# A physiologically-based flow network model for hepatic drug elimination II: variable lattice lobule models

- Vahid Rezania
^{1}, - Rebeccah Marsh
^{2}, - Dennis Coombe
^{3}and - Jack Tuszynski
^{4}Email author

**10**:53

https://doi.org/10.1186/1742-4682-10-53

© Rezania et al.; licensee BioMed Central Ltd. 2013

**Received: **28 May 2013

**Accepted: **28 August 2013

**Published: **5 September 2013

## Abstract

We extend a physiologically-based lattice model for the transport and metabolism of drugs in the liver lobule (liver functional unit) to consider structural and spatial variability. We compare predicted drug concentration levels observed exiting the lobule with their detailed distribution inside the lobule, and indicate the role that structural variation has on these results. Liver zonation and its role on drug metabolism represent another aspect of structural inhomogeneity that we consider here. Since various liver diseases can be thought to produce such structural variations, our analysis gives insight into the role of disease on liver function and performance. These conclusions are based on the dominant role of convection in well-vascularized tissue with a given structure.

## Keywords

## Background

The liver is the major organ responsible for the metabolism and detoxification of drugs. Within the liver, it is the hepatocytes which express a high level of drug-metabolizing enzymes and are primarily responsible for liver drug disposition. Drug access to hepatocytes is governed by transport processes in the well-vascularized liver tissue, and structural variability can obviously impact such transport. In this paper, we extend our physiologically-based lattice model for the transport and metabolism of drugs in the functional unit of the liver [1] (paper I) to consider spatial and structural variability and its impact on hepatic drug metabolism.

### The liver lobule and drug kinetics

#### Functional unit as a regular lattice model

The functional unit of the liver is termed the liver lobule [2] and is the smallest structural unit with complete hepatic functionality. Although various definitions of this unit have been proposed, we have chosen a symmetry element connecting the portal (arterial) tract with hepatic venules. In our first paper, we defined a base-case regular lattice structural model and explored the dynamics of competing convective, diffusive, and reactive processes acting on an injected drug (paclitaxol). Such simulations had the useful consequence of relating drug concentration levels found exiting the lobule to their detailed spatial distribution within the lobule, caused by competing processes. Tables one to four in our first paper detail the parameters chosen for our regular lattice model of the lobule. They provide a basis, and a point of contrast for the drug distributions obtained when some of these assumptions on lobule structure are relaxed, as discussed in this current paper.

### Structural variability

Structural variability of lobule units is expected to be the rule, even among the approximately 1500 lobules that make up the liver of one healthy human. Teutsch and colleagues [4] illustrate this aspect morphologically.

Diseased states can be expected to add a spectrum of additionally variability. The health of the liver can be compromised by viruses, hereditary diseases, and toxins such as alcohol [5]. Damage or death of the hepatocytes leads to inflammation of the liver, called hepatitis. Although zones of necrosis can form when adjacent cells die, this damage is to some extent reversible, since the liver has the ability to regenerate. Thus hepatitis is typically characterized by waves of cell death and regeneration, leading to a mixture of necrotic areas and nodules of new hepatocytes. Because the architecture of the liver is often compromised, some cells may not receive normal levels of blood supply. Furthermore, as inflammation progresses, fibrous tissue may replace the normal hepatocytes, resulting in the irreversible condition of cirrhosis. The damage can be compounded because the formation of necrotic zones increases the resistance to blood flow, and intra-hepatic shunts can occur in which blood vessels begin to bypass the liver altogether. Therefore, although the liver has the capacity to withstand and even correct a lot of damage, its ability to transport, absorb, and metabolize important nutrients and drug molecules can be compromised.

A Base case model, of necessity, required numerous assumptions on an appropriate idealization of the liver lobule structure. In our analysis, we will utilize the concept of a random permeability lattice to capture small vascular irregularities in healthy livers, and the more extreme effects of hypertension sustained due to hepatitis. Damage and scarring occurring with cirrhosis will be treated within the framework of percolation theory, such that the architecture of the liver is physically changed.

