Fractal properties of perfusion heterogeneity in optimized arterial trees: a model study

Molecular mechanisms of endothelial dysfunction in coronary microcirculation dysfunction

Coronary microvascular endothelial cells (CMECs) react to changes in coronary blood flow and myocardial metabolites and regulate coronary blood flow by balancing vasoconstrictors-such as endothelin-1-and the vessel dilators prostaglandin, nitric oxide, and endothelium-dependent hyperpolarizing factor. Coronary microvascular endothelial cell dysfunction is caused by several cardiovascular risk factors and chronic rheumatic diseases that impact CMEC blood flow regulation, resulting in coronary microcirculation dysfunction (CMD). The mechanisms of CMEC dysfunction are not fully understood. However, the following could be important mechanisms: the overexpression and activation of nicotinamide adenine dinucleotide phosphate oxidase (Nox), and mineralocorticoid receptors; the involvement of reactive oxygen species (ROS) caused by a decreased expression of sirtuins (SIRT3/SIRT1); forkhead box O3; and a decreased SKCA/IKCA expression in the endothelium-dependent hyperpolarizing factor electrical signal pathway. In addition, p66Shc is an adapter protein that promotes oxidative stress; although there are no studies on its involvement with cardiac microvessels, it is possible it plays an important role in CMD.

In silico simulation of hepatic arteries: An open-source algorithm for efficient synthetic data generation

BACKGROUND

In silico testing of novel image reconstruction and quantitative algorithms designed for interventional imaging requires realistic high-resolution modeling of arterial trees with contrast dynamics. Furthermore, data synthesis for training of deep learning algorithms requires that an arterial tree generation algorithm be computationally efficient and sufficiently random.

PURPOSE

The purpose of this paper is to provide a method for anatomically and physiologically motivated, computationally efficient, random hepatic arterial tree generation.

METHODS

The vessel generation algorithm uses a constrained constructive optimization approach with a volume minimization-based cost function. The optimization is constrained by the Couinaud liver classification system to assure a main feeding artery to each Couinaud segment. An intersection check is included to guarantee non-intersecting vasculature and cubic polynomial fits are used to optimize bifurcation angles and to generate smoothly curved segments. Furthermore, an approach to simulate contrast dynamics and respiratory and cardiac motion is also presented.

RESULTS

The proposed algorithm can generate a synthetic hepatic arterial tree with 40 000 branches in 11 s. The high-resolution arterial trees have realistic morphological features such as branching angles (MAD with Murray's law= 1.2 ± 1 . 2 o $ = \;1.2 \pm {1.2^o}$ ), radii (median Murray deviation= 0.08 $ = \;0.08$ ), and smoothly curved, non-intersecting vessels. Furthermore, the algorithm assures a main feeding artery to each Couinaud segment and is random (variability = 0.98 ± 0.01).

CONCLUSIONS

This method facilitates the generation of large datasets of high-resolution, unique hepatic angiograms for the training of deep learning algorithms and initial testing of novel 3D reconstruction and quantitative algorithms designed for interventional imaging.

Adaptive constrained constructive optimisation for complex vascularisation processes

Mimicking angiogenetic processes in vascular territories acquires importance in the analysis of the multi-scale circulatory cascade and the coupling between blood flow and cell function. The present work extends, in several aspects, the Constrained Constructive Optimisation (CCO) algorithm to tackle complex automatic vascularisation tasks. The main extensions are based on the integration of adaptive optimisation criteria and multi-staged space-filling strategies which enhance the modelling capabilities of CCO for specific vascular architectures. Moreover, this vascular outgrowth can be performed either from scratch or from an existing network of vessels. Hence, the vascular territory is defined as a partition of vascular, avascular and carriage domains (the last one contains vessels but not terminals) allowing one to model complex vascular domains. In turn, the multi-staged space-filling approach allows one to delineate a sequence of biologically-inspired stages during the vascularisation process by exploiting different constraints, optimisation strategies and domain partitions stage by stage, improving the consistency with the architectural hierarchy observed in anatomical structures. With these features, the aDaptive CCO (DCCO) algorithm proposed here aims at improving the modelled network anatomy. The capabilities of the DCCO algorithm are assessed with a number of anatomically realistic scenarios.

