20.4 Laboratory Mathematics and Instrumentation

20.4 Laboratory Mathematics and Instrumentation

Quick Answer: RhIG vials = (KB% / 100 × maternal blood volume) / 30, then round by the stem’s rule and add 1. Default maternal volume is 5,000 mL. One 300 µg vial covers 30 mL whole blood. One RBC unit raises a stable 70 kg adult about 1 g/dL Hb (3% Hct). Units to screen = units needed / antigen-negative frequency. CCI uses BSA and dose × 10^11. Instruments: serofuge, cell washer, 37 °C block, microscope, irradiator, automated analyzer — each with QC.

The June 9, 2026 outline puts VI.C Laboratory mathematics and VI.D Instrumentation together. Chapter 12.3 already ran one postpartum RhIG stem; Chapter 17.3 already defined CCI for refractoriness. This section is the general math toolkit and the boxes on the bench.

RhIG vials from a Kleihauer–Betke

One 300 µg (1500 IU) vial of Rh immune globulin neutralizes about 30 mL of D-positive whole blood or 15 mL of packed red cells. A 50 µg microdose is an early-pregnancy dose in some protocols, not the term FMH dose.

Fetal whole blood (mL) = (KB percent / 100) × maternal blood volume (mL).
If the stem omits blood volume, use 5,000 mL. That shortcut is FMH (mL) = KB% × 50.

Vials to cover the bleed = FMH / 30.
Then apply rounding and add 1 safety vial.

AABB-style rounding (use this unless the stem gives another rule): look at the decimal after dividing by 30. If it is < 0.5, round down; if it is ≥ 0.5, round up. Then add 1. Some stems say only “round up and add 1” — obey the stem. An already-whole number (3.0) just gets the extra vial (4).

Worked examples (volume 5,000 mL):

Flow cytometry / immunofluorescence FMH assays report mL of fetal red cells or a percent; convert to the same 30 mL whole-blood coverage before you add the safety vial. After RhIG the mother may type as weak D and the screen may show anti-D — do not call her D-positive and do not withhold the postpartum dose (Chapter 12.3).

Expected hemoglobin increment, units to screen, CCI, and conversions

Red-cell increment. In a stable ~70 kg adult who is not bleeding or hemolyzing, 1 RBC unit raises hemoglobin about 1 g/dL and hematocrit about 3%. A patient whose Hb is 6.8 g/dL and whose target is 9.8 g/dL needs about 3 units if those assumptions hold. Failure to increment is bleeding, hypersplenism, or hemolysis (including a delayed serologic/hemolytic reaction), not a reason to keep issuing the same type blindly. Pediatrics: 10–15 mL/kg packed red cells typically raises Hb about 2–3 g/dL. Do not apply the 1 g/dL adult rule to a 12 kg toddler.

Units to screen.
Number of units to screen = units needed / (antigen-negative frequency).
For several alloantibodies, multiply the antigen-negative frequencies if the antigens assort independently (the usual exam assumption). Use the Caucasian teaching frequencies unless the stem gives another population.

Common antigen-negative frequencies (Caucasian teaching set): K-neg 0.91, E-neg 0.70, c-neg 0.20, C-neg 0.30, e-neg 0.02, Fya-neg 0.34, Jka-neg 0.23, S-neg 0.45.

Need 2 units for a patient with anti-E and anti-K:
0.70 × 0.91 = 0.637.
2 / 0.637 = 3.14 → screen at least 4 units.

Need 3 units for anti-c and anti-K:
0.20 × 0.91 = 0.182.
3 / 0.182 = 16.5 → screen 17 units (and call the reference lab / rare-donor program rather than antigen-type your entire inventory at 02:00).

CCI (conceptual). Chapter 17.3 is the clinical home. Formula for this math section:

CCI = [(post − pre platelet count per µL) × BSA (m²)] / (platelets transfused × 10^11)

A 10–60 minute CCI < 5000–7500 on two occasions supports immune refractoriness. A single low 24-hour count is survival, not a one-hour increment. You need the labeled yield; “one apheresis bag” is not a number.

Conversions and percent solutions. 1 dL = 100 mL, so Hb 8 g/dL = 80 g/L. 37 °C = 98.6 °F; 1–6 °C storage is 33.8–42.8 °F. Weight/volume percent is grams per 100 mL: 0.9% NaCl = 0.9 g/100 mL = 9 g/L. Dilutions use C1V1 = C2V2. To make 250 mL of 5% albumin from 25% albumin: 25 × V1 = 5 × 250 → V1 = 50 mL of 25% plus 200 mL diluent. Do not treat a 5% bottle as if it were 25% just because both say “albumin.”

Instrumentation and instrument QC

Serologic centrifuge (serofuge). Immediate-spin, wash, and AHG buttons need different times. QC: timer accuracy, speed / RCF with a tachometer, and function (a known weakly positive cell still buttons and resuspends as the SOP describes). An extra 15 seconds is not “safer” — it can pack rouleaux into a fake agglutinate or change a calibrated IS ABO. Failed QC takes the head out of service (Chapter 16.2 / 20.1).

Cell washer. Automated saline fills and decants before AHG. QC the fill volume and the residual saline left after the last decant. Too much leftover saline dilutes AHG and causes false-negative IAT. Too little wash leaves unbound IgG and wastes AHG on leftover protein. A washer that “sounds normal” is not QC.

37 °C incubator / heat block. Daily thermometer check, usually 36–38 °C. A heat-block well can read 34 °C while the display says 37 °C — measure the well you actually use. Antibody screens incubated cold miss warm alloantibodies; overheated wells can hemolyze or cook complement.

Microscope. Use it when the SOP says so: mixed-field ABO (recent transfusion, A3, chimera, HPC transplant), distinguishing rouleaux from true agglutination, and immunofluorescence / FMH slides. Autofluorescence on an FMH slide can mimic a fetal cell if the filter set or quenching step is wrong — that is an instrumentation limitation, not extra FMH. Many modern SOPs forbid microscopic reading of antibody screens because it inflates false positives. If the stem says the tech read a 1+ screen only under oil, ask whether the SOP allowed it.

Irradiator. Cesium-137 or X-ray. Target dose is 25 Gy (2500 cGy) to the midplane of the canister, with no point < 15 Gy. Upper limits (often 50 Gy) protect the red-cell membrane. Dose mapping at installation and at defined intervals proves the field. The irradiation indicator on the bag only shows the unit entered a radiation field. It is not a personal dosimeter and it does not prove 25 Gy reached the midplane. QC: indicator on every irradiated unit, dose map current, turntable / cycle complete, and the DIN in the irradiator log. A unit with an unchanged indicator is not irradiated, even if someone “is sure it went in.”

Automated immunohematology analyzer (gel / column agglutination or solid-phase red-cell adherence). These are the Vision / Erytra-class columns and the solid-phase well systems. They give reproducible IAT and are the engine of reflex panels. Limitations you must be able to name:

• Mixed-field and some ABO subgroups are easy for a camera to under-call.
• Rouleaux, cold autoantibodies, and hemolysis need a human and often a tube reflex.
• Solid-phase can look “positive” with some warm autoantibodies and some HLA-ish noise; gel can miss some weak ABO isoagglutinins if the card and the sample are a mismatch.
• The analyzer does not replace an emergency group O decision while a discrepancy algorithm is running.

Analyzer QC: manufacturer-required controls each day of use, calibration/maintenance per the file, comparison with tube during validation, and a written reflex SOP. Do not report a patient type the instrument flagged as invalid just because the previous 40 types were fine.