Why Cell Burden and Tumor Volume Are Not the Same Quantity
Tumor volume is often the measurement available in a longitudinal study, while the biological quantity of interest may be the number of viable tumor cells. These are related, but they are not interchangeable. A measured mass can contain viable cells, necrotic tissue, extracellular space, stroma, and fluid. Its outer boundary may also be only approximately represented by an imaging measurement. A useful QSP model therefore makes the link between biological states and the recorded volume explicit rather than treating volume as a direct count of living cells.
The purpose here is to define a compartment-based mapping at a given time. It is not a model of how any of the compartments change over time. The mapping requires assumptions about how cell number translates into the gross volume of viable tissue and how the tissue compartments combine.
The distinction between cell number and viable-tissue volume matters because viable cells do not necessarily occupy every point in the region classified as viable tumor. Let \(v_c\) denote the mean physical volume of one viable tumor cell, and let \(\phi_v(t)\) be the fraction of the gross viable-tissue compartment occupied by viable cells. Under a homogeneous-compartment approximation, \(N_v(t)v_c\) is the volume of the cells themselves, while division by \(\phi_v(t)\) accounts for the spaces within that gross tissue region.
Here \(v_c\) has units of volume per cell, \(N_v\) is a cell count, and \(\phi_v\) is dimensionless, so \(V_v\), \(V_n\), \(V_o\), and \(V_T\) all have units of volume. A value of \(\phi_v=0.5\), for example, means that viable cells occupy half of the gross viable-tissue compartment under this definition. Necrotic tissue is represented by its gross compartment volume, including any internal space that belongs to that compartment. The same convention applies to \(V_o\). This definition prevents an additional “space” term from being counted again within those compartments.
Deriving the Compartment Equation
The equation follows from two separate accounting steps. First, if the viable cells have volumes \(v_1,\ldots,v_{N_v}\), their total physical volume is \(\sum_i v_i\). If \(v_c\) is the mean cell volume, this is represented as \(N_v v_c\). By definition, the viable-cell volume fraction is the cell volume divided by the gross volume of the viable-tissue compartment: \(\phi_v=N_vv_c/V_v\). Rearranging gives \(V_v=N_vv_c/\phi_v\). Second, the gross tumor is partitioned into disjoint compartments. Additivity of volume for disjoint regions gives \(V_T=V_v+V_n+V_o\).
This justification depends on the partition being mutually exclusive and collectively covering the volume counted as tumor. If a necrotic region is already included in a reported gross “viable-tumor” volume, adding it again would double count it. Likewise, if a protocol excludes surrounding fluid or stroma from the tumor boundary, those components must not be included in \(V_o\). The compartment definitions need to match the measurement protocol.
The simplest model may omit \(V_o\), but only if other included material is negligible or intentionally absorbed into a calibrated viable or necrotic compartment. In that case, \(V_T=V_v+V_n\). This simplification does not mean that stroma or fluid is biologically absent; it means the model does not represent it separately. If these components vary substantially, that simplification may distort the mapping from viable-cell number to total volume.
Worked Example: Mapping Cell Number to Total Volume
Consider a hypothetical tumor with \(8.00\times10^8\) viable cells. Suppose the mean cell volume is \(1.20\times10^{-6}\) mm\(^3\) per cell and the viable-cell fraction of the gross viable-tissue compartment is \(0.60\). Let necrotic tissue occupy \(350\) mm\(^3\) and other included material occupy \(250\) mm\(^3\). Assume these compartments are disjoint and that the stated fraction and mean cell volume apply throughout the viable compartment.
The physical volume of the viable cells is \(N_vv_c=(8.00\times10^8)(1.20\times10^{-6})=960\) mm\(^3\). The gross viable-tissue volume is therefore \(V_v=960/0.60=1{,}600\) mm\(^3\).
Adding the disjoint compartments gives \(V_T=1{,}600+350+250=2{,}200\) mm\(^3\), or \(2.20\) cm\(^3\), because \(1{,}000\) mm\(^3=1\) cm\(^3\). Under these assumptions, the model maps the specified viable-cell burden and tissue compartments to a total anatomical volume of \(2.20\) cm\(^3\). It does not imply that every tumor with this total volume has the same viable-cell count.
