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

2. 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

3. 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

4. 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

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

6. 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

7. 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

8. 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:

9. 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

10. 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

11. 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

12. Treatment 1: Blood Transfusion
Bilirubin concentration from 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

13. 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

14. 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

15. Treatment Simulations
No treatment
Phototherapy
Blood Transfusion
Medication

16. 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

17. 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

18. The end!