Crop growth follows remarkably strict mathematical principles. It is truly astonishing how precisely and predictably a plant responds to environmental factors such as light and temperature. Since 1989, this mathematical structure behind plant growth and development has fascinated us. The model we developed to describe it is called the Plant Balance Model, and it follows a cascade of processes.

Light input
In an indoor farm, LED light with a specific spectral composition and intensity (PPFD) is provided, carefully adjusted to the crop and to each stage of its development. By multiplying the light intensity by the duration of illumination per day, the Daily Light Integral (DLI) is calculated, expressed in moles of light per square meter per day.

This light falls on a canopy of leaves that intercepts somewhere between zero and one hundred percent of the incoming light. Gaps in the canopy are undesirable because valuable light is then lost. By placing young plants very close together and gradually reducing plant density as they grow larger, the goal is to achieve the highest possible level of light interception by the canopy. This is expressed as Light Interception (LI).

Of the intercepted light, a large portion is actually absorbed by the leaves, while a small part is reflected. The degree of reflection per unit of leaf area depends strongly on the intensity and spectral composition of the light. The Fraction absorbed (Fa) can be very high; with a mixture of blue and red light it often reaches around 95%.

Carbon fixation
Plants possess the unique ability to convert carbon dioxide and water into glucose and oxygen using light energy. Without plants, humans and animals would not survive—and vice versa. Plants absorb CO₂ and produce oxygen, while and animals consume oxygen and produce CO₂. The amount of carbon a plant can assimilate from CO₂ using a given amount of absorbed light is called the Quantum Yield (QY). It is expressed as moles of carbon assimilated per mole of absorbed light.

Plants also require energy simply to stay alive. This is known as maintenance respiration. Part of the glucose produced by photosynthesis is burned again within the plant to generate the energy required for maintenance processes. Only the glucose remaining after this maintenance cost can be used for growth. The percentage of absorbed carbon that is ultimately used for growth is called the Carbon Use Efficiency (CUE).

Biomass growth
As plants grow, carbon, hydrogen, oxygen and minerals are combined to form structural material known as dry matter. Plants then add water, resulting in leaves, stems, fruits and roots. Depending on the crop, the dry matter percentage of plant organs varies widely, ranging from only a few percent to several tens of percent of fresh weight. Lettuce, for example, contains about 4% dry matter and therefore approximately 96% water.

By multiplying the daily light integral per square meter (DLI) by the light interception of the canopy (LI) and the fraction of absorbed light (Fa), we know how much light per day per square meter is actually used for photosynthesis.

If this light quantity is multiplied by the Quantum Yield (QY) and the Carbon Use Efficiency (CUE), we can determine how much carbon per day is used for growth.

Yield allocation
When that amount of carbon is multiplied by the molecular weight of carbohydrate (k = 30 grams) and corrected for the mineral fraction (approximately 15% of the dry matter), we know how much growth—expressed as dry matter per square meter—takes place each day.

Not all parts of a plant are harvested. In tomato or cucumber crops only the fruits are harvested, while leaves, stems and roots are not. In lettuce and herbs all above-ground parts are harvested, while the roots remain. The percentage of dry matter allocated to harvestable plant parts is called the harvest index (HI). In tomatoes the harvest index is about 65%, in cucumbers about 70%, and in lettuce about 95%.

By multiplying the daily dry matter production by the harvest index, we know how many grams of dry matter per square meter per day are allocated to harvestable plant parts. By converting this dry matter to fresh weight, we can determine how much harvestable fresh weight is produced per day per square meter. The dry matter percentage of tomatoes ranges from about 4% to more than 10%, depending on the type and variety. Cucumbers typically contain between 2.5% and 4% dry matter, while lettuce contains around 4%.

Annual yield
To calculate the annual fresh yield per square meter, the daily production of harvestable fresh weight must be multiplied by the number of growing days per year. After transplanting a tomato plant whose first flowers are already visible, it typically takes about eight weeks before the first harvest. For cucumbers this period is usually only three weeks. Such crop-specific development times must be taken into account. The number of growing days and harvest days per year depends on the crop and on the time required to maintain and clean the indoor farm’s cultivation system.

The entire cascade can be summarized in the following equation:

PPFD × Photoperiod × LI × QY × CUE × Fa × k / (1 – mineral%) x HI / DM% × Annual Growing Days = annual yield expressed in kilograms of fresh product per square meter per year.

Part of the full equation (QY × CUE × Fa × k / (1 – mineral%)) is commonly referred to as LUE-intercepted. This parameter expresses the number of grams of dry matter a crop can produce per day per square meter from one mole of light actually intercepted by the canopy.

In practice, crop yield is therefore often calculated using the following formulation:

PPFD × Photoperiod × LI × LUE-intercepted × HI / DM% × Annual Growing Days = annual yield

The cascade described above represents what is known as the source capacity. In most agricultural systems—both open-field and greenhouse—there are usually sufficient plant organs (tomatoes, cucumbers, lettuce leaves) available, and yield is therefore determined by the source capacity.

However, in a well-controlled indoor farm the source capacity can become so high that there are not enough fruits or leaves available to convert this capacity into actual yield. This limitation is referred to as sink limitation. In high-performance indoor farming, the source capacity can quickly exceed the sink capacity, meaning that the latter becomes the limiting factor for yield.

Source – Sink
Within the Plant Balance Model, both capacities are continuously calculated and optimized in such a way that production costs are minimized. Sink capacity can be increased by raising plant and fruit densities and by accelerating crop development. Faster development can be achieved by increasing temperature, but this requires crop varieties (genetics) that are capable of developing more rapidly at higher temperatures without causing quality problems or growth disorders. Most varieties developed for greenhouse production are not suitable for such higher temperatures. This creates an important opportunity for breeding companies to develop new varieties that are better adapted to indoor farming systems.

Plant Performance
The yield of a crop depends on many different factors. Part of these factors is determined by genetics. There are large differences in yield potential between plant species, and within those species there are further differences between types, varieties, and cultivars. However, the method of calculation used in the Plant Balance Model is universally applicable to any crop.

An indoor farm must above all be a controlled environment, with the emphasis on control. This means that we can adjust the key environmental factors in order to help a crop perform optimally in both quantity and quality. Quality is a broad concept and includes, among other things, shelf life, but also nutritional value, shape, aroma, taste, and other visible and invisible characteristics.

There are also significant differences between varieties developed by breeding companies. We see genetics as the talent of a variety. A truss tomato, for example, may have the genetic ability to allocate between 60 and 65 percent of its dry matter production to harvestable fruit. The Harvest Index that is actually realized in practice, however, is the result of both talent and growing conditions.

In a well-designed indoor farm, those growing conditions can be controlled with great precision. Research within CERIN89 therefore focuses primarily on understanding which environmental controls must be adjusted, and to what extent, to optimize the factors within the Plant Balance Model.

The number of possible combinations of light spectrum, light intensity, temperature, vapor pressure deficit, air velocity, CO₂ concentration, irrigation, and fertilization is extremely large. Simply relying on trial and error is therefore not a practical approach.

The use of a complete mathematical model that can calculate both the source capacity and the sink capacity of a crop is therefore an essential tool in both designing research trials and evaluating their results.

The Plant Balance Model also makes it possible to calculate energy consumption per unit of product, to estimate production costs, to compare different cultivation scenarios, and to perform economic feasibility studies.