From Anatomical Volume to an Imaging Measurement
The model state \(V_T\) is an anatomical quantity. Imaging typically produces dimensions, a segmentation, or a reported volume estimate. A measurement equation connects that observation to the state. If image processing directly yields a volume and has been calibrated to the anatomical volume scale, a basic model is \(Y=V_T+\epsilon\), where \(Y\) is the measured volume and \(\epsilon\) is measurement error. If the measurement method has a known scale bias or offset, it can instead be represented as follows.
where \(a\) is a dimensionless scale factor, \(b\) is an offset in volume units, and \(\epsilon(t)\) is the residual measurement error, also in volume units. In an ideal calibration, \(a=1\) and \(b=0\). This equation assumes the error is adequately described on the original volume scale; if errors are mainly proportional rather than absolute, a multiplicative error model may be more appropriate. The error model should reflect the acquisition and processing method, not be chosen only for algebraic convenience.
If volume is calculated from three measured orthogonal diameters and the tumor is approximated as an ellipsoid, the geometric estimate is \(Y=\pi LWH/6\), with \(L\), \(W\), and \(H\) measured in the same length units. The ellipsoid approximation is a measurement convention, not a claim that the true tumor has a perfect ellipsoidal boundary. Irregular shapes, partial-volume effects, image resolution, and variation in segmentation can all contribute to the discrepancy between this estimate and anatomical volume.
Worked Example: Converting Imaging Dimensions to Volume
Suppose a hypothetical scan gives orthogonal tumor dimensions of \(18\) mm, \(14\) mm, and \(12\) mm. Assume an ellipsoid approximation and, for this calculation, an ideal scale calibration with no offset.
The estimated volume is \(Y=\pi LWH/6\). Substituting the measurements gives \(Y=\pi(18)(14)(12)/6=\pi(3{,}024)/6=504\pi\approx1{,}583\) mm\(^3\).
This is approximately \(1.58\) cm\(^3\). The result is an imaging-derived estimate of total volume, not a direct measurement of viable cells. Interpreting it as cell burden would additionally require the compartment mapping and its parameters, such as \(v_c\), \(\phi_v\), \(V_n\), and \(V_o\).
Identifiability and Alternative Mappings
A total-volume observation alone generally cannot determine the viable-cell count. The equation contains several contributions: viable cells, their volume fraction, necrotic tissue, and other included material. One measured value does not uniquely determine all these quantities. Fixing some parameters or collecting measurements that distinguish compartments can make the mapping more informative, but the assumptions need to be stated.
Worked Example: Equal Total Volumes, Different Viable Burdens
Consider two hypothetical tumors, each with total volume \(2{,}000\) mm\(^3\). For both, let \(v_c=1.00\times10^{-6}\) mm\(^3\) per cell and \(\phi_v=0.50\). Assume the listed compartments are disjoint.
For tumor A, let \(V_n=800\) mm\(^3\) and \(V_o=200\) mm\(^3\). The viable-tissue volume is \(2{,}000-800-200=1{,}000\) mm\(^3\). Thus \(N_v=V_v\phi_v/v_c=(1{,}000)(0.50)/(1.00\times10^{-6})=5.00\times10^8\) cells.
For tumor B, let \(V_n=300\) mm\(^3\) and \(V_o=200\) mm\(^3\). Its viable-tissue volume is \(1{,}500\) mm\(^3\), giving \(N_v=(1{,}500)(0.50)/(1.00\times10^{-6})=7.50\times10^8\) cells. The total volumes match, but the inferred viable-cell burdens differ by \(2.50\times10^8\) cells. A total-volume measurement by itself cannot distinguish these cases.
