Sinusoid-Scale and Organ-Scale Homogeneous Models.

Models without spatial resolution are frequently used if a zonation of metabolic capability is unknown or does not play a role, e.g., if a compound is not metabolized at all by a given organ. Such models are structured as compartmental models without spatial resolution of organs [20, 21], or combine multiple organs in one generic compartment [22, 23], and therefore assume that a description corresponding to a well-stirred setting is appropriate.

Sinusoid-Scale Heterogeneous, Organ-Scale Homogeneous Models.

Any metabolic process along a sinusoid introduces a spatial concentration gradient, particularly prominent for compounds subject to significant first-pass elimination [24]. Additionally, metabolic capabilities of neighboring cells along a sinusoid can be distributed inhomogeneously along the sinusoid. This can be due to gene/enzyme expression [25, 26] or zonal presence of other cell types [25]. Examples in this category are carbohydrate metabolism [25], ammonia detoxification [27], and cytochrome metabolism for which two cases are considered in this article. Zonated metabolization can also be caused by pathological conditions appearing zonally, two examples are considered in this article: necrosis after CCl₄ intoxication [28] and different states of steatosis [29]. Processes involving sinusoid-scale heterogeneity can be modeled by using sequential well-stirred compartments [4, 5, 30, 31] for the liver. Other approaches build a zonation into lobule-scale models containing multiple sinusoids and their surrounding cells (cf. [32] modeling lobular perfusion and [33, 34] modeling lobular perfusion and PK) or into multiphase continuum models (cf. [35, 36] modeling lobular perfusion and [37] modeling lobular perfusion and PK).

Organ-Scale Heterogeneity.

Metabolization may additionally be heterogeneous at the organ scale, due to differences between sinusoids at different position in the organ. This is mainly due to pathological conditions varying at this length scale. Examples include steatosis [29, 38], fibrosis due to nonalcoholic steatohepatitis [39] or in case of hepatitis C [40], cirrhosis [41], granuloma [41], and carcinoma [42, 43]. Another reason can be heterogeneous concentration sources, e.g., due to intrahepatic injection via catheters [44] or targeted drug delivery [45]. Appropriate modeling techniques for the case of organ-scale heterogeneity without additional zonation are given by multiple parallel well-stirred compartments per organ [20, 31, 46, 47]. Organ-scale porous-medium-based multiphase continuum models [8] can be used as well. These are, however, of limited practical applicability if also a sinusoid-scale heterogeneity is present as they require high spatial resolution in this case. A combination of multiple parallel and sequential well-stirred compartments [31] per organ can be used in case of organ-scale heterogeneity combined with zonation. Also a parallel use of recent cell-based [34] or multiphase porous-medium-based [37] lobule-scale models is conceivable.

In any case, the structure of the model should be chosen depending on what is relevant for the application being considered. It should sufficiently detailed to capture relevant effects, but no more detailed to avoid unnecessary need for parameters and potentially error-prone model assumptions, and to keep computational workload at bay. This in particular implies that it can be useful to consider different degree of detail for representing different organs or organ groups [23].

We point out that this section not meant to be an exhaustive literature review of PK modeling. For more detailed reviews, we refer to [4, 5, 31].

Comparison to Related Approaches.

Our approach has the same goal of multiscale liver modeling as the two recent models [37] dealing with glucose metabolism and [34] considering ammonia detoxification. These approaches differ on a technical level and thus have slightly different foci. The model in [37] considers homogenized multiphase flow in a 2D model lobulus and determines flow velocities as part of the simulation. The model in [34] uses a 3D slice as part of a lobulus with sinusoids and individual hepatocytes. In addition to performing metabolization, the hepatocytes in [34] can grow, divide, and migrate, which allows modeling regeneration in more detail. Consequently, these approaches provide more detail at the lobular level than the simplified 1D sinusoids used in our approach. This, however, implies a higher computational workload for simulation when using more detailed lobular models. While it is certainly possible to extend these to including organ-scale heterogeneity, i.e., many different lobule models with different parameters, efficiency then becomes a challenge.

On the cellular level, [37] considers model of glucose metabolism obtained by model reduction from a detailed kinetic model, whereas [34] and our approach use equations of a given structure with parameters representing biochemical properties and/or parameters fitted to experimental data. It should, however, be possible to use any of these models or also more detailed intracellular models in either approach. Similarly, the extrahepatic models differ, but could in principle be exchanged between the different approaches. In [37], liver inflow concentration profiles are used as model input, corresponding to an isolated perfused liver [56, 57] model. In [34], a three-compartment body-scale recirculation model is used. We consider an isolated perfused liver in one of the example applications presented below, the other two applications involve a multi-compartment body-scale recirculation model representing each individual organ in the respective human or murine body. Given appropriate model parameters, this permits a more detailed investigation of the role of the different organs for the compound being investigated. Detailed recirculation models increase the computational workload of the simulation. This is not a major challenge for our purpose as the additional workload for the extrahepatic model does not scale with the degree of hepatic organ-scale heterogeneity.

Model Structure.

Our model consists of four separate building blocks, each representing a distinct physiological spatial scale as described above, see Fig 2. The connection from body via organ and sinusoids to the cells, and back, is provided via the blood flow. Bile flow and lymph flow, such as considered in [36], are not part of our model as the flowing volumes are negligible compared to the blood flow.

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Fig 2. Model Overview.

The figure shows the four spatial scales present in our multiscale pharmacokinetics simulation framework. The connection between the scales is given by the blood flow, indicated by arrows. The ‘representative sinusoids’ are the central building blocks in our framework. For each of these, we solve an advection-reaction problem. (HA: hepatic artery; HV: hepatic vein; PV: portal vein.)

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Recirculation through the organism (body scale) outside the liver as well as possible accumulation and metabolization in other organs are considered in effective form and in particular not at the same degree of detail as for the liver. For this purpose, an ODE system describing the whole-body PBPK model for the substance under consideration is used. These ODE systems are solved numerically using standard backwards differentiation formula (BDF) techniques [58]. The recirculation takes into account the temporal delay of the blood flowing through the organism.

On the organ scale, the heterogeneity of metabolization or pathology parameters determines the number and size of regions of different parameter values needed. To represent the connection between the organism and the sinusoids, the hepatic vascular systems distributing the blood throughout the organ are taken into account. The concentrations flowing from the body to the liver are hereby transported to each representative sinusoid with a given temporal delay. The outputs of the representative sinusoids with the respective temporal delays are summed up and form the concentrations flowing from the liver back to the body. In both vascular systems, only advection is considered. In particular, exchange between red blood cells and plasma and any metabolization processes are omitted in the blood vessels. As velocities are constant along each edge and over time, an advection simulation as in [8] is not necessary here and it is sufficient to consider appropriate delay times for different paths through the vascular system. If the actual geometrical configuration of regions of the same sinusoid-scale parameter patterns does not need to be reflected in the simulation, in particular if no exact temporal correspondence between different representative sinusoids is necessary, the vascular systems do not need to be considered for the simulation. In this case, only the relative contribution of each representative sinusoid for the whole organ and possibly the respective temporal flow delay are needed. Let us, however, point out that the simulation results can still be mapped to the actual geometric position for interpretation or visualization if needed. Depending on the organ-scale parameter heterogeneity, the liver model contains several representative sinusoids.

