- Research article
- Open Access
Systems analysis of non-parenchymal cell modulation of liver repair across multiple regeneration modes
© Cook et al. 2015
- Received: 3 June 2015
- Accepted: 10 October 2015
- Published: 22 October 2015
A hallmark of chronic liver disease is the impairment of the liver’s innate regenerative ability. In this work we use a computational approach to unravel the principles underlying control of liver repair following an acute physiological challenge.
We used a mathematical model of inter- and intra-cellular interactions during liver regeneration to infer key molecular factors underlying the dysregulation of multiple regeneration modes, including delayed, suppressed, and enhanced regeneration. We used model analysis techniques to identify organizational principles governing the cellular regulation of liver regeneration. We fit our model to several published data sets of deficient regeneration in rats and healthy regeneration in humans, rats, and mice to predict differences in molecular regulation in disease states and across species.
Analysis of the computational model pointed to an important balance involving inflammatory signals and growth factors, largely produced by Kupffer cells and hepatic stellate cells, respectively. Our model analysis results also indicated an organizational principle of molecular regulation whereby production rate of molecules acted to induce coarse-grained control of signaling levels while degradation rate acted to induce fine-tuning control. We used this computational framework to investigate hypotheses concerning molecular regulation of regeneration across species and in several chronic disease states in rats, including fructose-induced steatohepatitis, alcoholic steatohepatitis, toxin-induced cirrhosis, and toxin-induced diabetes. Our results indicate that altered non-parenchymal cell activation is sufficient to explain deficient regeneration caused by multiple disease states. We also investigated liver regeneration across mammalian species. Our results suggest that non-invasive measures of liver regeneration taken at 30 days following resection could differentiate between several hypotheses about how human liver regeneration differs from rat regeneration.
Overall, our results provide a new computational platform integrating a wide range of experimental information, with broader utility in exploring the dynamic patterns of liver regeneration across species and over multiple chronic diseases.
- Liver regeneration
- Molecular regulatory mechanisms
- Systems biology
- Regeneration modes
- Chronic liver disease
The liver’s unique ability to regenerate allows partial liver resection to be a viable treatment option for patients with various liver diseases. Because the liver can regenerate even when most of its mass is lost (up to ~75 %), typical treatment for hepatocellular carcinoma involves resection of liver mass containing tumors. Patients with liver diseases such as cirrhosis can be treated similarly using partial liver transplant from a live donor, followed by liver regeneration in both donor and recipient. Regenerative ability is not equal in all livers, however. Surgeons have long been wary of transplanting fatty livers from organ donors because fatty livers regenerate insufficiently or not at all . Age, diet, and miRNA regulation have also been linked to the liver’s overall regeneration ability [2–6]. Additionally, many chronic diseases impair liver regeneration and repair following hepatic resection . It has even been postulated that inhibition of the liver’s natural repair ability contributes to the progress of steatohepatitis to cirrhosis in the liver .
Despite clinical relevance and advances in our understanding of the molecular mechanisms underlying regeneration, however, the organizational principles governing molecular regulation of liver regeneration remain unclear. To investigate these organizational principles, a computational model of liver regeneration was developed recently, taking into account growth factor (GF) signaling, cytokine signaling along the JAK-STAT pathway, and hepatocyte replication . Furchtgott, Chow, and Periwal employed this computational model to account for differential regeneration profiles after various degrees of partial hepatectomy. This model considered cell proliferation but not cell growth, thus limiting its ability to account for liver repair scenarios that involve hypertrophy in addition to hyperplasia.
