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Oxygen Advection and Diffusion in a Three Dimensional Vascular

Anatomical Network

Qianqian Fang1, Sava Sakadžić1, Lana Ruvinskaya1, Anna Devor1,2, Anders M. Dale2, and

David A. Boas1,*

1Martinos Center for Biomedical Imaging, Massachusetts General Hospital, 149 13th St, Charlestown,

Massachusetts, 02129, USA

2Department of Neurosciences, University of California, San Diego, CA, USA

Abstract

There is an increasing need for quantitative and computationally affordable models for analyzing

tissue metabolism and hemodynamics in microvascular networks. In this work, we develop a hybrid

model to solve for the time-varying oxygen advection-diffusion equation in the vessels and tissue.

To obtain a three-dimensional temporal evolution of tissue oxygen concentration for realistic

complex vessel networks, we used a graph-based advection model combined with a finite-element

based diffusion model and an implicit time-advancing scheme. We validated this algorithm for both

static and dynamic conditions. We also applied it to a complex vascular network obtained from a

rodent somatosensory cortex. Qualitative agreement was found with in-vivo experiments.

1. Introduction

Numerous optical methods are utilized to quantify tissue blood flow, blood volume, and oxygen

delivery, ranging from the macroscopic whole tissue level to the microscopic capillary network

level and sub-cellular level. Near-infrared spectroscopy (NIRS) and imaging utilize near-

infrared light to quantify the concentrations of oxygenated and deoxygenated hemoglobin over

cubic centimeters of tissue [1-3]. Visible light tissue reflectance imaged with a CCD camera

is frequently used to image superficial changes in hemoglobin concentrations with light

spectroscopy [4] and blood flow with laser speckle contrast [5,6], particularly in the

neurosciences. Confocal and multiphoton microscopy enable sub-micrometer resolution with

depth penetration of up to ∼100 μm and ∼500 μm or more, respectively, and are used to image

the three-dimensional (3D) vascular anatomy and red blood cell velocity within vessels [7,8],

and oxygenation of the blood [9-11]. Doppler optical coherence tomography is enabling rapid

volumetric imaging of flow within the vascular network over potentially cubic millimeters in

seconds [12-14]. There seems to be a unique opportunity for optics to provide a detailed

characterization of oxygen delivery to the tissue by combining these measurements of blood

flow, volume, and oxygenation with fluorescence-based measurements of cellular activity

[15,16] and metabolism [17,18].

The advanced analysis of near-infrared spectroscopy data from centimeters of tissue and visible

light imaging from hundreds of micrometers of tissue are yielding information about oxygen

metabolism [19-23]. While these estimates of oxygen metabolism appear precise, validating

the accuracy is challenged by the lack of a gold standard method. On the other hand, the

*Corresponding author: E-mail: dboas@nmr. mgh. harvard. edu.

OCIS codes: (180.6900) Three-dimensional microscopy.

NIH Public Access

Author Manuscript

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Published in final edited form as:

Opt Express. 2008 October 27; 16(22): 17530–17541.

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microscopic measures of individual microvessels and surrounding tissue provide more direct

measures of oxygen delivery, but detailed models are needed to provide an analysis framework

to quantify physiological parameters like vessel wall oxygen permeability, tissue oxygen

diffusion, and oxygen consumption that are not directly measurable. Further, due to the time

varying nature of oxygen delivery, particularly during brain activity, there is a need for models

that can handle dynamic, time-varying processes.

In this paper, we describe a detailed computational model of tissue oxygen delivery by a

microvascular network that will provide a framework for analyzing multimodal microscopic

measures to better quantify additional physiological parameters, and will enable more detailed

and realistic analyses of the accuracy of near-infrared spectroscopy and visible light imaging.

We start by describing our method for solving the time-varying oxygen advection diffusion

equation that estimates velocity within individual microvessel segments, oxygen flux across

the permeable vascular walls, and then oxygen diffusion and consumption within the tissue.

