 Research article
 Open Access
Modelbased virtual patient analysis of human liver regeneration predicts critical perioperative factors controlling the dynamic mode of response to resection
 Babita K. Verma^{1, 2},
 Pushpavanam Subramaniam^{2} and
 Rajanikanth Vadigepalli^{1}Email authorView ORCID ID profile
 Received: 8 May 2018
 Accepted: 2 January 2019
 Published: 16 January 2019
Abstract
Background
Liver has the unique ability to regenerate following injury, with a wide range of variability of the regenerative response across individuals. Existing computational models of the liver regeneration are largely tuned based on rodent data and hence it is not clear how well these models capture the dynamics of human liver regeneration. Recent availability of human liver volumetry time series data has enabled new opportunities to tune the computational models for humanrelevant time scales, and to predict factors that can significantly alter the dynamics of liver regeneration following a resection.
Methods
We utilized a mathematical model that integrates signaling mechanisms and cellular functional state transitions. We tuned the model parameters to match the time scale of human liver regeneration using an elastic net based regularization approach for identifying optimal parameter values. We initially examined the effect of each parameter individually on the response mode (normal, suppressed, failure) and extent of recovery to identify critical parameters. We employed phase plane analysis to compute the threshold of resection. We mapped the distribution of the response modes and threshold of resection in a virtual patient cohort generated in silico via simultaneous variations in two most critical parameters.
Results
Analysis of the responses to resection with individual parameter variations showed that the response mode and extent of recovery following resection were most sensitive to variations in two perioperative factors, metabolic load and cell death post partial hepatectomy. Phase plane analysis identified two steady states corresponding to recovery and failure, with a threshold of resection separating the two basins of attraction. The size of the basin of attraction for the recovery mode varied as a function of metabolic load and cell death sensitivity, leading to a change in the multiplicity of the system in response to changes in these two parameters.
Conclusions
Our results suggest that the response mode and threshold of failure are critically dependent on the metabolic load and cell death sensitivity parameters that are likely to be patientspecific. Interventions that modulate these critical perioperative factors may be helpful to drive the liver regenerative response process towards a complete recovery mode.
Keywords
 Liver regeneration
 Dynamic modeling
 Level of resection
 Metabolic load
 Cell death sensitivity
 Phase portrait
Background
Liver has the ability to fully regenerate post liver injury or surgical resection [1]. This process of regeneration takes place via a unique mechanism in which differentiated hepatocytes reenter the cell cycle to replenish lost cells [2]. This is followed by proliferation of nonparenchymal cells to eventually reconstitute the cell types in the liver tissue, and tissue remodeling to reestablish the lobular scale morphology [3]. This unique regenerative capacity enables majority of clinical interventions into liver disease via surgical resection, as well as live donor transplants and smallforsize liver transplants. While liver regeneration is robust in healthy individuals, chronic disease diminishes this regenerative capacity making the surgical interventions less than effective and can even lead to postsurgery liver failure in some patients [4, 5]. There is an unmet need for a prognostic tool that takes into account the patientspecific, diseasemodified, regenerative capacity and evaluates these in the context of known activation of molecular pathways and cellular events post resection. A limited set of studies focused on developing a mathematical model of the underlying cellular and molecular mechanisms as a potential solution to this problem [6–14]. These studies demonstrate that a unified set of cellular and molecular mechanisms, with associated parameters, can account for differences in the regenerative response to varying levels of resection [15], with failure above a threshold [16].
Majority of the computational modeling studies have focused on the relatively short time scale of regenerative response in the rat and mouse (hours to few days), and hence are not correctly tuned to account for the weekstomonths time scale relevant to the human regenerative response to liver resection [17]. A limited tuning to human time scales was performed using extremely low amount of clinical data available on liver volume at more than a year after the surgical resection [7, 8]. Hence, it is questionable whether the findings from the simulation and analysis of the rodentbased models are applicable to the human condition. Recently, time series data on volumetric changes in liver have become available in a relatively large number of patients that have undergone different levels of surgical resection of the liver [18]. This data on the dynamic changes in liver volume, presumably correlated with the tissue mass, enables an opportunity to tune the network model to match the time scales relevant to the human liver regeneration, and analyze the resultant computational model for different “virtual patients” based on their clinical tests and the effect on the regenerative outcome.