The representative sinusoids (sinusoidal scale) are used to model the actual exchange of compounds between the blood and the liver cells as well as the metabolization in the cells. For this purpose, the sinusoid is viewed as consisting of four subspaces: red blood cells (RBC), plasma, interstitium, and cells. The interstitium mainly represents the physiological space of Disse, the cells primarily refer to the hepatocytes. This means that the actual sinusoid only corresponds to the red blood cells and the plasma. However, to simplify terminology, we view the adjacent interstitium and cells as part of the representative sinusoid. The compound under consideration is exchanged between the subspaces and metabolized in the cellular subspace. This structure is chosen in analogy to the liver subcompartments in PBPK models [59] and was previously used in [8]. In addition, the compound concentration is subject to blood flow with constant flow velocity in the RBC and plasma subspaces. Exchange and metabolization at the cellular scale are described by cell-specific ODE systems with potentially different parameters to reflect sinusoid-scale heterogeneous patterns. Blood flow is represented by a one-dimensional advection PDE, so that we obtain an advection-reaction problem for each representative sinusoid. We will apply a combination of a Eulerian–Lagrangian Localized Adjoint Method (ELLAM) [60] and Runge–Kutta–Fehlberg 4th/5th order [61] time stepping to solve these numerically. Our approach considers sinusoids at their correct physiological length scale, even though one sinusoid in the model represents larger regions in the liver. In contrast to models using a series of well-stirred compartments without delay [20, 30, 47], modeling the blood flow by advection inherently represents the physiological delay of flowing through a sinusoid with a given velocity. This provides more accurate temporal resolution at sub-second or second time scales, which is a relevant time scale for some applications, e.g., [62].

Body-Scale Recirculation Model.

The body-scale PBPK model [59] describes the blood flow between the organs, and thus the transport of a compound through the body, as well as the exchange of compounds between subcompartments of the organs and the metabolization. These processes are described by a non-linear ODE system in several hundred variables.

On the body scale, it is necessary to consider a temporal delay of the recirculation. In certain cases, more than one initial pass of an injected compound can be observed experimentally, see, e.g., [62] for measurements of antipyrine in dogs or [63] for an MRI contrast agent in humans. An ODE model not taking into account the temporal delay of the blood flow between and through the different organs is generally not capable of reproducing second and further passes. This is not an issue for observations on the time scale of hours or days, but should not be ignored when simulating processes on the time scale of seconds to minutes.

In our model, we integrate this temporal delay of the recirculation as a delay τ_body for blood flowing between the lung and the systemic arterial blood pool, the latter supplying all organs in the model. This delay thus affects the compound mass flows in the red blood cells and the plasma from the lung to the arterial blood pool. This location in the PBPK model was chosen because all recirculating blood passes through the lung. The body delay τ_body was computed as follows: First, let τ_{rec total} denote the total recirculation time through the body obtained as total blood volume in the organism divided by the blood flow rate through the lung [59], resulting in (1)

We refer to the results section below for a discussion about how these values differ from the measurements in [63]. The recirculation time [64] and time between peaks can be expected to differ due to intra-organ retention of compounds. From the values τ_{rec total}, two transit times explained below are subtracted to obtain τ_body, namely the sinusoidal transit time, see Eq 6, and the transit time for the hepatic vascular systems, if present in the model, see Eqs 7 and 8.

Clearly, this recirculation delay model is not mechanistic, highly simplified, and does not take into account flow delays within or between the individual organs. In a similar manner as for the whole body, we could use specific transit times for each organ, namely dividing the respective blood volume by the respective flow rate. This would make the implementation more technical and increase computational costs for the simulations, but it is not clear to what extent the overall model accuracy would benefit. As this article focuses on the liver, we will leave the recirculation delay model open for future model refinement.

The ODE systems describing recirculation turned out to be very stiff. For solving stiff ODE systems, various numerical methods are available [65, 66]. Based on a brief performance evaluation, we chose a scheme using a BDF scheme available as part of CVODE solvers [58], based on [67], in the SUNDIALS library [68] to solve the recirculation ODE systems. This part of the simulation turned out not to be very time critical, so passing values between our simulation and an external ODE solver is no performance bottleneck.

Physiologically Based Pharmacokinetics Models Used for the Cellular Scale.

The recirculation ODE systems are expressed in terms of the molar amounts of the compound in different organs. For introducing spatial resolution, we use molar concentrations in the other building blocks of our multiscale model and as the interface to the recirculation model. Throughout this article, concentration is always meant as molar concentration in units mM = mol m⁻³, unless stated otherwise.

To describe the connection between organism and organ, we generally distinguish between concentrations c_f flowing in the blood and concentrations c_s in the surrounding tissue. Moreover, let denote the liver inflow concentration (‘body to liver’). We can then describe well-stirred flow, inter-subcompartmental exchange and intracellular metabolization in terms of compound concentrations as (2)where we omitted the dependency on time for a more concise notation. The flow rate α > 0 indicates which fraction of the blood volume in the organ is exchanged per time unit. The function E describes the exchange between the subspaces in the representative sinusoid, and m the metabolization. For multiple compounds, the exchange terms E typically apply separately for each compound. In contrast, the metabolization term m can model metabolic conversion of compounds to the respective metabolites and thus involve concentrations of multiple compounds.

Throughout this section, we will consider a model structure consisting of red blood cells and blood plasma as subcompartments subject to flow. Interstitium and cells will be considered as stationary subcompartments. The corresponding molar concentrations of the compound being considered are denoted by c_rbc, c_pls, c_int, and c_cel, respectively. In this setting, the cellular subcompartment summarizes the contribution of all cells to the exchange and metabolization of the compound. In the liver, the cellular subcompartment mainly represents hepatocytes, spatial patterns of which we focus on throughout this article.

This model structure applies to two of the applications below, midazolam and caffeine, and serves as the example for the presentation of the methods. If only passive, gradient-driven exchange of compounds takes place, we can write E as the multiplication by a 4 × 4 matrix of the form (3) as used in own earlier work [8]. Here, φ_{{rbc, pls, int, cell}} are the volume fractions specified below in Eq 5, κ_rbc,pls, κ_pls,int, κ_int, and κ_cell are dimensionless partition coefficients describing the equilibrium state of molar concentrations at which the respective individual exchanges vanish, P_rbc,pls, P_pls,int, and P_int,cell are the local effective permeabilities [s⁻¹] between the different subspaces of the representative sinusoid. The metabolization m is, in our case, described by Michaelis–Menten kinetics [69] of the form [8] (4) with parameters describing the maximum metabolization rate [s⁻¹] and describing the molar concentration [mM] at which half is attained. In this form, only the removal of a single compound from our system is represented. More complex functions E and m are required if several compounds are considered, as demonstrated in the insulin example in the results section.