After validating our solver, we demonstrate its application by calculating oxygen delivery by

a microvascular network acquired in rodent somatosensory cortex by two-photon microscopy

over a 230 × 230 × 450 μm3 volume of tissue.

2. Methods

2.1 Models and Domain Discretization

The transport and consumption of oxygen in the vessel network and tissue are characterized

by the advection-diffusion equation [24], which can be expressed as the following equations:

(1)

(2)

where CF and CB are the free and bounded oxygen concentrations (μM), respectively; CT =

CB + CF denotes the total oxygen concentration, denotes the velocity (m/s), DO2 is the oxygen

diffusion coefficient (m/s2), and OC is the tissue oxygen consumption rate (μM/s). Equation

(2) denotes the equilibrium between the hemoglobin bound oxygen and free oxygen where

H denotes hematocrit, CHb is the hemoglobin concentration within a red blood cell, SO2(CF)

is the hemoglobin oxygen saturation and it is in equilibrium with the concentration of free

oxygen, CF, as described by the oxygen disassociation curve. We use an invertible form for

the oxygen disassociation curve given by [25]. We note that under physiological conditions,

all of these parameters can vary in space and time.

Certainly, one could solve Eq. (1) using a single numerical approach, such as a finite element

(FE) or a finite difference (FD) method, by applying a full spatial discretization of the tissue

volume, the exterior surface and the interior volume of vessels. Because of the small length-

scale of the vessel wall, the meshing and computational expense can be prohibitive with

increasing complexity of the vessel networks. Instead, we devised a hybrid model that includes

a graph-based one-dimensional (1D) oxygen advection model within the vessel, a 1D oxygen

flux conservation model across the vessel wall, and an FE-based 3D oxygen diffusion model

within the tissue. An implicit (Crank-Nickson) [26] scheme was implemented to compute the

updates at each time step. A schematic of the problem domain decomposition and our spatial

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discretization scheme is shown in Fig. 1. The computational space is separated into three

domains to characterize three distinct physical and physiological processes:

I. vessel network: inside the vessel, the oxygen transport is represented by the advection

equation subject to the equilibrium of the free and hemoglobin bound oxygen:

(3)

An oriented graph representation was created to represent the vessel network: the

axial line of each vessel was extracted and discretized by nodes and oriented edges

(see Fig. 1). The connection (including branches) of the vessel network can be

naturally expressed by the adjacency matrix [27] of the graph. We calculated a vessel

volume associated with each vessel node and assumed uniform oxygen distribution

within the volume.

II. vessel wall: the oxygen flux across a thin vessel wall is a diffusion process. To

simplify analysis, we only discretized the exterior of vessel walls by a triangular

surface mesh. Because the diffusion profile within the vessel wall layer is not the

primary interest of this study, we used an oxygen conversation relationship based on

Fick's law [28]:

(4)

where Ji is the oxygen flux (mol/m2/s) across the vessel wall at the ith vessel wall

node, Ai and Vi are the associated vessel wall area and tissue volume for this node.

The right-hand-side of Eq. (4) represents the extracted oxygen from the vessel per

unit time. The flux, Ji, is computed by

(5)

where Kw is the vessel wall permeability, w is the vessel wall thickness, CF,i and

CF,i0 are the free oxygen concentrations at the exterior and interior surfaces of the

vessel, respectively; α is the Bunsen solubility coefficient and equals 1.27 × 10-15

μmol/(μm3 mmHg) [28].

III. tissue: the free oxygen that is transferred across the vessel wall diffuses through the

whole tissue volume governed by the diffusion equation

(6)

where DO2 is the tissue oxygen diffusion coefficient and the consumption of oxygen

is governed by OC. The tissue volume is discretized by a tetrahedron mesh and is

truncated by a bounding box.

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In the following subsections, we detail our numerical implementation for solving the above

models for each domain.

