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A Solution Tool for Clinical Challenges in the Prognosis of Liver Injury

The Power of Mathematical Assessment via Modeling and Inference Approaches

By Aditi Ghosh and Anuj Mubayi

Acute liver failure (the loss of liver function) occurs rapidly over days or weeks and usually affects people without preexisting liver disease. Liver trauma due to drug overdose—such as an excessive dependence on acetaminophen (APAP)—is one possible source of acute liver injury. However, the most frequent cause of liver injury is ischaemic hepatitis (IH), which occurs due to centrilobular liver cell necrosis that stems from an underlying condition like shock, trauma, or surgery. We use mathematical modeling techniques, statistical inference, and inverse problem approaches to study the dynamics of drug-induced acute liver injury (a serious but preventable life-threatening clinical manifestation) and heart-attack-influenced IH liver injury [1, 2].

Figure 1. Flowchart of the Case 1 model variables and their interactions.
APAP is an over-the-counter analgesic that is commonly known as Tylenol and widely used throughout the world. More than 28 billion doses are distributed annually in the U.S., and acute liver injury due to APAP overdose accounts for approximately 56,000 emergency room visits, 26,000 hospital admissions, and 500 deaths in the country each year [3, 4, 5]. A major challenge for practitioners is to accurately know a patient’s patterns of alcohol use—which drastically impact the damage from drug overdose—in order to carry out precise prognosis and prescribe proper treatment for controlling liver deterioration. We extend an existing model [5] to evaluate alcohol patterns’ effects on APAP overdose-related liver degradation. Our goal was to: (i) understand alcohol’s impact on the dynamics of APAP-induced liver injury, (ii) quantify alcohol intake levels in patients, and (iii) provide a method of tracking the extent of liver damage in real time that results from a patient’s alcohol consumption history under distinct treatment regimes. 

IH, on the other hand, accounts for 10 percent of all patients who are admitted to hospital intensive care units. Low oxygen extraction by hepatocytes, hepatic blood perfusion, systemic arterial hypoxemia, and/or venous congestion all lead to IH. Usually the only way to determine a case of IH is to rule out all other possible conditions for liver injuries. Although medical practitioners traditionally consider certain liver biomarkers at the time of admission, a lack of proper quantitative methods prevent the prediction of biomarker trajectories — which are necessary for determining treatment profiles for IH patients. 

Several clinical scoring systems (e.g., Roussel Uclaf Causality Assessment Method and the Clinical Diagnostic Scale) have gradually emerged for liver injury diagnosis, but certain serious inherent deficiencies currently limit their predictive value. Our study is therefore the first of its kind to fill this gap and provide an accurate novel prognosis method that produces real-time updates of biomarkers that are based on other patient characteristics to ultimately improve tools for diagnosis and clinical management. We aimed to identify mechanisms that can alter associated biomarkers and reduce the density of damaged hepatocytes, thus minimizing the chances of IH and optimizing treatment efficacy.

Figure 2. Flowchart of Case 2 model variables and their interactions.
We now present our mathematical modeling process and its results in the context of the two types of liver injuries: alcohol-influenced APAP-overdose-related liver injury (Case 1) and cardiac failure-initiated IH (Case 2). 

Case 1 

Researchers believe that alcohol (EtoH) intake significantly affects APAP-induced hepatotoxicity. In fact, various studies indicate that alcohol regulates the formation of a toxic compound NAPQI (via the induction of cytochrome enzymes such as P450, CYP2E1, CYP1A2, and CYP3A4). In our mathematical model—a dynamical system—we incorporated select key mechanisms that are known to occur inside the hepatocyte cells via critical bioindicators, which are represented as variables in the model (see in Figure 1). Patients with chronic alcoholism are at higher risk of developing APAP-induced liver injury due to the induction of CYP2E1. In contrast, we see a protective effect of alcohol ingestion in those who drink occasionally due to the inhibition of CYP2E1. Alcohol ingestion thus impacts NAPQI formation in a manner that depends on the amount of consumption and the time lag between alcohol and APAP ingestion. 

To model the effect of alcohol on APAP-induced liver injury, we connected the Model for Acetaminophen-induced Liver Damage (MALD) [5] to a physical pool of CYP2E1 enzymes (see Figure 1). We assume that a single physical pool of CYP2E1 rapidly degenerates after alcohol enters the body at a certain degradation rate. We also updated the original MALD to include the mechanisms of alcohol metabolism. APAP is metabolized via glucuronidation, and sulfation occurs due to the overdose of APAP and CYP2E1 enzymes that is initiated by alcohol intake. Sulfation of APAP follows Michaelis-Menten kinetics, and the induction/inhibition of CYP2E1 via alcohol follows Hill kinetics

In the case of occasional alcohol use, we consider three different scenarios: (i) when alcohol is ingested within a range of one to seven days before the ingestion of APAP, (ii) when APAP is simultaneously ingested with alcohol, and (iii) when alcohol is ingested within a range of one to seven days after the ingestion of APAP. We also consider a patient who uses alcohol chronically. We obtain the minimum hepatocyte ratio (MHR), which is the ratio of the minimum hepatocyte count in scenario wherein alcohol is ingested along with APAP to the minimum hepatocyte count when APAP is ingested without alcohol. 