In this study, we build on our previous computational model of rat liver regeneration [8] and tune the parameters to the human time scale, by utilizing select data from a recently available clinical data set [18]. We initially tuned the model parameters based on constrained optimization. Subsequently, we considered a range of parameter variations corresponding to different “virtual patients” to analyze the modelpredicted response dynamics, and evaluated the directional influence of each of the parameters on the liver mass outcome following resection. Through this approach, we identified a subset of virtual clinical cohort of patients that exhibit recovery or failure, depending on the corresponding parameter values. We analyzed sensitive parameters individually and in combination to delineate the parameter subspaces that correspond to three response modes  full recovery, partial recovery, failure. We utilized phase plane analysis [19, 20] to compute a threshold of resection that separates the recovery versus failure response modes exploiting their basins of attraction. We assessed how this failure threshold varies for different “virtual patients”. Phase plane analysis demonstrated that the variation in response of this cohort of patients yields distinct phase planes with either one sink of failure or two sinks corresponding to recovery and failure. This patientspecific parameterdependent shift in the multiplicity of the system may help with decision making on whether liver surgery is a viable option in certain cases. For the case of two sinks, recovery and failure, our modelbased analysis suggests the level of resection that is likely to be safe.
Results
Tuning the model for human time scale of response based on patient data
Although the Cook et al. [8] model characterizes the important attributes of rodent liver regeneration, it fails to capture the liver regeneration time scale accurately in humans, largely attributed to the lack of availability of human time series data when the original model was tuned. We used our modified model and estimated the human time scale relevant parameters using patient data from Yamamoto et al. [18] (see Methods). We utilized liver volume time series data from a set of 27 patients that showed recovery following resection, with sufficient number of time points in the regeneration profile to permit model tuning and analysis of dynamics. We pursued an optimizationbased approach that allowed variations in all the parameters to identify the tuning with best fit to the clinical data. We compared the results of our parameter tuning approach to that of Young et al. [21] and Cook et al. [8] approaches for translating the model parameters across species based on body mass. In case of Young et al. [21], only the metabolic load (M) is tuned, whereas Cook et al. [8] approach also modified the relative cell mass growth constant (k_{G}). We applied the three parameter tuning approaches to each of the 27 patient time series data sets and compared the results.
An illustrative case of modelderived liver regeneration profiles comparing the three parameter tuning approaches is shown in Fig. 1b. In this case (patient ID 71), the two regeneration profiles derived from scaling the metabolic load qualitatively capture the clinically observed trend in liver recovery, but do not match the observed time scale and longterm recovery level. By contrast, the multiparameter optimization approach achieved a better fit to the observed dynamics and longterm recovery level with a lower residual error. The optimization approach resulted in variations in multiple parameters, five of which were significantly scaled in translating the rodent model to the human case (Fig. 1c). Even when optimizing all the parameters for best fit to data, M and k_{G} were two of the top three parameters that were altered from baseline ratbased parameters. Interestingly, the metabolic load value was similar across the three approaches for translating the rat model to the human case (Fig. 1c). The cell growth parameter was also tuned similarly between Cook et al. [8] approach and the multiparameter optimization (Fig. 1c). In the latter case, three additional parameters were varied in order to achieve the best fit: cell death sensitivity parameter θ_{ap}was higher, and immediate early signaling parameters, \( {\mathrm{K}}_m^{STAT3} \) and κ_{JAK} were lower in the human case compared to the rodent model.
We compared the optimizationderived and the metabolic scalingbased parameter values across the 27 patients. In the case of multiparameter optimization, the parameters were tuned individually to each of the 27 time series profiles. In the case of metabolic scalingbased tuning, we utilized the patientspecific BMI and derived body mass based on the average height of Japanese male and female, as appropriate (see Methods). For the Cook et al. [8] case, the relative cell mass growth constant (k_{G}) was modified to 6.5675e4, as per Cook et al. [8]. Our results indicate that the metabolic scaling parameter was tuned very similarly across all three approaches, yet with dissimilar fit to data as noted by the differences in the residual error (Additional file 1: Table S1). We observe that the metabolic scaling approaches of Young et al. [21] and Cook et al. [8] capture the regeneration dynamics only in a subset of the cases, and the multiparameter optimization improved the fit to the patient time series data in most cases (Additional file 1: Table S1). This improved fit in a subset of the patient cases was achieved by variations in additional parameters. A representative case of the model fit to data where the optimizationbased approach yielded similar parameters as the metabolic scaling approach is shown in Additional file 2: Figure S1. This is in contrast to the ID 71 case shown in Fig. 1, where several parameters were altered to translate the model from rat to human case.