Vessel Blood Advection—Equation (3) is solved in two steps for each time advance. We

first solve the two advection equations in Eq. (3). Integrated within the associated volume of

each vessel node, the advection equations lead to the conservation equations of free and bound

oxygen. Assuming that the adjacency matrix for the vessel graph is K, for the ith vessel node

(excluding the vessel end nodes) the conservation of oxygen implies that

(7)

where Nv is the total number of vessel nodes; ki,j is the (i,j)th element of K with a value of 0 if

nodes i and j are not adjacent, 1 for in-flow from node j to i, and -1 for out-flow from node i

to j; di,j is the distance between the ith and jth nodes and Ci and Cj are the oxygen concentrations

(free or hemoglobin bound); vi,j is the velocity, Vi and Vj are the associated vessel volumes of

the two nodes. An implicit finite difference scheme is used for the time-derivative term in (7).

This results in a linear equation with a form of

(8)

where θ ∈ [0,1] is a constant, Δt is the time step and n is the time index. This can be rewritten

in matrix form as

(9)

The equilibrium of CF and CB is achieved by solving

(10)

subject to the conservation of CT.

Oxygen Flux Across Vessel Wall—From Eq.(4), we can get

(11)

which can be converted into an implicit time update as

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(12)

The matrix form of Eq. (12) can be written as a linear equation defined between the vessel wall

and axial nodes:

(13)

Tissue Oxygen Diffusion—To accommodate for the arbitrary shape of a complex vessel

network, we used a tetrahedral mesh to discretize the tissue region outside the vessel. A finite

element method (FEM) coupled with implicit time-stepping is used to solve for the diffusion

of oxygen in the tissue.

Using the Galerkin's method, the FEM weak form of Eq. (6) can be written as [26]

(14)

where φi and φj are the basis functions in a tetrahedral element; ∂Ω is the surface of the tissue,

and n̂ is a unit vector pointing to the normal direction of a surface element; < • > denotes volume

integration inside an element. After applying the implicit method to discretize the time

derivative in Eq. (14), we obtain

(15)

where the elements of matrices Ad and Bd and vector R are, respectively, defined by

Lumped System—The discretized advection equation, Eq. (9), flux equation, Eq. (13) and

diffusion equation, Eq. (15), are defined on the vessel axial nodes, vessel surface/axial nodes

and vessel surface/tissue nodes, respectively. In order to solve the entire domain, we can

assemble these equations into a lumped system by assigning a global index for the nodes located

in three regions. Because of the nonlinear nature of Eq. (10), we solve for the advection

separately. We combine Eq. (13) and Eq. (15) as

(16)

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Then, we solve for the time evolution of the oxygen concentration (including both free and

hemoglobin bound oxygen) by solving Eqs. (9), (10), and (16) sequentially.

We also note that the unconditional stability of implicit time-stepping may no longer hold due

to the presence of Eq. (10). A rigorous derivation of the stability condition for this model is

beyond the scope of this paper. We confirmed the stability of our solution by obtaining identical

solutions with smaller time steps.

Boundary Conditions—Equations (9) and (16) must be solved with proper boundary

conditions. The boundary nodes include all the nodes on the exterior surface of the domain,

which are either tissue surface nodes or the in-flow/out-flow vertices of vessel axes. For the

tissue surface nodes, we used a mirror boundary condition. i.e.

(17)

This effectively eliminates the surface integration term in Eq. (14) and the expression for the

elements of the vector R is simplified to

For the vessel axes end nodes, the flow rates at all in-flow peripheral points are specified (type-1

boundary condition) and a sink condition is used for the out-flow peripheral points.

