Neonatal Jaundice – Its Mathematical Model and Treatments Presented by: Hannah Authors: Tram Hoang, Shazia Khan, Lorena Ortiz Department of Mathematics California State University Fullerton

Under normal circumstances… Spleen filters out old/damaged blood cells Old red blood cells produce bilirubin – also known as fat soluble Waste product is carried to the liver Converted into a water-soluble form of bilirubin This is then excreted easily through urine, stool, and sweat glands Spleen removes old/damaged red blood cells Fat-soluble is toxic Water-soluble is not

Neonatal Jaundice: causes and dangers Accumulation of bilirubin happens when is it not excreted properly Common causes: Healthy newborn has high volume of red blood cells Translates to high volume of bilirubin Spreads to skin and eyes Premature infant Premature liver cannot convert bilirubin fast enough Lack of normal floral bacteria in gastrointestinal track Normal function is to aid in breaking down bilirubin to prevent reabsorption In extreme cases: infant can develop Kernicterus Major problem in 50s-70s – less emphasis in the 90s but has become more frequent again Baby 8 weeks after birth with hemolytic disease Kernicterus – poor feeding, high pitched crying, slow motor skills (can take up to 5 years to walk) Second phase: backward arching of neck, fever After first year: tremors, hearing loss (damage to cochlear in brainstem), limitation of upward gaze (eye movement problems) Athetosis Hearing loss may be the only symptom in some children

Physiologic jaundice vs. Pathologic jaundice Rapid destruction of fetal red blood cells Bilirubin produced is reabsorbed from the intestinal track and sent back to the liver in the form of unconjugated (toxic) bilirubin Pathologic Jaundice Affects infant’s health within first 24 hours of life Main cause: Hemolytic disease Fetal and neonatal blood cells are exposed to fetus too early in development Bilirubin accumulates Physiologic Jaundice occurs in ~50% of infants and 80% of preterm infants Paper looks at physiologic jaundice since it is more prevalent and usually curable – not a disease

Treatments Blood transfusion Phototherapy Medication Infant is exposed to fluorescent light that causes chemical change Medication Can either assist in excretion or block production

Tin-mesoporphyrin (Sn-MP) Blocks production of bilirubin Has been shown to be more effective for reducing the time it takes to get bilirubin back to a normal level Reduces need for phototherapy Diminishes the oxidative metabolism of bilirubin in neural tissues Requires less dosage than alternative medication

Compartment Mass Transport Model Three compartments: bloodstream, body surface, and liver/bile C1 = hourly creation of bilirubin x1, x2, x3 = concentration of bilirubin in blood, body surface, and liver at time t x4 not included because it does not flow back into the body aij = constants of proportionality i = connected compartment j = destination compartment C1 C1 = concentration per hour

Mass Balance Law Model was derived using this law Body surface: Used matlab to solve numerically C1 = hourly creation of bilirubin x1, x2, x3 = concentration of bilirubin in blood, body surface, and liver at time t aij = constants of proportionality i = connected compartment j = destination compartment By accounting for material entering and leaving a system, mass flows can be identified which might have been unknown, or difficult to measure without this technique. The exact conservation law used in the analysis of the system depends on the context of the problem, but all revolve around mass conservation, i.e. that matter cannot disappear or be created spontaneously Blood: Liver:

Assumptions A baby’s ability to excrete bilirubin from the liver into the stool at birth is P0 Percentage of the normal functioning liver The liver improves over a period of time t at maturity rate rm Jaundice is diagnosed by measuring concentration by a complete blood count The model gives measurements of bilirubin levels at the surface and in the liver Not realistically known Larger rm = faster recovery rate (1-P0) = proportion of bilirubin not excreted via stool Graph has fixed maturity rate

Time Dependence Larger maturity rate rm = faster recovery rate This causes some parameters to become time dependent Transport rates of bilirubin concentration out of the liver Rate of excretion via stool Rate of transport out of the liver back to blood compartment A43 – liver to outside environment A13 – liver back to blood x1, x2, x3 = concentration of bilirubin in blood, body surface, and liver at time t aij = constants of proportionality i = connected compartment j = destination compartment A baby’s ability to excrete bilirubin from the liver into the stool at birth is P0 P0 = Percentage of the normal functioning liver The liver improves over a period of time t at maturity rate rm

Model with no treatment Bilirubin concentration is increased in each compartment over 72 hours Alarming level of concentration identified around hour 10 Without treatment, conditions stay like this for 24 hours Goal: to bring concentration down as quickly as possible

Treatment 1: Blood Transfusion Bilirubin concentration from 15-20 mg/dl requires immediate medical intervention Causes blood to flow in and out of the blood compartment New blood carries some bilirubin into the blood compartment Removal of blood carries bilirubin out of the body Transfusion rate Treatment on/off Treatment is on when delta(t) = 1, off when delta(t) = 0 C1 = hourly creation of bilirubin x1, x2, x3 = concentration of bilirubin in blood, body surface, and liver at time t aij = constants of proportionality i = connected compartment j = destination compartment Creation of bilirubin Volume of blood in the body Bilirubin concentration in new blood

Treatment 2: Phototherapy Bilirubin levels from 1—15 mg/dl is considered a mild case Bilirubin is bleached by the light, and becomes water-soluble Structure of bilirubin is changed and the excretory capacity of sweat glands is increased Important influences on effectiveness: Power output of light Surface area exposed to the light Blue-green spectrum Important to stop treatment when concentration falls below a certain level or there will be negative effects C1 = hourly creation of bilirubin x1, x2, x3 = concentration of bilirubin in blood, body surface, and liver at time t aij = constants of proportionality i = connected compartment j = destination compartment treatment Intensity of bililight Rate out of body surface to outside environment

Treatment 3: Medication Infants do not need phototherapy after use of Sn-MP Decreases bilirubin production rate Sn-MP is administered with dose of N3 every D hours C1 = hourly creation of bilirubin x1, x2, x3 = concentration of bilirubin in blood, body surface, and liver at time t aij = constants of proportionality i = connected compartment j = destination compartment Models behavior of bilirubin level in response to drug

Treatment Simulations No treatment Phototherapy Blood Transfusion Medication

Treatment Effectiveness & Conclusion Blood Transfusion and Phototherapy Blood Transfusion and Medication Blood Transfusion, Phototherapy, and Medication Concentration mg/dl Treatments All equalize to the same level at the 120th hour Time Time Time Combination of all three makes each treatment require less time and the bilirubin concentrations to level off more steadily

Conclusion & Extensions Model is able to estimate the behavior of bilirubin concertation in relation to each treatment Need data and estimated parameters to improve accuracy of the model Studying effectiveness of medication and lasting effect of the drug can help improve model behavior for these drugs Would like to predict the liver function to provide/suggest optimal treatment depending on infant’s condition

The end!