Why Reconcile Volumes With Body Mass?
A PBPK physiology table may contain volumes for organs and tissues, while the reference individual is described by a whole-body mass. Checking whether those entries can plausibly coexist is a useful bookkeeping step: it can reveal an omitted compartment, an overlap between entries, or a mismatch in how volume or density was defined. It is not a test that every organ value is correct. A total that agrees with body mass can still conceal compensating errors.
The key is to convert each volume to mass using a density that corresponds to that volume’s definition. Then ask whether the entries describe distinct parts of the same body, at compatible conditions, and whether together they cover the same material included in whole-body mass.
Units and the Accounting Boundary
Mass density is mass per unit volume. Common units include kg/L and g/mL; these are numerically equivalent because 1 kg/L = 1 g/mL. Thus, multiplying a volume in litres by a density in kg/L gives mass in kilograms. Likewise, multiplying a volume in millilitres by a density in g/mL gives mass in grams. Convert the result if it is to be compared with a body mass in different units.
“Volume” is not a single interchangeable quantity. An organ volume might be a gross anatomical volume, a volume of tissue excluding blood, or a value that includes vascular contents. A density might describe wet tissue, an organ’s apparent density at a specified inflation state, or another explicitly defined material basis. Use the density appropriate to the measured volume, not simply a familiar density for a related material.
Before adding entries, define the accounting boundary. For example, a budget might include the whole body, including blood and gastrointestinal contents, or it might exclude a particular content. A whole-body mass and the summed compartments must use the same boundary. Entries also need to be non-overlapping: a whole-organ mass and a separately listed amount of blood within that organ cannot both be counted as independent components.
The additive mass equation follows from ordinary mass conservation. If the body is partitioned into disjoint compartments, its total mass is the sum of their masses. If a compartment’s mass is density times volume on a consistent basis, substitution gives the volume-density sum. This reasoning does not require the body’s total volume to equal the sum of the listed organ volumes: omitted tissues, contents, and spaces may remain. It requires mass contributions to be counted once, not that every volume entry be exhaustive.
Here \(R\) is positive when listed compartments fall short of the reference mass and negative when their calculated mass exceeds it. A useful descriptive measure is the residual as a fraction of reference body mass, \(100R/M\) percent. This makes the size of a discrepancy easier to describe across bodies of different sizes, but it does not by itself establish whether the discrepancy is acceptable.
Worked Example: A Reference-Adult Mass Budget
Consider a hypothetical 70.0 kg reference adult. Suppose a table provides the following deliberately simplified, non-overlapping component volumes and matching densities. Treat the listed components as excluding blood and as not overlapping one another.
| Component | Volume (L) | Density (kg/L) | Calculated mass (kg) |
|---|---|---|---|
| Liver | 1.50 | 1.05 | 1.575 |
| Brain | 1.40 | 1.04 | 1.456 |
| Heart | 0.300 | 1.06 | 0.318 |
| Kidneys | 0.300 | 1.05 | 0.315 |
| Lungs | 1.20 | 0.300 | 0.360 |
| Skeletal muscle | 28.0 | 1.04 | 29.12 |
| Adipose tissue | 14.0 | 0.920 | 12.88 |
| Bone | 5.00 | 1.80 | 9.00 |
| Blood | 5.00 | 1.06 | 5.30 |
For the liver, for example, \(m=1.50\text{ L}\times1.05\text{ kg/L}=1.575\text{ kg}\). Applying the same calculation to each row and adding gives:
The residual is \(R=70.0-60.324=9.676\text{ kg}\), or \(100(9.676/70.0)=13.82\%\) of the reference mass. Under the stated assumptions, the table accounts for 60.324 kg and leaves 9.676 kg unaccounted for. This can be a plausible signal that other tissues or contents have not been listed; it is not evidence that the residual is a single anatomical compartment. The calculation is only as interpretable as the hypothetical non-overlap and density assumptions.
Interpreting the Residual
A residual is an accounting result, not an automatic diagnosis. A positive residual can arise because the table omits components, uses volumes for only selected tissues, or applies densities that do not match the recorded volume definitions. A negative residual can arise because compartments overlap, a component has been counted twice, or a density or unit has been misapplied. Rounding also affects the total, although it is unlikely to explain a large discrepancy.
The strongest check is structural: confirm the meaning and boundary of every entry before judging the size of \(R\). If the inputs are intended to form a complete partition, a non-negligible residual calls for investigation. If the table intentionally contains only selected organs, a positive residual is expected. There is no universal percentage threshold that turns a residual into proof of error; its interpretation depends on the stated scope, measurement basis, and precision.
Worked Example: Detecting Blood Double Counting
Suppose a hypothetical 70.0 kg reference body has listed masses of 1.60 kg for liver, 1.40 kg for brain, 0.32 kg for heart, 0.32 kg for kidneys, 0.70 kg for lungs, 29.0 kg for muscle, 13.0 kg for adipose tissue, 9.00 kg for bone, 9.36 kg for other tissues, and 5.30 kg for whole-body blood. At first, these appear to sum exactly to body mass:
Now suppose the organ and tissue masses were defined as wet masses that already include 0.75 kg of blood within those components, while the 5.30 kg blood entry represents all blood in the body. The entries overlap: the same 0.75 kg appears once inside the organ and tissue masses and again in whole-body blood. A non-overlapping total is therefore \(70.00-0.75=69.25\text{ kg}\), leaving \(70.0-69.25=0.75\text{ kg}\) unaccounted for relative to the reference mass.
