlate anatomical features, stenosis and the presence of fibrosis, with functional ones, cardiac perfusion. Altogether, the results suggest that the model can support the process of non-invasive cardiac perfusion quantification.

Pixel-wise quantification of myocardial perfusion using spatial Tikhonov regularization

Quantification of myocardial perfusion by contrast-enhanced cardiovascular magnetic resonance imaging (CMR) aims for an observer independent and reproducible risk assessment of cardiovascular disease. Currently, the data used for the pixel-wise analysis of cardiac perfusion are either filtered prior to a fitting procedure, which inherently reduces the spatial resolution of data; or all pixels are considered without any regularization or prior filtering, which yields an unstable fit in the presence of low signal-to-noise ratio. Here, we propose a new pixel-wise analysis based on spatial Tikhonov regularization which exploits the spatial smoothness of the data and ensures accurate quantification even for images with low signal-to-noise ratio. The regularization parameter is determined automatically by an L-curve criterion. We study the performance of our method on a numerical phantom and demonstrate that the method reduces significantly the root-mean square error in the perfusion estimate compared to a non-regularized fit. In patient data our method allows us to recover the myocardial perfusion and to distinguish between healthy and ischemic regions.

Multiphysics and multiscale modelling, data-model fusion and integration of organ physiology in the clinic: ventricular cardiac mechanics

With heart and cardiovascular diseases continually challenging healthcare systems worldwide, translating basic research on cardiac (patho)physiology into clinical care is essential. Exacerbating this already extensive challenge is the complexity of the heart, relying on its hierarchical structure and function to maintain cardiovascular flow. Computational modelling has been proposed and actively pursued as a tool for accelerating research and translation. Allowing exploration of the relationships between physics, multiscale mechanisms and function, computational modelling provides a platform for improving our understanding of the heart. Further integration of experimental and clinical data through data assimilation and parameter estimation techniques is bringing computational models closer to use in routine clinical practice. This article reviews developments in computational cardiac modelling and how their integration with medical imaging data is providing new pathways for translational cardiac modelling.

Clinical Diagnostic Biomarkers from the Personalization of Computational Models of Cardiac Physiology

Computational modelling of the heart is rapidly advancing to the point of clinical utility. However, the difficulty of parameterizing and validating models from clinical data indicates that the routine application of truly predictive models remains a significant challenge. We argue there is significant value in an intermediate step towards prediction. This step is the use of biophysically based models to extract clinically useful information from existing patient data. Specifically in this paper we review methodologies for applying modelling frameworks for this goal in the areas of quantifying cardiac anatomy, estimating myocardial stiffness and optimizing measurements of coronary perfusion. Using these indicative examples of the general overarching approach, we finally discuss the value, ongoing challenges and future potential for applying biophysically based modelling in the clinical context.

Measurement and modeling of coronary blood flow

Ischemic heart disease that comprises both coronary artery disease and microvascular disease is the single greatest cause of death globally. In this context, enhancing our understanding of the interaction of coronary structure and function is not only fundamental for advancing basic physiology but also crucial for identifying new targets for treating these diseases. A central challenge for understanding coronary blood flow is that coronary structure and function exhibit different behaviors across a range of spatial and temporal scales. While experimental studies have sought to understand this feature by isolating specific mechanisms, in tandem, computational modeling is increasingly also providing a unique framework to integrate mechanistic behaviors across different scales. In addition, clinical methods for assessing coronary disease severity are continuously being informed and updated by findings in basic physiology. Coupling these technologies, computational modeling of the coronary circulation is emerging as a bridge between the experimental and clinical domains, providing a framework to integrate imaging and measurements from multiple sources with mathematical descriptions of governing physical laws. State-of-the-art computational modeling is being used to combine mechanistic models with data to provide new insight into coronary physiology, optimization of medical technologies, and new applications to guide clinical practice.