Case Study 2 — How Fast Is the Tumor Growing? Related Rates in Medical Imaging

Field: Medicine / oncology and diagnostic radiology Calculus used: Related rates for a spherical volume, with implicit differentiation of the volume–radius relation (§8.7, §8.14)

What the scan actually measures

A radiologist reading a series of CT scans does not measure a tumor's volume. She measures its radius — or, more precisely, a diameter she can place calipers on across the largest cross-section the scanner reveals. Volume is hidden; the machine reports a length. Yet the quantity that matters clinically — the one that decides whether a treatment is working, whether a "stable" lesion is quietly doubling, whether to escalate to surgery — is how fast the volume is changing.

Bridging the two is a related-rates problem, and it is one oncologists confront on every follow-up scan. The tumor is modeled, to good approximation for many solid masses, as a sphere. Its volume and radius are locked together by

$$V = \frac{4}{3}\pi r^3,$$

a relation that holds at every instant of the tumor's growth. The radiologist measures $r$ and its change $\dfrac{dr}{dt}$ from one scan to the next; the clinician needs $\dfrac{dV}{dt}$. The chain rule converts one into the other — and, as we will see, the conversion factor is itself a clinically meaningful quantity.

This case study follows one patient's two scans, computes the volume growth rate the way a tumor-board calculation actually proceeds, and then confronts the subtler question the bare number cannot answer.

Two scans, eight weeks apart

A patient with a solid hepatic lesion is scanned, treated, and rescanned. The measured radii are:

  • Scan 1 (week 0): $r = 1.5$ cm.
  • Scan 2 (week 8): $r = 1.8$ cm.

Over $8$ weeks, the radius grew by $0.3$ cm. The average radial growth rate is

$$\frac{\Delta r}{\Delta t} = \frac{0.3 \text{ cm}}{8 \text{ weeks}} = 0.0375 \text{ cm/week}.$$

We will treat this measured average as the instantaneous radial rate $\dfrac{dr}{dt}$ at the second scan — the standard clinical approximation when only two time points are available. (With three or more scans, one fits a growth curve and reads off a true instantaneous slope; the related-rates machinery is identical.)

The Key Insight. The tumor's radius is growing at a modest, almost reassuring $0.0375$ cm/week — less than half a millimeter. A clinician who looks only at the radius might call the lesion "slowly progressing." But volume scales as the cube of radius, and the related rate $\dfrac{dV}{dt}$ will tell a far more alarming story. This is the entire reason the calculus matters here: the eye reads a length, but the biology lives in a volume.

Apply the seven-step method (§8.7). The changing quantities are $r$ and $V$; the relating equation is the sphere volume $V = \tfrac43\pi r^3$. Differentiate both sides with respect to time, treating $r$ as a function of $t$ and applying the chain rule:

$$\frac{dV}{dt} = \frac{4}{3}\pi \cdot 3r^2 \cdot \frac{dr}{dt} = 4\pi r^2\,\frac{dr}{dt}.$$

This is the §8.14 result, and it carries a beautiful interpretation: the conversion factor between radial rate and volume rate is exactly $4\pi r^2$, the tumor's surface area. Geometrically, growing the radius by an increment $dr$ wraps a thin shell of thickness $dr$ around the whole surface; the volume added is (surface area) $\times$ (shell thickness). The chain rule has rediscovered the shell, a fact Chapter 18 will turn into a full method of integration.

Substituting the second scan

At scan 2, $r = 1.8$ cm and $\dfrac{dr}{dt} = 0.0375$ cm/week. Substitute:

$$\frac{dV}{dt} = 4\pi (1.8)^2 (0.0375) = 4\pi (3.24)(0.0375).$$

Compute the constants: $4 \times 3.24 \times 0.0375 = 0.486$, so

$$\frac{dV}{dt} = 0.486\pi \approx 1.527 \text{ cm}^3/\text{week}.$$

Reading the answer the way a clinician must

The radius grows at $0.0375$ cm/week — practically invisible. The volume grows at about $1.53$ cm³ per week. To feel what that means, compute the volumes directly:

$$V_1 = \tfrac43\pi(1.5)^3 = \tfrac43\pi(3.375) \approx 14.14 \text{ cm}^3, \qquad V_2 = \tfrac43\pi(1.8)^3 = \tfrac43\pi(5.832) \approx 24.43 \text{ cm}^3.$$