Model variables and parameters and their approximate biological correlates
Nominal or Starting Value
Approximate Biological Correlate
Relative nutrient and toxin delivery/absorption rate in the liver
Growth rate of hepatocyte mass [mass equivalent doublings/min]
Rate at which non-parenchymal cells (primarily Kupffer cells) are able to modify the cytokine milieu post-PHx
Rate of cytokine degradation
Maximum JAK activation rate
JAK Michaelis concentration
Rate of JAK degradation
Relative concentration of monomeric STAT3 in the liver
Maximum STAT3 phosphorylation rate
pSTAT3 Michaelis concentration
Rate of pSTAT3 dephosphorylation
Maximum SOCS3 activation rate
SOCS3 Michaelis concentration
Rate of SOCS3 degradation
SOCS3 Inhibition effect on STAT3 phosphorylation
Maximum IE gene activation rate
IE gene Michaelis concentration
Rate of IE gene degradation
Rate of ECM degradation by MMPs
Rate of constitutive ECM degradation
Rate at which non-parenchymal cells (primarily hepatic stellate cells) directly & indirectly produce growth factors post-PHx
Rate of growth factor degradation
Rate of growth factor absorption/binding to the ECM
Maximum rate of hepatocyte transition from Quiescence to Primed [cells/min]
Maximum rate of hepatocyte transition from Primed to Replicating [cells/min]
Maximum rate of hepatocyte transition from Replicating to Quiescence [cells/min]
Rate of hepatocyte progression through the cell cycle [doublings/min]
Requiescence rate of Primed hepatocytes [cells/min]
Apoptosis rate of damaged hepatocytes
Rate of release of matrix bound factors during ECM remodeling
Degradation rate of matrix bound factors once they are released from the ECM
Nominal or Starting Value
Approximate Biological Correlate
Fraction of hepatocytes in the Quiescent state
Fraction of hepatocytes in the Primed state
Fraction of hepatocytes in the Replicating state
Cytokine microenvironment of the liver
Relative levels of activated receptors for cytokine signals in hepatocytes
Relative levels of phosphorylated STAT-3 compared to monomeric STAT-3 or other downstream effectors of cytokine signaling (i.e. NF-κB)
Relative levels of SOCS3 or other inhibitors of cytokine signaling
Relative levels of immediate early genes induced in hepatocytes (e.g. cFOS, cJUN, and AP-1)
Relative bioavailability of growth factors promoting hepatocyte proliferation
Relative levels of extracellular matrix buildup of matrix composed of collagens inhibitory to regeneration
Relative levels of matrix bound factors priming hepatocytes
Relative levels of free matrix bound factors that were initially bound by ECM
Additionally, we extended the model further to consider explicitly the contributions of initially matrix-bound factors, MBFs, (including growth factors and potentially WNT precursors), that are liberated from the matrix during remodeling post-PHx. These factors likely contribute to the quiescent-to-primed transition that hepatocytes undergo during the priming phase and may be equally as important as the early, predominantly Kupffer cell-produced cytokine microenvironment in priming hepatocytes for entry to the cell cycle (Fig. 1b, gray portion).
Our extended computational model allowed us to investigate several issues outstanding in the field of liver regeneration. We analyzed the extended model to determine the relative contributions of the predominantly Kupffer cell-produced cytokine microenvironment and the ECM-liberated signals to prime hepatocytes to enter the cell cycle. We simulated the extended model over a wide range of parameter values and identified parameter sets giving rise to distinct modes of liver regeneration and common or unique molecular regulation of liver regeneration dynamics. We next questioned what organizational principles regulate the biology of liver regeneration. Our model-based analyses revealed how altering regulatory balances can shift the liver into distinct, clinically relevant regeneration modes. We analyzed multiple published regeneration profiles to identify common organizational principles underlying liver regeneration across distinct tissue response phenotypes. We then predicted which molecular signaling dysregulation may account for altered liver regeneration profiles in multiple species and disease scenarios. We also used our model to compare several hypotheses about differences in regeneration between humans and rats, and suggest measurements that can be used to test these hypotheses. We hope that understanding how organizational principles work together to govern the dynamics of liver regeneration and repair will provide unique insights into liver disease progression, suggest further avenues of research for targeted therapy for chronic liver diseases, and provide insights into treatments to promote liver regeneration after surgical resection.
Our computational model extends the model previously published by Furchtgott, Chow, and Periwal by adding terms describing the contributions of cell growth and initially matrix-bound factors to liver regeneration following resection . Our computational model consists of 11 ODEs (described in detail in the Methods section), 43 parameters (Table 1), and 12 variables representing molecular levels and cell abundances (Table 1). The Matlab code used for this study is available as supplemental information in Additional file 2. All variables representing molecular levels, except matrix bound factors (MBF), have an initial steady-state level of 1 and any change thereafter is a fold-change over baseline. Determination of MBF initial level is described in the following section. The initial level of quiescent hepatocytes is 1, while initial levels of primed and replicating hepatocytes are 0. All simulations were performed using Matlab (Mathworks, Natick, MA).