Optimal values of the parameters corresponding to patient ID71 from Yamamoto et al. [18]
Parameter  Optimized value  Parameter  Optimized value 

M  5.8206  k_{deg}  6.9843 
k_{IL6}  1.4528  κ_{ECM}  32.9924 
κ_{IL6}  0.6878  k_{GF}  0.1014 
V_{JAK}  20,000  κ_{GF}  0.2016 
\( {\mathrm{K}}_{\mathrm{M}}^{\mathrm{JAK}} \)  9999.9999  k_{up}  0.0589 
κ_{JAK}  0.1695  k_{QP}  0.0072 
[proSTAT3]  1.9108  k_{PR}  0.0045 
V_{ST3}  749.9994  k_{RQ}  0.0520 
\( {\mathrm{K}}_{\mathrm{M}}^{\mathrm{ST}3} \)  0.1715  k_{prol}  0.0232 
κ_{ST3}  0.0828  k_{req}  0.0912 
V_{SOCS3}  24,000.0000  θ_{req}  7.9493 
\( {\mathrm{K}}_{\mathrm{M}}^{\mathrm{SOCS}3} \)  0.0006  β_{req}  2.9285 
κ_{SOCS3}  0.3173  k_{ap}  0.0982 
\( {\mathrm{K}}_{\mathrm{I}}^{\mathrm{SOCS}3} \)  0.0153  θ_{ap}  0.0321 
V_{IE}  249.9992  β_{ap}  0.0045 
\( {\mathrm{K}}_{\mathrm{M}}^{\mathrm{IE}} \)  17.9736  k_{G}  0.0007 
κ_{IE}  4.9595 
Analysis of virtual patients by varying individual parameters
We analyzed the individual effect of all the 33 model parameters on the longterm liver mass recovery fraction after two and half years post hepatectomy. Starting with the human time scale optimized parameter vector from the previous section, we varied one parameter at a time to develop a distribution of “virtual patients”. We simulated the liver regeneration profiles for each patient over the range of each varied parameter, while keeping the remaining parameters at their optimized values. We use the term “virtual patient” since all the parameter values are not patientspecific, but rather that these parameters take values within biologically reasonable bounds around the human time scale optimized value. We use this approach rather than the local sensitivity analysis in which a parameter is only marginally varied one at a time, since we are interested in the directionality of the influence of each parameter on the liver mass outcome over a wide range of parameter values.
Model parameters categorized according to the effect on liver response profile, leading to recovery alone, or causing either recovery or failure depending on the parameter value
Only recovery  Both recovery and failure  

Sensitive  Insensitive  Sensitive  Insensitive  
Improves recovery  Decelerate recovery  
\( {\displaystyle \begin{array}{l}{k}_{IL6},\left[ prosSTAT3\right],\\ {}{\kappa}_{SOCS3},{K}_I^{SOCS3},{V}_{IE},\\ {}{k}_{\mathrm{deg}},{k}_{GF},{k}_{QP},{k}_{PR},\\ {}{k}_{prol},{\beta}_{req},{k}_G\end{array}} \)  \( {\displaystyle \begin{array}{l}{\kappa}_{IL6},{K}_M^{JAK},{\kappa}_{JAK},\\ {}{K}_M^{ST3},{V}_{SOCS3},\\ {}{K}_M^{SOCS3},{K}_M^{IE},{\kappa}_{IE},\\ {}{\kappa}_{ECM},{\kappa}_{GF},{k}_{up},\\ {}{k}_{RQ},{k}_{req}\end{array}} \)  \( {\displaystyle \begin{array}{l}{V}_{JAK},{V}_{ST3},{\kappa}_{ST3},\\ {}\;{\theta}_{req},{k}_{ap}\end{array}} \)  M, β_{ap}  θ _{ ap} 
We sought to characterize [8] the differences between the present study and Cook et al. model [8] in terms of the differences in parametric sensitivity. We performed a similar analysis of directional influence of all the model parameters on liver regeneration outcome in a cohort of virtual patients using Cook et al. model [8]. Our results indicate that majority of the parameters showed similar effects on the liver response in Cook et al. model [8] and in the present model (Additional file 5: Table S2). Due to numerical difficulties in integrating the Cook et al. model [8] for certain parameter values, the effects of variation in the parameters M, k_{IL6}, k_{prol}, θ_{ap}, β_{ap} were evaluated in a more restricted range than the present model. These issues were overcome in the present model by consideration of senescent fraction in accounting for overall tissue mass (parameter ε in model equations below). [8]
Owing to the high sensitivity of our model to M and β_{ap} (Fig. 2de) as well as physiological relevance of increased hepatocellular metabolic activity and cell death (linked to M and β_{ap}, respectively) in the context of liver regeneration [22], we examined the effect of these parameters on the liver mass fraction response to resection in a cohort of virtual patients.
