Why mixtures are different (components sum to 1)
A mixture design doe is needed whenever your factors are the proportions of components that must add up to a fixed total, usually 100% or a fraction of 1. The defining constraint is that the components are not independent: if a medium is made of three carbon sources and you increase glucose from 30% to 40%, the other two must fall by 10% combined. That dependency breaks the assumptions of an ordinary full factorial design, which assumes each factor can be set independently of the others.
Because the proportions sum to one, the experimental region is not a cube but a simplex — a triangle for three components, a tetrahedron for four. Every valid blend is a point inside that simplex. The response is fitted with a special form of polynomial introduced by Henry Scheffé in 1958, which drops the intercept and pure-quadratic terms (they are not estimable under the constraint) and replaces them with cross-product terms that capture how components blend.
This is exactly the situation in media formulation and buffer formulation. The real question is rarely "how much of each ingredient in absolute terms" but "what is the best blend" — what ratio of amino acids, what proportion of buffering salts. A mixture design answers that question directly, where a factorial design would waste runs exploring impossible combinations that violate the sum-to-one constraint.
Simplex-lattice vs simplex-centroid
The two foundational mixture designs are the simplex-lattice and the simplex-centroid. Both place experimental blends at structured points inside the simplex; they differ in which blends they choose.
A simplex-lattice design, written {q, m} for q components and degree m, spaces points evenly across the simplex so you have just enough runs to fit a polynomial of order m. For three components a {3, 2} simplex-lattice has 6 points — the three pure components and the three 50:50 binary blends — enough to fit a second-order Scheffé model. Raising m to 3 adds the one-third/two-third blends and the centroid, giving more resolution for strongly curved responses at the cost of more runs.
A simplex-centroid design for q components uses all 2q−1 non-empty subsets at equal proportions: every pure component, every 50:50 binary blend, every one-third ternary blend, and so on up to the overall centroid. For three components that is 7 points. The simplex-centroid is the better default when you expect every component to contribute something, because it always includes the all-components centroid blend that real media and buffers actually use.
| Design | Points (3 components) | Points (4 components) | Blends included |
|---|---|---|---|
| Simplex-lattice {q,2} | 6 | 10 | Pure + binary (50:50) blends |
| Simplex-lattice {q,3} | 10 | 20 | Pure + ⅓/⅔ binary + ternary blends |
| Simplex-centroid | 7 | 15 | All subsets at equal proportions + centroid |
For most early-stage media formulation work, a simplex-centroid or a {q, 2} simplex-lattice with a few extra interior check points is the practical starting choice. As with response-surface work, add several replicates of the overall centroid to estimate pure error and test for lack of fit.
Constrained mixtures
Pure simplex designs assume every component can range from 0% to 100%, but real formulations almost never allow that. A medium cannot be 100% trace-metal solution, and a buffer needs a minimum salt concentration to buffer at all. A constrained mixture design adds lower and upper bounds to each component, which shrinks the full simplex to a smaller polygon inside it.
When the bounds are tight, the feasible region becomes an irregular sub-polygon, and the evenly spaced simplex-lattice points no longer fall inside it. The standard fix is an extreme-vertices design (sometimes called a D-optimal mixture design), which places runs at the vertices and edge midpoints of the constrained region and selects an efficient subset. Most software — and the free generator below — computes these vertices for you once you enter the bounds.
Constraints are the norm, not the exception, in buffer formulation: you might fix total ionic strength while varying the ratio of two buffering species and a salt, all within bounds that keep the buffer in its useful pH range. Defining sensible bounds before you design is the single most important step — bounds that are too wide waste runs on infeasible blends, while bounds that are too tight can hide the optimum.
Mixture + process variables
Sometimes the blend is only half the story. A mixture-process variable design crosses the mixture components with one or more independent process variables — temperature, pH, total medium concentration, or feed timing — so you can ask whether the best blend changes with the process conditions. A related variant, the mixture-amount design, varies the total amount of the mixture as well as its proportions.
These combined designs are powerful but expensive: crossing a 7-point simplex-centroid with two levels of a process variable already doubles the run count. They are best kept for the final optimisation, once screening and a pure mixture design have identified the critical components. For the broader screen-then-optimise logic that decides when to invest in the larger design, see the DOE for bioprocess optimization guide, and for choosing the optimisation design itself, the response surface methodology article.
Worked media example
Mixture designs shine in cell culture media formulation, where the components are amino acids, sugars, salts, and trace elements whose proportions sum to a fixed total. The example below shows a three-component carbon-source blend, the kind of study that follows a Plackett-Burman screen of many components.
Worked example: 3-component carbon-source blend for CHO titer
A screen has identified three carbon/energy sources as critical. You want the blend (summing to 100% of the carbon pool) that maximises mAb titer: A = glucose, B = galactose, C = mannose. You run a 7-point simplex-centroid design (3 pure, 3 binary 50:50, 1 ternary centroid) plus 2 centre replicates.
| Run | A (glu) | B (gal) | C (man) | Titer (g/L) |
|---|---|---|---|---|
| 1 | 1.00 | 0 | 0 | 2.4 |
| 2 | 0 | 1.00 | 0 | 1.9 |
| 3 | 0 | 0 | 1.00 | 1.6 |
| 4 | 0.50 | 0.50 | 0 | 3.5 |
| 5 | 0.50 | 0 | 0.50 | 2.6 |
| 6 | 0 | 0.50 | 0.50 | 2.1 |
| 7 | 0.33 | 0.33 | 0.33 | 3.2 |
Reading it: the 50:50 glucose/galactose blend (run 4, 3.5 g/L) beats every pure component — a classic synergistic blending effect that a one-factor-at-a-time approach would never reveal. Fitting the Scheffé model and mapping the response onto the triangle places the optimum near 55% glucose, 40% galactose, 5% mannose at ~3.6 g/L. A confirmation run at that blend closes the loop.