Extended model predicts the importance of Kupffer cell-mediated signaling during the priming phase
The importance of direct intercellular signaling leading to IE gene expression has been widely studied. Direct interventions to intercellular signaling have been shown to impact liver regeneration dynamics significantly . Whereas, the effects of matrix bound factors (MBF) are less well appreciated but appear to have a more subtle effect on regeneration dynamics . Therefore, we reasoned that the effects of MBF are likely less than the effects of IE genes on driving regeneration. We tested model behavior if the effects of MBFs are just as important to regeneration as IE gene effects. Rather than match parameters for MBF signaling to a particular MBF (i.e. WNT) we tuned the model parameters initial MBF levels, production rate, and degradation rate such that the relative magnitude of the priming signal from initially MBF signaling and IE gene production were of the same order of magnitude during the timeframes when they were contributing to hepatocyte priming (Additional file 3: Figure S2). Table 1 contains the parameters that correspond to this phenotypic behavior. This parameter choice relies on the assumption that MBF signaling is as important as IE gene production to induce hepatocyte priming, and MBFs are depleted following the priming phase. Unbinding of MBF peaked approximately 45 min post-PHx and lasted over the duration of the priming phase (6 h post-PHx), while IE gene levels peaked close to 3 h post-PHx and remained high throughout the early stages of liver regeneration (>12 h post-PHx). We found that including MBF signaling altered the dynamic mass recovery only slightly, leading to a sustained offset in mass recovery compared to the case without MBF signaling (Additional file 4: Figure S3). The effect of MBF signaling in our model is slight most likely because the duration of MBF signaling is shorter than the duration of cytokine signaling. Because of the negligible effect that MBF signaling had on liver regeneration dynamics, we excluded its contributions from the subsequent model analyses.
Extended model with cell growth better accounts for rat liver regeneration profile
Exploring the state space of liver regeneration reveals distinct regeneration modes
Sensitivity analysis reveals that molecular and physiological regulation strongly affects dynamic mass recovery
We investigated how the inclusion of cell growth modified the dynamic sensitivity of the metabolic demand parameter (M). When cell growth is not considered, increased metabolic demand was inhibitory to liver recovery during the first 53 h post-PHx, largely due to increased hepatocyte apoptosis (Fig. 5b). After 53 h, increased metabolic demand enhanced regeneration. With cell growth considered, the initial inhibitory effect of increasing metabolic demand lasted only for the first 43 h post-PHx, after which it enhanced mass recovery but to a lower extent than the model without cell growth (Fig. 5c). The inclusion of cell growth also allowed us to recognize a potential dynamic antagonism between metabolic demand and cell growth rate. Early post-PHx, hepatocyte growth was a positive contributor to liver regeneration, while metabolic demand negatively affected progression of regeneration. At this early time, metabolic demand acted in hepatocytes predominantly to induce apoptosis in damaged cells through high metabolic load, causing reduced liver mass. After approximately 43 h post-PHx, high metabolic load induced high response in non-parenchymal cells causing increased priming and regeneration. From approximately 43 to 87 h post-PHx, metabolic load and hepatocyte growth acted synergistically to promote liver regeneration. Near the termination stage of liver regeneration, however, hepatocyte growth inhibited liver regeneration by inducing hepatomegaly and decreasing the driving force for regeneration.
Paired parameter analysis reveals control principles governing the network balances driving liver regeneration
When we investigated the relationship between IL-6 and GF production and degradation, we found that relatively slight increases to both IL-6 (Fig. 6b) and GF (Fig. 6c) production rate increased mass recovery, while degradation had to increase much more to cause an equivalent magnitude decrease in mass recovery. This antagonism was more pronounced in IL-6 balance, but was relatively subtle in GF balance. For further visualization of the effects of GF production and degradation balance, see Additional file 12: Figure S11. These relationships reveal an organizational principle whereby production of molecules acts as a means of achieving coarse-grained control of molecular levels while degradation acts to achieve fine-tuned control. These results suggest that non-parenchymal cells may act predominantly as coarse-grained controllers of liver regeneration, while hepatocyte responsiveness and miRNA or other regulation may act to achieve fine-tuned control of liver regeneration.
Translating among species using the computational model
We tested whether translating among species can potentially be achieved simply by adjusting model parameters in the extended computational model. Prior to simulation, we sought to identify which parameters likely change among species. The cell cycle duration is known to be fairly consistent across mammalian species; therefore, we maintained this parameter at nominal levels [17–20]. Similarly, the JAK-STAT pathway is understood to be ubiquitous in mammalian species. Therefore, we maintained JAK-STAT signaling pathway parameters constant across species. Additionally, while the physiological parameters used to approximate multiple pathways may indeed change between species, there is little reason to believe that the essential mechanisms of these pathways differ any more than the JAK-STAT signaling pathway does. Therefore we maintained the physiological parameters at nominal levels as well. This assumption of consistent pathway behavior across species does not take into account any differences in network dynamics caused by species-specific molecular dynamics, for example rat IL-6 half-life in rat macrophages compared to human IL-6 half-life in human macrophages.