Shift in the regeneration modes of virtual patients with varying metabolic load and cell death sensitivity for different levels of resection
We analyzed the impact of intrinsic perioperative factors, metabolic load and cell death sensitivity, as well as the extrinsic factor of different levels of resection, on the regeneration modes of a cohort of virtual patients. We examined the modelpredicted temporal response of an individual patient’s liver mass post hepatectomy by focusing on the effect of variation in specific parameters that can cause liver failure or recovery, depending on the parameter values. Our parameter correlation analysis indicated that the metabolic load (M), cell death sensitivity (β_{ap}) parameters did not vary in a correlated manner with any of the model parameters across the patient data set (Additional file 4: Figure S3). Hence, in the present manuscript, we generate and analyze virtual patient cohorts that span the entire range for the subset of parameters considered without excluding any parameter subspaces, following similar practice in literature [23].
Subsequently, we simulated a cohort of virtual patients to analyze the combined effect of variation in M as well as β_{ap} on liver regeneration. We generated a cohort of 1000 virtual patients based on simultaneous variations in the parameters, M and β_{ap}, while the remaining 31 parameters were fixed at their optimized values. Simulations revealed that the parameter space can be partitioned into three distinct regions corresponding to different response modes (Fig. 4c). At low values of both the critical parameters, resection leads to a suppressed response mode. At high values of either or both of these critical parameters, resection leads to liver failure. It is likely that liver failure occurs in the case of high metabolic load as the remnant cells are unable to meet the high functional requirements. The intermediate region corresponds to normal growth enveloped by the suppressed growth mode. These model predictions indicate that complete liver mass recovery requires a balance between the two intrinsic perioperative factors (metabolic load and cell death sensitivity). To evaluate the impact of these intrinsic perioperative factors on the liver regeneration in rats we analyzed the combined effect of variations in metabolic load and cell death sensitivity (M and β_{ap}) in our model for 33.3% hepatectomy with the rat specific parameters from Cook et al. [8]. We found that the distribution of the suppressed, normal and failure response modes in the parameter space of M and β_{ap} is similar in the rat to that in human case, even though the specific values of these parameters are different for the two species (Additional file 6: Figure S4). Suppressed mode was observed for low metabolic load and cell death sensitivity, while failure mode occurred at the other extreme of high values for these factors. The results suggest that normal recovery mode likely depends on a balance between metabolic load and cell death sensitivity in both rat and human liver resection scenarios.
Determining the safe level of resection based on the metabolic load (M), an intrinsic perioperative factor
The phase portrait for the virtual patients under study captures the two attractors of liver failure and liver recovery (Fig. 6a). The phase plane analysis demonstrated that the trajectories for high levels of resection progress towards the liver failure attractor, whereas trajectories for low levels of resection converge to the attractor of liver recovery. The phase plane shows a clear demarcation between the safe and the unsafe level of resection for liver surgery. Figure 6ac shows the phase plane for increasing values of metabolic load. The results indicate that the span of the attractor for liver recovery progressively decreases with increasing metabolic load, and finally vanishes, leading to liver failure irrespective of the level of resection. This suggests a change in the multiplicity of the system, which may have clinical implications in considering the surgical options. We utilized the phase plane analysis to examine the threshold of liver failure for a given virtual patient. Our analysis indicates an inverse monotonic relationship between threshold of liver failure and metabolic load, such that higher the metabolic load lower is the level of maximum resection that can still lead to recovery, i.e., higher requirement for safe level of remnant liver mass post resection (Fig. 6d).