Computational feasibility of simulating whole-organ vascular networks

The human body contains approximately 20 billion blood vessels, which transport nutrients, oxygen, immune cells, and signals throughout the body. The brain's vasculature includes up to 9 billion of these vessels to support cognition, motor processes, and myriad other vital functions. To model blood flowing through a vasculature, a geometric description of the vessels is required. Previously reported attempts to model vascular geometries have produced highly-detailed models. These models, however, are limited to a small fraction of the human brain, and little was known about the feasibility of computationally modeling whole-organ-sized networks. We implemented a fractal-based algorithm to construct a vasculature the size of the human brain and evaluated the algorithm's speed and memory requirements. Using high-performance computing systems, the algorithm constructed a vasculature comprising 17 billion vessels in 1960 core-hours, or 49 minutes of wall-clock time, and required less than 32 GB of memory per node. We demonstrated strong scalability that was limited mainly by input/output operations. The results of this study demonstrated, for the first time, that it is feasible to computationally model the vasculature of the whole human brain. These findings provide key insights into the computational aspects of modeling whole-organ vasculature.

Neglected interstitial space in malaria recurrence and treatment

The interstitial space, a widespread fluid-filled compartment throughout the body, is related to many pathophysiological alterations and diseases, attracting increasing attention. The vital role of interstitial space in malaria infection and treatment has been neglected current research efforts. We confirmed the reinfection capacity of parasites sequestrated in interstitial space, which replenish the mechanism of recurrence. Malaria parasite-infected mice were treated with artemisinin-loaded liposomes through the interstitial space and exhibited a better therapeutic response. Notably, compared with oral administration, interstitial administration showed an unexpectedly high activation and recruitment of immune cells, and resulted in better clearance of sequestered parasites from organs, and enhanced pathological recovery. The interstitial route of administration prolongs the blood circulation time of artemisinin and increases its plasma concentration, and may compensate for the inefficiency of oral administration and the nanotoxicity of intravenous administration, providing a potential strategy for infectious disease therapy.

Overview of mathematical modeling of myocardial blood flow regulation

The oxygen consumption by the heart and its extraction from the coronary arterial blood are the highest among all organs. Any increase in oxygen demand due to a change in heart metabolic activity requires an increase in coronary blood flow. This functional requirement of adjustment of coronary blood flow is mediated by coronary flow regulation to meet the oxygen demand without any discomfort, even under strenuous exercise conditions. The goal of this article is to provide an overview of the theoretical and computational models of coronary flow regulation and to reveal insights into the functioning of a complex physiological system that affects the perfusion requirements of the myocardium. Models for three major control mechanisms of myogenic, flow, and metabolic control are presented. These explain how the flow regulation mechanisms operating over multiple spatial scales from the precapillaries to the large coronary arteries yield the myocardial perfusion characteristics of flow reserve, autoregulation, flow dispersion, and self-similarity. The review not only introduces concepts of coronary blood flow regulation but also presents state-of-the-art advances and their potential to impact the assessment of coronary microvascular dysfunction (CMD), cardiac-coronary coupling in metabolic diseases, and therapies for angina and heart failure. Experimentalists and modelers not trained in these models will have exposure through this review such that the nonintuitive and highly nonlinear behavior of coronary physiology can be understood from a different perspective. This survey highlights knowledge gaps, key challenges, future research directions, and novel paradigms in the modeling of coronary flow regulation.

Mathematical synthesis of the cortical circulation for the whole mouse brain-part I. theory and image integration

Microcirculation plays a significant role in cerebral metabolism and blood flow control, yet explaining and predicting functional mechanisms remains elusive because it is difficult to make physiologically accurate mathematical models of the vascular network. As a precursor to the human brain, this paper presents a computational framework for synthesizing anatomically accurate network models for the cortical blood supply in mouse. It addresses two critical deficiencies in cerebrovascular modeling. At the microscopic length scale of individual capillaries, we present a novel synthesis method for building anatomically consistent capillary networks with loops and anastomoses (=microcirculatory closure). This overcomes shortcomings in existing algorithms which are unable to create closed circulatory networks. A second critical innovation allows the incorporation of detailed anatomical features from image data into vascular growth. Specifically, computed tomography and two photon laser scanning microscopy data are input into the novel synthesis algorithm to build the cortical circulation for the entire mouse brain in silico. Computer predictions of blood flow and oxygen exchange executed on synthetic large-scale network models are expected to elucidate poorly understood functional mechanisms of the cerebral circulation.