2.2 Two-photon microscopy measurements of rodent microvasculature network

Sprague Dawley rats (250-320 g) were initially anesthetized with isofluorane and ventilated

with a mixture of air and oxygen during surgical procedures. Cannulas were inserted in the

femoral artery and vein and heart rate, blood pressure, body temperature and blood gases were

monitored. A cranial window 3 mm × 3 mm in size was opened in the parietal bone and dura

was removed. The window was filled with agarose and sealed with a coverslip. During the

measurement isofluorane was discontinued and anesthesia was maintained with 50 mg/kg

intravenous bolus of alpha-chloralose followed by continuous intravenous infusion at 40 mg/

(kg h). All experimental procedures were approved by the Massachusetts General Hospital

Subcommittee on Research Animal Care.

Blood plasma was labeled with Dextran-conjugated fluorescein (FD 2000S, Sigma-Aldrich;

500 nM concentration in blood). Imaging of the cortical vasculature was performed with a two-

photon microscope (Ultima, Prairie Technology Inc.) at 800 nm and with an Olympus 40X

water immersion objective (NA = 0.8). The individual 256 × 256 pixels frames in the X-Y planes

perpendicular to the optical axis spanned 230 μm × 230 μm. They were obtained as an average

of four images in order to smooth the appearance of the red blood cells (not labeled with the

dye) and to improve further image analysis. The step size between frames along the optical

axis was set to Δz = 1 μm and adjustment of optical power was made continuously with the

imaging depth to address increased scattering and absorption along the optical path-length. An

image stack was acquired over a depth of 450 μm.

2.3 Graphing of the vessel network

Our two-photon microscopy images of the vascular network have a high contrast-to-noise ratio

enabling easy identification of the vascular networks. Rather than employing an automatic

segmentation procedure for graphing the vascular network, we manually graph the network

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using custom software to accelerate the process. The software enables us to place node points

and connecting edges along the vessels. Vessel diameter is estimated at each node by

thresholding the image at a low value of approximately 2% of the maximum image intensity,

but still above the image noise, considering lines through the node point oriented every 3

degrees in the local X-Y plane perpendicular to the vessel axis, and finding the minimum

distance from vessel edge to vessel edge. At the graph end points we need to establish boundary

conditions for velocity or blood pressure. We set the blood pressure boundary condition for

the major arterioles and venules and use a zero velocity boundary condition for minor vessel

branches that do not connect back to our draining vein. The vascular resistance for each edge

between 2 node points is given by Poiseuille's law

(18)

where η is the viscosity of blood (η = 15×10-6 mmHg s [29]), l is the length of the vessel, and

d is the vessel diameter. Given the resistor network and boundary conditions on blood pressure

and velocity, it is straight-forward to solve the set of simultaneous equations for blood pressure

and blood velocity at every node point. We follow the procedure we described previously, in

[30].

3. Results

In this section, we first present our simulation results for a simple vessel network: a single

vessel across the tissue. Using these results, we validate our model proposed in the previous

section for both steady-state and dynamic simulations. In the second half of this section, we

apply our analysis to a more complex vessel network extracted from a two-photon microscopy

scan of a rodent somatosensory cortex. These simulations further demonstrate the accuracy

and scalability of the proposed model. We note that, while our calculations are for the

concentration of oxygen, we report the partial pressure of oxygen (PO2) in our figures as it is

commonly reported in experimental papers. The PO2 is related to the oxygen concentration,

C, by PO2 = C/α.

3.1 Validation

The tissue that we are simulating is 100 × 100 × 100 μm3 in size and contains a single blood

vessel with diameter of d = 10 μm. The vessel is passing through the center of the tissue and

parallel to the x-axis. Unless otherwise stated, for this simple simulation the velocity of blood

inside the vessel is 400 μm/s, hematocrit is set to 0.1, and PO2 = 90 mmHg at the input boundary

of the vessel. The vessel wall permeability is Kw = 1.115×10-12 μmol/(μm s mmHg) and wall

thickness is 1 μm. The oxygen diffusion coefficient is DO2 = 2.4×106 μm2/s. Oxygen

consumption is 41.7×10-16 μmol/(μm3s), or 10% of the baseline value for the rat cortex [31].