In parallel, we analyzed the effect of the intrinsic perioperative factor of cell death sensitivity (β_{ap}) on the threshold of failure. Our results indicate that cell death sensitivity has a significant effect on the safe level of resection (Additional file 7: Figure S5AC), with a similar inverse monotonic relationship between the level of cell death sensitivity and threshold of failure (Additional file 7: Figure S5D).
Determining the safe level of resection based on variations in both metabolic load and cell death sensitivity (M and β_{ap})
Thus far, we analyzed the threshold of liver failure (conversely, the safe level of resection) in a cohort of virtual patients based on individual variations in either metabolic load (M) or cell death sensitivity (β_{ap}). Both these parameters had a similar effect on the threshold of failure of the virtual patients. In this section, we investigated how the threshold of failure changes in virtual patients based on simultaneous variations in these two intrinsic perioperative parameters. We simulated the dynamic model by varying the two parameters (M and β_{ap}) using a Sobol sample of size 50 × 50, and the remaining parameters were fixed at human optimized value for twothirds resection (Fig. 1b; Table 1). Sobol sampling of size 50 × 50 corresponds to metabolic load and cell death sensitivity such that each value of metabolic load is paired with all the values of cell death sensitivity and viceversa. This approach yielded a cohort of 2500 virtual patients.
Discussion
We started with a quantitative model of liver regeneration response to resection and finetuned the parameters to account for a normal liver recovery profile at humanrelevant time scales. We built on this initial simulation and analyzed the distribution and modes of response of a virtual patient cohort to varying level of resection, which potentially account for differences due to disease etiology, patient demographics, and perioperative conditions. Notably, the range of parameter variation covered a full span of an individual virtual patient’s potential response along three distinct modes: accelerated growth, slower recovery, and failure. Our approach differs from that of a population of models (POM) approach [24], in which the objective is to account for the distribution of responses in clinical data, and the response of a virtual patient is accepted or rejected based on a specified tolerance limit. By contrast, our approach is targeted at characterizing the entire range of responses to analyze the distinct modes of response across all virtual patients. Such an unbiased approach has been pursued in other studies with informative results on parameter subspaces that distinguish qualitatively different patient responses [23]. Our Sobol samplingbased wide range of simulations led us to identify subsets of critical parameters and their combinations that govern the transitions in the response of virtual patients to varying levels of resection. Our approach to accounting for dynamics of human liver regeneration response to resection is different from that of Yamamoto et al. [18] study from which we utilized the liver volumetric data to tune the computational model parameters. Yamamoto et al. [18] model was based on modification of a logistic model whose response is largely governed by the sign of the initial rate of response to resection (positive value leading to recovery, and negative value leading to failure). This simplified representation allowed the development of a discriminant function to correlate the rate of liver regeneration to the pre and perioperative clinical factors, and then predicted the outcome based on a binary classification of the initial rate of liver regeneration being positive or negative. By contrast, we utilized a multiscale network model that contains a relatively more detailed representation of molecular interactions and cellular functional states, and tuned the parameters of the model to account for the observed timescales of human liver regeneration. We analyzed the influence of key parameters on the outcome based on the level of resection, and examined the quantitative relationship between variations in two key parameters and the threshold of failure.
Our study considered a cohort of virtual patients based on two critical parameters: metabolic load and cell death sensitivity. These two modelpredicted critical parameters are perioperative factors and have the potential to be estimated from patientspecific clinical information. For instance, metabolic load can be empirically related to mass, body mass index [25], and further modified based on other factors such as age, gender, etc. of the patient [26, 27]. We expect that cell death sensitivity can be related to disease etiology, patient’s medical history, and perioperative conditions such as blood loss (Yamamoto et al. [18]). It is reasonable to expect that certain patients may be more sensitive to injury than others depending on the advanced versus earlier stage of the underlying liver disease. In addition, aging also has a significant impact on cell death sensitivity [28, 29]. On the contrary, healthy liver donor transplant is likely less sensitive to increase in metabolic demand per unit of tissue and cell death sensitivity as compared to liver of a patient with underlying chronic disease or a patient being operated to treat a metastatic tumor. Understanding the combinatorial effect of these two intrinsic parameters, which likely vary from patienttopatient, on the mode of response to injury can help with better characterization of the perioperative conditions under which liver surgery can lead to a successful recovery or failure.
