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Blood Bank

Laboratory Mathematics

Unlike Clinical Chemistry, where mathematics focuses on concentration curves and unit conversions, mathematics in Blood Bank is primarily focused on probability, statistics, and dosage calculations for immune prophylaxis. These calculations are critical for inventory management (ensuring enough blood is available for patients with antibodies) and for preventing Rh hemolytic disease of the fetus and newborn (HDFN). Mastery of these formulas is essential for the laboratory scientist

Probability & Antigen Frequency

One of the most common logistical challenges in Blood Bank is finding compatible blood for a patient with multiple unexpected antibodies. The laboratory scientist must calculate the number of units that must be screened to yield a specific number of antigen-negative units. This relies on knowing the prevalence of antigens in the donor population

Calculating the Number of Units to Screen

To determine how many random donor units must be antigen-typed to find a specific number of compatible units, the laboratory scientist uses the frequency of antigen-negative individuals in the population

  • The Formula \[ \text{Units to Screen} = \frac{\text{Number of Compatible Units Needed}}{\text{Frequency of Antigen-Negative Individuals}} \]
  • Step 1: Determine the percentage of the population that is negative for the antigen. For example, if the K (Kell) antigen is present in 9% of the population, then 91% (0.91) are K-negative
  • Step 2: Divide the number of units requested by the negative frequency (in decimal form)
  • Example: A patient needs 2 units of K-negative blood \[ \frac{2}{0.91} = 2.19 \] Since you cannot screen a fraction of a unit, you round up. You would need to screen at least 3 units to statistically find 2 compatible ones

Multiple Antibodies (Combined Probability)

When a patient has multiple antibodies (e.g., Anti-E and Anti-K), the probability of finding a unit negative for both antigens is determined by multiplying their individual negative frequencies

  • Step 1: Identify the antigen-negative frequency for each antibody
    • E antigen is present in ~30% of the population; therefore, 70% (0.70) are E-negative
    • K antigen is present in ~9% of the population; therefore, 91% (0.91) are K-negative
  • Step 2: Multiply the negative frequencies to find the combined probability \[ 0.70 \times 0.91 = 0.637 \text{ (or } 63.7\% \text{ of the population is negative for both)} \]
  • Step 3: Apply the screening formula. If the patient needs 4 units: \[ \frac{4}{0.637} = 6.27 \] The laboratory scientist would need to pull approximately 7 units from the shelf to find 4 that are negative for both antigens

Rh Immune Globulin (RhIG) Dosage Calculation

RhIG (RhoGAM) is administered to Rh-negative women to prevent sensitization to the D antigen after exposure to Rh-positive fetal blood. The standard dose (300 \(\mu\)g) protects against a bleed of 30 mL of Whole Blood (or 15 mL of Packed Red Blood Cells). When a Fetal-Maternal Hemorrhage (FMH) exceeds this amount, a calculation based on the Kleihauer-Betke (KB) test or Flow Cytometry is required to determine the number of vials needed

The Calculation Steps

  1. Calculate the Volume of the Hemorrhage: The KB test reports the percentage of fetal cells. To convert this to a volume, multiply the percentage by the estimated maternal blood volume (assumed to be 5000 mL) \[ \text{Volume of FMH (mL)} = \% \text{ Fetal Cells} \times 50 \] (Note: The factor 50 is derived from 5000 mL / 100)

  2. Calculate the Number of Vials: Divide the volume of the hemorrhage by 30 (since one vial covers 30 mL of whole blood) \[ \text{Raw Vials} = \frac{\text{Volume of FMH}}{30} \]

  3. Apply Safety Margin Rules (Rounding): This is the most critical step and is unique to Blood Bank rules to ensure overdose rather than underdose

    • If the number to the right of the decimal point is < 5: Round down and add 1 vial
    • If the number to the right of the decimal point is \(\ge\) 5: Round up and add 1 vial

Example Calculation

A Kleihauer-Betke stain reveals 1.3% fetal cells

  1. Volume: \(1.3 \times 50 = 65 \text{ mL}\) of fetal whole blood
  2. Raw Vials: \(65 / 30 = 2.17\) vials
  3. Rounding: The number is 2.17. The decimal (.17) is less than.5, so we round down to 2
  4. Add Safety Margin: \(2 + 1 = 3\) vials

Alternative Scenario: If the calculation yielded 2.8 vials, we would round up: to 3, and then add 1, for a total of 4 vials

Hardy-Weinberg Principle

While primarily a genetics concept, the Hardy-Weinberg equation is used in Blood Bank to estimate gene frequencies based on phenotype prevalence. This is useful when determining the likelihood of a person being homozygous or heterozygous for a specific antigen (e.g., determining the likelihood of a father being homozygous for D to predict HDFN risk)

  • The Equation \[ p^2 + 2pq + q^2 = 1 \]
    • \(p\) = frequency of the dominant allele (e.g., \(Jk^a\))
    • \(q\) = frequency of the recessive allele (e.g., \(Jk^b\))
    • \(p^2\) = frequency of the homozygous dominant genotype (\(Jk^a/Jk^a\))
    • \(2pq\) = frequency of the heterozygous genotype (\(Jk^a/Jk^b\))
    • \(q^2\) = frequency of the homozygous recessive genotype (\(Jk^b/Jk^b\))
  • Application: If you know the percentage of the population that lacks an antigen (the null phenotype, representing \(q^2\)), you can find the frequency of the gene (\(q\)) by taking the square root
    • Example: 16% of the population is Rh-negative (dd)
    • \(q^2 = 0.16\)
    • \(q \text{ (frequency of d gene)} = \sqrt{0.16} = 0.4\)
    • Since \(p + q = 1\), then \(p \text{ (frequency of D gene)} = 1 - 0.4 = 0.6\)

Serial Dilutions & Titers

Titration is a semi-quantitative method used to determine the concentration of an antibody. This is often used in prenatal studies to monitor the strength of maternal antibodies (e.g., Anti-D) that could cause HDFN

  • The Dilution Factor Most Blood Bank titrations are two-fold serial dilutions. Tube 1 is undiluted (1:1), Tube 2 is 1:2, Tube 3 is 1:4, Tube 4 is 1:8, etc
  • Determining the Titer The titer is reported as the reciprocal of the highest dilution that produces a macroscopic (1+) agglutination reaction
    • If the last positive tube was the 1:64 dilution, the Titer is 64
  • Calculating Total Volume/Solute \[ \text{Concentration} = \frac{\text{Volume of Solute}}{\text{Total Volume (Solute + Solvent)}} \] To make a 1:10 dilution of serum in saline with a total volume of 100 \(\mu\)L:
    • 1 part serum + 9 parts saline = 10 parts total
    • 10 \(\mu\)L serum + 90 \(\mu\)L saline = 100 \(\mu\)L total

Component Quality Control Calculations

Blood components must meet specific FDA/AABB standards, requiring periodic mathematical verification

  • Leukoreduction Efficiency To verify that a unit is leukoreduced, the absolute WBC count must be \(< 5.0 \times 10^6\) \[ \text{Total WBCs} = (\text{WBC count}/\mu\text{L} \times 1000) \times \text{Volume of Unit (mL)} \]
  • Factor VIII in Cryoprecipitate Cryoprecipitate must contain \(\ge 80\) IU of Factor VIII \[ \text{Total Factor VIII} = (\text{Factor VIII concentration}) \times (\text{Volume of bag}) \] Care must be taken with units; if the concentration is per mL, multiply by the volume in mL