Morphometric Reconstruction of Coronary Vasculature Incorporating Uniformity of Flow Dispersion

Experimental limitations in measurements of coronary flow in the beating heart have led to the development of in silico models of reconstructed coronary trees. Previous coronary reconstructions relied primarily on anatomical data, including statistical morphometry (e.g., diameters, length, connectivity, longitudinal position). Such reconstructions are non-unique, however, often leading to unrealistic predicted flow features. Thus, it is necessary to impose physiological flow constraints to ensure realistic tree reconstruction. Since a vessel flow depends on its diameter to fourth power, diameters are the logical candidates to guide vascular reconstructions to achieve realistic flows. Here, a diameter assignment method was developed where each vessel diameter was determined depending on its downstream tree size, aimed to reduce flow dispersion to within measured range. Since the coronary micro-vessels are responsible for a major portion of the flow resistance, the auto regulated coronary flow was analyzed in a morphometry-based reconstructed 400 vessel arterial microvascular sub-tree spanning vessel orders 1–6. Diameters in this subtree were re-assigned based on the flow criteria. The results revealed that diameter re-assignment, while adhering to measured morphometry, significantly reduced the flow dispersion to realistic levels while adhering to measured morphometry. The resulting network flow has longitudinal pressure distribution, flow fractal nature, and near-neighboring flow autocorrelation, which agree with measured coronary flow characteristics. Collectively, these results suggest that a realistic coronary tree reconstruction should impose not only morphometric data but also flow considerations. The work is of broad significance in providing a novel computational framework in the field of coronary microcirculation. It is essential for the study of coronary circulation by model simulation, based on a realistic network structure.

Fractal characteristics of the microvascular network: A useful index to assess vascularization level of porous silk fibroin biomaterial

The neovascularization of biomaterials for tissue engineering is not only related to growth of capillaries but also involves appropriate hierarchy distribution of the microvessels. In this study, we proposed hierarchy distribution contrast method which can assess vascular transport capacity, in order to examine the hierarchy distribution of the neovessels during vascularization of the porous silk fibroin biomaterials implanted into rats and its evolution. The results showed that the fractal characteristics appeared toward the end of the vascularization stages, and the structure of the microvascular network after 3 weeks of implantation was similar to the fractal microvascular tree with bifurcation exponent x = 3 and fractal dimension D = 1.46, which became a sign of maturation of the regenerative vasculature. © 2017 Wiley Periodicals, Inc. J Biomed Mater Res Part A: 105A: 2276-2290, 2017.

Compound speckle model detects anti-angiogenic tumor response in preclinical nonlinear contrast-enhanced ultrasonography

PURPOSE

This paper proposes a method for analyzing the first-order speckle statistics of nonlinear contrast-enhanced ultrasound images from tumors.

METHODS

Contrast signal intensity is modeled as a compound distribution of exponential probability density functions with a gamma weighting function. The gamma probability weighting function serves as an approximation for log-normally distributed flow velocities in a vascular network. The model was applied to sub-harmonic bolus-injection images acquired from a mouse breast cancer xenograft model treated with murine version bevacizumab.

RESULTS

The area under curve produced using the compound statistical model could more accurately discriminate anti-VEGF-treated tumors from untreated tumors than conventional contrast-enhanced ultrasound image processing. This result was validated with gold standard histological measures of microvascular density. Fractal vessel geometry was estimated using the gamma weighting function and tested against micro-CT perfusion casting. Treated tumors had a significantly lower vascular fractal dimension than control tumors. Vascular complexity estimated using the ultrasound compound statistical model performed similarly to micro-CT fractal dimension for discriminating treated from control tumors.

CONCLUSION

The proposed technique can quantify tumor perfusion and provide an index of vascular complexity, making it a potentially useful addition for clinical detection of vascular normalization in anti-angiogenic trials.

Three-dimensional pulmonary perfusion MRI with radial ultrashort echo time and spatial-temporal constrained reconstruction

PURPOSE

To assess the feasibility of spatial-temporal constrained reconstruction for accelerated regional lung perfusion using highly undersampled dynamic contrast-enhanced (DCE) three-dimensional (3D) radial MRI with ultrashort echo time (UTE).

