Dilution Math

Parts and dilution factor

A dilution ratio counts parts of the final mixture. A 1:5 dilution contains one part specimen and four parts diluent, giving five total parts and a dilution factor of 5. A specimen-to-diluent ratio describes the components separately. Prepare the dilution from those stated volumes and factor.^{11 Bishop ML, et al. Clinical Chemistry. 2023.}

Written as 1 + 5, the notation means one part specimen plus five parts diluent. That is six finished parts with a factor of 6.

For a chosen final volume:

specimen volume = final volume ÷ total parts

diluent volume = final volume − specimen volume

The Dilution station applies these relationships while a 1:5 dilution is prepared, mixed, measured, and used to calculate the patient result. The printable dilutions and dilution factors guide keeps the volume and result calculations together for use alongside the bench.

Concentration and patient-result calculation

The diluted concentration depends on the fraction of the finished mixture occupied by specimen:

diluted concentration = original concentration × specimen volume ÷ final volume

This form also applies when a mixture misses its intended final volume. The dilution factor used to calculate the patient result must match the mixture actually made.

calculated patient result = measured diluted result × dilution factor

Measuring intervals

The analytical measurement range (AMR) is the interval a method measures directly without dilution, concentration, or another pretreatment outside its usual assay process. Reportable range may also refer more broadly to values made reportable through dilution.^{22 College of American Pathologists. Analytical Measurement Range.}

CLSI calls this directly measured interval the analytical measuring interval and the interval extended by an established dilution scheme the extended measuring interval. The measurement made on the diluted aliquot must fall inside the analytical measuring interval, and the calculated original result must remain inside the extended measuring interval that the scheme established.^{33 Clinical and Laboratory Standards Institute. Establishing and Verifying an Extended Measuring Interval… 2018.}

A manufacturer may establish the dilution scheme and its claims, which the laboratory then verifies. When those claims are inadequate, the laboratory may establish and verify a local scheme. The scheme specifies the diluent and permitted ratios. Precision across the interval is part of establishing or verifying the scheme, particularly at high dilution ratios. ^{33 Clinical and Laboratory Standards Institute. Establishing and Verifying an Extended Measuring Interval… 2018.}

How a laboratory evaluates the interval is covered under Recovery, interference, linearity, and reference intervals.

Concentration before a serial dilution

Concentration describes how much dissolved substance is present in a given volume of solution. A concentration of 100 mg/L means 100 mg in each liter. A smaller, well-mixed portion has the same concentration: 2,000 µL (0.002 L) contains 0.2 mg. The amount is smaller because the volume is smaller. A diluent lowers concentration by adding volume without adding the substance being measured. The dilution equation follows from keeping the transferred dissolved amount unchanged while increasing its volume.^{44 Flowers P, et al. Molarity. 2019.}

The Dilution station's serial shelf uses a model dissolved substance starting at 100 mg/L. It assumes additive volumes, complete transfers, and no reaction, loss, or evaporation. The colored liquid and magnified dots are schematic aids. Every comparison box represents 100 µL; a dot represents 0.1 µg. Below 1 mg/L, no whole dot fits, but the numeric concentration remains above zero.

Withdrawal and mixing

An aliquot is a measured portion of solution. In a well-mixed tube, every portion has the same amount per unit volume. Withdrawing half the liquid therefore removes half the dissolved amount, leaving the concentration unchanged. A 1,000 µL tube at 50 mg/L contains 50 µg; removing 500 µL leaves 25 µg in 500 µL, still 50 mg/L. These values follow from the uniform composition of a solution and the relationship between concentration, amount, and volume.^{44 Flowers P, et al. Molarity. 2019.}

Mix the transferred solution with its prepared diluent before taking the next aliquot. Until mixing is complete, the concentration at the pipette tip may not match the tube's average concentration, so another transfer is not reliable.

Volume prepared is the combined transfer and diluent volume when the tube is made. Volume remaining is what stays after withdrawals. For example, transferring 500 µL into 500 µL diluent prepares 1,000 µL; withdrawing 500 µL for the next tube leaves 500 µL. This withdrawal changes neither the completed dilution factor nor the concentration of the remaining solution.

Serial factors multiply

Each serial step uses the preceding mixed tube as its source. If a twofold step halves the concentration and the next step halves it again, the second tube holds one half of one half, or one quarter, of the original concentration. The total factor is 2 × 2 = 4. Adding factors would not describe these successive reductions.^{55 Kingston University. Concentration of solutions. 2021.}

Step factor = prepared volume ÷ transferred volume

Destination concentration = source concentration ÷ step factor

Total factor from the original = product of the completed step factors

With 100 mg/L as the starting concentration, the worked bench series give:

Unequal steps and volume errors

The same calculation applies when step sizes differ. Starting at 100 mg/L, factors of 2, 5, and 10 give 50, 10, and 1 mg/L. The second tube's factor is 5 relative to its source and 2 × 5 = 10 relative to the original. A step factor and a total factor answer different comparisons.

A wrong transfer changes its actual step factor and every concentration downstream. For example, 200 µL source added to 900 µL diluent gives factor 1,100 ÷ 200 = 5.5. The intended step was 100 µL source plus 900 µL diluent, factor 10. Two later tenfold steps produce total factor 5.5 × 10 × 10 = 550 and a concentration of 100 ÷ 550, about 0.1818 mg/L. The incorrect transfer therefore propagates a concentration error through every downstream dilution, even when later transfers are performed correctly. To correct the preparation, repeat the affected step and every transfer after it.

Applying a dilution scheme

The established dilution scheme specifies the diluent and permitted ratios used to extend the measuring interval.^{33 Clinical and Laboratory Standards Institute. Establishing and Verifying an Extended Measuring Interval… 2018.} The dilution factor used to calculate a patient result must match the mixture that was prepared. Patient testing follows the applicable assay instructions and laboratory procedure. Before you multiply, confirm that the factor describes the mixture you actually made.