^{*}

Conceived and designed the experiments: JW IS. Performed the experiments: JW. Analyzed the data: JW. Contributed reagents/materials/analysis tools: JW. Wrote the paper: JW IS.

The authors have declared that no competing interests exist.

The liver plays a key role in removing harmful chemicals from the body and is therefore often the first tissue to suffer potentially adverse consequences. To protect public health it is necessary to quantitatively estimate the risk of long-term low dose exposure to environmental pollutants. Animal testing is the primary tool for extrapolating human risk but it is fraught with uncertainty, necessitating novel alternative approaches. Our goal is to integrate

Virtual tissues are emerging as a powerful tool for computational biology. By encoding known biology into a simulation of tissue function, gaps in knowledge can be identified. As a simulation of tissue function,

As the number of man-made environmental chemicals continues to grow, there is an urgent need to develop effective tools to test their potential risk to humans. The number of environmental chemicals that are produced in substantial quantities now numbers approximately 10,000

Without appropriate context,

Our goal is to establish an ^{5} to 10^{6} functional units called lobules

Blood flows past sheets of hepatocytes through the sinusoids and into the central vein. Image adapted from an original by Amber Goetz, first published in

Tissue dosimetry is traditionally estimated using physiologically-based pharmacokinetic (PBPK) models. A PBPK model consists of a system of ordinary differential equations (ODEs) for the concentration of a compound (or compounds) in different tissues. Typically some key tissues are treated as separate compartments for which a tissue-specific concentration is calculated, while other tissues are modeled using aggregate compartments. More complicated dynamics within a tissue, such as diffusion or membrane transport, are often modeled with additional sub-compartments but each sub-compartment is well-mixed. The equations are parameterized by subject- or species-specific physiologic parameters such as cardiac output and tissue volumes as well as compound-specific parameters such as diffusion/transport rates and tissue-specific plasma to tissue “partition coefficients” corresponding to the assumption of a rapidly-established equilibrium between concentration of compound stored in the tissue and the concentration of compound in the plasma flowing through the tissue. PBPK models relate the concentration of compounds inhaled or ingested from the environment to internal tissue doses

In addition to the well-mixed approach, the parallel tubes model of liver function has often been used to calculate

Though

The first multi-compartment geometric model of the liver was developed by Andersen et al.

Ierapetritou et al.

Hunt et al.

We have implemented a microdosimetry model that relates whole-body chemical exposures to cell-scale concentrations. Our objective was to develop the framework for simulating the microanatomic distribution of various environmental chemicals in a canonical lobule for extended periods of time ranging from hours to months. This required an approach that is quantitative, efficient in computational resources, and sufficiently flexible to account for anatomic changes (due to chemical insult or other factors)

The two dimensional morphologic characteristics of the mammalian hepatic lobule were represented as a discrete connectivity graph, in which the edges captured spatial proximity. The two main anatomic entities considered are hepatocytes, the parenchymal cells responsible for the metabolism of chemicals, and vasculature,

A simplified geometry of the lobule was defined using the following morphologic parameters: the number portal triads (defining the vascular inputs), the branching factor of the sinusoids, the number of sinusoids entering the central vein, and the sizes of sinusoids, hepatocytes, and the lobule. The graphical model of the lobule was constructed algorithmically using these parameters and visualized spatially (

First, sinusoids outward from the central vein (i). In addition to small random variations in the direction of propagation, the sinusoids branch into two sinusoids pointed away from the central vein with probability P_{br} (ii). Multiple sinusoids are started from the central vein in an attempt to fill space (iii). Portal “triads” consisting of arterioles and venules through which blood enters the lobule are added to the perimeter of the lobule and connected to the vasculature (iv). Finally, the sinusoids are lined with hepatocytes as space allows (v).

