20.3 Laboratory Mathematics and Instrumentation

20.3 Laboratory Mathematics and Instrumentation

Quick Answer: Dilutions use C1V1=C2V2; a serial 1:10 is 1 mL into 9 mL, repeated. Molarity is mol/L; normality is eq/L. Standard curves interpolate concentration from absorbance in the linear range. Mean, median, and mode describe a set; a confidence interval estimates where the true mean lies. Sensitivity and specificity come from a 2x2 table; PPV and NPV move with prevalence. Know brightfield, phase, and fluorescence microscopy, why RCF is not RPM, and the QC logic of spectrophotometers, thermocyclers, continuous-monitoring blood-culture instruments, MALDI-TOF MS, and automated AST systems.

Official outline IV.D Laboratory Mathematics and IV.E Instrumentation (guideline 2025-09-25) close Domain IV (Laboratory Operations, 10–15% of M). Quality-control policy and Westgard-style rules are Chapter 19; this section is the arithmetic and the machines those QC rules sit on. IV.F administration remains SM-only.

Concentration, volume, and C1V1=C2V2

The workhorse equation is C1 × V1 = C2 × V2, with concentration and volume in the same units on both sides.

Worked dilution. You need 500 mL of 1% (v/v) working bleach. The stock bottle is 10% sodium hypochlorite.

• C1 = 10%, V1 = unknown, C2 = 1%, V2 = 500 mL
• V1 = (C2 × V2) / C1 = (1 × 500) / 10 = 50 mL of 10% stock
• Diluent = 500 − 50 = 450 mL of water (add bleach into water, not water into concentrated bleach)

That 1% working solution is an exam-clean C1V1 number. Many blood-spill SOPs instead use a 1:10 household bleach (about 5–8% stock → about 0.5% available chlorine), prepared fresh (20.1). Do not confuse the SOP recipe with the algebra: the equation does not care which percent pair you are given, only that the units match.

A second form: prepare 100 mL of 0.85% saline from 5% saline. V1 = (0.85 × 100) / 5 = 17 mL of 5% saline plus 83 mL water. Percent for solids is usually w/v (g/100 mL): 5% NaCl is 5 g in 100 mL, or 50 g/L. Do not mix % w/v with molarity without converting.

Serial 1:10 dilutions

A 1:10 (10-fold) dilution is 1 part specimen + 9 parts diluent (1 mL + 9 mL, or 0.1 mL + 0.9 mL). Repeating that dilution is a serial 1:10:

If 0.1 mL of tube 3 is plated and 42 colonies grow, CFU/mL of the original = colonies × (1 / volume plated in mL) × dilution factor = 42 × (1 / 0.1) × 1,000 = 42 × 10 × 1,000 = 4.2 × 10^5 CFU/mL. This is quantitative-urine and quantitative-tissue logic. A 1:2 is 1 part + 1 part; a 1:5 is 1 + 4. The dilution factor is the reciprocal (a 1:10 has factor 10).

Molarity and normality at exam-usable level

Molarity (M) = moles of solute per liter of solution. Moles = grams / molecular weight. Molecular weight of NaCl is 58.4 g/mol, so 0.85% saline (8.5 g/L) is 8.5 / 58.4 ≈ 0.15 M (about 150 mM NaCl, physiologic saline).

Normality (N) = equivalents per liter. For monoprotic acids and bases (HCl, NaOH), 1 M = 1 N. For H2SO4 (two replaceable hydrogens), 1 M = 2 N. If a stem gives 0.1 N HCl and 0.1 M HCl, they are the same; 0.1 N H2SO4 is 0.05 M. Prepare 250 mL of 0.2 M NaCl: grams = M × MW × liters = 0.2 × 58.4 × 0.250 = 2.92 g brought to 250 mL.

Standard curves

A standard curve plots a measurable signal (usually absorbance) against known concentrations, then reads unknowns off the line. Beer's law: A = εlc (absorbance = absorptivity × path length × concentration) in the linear range. Blank with reagent or diluent at the same wavelength. Use at least three to five standards spanning the expected unknown. Do not report an unknown that plots above the highest standard without diluting it back into range. An R² near 1.00 supports linearity; a bent curve means the linear range was exceeded or the wavelength is wrong. Spectrophotometer QC (below) exists so the curve is physically meaningful.

Mean, median, mode, and confidence intervals

For the set 2, 3, 3, 5, 12:

• Mean (average) = (2+3+3+5+12)/5 = 5
• Median (middle of the ordered set) = 3
• Mode (most frequent) = 3

The mean is pulled by outliers (the 12); the median is robust. QC programs often watch the mean of control values; a single wild result is still investigated (Chapter 19).

A confidence interval (CI) is a range that, with stated probability (commonly 95%), contains the true population parameter. For a mean, 95% CI ≈ mean ± 1.96 × (SD / √n), where SD/√n is the standard error. A wider CI means less precision (small n or large SD). The exam uses CI conceptually: a 95% CI that does not overlap a reference value is the statistical way of saying different. Coefficient of variation CV% = (SD / mean) × 100 is the usual precision metric on instruments.

