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Hematology

Concentration, Volume, & Dilutions

Competence in laboratory mathematics is a fundamental requirement for the laboratory scientist, particularly in Hematology where specimen integrity and linearity limits frequently require manual manipulation of samples. While modern hematology analyzers perform complex calculations automatically, the laboratory scientist must be proficient in manual calculations to prepare reagents, troubleshoot out-of-range results, and correct for preanalytical variables. The most common mathematical applications in this department involve the preparation of solutions, the calculation of dilutions for samples exceeding the analyzer’s linearity, and specific correction formulas for interference

Volume & Concentration Basics

Understanding the relationship between solute (the substance being dissolved) and solvent (the dissolving medium) is the basis of all reagent preparation. In Hematology, this is most often applied to the preparation of stains (e.g., Wright-Giemsa), buffers, and cleaning solutions

  • Percent Solutions: This is the most common expression of concentration in the hematology laboratory. It allows for the quick preparation of solutions without complex molar mass calculations
    • Weight/Volume (w/v): The number of grams of solute per 100 mL of solution. For example, a 0.85% Saline solution contains 0.85 grams of NaCl dissolved in a total volume of 100 mL of deionized water
    • Volume/Volume (v/v): The number of milliliters of solute per 100 mL of solution. This is used when two liquids are mixed. For example, a 70% ethanol solution is made by mixing 70 mL of absolute ethanol with enough water to bring the total volume to 100 mL
  • The \(C_1V_1 = C_2V_2\) Equation: This formula is essential for changing the concentration of a solution (e.g., diluting a stock solution to a working solution)
    • \(C_1\): Initial Concentration (Stock)
    • \(V_1\): Initial Volume (How much stock do you need?)
    • \(C_2\): Final Concentration (Desired working concentration)
    • \(V_2\): Final Volume (Total amount desired)
    • Application: If a procedure requires 100 mL of 10% bleach for decontamination, and you have 100% stock bleach, the equation allows you to calculate that you need 10 mL of stock bleach diluted with 90 mL of water

Dilutions

A dilution represents the ratio of the volume of the substance being diluted (the sample) to the total volume of the final solution. Mastery of dilutions is critical in Hematology when a patient’s cell count exceeds the Analytical Measurement Range (AMR) or “linearity” of the instrument. For example, if an analyzer can only count up to 100,000 WBCs/µL, and a leukemia patient has 250,000 WBCs/µL, the sample must be diluted, re-run, and the result mathematically corrected

The Components of a Dilution

  • Sample Volume: The amount of patient specimen (solute)
  • Diluent Volume: The amount of fluid used to dilute the sample (usually Isoton or Saline)
  • Total Volume: Sample Volume + Diluent Volume
  • The Dilution Fraction: Expressed as \(\frac{\text{Sample Volume}}{\text{Total Volume}}\). A common error is expressing the dilution as sample-to-diluent rather than sample-to-total

Calculating the Dilution

To calculate a dilution, the laboratory scientist uses the formula: \[ \text{Dilution} = \frac{\text{Parts Sample}}{(\text{Parts Sample} + \text{Parts Diluent})} \]

  • Example: If you mix 100 µL of whole blood with 900 µL of saline:
    • Sample = 100 µL
    • Diluent = 900 µL
    • Total Volume = 1000 µL
    • Dilution = \(100 / 1000\) = 1:10 dilution

Distinction: Dilution vs. Ratio

It is vital to distinguish between a dilution and a ratio, as they are often confused in laboratory instructions

  • Dilution (1:10): Means 1 part sample in 10 parts total (1 part sample + 9 parts diluent)
  • Ratio (1:10): Means 1 part sample to 10 parts diluent. This would result in a total volume of 11 parts, creating a 1:11 dilution

The Dilution Factor (DF)

The Dilution Factor is the reciprocal of the dilution. It is the number used to multiply the result obtained from the diluted sample to determine the actual concentration in the original patient specimen