### Zonation and elimination kinetics

Zonation is a well-known feature of many metabolic processes occurring in the liver lobule [6], including carbohydrate [7] and nitrogen [8] metabolism, such that some processes are up-regulated near the periportal zone while others are up-regulated near the perivenal side of the lobule. Drug metabolism and drug metabolizing enzymes also show similar zonation features [9]. Our focus will be on the distribution of the cytochrome P450 (CYP) enzyme in particular.

Zonation has been attributed primarily to non-uniform distribution of O_{2} across the lobule [10], with the periportal zone experiencing relatively high concentrations of O_{2} while the perivenal zones see near hypoxic levels of O_{2}. The distribution of other factors (e.g. growth factors) has also been shown to play a role. Indeed, injection of specific xenobiotic factors has been utilized to alter zoned enzyme expression levels [11–13]. A future application of our model might be to track O_{2} distribution and metabolism across the lobule. Here, for such a small molecule, molecular diffusion can be expected to play a dominant role.

Following our earlier work, drug uptake and elimination (i.e. conversion to metabolized product) is viewed as a single-step saturable process following Michaelis-Menten kinetics [1].

## Model and methods

MATLAB [14] was used to generate multiple realizations of spatial sinusoid permeability variations drawn from a selected sinusoid permeability distribution function (uniform, normal, or log-normal). We are interested in the sensitivity to the spread (i.e. variation) of the distribution of this function, and we will use the term “Random Lattice” to signify our studies in this portion of the work.

Specifically, random permeabilities between [*a*, *b*] (=*K*_{sin} [1-σ, 1 + σ] in our case) was produced by the uniform random generator in MATLAB. Every representation is created independently, multiplied by the sinusoidal permeability *K*_{sin} and then averaged over the number of representations. Based on the central limit theorem, the average value over the sample size *N* approaches the mean value (*a + b*)/2 (=*K*_{sin} in our case) with standard deviation ~ 1/*N*^{1/2}( ~ σ /*N*^{1/2} in our case). This means the mean value is independent of σ. Permeability of tissue sites was set to constant *K*_{tis} = 7.35e-2 μm^{2} after the sinusoidal permeability was set and averaged over the number of representations.

*K*

_{sin}= 1.125 μm

^{2}and σ = 0.75 for 1, 10, 100 and 1000 realizations, respectively. The first peak represents a permeability of tissue value

*K*

_{tis}= 7.35e-2 μm

^{2}. As expected, uniform distributed numbers tend toward the mean value. After

*N*= 1000 sampling, only two peaks are left, one for the tissue and one for the sinusoids, similar to the fixed tissue-sinusoid permeability case.

*X*

_{i}. Then by comparing the value at each site with a given percolation probability P

_{c}, the value of that site will change as follows:

All X_{i} values were stored in a separate matrix for averaging purpose. Then a new set random number was chosen for each site and the above procedure was repeated up to desired number of samples. At the end, an averaged value found over all representations was calculated for each site. The permeability of tissue sites were again set to the constant 7.35e-2 μm^{2} and the permeability of corner sinusoidal sites set to *K*_{sin}.

*K*

_{sin}= 1.125 μm

^{2}and

*P*

_{c}= 0.7 for 1, 10, 100 and 1000 realizations, respectively. The first peak represents a permeability of tissue value of

*K*

_{tis}= 7.35e-2 μm

^{2}. As expected, the top panel is just for one representation that values are only 7.35e-2 μm

^{2}or 1.125 μm

^{2}. Adding more representations and averaging over them produces a second peak close to the

*P*

_{c}value (here 0.7). This is true for any value of

*P*

_{c}. As a result, the fixed and averaged random cases are similar to the extreme case of percolation with

*P*

_{c}= 1. The third peak appearing in Figure 3 has value close to

*P*

_{c}

*K*

_{sin}.