We considered an approach where all molecular driving events were maintained constant between species, leaving the metabolic demand parameter and the cell growth rate parameter as the only ones available for modification. It has been shown that metabolic demand of an organism is proportional to the mass of the organism raised to an exponential power (estimated to be between 2/3 and 3/4); this is true for both plants and animals and appears to be an organizing principle of biology [21–23]. The metabolic demand parameter is a lumped parameter approximating extrinsic signals that occur in parenchymal and non-parenchymal cells and intrinsic hepatocyte capacity to respond to these signals; however, a portion of these signals may be caused by increased nutrient and toxin flux. Therefore, this term represents, at least in part, a metabolic response to these fluxes, which may vary among species according to overall mass. Lumping extrinsic and intrinsic drivers of regeneration into one parameter makes it difficult to simulate experiments where hepatocytes from one species are transplanted into another, but such a technique is appropriate when considering each species individually . In addition to metabolic demand potentially changing across species, it is possible that cell growth rate may also differ across species. We were able to find no studies reporting grossly observable differences in cell growth rates, while several studies have suggested that the cell growth rate across species appeared to be fairly similar among mammalian species [25, 26]. These results led us to believe that cell growth rate likely changes among mammalian species, but that change is likely not orders of magnitude different. Therefore, we changed the cell growth rate and metabolic demand parameters across species in our model to simulate regeneration in multiple species.
We fit regeneration profiles of rats, mice, and humans by simultaneously changing only the hepatocyte growth rate and metabolic demand parameters and minimizing the sum of squared error between experimental data and simulation output. For rats and mice, the growth rates estimated using this least squares approach were fairly similar (G = 3.5x10−4 and 9.7x10−4 mass equivalent doublings/min, respectively). The optimum fit for humans, however, resulted in a much higher estimated growth rate (G = 2.5x10−2 mass equivalent doublings/min). This estimation is inconsistent with literature suggesting cell growth rate is fairly similar among mammalian species [25, 26]. We therefore constrained human hepatocyte growth rate to the average of rat and mouse growth rates (G = 6.6x10−4 mass equivalent doublings/min) (Additional file 13: Table S1) and changed only the metabolic demand parameter to fit human regeneration data.
We have shown that it is possible that the difference in time necessary to regenerate fully is due predominantly to the differential functional demands of the liver across species. Rodents, which live in an environment more prone to infection and liver injury, may require a higher metabolic demand (a component of which is the nutrient delivery per cell) to maintain healthy liver function than humans, which live in a relatively clean environment. Because blood flow and overall nutrient delivery does not change following PHx, a smaller number of cells are receiving a relatively increased nutrient delivery in all species. It is possible that post-PHx the relative increase in metabolic demand per cell—and therefore the driving force for regeneration—may be higher in rodents than in humans.
Summary of parameters used to simulate alternate hypotheses of how human liver regeneration differs from rat liver regeneration
Hyp 1: Altered cytokine response
Hyp 2:Altered GF storage and ECM balance
Hyp 3: Altered state transition rate
Hyp 4: From Periwal et al. 
Hyp 5: Reduced metabolic demand
kIL6 = 0.1435
κdeg = 4.955
kQP = 1.4x10−3
kQP = 1.1x10−3
M = 20.8217
κIL6 = 0.4942
κECM = 56.30
kPR = 1.5x10−3
kPR = 2.6x10−3
G = 3.474x10−4
VJAK = 1.364x103
kGF = 3.288x10−3
kRQ = 70.9x10−3
kRQ = 135x10−3
Km JAK = 7.565x103
κGF = 2.139x10−3
kreq = 4.17x10−3
κJAK = 0.0398
kup = 0.1008
kap = 4.17x10−3
[STAT3] = 2.031
kprol = 8.33x10−3
kSTAT3 = 1.109x103
Km STAT = 0.5178
κSOCS = 0.1682
KI SOCS3 = 0.0569
kIE = 18.60
Km IE = 88.13
κIE = 1.148
Patterns of molecular regulation (30 days) and mass recovery (14 days) that could differentiate hypotheses of mechanisms underlying liver regeneration in humans
IL-6 / Inflammation
(1) Altered Inflammation
(2) Altered ECM remodeling and GF storage
(3) Altered transition times
(4) Parameter changes assumed in Periwal et al. 