The graphical model of the lobule was generated iteratively (the algorithm is described in the

The approach described above is flexible, allowing the generation of diverse lobular topologies through which flows can be simulated. Five basic morphologies were examined, as depicted in _{br}

They are: a) one portal triad, no branching or noise, b) six portal triads with noise and additional sinusoids, c) six portal triads, 10% chance of branching, d) three portal triads with 10% branching, and e) six portal triads with 5% chance of branching. Though the overall layout (middle column) can be compared qualitatively with physiology, we evaluate these geometries by comparing the flow (left-hand column) predicted for a rat with _{max} with the prediction for a well-mixed compartment with equivalent metabolic clearance (heavy dashed line). Comparison of profiles b-e with profile a provides an approximate comparison to a parallel tubes prediction. The solid line indicates the mean for multiple lobules and sinusoids, while the shading indicates the 95% quantile (variability).

Miller et al. (1979) observed that the branching of sinusoids is greater near the portal triad than near the central vein

Blood circulation through the graphical model of the vasculature was simulated as a network flow (^{μ}_{i} (see

Spatial proximity between sinusoids within simulated lobule (a) was used to generate connectivity graphs (b), which are aggregated (c) in order to solve for flow from the portal triads to the central vein using ODEs.

Symbol | Definition |

P_{br} |
Sinusoidal Branching Probability |

sinusoid graph consisting of vertices (nodes) V and edges E | |

^{μ}_{ij} |
Micro flow rate (L/h) across from node |

_{i} |
Total flow into node |

Q^{μ}_{art} |
Micro flow rate (L/h) through each arteriole |

Q^{μ}_{art} |
Micro flow rate (L/h) through each venule |

R_{liv∶lob} |
Ratio of liver to lobule volume |

Q_{gut} |
Flow rate (L/h) through gut tissue |

Q_{liv} |
Flow rate (L/h) of arterial blood into liver |

^{μ}_{i} |
Concentration of within aggregate sinusoid |

C_{liv} |
Concentration for a well-mixed liver compartment |

Concentration averaged over the lobule | |

Maximum concentration averaged over the lobule | |

^{μ}_{i,max} |
Maximum concentration within aggregate sinusoid |

t_{max} |
Time at which maximum average concentration is reached |

Mass-balanced flow through the aggregate graph was determined by solving for the flow across each edge of the sinusoid graph _{ij} requires ^{μ}_{ij}

We made use of the hemodynamical equivalent of Ohm's law _{i} is the pressure at node _{ij} could be determined using schemes such as the cross-sectional area of each branch. Hemodynamics provides |E| additional constraints, but introduces |V| additional unknown pressures P_{i}. Together with mass balance we have |E|+|V| constraints for |E|+|V| unknowns. This system of equations can be represented with a matrix and, given source flows and outlet pressure, can be solved by diagonalization. Since we are not currently interested in sinusoidal pressure, R and the outlet pressure were taken as one. This assumption does not effect the quantitative values of ^{μ}_{ij}_{i}.

As can be seen in

To evaluate the appropriateness of these assumptions and the suitability of the approach to arbitrary graphical structures, we return to

Oral dose | 10 µMol |

Number of Lobules per Ensemble Analyzed | 50 |

Agent-based model steps per Iteration | 8 |

time per iteration | 0.2 h |

Total hours simulated | 5 |

Number of Portal Triads | 6 |

Number of Sinusoid starts at central vein | 6 |

Sinusoidal Branching Probability _{br} |
10% |

Radius of Lobule | 15 hepatocytes |

diameter of hepatocyte | 100 µm (assumed) |

Thickness of lobule | 23.5 µm |

Diameter of sinusoid primitive | 25 µm |

The final step needed to determine the concentration ^{μ}_{i}^{μ}_{art} = Q_{liv}/R_{liv∶lob}/N_{PT} and a venule flow Q^{μ}_{ven} = Q_{gut}/R_{liv∶lob}/N_{PT} where R_{liv∶lob} is the ratio of liver to lobule volume and N_{PT} is the number of portal triads per central vein. Concentrations within the lobule are determined by solving_{i}