The determination of threshold of liver failure through a phase portrait approach is analogous to detection of tipping point in a complex dynamical system. Tipping point is a “point of no return” that results in a transition from the state of normal functioning to a catastrophic state [30]. Examples of tipping point are widespread, such as extinction of species in ecological systems, and heavy load on electrical grids or internet, etc. Once such a catastrophic state is attained, the system collapses and there is no going back. In the case of resection, the tipping point corresponds to the threshold of resection beyond which the liver cannot recover and will always progress towards reduced mass and failure. Our analysis suggests that the tipping point after resection is dependent on a combination of the level of metabolic load and the extent of cell death sensitivity [31].
We emphasize that the network modeling approach presented in this study is a post hoc analysis of the dynamics of human liver regeneration. Additional work on identifying patientspecific parameters and developing parameter signatures corresponding to different patient groups based on demographics, disease etiology, etc., so that the dynamic modeling can serve as a pre hoc predictive tool that can aid in clinical decision making. Availability of detailed perioperative clinical information opens new opportunities for developing a categorical (e.g., classificationbased) or a quantitative relationship between these physiological parameters (M and β_{ap}) and patientspecific clinical parameters [18], aiding generalization of the dynamic modeling approach. For example, if the model prediction suggests a likelihood of liver failure following resection, interventions such as portal vein embolization to induce regeneration and enhance preresection liver mass [32], preoperative dietary restriction [33] and nutritional changes [34] to reduce the risk of ischemiareperfusion injury, as well as preoperative reduction in systemic inflammation [35, 36], so as to modulate the metabolic load and cell death sensitivity parameters and thereby shift the likely response to the region of full recovery of liver mass. In addition, the phase portrait technique can be employed to further aid in predicting the likely safe level of resection.
Thus far, the computational modeling efforts by us and others have considered liver as a uniform tissue in a single lumped compartment [6, 8, 9]. Opportunities exist for computational modeling approaches that explicitly consider multiple lobes of differing size with potentially distinct responses to resection [37]. In such a scenario, the metabolic load and cell death sensitivity parameters may need to be considered as heterogeneous across liver tissue. Noninvasive imaging techniques, which are regularly employed in the clinic to obtain whole organ physiological and functional parameters, can aid in evolving the dynamic models in such potentially fruitful directions.
Conclusions
In the present study, we have extended and finetuned a network model of liver regeneration to predict the dynamics of human liver response to resection. Analysis of the computational model helped us identify two crucial factors associated with the metabolic load and cell death post resection, which can control the dynamics of liver regeneration response. Our simulations indicate that the balance between these two factors is critical to drive the response towards liver recovery or failure. We evaluated the distribution of responses in a cohort of virtual patients, and analyzed the responses using phase plane analysis to identify how the threshold of liver failure varies as a function of the modelpredicted critical factors. Our analysis demonstrates a modelbased approach to estimate the safe level of resection to increase the likelihood of recovery. These results serve as a basis for future efforts focused on relating the two modelpredicted critical factors to patientspecific pre and perioperative clinical parameters to aid in clinical decision making.
Methods
Computational model
We employed the mathematical model of Cook et al. [8] which is an extension of the model proposed by Furchtgott et al. [6]. This model describes the hepatocyte growth after partial hepatectomy. In this model, hepatocytes enter the cell cycle and is assumed to exist in one of the three states quiescent (Q), priming (P) or replicating (R) via a cascade of signals from cytokines and growth factors [38, 39]. We assume the liver to be a lumped system, and consequently, the molecular changes in hepatocytes are considered to be spatially homogeneous throughout the remnant liver mass in the present lumped parameter model.

Cellular states:

Molecular factors:

Relative cell mass:

Initial conditions:
Simulation and parameter optimization
The Matlab code used for model simulation in this study is available as supplemental information in Additional file 8. Simulations were performed in Matlab using ode15s. The initial guess values of the parameters for optimization were based on the values given in Cook et al. [8] for human population. The parameters of the model were optimized using the sparse regularization technique of elastic net, which is a combination of ridge regression and Lasso [40, 41]. We sampled the parameter space using Sobol sampling [42, 43] with a few parameters varied over a tenfold range and the remaining parameters varied within a twofold range around the initial value (Additional file 9: Table S3). The parameter ranges were so chosen to avoid the numerical integration error since the system of equations for the model is stiff. The Matlab code for parameter optimization is available as supplemental information in Additional file 10.