(5) Lower metabolic demand
Predicting effects of chronic disease on liver repair following partial hepatectomy
Summary of predicted disease effects on liver regeneration
(1) Non-alcoholic Steatohepatitis
(2) Alcoholic Steatohepatitis
(3) Toxin-induced Cirrhosis
(4) Allotaxin-induced Diabetes
Delayed & low
Adaptation to chronic diseases also appears to influence the liver’s ability to recover a normal baseline function after an acute challenge. At long times post-PHx, NASH was characterized by sustained high levels of GF signaling, ASH was characterized by sustained high levels of IL-6 and reduced ECM accumulation, and diabetes was characterized by reduced ECM accumulation (Additional file 16: Figure S13 Additional file 17: Figure S14, and Additional file 19: Figure S16). Cirrhosis, on the other hand, was characterized by all molecular levels returning to baseline (Additional file 18: Figure S15). Our prediction of a sustained high inflammatory response in ASH simulations is consistent with previous reports of relatively high levels of inflammatory molecules found in the serum of patients with ASH . This result suggests that one of the fundamental mechanisms of disease progression between ASH and NASH may be a difference in inflammatory response of non-parenchymal cells.
Although our model simulations showed that altered non-parenchymal cell behavior is sufficient to cause impaired regeneration dynamics that are consistent with NASH, ASH, diabetes, and cirrhosis, parenchymal cells likely also contribute to impaired regeneration. We therefore tested whether alterations in hepatocyte response to non-parenchymal cells are sufficient to explain altered regeneration in these same disease phenotypes by changing parameters related to hepatocyte response to non-parenchymal cells (14 out of 33 parameters, Additional file 20: Table S3). We found that for NASH, ASH, and cirrhosis, alterations in hepatocyte response to non-parenchymal cells was also sufficient to explain altered regeneration in these disease phenotypes (Additional file 21: Figure S17A-C). Altering these hepatocyte response parameters was insufficient to explain diabetes-impaired regeneration dynamics (Additional file 21: Figure S17D). In all cases, the previous set of parameters (Additional file 15: Table S2) gave lower mean squared error (MSE) than the hepatocyte-specific parameter alterations (Additional file 20: Table S3). It was interesting to note that the parameter sets used to simulate NASH and ASH eventually resulted in liver failure, with hepatocyte numbers continuing to decrease as the simulation progressed. The results of these simulations, together with the simulations altering non-parenchymal cell behavior and experiments from literature, suggest that disease conditions likely alter the dynamic function of non-parenchymal cells and hepatocytes during liver regeneration. Therefore when investigating liver disease states and response to surgical interventions, a systems-based approach that explicitly accounts for cell-cell interactions is necessary to account for the underlying processes fully.
Our study provides an investigation into the organizational principles and molecular regulation underlying liver regeneration following resection across multiple species and disease states. Our study identified altered modes of regeneration and investigated disease states that cause regeneration to follow these altered modes. This study, however, only addresses surgical resection of the liver and has not been applied to drug-induced liver injury (DILI). Because similar archetypal processes also likely govern liver regeneration following DILI, it is possible that some of the results of our modeling study can be generalized to inform principles underlying regeneration following DILI as well. The altered regeneration dynamics following DILI indicate that additional processes need to be added to the model to accurately capture the complete physiology (for example, clearance of injured or necrotic hepatocytes and immune cell infiltration).
This study investigated liver regeneration through a computational model involving archetypal signaling pathways that represent classes of molecular signaling. Therefore, the simulations in this study suggest relative balances and timing of molecular signals that may be deregulated in disease or altered across species. We have used this approach in a previous study to investigate the molecular factors governing the altered liver regeneration dynamics caused by ablation of the gene adiponectin (Adn). Our modeling approach suggested that the delay and acceleration of regeneration observed in Adn−/− mice was caused by decreased priming in hepatocytes (seen as decreased STAT3 phosphorylation during the first 6 h post-PHx) and enhanced growth factor signaling (observable by 20–40 h post-PHx) . We then measured STAT3 phosphorylation and growth factor levels in liver lysates of Adn−/− mice and found reduced STAT3 phosphorylation at 3 and 6 h post-PHx coupled with high levels of ANG-1, FGF-2, and HGF proteins from 6 to 42 h post-PHx.