The volume fractions φ_{{rbc,pls,int,cell}} are specific for different species, in our case the volume fractions for humans [59] are (5a) where as volume fractions for mice [59] are (5b) The exchange and metabolization parameters P_⋆, κ_⋆, , and depend on compound and species and are given below for our example applications.

The typical transit time τ_typ for the sinusoidal scale is obtained as the ratio of the sinusoidal volume fraction φ_sin = φ_rbc + φ_pls over the total liver blood flow, both values taken from [59]. Thus, τ_typ satisfies the relation α = 1/τ_typ for the flow rate α from Eq 2, its value is (6) For a discussion of the precise form of the flow, we refer to Eq 9 and its description. We are aware that this transit time is not in good agreement with experimental data for sinusoidal flow velocities for humans [70] or rats [71, 72], using lobular diameters [1, 7] as a lower bound on sinusoidal length. Due to the large variability of the experimental velocity measurements, we decided using the values in Eq 6 for τ_typ consistent with the underlying whole-body models [59]. These values can, however, easily be exchanged if a better estimate is available.

Blood Flow Model.

For simplicity, we only consider a single supplying vascular system denoted by SV, comprising the physiologically present HA and PV, and already compute the mixed concentration of arterial and portovenous blood at the liver inflow. This is clearly a simplification and would need to be modified if we wanted to consider locally varying relative contributions of the two blood types [73, 74]—for which, however, we would need appropriate data. As another simplification, we assume constant flow velocities across vascular cross sections satisfying Poiseuille’s law [75], which reduces our flow model to flow on a branching one-dimensional domain. The Fåhræus–Lindqvist effect [76], the decrease of the effective viscosity of blood in blood vessels of radius ≤ 150 μm, is accounted for in the flow velocities. As we do not focus on the local flow in the vascular structures, we do not consider the precise velocity profile across cross sections or near bifurcations. Moreover, we assume that no metabolization and no exchange between plasma and red blood cells takes place in the vascular systems. This simplifies the model so that we only need to take into account advection by blood flow for which explicit formulas can be given. For this purpose, concentrations in the plasma and the red blood cells as well for all compounds present in the blood flow can be treated in the same way as explained next, and we omit the respective indices for notational convenience.

Model: For a given point x in the supplying vascular system, we can compute a temporal delay τ^SV(x) between the inflow and x, so the concentration in the SV at position x and at time t can easily be computed from the liver inflow concentration as (7) This allows computing the inflow concentrations for all representative sinusoids by substituting the respective connection point for x.

For a point x in the draining vascular system, this is slightly more technical since the flow from multiple representative sinusoids contributes to the concentration at x, each with potentially different delay. Let R(x) denote all these representative sinusoids and τ^HV(r, x) the temporal delays for all r ∈ R(x). Moreover, let w_{r, x} denote the relative contributions of the flows to the flow at x, determined from flow velocities and cross section areas of the respective edges. Then the concentration in the HV at position x and at time t is given as a weighted, delayed sum of outflow concentrations of representative sinusoids r, namely (8) to obtain the concentration at point x and time t. The liver outflow concentration c_BL (‘liver to body’) is then easily obtained by evaluating Eq 8 at the root of the HV.

Implementation: The evaluation of Eqs 7 and 8 does not require sophisticated simulation techniques. The approach merely makes it necessary to store concentration history at the transfer points, but at no other locations. Thus, a spatial discretization of the concentrations in the vascular systems is not necessary. Storage is only needed for a limited period of time, long enough so that appropriate interpolation is possible for the maximal transition time, which limits memory consumption. In contrast, spatial concentration profiles for the vascular trees as well as data for advection simulation is not necessary at all, saving memory and computation time compared to a full-featured advection simulation for the vascular systems as performed in [8]. Discrete time points are given by the outer simulation loop and may require piecewise first-order polynomial interpolation of concentrations in time.

Obtaining Realistic Geometric Models of Organ and Vasculature.

A realistic 3D geometric model of a human liver and its vascular structures was obtained by applying the same workflow described in [77] for in vivo μCT scans of mouse livers. In summary, an abdominal in vivo MRI scan of a healthy 31-year old male Caucasian was used to segment the liver and its vascular structures [78]. The latter were subsequently skeletonized and converted to a graph representation using a semi-automatic procedure [78]. The vascular graphs were simplified to strictly bifurcative trees with cylindrical edges as described in [79]. They were refined algorithmically [80] to the desired degree of detail as described in [79]. For this purpose, the same set of 10 000 end points for the supplying and draining vascular systems were used. Each of these points represents groups of about 5³ physical lobuli, leading to a total of 1.25 million lobuli, a realistic number [1]. In each group, we assume the same metabolic properties for all sinusoids. The underlying organ mask an the resulting vascular system are visualized in Fig 3 and provided as S1 Dataset.

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Fig 3. Human Liver Vascular Dataset.

The image shows a visualization of a human liver shape including algorithmically refined vascular structures. The supplying vascular system, comprising portal vein and hepatic artery, is shown in red, the draining vascular system (hepatic vein) in blue.

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For a fully lobular resolution, one would have to take into account how the blood supplied by one portal field is distributed to multiple surrounding central veins, an effect we neglect in the approach here. We could, however, easily incorporate it for applications where such a more detailed draining pattern is of importance.

Our applications involving mouse models presented here did not require 3D geometric models as these examples did not include organ-scale heterogeneity. Such geometric models are available as supporting information to [8] and [77] and could easily be included in the simulations if organ-scale heterogeneity needs to be included.

Sinusoidal Blood Flow and Pharmacokinetics Model.

For the sinusoidal scale, we again start with the general model form as in Eq 2 distinguishing flowing and stationary concentration. The 1D advection-reaction for one representative sinusoid can then be expressed as (9) where E and m, as above, denote the exchange of concentrations between the subspaces and the metabolization, respectively, and v is the one-dimensional flow velocity. Boundary values at the inflow are those provided by the supplying vascular system, outflowing values at the other end of the sinusoid are computed and passed to the draining vascular system. In the current model version, we in particular neglect diffusion outside the cells, in particular in the space of Disse, as well as exchange between neighboring cells due to gap junctions [81].

For the reaction part of Eq 9, multiple compounds may play a role. On the one hand, one compound may change the exchange or metabolization behavior for another compound if the underlying bio-chemical processes involve such an interdependence. On the other hand, metabolization itself can produce metabolites (i.e., other compounds) from compounds.