The apparent exact agreement was not a successful reconciliation; it resulted from counting some blood twice while still leaving another amount unrepresented. A consistent budget could instead use organ and tissue masses excluding blood and add whole-body blood once, or retain blood within the organ/tissue entries and omit the separate whole-body blood entry. The correct choice depends on how the input values are defined.
Density and Volume Must Share a Basis
Even when units match, the physical definitions may not. For instance, a lung volume measured with air present is not the same volume basis as a volume of lung tissue with air excluded. Multiplying an air-inclusive organ volume by a density defined for solid tissue will generally produce a misleading mass. Conversely, using a low apparent density that already reflects an air-inclusive volume with a tissue-only volume can distort the result in the other direction.
Worked Example: A Density-Basis Mismatch
Imagine a hypothetical inflated lung volume of 1.20 L and a matching apparent density of 0.300 kg/L. The implied mass is \(1.20\times0.300=0.360\text{ kg}\). The density here is stipulated to match the air-inclusive gross volume. If someone instead multiplies that same 1.20 L by a tissue density of 1.05 kg/L, the result is \(1.20\times1.05=1.260\text{ kg}\). The arithmetic is valid, but the pairing is not: the density does not describe the same volume basis.
As a unit check, the first product has units \(\text{L}\times\text{kg/L}=\text{kg}\), but dimensional consistency alone cannot verify that the definitions are compatible. The remedy is to identify whether the volume includes air or other contents and select a density measured or defined for that same basis. If the basis cannot be established, the calculated mass should be treated as uncertain rather than as a reliable reconciliation.
Practical Checks and Alternatives
A basic mass-budget check is most informative when performed systematically. First, put body mass, volumes, and densities into compatible units. Next, record whether each entry includes blood, fluid, air, or contents and whether it overlaps another entry. Then calculate component masses and the residual. Finally, compare the residual’s size and direction with what the list is intended to cover. Keep unrounded values for summation and round the reported result to a sensible precision.
Specify the reference body mass and what material it includes, such as blood or gastrointestinal contents.
Confirm volume units, density units, compatible definitions, and whether components overlap.
For each entry use \(m_i=\rho_iV_i\), then calculate \(R=M-\sum_i m_i\).
Decide whether the residual is consistent with omitted material or points to a definition, unit, or overlap problem. Agreement alone does not validate every input.
This check complements, rather than replaces, other ways of assessing physiological inputs. A volume-only check compares volumes with a defined whole-body volume but does not establish mass unless a compatible density is available. A mass budget directly compares mass contributions with body mass, but depends on the density and accounting assumptions. Directly measured whole-body mass provides a useful target, while independently measured component masses can help locate a discrepancy; neither method makes inconsistent compartment definitions additive.
The approach also has limits. Some entries may be estimates rather than measurements on the same individual, and densities may vary with tissue composition or measurement conditions. A residual can therefore combine real omissions, measurement differences, and approximation. This check is a plausibility and bookkeeping tool, not a method for deriving missing anatomy or proving the accuracy of a reference physiology table.
Common Pitfalls
- Adding volumes and calling the result body mass. Volumes have units of volume, not mass. Multiply each volume by a compatible density before comparing it with body mass.
- Assuming a matching unit is a matching definition. A product can have units of kilograms and still be physically inappropriate if the density and volume refer to different material or conditions.
- Counting blood twice. Check whether organ masses include vascular blood before adding a separate blood compartment.
- Treating a positive residual as a named tissue. The residual identifies an amount not represented by the entries; it does not identify its anatomical source.
- Treating a zero residual as proof of correctness. Overlaps and omissions can cancel, so agreement is necessary for a complete budget but not sufficient to validate its inputs.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A hypothetical tissue volume is 2.00 L and its compatible density is 1.10 kg/L. What is its mass contribution?
- A body mass is 20.0 kg and non-overlapping listed components sum to 18.5 kg. Calculate the residual and its percentage of body mass. What can you conclude?
- Why might adding a whole-body blood mass to organ masses that already include vascular blood give an invalid total?
- A volume is in millilitres and a density is in kg/L. What should be checked or converted before multiplying?
Solutions. (1) \(m=2.00\text{ L}\times1.10\text{ kg/L}=2.20\text{ kg}\). (2) \(R=20.0-18.5=1.50\text{ kg}\), and \(100(1.50/20.0)=7.50\%\). The listed components leave 1.50 kg unaccounted for; the result alone does not identify the source. (3) The blood within the organs would be counted both as part of organ mass and again as whole-body blood, violating the requirement that compartments be non-overlapping. (4) Convert to compatible volume units, such as litres, or convert density to compatible units, such as kg/mL, and confirm that the density and volume refer to the same physical basis.