Zonation effects in the lobule have been treated here by simply varying the spatial expression of the active CYP enzyme. Thus hepactocytes are assumed to be located throughout the lobule tissue as in the Base case model, but the expression of CYP is assumed to be regionally-biased at three levels (zero-, mid-, and full-expression). Two extremes are explored, with the zero expression zone located upstream (i.e. surrounding the arteriole injection site) or downstream (i.e. surrounding the venuole production site). Recall that the Base case model assumed full CYP expression in all hepatocytes of the lobule.

The simulations were performed using the STARS advanced process simulator [15] designed by the Computer Modelling Group (CMG) Ltd. in Calgary, Alberta, to model the flow and reactions of multiphase, multicomponent fluids through porous media [16, 17]. Specific biomedical applications of STARS include modeling reactive flow processes in cortical bone [18–21] and in the intervertabral disk [22].

## Results

### Tissue permeability sensitivities on drug transport

**Geometric average ranking of random permeability lattices**

Parameter | SS Flow Rate (cm | Geometric Average Rate (cm |
---|---|---|

Try1 | 1.68366e-6 | 1.9009e-6 |

Try2 | 1.71368e-6 | 1.9246e-6 |

Try3 | 1.62196e-6 | 1.9568e-6 |

Try4 | 1.94859e-6 | 1.9554e-6 |

Try5 | 1.99852e-6 | 1.9891e-6 |

Try6 | 1.70890e-6 | 1.9119e-6 |

Try7 | 1.09153e-6 | 1.9924e-6 |

Try8 | 1.34926e-6 | 1.9038e-6 |

Try9 | 1.61186e-6 | 1.9246e-6 |

Try10 | 1.10223e-6 | 1.9148e-6 |

Try11 | 1.21848e-6 | 1.9365e-6 |

Base Case | 2.44415e-6 | 2.4415e-6 |

### Sinusoid random permeability sensitivities on drug transport

Effects of sinusoidal permeability on paclitaxol drug transport can be explored in the same way as those of tissue permeability presented in the above section. Specifically, representative upper and lower sinusoid permeability values could be selected around the Base case value (*K*_{sin} = 1.125 μm^{2}), and their effects on drug transport could be predicted. Such an approach could also illustrate the effects of angio-sensitive molecules causing uniform vascular restriction or dilation of sinusoidal pathways.

*p*, while Figure 5b shows the corresponding paxlitaxol production history for each representation. These figures demonstrate that higher flow rates generate earlier paclitaxol production. This method also allows an assessment of the spread of expected behavior by focusing on the 10%, 50%, and 90% cases. For the 11 realizations studied here, these three cases correspond to Try11, Try2, and Try4, respectively. Figures 8a-c show the permeability distribution profiles generated by our algorithm for these three cases.

*p*as follows:

*F*is a constant network geometric factor that assumes a constant viscosity. For each realization, the effective network permeability can be estimated by a geometric average of all sinusoid and tissue values in a given realization [23] as follows:

Here the power “w” is a best fit parameter. This equation reduces to both arithmetic and harmonic averages when the power “w” is 1 or −1 and becomes the geometric average when “w” is 0.

This requires a root finding method, such as Newton’s method, to find the appropriate answer. Newton’s method requires an initial estimate, appropriately chosen here as the geometric average. Here, we investigated both the geometric and power law averaging techniques, but with limited success.

As an illustration, Table 1 summarizes the geometric mean ranking of the 11 realizations used to generate the drug production responses of Figure 5. Table 1 additionally quotes the flow rate obtained from our Base case (constant sinusoid permeability) model in our first paper. This rate is significantly higher than the flows predicted from any realization, even though individual permeability values were drawn symmetrically from a distribution around the same average permeability. This illustrates that flow paths within the lobule are at least partly in series, especially because of the diverging/converging flow geometry of the model, and hence a harmonic component of flow resistances can be expected (i.e. any lower permeability elements will tend to impart an overall higher resistance to flow).