METHODS

A combined strategy was used to accelerate DCE MRI for 3D pulmonary perfusion with whole lung coverage. A highly undersampled 3D radial UTE MRI acquisition was combined with an iterative constrained reconstruction exploiting principal component analysis and wavelet soft-thresholding for dimensionality reduction in space and time. The performance of the method was evaluated using a 3D fractal-based DCE digital lung phantom. Simulated perfusion maps and contrast enhancement curves were compared with ground truth using the structural similarity index (SSIM) to determine robust threshold and regularization levels. Feasibility studies were then performed in a canine and a human subject with 3D radial UTE (TE=0.08 ms) acquisition to assess feasibility of mapping regional 3D perfusion.

RESULTS

The method was able to accurately recover perfusion maps in the phantom with a nominal isotropic spatial resolution of 1.5 mm (SSIM of 0.949). The canine and human subject studies demonstrated feasibility for providing artifact-free perfusion maps in a simple 3D breath-held acquisition.

CONCLUSION

The proposed method is promising for fast and flexible 3D pulmonary perfusion imaging. Magn Reson

Arterial input function in perfusion MRI: a comprehensive review

Cerebral perfusion, also referred to as cerebral blood flow (CBF), is one of the most important parameters related to brain physiology and function. The technique of dynamic-susceptibility contrast (DSC) MRI is currently the most commonly used MRI method to measure perfusion. It relies on the intravenous injection of a contrast agent and the rapid measurement of the transient signal changes during the passage of the bolus through the brain. Central to quantification of CBF using this technique is the so-called arterial input function (AIF), which describes the contrast agent input to the tissue of interest. Due to its fundamental role, there has been a lot of progress in recent years regarding how and where to measure the AIF, how it influences DSC-MRI quantification, what artefacts one should avoid, and the design of automatic methods to measure the AIF. The AIF is also directly linked to most of the major sources of artefacts in CBF quantification, including partial volume effect, bolus delay and dispersion, peak truncation effects, contrast agent non-linearity, etc. While there have been a number of good review articles on DSC-MRI over the years, these are often comprehensive but, by necessity, with limited in-depth discussion of the various topics covered. This review article covers in greater depth the issues associated with the AIF and their implications for perfusion quantification using DSC-MRI.

Cerebral microcirculation and oxygen tension in the human secondary cortex

The three-dimensional spatial arrangement of the cortical microcirculatory system is critical for understanding oxygen exchange between blood vessels and brain cells. A three-dimensional computer model of a 3 × 3 × 3 mm(3) subsection of the human secondary cortex was constructed to quantify oxygen advection in the microcirculation, tissue oxygen perfusion, and consumption in the human cortex. This computer model accounts for all arterial, capillary and venous blood vessels of the cerebral microvascular bed as well as brain tissue occupying the extravascular space. Microvessels were assembled with optimization algorithms emulating angiogenic growth; a realistic capillary bed was built with space filling procedures. The extravascular tissue was modeled as a porous medium supplied with oxygen by advection-diffusion to match normal metabolic oxygen demand. The resulting synthetic computer generated network matches prior measured morphometrics and fractal patterns of the cortical microvasculature. This morphologically accurate, physiologically consistent, multi-scale computer network of the cerebral microcirculation predicts the oxygen exchange of cortical blood vessels with the surrounding gray matter. Oxygen tension subject to blood pressure and flow conditions were computed and validated for the blood as well as brain tissue. Oxygen gradients along arterioles, capillaries and veins agreed with in vivo trends observed recently in imaging studies within experimental tolerances and uncertainty.

Analysis and algorithmic generation of hepatic vascular systems

A proper geometric model of the vascular systems in the liver is crucial for modeling blood flow, the connection between the organ and the rest of the organism. In vivo imaging does not provide sufficient details, so an algorithmic concept for extending measured vascular tree data is needed such that geometrically realistic structures can be generated. We develop a quantification of similarity in terms of different geometric features. This involves topological Strahler ordering of the vascular trees, statistical testing, and averaging. Invariant features are identified in human clinical in vivo CT scans. Results of the existing "Constrained Constructive Optimization" algorithm are compared to real vascular tree data. To improve bifurcation angles in the algorithmic results, we implement a postprocessing step calibrated to the measured features. This framework is finally applied to generate realistic additional details in a patient-specific hepatic vascular tree data set.