Clearly, the blood flow through sinusoids is probably better described as an intermittent, corpuscular flow involving a non-Newtonian fluid through sinusoids of complex geometry that is not much larger than the red blood cells. We refer to Video S1 of [82] for an illustration of this blood flow. However, for a representation of the average flow behavior, as intended here, we believe this strong simplification to constant flow velocity to be an appropriate description.

Discretization and Implementation.

As in [8], we discretize the advection and reaction parts of Eq 9 separately. This avoids a rather technical implementation of a numerical scheme capable of treating both processes simultaneously, even though this is possible even for a nonlinear reaction term [83]. The simulation moreover does not require a small time step appropriate for both processes, but needs appropriate time steps for the two processes separately and an appropriate synchronization strategy.

We do not resolve the internal spatial structure of hepatocytes and thus assume them to be well-stirred. It is thus a natural choice to use a resolution of one grid point per hepatocyte for the spatial discretization of the representative sinusoids. This is also the spatial resolution at which sinusoid-scale parameter heterogeneity needs to be given.

Advection: While pure 1D advection problems with constant velocity can be solved analytically, this is no longer possible if a reaction term is present as well, so it needs to be addressed numerically. The simplest implementation of time stepping for the advection part of Eq 9 could be obtained by choosing the time step such that the velocity is one grid cell per time step. This, however, would not allow joint time steps if velocities in different representative sinusoids differ, and might be too large a time step for synchronizing with fast reactions, so we need a numerical advection scheme that can deal with arbitrary velocities. An ideal such discretization scheme would conserve mass and neither introduce numerical diffusion nor create spurious oscillations [84]. However, state-of-the-art methods do not simultaneously satisfy all three properties, so a compromise is needed. For our purposes, we particularly focus on mass conservation and avoiding spurious oscillations, which might result in overshoots and negative concentrations, so we need to accept numerical diffusion effects.

We use a 1D Eulerian-Lagrangian Localized Adjoint Method (ELLAM) [60], going back to [85] with more details on mathematical analysis in [86]. This method was preferred over other discretization schemes for advection [87] as it prevents numerical artifacts [88]. To prevent artificial oscillations, mass lumping as suggested in [89, 90] was used. While ELLAM could, in principle, treat more than advection simultaneously [83, 91, 92], also for non-linear reactions [93], we treat flow, exchange/metabolization, and recirculation separately by appropriate methods. The scheme was implemented in custom C++ code based on own earlier work, Section 1 in the supporting information Text S1 of [8] and [94]. In summary, the ELLAM scheme for advection amounts to solving a linear system of equations to update the concentrations for the compounds present in the blood flow in each time step.

Reaction For the reaction part of Eq 9, we observed no issues due to stiffness of the ODE systems. So we chose a standard Runge-Kutta-Fehlberg 4th/5th order (RKF45) [61] scheme that automatically adapts the time step size. This method is recommended as a ‘good general-purpose integrator’ in the GNU Scientific Library [95]. The RKF45 performed well in this case, so we do not need to apply more sophisticated methods as recommended in [96, 97]. We used a custom C++ implementation of an RKF45 scheme from [8] to avoid data exchange with external libraries as these ODE solution steps are rather time-critical in the simulation.

One-Dimensional Representation of Sinusoids.

To account for increasing sinusoid diameter from portal fields to central veins in a 1D representation of the blood flow in 3D lobuli, we need to find an appropriate representation. As variations of the processes in longitudinal direction of the lobulus are of minor importance, we can restrict our view to 2D cross sections of lobuli. Mathematically, the goal of this subsection is to describe how parameters of the PBPK liver submodel need to be adapted. This adaption will involve scaling some of the parameters and replacing the constant volume fractions φ_i, i ∈ {rbc, pls, int, cell}, by volume fractions ψ_i(λ) depending on the position λ along the sinusoid.

For this purpose, our approach is to consider one physiological sinusoid starting from the portal field. As it combines with other sinusoids towards the central vein, the blood flow from the one, portally starting, sinusoid is only a certain fraction between 0 and 1 of the blood flow through the centrally ending sinusoid. For the lack of more detailed data, we assume that the flow velocity does not change along the physiological sinusoids. Moreover, we assume that the physiological sinusoids are surrounded by an interstitial and a cellular layer of constant width, independent of the sinusoid radius.

Radii of periportal and pericentral sinusoids in mice are r_sin,pp = 4.4 μm and r_sin,pc = 6.85 μm [98], respectively. This range is similar to sinusoidal diameters of 7 and 15 μm reported in [1], so we will use the following formulas for both humans and mice. Assuming a linear increase of the cross section area, the sinusoidal radius can be computed as (10a) (10b) where λ indicates the position along the sinusoid in units of sinusoid length, i.e., λ = 0 at the periportal end and λ = 1 at the pericentral end.

For the thickness of the interstitium and the representative cellular layer, denoted by w_int and w_cell, we assume that the volume fractions for mice from Eq 5b correspond to the average sinusoidal cross-section area, i.e., to the radius r_{sin, avg} = r_sin(0.5) = 5.757 μm. We then compute (11a) (11b) Note that these values refer to volumes representative for the physiological interstitium and cells and are not meant as their precise actual sizes, in particular for the cells. The radii and these thicknesses are illustrated in Fig 4.

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Fig 4. Sketch for the 1D Representation of Sinusoids.

The volume rendering in the middle illustrates our assumption of multiple periportally starting sinusoids contributing to a thicker pericentrally terminating sinusoid. The width of the interstitial and cellular layer is assumed to remain constant along the sinusoid. The sinusoidal cross-section sketches on the left and right show that this also has an effect on the respective surface areas, which has an influence on the respective effective permeabilities.

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Using Eqs 10 and 11, the variation of the volume fractions along the representative sinusoid can be expressed as (12a) (12b) (12c) (12d) (12e)

The contribution of the periportally starting sinusoid to the representative sinusoid at position λ is given by the ratio of the cross section areas, (13) We refer to Fig 4 for a sketch of this interpretation of the contributions. The inverse of this contribution is needed when determining the represented volume by one section of the representative sinusoid. In addition, permeabilities relating to surface areas need to be scaled by a factor derived from ρ(λ), taking into account the circumference of the representative sinusoid at position λ and the fraction of represented there, (14) This factor is multiplied by the permeability which we assume to be given for the middle of the sinusoid.

These adaptions imply a modification of the exchange term Eq 3 to (15) whereas the metabolization term Eq 4 is unaffected.

One-Dimensional Representation of Sinusoid-Scale Heterogeneity.