Eventually we settled on a steady state numerical method to rank realizations. With this approach, we simulated one timestep (with a value large enough to represent steady flow-1.0e-4 min or larger, see Figure 5) of a non-reactive case for each realization. Our (fully implicit) numerical method is extremely stable to such large timesteps and allows the calculation. This would represent one fixed time point for each realization from Figure 5, for ranking purposes.

Rather than the 11 realizations used here for practicality, a more statistically consistent sampling set would include, for example, 101 or 1001 realizations. Utilizing the same methodology of ranking realizations, the 10%, 50%, and 90% cases can again be selected for further detailed analysis. This is the recommended method for analyzing statistical realizations.

### Sinusoid percolation connectivity sensitivities on drug transport

More extreme disturbances in lobule flow conductivity can occur as connectivity is complete (i.e. flow paths are connected). This may be expected to occur with more severe liver damage and disease, such as hepatitis and increased fibrosis. Mathematically, such instances can be appropriately analyzed via percolation theory, with extreme effects seen as the percolation limit is reached.

*P*

_{c}= 1 on the Base case lattice. One sees a trend of decreasing flow and later drug production as the

*P*

_{c}value is reduced. Each curve represents an average of 100 realizations at a specific percolation value. These realizations were generated as described in the Methods section above. Further analysis of percolation behavior at higher

*P*

_{c}values could be done as with the random permeability treatments; various realizations for a given

*P*

_{c}level could be ranked, and single realizations corresponding to 10%, 50%, and 90% cumulative probability could be assessed dynamically. However, as the

*P*

_{c}level is decreased, the use of geometric mean or effective permeability methods becomes less reliable. As the percolation limit is approached (

*P*

_{c}= 0.5 for a 2D lattice), the average flow decreases to zero, and each realization can fluctuate wildly in predicted flow.

*P*

_{ c }= 0.55. The corresponding flow across this lattice is shown in Figure 11b. It is obvious that the steady state flow rate is 100 times smaller than the Base case level, as is the time to reach steady state. Figure 11c also illustrates the steady state flow velocity distribution across the lobule, showing the many “dead” regions of flow.

Here, *P* is the dimensionless Peclet number, expressed as convective flow velocity times length divided by the dispersion coefficient, while *t*_{R} is the dimensionless time, defined as time divided by the time required to get a dimensionless concentration *C*_{R} = 0.5. It should be emphasized that the Sauty solution is for one-dimensional flow while the real flow network here is two-dimensional with the possibility of internal cross flows. For our network, the length of interest is the diagonal distance of 0.106 cm, and the injected concentration is 1.8e-8 (mole fraction), implying a half concentration of 9.0e-9. Fitting the profile should allow estimates of the convective velocity and the effective dispersion for each case.

**Production profile fit to sauty**[3] **analytic convection-diffusion profile**

C | t | P | P | K | K | χ | |
---|---|---|---|---|---|---|---|

Reg_nodiff | 8.99e-9 | 0.2194 min | 7.656 | 0.1306 | 1.306e-3 | 1.306e-3 | 1.83e-9 |

Reg_diff | 9.00e-9 | 0.2228 min | 6.503 | 0.1538 | 1.538e-3 | 1.538e-3 | 1.90e-9 |

Perc_nodiff | 8.56e-9 | 8.7990 min | 4.069 | 0.2458 | 2.548e-3 | 6.370e-5 | 66.2e-9 |

Perc_diff | 9.02e-9 | 18.390 min | 3.609 | 0.2771 | 2.771e-3 | 6.928e-5 | 1.11e-9 |

### Zonation and effects on drug elimination

Two zonation cases were analyzed: the upstream (normal) case, where CYP expression is active primarily near the drug inlet zone, and the downstream (reversed) case, where CYP expression is active primarily near the drug outlet zone. Typically, CYP expression is expected to follow the latter case, but both are analyzed for comparison.