Model analysis of local oxygen delivery with liposome-encapsulated hemoglobin

Liposome-encapsulated hemoglobins (LHs) are comparable to red blood cells (RBCs) in terms of oxygen (O(2))-carrying capacity. The smaller particle size of LHs than of platelets allows their homogeneous dispersion in circulating plasma. In this study, we evaluated the effect of LH transfusion on arterial O(2) delivery through vascular trees by simulation. A mathematical model was established on the basis of the coronary arterial anatomy, the conservation of flow and RBC flux, and Poiseuille's law. The Fåhraeus-Lindqvist, Fåhraeus, and phase separation effects were considered in the model. By assuming steady perfusion, the arterial flow and O(2) delivery were calculated for five model trees undergoing the isovolumic replacement of RBCs (0.3 mg hemoglobin (Hb)/mL) with LHs (0.2 mg Hb/mL) or a plasma volume expander (PVE). The RBC-LH exchange increased both the total flow and the total O(2) flux but had almost no effect on the relative distribution of O(2) flux. In contrast, the RBC-PVE exchange decreased the total O(2) flux and increased the proportion of regions receiving a relatively low O(2) supply. Thus, LH transfusion may compensate for an enhanced bias in RBC-associated O(2) flux under hemodilution and is expected to be beneficial for both total and local O(2) delivery.

Relationship between local perfusion and FFA uptake in human skeletal muscle—no effect of increased physical activity and aerobic fitness

We investigated heredity-independent effects of increased physical activity and aerobic fitness on skeletal muscle free fatty acid (FFA) uptake, perfusion, and their heterogeneity at rest and during exercise. Also, the relationship between local skeletal muscle FFA uptake and perfusion was studied. Nine young adult male monozygotic twin pairs with significant difference in physical activity [229 min (SD 156) average time spent for conditioning exercise per week in more and 98 min (SD 71) in less active twins, P = 0.013] and aerobic fitness [18% (SD 10) difference in maximum O2 uptake] between brothers were studied using positron emission tomography. Submaximal knee-extension exercise increased perfusion, FFA uptake, and oxygen uptake in quadriceps femoris muscles 6–10 times compared with resting values ( P < 0.001). More active twins tended to utilize more oxygen, while no differences were found in muscle perfusion or FFA uptake between groups. Mean perfusion and FFA uptake correlated strongly at a whole muscle level, both at rest ( r = 0.97, P = 0.03 in more and r = 0.98, P = 0.02 in less active twins) and during exercise ( r = 0.99, P = 0.01 and r = 0.94, P = 0.06), but at the voxel level (87 mm3) correlation was only moderate during exercise [ r = 0.73 (SD 0.08) vs. r = 0.74 (SD 0.10), P = 0.92] and weak at rest [ r = 0.28 (SD 0.13) vs. r = 0.33 (SD 0.21), P = 0.58]. Exercise decreased both perfusion and FFA uptake heterogeneity within the muscles ( P < 0.001) similarly in both groups. In conclusion, long-term history of moderately increased physical activity tends to enhance muscle oxidative metabolism, but it does not have any significant influence on the FFA uptake or perfusion rates or their heterogeneity in skeletal muscle. Submaximal knee-extension exercise decreases heterogeneity of muscle FFA uptake and perfusion and improves matching between local muscle perfusion and FFA uptake. Thus it seems that the genetic influence is more important to determine the heterogeneity of perfusion and FFA uptake in skeletal muscle than exercise training.

Muscle Perfusion and Metabolic Heterogeneity

Recent developments in noninvasive imaging techniques have enabled the study of local changes in perfusion and metabolism in skeletal muscle as well as patterns of heterogeneity in these variables in humans. In this review, the principles of these techniques along with some recent findings on functional heterogeneity in human skeletal muscle will be presented.

Nitric oxide and prostaglandins influence local skeletal muscle blood flow during exercise in humans: coupling between local substrate uptake and blood flow