A zonation of metabolic capabilities is typically determined by staining enzymes in histological slices. These images are evaluated by measuring the total cross-section area of different lobuli and the corresponding areas of different staining result. For an enzyme near, e.g., the central veins, the analysis thus allows defining a pericentral area ratio A_pc/A_∅, whereas we need a pericentral length ratio l_pc/l_{rep. sin.} for the representative sinusoid models. We view lobuli as cylinders around their respective central vein, which is certainly a simplification from 3D to 2D, but a suitable approximation of the actual polygonal shape of lobuli [99]). Then, we can easily convert from central circular area to central length via (16) This conversion also applies when considering the zonation of pathological changes as these can be determined similarly, see, e.g., Fig 1, or, more generally, any sinusoid-scale parameter heterogeneity.

Let us point out that there is no unique ‘periportal’ or ‘pericentral’ zone, they can in particular not necessarily be identified with zones 1 or 3 in the notation of [29, 100]. The size of such zones depends on the process considered. In the caffeine example considered in the results section, there are two pericentral regions: one of fixed size where the enzyme Cyp1a2 is expressed, another one of variable size which is affected by necrosis.

From Well-Stirred to Representative Sinusoid Models.

Frequently, the PK parameters are fitted for a well-stirred organ-scale model rather than a spatially resolved representative sinusoid model already taking into account the adaptions described by Eqs 15 and 16. In this case, they cannot be used immediately for the cellular scale in representative sinusoids. Our approach is to rescale parameters dominantly causing the differences such that the liver outflow concentrations as model output from the organ-scale homogeneous representative sinusoid model are close to outflow concentrations from the well-stirred organ-scale model. No exact match can be expected here, the sinusoid-scale model is designed to represent the transit time and also exhibits some mixing and other effects due to the advection simulation.

This adaption should clearly be independent of the inflow profile considered. If we were dealing with a linear time-invariant (LTI) system, we could simply determine the impulse responses, i.e., the model output when using a Dirac δ impulse as input, of the two models and use these for comparison. The ODE systems considered here, however, are not necessarily linear, so standard methodology for LTI systems (see, e.g., Chapter 2 in [101]) cannot be applied. Instead, we make use of the following pragmatic approach. Assuming that the system behaves approximately linearly for our ranges of input, we consider an inflow profile roughly approximating a δ impulse. We then fit a scaling factor for the dominating coefficients in the ODE system so that an appropriate deviation is minimized. We will specify this in the results section for our insulin example. This optimization is performed by iterative interval nesting until a given tolerance is reached.

Simplified Steatosis Model.

Steatosis is a common liver disease in humans, it is often caused by alcohol abuse, diabetes, protein malnutrition, obesity or the consequence of other pathological conditions [102]. In steatotic livers, lipids accumulate in the cellular subspace [103]. Most often, this happens pericentrally, but the lipid deposition can also occur in the periportal zone, see [104] and the references therein. In addition, an organ-scale heterogeneity between different sinusoids can be observed [105].

Model Perturbation: We represent the effect of steatosis as a change in the equilibrium between cells and interstitium via the respective partition coefficient κ_{cell, healthy} = κ_cell in the PBPK equation Eq 3 in the same manner as presented in [8]: Let Δs be the lipid accumulation due to steatosis, then κ_cell(Δs) is computed from κ_{cell, healthy} via (17) where log P is the lipophilicity of the compound considered. Clearly, this is a strongly simplified representation of steatosis. It explicitly ignores any other effects of steatosis on the PK parameters such as on metabolic capabilities of the cells [106] and intra-subcompartmental permeabilities [107], any changes in microcirculation [108, 109], organ size (as shown in Table 6 in [110]), and any other effects of steatosis on the organ and the whole organism.

Synthetic Steatosis Data: The steatosis data used here are synthetic datasets based on experimental observations from the literature. Besides the healthy case (no steatosis), we consider three types of zonation: predominantly periportal (similar to ‘zone 1’ in the terminology of Fig 1 in [29]), predominantly pericentral (‘zone 3’), and non-zonal (‘panacinar’ or ‘azonal’). For this purpose, we assume a total fat accumulation of 9.2% of the liver volume, corresponding to stage 2 steatosis as observed in [111]. We assign pseudo-random [112] steatosis values Δs to each zone, uniformly distributed in the interval 0.092 ⋅ (ρ(λ))⁻¹ ⋅ ζ_z(λ) ⋅ [(1 − 0.69), (1 + 0.69)] where 0.69 is the coefficient of variation reported in [111], the factor ρ(λ) from Eq 14 in the denominator cancels when computing the total lipid content, and ζ_z(λ) controls the zonation via (18) Examples for these zonated states of steatosis are visualized in Fig 5.

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Fig 5. Steatosis Heterogeneity.

The left images show four examples of different synthetic zonated patterns of steatosis in humans, visualized on a color scale from violet to yellow, corresponding to 0% to 27.5% steatotic lipid accumulation. The three steatotic states correspond to the same total lipid accumulation of 9.2%. The volume rendering on the right visualizes the organ-scale gradient ζ_h with a difference of 7.96% lateral direction for a human liver.

https://doi.org/10.1371/journal.pone.0133653.g005

For introducing an organ-scale heterogeneity, we multiply the ranges above by an additional factor (19) where x_min and x_max are the smallest and largest x coordinate of the organ. This factor leads to a gradient in the steatosis profile with a difference of 2 ⋅ 3.98% between the leftmost and rightmost point, given by the smallest and largest x coordinate, respectively. The value 3.98% is the mean maximum difference between left and right liver as reported in [105] for type-2 diabetic patients, where lower steatosis values are present on the left. A volume rendering in Fig 5 shows this macroscopic lateral steatosis gradient.

Simplified Regeneration Model.

A frequently used experimental protocol to study toxic liver damage is the administration of CCl₄ in animals [28]. It induces a necrotic pericentral zone [34], similar to effects of acetaminophen overdoses [113], a frequent cause for acute liver failure in humans [114].

Model Perturbation: In a similar manner as in [8], we represent the effect of the necrotic area by replacing the cellular volume by interstitial space, changing the volume fractions in the PK equation Eq 3. As the actual metabolization only takes place in the cellular volume, this effectively prevents metabolization in the necrotic zone. Let us point out that this is again a strongly simplified model. It explicitly ignores any interaction of the CCl₄ and its metabolites with the compound, any effects of the fragments of the dead cells, any differences in the metabolic capabilities of the regenerating cells to those of the cells natively at the respective position, and any additional influence of the CCl₄ administration on the whole organism. For a more detailed model of the interplay of regeneration and metabolism we refer to [115].

Synthetic Necrosis Data: A CCl₄ dose of 1.6 mg g⁻¹ body weight leads to necrotic areas reported in Fig 1S (manually derived data) in the supporting material to [34], with piecewise affine-linear interpolation in time and the assumption of full regeneration after 7 days. We use this data as the basis for defining a time-dependent necrotic zone in our representative sinusoids, converting, as explained in Eq 16, from pericentral area as visible in the histological images to pericentral length in the representative sinusoid model. The evolution of the necrotic area as input for our simulations is shown in Fig 6. A necrosis value between 0% and 100% is proportionally mapped to a change of volume fractions as described above.