In eight weeks the tumor's volume rose from about $14$ cm³ to about $24$ cm³ — a 73% increase in burden from a radius change that looked trivial. The related rate $\dfrac{dV}{dt} = 4\pi r^2\,\dfrac{dr}{dt}$ is what makes that disconnect quantitative and, crucially, prospective: at the current surface area, every additional half-millimeter of radius now adds far more volume than it did when the tumor was small, because $4\pi r^2$ has grown with $r$.

Common Pitfall. A natural mistake is to estimate volume growth by scaling the radius growth linearly — "the radius went up $20\%$, so the volume went up about $20\%$." That reasoning ignores the $r^2$ factor in $\dfrac{dV}{dt}$ and the cubic relationship behind it. The radius rose $20\%$ ($1.5 \to 1.8$), but the volume rose $73\%$. Whenever a measured length feeds a power-law volume or area, the related rate must carry the derivative's power factor — here the surface-area factor $4\pi r^2$. Dropping it understates growth dangerously.

The conversion factor is the diagnosis

Notice what the formula $\dfrac{dV}{dt} = 4\pi r^2\,\dfrac{dr}{dt}$ predicts about the future. Suppose the radial rate held steady at $0.0375$ cm/week. Then the volume rate would keep climbing, because $4\pi r^2$ grows as the tumor enlarges. At $r = 1.8$ cm the volume rate is $1.53$ cm³/week; at $r = 2.5$ cm it would be

$$4\pi(2.5)^2(0.0375) = 4\pi(6.25)(0.0375) = 0.9375\pi \approx 2.95 \text{ cm}^3/\text{week},$$

nearly double — from the same radial growth. A constant-looking radius trend masks an accelerating volume burden. This is precisely why oncology guidelines (e.g., RECIST) track tumor dimensions over time but interpret them through volumetric reasoning: the related rate translates a benign-seeming length trend into the true tempo of disease.

Real-World Application — RECIST and volumetric response. The Response Evaluation Criteria in Solid Tumors (RECIST) traditionally classify response by changes in the longest diameter, but modern volumetric analysis uses exactly the $V = \tfrac43\pi r^3$ relation (and its ellipsoidal generalizations) to compute true volume change. A "$30\%$ decrease in diameter" — RECIST's partial-response threshold — corresponds to a $(0.70)^3 \approx 0.34$ volume, i.e. a $66\%$ volume reduction. The related-rate factor $4\pi r^2$ is what converts the diameter the radiologist measures into the volume the oncologist treats.

A second geometry: contrast washing out

The same calculus governs a different imaging task. In a contrast-enhanced study, a roughly spherical region of injected contrast agent shrinks in apparent radius as the agent diffuses and washes out. If the imaged radius is decreasing at, say, $\dfrac{dr}{dt} = -0.2$ cm/min when $r = 1.0$ cm, then the rate at which the enhancing volume disappears is

$$\frac{dV}{dt} = 4\pi r^2\,\frac{dr}{dt} = 4\pi(1.0)^2(-0.2) = -0.8\pi \approx -2.51 \text{ cm}^3/\text{min}.$$

The negative sign reports washout — volume leaving. Radiologists use this washout rate to characterize tissue: malignant and benign lesions wash contrast in and out at different speeds, and the rate $\dfrac{dV}{dt}$ (or its surrogate, the rate of signal-intensity change) is a diagnostic fingerprint. One relating equation, $V = \tfrac43\pi r^3$, differentiated once, serves both the growth question and the washout question — they differ only in the sign of $\dfrac{dr}{dt}$.