  • Calculation: If the dilution is 1:10, the Dilution Factor is 10
  • Application
    1. A patient sample flags for “Exceeds Linearity” on the WBC count
    2. The laboratory scientist performs a 1:2 dilution: (mixing equal parts blood and saline). Note: 1 part blood + 1 part saline = 2 parts total
    3. The diluted sample is analyzed, giving a result of 60.0 x \(10^9\)/L
    4. Math: \(60.0 \times 2 (\text{DF}) = 120.0 \times 10^9\)/L
    5. The value 120.0 is reported to the physician

Serial Dilutions

A serial dilution is a sequence of dilutions where the dilution factor is the same in each step. While less common in routine CBCs, this is frequently used in Coagulation (for factor assays) or Immunohematology (for antibody titers)

  • Process: A specific amount of diluent is placed in a series of tubes. Specimen is added to the first tube, mixed, and then an aliquot is transferred to the second tube, and so on
  • Compound Dilution Calculation: The total dilution of any tube in the series is the product of the individual dilutions of all the tubes leading up to it
    • Example: If Tube 1 is a 1:10 dilution, and you take 1 part of Tube 1 and add it to 9 parts diluent in Tube 2 (another 1:10 dilution):
    • Total Dilution of Tube 2 = \(\frac{1}{10} \times \frac{1}{10} = \frac{1}{100}\)

Specific Hematology Correction Formulas

Beyond standard chemical dilutions, Hematology requires specific mathematical corrections to account for interfering substances or anticoagulant variables

Corrected WBC Count (for nRBC Interference)

Nucleated Red Blood Cells (nRBCs) are roughly the same size as lymphocytes and contain a nucleus. Many impedance-based analyzers cannot distinguish them from WBCs, leading to a falsely elevated WBC count. While modern analyzers usually flag and correct this, manual correction is sometimes required

  • The Formula \[ \text{Corrected WBC} = \frac{\text{Uncorrected WBC} \times 100}{(\text{Number of nRBCs per 100 WBCs} + 100)} \]
  • Application: If the analyzer reports a WBC count of 15.0 x \(10^9\)/L, and the manual differential reveals 50 nRBCs per 100 WBCs:
    • Calculation: \((15.0 \times 100) / (50 + 100)\)
    • Calculation: \(1500 / 150\)
    • Corrected WBC: 10.0 x \(10^9\)/L

Sodium Citrate Adjustment (Platelet Clumping)

When a patient reacts to EDTA (causing pseudothrombocytopenia/platelet clumping), the sample must be redrawn in a Sodium Citrate (Light Blue) tube. Citrate tubes contain a significant volume of liquid anticoagulant (1 part anticoagulant to 9 parts blood), which dilutes the blood more than a spray-dried EDTA tube

  • The Correction Factor: Because the blood is diluted by 10% in the citrate tube, the counts obtained from the analyzer must be multiplied by 1.1 to approximate the true value in undiluted blood
  • Application
    1. Citrate tube Platelet count = 100,000 /µL
    2. Math: \(100,000 \times 1.1 = 110,000\) /µL
    3. Report 110,000 /µL with a comment “Result from Sodium Citrate tube corrected for dilution.”

Hemocytometer (Neubauer) Calculation

Although manual cell counts are rare, they are still the reference method for extremely low counts (e.g., body fluids) or when analyzers are down. The standard calculation determines the number of cells per microliter (\(\mu\)L)

  • The Formula \[ \text{Total Cells}/\mu\text{L} = \frac{\text{Cells Counted} \times \text{Dilution Factor} \times \text{Depth Factor}}{\text{Area Counted (mm}^2)} \]
    • Note: The standard depth of a Neubauer chamber is 0.1 mm, so the “Depth Factor” (reciprocal) is 10. Alternatively, the formula is often simplified as: \[ \frac{\text{Cells Counted} \times \text{Dilution Factor}}{\text{Volume of Chamber (Area} \times \text{Depth)}} \]
  • Standard Areas for Peripheral Blood Counts
    • WBCs are typically counted in the 4 corner squares (\(4 \text{ mm}^2\) total area)
    • RBCs are typically counted in the center square (\(0.2 \text{ mm}^2\) total area)
  • Standard Areas for Body Fluid Counts
    • WBCs (nucleated cells) and RBCs are counted in as many squares to reach ≥ 100 for each cell type (nucleated or RBCs), then the total area is calculated