Planning for Rare Cell Populations

Why Rare-Population Planning Needs Two Counts

A population can be biologically important yet appear in only a small fraction of cells. Planning then has two linked questions: how many usable cells should be obtained from each donor, and how many donors should be included? A large total cell count does not by itself answer either question. If most cells come from a few donors, the data may contain many cells but little chance of observing the population in several independent donors.

The key complication is that prevalence may differ among donors. A population with an average prevalence of 0.5% might be near that level in every donor, or it might be common in some donors and nearly absent in others. Those situations have the same average prevalence but different probabilities of detection. The calculations below concern observing a population, not the power of a later test for a condition effect.

This model treats the sampled cells from a donor as draws from that donor's cell population. It assumes the \(m\) cells are sampled without strong cell-level dependence, the target population can be identified with adequate accuracy, and donors are independent draws from the donor population relevant to the study. If cell capture or classification is imperfect, the prevalence used for planning should reflect the probability that a target cell is both captured and recognized, or that loss should be represented explicitly.

Cells per Donor When Prevalence Is Fixed

First suppose every donor has the same known prevalence \(p\). Then \(K_i\sim\operatorname{Binomial}(m,p)\), and the probability of observing no target cells in a donor is \((1-p)^m\). Therefore, the probability of at least one detection in that donor is \(1-(1-p)^m\). To require a detection probability of at least \(1-\alpha\), choose the smallest integer \(m\) satisfying the resulting inequality.

Both logarithms in the ratio are negative, so the ratio is positive. The integer requirement is obtained by rounding the bound upward. This is a per-donor detection target: it does not guarantee detection in every donor, nor does it guarantee a specified total number of target cells.

Allowing Prevalence to Vary Across Donors

When donor prevalences vary, use a distribution for \(P_i\), rather than substituting only its mean into the fixed-prevalence formula. A convenient model is \(P_i\sim\operatorname{Beta}(a,b)\), independently across donors. Its mean is \(\mu=a/(a+b)\). The parameters \(a\) and \(b\) control the spread as well as the mean; knowing \(\mu\) alone is not enough to determine detection reliability.

A simpler alternative for transparent planning is to specify a discrete mixture of plausible donor prevalences. If prevalence is \(p_j\) with probability \(w_j\), where the weights sum to one, then \(q_m=\sum_j w_j[1-(1-p_j)^m]\). This explicitly describes the donor variation being assumed and can be useful when pilot information suggests distinct prevalence levels.

The effect of variation has a useful justification. For \(m\ge 2\), the function \(f(p)=(1-p)^m\) is convex on \(0\le p\le 1\), since \(f''(p)=m(m-1)(1-p)^{m-2}\ge 0\). Jensen's inequality therefore gives \(\operatorname{E}[(1-P_i)^m]\ge (1-\operatorname{E}[P_i])^m\). Consequently, at a fixed mean prevalence, variation increases the average probability of a zero-cell donor and reduces the average detection probability compared with assuming every donor has the mean prevalence. This result concerns detection in a donor; it does not say that variation reduces the expected total number of target cells.

Choosing the Number of Donors

Once the per-donor detection probability \(q_m\) is specified, the donor-level planning calculation is straightforward if donors are independent and share that probability. Let \(R\) be the number of donors in which at least one target cell is observed. Then \(R\sim\operatorname{Binomial}(D,q_m)\). A study can choose \(D\) to make the probability of detecting the population in at least \(r\) donors exceed a target probability.

For a beta prevalence model, this formula applies after integrating over the prevalence distribution to obtain \(q_m\), provided donor prevalences are independent and identically distributed. If donor groups have different prevalence distributions, calculate their detection probabilities separately; a single binomial model with one common \(q_m\) may not be appropriate.

Expected Counts, Diagnostics, and Alternatives

With \(m\) cells per donor and mean prevalence \(\mu=\operatorname{E}[P_i]\), the expected total number of observed target cells across \(D\) donors is \(Dm\mu\). This follows because \(\operatorname{E}[K_i]=m\operatorname{E}[P_i]=m\mu\). But this expected count does not determine the probability of meeting a practical target, such as detecting at least one cell in several donors or obtaining at least a specified total count. For those targets, use the relevant count distribution or simulate the complete donor-and-cell sampling process.

A Poisson approximation can simplify the fixed-prevalence calculation when \(p\) is small and \(m\) is large: \(P(K_i=0)=(1-p)^m\approx e^{-mp}\), so \(q_m\approx1-e^{-mp}\). With variable prevalence, the corresponding approximation averages \(e^{-mP_i}\) over donors; it is not generally \(1-e^{-m\mu}\). For a required total number of cells, a beta-binomial or simulation-based calculation can account for prevalence heterogeneity and the variation in counts among donors.

Useful diagnostics and planning checks include the following:

• Compare the assumed prevalence distribution with pilot or prior donor-level data. A mean without a measure of between-donor spread cannot establish reliable donor detection.
• Check that \(m\) means usable cells per donor, not merely cells loaded, sequenced, or retained before quality control. If usable yield varies, model that variation or plan for a conservative yield.
• Examine whether the rare population is consistently identifiable. Low sensitivity lowers effective detection, while false-positive classifications can make apparent detection unreliable.
• Check independence and representativeness assumptions. Related donors, clustered recruitment, or a study population that differs from the target population can invalidate the simple donor-binomial calculation.
• Report the target explicitly: probability of at least one detected cell per donor, number of detected donors, or total number of cells. These are different design goals.

The homogeneous binomial model is the simplest alternative and is adequate when prevalence is genuinely similar across donors or when a conservative design has been checked against plausible heterogeneity. The beta-binomial model represents continuous donor variation with a compact two-parameter distribution; a discrete mixture represents specified scenarios directly. Simulation is more flexible when cell yields, prevalence, classification sensitivity, or donor-specific features vary together, but it depends on the accuracy of the simulated assumptions. No model can compensate for an unrepresentative pilot or an incorrectly defined target population.