Where the model bends — and the honesty calculus demands

The sphere is an idealization. Real tumors are irregular; a more faithful model is an ellipsoid $V = \tfrac43\pi abc$ with three semi-axes, and the related rate becomes a sum of three terms (a product rule on $abc$), each carrying its own axis's growth rate. Measurement noise matters too: if the radius is known only to $\pm 0.1$ cm, a single-scan $\dfrac{dr}{dt}$ inherits real uncertainty, and the computed $\dfrac{dV}{dt}$ should be reported as a range, not a false-precision decimal. The §8.8 ladder warned that a related rate "faithfully reports when a model breaks, if you read the answer honestly." Here, honesty means: the calculus is exact, but it is exact about a sphere, and the clinician must judge how spherical the tumor really is. The mathematics is a lens, not an oracle.

Discussion Questions

  1. The radius grew $20\%$ but the volume grew $73\%$. Derive the exact relationship: if the radius scales by a factor $k$, by what factor does the volume scale? Why does this make small radius changes clinically significant for large tumors?

  2. The conversion factor $4\pi r^2$ is the tumor's surface area. Explain in words why "rate of volume growth = surface area × rate of radial growth," and connect it to the shell idea that Chapter 18 develops.

  3. For the same constant radial rate $\dfrac{dr}{dt}$, $\dfrac{dV}{dt}$ increases as the tumor grows. Is a constant radial growth rate therefore good news or bad news for the patient? Explain using the formula.

  4. In the washout example, $\dfrac{dr}{dt} < 0$ produced $\dfrac{dV}{dt} < 0$. What would $\dfrac{dr}{dt} = 0$ mean physically, and what does it say about a lesion whose imaged radius is stable across scans?

  5. Real tumors are ellipsoids. Differentiate $V = \tfrac43\pi abc$ with respect to $t$ (treating $a$, $b$, $c$ as functions of time) and explain why measuring only one axis can badly misestimate $\dfrac{dV}{dt}$.

Your Turn

  1. A spherical lesion has $r = 2.0$ cm and is measured to be growing radially at $0.05$ cm/week. Compute $\dfrac{dV}{dt}$ and the percentage volume increase you would expect over the next $4$ weeks.

  2. Suppose a treatment shrinks a tumor's diameter by exactly $30\%$. Using $V = \tfrac43\pi r^3$, compute the percentage volume reduction and compare it to the RECIST "partial response" intuition.

  3. The radius measurement carries uncertainty $\pm 0.1$ cm. With $r = 1.8 \pm 0.1$ and $\dfrac{dr}{dt} = 0.0375$ cm/week, compute the range of plausible $\dfrac{dV}{dt}$ values. How much does measurement noise widen the answer?

Annotated Further Reading

  • Eisenhauer, E. A., et al. (2009). "New response evaluation criteria in solid tumours: revised RECIST guideline (version 1.1)," European Journal of Cancer 45(2). The definitive RECIST reference; read it to see how clinical practice handles the diameter-to-volume translation the case study formalizes.

  • Prokop, M., and Galanski, M. (2003). Spiral and Multislice Computed Tomography of the Body, Thieme. A radiology standard; its volumetry chapters show the real measurement pipeline behind the idealized sphere.

  • Stewart, J. (2020). Calculus: Early Transcendentals (9th ed.), Cengage, §3.9. The related-rates section; its inflating-balloon problems are the direct mathematical twins of this tumor-growth calculation.

  • Strang, G., and Herman, E. (2016). Calculus, Volume 1, OpenStax (free), §4.1 "Related Rates." A freely available treatment with worked spherical-volume examples; good for additional practice on the $\dfrac{dV}{dt} = 4\pi r^2\dfrac{dr}{dt}$ pattern.

Connections

  • §8.14 — the expanding-sphere related rate ($dV/dt = 4\pi r^2\,dr/dt$, the identity $dV = S\,dr$) that this case study applies clinically.
  • §8.7 — the seven-step method followed throughout.
  • Chapter 18 — the shell interpretation of $4\pi r^2\,dr$ becomes the shell method of integration.
  • Chapter 39 — the modeling portfolio's biology track extends this to tumor-growth differential-equation models (e.g., Gompertz growth).