Why Dose Inputs Need Their Own Unit Check
A dose can be reported as an amount, such as 250 mg, while a model’s equations describe how an amount changes over time. Connecting those representations requires care: an instantaneous bolus is not the same quantity as a constant infusion rate, and a rate expressed per hour is not numerically interchangeable with a rate expressed per day. A misplaced time conversion can change a simulated input by a factor of 24, 60, or 86,400.
The earlier material on state-variable units and dimensional analysis established the basic check: quantities added together must have compatible units. Here the focus is narrower: express bolus doses, oral input rates, and intravenous infusions in units compatible with a model’s chosen time axis. The same unit logic applies whether the model tracks drug in one compartment or several.
Bolus Amounts and Continuous Input Rates
For an instantaneous bolus, the model records an amount increment at the administration time. The amount \(D\) remains in mg whether the simulation clock is recorded in seconds, hours, or days. Changing the time unit does not convert the dose amount. In a numerical implementation, the bolus is usually handled as an event that increments the relevant amount state by \(D\), rather than as a continuous rate acting over a small time step.
A constant-rate input is different. If an amount \(D\) is delivered uniformly over a duration \(T\), the rate is the amount divided by the duration:
The time unit in the denominator of \(R_{\mathrm{in}}\) must agree with the time unit used for \(T\) and the model’s time axis. For example, if the state is an amount in mg and time is in hours, its input-rate term must be in mg/h. If time is in days, the corresponding term must be in mg/day.
For oral administration, the dose may be represented as an amount placed into a gut or absorption state at administration. Alternatively, a model may represent oral delivery as an input rate, for example when the formulation releases drug uniformly over a specified period. These representations are not automatically equivalent: a bolus into a gut state is an instantaneous amount there, while a constant oral release is an amount delivered over time. An input rate into a gut state also need not equal the rate at which drug appears in systemic circulation.
Worked Example: Converting an Oral Release Amount to a Rate
Suppose a hypothetical oral formulation releases 120 mg uniformly into a modeled gut state over 8 h. The model time unit is hours. Find the input rate and verify the amount delivered.
Because release is assumed constant over the full 8 h, use \(R_{\mathrm{in}}=D/T\):
Over 8 h, the rate supplies:
The units cancel to mg, as required for an amount. This calculation assumes uniform release throughout the stated interval; it does not assert that systemic absorption is constant.
Changing the Time Unit of a Rate
A rate is a quantity per unit time. When the chosen time unit changes, its numerical value must change so that the physical amount delivered over any fixed interval remains the same. The conversion factor is the number of old time units contained in one new time unit.
For example, one day contains 24 hours. A rate of 10 mg/h therefore corresponds to \(10\times24=240\) mg/day, not \(10/24\) mg/day. The latter value would describe a much smaller physical input per day. Conversely, one minute is \(1/60\) of an hour, so a rate in mg/h is multiplied by \(1/60\) to express it in mg/min.
Why the Conversion Rule Works
The conversion follows by requiring that the amount delivered over the same physical interval is unchanged. Let one new time unit contain \(c\) old time units. An interval of one new unit therefore lasts \(c\) old units. The old rate delivers \(R_{\mathrm{old}}c\) amount over that interval, so the rate expressed per new unit must have that numerical value.
The units in the first line cancel from old time units to amount, leaving an amount. Since the interval is one new time unit, that amount is numerically the rate per new time unit. This derivation also explains why rates and amounts behave differently: an amount does not have a time denominator to convert.
Worked Example: Converting an Infusion Rate from Hours to Days
A hypothetical intravenous infusion supplies 10 mg/h. Express the same rate in mg/day and verify the amount supplied over 3 h.
There are 24 h in one day, so the number of old units (hours) in one new unit (day) is 24:
Over 3 h, the infusion supplies \(10\ \mathrm{mg/h}\times3\ \mathrm{h}=30\ \mathrm{mg}\). Since 3 h is \(3/24=0.125\) day, the converted rate gives the same result:
The two calculations agree, and the time units cancel in each. The infusion’s physical delivery has not changed; only the unit used to express its rate has changed.
Matching a Dose, a Duration, and the Model Clock
A reliable input calculation starts by writing the dose amount, delivery duration, and model time unit with their units. If the duration is given in minutes but model time is in hours, convert the duration or the rate before entering the input. The product of a constant rate and its duration should recover the intended dose amount, with compatible time units.
Decide whether the model receives an instantaneous amount or a continuous input over time. Do not assign an amount-per-time value to a bolus event without defining a finite delivery duration.
Record the amount, duration, rate, and simulation time unit, such as mg, 30 min, mg/h, and hours.
Express the duration or rate using the model’s time unit. For a rate conversion, multiply by the number of old time units in one new time unit.
For a constant input, multiply the rate by the matching duration. Confirm that the result has units of amount and equals the intended amount.