Synergic action of nitric oxide (NO) and prostaglandins (PG) in the regulation of muscle blood flow during exercise has been demonstrated. In the present study, we investigated whether these vasodilators also regulate local blood flow, flow heterogeneity, and glucose uptake within the exercising skeletal muscle. Skeletal muscle blood flow was measured in seven healthy young men using near-infrared spectroscopy and indocyanine green and muscle glucose uptake using positron emission tomography and 2-fluoro-2-deoxy-d-[18F]glucose without and with local blockade of NO and PG at rest and during one-legged dynamic knee-extension exercise. Local blockade was produced by infusing nitro-l-arginine methyl ester and indomethacin directly in the muscle via a microdialysis catheter. Blood flow and glucose uptake were measured in the region of blockade and in two additional regions of vastus lateralis muscle 1 and 4 cm away from the infusion of blockers. Local blockade during exercise at 25 and 40 watts significantly decreased blood flow in the infusion region and in the region 1 cm away from the site of infusion but not in the region 4 cm away. During exercise, muscle glucose uptake did not show any regional differences in response to blockade. These results show that NO and PG synergistically contribute to the local regulation of blood flow in skeletal muscle independently of muscle glucose uptake in healthy young men. Thus these vasodilators can play a role in regulating microvascular blood flow in localized regions of vastus lateralis muscle but do not influence regional glucose uptake. The findings suggest that local substrate uptake in skeletal muscle can be regulated independently of regional changes in blood flow.

Comparing microsphere deposition and flow modeling in 3D vascular trees

Blood perfusion in organs has been shown to be heterogeneous in a number of cases. At the same time, a number of models of vascular structure and flow have been proposed that also generate heterogeneous perfusion. Although a relationship between local perfusion and vascular structure has to exist, no model has yet been validated as an accurate description of this relationship. A study of perfusion and three-dimensional (3D) arterial structure in individual rat kidneys is presented, which allows comparison between local measurements of perfusion and model-based predictions. High-resolution computed tomography is used to obtain images of both deposited microspheres and of an arterial cast in the same organ. Microsphere deposition is used as an estimate of local perfusion. A 3D cylindrical pipe model of the arterial tree is generated based on an image of the arterial cast. Results of a flow model are compared with local microsphere deposition. High correlation (r(2) > 0.94) was observed between measured and modeled flows through the vascular tree segments. However, the relative dispersion of the microsphere perfusion measurement was two- to threefold higher than perfusion heterogeneity calculated in the flow model. Also, there was no correlation in the residual deviations between the methods. This study illustrates the importance of comparing models of local perfusion with in vivo measurements of perfusion in the same biologically realistic vascular tree.

Application of a random network with a variable geometry of links to the kinetics of drug elimination in healthy and diseased livers

This paper discusses an application of a random network with a variable number of links and traps to the elimination of drug molecules from the body by the liver. The nodes and links represent the transport vessels, and the traps represent liver cells with metabolic enzymes that eliminate drug molecules. By varying the number and configuration of links and nodes, different disease states of the liver related to vascular damage have been simulated, and the effects on the rate of elimination of a drug have been investigated. Results of numerical simulations show the prevalence of exponential decay curves with rates that depend on the concentration of links. In the case of fractal lattices at the percolation threshold, we find that the decay of the concentration is described by exponential functions for high trap concentrations but transitions to stretched exponential behavior at low trap concentrations.

Error estimation of geometrical data obtained by histomorphometry of oblique vessel sections: a computer model study

The errors of radius and wall thickness of a single vessel due to oblique sectioning in histomorphometry are expressed as a function of the circular shape factor (CSF) of the section's lumen, assuming cylindrical geometry and the absence of tissue deformation. Using computer model trees generated by constrained constructive optimization, mean errors are estimated for an ensemble of vessel segments. A geometrical exclusion criterion for segments cut too obliquely is defined on the basis of a CSF-cutoff value. It is shown that CSF-values ranging from 0.95 to 0.9 are reasonable choices for a cutoff and lead to mean errors of the same order of magnitude (9.6% [9.3%] to 15.4% [14.8%] for the radius [wall thickness]) as errors due to histological tissue processing.

Optimized arterial trees supplying hollow organs

Computer models of arterial trees can be generated from optimization principles using the algorithm of constrained constructive optimization (CCO). Up to now this algorithm could handle only tissue areas of convex shape, without concavities. CCO is now generalized to cope also with non-convex organ shapes, possibly featuring external as well as internal concavities. This allows the modeling of a much larger class of interesting real arterial systems. The concept of a generalized domain-potential was developed to represent arbitrary non-convex shapes mathematically and incorporate them as boundary conditions to optimization. Domain-potentials may be derived from analytical representations as well as from finite element triangulations obtained from organ images. To demonstrate the feasibility of the concept, the optimized growth of an arterial tree model is confined to some part of an elliptical shell, representing the free wall of the left ventricle of the heart.