There are three common modelling choices, each with a different information requirement.
| Approach | Mapping | When it is useful |
|---|---|---|
| Compartment-based | Explicitly represents viable, necrotic, and other volumes. | When compartment information or defensible assumptions are available. |
| Fixed-fraction approximation | Uses fixed values for \(\phi_v\) and, if needed, the fractions assigned to necrotic and other material. | When the data do not support time-varying fractions and the approximation is adequate for the intended use. |
| Empirical calibration | Relates an observed volume to a cell-burden measure using fitted calibration parameters. | When paired measurements support calibration in a defined setting, without requiring every compartment to be modelled separately. |
A fixed-fraction or empirical relationship can be simpler, but simplicity does not remove the need to check its domain of validity. A calibration from one imaging method, tumor type, or disease stage may not apply to another. Conversely, a more detailed compartment model is not automatically more reliable: if its parameters cannot be measured or constrained, it can create an appearance of biological precision without identifying the cell burden.
Diagnostics, Assumptions, and Common Pitfalls
Useful checks begin with definitions and units. Confirm that each volume is in the same units, that \(0<\phi_v\leq1\), and that the resulting compartments are nonnegative. Check that the sum of modelled compartments corresponds to the boundary used by the imaging protocol. If repeated measurements are available, inspect differences between measured and predicted volumes for systematic bias, dependence on tumor size, or time-varying patterns. Such patterns can indicate that a constant scale factor, fixed volume fraction, or assumed boundary convention is inadequate.
When possible, compare the mapping with independent information, such as histological estimates of viable and necrotic fractions or a second imaging measure that is sensitive to tissue composition. This is not a guarantee of accuracy: the independent measurement has its own sampling and measurement limitations. It can, however, reveal an important mismatch that a total-volume fit alone would conceal. A good fit to total volume does not prove that the model has correctly separated viable from necrotic tissue.
- Wrong reasoning: “The tumor volume is \(2\) cm\(^3\), so its viable-cell volume is \(2\) cm\(^3\).”
Correction: Total volume may include necrotic tissue and other material. The viable-cell physical volume is \(N_vv_c\); the gross viable-tissue volume is \(N_vv_c/\phi_v\). - Wrong reasoning: “I should add a space volume to the total after using \(V_v=N_vv_c/\phi_v\).”
Correction: The gross viable-tissue volume already accounts for space within that compartment. Add only separately defined, nonoverlapping compartments. - Wrong reasoning: “A calibrated total-volume measurement uniquely identifies viable-cell burden.”
Correction: Calibration can relate measurement to total anatomical volume; additional information or assumptions are needed to separate viable cells from the other contributions.
The central limitation is structural: the mapping depends on cell size, packing fraction, compartment boundaries, and measurement calibration. These may vary across tumors or over time. If they are held fixed, that is an explicit modelling assumption, not a consequence of the total-volume data. Use the simplest mapping that answers the scientific question, and avoid interpreting a total-volume observation as a direct measure of viable-cell number when its component volumes are not identified.
Check Your Understanding
Answer each question, then compare your reasoning with the brief solutions.
- A viable compartment contains \(4.00\times10^8\) cells with mean cell volume \(1.00\times10^{-6}\) mm\(^3\) per cell and \(\phi_v=0.40\). What is its gross volume?
- Why can necrotic tissue not be added separately if it is already included in the defined viable-tissue compartment?
- An ellipsoid has measured dimensions \(10\), \(8\), and \(6\) mm. What is its estimated volume in mm\(^3\)?
- Two tumors have equal measured total volume but may have different viable-cell burdens. State one reason this can occur.
- What does a nonzero offset \(b\) represent in \(Y=aV_T+b+\epsilon\), and what are its units?
Solutions: (1) \(V_v=(4.00\times10^8)(1.00\times10^{-6})/0.40=1{,}000\) mm\(^3\). (2) Adding it again would count the same physical region twice, violating the disjoint-compartment assumption. (3) \(Y=\pi(10)(8)(6)/6=80\pi\approx251.3\) mm\(^3\). (4) The tumors can differ in necrotic volume, other included material, or viable-cell fraction. (5) \(b\) is a fixed measurement offset and has units of volume, such as mm\(^3\).