Parameter | Value | Source |

Qcard | 336 L/h | |

Qgut | 66 L/h | |

Qliv | 18 L/h | |

Qgfr | 7.5 L/h | |

Qrest | 252 L/h | |

Bodyweight | 70 kg | assumed |

Lean Fraction of BW | 0.7 | |

Vart, Vven | 0.025 L/kg lean bw | |

Vgut | 0.0165 L/kg bw | |

Vliv | 0.035 L/kg lean bw | |

Vlung | 0.27 L | |

Vrest | 0.6 L/kg bw – (Vart+Vven+Vgut+Vliv+Vlung) | |

k_{ad}, k_{inh}, K_{rest∶plas}, K_{liv∶plas}, K_{gut∶plas}, R_{blood∶plas}, f |
1 | assumed |

For a completely physiologic, three-dimensional lobule R_{liv∶lob} would be equal to the number of lobules in the liver – approximately 10^{6} _{liv∶lob}, the ratio of the total volume of the liver to the total volume of the sinusoidal spaces and hepatocytes in the simulated lobule, to be approximately 10^{8}, which is roughly 100 times greater than the physiologic value. We expect a greater value for two reasons: First, many components of the lobule other than the sinusoidal spaces and hepatocytes, such as endothelial and stellate cells, extracellular space, and bile ducts, contribute to the volume of the lobule. Including these additional components, and therefore increasing the volume of the simulated lobule, will reduce R_{liv∶lob}. Second, each simulated lobule is assumed to have a thickness equal to a sinusoidal diameter (23.5 µm _{liv∶lob} and the actual number of lobules indicates that 100 simulated lobules are currently needed to fill the space of a single physiologic lobule.

Our (quasi-)two-dimensional simulated lobule is assumed to have a thickness equal to a sinusoidal diameter (23.5 µm _{liv∶lob}, the ratio of the volume of the whole liver to the volume of single lobule.

Using the lobule geometries (given in _{liv} for a PBPK model with a well-mixed liver compartment with equivalent metabolic clearance _{liv∶lob}). It is important to note that the overall pharmacokinetics depends on the lobule layout because the effective number of lobules R_{liv∶lob} is determined by volume alone and therefore the total clearance of the liver depends on the number of hepatocytes relative to the volume of the lobule.

Though the overall clearance varied with geometry, the impact of different geometries on the average concentration in the lobule was small. As shown in _{liv} predicted for the appropriate CL. We find that in all cases the predicted average concentration slightly exceeds the well-mixed PBPK prediction, but that otherwise the pharmacokinetics are very similar.

The ensemble average for all five lobules is very similar to the well-mixed lobule prediction (indicated by the dashed line) however the different morphologies produce different whole-liver clearances because the number of hepatocytes as a fraction of the volume of the simulated lobule is geometry-dependent.

Plotted on the right-hand side of _{max} – the time at which the lobule reaches maximum average concentration,

Geometry had a much greater impact on the variability in predicted concentrations

To test whether a continuum approximation (ODEs) was appropriate for modeling mass transfer in the sinusoidal graph we estimated the number of molecules at a hepatocyte. If the number of molecules at higher concentrations is not large enough a stochastic approach

The maximum concentration in the tissue following a dose is a commonly used measure of tissue exposure in pharmacokinetics. For the simulated lobule a local C^{μ}_{i,max} can be calculated for each hepatocyte as a result of different sinusoids receiving different concentrations. ^{μ}_{i,max} experienced by all the hepatocytes in an ensemble of fifty lobules with intrinsic hepatic metabolic clearance of 10 µL/min/million hepatocytes. The values have been normalized to the C_{max} predicted for a well-mixed liver with the same overall metabolic clearance (indicated but the solid line). The peak for the distribution is in excess of the well-mixed prediction, while the breadth is quite wide, indicating that at this rate of metabolism some hepatocytes receive exposures nearly 40% greater than would be predicted for a well-mixed liver while others receive almost no exposure.