A mesh with 25554 nodes and 120942 tetrahedral elements was created using our customized

mesh generator. The meshing process took about 14 seconds on a 1.8GHz Intel Xeon (64bit)

PC and building the matrices in Eqs. (9) and (16) took about 11 seconds. Subsequently, we

solved Eqs. (9), (10) and (16) for about 1.3 seconds per time step.

In Fig. 2(a) we show a contour plot of the steady state PO2, obtained for the above simulation

parameters in X-Y plane that contains the vessel axis. We see a high PO2 for the inflowing

blood on the left (starting at 90 mmHg), diminishing as the blood flows across the vessel to a

minimum of 64 mmHg indicated by the last contour on the right. In Fig. 2(b), (c), and (d), we

plot the PO2 along the interior of the vessel for various values of the blood velocity, hematocrit,

and tissue oxygen consumption, respectively. As we increase velocity from 0.4 mm/s to 0.8

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mm/s and 1.6 mm/s, we see that the PO2 is increasing as more oxygen is being delivered by

the blood per unit time, but the oxygen extraction by the tissue is constant, and thus the oxygen

extraction fraction is expected to decrease. Similarly, as we increase the hematocrit from 0 to

0.2 we see that PO2 increases as more oxygen is delivered by the blood. When tissue oxygen

consumption is increased, the PO2 in the blood decreases, as expected. The difference between

the oxygen flowing into the volume per unit time and the oxygen flowing out should be equal

to the tissue volume integral of the oxygen consumption. Assuming the uniform oxygen

consumption (OC, in μM/s), the oxygen conservation requires

(19)

where v is the velocity of blood, CT,in and CT,out are the total concentrations of oxygen at the

input and output boundaries of the blood vessel, respectively, and Vtis is the tissue volume. The

left-hand-side of Eq. (19) denotes the total oxygen extracted from the vessel per unit time while

the right-hand-side denotes the quantity of oxygen consumed by the tissue.

To validate the steady-state solution obtained by our finite element model, we calculated the

PO2 distributions under various velocities, hematocrit rates and OC values, and the solutions

are shown in Fig. 2. The expected conservation relationship, i.e. Eq. (19), holds for all the

steady-state simulations.

The second step is to validate the model for dynamic processes. In these simulations, we

separately explored the dynamics of oxygen advection within the vessel, flux across the vessel

membrane and diffusion in the tissue and compared the calculations with simple analytic

solutions. These results are shown in Fig. 3 and detailed below.

Oxygen advection within the vessel was isolated from oxygen diffusion by setting the vessel

wall permeability Kw = 0. We started with no oxygen in the vessel and introduced 90 mmHg

of oxygen pressure at the input of the vessel ramping down to 0 mmHg at 40 μm into the vessel.

Fig. 3(a) shows the oxygen pressure profiles along the vessel at 20 ms time intervals. We see

that the oxygen wavefront is advancing almost 8 μm every 20 ms, consistent with the velocity

of 0.4 mm/s. It is known that a two-level implicit advection solver produces solutions that

underestimate the true velocity at high spatial frequencies and accurately estimate the velocity

at low spatial frequencies [26]. This is evident in Fig. 3(a) with smoothing of the edges at 90

mmHg and 0 mmHg as the oxygen advects along the vessel. We note that this is an intrinsic

limitation of the discretization scheme, and although we have implemented the most accurate

solver, it places limits on the accuracy of our solution for oxygen content varying with high

spatial frequency along a vessel. As noted in Ref. [26], this limitation can be reduced by using

a refined time step or smaller grid spacing.