Our study suggests several organizational principles of regeneration. Initiation of regeneration appears to be governed by the number of hepatocytes entering the priming phase, which in turn is largely driven by the inflammatory response (modeled as IL-6 signaling). The computational model simulations further suggest that IL-6 signaling activity is amplified at the level of STAT-3 phosphorylation, so that small changes in inflammatory response can cause large changes to STAT-3 phosphorylation and significantly alter the regeneration profile. The timing and magnitude of GF response appears critical to replication, with low or late GF response suppressing overall regeneration. Our results led us to predict that chronic diseases impair liver regeneration through a combination of deficient inflammatory signaling and growth factor bioavailability. We further predicted that these deficiencies are shared between non-parenchymal cell activation and hepatocyte responsiveness to extracellular stimuli.
Our approach allowed us to investigate several hypotheses about how regeneration differs between rats and humans. By maintaining molecular and phenomenological parameters constant across species and modifying metabolic load and hepatocyte growth rate, we were able to fit experimental regeneration profiles across species. This approach has the benefit of conserving hepatocyte-related signaling pathways including the JAK-STAT signaling kinetics across species. These results revealed that regenerative capacity is likely related to animal mass, with larger species having fewer energetic resources to devote to regeneration. This explanation is consistent with identification of peak regeneration in pigs and dogs occurring later than in rats and mice (3 days post-PHx in pigs and dogs, as opposed to 1 day in rodents) . Alternate hypotheses about differences between rat and human liver regeneration dynamics, however, offer different predictions about dynamic tissue behavior post-PHx. We predicted that tissue biopsies and scans taken at two weeks post resection or molecular measurements at one month post resection in humans could differentiate between these hypotheses.
Another factor governing the length of regeneration time is how rapidly hepatocytes are able to increase their functional mass to compensate for lost tissue. Large mass may not be beneficial to liver repair if much of the extra mass does not contribute to liver function; therefore, the mass regained in this simulation can be seen as functional mass increase that contributes to liver function. As opposed to the metabolic demand parameter, hepatocyte growth rate was not related to animal mass. Growth rate may therefore be governed by other factors, such as maximum glucose metabolic flux possible, mitochondrial activity and number of mitochondria, and the relative amount of nutrients available post PHx. By incorporating cell growth, the model proposed in this work was able to capture the rapid increase in tissue mass humans are capable of, up to 70 % of liver mass restored by 30 days after 70 % PHx . Experiments measuring growth rates of hepatocytes in vitro or further hepatectomy experiments performed using pigs or other species can be used to test and refine the simple relationship proposed between metabolic demand and body mass.
According to our analysis, the number of parameters that need to be changed to translate across species is relatively small (a minimum of two). Futhermore, the minimum set of parameters changed were physiological parameters, M and G. This does not mean that there are no differences in molecular regulation across species; it does, however, suggest that the differences are the result of similar processes across species responding to species-specific physiology. This results in altered molecular and regeneration dynamics across species. In contrast, we changed multiple parameters, including parameters related to molecular signaling, to simulate disease effect on liver regeneration. Taken together, these results suggest that biological processes behaving normally can account for differences across species but cannot account for disease effects on regeneration phenotypes.
Although the model describes fairly well experimental data, the model description of the cell cycle does not contain specific phases of the cell cycle. The rate of cell proliferation in the model contains all the steps from exit from the G0 phase to a complete cell division. Therefore, this rate also includes any additional time taken for a quiescent hepatocyte to dedifferentiate, divide, and redifferentiate. Little is known about how long any dedifferentiation and redifferentiation takes or if the time needed for these processes varies across species. Therefore, the overall rate of cell proliferation may vary between species. Although we did not explicitly address this possibility in the current study, further studies could explore this as a potential contribution to the difference in peak hepatocyte replication times between rats and mice.
Parametric sensitivity analysis of the computational model revealed that regeneration is dynamically controlled and that not all factors respond the same across all times. This result coupled with the pulsatile sensitivity analysis recently performed on the original model proposed by Furchtgott et al.  indicates that treatments designed to improve regenerative ability during chronic disease or following liver transplant may need to be dynamic as well . Extending the results of simulations of chronic disease states in rats to the human model may assist in scheduling treatments for patients suffering from chronic diseases post-transplantation to maximize regeneration. For example, during the first week (the apparent priming phase in humans) it might be necessary to renormalize hepatocyte response to inflammation signals while later treatments (replication phase) may need to increase growth factor levels.