Ito and Houston

The shaded region indicates the 95% interval.

Heterogeneity within the lobule is dynamic

We conducted a preliminary analysis of the cellular effects due to microdosimetry using a simple agent-based model for hepatocytes. Each agent was defined by a fixed, identical xenobiotic metabolism rate, and functional states that were updated at each time step via state transition rules. A simple approach was used to encode probabilistic state transition rules conditioned on inputs from the agent environment. Future cellular models will be able to take better advantage of the freedom to proliferate and move provided by this approach since flow for a new arrangement can be determined rapidly by updating the sinusoid and contact graphs. Here we considered normal hepatocytes and cell death following exposure to threshold cytotoxic concentration. The ABM was integrated with the sinusoidal flow model with each being updated alternately. We simulated twelve minutes of the flow followed by eight iterations of the ABM – intended to be sufficiently small time periods for each model to respond realistically to changes in the other. Experimental verification will be needed to determine the appropriate time scales.

Given the current cellular model and the predicted increase in variability with metabolism rate shown in

For a well-mixed lobule, a threshold in excess of maximum lobule concentration should have no effect. Instead, as shown in

For a rapidly metabolized compound (upper curve) variability in exposure causes some apoptosis in the spatially-extended lobule. The shaded region indicates the 95% interval.

We have described a microdosimetry model to relate environmental exposures to cellular exposures. This is only a step toward developing virtual tissues that can predict the

The liver lobule is known to be spatially heterogeneous

A model for a spatially-extended hepatic lobule sets the stage for investigating emergent behavior in models of hepatocyte function. If the action of hepatocytes creates spatial variation across the lobule then any cellular dynamic response that depends on chemical or nutrient concentration may in turn be altered, which could be a prelude to zonal patterns of biological functions. More extreme effects, such as central lobular necrosis, may be due to the transformation of the compound via metabolism into a more potent compound or zone-dependent variation in sensitivity of the hepatocytes.

In contrast to the well-stirred model of the liver, the simulated lobule provides a means of accessing a variety of inter- and intracellular dynamics. Though the results we obtain are in some respects similar to previous models, we gain the additional capability of allowing hepatocyte-specific dosimetry as well as the potential to alter lobule geometry,

In contrast to computationally intensive, spatially continuous approaches such as fluid dynamics, this graph-theoretic approach has hopefully sacrificed little physiologic detail but gained a great deal in terms of computational efficiency. Calculating hemodynamical flow on a graph allows rapid determination of flow given minimal boundary conditions, which will be especially useful for recalculating flow as morphology changes (e.g. lesion formation) or as individual sinusoids are temporarily blocked (e.g. Kupffer cells). A faster dosimetry model allows the focus to center on cellular phenotypes, which are the key to modeling disease pathogenesis. A computationally-tractable approach allows for simulating the long run times associated with sub- and chronic toxicity studies as well as simulating large populations.

We evaluated our approach to hepatic blood flow in three ways. First, we qualitatively tuned the appearance of the lobule to match actual physiology. Second, we compared the predicted pharmacokinetics for our spatially-extended lobule with traditional approaches, finding regimes in which our approach reduced to the well-mixed liver and the parallel tubes model. Third, we quantitatively compared the flow predicted for a rat with observations made

This work addresses the dose portion of the dose-response curve, allowing assessment of how changes in exposure impact the hepatic lobule. The greater body of work remains with modeling response. Sufficiently complex models for hepatocellular dynamics, and eventually models for additional cell types, especially the Kupffer cells responsible for inflammatory responses, must be developed before we arrive at a useful model for homeostatic liver function. It remains to be seen whether three-dimensionality or even a departure from the classical lobule paradigm to simulate multiple lobules will be needed.