Reading a ternary plot
The natural way to read a three-component mixture design doe is a ternary plot — the same triangular simplex, now overlaid with response contours. Each corner is 100% of one component; the opposite edge is 0% of it. Any interior point is a blend, and you read its three proportions by following the grid lines parallel to each edge back to the axes. The optimal blend is the point inside the innermost closed contour.
A response that peaks along an edge or in the interior, away from the pure-component corners, is the signature of a synergistic blend — components that work better together than alone. A response that is highest at a corner means one component dominates and blending hurts. Reading these patterns straight off the ternary plot is what makes mixture designs so intuitive for formulation work.
Build a mixture design free
You do not need an expensive statistics package to run a mixture design doe. A free design of experiments calculator builds simplex-lattice and simplex-centroid designs, handles component bounds for constrained mixtures, randomises the run order, and exports a blend sheet ready for the bench.
Enter your components and any lower/upper bounds, choose simplex-lattice or simplex-centroid, and the tool returns the full set of blends. After the runs, fit the Scheffé model and read the optimum from the ternary plot. Because the same generator builds screening and response-surface designs too, you can screen many medium components, optimise the critical few as a mixture, and never switch tools — a complete free screening-to-formulation workflow for media and buffer formulation.
Build a mixture design now
A free, no-install DOE generator: simplex-lattice and simplex-centroid designs, component bounds, randomised run order, and blend-sheet export.
Frequently Asked Questions
What is a mixture design in DOE?
A mixture design is a DOE in which the factors are the proportions of components that must sum to a constant (usually 100% or 1). Because the proportions are not independent — raising one means lowering another — you cannot use an ordinary factorial design. Instead a mixture design places experiments inside a simplex (a triangle for three components, a tetrahedron for four) and fits a special Scheffé polynomial. It is the standard approach for media formulation and buffer formulation problems.
When do I need a mixture design instead of a factorial design?
You need a mixture design whenever your factors are fractions of a whole that must add up to a fixed total — for example, the proportions of carbon sources in a medium or the buffer salts in a formulation. In that case the components are constrained, so changing one forces a change in another, and a normal factorial design is invalid. If your factors are independent process settings such as temperature, pH and feed rate, use a factorial or response-surface design instead.
What is the difference between a simplex-lattice and a simplex-centroid design?
A simplex-lattice design spaces points evenly across the simplex at a chosen degree m, giving enough runs to fit a polynomial of that order; a {3,2} simplex-lattice for three components has 6 points. A simplex-centroid design uses the pure components, all binary 50:50 blends, the ternary centroid and so on — 7 points for three components. Simplex-centroid is the better default when you expect every component to contribute, while a higher-degree simplex-lattice gives more resolution for strongly curved responses.
How do you read a ternary mixture plot?
A ternary plot is a triangle where each corner is 100% of one component and the opposite edge is 0% of that component. Any point inside represents a blend whose three proportions sum to 100%. Contour lines drawn on the triangle show the predicted response, so the optimal blend is the point inside the innermost contour. You read the proportions of a point by following the grid lines parallel to each edge back to the three axes.
Can a mixture design include process variables like temperature?
Yes. A mixture-process (or mixture-amount) design crosses the mixture components with one or more independent process variables such as temperature, pH or total medium concentration. This lets you ask not only what the best blend is, but whether the best blend changes with the process conditions. These combined designs need more runs than a pure mixture design, so they are usually reserved for the final optimisation once the key components are known.
Is mixture design useful for cell culture media optimization?
Very. Cell culture media and feeds are mixtures by definition — amino acids, sugars, salts and trace elements whose proportions sum to a fixed total — so a mixture design doe answers the real question of which blend maximises growth or titer, rather than treating each component as an independent factor. In practice teams first screen many components with a Plackett-Burman design, then optimise the proportions of the critical few with a simplex-lattice or simplex-centroid mixture design.
Related Tools
- DOE Experiment Generator — Build simplex-lattice, simplex-centroid, screening and response-surface designs free in the browser.
- Media & Feed Estimator — Cost the medium and feed blends you compare in a mixture design study.
- Buffer Calculator — Work out the salt proportions and pH for the buffer formulation blends you test.
References
- Scheffé, H. (1958). Experiments with mixtures. Journal of the Royal Statistical Society: Series B, 20(2), 344–360. DOI: 10.1111/j.2517-6161.1958.tb00299.x
- Scheffé, H. (1963). The simplex-centroid design for experiments with mixtures. Journal of the Royal Statistical Society: Series B, 25(2), 235–263. DOI: 10.1111/j.2517-6161.1963.tb00506.x
- Mandenius, C.F. & Brundin, A. (2008). Bioprocess optimization using design-of-experiments methodology. Biotechnology Progress, 24(6), 1191–1203. DOI: 10.1002/btpr.67