9.3 Laboratory Mathematics: Dilutions, Concentration, Molarity & Standard Curves

Laboratory Mathematics: Dilutions, Concentration, Molarity & Standard Curves

The ASCP BOC lists laboratory mathematics as an explicit Laboratory Operations sub-topic covering concentration, volume and dilutions, molarity and normality, standard curves, descriptive statistics, and diagnostic predictive values. It separately states that examinees must be able to perform corrected WBC counts when more than 10 nRBCs are present, manual hemocytometer counts, red cell indices, and absolute counts from relative values. Calculators are provided on screen; the failure mode is method, not arithmetic.

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1. Dilutions

Notation

A dilution expresses part solute in total volume. A 1:10 dilution is 1 part specimen plus 9 parts diluent, giving 10 parts total. This is the single most common error: 1:10 is not 1 part plus 10 parts.

• Dilution factor is the reciprocal of the dilution. A 1:10 dilution has a dilution factor of 10, so the measured value is multiplied by 10 to recover the original concentration.
• Worked example. A leukocyte count is too high to read directly, so 0.1 mL of blood is added to 1.9 mL of diluent. Total volume is 2.0 mL, so the dilution is 0.1/2.0 = 1:20 and the dilution factor is 20. If the diluted count reads 3.9 x 10^9/L, the reported count is 3.9 x 20 = 78 x 10^9/L.

Serial Dilutions

In a serial dilution each tube is diluted from the previous one, so the dilutions multiply. Four successive 1:5 steps give a final dilution of 1:5 x 1:5 x 1:5 x 1:5 = 1:625. The general rule is that the final dilution equals the product of the individual dilutions.

The Dilution Equation

$C_1V_1 = C_2V_2$ To prepare 500 mL of 0.5 molar sodium chloride from a 2.0 molar stock: $(2.0)(V_1) = (0.5)(500)$, so $V_1 = 125$ mL of stock brought to 500 mL with water.

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2. Concentration Expressions

Relationship to remember: $N = M \times$ valence. A 0.5 M solution of calcium chloride, whose calcium ion has a valence of 2, is 1.0 N with respect to calcium.

Worked molarity example. How many grams of sodium citrate (gram molecular weight 294.1) are needed to make 250 mL of a 0.109 molar solution, the concentration of the 3.2% citrate coagulation tube? Moles = 0.109 mol/L x 0.250 L = 0.02725 mol. Grams = 0.02725 x 294.1 = 8.0 g.

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3. Standard Curves and Beer's Law

Many hematology assays are photometric: hemoglobin by the cyanmethemoglobin or sodium lauryl sulfate method, chromogenic anti-Xa, and fibrinogen by some methods. All rest on the Beer-Lambert law:

$A = \varepsilon \times b \times c$

Absorbance is directly proportional to concentration when path length and molar absorptivity are constant, so a plot of absorbance against concentration is a straight line through the origin.

• Constructing the curve. Analyze a series of standards of known concentration, plot absorbance on the y-axis against concentration on the x-axis, and fit a line by least-squares regression. The correlation coefficient (r) should approach 1.00, and the y-intercept should approach zero.
• Using the curve. Read the unknown's absorbance on the y-axis and interpolate down to the x-axis. Never extrapolate beyond the highest standard: the relationship flattens at high concentration as the sample stops transmitting light proportionally, and a result read off the flat region is falsely low. Dilute and re-assay instead, then multiply by the dilution factor.
• Deviations from linearity arise from stray light, turbidity or lipemia, a non-monochromatic light source, chemical association or dissociation of the chromophore, and analyte concentrations above the linear range.
• Calibration curve versus calibration verification. The curve establishes the relationship; calibration verification confirms periodically that the established curve still holds across the reportable range.

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4. The Four Calculations Named in the Content Guideline

a. Corrected WBC Count for Nucleated Red Cells

Nucleated red cells are not lysed in the WBC channel and are counted as leukocytes. The ASCP BOC expects correction when more than 10 nRBCs per 100 WBCs are seen on the differential:

$\text{Corrected WBC} = \frac{\text{Uncorrected WBC} \times 100}{100 + \text{nRBC per 100 WBC}}$

Example: uncorrected WBC 22.0 x 10^9/L with 45 nRBC/100 WBC gives (22.0 x 100) / 145 = 15.2 x 10^9/L.

b. Manual Hemocytometer Count

$\text{Cells}/\mu\text{L} = \frac{\text{Cells counted} \times \text{Dilution factor} \times \text{Depth factor (10)}}{\text{Area counted (mm}^2)}$

Example: 240 leukocytes counted in the four large corner squares (4 mm$^2$) at a 1:20 dilution gives (240 x 20 x 10) / 4 = 12,000/uL.

c. Red Cell Indices

$\text{MCV (fL)}=\frac{\text{Hct (\%)}\times 10}{\text{RBC }(10^{12}/\text{L})} \qquad \text{MCH (pg)}=\frac{\text{Hgb (g/dL)}\times 10}{\text{RBC }(10^{12}/\text{L})} \qquad \text{MCHC (g/dL)}=\frac{\text{Hgb (g/dL)}\times 100}{\text{Hct (\%)}}$

Example: Hgb 9.0 g/dL, Hct 27%, RBC 3.60 x 10^12/L. MCV = 270/3.6 = 75 fL; MCH = 90/3.6 = 25 pg; MCHC = 900/27 = 33.3 g/dL. Microcytic and hypochromic by MCV and MCH, with an MCHC still inside the 32-36 g/dL interval.

d. Absolute Counts from Relative Values

$\text{Absolute count} = \text{Total count} \times \frac{\text{Percentage}}{100}$

Example: WBC 12.4 x 10^9/L with 4% eosinophils gives an absolute eosinophil count of 0.50 x 10^9/L, above the 0-0.3 x 10^9/L reference interval, so this is a true eosinophilia. The same rule converts a reticulocyte percentage to an absolute reticulocyte count: reticulocytes (%) x RBC (10^12/L) x 10.