To establish the safety of a compound one ideally finds the dose-response curve for various toxicity endpoints, so that an acceptable level of exposure can be determined. Currently the gold standard of toxicology is animal testing, but the need and desire for i

The multiscale approach describe here is intended to be fast and verifiable, and would allow the determination of whether an observed

Histopathology images have long been used to obtain information on microanatomic regions, vasculature, individual cells, cell types, and cell phenotypes from two- and three-dimensional images. Though traditionally time-intensive, advances in automated extraction of information from histopathology images are making it possible to analyze these images at a single cell resolution

True variability in the response of a given hepatocyte is either a product of independent microdosimetry and cell variability, or is a function of the two, depending on the degree of correlation. To determine the significance of a chemical perturbation it is not enough to understand the cellular dynamics, but also the context in which those dynamics exist – i.e., microdosimetry.

We have implemented a microdosimetry model for relating whole-body chemical exposures to cell-scale concentrations. The model is written in the freely available statistical language R, version 2.8.1

Given morphologic parameters N_{t}, the number of portal triads; N_{s}, the number of sinusoids per source/sink; P_{branch}, the probability of a sinusoid branching; and D_{max}, the size of the lobule, and calculating θ_{CV} is the angle to the central vein, given current position:

Place central vein

For each of N_{s} sinusoids:

Select initial angle θ^{0}_{s}

Place sinusoidal primitive on edge of central vein at θ^{0}_{s}

Call the sinusoid placement algorithm (SPA) with θ_{s} = θ^{0}_{s}

Increment θ^{0}_{s} approximately 2π/N_{s}

For each of N_{t} portal triads:

Select initial angle θ_{t}

Place a periportal vein at angle θ_{t} and distance 0.8*D_{max}

For each of N_{s} sinusoids:

Select initial angle θ^{0}_{s} = θ_{CV}

Place a sinusoid primitive on edge of the periportal vein at θ^{0}_{s}

Call SPA with θ_{s} = θ^{0}_{s}

Increment θ^{0}_{s} approximately 2π/N_{s}

Place an arteriole randomly at the edge of the periportal vein

For each of N_{s} sinusoids:

Select initial angle θ^{0}_{s} = θ_{CV}

A sinusoid primitive is placed at the edge of the arteriole at θ^{0}_{s} and the SPA is called with θ_{s} = θ^{0}_{s}

θ^{0}_{s} is incremented approximately 2π/N_{s}

Calculate the potential position of the next sinusoidal primitive using θ_{s}

If either the distance from the central vein exceeds D_{max} or the potential location overlaps with a previously placed sinusoid, then

Return

If a randomly drawn number [0,1] is less than P_{branch}, then

Randomly select θ'_{s} from the interval [θ_{CV}−π/2, θ_{CV}+π/2]

Call SPA with angle θ_{s} = θ'_{s}

θ_{s} is randomly perturbed

Call SPA

The aggregation process is performed using the following algorithm:

All sinusoid primitive nodes are assigned a corresponding aggregate node (CAN) initially set to NULL

For each sinusoid node I adjacent to the central vein, if the CAN is NULL, then,

if the number of sinusoid neighbors N^{i}_{n} = 1, a “dead end” CAN is created,

if N^{i}_{n} = 2, a “straight” CAN is created

if N^{i}_{n}>2 a “branch” CAN is created

For each neighbor j, If N^{i}_{n} = N^{j}_{n} then,

The CAN for j is set to the CAN for i unless the CAN is a straight node already consisting of 5 sinusoid primitive nodes

Step 2d is called recursively for node j

Repeat step 2 for each arterial and venous source

For each branch CAN i,

For each neighbor j if j is also a branch then CAN i is absorbed into CAN j

For each straight CAN i, if there is now only one neighbor and that neighbor is a branch CAN, merge i with the neighbor

For each branch CAN i, if there is only one neighbor convert I into a dead end CAN

We thank Ann Marie Pitruzzello, R. Woodrow Setzer, John Jack, and Christopher Basciano for helpful discussions and comments.