Flux of oxygen across the vessel wall was validated by fixing the vessel PO2 at 90 mmHg,

initiating the tissue PO2 at 0 mmHg, setting the tissue oxygen consumption to 0, and increasing

the oxygen diffusion coefficient within the tissue by a factor of 1000 such that the system

dynamics would be limited by the vessel wall permeability Kw. In Fig. 3(b), we plot the tissue

temporal evolution of PO2, which is increasing to 90 mmHg due to oxygen flux across the

permeable vessel wall. When we reduce Kw by a factor of 2, we see that the rate of increase

for the tissue PO2 is reduced. The rate of change in the total oxygen content of the tissue volume

is given by Fick's law [28] as

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(20)

where NO2 is the number of moles of oxygen in the vessel or tissue, d is the diameter of the

vessel, l is the vessel length, w is the vessel wall thickness, and we assume that any oxygen

that crosses the vessel wall distributes uniformly throughout the tissue volume effectively

immediately (i.e., the diffusion time constant is much shorter than the vessel wall flux time

constant). Under these conditions, the solution of Eq. (20) is a simple exponential. Note that

(21)

where x indicates either tissue (tis) or vessel (ves). The solution of Eq. (21) is shown in Fig. 3

(b) and agrees well with the solution of our finite element model.

Oxygen diffusion was validated by setting oxygen consumption to 0 and initiating the tissue

PO2 to 0 mmHg except for a single volume element far from the boundaries of the tissue volume

that was given a unit quantity of oxygen. This oxygen was then diffused with our finite element

model. The early responses of diffusion after 20 (blue), 25 (green), and 30 ms (red) are plotted

in Fig. 3(c) and they roughly agree with the analytic solution of the diffusion equation for a

homogeneous infinite medium. We also computed the finite element solution reducing the

diffusion coefficient by a factor of 2 and plot the results at 40 (blue), 50 (green), and 60 ms

(red) and observed the expected DO2t scaling of the diffusion equation. That is, if we half the

diffusion coefficient and double the time, the analytic solution [32] predicts the same solution.

Tissue oxygen consumption was validated by setting PO2,tis to 90 mmHg throughout the tissue

volume, inhibiting oxygen flux across the vessel wall by setting Kw = 0, and increasing oxygen

consumption. Our simulation result (not shown) demonstrates a linear decrease in oxygen

concentration in the tissue with time, as we expected.

3.2 Application to a microvessel network

After validating the steady state and dynamic solutions of our oxygen advection-diffusion

solver, we proceed in solving for the delivery of oxygen to tissue by a realistic microvessel

network. We obtain a volumetric image of the microvessel network from the somatosensory

cortex of a rat using two-photon microscopy as described in the methods. The volumetric image

spanning 230 × 230 × 450 μm3 is shown in Fig. 4(a). In this image, we identify a 25 μm in

diameter surface pial arteriole in the upper right delivering oxygenated blood to the tissue and

a draining ascending venule in the upper left of the image, connected by the intervening

capillary network.

We manually graphed this vessel network in under 2 hours using custom software to accelerate

the process, by placing node points and connecting edges along the vessels. The diameter of

the vessel at each node point, the blood pressure, and velocity of blood was calculated as

described in the methods. The velocity at each node in the vascular graph was used in solving

the oxygen advection diffusion equation to obtain the partial pressure of oxygen at every node

point in the vessels and throughout the tissue. The vessel in-flow and out-flow points (i.e., the

boundary points as mentioned in Section 2.1) were manually labeled and used for the

subsequent advection modeling. The generated 3D mesh for this case contains 138685 nodes

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and 607406 elements. The total time for mesh generation, calculating the matrices and solving

for each time step are 136, 62 and 6 seconds, respectively.

In Fig. 4(b) we plot the graphed vascular network identifying the arterioles (red), capillaries

(green), and venules (blue). We see that the surface pial arteriole branches twice. The left-hand

branch propagates along the surface to the lower left of the image and then descends 450 μm

to the bottom of the image without branching. The right-hand branch descends into the tissue

at the lower right. This descending arteriole quickly branches a few times leading to a capillary

network that connects with an ascending venule on the upper left of the image. The velocity

of blood along the vascular network is shown in Fig. 4(c). The velocity has a maximum of 10

mm/s in the pial arteriole and reaches a minimum of ∼1 mm/s in the capillaries and draining

venules. The partial pressure of oxygen along the network is graphed in Fig. 4(d), showing the

expected decrease from feeding arteriole to draining venule.