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Fig 6. Necrosis and Regeneration.

The plot shows how the spatial extent of the necrotic region evolves according to our model of the effect of CCl₄ intoxication along a representative sinusoid of a mouse liver. The representative hepatocytes are separated by vertical black lines in this plot, A color range from white to red indicates zero to full necrotic damage of the respective representative hepatocyte. Necrosis develops during the first day, until a maximally necrotic state is attained. Subsequent regeneration starting on the second day leads to a shrinkage of the necrotic region until the end of day 7.

https://doi.org/10.1371/journal.pone.0133653.g006

Pharmacokinetics Model.

Based on Fig 1 in [119], we estimated the size of the pericentral region of CYP3A4 expression to be be slightly over 50% of the lobular cross-section area. By Eq 16, this pericentral area corresponds to 14 of 19 hepatocytes performing the metabolization.

We used the available software [59] to establish a human PBPK model for midazolam based on physiochemical parameters from [120] and the Human Metabolome Database [121] as well as to determine the partition coefficients κ and the permeabilities P listed in Table 2. The parameters , , and log P were fitted to data manually derived from Fig 2 in [122].

This model was parametrized for a 19-zonal liver model matching our representative sinusoid model for humans, the parameters could thus be used without further adaption.

Simulation Results: Influence of the Body Delay.

First, we represented the healthy liver by a single representative sinusoid and used an intravenous bolus injection of 11.42 mg as in the experiments underlying the data used above for fitting. Here, we determined the simulated midazolam concentrations in the plasma entering the liver via the portal vein, comparing a recirculation model as described above to one without delay during the recirculation.

Results for this simulation in Fig 7 show that, without recirculation delay, our model failed to predict reoccurring peaks of the second and third pass. With nonzero recirculation delay, these peaks were captured with a physiologically reasonable temporal delay. In our case, we observed a temporal spacing of about 75 seconds between the first two peaks. This is clearly longer than a time span of approximately 23 s which can be estimated from Fig 4 in [63]. The literature data is for a different compound with potentially different PK characteristics throughout the body, which could account for part of the difference. Mainly, the difference is due to our recirculation model not yet accurately representing the flow delays by different organs and for different paths through the body. However, a more accurate recirculation delay is beyond the scope of this article where we focus on a detailed liver model and consider the simplified recirculation delay sufficient for our purposes. In particular, we did not simply pick our recirculation delay τ_{rec total} from Eq 1 such that the time span between two peaks matches the literature values. Such an approach would merely hide the inaccuracy described and would probably lead to incorrect amplitude of the second peak because flow delays by slower paths through the body would not be represented correctly and the circulation speed of the entire mass would be overestimated.

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Fig 7. Influence of the Body Delay.

For an intravenous bolus injection in our human whole-body model with a liver described by a single representative sinusoid, the plot shows the simulated midazolam concentration in the blood plasma at the liver inflow. Setting the temporal delay of the recirculation to zero (dashed line) shows that the delay in the model is indispensable for a correct prediction of recurring peaks for a second and third pass, indicated by circled numbers in the plot.

https://doi.org/10.1371/journal.pone.0133653.g007

Results for the Zonated, Organ-Scale Homogeneous Model.

We next assumed infusion of the same dose of midazolam into the blood plasma flowing through the portal vein within 5 seconds. In this case, we determined the simulated spatio-temporally resolved concentration profiles along the sinusoid in the healthy, predominantly periportal, predominantly pericentral, and non-zonal steatotic cases.

In Fig 8, we can clearly observe the influence of the different steatosis patterns on the spatio-temporal midazolam distribution. As expected from Eq 17, a higher accumulation of midazolam was predicted for the steatotic regions. In the plots, the apparent velocity of the peak is particularly noteworthy with an apparent transit time of about 200 seconds for the healthy state, and slower apparent velocity for the steatotic states. We emphasize that this time scale is different from the blood flow velocity, for which the transit time is 13.6 s as given in Eq 6, i.e., significantly shorter and according to our assumptions in particular independent of the steatosis.

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Fig 8. Spatio-Temporal Midazolam Concentration Profiles.

The surface plots show the spatio-temporal evolution of the midazolam concentrations in the blood plasma and the hepatocytes along representative sinusoids assuming an infusion of duration 5 seconds into the portal vein, comparing the healthy reference case with three different steatotic cases with the same total amount of lipid accumulation. While the height in the graph covers the total concentration ranges, the color highlights differences in a lower range of concentrations, emphasizing the differences between the four cases. In addition, the steatosis patterns along the sinusoids are shown below the cellular concentrations. Differences in the transit time of the peak are due to different extent of storage and release of the midazolam due to the steatotic lipid accumulations. This should not be mistaken for the blood flow transit time, which is 13.6 s for all four cases and thus much shorter than the peak transit time.

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The simulation results here clearly depend on how the randomized steatosis patterns were chosen, i.e., on the seed value of the pseudo-random number generator. While it would be reasonable to investigate how sensitive the results are to the seed, and more generally to all the model parameters, such a sensitivity analysis is beyond the scope of the present exemplary use case.

Results for the Zonated, Organ-Scale Heterogeneous Model.

Moreover, we considered a liver model using 10 000 representative sinusoids with the vasculature as shown in Fig 3. In this case, each representative sinusoid had its individual steatosis pattern (predominantly periportal predominantly pericentral, non-zonal), additionally the lateral organ-scale gradient as given in Eq 19 was present. Due to the large number of sinusoids, the influence of the individual patterns, as discussed above, averaged out. For these simulations, we again assumed an infusion into the blood plasma flowing through the portal vein within 5 seconds. We then determined the simulated midazolam concentrations in the systemic venous blood plasma. We again compared healthy and the three steatotic cases, where we now additionally considered an organ-scale heterogeneity of the steatosis according to Eq 19.

In Fig 9, differences between the three steatotic states during the first minutes can be observed. This shows a key property of our spatially resolved model which is capable of distinguishing different spatial patterns with the same total steatotic lipid accumulation. In these liver outflow curves, two concentration maxima can be observed. The first peak corresponds to the first pass after the blood flow transit time of the organ, which is the same for all four cases. The second peak corresponds to the different apparent peak velocities for the four cases observed in Fig 8. This shows that the effect of different apparent peak velocities is also present in the superposition of 10 000 different steatosis patterns of the four types (healthy, predominantly periportal, predominantly pericentral, non-zonal) and in particular not an artifact of the specific single patterns used for the simulations shown in Fig 8.

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Fig 9. Organ-Scale Midazolam Concentration Profiles.