Metabolic scaling from rat to human based on body mass
In addition to M, we also modified the relative cell mass growth constant (k_{G}) to 6.5675e4 when simulating the model according to the parameter tuning published by Cook et al. [8] to translate from rat to the human case.
Virtual patient cohort generation
In order to identify the critical factors controlling both the mechanism of liver recovery and failure, we generated in silico cohorts of virtual patients [23, 45] starting with specific patient data (ID71) from Yamamoto et al. [18] and introducing wide variation in specific parameters as detailed in the Results. We utilized a Sobol sampling approach that yields a spacefilling sample with little bias [43]. The modelpredicted critical factors were first varied one at a time and then in combination to analyze their influence on the individual patient liver response to surgical resection. The Matlab code for virtual patient cohort analysis is available as supplemental information in Additional file 11.
Decision boundary and threshold of failure
The decision boundary demarcating the normal liver growth from the other classes of liver response were drawn using a support vector machine approach with a third order polynomial kernel using fitcsvm function in Matlab with BoxConstraint in the range of 1 to 10^{4}, for different levels of resection [46]. For each virtual patient, the threshold of failure was calculated by evaluating the response to varying level of resection, ranging from 5 to 90%. The response was considered as liver failure if the liver mass fraction was below 0.1 at the 2 year time point postsurgery. Responses of a virtual cohort of patients were simulated for different levels of resection to identify the threshold of resection beyond which any given virtual patient undergoes liver failure.
Response mode analysis
The in silico generated regeneration profiles of different virtual patients are classified as normal growth when the liver mass fraction at 2.5 years postsurgical resection is within 0.9–1.1 and as suppressed mode for fraction below 0.9 for recovered patients. Liver failure modes are those where the liver mass fraction is below 0.1 after a 2 year post surgery. The unresponsive mode corresponds to the case where the virtual patient does not show any change in the liver mass fraction after surgery.
Clinical data set
The data used for the present work has been obtained from Yamamoto et al. [18], which contained information on liver volume post liver resection in 196 patients. We analyzed the data to identify the subset of patients who recovered fully i.e., the patients whose final liver volume was in the range of 90 to 100% of the preoperative liver volume. There were a total of 101 patients whose liver volume recovered fully. These patients were further categorized based on the temporal profile of liver growth. Some patients showed delayed liver growth, while others exhibited suppressed but continuous liver growth.
Model reproducibility
Simulations presented in the current work were reproduced independently by a laboratory colleague, not associated with the study, who developed new Matlab code based on the model equations and parameter values included in this manuscript. See Additional file 12: Figure S6 for details. The original and reproduced model are provided in the Additional files 1 and 13.
Declarations
Acknowledgements
BKV thanks Mr. Aalap Verma and Ms. Abha Belorkar for discussions on the mathematical modeling and statistical understanding of this model. We thank Mr. Timothy Josephson for independently reproducing the computational modeling results presented in this study, and providing the Matlab code for the reproduced model included in the supplementary material.
Funding
Financial support for this work and publication costs for open access were provided by National Institute on Alcohol Abuse and Alcoholism grant R01 AA018873, National Institute of Biomedical Imaging and Bioengineering grant U01 EB023224, and National Science Foundation grant EAGER 1747917. The funding sponsors had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, and in the decision to publish the results.
Availability of data and materials
Matlab code for model simulation, parameter optimization, and virtual patient cohort analysis are provided in the Additional files 8, 10, 11. Matlab code for reproduced model is available as Additional file 13. Parameter values are included in Table 1. Model specification and description are included in the main text.
Authors’ contributions
Conceived the study  RV; designed the in silico experiments  BKV, PS, RV; performed simulations  BKV; analyzed and interpreted the results  BKV, PS, RV; wrote the manuscript  BKV, PS, RV. All authors read and approved the final manuscript.
Ethics approval and consent to participate
Not applicable.
Consent for publication
Not applicable.
Competing interests
The authors declare that they have no competing interests.
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Authors’ Affiliations
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