The median velocity, blood pressure, and partial pressure of oxygen versus vessel diameter are

shown in Fig. 5. Our pressure distribution compares well with a survey of experimental data

[33]. Our velocity and partial pressure of oxygen, however, drop below experimental data

[33,34] in the capillaries and venules. This is likely a result of our incomplete vascular graph

not accounting for the surrounding microvessels that are providing more blood into the draining

vein and delivering more oxygen to the tissue. We will correct this in the future by considering

a vascular graph over a larger volume of tissue in order to obtain a more completely connected

graph in the central regions.

The partial pressure of oxygen within the tissue is shown in Fig. 6 in slices at different depths

and in a movie of the volumetric data. From these data, we see the high tissue PO2 around the

arterioles dropping off rapidly with distance, while the lower PO2 around the capillaries is

much more spatially uniform. In Fig. 7 we plot the drop in PO2 versus distance from selected

arterioles, capillaries, and venules. These results qualitatively agree with experimental in-

vivo measurements [34].

4. Summary

We have described a hybrid numerical algorithm and a general work-flow to model the 3D

time-varying oxygen advection diffusion process in an anatomical microvascular network. This

framework provides a quantitative and computationally feasible approach for modeling the

dynamic processes in the microvessels in the tissue, which is becoming increasingly important

in studies of cerebro-vascular physiology and tumor physiology. The proposed algorithm is

structurally generic and readily applicable to vessel networks with varying degrees of

complexity. The graph representations of the vessel network and finite-element modeling also

provide a flexible spatial discretization scheme, which is important to balance computational

expense and accuracy. Solving the 3D dynamic diffusion-advection process not only makes

this framework useful for modeling tissue oxygen delivery and consumption, but is also

extendable to include other molecules such as NO and CO2, or to model the pharmacokinetic

processes in drug delivery. Future efforts for this framework include automatic graphing of the

vessel network from 3D microscopic images and an in-depth understanding of the stability

condition. We will also apply this method to full-sized vessel networks to characterize the

microscopic details of oxygen delivery during brain activation and to validate macroscopic

NIRS and functional magnetic resonance imaging (fMRI) measurements of oxygen delivery.

Acknowledgments

The authors would like to acknowledge funding support from NIH R01NS057476, P01NS055104 and R01NS051188,

and from the Air Force Office of Scientific Research, Medical Free Electron Laser Program Contract

FA9550-07-1-0101. The authors would also like to acknowledge the assistance of Shuai Yuan, Mark Shalinsky, and

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Weicheng Wu for in vivo experiments, and the help from Wes Turner (Kitware Inc.) and Scott Dineen (OSA) for

preparing the interactive datasets.

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Fig. 1.

The decomposition of the problem domain.

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Fig. 2.

Model validations with a single vessel: (a) PO2 distribution across an X-Y plane along the

vessel; (b-d) PO2 along the vessel for different values of (b) blood velocity, (c) hematocrit, and

(d) oxygen consumption.

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Fig. 3.

Validation of dynamic processes with a single vessel. (a) PO2 profiles along the vessel at 20

ms intervals for isolated advection. The modeled PO2 profile is given by the solid line. The

star represents the expected advancement of oxygen pressure profile of 8 μm/s. (b) Tissue

PO2 evolution for validation of vessel wall permeability. The dots and dashed line show our

model results for Kw and Kw/2 respectively, and the solid line is the analytic prediction. (c)

Tissue PO2 profiles at different times for validation of diffusion (see text for more details).

The solid line is the analytic solution of the diffusion equation and dots and dashed line are

our model results for DO2 and DO2/2 respectively.

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