For an assumed infusion of midazolam into the portal vein within 5 seconds, and our human whole-body model with a liver described by 10 000 representative sinusoids, the plot shows the midazolam concentrations in the blood plasma flowing out of the liver. In these model predictions, the different steatotic cases lead to differences during the first minutes. Afterwards, only the healthy state leads to concentrations distinct from the steatotic cases.

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For longer simulations, only the difference between the healthy and steatotic cases can be observed. However, the quantitative difference mainly depends on the extent of steatosis, which is not the focus here. Still, the relatively small difference between the healthy and the steatotic states fits to the clinical finding in [123] that morbid obesity does not lead to changes in Midazolam clearance. Despite the qualitative similarity, this comparison is clearly limited. While obesity and steatosis are associated, the latter was not reported for the individuals considered in [123]. Moreover, our model did not take into account any aspects of obesity besides the hepatic steatosis.

Pharmacokinetics Model.

For caffeine, we used the PBPK model from [19] determined for an intravenous bolus injection of 5 mg caffeine per kg body weight. Caffeine is metabolized by Cyp1a2 [127], an enzyme only expressed in the pericentral region [12], which we estimated to be about 58% of the lobular cross-section area based on Fig 3 in [128]. By Eq 16, this area of enzyme expression was translated to 9 of 12 pericentral hepatocytes performing the metabolization, the 3 periportal hepatocytes do not. We hence used the enzyme activity parameters from the PBPK model assigned above for the 9 pericentral hepatocytes and set the term m from Eq 2 to zero for the 3 periportal hepatocytes. The intra-subcompartmental exchange took place in all 12 representative hepatocytes in the model, i.e., E(λ) from Eq 15 is present for all of them. Despite a direct zonated translation to the hepatocytes, an adaption from well-stirred to representative sinusoid models as described above turned out not to be necessary for this model.

Regeneration after CCl₄ intoxication was built into the model as described above by synthetic necrosis data. The necrotic region was also located pericentrally and at its maximum size covered most of the metabolizing region. Besides those two zonated effects, no additional organ-scale heterogeneity was present in the model, it is thus sufficient to represent the liver by one representative sinusoid. This model included recirculation with a recirculation delay of τ_body = 15.5 seconds as explained above.

Simulation Results.

In our simulations, we considered an intravenous bolus injection of 5 mg caffeine per kg body weight, as in the underlying experimental data of the model. The injection was assumed to take place at different time points after inducing CCl₄ damage, from 0 days, when necrosis starts to develop, to 7 days, when the fully recovered state according to our damage model is attained. After the injection, we simulated one day of blood flow and caffeine clearance in the model, involving spatially and temporally resolved caffeine concentrations for the representative sinusoid and temporally resolved concentrations for the other compartments in our body-scale model. From these values, we computed the total amount of caffeine present in the body and in particular the half-life t_1/2, i.e., the time it took for half of the initial caffeine to be cleared from the body, cf. elimination half-life in [129]. While this half-life is easy to compute in the model by summing up caffeine amounts in all compartments, it is not easily accessible experimentally. Instead, many studies determine the half-life of the compound in the systemic blood plasma, leading to slightly different results.

The caffeine concentrations in the plasma in the venous blood pool show a second and weak third peak after about 30 and 60 seconds, see the small line plot in Fig 10. Later peaks are of diminishing amplitude and not visible in the curves. These recurring peaks are superpositions of the delays due to the liver model and the recirculation as well as an exchange between subcompartments of all organs, the spatially resolved liver and the well-stirred remaining organs. The peaks thus take longer than the total recirculation time for mice of τ_{rec total} = 19.72 s, see the discussion of the recirculation time scale in the human midazolam case above. This effect qualitatively occured regardless of the necrotic state of the model. On the longer time scale of one day, the necrotic state of the model had a major influence, see large line plot in Fig 10: for injection at day 0, the necrosis developed and slowed down the decrease of caffeine concentration in the plasma in the venous blood pool. For injection at day 1, the slowest concentration decrease can be observed. Regeneration lead to an increasingly larger metabolizing region for the injections at days 2 to 7, leading to faster decrease of the caffeine concentrations, until the fully regenerated state was reached for the injection at day 7.

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Fig 10. Caffeine Clearance After CCl₄-Induced Necrosis.

The plot shows the caffeine concentrations in the plasma in the venous blood pool for 24 hours after intravenous bolus injection at different time points during the necrosis development and regeneration process induced by CCl4. The influence of the necrotic damage occurred during day 0, leading to slowest decrease of caffeine concentration in the venous blood for the injection at day 1. Regeneration during days 1 to 7, according to our necrosis model from Fig 6, lead to increasingly faster decrease of the caffeine concentrations for the later injections, until the fully regenerated state was reached for the injection at day 7. The inset shows the second pass after the first recirculation cycle, in this case after 30.2 s. The table lists the half-life of caffeine in the organism, i.e., the first decrease by 50%, if it occurs within one day after the administration. These numbers confirm the observations from the plot which only involves the venous blood pool.

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Correspondingly, a change in the half-lives listed in Fig 10 can be observed. The maximally necrotic state at day 1 (t_1/2 = 76.2 minutes) and the healthy state at day 7 (t_1/2 = 20.2 minutes) differ by a factor of 3.77. This fits to the experimental findings in [130]. These experiments involved rats one day after inducing necrosis by CCl₄ compared to control rats, thus a different species. Furthermore, different doses of 0.25 mg CCl₄ per kg body weight and 5 mg caffeine per kg body weight were used as well as a different notion of half-life computed based on concentrations in the systemic blood plasma. Still, the half-lives reported in Table 1 in [130] are 1.68 hours (control) and 6.29 hours (day 1) and thus differ by a factor of 3.74, surprisingly close to our finding. Computing half-lives in a similar way as in [130], i.e., via an elimination rate constant obtained by logarithmic regression analysis to the concentration in the venous blood pool in the range [30, 300] minutes, we obtain 28.5 and 76.8 minutes, respectively. This is, on the one hand, in the same range as our half-lives t_1/2 for the entire body. On the other hand, they differ by a factor of 2.69, which is again close to 3.74 obtained from [130].

Insulin Model.

The dynamic model of cellular insulin binding consists of eight ordinary differential equations describing the processes shown in Fig 11 by chemical mass action laws. For parameter estimation, flow cytometry (FACS) measurements of labelled insulin were used. The dose-response 15 minutes after stimulation as well as time course data (t = 0, 1, 2, 5, 15, 30 min) for 10 nM, 100 nM and 1000 nM were repeatedly measured and averaged [138]. A selection of the FACS data is shown Fig 11. In addition, the dose- and time dependency of insulin depletion in the medium was evaluated by an enzyme-linked immunosorbent assay (ELISA).