Sensitivity, specificity, PPV, NPV — worked 2x2

Build the table from a gold standard:

• Sensitivity = TP / (TP + FN) — among the diseased, the fraction the test catches. A negative highly sensitive test helps rule out.
• Specificity = TN / (TN + FP) — among the nondiseased, the fraction correctly called negative. A positive highly specific test helps rule in.
• PPV = TP / (TP + FP) — given a positive test, the probability of disease. Falls when prevalence falls.
• NPV = TN / (TN + FN) — given a negative test, the probability of no disease. Falls when prevalence rises.

Worked 2x2. A new stool antigen is compared with culture-plus-PCR on 200 specimens. 40 patients are truly infected and 160 are not.

• TP = 36, FN = 4 → sensitivity = 36/40 = 90%
• FP = 16, TN = 144 → specificity = 144/160 = 90%
• PPV = 36 / (36 + 16) = 36/52 ≈ 69%
• NPV = 144 / (144 + 4) = 144/148 ≈ 97%

Sensitivity and specificity are both 90%, but PPV is only 69% because 16 false positives are not rare relative to 36 true positives at this 20% prevalence. Move the same test into a 1% prevalence population and PPV collapses further—the classic screening trap. Do not call PPV sensitivity.

Instrumentation the outline names

Microscope. Brightfield is the Gram stain, AFB stain, and most stained smears; oil-immersion 100× is required to resolve bacteria. Phase contrast converts optical-path differences into contrast so unstained wet mounts (urine, some parasite preps, colony wet mounts) are visible. Fluorescence uses excitation/emission filters: auramine-rhodamine (AFB), calcofluor white (fungal cell wall), DFA, and some FISH. Auramine is not a brightfield stain. Daily: clean oil, check bulbs, and align Köhler illumination where required. A stage micrometer calibrates ocular micrometers for parasite and fungal measurements.

Centrifuge. RPM is what the tachometer reads. RCF (relative centrifugal force, × g) is what actually sediments particles. RCF = 1.12 × r(mm) × (RPM/1000)^2. The same RPM on two rotors is not the same g if the radii differ. Protocols that say 1,500 × g for 10 minutes must be converted to RPM for that rotor. Balance tubes, use sealed cups for infectious loads, and include the centrifuge in preventive maintenance.

Spectrophotometer. Measures absorbance or transmittance at a selected wavelength. Used for some biochemical IDs, nucleic-acid concentration (A260/A280), and any assay that builds a standard curve. QC: wavelength accuracy (didymium or holmium filters), photometric accuracy, and stray-light checks on the manufacturer's schedule; blank every run.

Thermocycler (PCR). Cycles denaturation, annealing, and extension temperatures for nucleic-acid amplification. Real-time platforms add fluorescence detection. Contamination control is part of use: unidirectional workflow, closed tubes, and enzymatic UNG where designed. PM: temperature verification with a probe or the manufacturer's thermocouple kit; do not run patient PCR on a cycler that failed its temperature check.

Continuous-monitoring blood-culture system. BACTEC, BacT/ALERT, VersaTREK and equivalents detect CO2, pH, or pressure change and flag a bottle as positive. The technologist then Gram stains and subcultures; the instrument does not identify the organism. Fill-volume QC, bottle-lot QC, and delayed-entry rules are part of PM. Organisms recovered are Chapter 5.

MALDI-TOF MS. Matrix-assisted laser desorption/ionization time-of-flight mass spectrometry generates a protein fingerprint compared with a database. It identifies colonies in minutes. It does not produce MICs. Limitations include Shigella versus E. coli and S. pneumoniae versus mitis-group streptococci. Do not spot live BSL-3 or rule-out organisms on an open-bench target (13.2, 14.3). Calibration with a bacterial test standard is the daily QC.

Automated antimicrobial susceptibility system. VITEK, MicroScan, Phoenix and similar platforms are automated broth microdilution (or a closely related MIC method). They report MIC and interpretative categories. QC uses CLSI ATCC strains on the required schedule. Inducible resistance, some fastidious organisms, and unusual phenotypes still need a manual method (Chapter 12). An instrument call of susceptible does not override a contradictory D-test or carbapenemase screen.

Preventive maintenance and QC of instruments

Preventive maintenance is the scheduled work that keeps the device inside specifications: cleaning, lubrication, filter changes, lamp replacement, software updates, and manufacturer-required function checks. QC is the run-time evidence that the device is measuring correctly today: temperatures of incubators, refrigerators, and freezers recorded each day of use; microscope cleanliness; spectrophotometer blanks and standards; MALDI calibrant; AST ATCC organisms; blood-culture bottle fill-volume audits; thermocycler temperature checks.

Out-of-range QC stops patient testing on that device until corrective action and acceptable QC are documented (Chapter 19). Sticker dates on a centrifuge or BSC certification are not decorative—an expired certification means the device is not in service.