Worked Example: A Short Infusion Entered on an Hour-Based Clock
A hypothetical intravenous dose of 300 mg is infused uniformly over 30 min. The model time unit is hours. Calculate the input rate in mg/h and check the result using both hours and minutes.
First convert the duration: \(30\ \mathrm{min}\times(1\ \mathrm{h}/60\ \mathrm{min})=0.5\ \mathrm{h}\). Then calculate:
The hour-based check gives \(600\ \mathrm{mg/h}\times0.5\ \mathrm{h}=300\ \mathrm{mg}\). In minute units, the same rate is \(600/60=10\ \mathrm{mg/min}\), because one minute is \(1/60\) hour. Thus \(10\ \mathrm{mg/min}\times30\ \mathrm{min}=300\ \mathrm{mg}\). Both representations describe the same infusion.
If the model instead received a bolus, the input would be a 300 mg amount event, not a 600 mg/h rate. The rate calculation is appropriate here because the dose is assumed to enter uniformly over a nonzero 30-minute interval.
Rate Constants and Diagnostic Checks
A first-order rate constant also depends on the model time unit. For example, if a model uses \(k=0.2\ \mathrm{h}^{-1}\), then the equivalent value per day is \(0.2\times24=4.8\ \mathrm{day}^{-1}\). This conversion preserves the rate process because the numerical time coordinate has changed. It does not mean that the physical process has become faster.
Several quick checks can reveal an inconsistent input before simulation:
- Dimensional check: a rate term should have units of amount/time, and rate multiplied by duration should have units of amount.
- Magnitude check: a rate in mg/day is numerically 24 times a rate in mg/h for the same physical delivery. A result smaller by 24 when converting hours to days signals an inverted factor.
- Independent duration check: compute the delivered amount using the original rate and duration, then repeat using the converted rate and converted duration. The amounts should agree.
- Representation check: confirm whether the model expects a bolus event, a constant input rate, or a rate into a particular modeled state. These are not interchangeable descriptions.
The conversion assumes exact unit relationships and, for a constant-rate infusion, uniform delivery over the specified duration. A formulation with changing release, interruptions, or a nonconstant infusion requires a time-varying rate rather than one constant value. Also, the dose amount administered and the amount assigned to a particular model state must be consistent with the model’s specified input pathway; a correct time conversion cannot fix an incorrectly chosen state or input amount.
Common Mistakes and Alternatives
A frequent error is to multiply or divide by a time conversion factor without checking the direction. The reliable question is: how many old time units fit in one new time unit? When converting mg/h to mg/day, one new day contains 24 old hours, so multiply by 24. When converting mg/h to mg/min, one new minute contains \(1/60\) old hours, so multiply by \(1/60\).
Another mistake is to treat an amount as though it were already a rate. A 100 mg bolus remains 100 mg; it does not become 100 mg/h merely because the simulation time is hours. If the same 100 mg is delivered over 2 h at a constant rate, then the rate is \(100/2=50\) mg/h. A finite numerical time step does not by itself define a pharmacologically intended infusion duration.
The main alternatives are event-based bolus input and continuous-rate input. Use a bolus representation when the model assumes the amount enters instantaneously relative to the timescale of interest. Use a constant rate when the input is delivered uniformly over a stated interval. Use a time-varying rate when delivery changes over time. The choice is about the assumed input process, not simply which representation is easiest to calculate.
Check Your Understanding
Answer each question, then compare with the solutions below.
- A hypothetical 90 mg oral formulation releases drug uniformly over 6 h. What is the input rate in mg/h?
- Convert 4 mg/h to mg/day. Explain which conversion factor is used.
- A hypothetical infusion delivers 240 mg uniformly over 40 min. What is its rate in mg/min and in mg/h?
- A 75 mg bolus is entered into a model whose time unit is days. Does the bolus amount change? Explain.
- A model uses a first-order rate constant of \(0.15\ \mathrm{h}^{-1}\). Express it in day\(^{-1}\).
Solutions. (1) \(90\ \mathrm{mg}/6\ \mathrm{h}=15\ \mathrm{mg/h}\). The check is \(15\ \mathrm{mg/h}\times6\ \mathrm{h}=90\ \mathrm{mg}\). (2) One day contains 24 hours, so \(4\ \mathrm{mg/h}\times24\ \mathrm{h/day}=96\ \mathrm{mg/day}\). (3) In minutes, \(240\ \mathrm{mg}/40\ \mathrm{min}=6\ \mathrm{mg/min}\). Since an hour contains 60 minutes, \(6\ \mathrm{mg/min}\times60\ \mathrm{min/h}=360\ \mathrm{mg/h}\). (4) No. It remains 75 mg because an amount has no time denominator. (5) One day contains 24 hours, so \(0.15\ \mathrm{h}^{-1}\times24\ \mathrm{h/day}=3.6\ \mathrm{day}^{-1}\).