Our model-based approach offers unique insights into the mechanisms of liver disease progression in the context of chronic disease; however, there are several limitations inherent to this approach. The first limitation is that only the JAK-STAT signaling pathway is explicitly considered in this model. Although this pathway has been shown to be critical for a normal repair phenotype, even a hepatocyte-specific STAT-3 knockout does not completely inhibit regeneration . In this genotype, signaling through ERK compensates for the lack of STAT-3. The importance of the liver’s repair mechanism ensures that multiple compensatory signaling pathways are available to act . Our model can be extended to include additional signaling pathways to account for compensatory signaling and cross-talk. We note, however, that the present simplification involving cell phenotype transitions sufficiently captures major features of the liver regeneration process. Such simplified models have led to important insights into biological regulation in other contexts as well [13, 37, 38].
Another limitation is that the current model takes into account only linear responses of non-parenchymal cells during liver repair. Many reviews highlight the important role of timing of non-parenchymal cell signaling during liver repair [10, 11]. For instance, the critical contribution of non-parenchymal cells has been demonstrated using animals where Kupffer cells have been depleted, thereby significantly delaying regeneration following hepatectomy . The current simulations suggest that Kupffer cells are largely responsible for priming hepatocytes. Hepatic stellate cells appear to be the main regulator of hepatocyte regeneration, governing both proliferation through control of growth factor bioavailability and termination of regeneration through ECM production and degradation. Therefore, moving towards a more comprehensive computational model of liver repair in health and chronic disease requires inclusion of alternative regulatory mechanisms within hepatocytes, as well as the activation and signaling of non-parenchymal cells. To facilitate this integration, one could consider the existing models of macrophage or Kupffer cell activation and hepatic stellate cell activation. For instance, macrophage activation has been studied using a computational model of the cytokine-mediated pathways [39, 40]. Specific to the liver, our group has recently developed a computational model of cytokine-mediated hepatic stellate cell activation that incorporates multiple pathways with cross-talk as well as microRNA mediated regulation [39, 40].
Our computational model was able to match liver regeneration profiles across multiple chronic disease models and across species. This modeling framework can act as a tool to translate results from rodent experiments to clinically actionable hypotheses in primates or humans. Our study suggests that liver regeneration is dynamically controlled by factors produced by non-parenchymal cells. Inflammatory signaling (predominantly from Kupffer cells) governs the priming response of hepatocytes, while growth factors (predominantly produced by hepatic stellate cells) govern hepatocyte entry into the cell cycle. The synchronicity of hepatocyte entry into the cell cycle is governed by both growth factor levels and timing as well as proliferation rate of hepatocytes. These findings underscore the importance of non-parenchymal cells to recovering the liver’s repair ability from a diseased state. Therefore, future computational work should explicitly take contributions from non-parenchymal cells into account.
The parameters β and θ in each of these equations are tuned so that when metabolic load is high, σap is high; conversely, when [GF] is high, σreq is low. Therefore, when cells are highly stressed (high metabolic load), apoptosis occurs at a high rate; when GFs are available, cells remain in the “Replicating” state.
Where [proSTAT3] represents the concentration of monomeric STAT-3 available to dimerize following IL-6 signaling. It should be noted that in the original model our [IL-6] term representing cytokine signaling was called [TNF]. Cannonically, TNF signals through the NF-κB cascade, while IL-6 signals through the JAK-STAT cascade. Table 1 states that the approximate biological correlate of the [IL-6] variable in the model is the “Cytokine microenvironment of the liver”. As previously described in  and in , the [IL-6] variable should be considered a lumped variable representing the physiological impact of general cytokine signaling rather than an exact analogue to IL-6 protein levels. Therefore, we used the name [IL-6] for this variable with parameters derived from TNF.
Where G represents the relative cell mass, which is initially set to 1.
Where MBFFree represents the signaling factors released from matrix and κMBF is the degradation rate of MBFFree once they have been released.
All simulations were performed in Matlab (Mathworks, Natick, MA). Model equations were set up to prevent molecular levels from becoming negative; however, some parameter sets combined with the integration tolerances of ode15s led to GF levels becoming negative at longer simulation times (greater than 150 h). These impossible GF levels did not significantly impact the regeneration profile because most of the growth had concluded by the time GF became negative. Because of these numerical instabilities, however, GF levels were constrained to a minimum of 1.