**Relative CYP levels in ideal zonation cases**

Case | Relative CYP Levels | (Upstream/Middle/Downstream) |
---|---|---|

Average zonation | 0.75/0.50/0.25 | (U/M/D) |

Extreme zonation | 1.00/0.50/0.00 | (U/M/D) |

Average reverse zonation | 0.25/0.50/0.75 | (U/M/D) |

Extreme reverse zonation | 0.00/0.50/1.00 | (U/M/D) |

### Fractal behaviour in the liver-an alternate perspective

Previously, to take into account organ heterogeneity and simulate enzyme kinetics in disordered media, lattice models have been introduced by several investigators. Berry [28] performed Monte Carlo simulations of a Michaelis-Menten reaction on a two-dimensional lattice with a varying density of obstacles to simulate the barriers to diffusion caused by biological membranes. He found that fractal kinetics resulted at high obstacle concentrations. Kosmidis et. Al. [29] performed Monte Carlo simulations of a Michaelis-Menten enzymatic reaction on a two-dimensional percolation lattice at criticality. They found that fractal kinetics emerged at large times.

Previously [30], we developed a network model of the liver consisting of a square lattice of vascular bonds connecting two types of sites that represent either sinusoids or hepatocytes. Random walkers with a drift velocity explored the lattice and were removed with a set probability from hepatocyte sites. To simulate different pathological states of the liver, random sinusoid or hepatocyte sites were removed. For a lattice with regular geometry, it was found that the number of walkers decayed according to an exponential relationship. For a percolation lattice with a fraction *p* of the bonds removed, the decay was found to be exponential for high trap concentrations but transitioned to a stretched exponential at low trap concentrations.

These models are all basically random walk models (emphasizing diffusive flow), and the lattices are abstract representations of the geometry of the space. Here we have created network models that incorporate realistic anatomical and physiological properties of the liver as well as emphasizing the consequent convective flow behavior of the well-perfused liver lobule. Here the fractal behavior of the liver lobule results from flow inhomogeneities.

## Conclusion

Together with our previous paper, this work presents a useful framework for analyzing the coupled and competing flow processes (convection/diffusion/reaction) that determine drug propagation and availability in a well-vascularized tissue such as a liver lobule. With our numerical approach, we have addressed both multidimensional spatial aspects as well as transient and steady state time behavior. In this paper, we have emphasized the impact of structural variability (including enzyme zonation) on non-uniform spatial distributions of drug and drug-metabolites occurring across the lobule. In particular, our non-dispersive models are shown to be more sensitive to such structural variations. We have also illustrated several techniques to quantify and analyze the role of such spatial variability.

Our network models including dispersive effects often correspond to the “well-stirred” compartment models [31], such that relatively uniform steady-state concentration levels occur throughout the lobule (if one ignores the smaller inlet mixing zone). Conversely, simulations on our network models without explicit dispersive mixing often correspond to modified “parallel tube” models [32], in which observed concentration profiles change along the length of the tubes (i.e. sinusoids). (Here our modified tube network structure allows cross-sinusoidal flow as well).

Aging and liver cirrhosis have been found to effect the transfer of drugs and metabolites from sinusoids to tissue, rather than directly effect the intrinsic metabolic process of hepatocytes [33, 34]. Recognizing that the dispersive mixing terms in our models allow easy permeation across the sinusoid-tissue interface, as well as improved intracellular transport, our calculated concentration profiles “with diffusion” can be viewed as representative of healthy liver behavior, while our non-diffusive profiles can be interpreted as representing aged or cirrhotic liver behavior.

A subsequent paper will expand this analysis to include sensitivities associated with variations in realistic lobule structure obtained from lobule images, which could even more realistically reflect extents of liver damage.

## Declarations

### Acknowledgements

J.A.T. acknowledges funding support for this project from NSERC. The Allard Foundation and the Alberta Advanced Education and Technology.

## Authors’ Affiliations

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