Figure 3. Hepatocyte death with varied oxygen levels with no treatment.
We found that \(\rm{MHR}<1\) when a patient ingests alcohol one to seven days before APAP, thereby causing more liver damage than a single APAP overdose. \(\rm{MHR}>1\) in the case of simultaneous ingestion of alcohol and APAP or ingestion of alcohol a day after APAP overdose, both of which cause either no liver damage or less damage than the result of a single APAP overdose. The MHR for alcohol ingestion one to seven days after APAP overdose is nearly one, which indicates that the consumption of alcohol a day later does not affect the liver injury that already incurred. In the case of an APAP overdose patient who is a chronic alcohol user, hepatoxicity in the liver increases with the amount of alcohol intake if CYP2E1 is constant. Using synthetic patient data, we also verified the hypothesis that alcohol consumption increases susceptibility to APAP toxicity with chronic alcohol intake. 

Case 2

Next, we studied the role of biomarkers in changing the density of damaged hepatocytes and explored the probability of IH. We validated our corresponding mathematical model with patient-level data from the existing literature; our model analysis provides an approach to predict the level of biomarkers based on variations in the body’s systemic oxygen due to various factors like cardiac failure. 

We employ a logistic function over time to capture the changing oxygen concentration (\(\rm{O}2\), which is represented by \(\rm{O}\)) in the body. This method better fits the physiology, as it takes time for the liver to regain oxygenated blood due to treatment intake. Our mathematical model considers a coupled dynamic of healthy hepatocytes (represented by \(\rm{H}\)), hepatocytes that are damaged by IH (represented by \(\rm{Z}\)), and relevant biomarkers such as AST, ALT, and LDH (see Figure 2). Physicians often use the levels of these biomarkers to determine the type or severity of liver injury. We assume that cardiac injury decreases blood flow to the liver and causes a lack of oxygen, and that any hepatocyte death is the result of necrosis due to insufficient adenosine triphosphate (\(\rm{ATP}\), which is represented by \(A\)). This assumption excludes hepatocyte death by any other processes.

Figure 4. The estimated time to reach the critical 30 percent hepatocyte level for irreversible damage given the initial oxygen level.
We predict the real-time peak levels of different biomarkers—namely AST, ALT, and LDH—due to IH liver injury at various oxygen levels and treatment regimes. Our study revealed that with no treatment and 30 percent initial oxygen (at the time of diagnosis), it takes approximately 12 hours for 70 percent of the hepatocytes to die (see Figure 3). However, it takes closer to 36 hours to reach the same percentage with 80 percent initial oxygen (see Figure 4). These results show that IH liver injury becomes irreversible in 12 to 36 hours. The treatment time is also a crucial factor in determining the extent of liver damage. We found that delaying treatment for just four hours causes the damage to extend from 40 percent to more than 60 percent for the most common initial oxygen level. 

Future Work 

Because our present model only considers IH due to cardiac failure, we aim to consider IH that is caused by other factors in the future. This extension may require the incorporation of other forms of cell deaths, such as apoptosis. As with any modeling study, our work also has limitations — such as the need for more thorough validation via data that captures different types of patients with various preexisting conditions and demographic factors. Our current work includes development of physician-friendly machinery: a dashboard that uses patients’ biomarker readings at the time of admission in the aforementioned mechanistic models to assist practitioners in timely diagnosis and prognosis for patients that may experience liver failure in the future.

[1] Ghosh, A., Berger, I., Remien, C.H., & Mubayi, A. (2021). The role of alcohol consumption on acetaminophen induced liver injury: Implications from a mathematical model. J. Theor. Biol., 519, 110559.
[2] Ghosh, A., Onsager, C., Mason, A., Arriola, L., Lee, W., & Mubayi, A. (2021). The role of oxygen intake and liver enzyme on the dynamics of damaged hepatocytes: Implications to ischaemic liver injury via a mathematical model. PLOS One, 16(4), e0230833.
[3] Nourjah, P., Ahmad, S.R., Karwoski, C., & Willy, M. (2005). Estimates of acetaminophen (paracetomal)-associated overdoses in the United States. Pharmacoepidem. Drug Safety, 15(6), 398-405
[4] Reddyhoff, D., Ward, J., Williams, D., Regan, S., & Webb, S. (2015). Timescale analysis of a mathematical model of acetaminophen metabolism and toxicity. J. Theor. Biol., 386, 132-146.
[5] Remien, C.H., Adler, F.R., Waddoups, L., Box, T.D., & Sussman, N.L. (2012). Mathematical modeling of liver injury and dysfunction after acetaminophen overdose: Early discrimination between survival and death. Hepatology, 56(2), 727-734.

  Aditi Ghosh is a tenure-track assistant professor at the University of Wisconsin-Whitewater and will be joining Texas A&M University-Commerce as a tenure-track assistant professor in the fall of 2021. Her research interests are in mathematical biology and her work has been critical in involving and preparing undergraduate students for cutting-edge practical research and prestigious national mathematical modeling competitions.  
Anuj Mubayi is an associate director in PRECISIONheor’s Advanced Modeling Group. He is an applied and computational mathematical scientist with more than 10 years of experience working on modeling problems that are of interest to the public health communities, such as the design and evaluation of cost-effective intervention programs in the healthcare sector.
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