Transforming published data on liver regeneration into fractional recovery of tissue mass
High fructose-induced steatosis (NASH) and Controls
Liver regeneration rate is the fractional mass recovery minus the remnant liver fractions; therefore, we added 30 % to the reported liver regeneration rate to convert liver regeneration rate to fractional mass recovery.
Ethanol-induced steatosis (ASH)
In the study by Yang et al. , Sprague–Dawley rats (125 g body weight) were fed either a liquid ethanol diet (355 kcal ethanol, 115 kcal carbohydrates, 360 kcal fat, and 180 kcal protein per liter) or a control diet (470 kcal carbohydrates, 360 kcal fat, and 180 kcal protein per liter) for a period of five weeks. After five week adaptation to these diets, rats were anesthetized using ether and a 70 % PHx was performed. Rats were sacrificed at 24 h and 48 h post-PHx, and liver weight was measured. The data presented by Yang et al.  were given in percentage of initial weight at 24 and 48 h post-hepatectomy . We assumed that the initial % of initial liver weight was 30 % because a 70 % PHx was performed. Therefore, to convert from % initial liver weight to fractional recovery, we divided % initial liver weight by 100 %. Although we imposed no further constraints on regeneration in rats with ASH, based on observations of 3H-thymadine incorporation from previous studies, we surmise that it is unlikely that significant hepatocyte replication occurs beyond 48 h post-hepatectomy in alcohol-fed rats .
In the study by Kaibori et al. , 6 week old male Sprague Dawley rats (150-200 g body weight) were injected with thioacetamide (4 % thioacetamide at 20 mg/100 g body weight) thrice weekly for 10 weeks. The rats were then kept for an additional 3 weeks to allow for thioacetamide washout. Cirrhosis was then confirmed by histology. Following development of cirrhosis, rats were anesthetized with ether and 45 % partial hepatectomy was performed. Rats were sacrificed and their livers were excised and weighed at 1, 2, 3, 5, and 7 days post-PHx .
Therefore, the only conversion necessary to convert liver regeneration rate to fractional mass recovery was to divide by 100 %.
Toxin-induced type 1 diabetes
In the study by Johnston et al. , diabetes was induced in male Wistar rats (200-300 g body weight) by administering a single dose of streptozotocin (65 mg/kg body weight) injected into the tail vein under light anesthesia (ether). Rats then received 0.28 M glucose to drink. Partial hepatectomy was performed five days following streptozotocin administration. During recovery, rats were sacrificed and dry liver weight was measured at 12, 24, and 48 h post-resection. The data reported in this study were given in liver dry weight percent of total body weight.
Thus, equation 19 can be solved for baseline dry liver to body weight percentage by inserting equation 21 into equation 19 to yield baseline dry liver to body weight percentage was 1.16 % in the rats used in this study. A 70 % PHx yields a starting dry liver to body weight percentage of 0.348 % corresponding to a fractional recovery of 0.3. All data in this study were therefore scaled by a factor of 0.3/0.348 % to convert the dry liver to body weight percentage to fractional recovery .
Previous studies have suggested that alloxan-induced diabetic rats showed a delay in regeneration but that diabetes did not suppress overall recovery . We therefore constrained recovery at 300 h post-PHx in diabetic rats to be the same as for wild-type rats.
Mouse liver regeneration
Male mice aged 8–12 weeks (129S1) were fed standard mouse chow ad libitum. Mice were anesthetized by pentobarbital and 70 % PHx was performed. The data from Shu et al.  for control mice were given in liver to body weight ratio. To convert these data to fractional recovery, these data were scaled by 0.3 divided by initial value for liver-to-body weight ratio.
Human liver regeneration
The data presented by Periwal et al.  were already given as the fraction of original liver volume, hence requiring no conversion. Similarly, the data presented by Pomfret et al.  were given in percent regeneration, which is defined as remnant volume divided by original volume (x100 %). No conversion was required for these data as well.
Mass(t) represents the nominal mass fraction of hepatocytes at any given time, t, and ΔMass(t) is the deviation from nominal caused by the parameter change. The result is a dynamic parametric sensitivity, showing how the profile of liver regeneration responds to changes in parameters as a function of time.
Where μ and σ2 were estimated from the residuals for each model.
The ltestratio function in Matlab was used to compare the likelihood of the two models.
Availability of supporting data
No datasets were generated in this study. The model used in this study is available as a supplemental file.
We thank Dr. Jan Hoek for helpful discussion about the model and manuscript. This work was supported by National Institutes of Health grants R01 AA018873, R21 AA022417, T32 AA007463, and F31 AA023445. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.
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