The structure of the insulin model is different from the previous two models. The extracellular medium from the experimental setup is replaced in the simulations by the plasma subject to blood flow. The plasma is assumed to be in instantaneous concentration equilibrium with the interstitium (not subject to blood flow). Moreover, red blood cells are here assumed not to be part of the insulin exchange and we thus have no insulin concentration for the red blood cells in our model. This allows us to define an extracellular volume fraction φ_ext = φ_pls + φ_int where φ_pls and φ_int are given in Eq 5b. According to these flow assumptions, we scaled the typical transit time as τ_typ ⋅ φ_ext/φ_pls = 15.3 s.

To account for the two different entities of hepatocytes, the cellular subcompartment consists of two volumes φ_cell,l = η_l φ_cell and φ_cell,h = η_h φ_cell for the low-binding and high-binding cells, respectively, that add up to φ_cell from Eq 5b. Additional processes such as insulin binding by receptors require further concentrations in the model, leading to a form of the equations different from Eqs 3 and 4.

Parameters in the cellular insulin model were originally fitted for the experimental setting with an unphysiological ratio of extracellular to cellular volume. This fit is shown in Fig 11. The parameters were subsequently rescaled and converted to the volume fractions in the physiological setting. For scaling the parameters derived in the well-stirred experimental environment to spatially resolved representative sinusoids, we applied the procedure described above. For this purpose, we first implemented a preliminary organ-scale model by introducing a flow term as the α in Eq 2 and defined a preliminary representative sinusoid model using these parameters. For this purpose, the position-dependent volume fractions ψ_ext, ψ_cell,l, and ψ_cell,h are computed in terms of the ψ_{{pls,int,cell}} defined in Eq 12 as (20) We could observe that the receptor binding and unspecific binding are the fastest processes described by the ODE system. Hence, we determined an adaption factor ω for these binding parameters, considering the peak-above-baseline amplitude as the deviation to be minimized when determining an optimal ω = 0.204. This factor is part of the parameters K_{on,off} and K_{a,d},{l,h} in the final organ-scale model (21) where c_I denotes the concentration of free insulin in the plasma and interstitium, c_B,u the concentration of unspecifically bound insulin, c_R,{l,h} are the concentrations of insulin receptors, c_IR,{l,h} those of bound insulin and c_IR,{l,h} those of bound internalized insulin, the latter three in the low-binding and high-binding case, respectively. Permeability-type parameters are scaled as before using σ(λ) defined in Eq 14. The dependency of the concentrations on space and time was omitted here such as not to overload the notation. The relations between the concentrations are illustrated in the diagram in Fig 11.

The constants used in Eq 21 are (22)

The unspecific binding expressed in Eq 21 by the concentration c_B,u is viewed as (a) free insulin which is (b) binding to the cells. Due to (a), it is a concentration in the extracellular space, hence ψ_ext(λ) in the denominator. Due to (b), binding capability scales with the cellular surface area, hence the factor σ(λ) in the numerator.

Results of the Parameter Study.

As a parameter study, we considered all 4096 possible configurations of 12 low- or high-binding cells along the representative sinusoid to investigate how the configuration affects the outflowing insulin concentration. As each position along the representative sinusoid represents a different contribution to the total volume, the factors ρ(λ) as defined in Eq 13 needed to be taken into account when computing the total fraction of the two cell entities for all possible configurations as η_{l,h} = ∑_λ ρ(λ) ⋅ ψ_cell,{l,h}(λ). This in particular implies that there are more distinct values for η_{l,h} than merely multiples of 1/12. Many of the 4096 possible configurations lead to fractions of low-binding cells highly different from the experimentally observed ratio η_l,obs = 0.606. Even though we put special focus to ratios close to η_l,obs in the presentation of the results all possible combinations were studied to be able to comprehensively investigate the impact of spatial configurations.

Insulin secretion by the pancreas is pulsatile, where frequency and amplitude can vary and in particular depend on the organism and on the glucose level in the blood. The typical periodicity is in the range of 5 to 15 minutes [132]. We considered an inflowing plasma concentration composed of a basal level of 1 nM in both HA and PV and an oscillatory secretion component into the PV inflow with a period of 720 s (12 minutes) and a peak-above-baseline amplitude of 9 nM. This results in a total inflow of (23) with half life times τ₁ = 60 s and τ₂ = 10.8 s and a scaling factor 1.752/2.102 being the relative contribution of the PV for the total blood flow [59]. The normalization factor I_max is given by the maximum (24) of the unnormalized product of the two exponentials in square brackets in Eq 23. The profile I(t) was continued periodically for t ≥ 720 s. Its range of values is chosen to match the experimental setting for which the model was parametrized, it is in particular not meant to approximate physiological insulin concentrations.

Initial concentrations were obtained by running the simulation for the well-stirred organ-scale model for 1.8 ⋅ 10⁷ s (5000 hours). For this purpose, initial receptor concentrations of c_R,l = 5.686 ⋅ 10⁻² mM and c_R,h = 1.366 ⋅ 10⁰ mM from the parameter fit and zero initial concentrations for the remaining states, as well as the average of I(t) from Eq 23 as inflow concentration were used. This resulted in (25)

In Fig 12, we show the dependency of the peak-above-baseline amplitude of the outflowing insulin concentration on the fraction of low-binding cells for all 4096 cases, i.e., for the full range of 0 ≤ η_l ≤ 1. Additionally, we plotted the outflowing concentration time curves for configurations with 0.581 ≤ η_l ≤ 0.631 low-binding cells, the range η_l,obs ± 0.025. We can observe that there is a relatively wide range of outflowing peak-above-baseline amplitudes. The obtained results indicate that not only the amount of the two entities of cells, but also their spatial configuration along the sinusoids determine the total insulin binding and degradation in the liver. Despite more experimental work being necessary in order to corroborate theses observations, it is clear that the insulin outflow from the liver is determined by the existence of both entities of cells able to bind different amounts of the hormone as well as by their spatial configuration along the sinusoid. The latter seems to be a new additional diversity factor in the liver independent from the well-known metabolic heterogeneity related to the position of the cells along the sinusoid.

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Fig 12. Parameter Study for the Insulin Model.

The left plot shows the relation between the fraction of low-binding cellular volume and the peak-above-baseline amplitude of the outflowing insulin concentration obtained by the simulation for 4096 different spatial configurations of low-binding and high-binding cells. Generally, a higher fraction of low-binding cells implies less insulin uptake by the cells and thus a higher simulated outflowing concentration. The scattering, however, clearly shows that it is not a strict functional dependency. The shaded area in the scatter plot corresponds to the range η_l = 0.606 ± 0.025 near the observed fraction of low-binding cells. It comprises 436 cases, for which the corresponding curves of outflowing insulin concentration are shown in the right plot. The individual lines are colored according to the low-binding cellular volume, cf. the shaded area in the scatter plot. The fact that these are not spectrally ordered from blue to red again shows that the simulation result does not only depend on the low-binding cellular volume fraction but also on the actual spatial configuration.

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